text:
Nonstandard functional interpretations and categorical models

# Nonstandard functional interpretations and categorical models

Amar Hadzihasanovic111Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, OX1 3QD Oxford. Email address: amarh@cs.ox.ac.uk. Supported by an EPSRC Doctoral Training Grant.  and Benno van den Berg222Institute for Logic, Language and Computation, Universiteit van Amsterdam, P.O. Box 94242, 1090 GE Amsterdam. Email address: bennovdberg@gmail.com. Supported by the Netherlands Organisation for Scientific Research (NWO).
4 February 2014
###### Abstract

Recently, the second author, Briseid and Safarik introduced nonstandard Dialectica, a functional interpretation that is capable of eliminating instances of familiar principles of nonstandard arithmetic - including overspill, underspill, and generalisations to higher types - from proofs. We show that, under few metatheoretical assumptions, the properties of this interpretation are mirrored by first order logic in a constructive sheaf model of nonstandard arithmetic due to Moerdijk, later developed by Palmgren. In doing so, we also draw some new connections between nonstandard principles, and principles that are rejected by strict constructivism.

Furthermore, we introduce a variant of the Diller-Nahm interpretion with two different kinds of quantifiers (with and without computational meaning), similar to Hernest’s light Dialectica interpretation, and show that one can obtain nonstandard Dialectica from this by weakening the computational content of the existential quantifiers – a process we call herbrandisation. We also define a constructive sheaf model mirroring this new functional interpretation and show that the process of herbrandisation has a clear meaning in terms of these sheaf models.

## 1 Introduction

The focus of this paper stands at a confluence of two quite different paths in mathematical logic.

On one end, there is nonstandard arithmetic, and analysis: a subject that has been an upshot of classical model theory, and even after it was recognised that it was amenable to a syntactic treatment, as in Nelson’s internal set theory [19], it mostly remained within the boundaries of classical set theory. On the other end, there is the markedly proof-theoretic topic of functional interpretations, stemming from Gödel’s Dialectica interpretation [8]; and, in particular, its recent revival through the programme of proof mining [13].

Where these ends meet, is in a general inclination towards the constructivisation of mathematics. The first explicit model of nonstandard analysis, due to Schmieden and Laugwitz [28], was actually fully constructive, but had a quite weak transfer property. On the other hand, Robinson’s model of nonstandard arithmetic [27], and subsequent ones which were elementary extensions of the standard model, were built from nonconstructive objects, such as nonprincipal ultrafilters of sets. Even in the syntactic approach, it was soon realised that many useful principles led to instances of the excluded middle. But did nonstandard analysis really have nothing to offer to constructive analysts?

Not everyone was convinced, including, notably, Per Martin-Löf, who pushed the question in the early 1990s: first, Erik Palmgren succeeded in building a model with a restricted, yet useful transfer principle [22]; then, in 1995, Ieke Moerdijk described the first constructive model of nonstandard arithmetic with a full transfer principle - a topos of sheaves over a category of filters [18]. Later, by working in this topos, Palmgren provided simplified, nonstandard proofs of several theorems of constructive analysis, and so demonstrated the usefulness of this model [23, 24, 25, 26].

But if nonstandard proofs do provide some constructive information, we might as well try to extract it in an automated fashion. In 2012, the second author, Briseid and Safarik succeeded in defining a functional interpretation, nonstandard Dialectica [3], which could eliminate nonstandard principles from proofs of intuitionistic arithmetic in all finite types, enriched, à la Nelson, with a predicate , “ is standard”, for all types ; also yielding a proof of conservativity of these principles over the base system. Section 2 is a review.

Now, some of the principles validated by nonstandard Dialectica were known to hold in Moerdijk’s topos - including a form of Nelson’s idealisation axiom, an underspill principle, and the undecidability of the standardness predicate. Our first aim was to investigate how deep this connection would go.

And a deep connection it is: with the exception of one principle, which requires an assumption about the metatheory, all the characteristic principles of nonstandard Dialectica are true in the topos model, for free. Section 3 is devoted to showing this. During this investigation, we also chanced upon two new principles, sequence overspill and sequence underspill, which appear to be more natural equivalents of principles that have been taken into consideration, earlier, in the context of proof-theoretic nonstandard arithmetic. We map their relation to other familiar principles from nonstandard and constructive analysis in Section 2.

Several characteristic principles of nonstandard Dialectica have a peculiarity: they are herbrandised. This is explained in more detail in Section 4; in short, where “traditional” functional interpretations would produce a single witness of an existential statement, these principles produce a finite sequence of potential witnesses, of which at least one is an actual witness. This property destroys the computational meaning of intuitionistic disjunction, yet seems unavoidable in the interpretation of nonstandard arithmetic.

The categorical analysis of nonstandard Dialectica supplied a very convenient way of “de-herbrandising”, through a simple change in the Grothendieck topology, down from finite covers to singleton covers. Full transfer is lost - in the new topos, disjunction is stronger than in the metatheory - as well as the link to nonstandard arithmetic; but the de-herbrandised principles induce a new functional interpretation, which we call uniform Diller-Nahm, and is the main focus of Section 4.

Uniform Diller-Nahm can be seen as an extension of the Diller-Nahm variant of the Dialectica interpretation [7], and has some striking similarities to light Dialectica [9], a variant of Dialectica with two different kinds of quantifiers - computational, and non computational - introduced in 2005 by Mircea-Dan Hernest, for the purpose of more efficient program extraction from formal proofs. Yet, irrespective of its technical value, the characteristic proof system of uniform Diller-Nahm might have a dignity of its own.

In 1985, Vladimir Lifschitz proposed a simple extension of Heyting arithmetic, where a distinction could be made between calculable, and non calculable natural numbers [15]; a synthesis of classical and intuitionistic arithmetic. Under the interpretation of the predicate as “ is calculable”, the proof system of uniform Diller-Nahm seems to be well-suited for Lifschitz’s intended calculus. This is also discussed in Section 4.

Finally, in Section 5, we survey some open questions.

Note. This work is based on research done by the first author, under the supervision of the second author, in partial fulfillment of the requirements for the degree of Laurea Magistrale in Mathematics at the University of Pavia.

## 2 The nonstandard Dialectica interpretation

We start by briefly recalling the definition of the system E-HA, as introduced in [3]; we refer to the original paper for a detailed presentation.

### 2.1 The system E-HAω∗st

We take E-HA to be an extension of the system called E-HA in [29], with additional types and constants for handling finite sequences. More precisely, the collection of types is generated by the inductive clauses

• is in ;

• if , are in , then and are in ;

and, for all types in , we have constants (empty sequence), (prepending operator), and (list recursor), with defining axioms

 SA:∀s:σ∗(s=⟨⟩σ∨∃x:σ∃s′:σ∗(s=Cxs′)), [l]{Lσ,τxy⟨⟩τ=σx,Lσ,τxy(Czs)=σy(Lσ,τxys)⟨z⟩,x:σ,y:σ→τ→σ,z:τ,s:τ∗,

where is the “singleton” .

• Notation. We use (and ) as variables of sequence type.

This system has an extensionality axiom

 ∀f,g:σ→τ(f=σ→τg↔∀x:σfx=τgx)

for all types , .

Using the projectors and combinators from the language of E-HA, it is possible, already in the latter system, to introduce a coding of finite sequences of elements of any type, as in [12, p. 59]; therefore, E-HA is a definitional, hence conservative, extension of E-HA. However, finite sequences seem to be quite ubiquitous in arguments of nonstandard arithmetic, mostly due to the expanded notion of “finiteness” in a nonstandard model; so it seems preferable to have them built into our syntax.

Since every type is provably inhabited, we can conservatively add for every type a constant . Using the list recursor, one can define all the basic operations on finite sequences one needs in practice.

1. A length function , satisfying

 |⟨⟩σ|=0,|Cas|=S|s|,

for , .

2. A projection function of type , satisfying

 (⟨⟩σ)i =∅σfor all i, (Cas)0 =a, (Cas)Si =si.
3. A concatenation operation , such that

 ⟨⟩σ⋅t=t,Cas⋅t=Ca(s⋅t).

As expected, concatenation is provably associative, so we will iterate it without bothering with brackets.

The following, easy properties are all established in [3].

1. E-HA

2. E-HA

###### Proof.

Let . By the sequence axiom , either or for some , . If , the latter case leads to a contradiction, for .

If , then the former case leads to a contradiction, and we have proven the directions left to right. The converses are immediate. ∎

###### Proposition 2.2.

E-HA proves the induction schema for sequences

 IA∗:(φ(⟨⟩σ)∧∀x:σ∀s:σ∗(φ(s)→φ(Cxs)))→∀s:σ∗φ(s).
###### Proof.

Suppose and . By the previous lemma,

 ∀s:σ∗(|s|=0→φ(s)).

Fix , and assume . Let be of length . Again by the previous lemma, for some , and of length , and holds by hypothesis. Therefore, holds as well; and we have proved

 ∀s:σ∗(|s|=n→φ(s))→∀s:σ∗(|s|=Sn→φ(s)).

By ordinary induction, it follows that . ∎

###### Definition 2.3.

Let . We say that and are extensionally equal, and write , if

 |s|=|t|∧∀i<|s|(si=ti).

E-HA

###### Proof.

By induction for sequences. Suppose . If , then , so, by Lemma 2.1, .

Otherwise, for some . Then for ; again, by Lemma 2.1, for some . But , and ; by the inductive hypothesis, . Therefore, . ∎

Since finite sequences will be used as a replacement for finite sets, we will borrow some set-theoretic notation.

###### Definition 2.5.

Let , . We define the abbreviations

1. ( is an element of );

2. ( is contained in ).

We will drop subscripts in most occasions. We also extend the relation to sequence-valued functionals, pointwise: for ,

The relation determines a preorder, provably in E-HA.

In the definition of the nonstandard Dialectica translation, one needs a form of application for finite sequences - and an associated form of -abstraction - that is monotone in the first component, with respect to the preorder we just defined.

###### Definition 2.6 (Finite sequence application and abstraction).

Let , , . Then

 s[a] :=(s0a)⋅…⋅(s|s|−1a):τ∗, Λx:σ.t :=C(λx:σ.t)⟨⟩:(σ→τ∗)∗.

The new application and abstraction are interdefinable with the usual ones. In fact, we have the following, easy compatibility result.

###### Proposition 2.7.

E-HA proves that for all , ,

 (Λx:σ.s)[a]=(λx:σ.s)a=s[a/x].
###### Lemma 2.8.

E-HA proves that for all , ,

 s⊆s′→s[a]⊆s′[a].
###### Proof.

See [3, Lemma 2.22]. ∎

Since we do not have product types, we will often work with tuples of types and of terms, for which we follow the conventions of [12]; the following is a brief summary.

• Notation. We write , for tuples of types and terms. stands for the empty tuple. We write

 fx––:=(…(fx0)x1)…)xn,

with the appropriate types; while, if , stands for . We will have, correspondingly,

 λx––.f––:=λx––.f0,…λx––.fm,

and the same for finite sequence application.

Relations distribute as expected: for instance, if , with the same length and types as ,

 x––=σ––y–:=n⋀i=0xi=σiyi;

and if is a tuple of sequences,

 x––∈σ––s–:=n⋀i=0xi∈σisi.

Most of the results we have listed so far are easily extended to tuples of terms; in particular, those concerning finite sequence application and abstraction.

We now lay the syntactic groundwork for doing nonstandard arithmetic in our system.

###### Definition 2.9.

The system E-HA is an extension of E-HA, whose language includes a (unary) predicate , , for all types of ; and the external quantifiers , .

• Notation. Following Nelson, so-called internal formulae - those in the language of E-HA - are always denoted with small Greek letters, and generic, external formulae with capital Greek letters.

The following axioms are added to those of E-HA:

1. the defining axioms of the external quantifiers:

 ∀stx:σΦ(x)↔∀x:σ(stσ(x)→Φ(x)), ∃stx:σΦ(x)↔∃x:σ(stσ(x)∧Φ(x));
2. axioms for the standardness predicate:

 [l]stσ(x)∧x=σy→stσ(y),stσ(a)for all \emph{closed} a:σ,stσ→τ(f)∧stσ(x)→stτ(fx);
3. the external induction schema:

 IAst:(Φ(0)∧∀stx:0(Φ(x)→Φ(Sx)))→∀stx:0Φ(x).

Since it is part of E-HA, the system E-HA also contains, besides the external induction schema, an “internal” induction schema , which is assumed to hold for internal formulae only.

So far, there is nothing inherently nonstandard about the system we have defined. In fact, one could interpret as , and all the new axioms would be provable in E-HA. This simple fact also implies that E-HA is a conservative extension of E-HA.

However, there are some simple results, of the kind we would expect from a “standardness property”, that can already be proved.

###### Proposition 2.10.

For every formula , E-HA proves

 Φ(x)∧x=y→Φ(y).
###### Proof.

Easy induction on the logical structure of , utilising the fact that the standardness predicate is extensional. ∎

###### Proof.

Apply external induction to the formula . ∎

Basically anything one can get from standard sequences is standard.

1. E-HA

2. E-HA

3. E-HA

4. E-HA

5. E-HA

###### Proof.

Everything follows from the standardness axioms, coupled with the fact that the list recursor is standard. ∎

A simple consequence of the lemma is that the operations of sequence application and abstraction, as defined in the previous section, preserve standardness.

###### Corollary 2.13.
1. E-HA

2. E-HA

Finally, we prove that finite sequences of standard elements are standard; the converse is already a consequence of Lemma 2.12.(a)-(b).

###### Lemma 2.14.

E-HA proves that

 ∀s:σ∗(st(|s|)∧∀i<|s|st(si)→st(s)).
###### Proof.

Suppose is finite, and that, for all , is standard. By an iteration of Lemma 2.12.(d), is also standard. Clearly, and are extensionally equal; by Corollary 2.4, . Thus, is standard. ∎

This, in turn, is used to prove an external induction schema for sequences.

###### Proposition 2.15.

E-HA proves the external induction schema for sequences

 IA∗st:(Φ(⟨⟩σ)∧∀stx:σ∀sts:σ∗(Φ(s)→Φ(Cxs)))→∀sts:σ∗Φ(s).
###### Proof.

From the previous lemma, one obtains that if and is standard, then and are also standard. Then one argues precisely as in Proposition 2.2, applying external instead of ordinary induction. ∎

The linguistic blocks are in place for the definition of the nonstandard Dialectica interpretation.

### 2.2 The Dst translation

###### Definition 2.16.

To every formula of the language of E-HA, with free variables , we associate inductively its nonstandard Dialectica translation

 Φ(a––)Dst=∃sts–∀sty–φDst(s–,y–,a––),

where is internal, and all the variables in are of sequence type.

• , for internal atomic;

• .

Let , :

• ;

• ;

• ;

• ;

• ;

• ;

• .

The idea is that, in the interpretation, realisers should be finite sequences of potential realisers, of which at least one is an actual realiser. Hence, if is a valid realiser, then any with should work as well. That this is the case is guaranteed by the following proposition.

###### Definition 2.17.

A formula is upwards closed in if

 Φ(s)∧s⊆s′→Φ(s′).
###### Proposition 2.18.

Let be a formula of E-HA, . Then proves that is upwards closed in :

 \emph{E-HA}ω∗⊢φ(s–,y–,a––)∧s–⊆s–′→φ(s–′,y–,a––).
###### Proof.

By induction on the logical structure of , using Lemma 2.8 in the clauses for and . ∎

In [3], the nonstandard Dialectica interpretation was given a characterisation in terms of five principles. We provide here an alternative characterisation, which keeps the following three principles from the former.

1. The herbrandised axiom of choice:

 HACst:∀stx:σ∃sty:τΦ(x,y)→∃st(f:σ→τ∗)∗∀stx:σ∃y∈f[x]Φ(x,y).
2. The herbrandised independence of premise principle:

 HIPst∀:(∀stx:σφ(x)→∃sty:τΨ(y))→∃stt:τ∗(∀stx:σφ(x)→∃y∈tΨ(y)).
3. The principle called non-classical realisation in [3] - which, as we will see in Section 4, could also be called herbrandised nonstandard uniformity:

 NCR:∀y:τ∃stx:σΦ(x,y)→∃sts:σ∗∀y:τ∃x∈sΦ(x,y).

In addition to these, the former characterisation had idealisation

 I:∀sts:σ∗∃y:τ∀x∈sφ(x,y)→∃y:τ∀stx:σφ(x,y),

whose dual

 R:∀y:τ∃stx:σφ(x,y)→∃sts:σ∗∀y:τ∃x∈sφ(x,y)

is clearly a consequence of , and the herbrandised generalised Markov’s principle

 HGMPst:(∀stx:σφ(x)→ψ)→∃sts:σ∗(∀x∈sφ(x)→ψ).

We will replace them as follows.

###### Definition 2.19.

Let . We say that is a hyperfinite enumeration of the type if

 ∀stx:σ(x∈s).

We define, for all types , a predicate

 hyper(s):=∀stx:σ(x∈s),

as well as quantifiers ranging over hyperfinite enumerations, with defining axioms

 ∀hyps:σ∗Φ(s):=∀s:σ∗(hyperσ(s)→Φ(s)), ∃hyps:σ∗Φ(s):=∃s:σ∗(hyperσ(s)∧Φ(s)).

The most basic nonstandard principles are, arguably, overspill and underspill in the type of natural numbers:

 OS0: ∀stn:0φ(n)→∃n:0(¬st(n)∧φ(n)), US0: ∀n:0(¬st(n)→φ(n))→∃stn:0φ(n).

These principles are almost invariably used with formulae of the form , stating that a certain property holds up to a number . From the assumption , which says that holds up to any standard natural number, allows one to derive that holds up to some nonstandard (infinite) number .

From , one can obtain a hyperfinite enumeration of the natural numbers, so that ; and, in a way, it is this fact - that induces a hyperfinite enumeration - that is relevant to the argument, rather than being nonstandard. This suggests the following generalisation of overspill and underspill to all finite types.

We introduce the principle of sequence overspill

 OS∗:∀sts:σ∗φ(s)→∃hyps:σ∗φ(s),

and its dual, sequence underspill

 US∗:∀hyps:σ∗φ(s)→∃sts:σ∗φ(s).

E-HA.

###### Proof.

Assume , and suppose . Let be a standard sequence of sequences; then is again standard, so holds. Furthermore, by construction, for all , ; in other words,

 ∀stt:(σ∗)∗∃s:σ∗∀t′∈t(t′⊆s∧φ(s)).

By idealisation, we obtain

 ∃s:σ∗∀stt:σ∗(t⊆s∧φ(s)).

It remains to prove that , an easy consequence of Lemma 2.12.

Conversely, assume , and suppose . By sequence overspill, it follows that

 ∃y:τ∃hyps:σ∗∀x∈sφ(x,y),

which implies

 ∃y:τ∀stx:σ∗φ(x,y).

This concludes the proof. ∎

Several consequences of are listed in [24] and in [3], which, by the previous proposition, are also consequences of . For us, it is particularly relevant that implies an external version of the lesser limited principle of omniscience, a nonconstructive principle well-known in the area of reverse mathematics, see e.g. [10]; namely,

 LLPOst:∀stx,y:σ(φ(x)∨ψ(y))→(∀stx:σφ(x)∨∀stx:σψ(x)).

E-HA.

###### Proof.

Suppose . We prove by external sequence induction that

 ∀sts:σ∗(∀x∈sφ(x)∨∀x∈sψ(x)). (1)

For , is vacuously true. Suppose it is true for some arbitrary, standard , and pick any standard . We want to show .

Suppose (the case where is true is similar). Since

 ∀stx,y:σ(φ(x)∨ψ(y)),

we have

 ∀b∈Cas(φ(a)∨ψ(b));

since is a finite sequence, we can run through all and see whether holds. If so, then holds and we are done; otherwise, we will get that holds for all and we again achieve the desired disjunction. Now, applying sequence overspill to (1) gives

 ∃hyps:σ∗(∀x∈sφ(x)∨∀x∈sψ(x)),

which implies . ∎

Notice that alone would have sufficed to prove the restriction of to type 0.

Since is equivalent to , it would make sense if were equivalent to ; yet things are not so simple. In fact, only one implication seems to hold.

E-HA.

###### Proof.

Suppose . Then

 ∀hyps:σ∗∀y:τ∃x∈sφ(x,y),

which, by sequence underspill, implies . ∎

What is missing, in order to obtain an equivalence, is precisely the last characteristic principle.

E-HA.

###### Proof.

Suppose . Then

 ∃hyps:σ∗∀x∈sφ(x)→ψ,

which is intuitionistically equivalent to

 ∀hyps:σ∗(∀x∈sφ(x)→ψ).

An application of sequence underspill leads to the conclusion. ∎

We now complete the characterisation of .

E-HA.

###### Proof.

Suppose ; that is,

 ∀s:σ∗(∀stx:σ(x∈s)→φ(s)).

By the herbrandised generalised Markov’s principle, this is equivalent to

 ∀s:σ∗∃stt:σ∗(t⊆s→φ(s));

which, by realisation and intuitionistic logic, implies

 ∃stt:(σ∗)∗∀s:σ∗(∀t′∈t(t′⊆s)→φ(s)).

Take a standard as in (2.2), and pick . By Lemma 2.12, is standard, and, for all , ; therefore, it holds that . We thus prove

 ∃sts:σ∗φ(s),

and the sequence overspill principle. ∎

Replacing with a contradiction, e.g. , and choosing a negated , we see that - hence, as well - implies an external version of Markov’s principle, another noted principle that is rejected by strict constructivism:

 MPst:(∀stx:σ(φ(x)∨¬φ(x))∧¬¬∃stx:σφ(x))→∃stx:σφ(x).

This is another instance of a principle whose nature appears markedly nonstandard, forcing a nonconstructive mode of reasoning.

###### Theorem 2.25 (Soundness of the nonstandard Dialectica interpretation).

Suppose

 \emph{E-HA}ω∗st+OS∗+US∗+NCR+HACst+HIPst∀+Δint⊢Φ(a––),

where is a set of internal sentences. Let . Then from the proof we can extract a tuple of closed terms such that

 \emph{E-HA}ω∗+Δint⊢∀y–φDst(t–,y–,a––).
###### Proof.

This is [3, Theorem 5.5], coupled with the fact that over E-HA. We provide explicit realisers for the new principles.

The interpretation of is

 ∃stS∀sts′(∀s∈S[s′]φ(s)→∃s(s′⊆s∧φ(s))),

and we can take .

The interpretation of is

 ∃stT∀sts′′(∀s∃s′∈s′′(s′⊆s→φ(s))→∃t∈T[s′′]φ(t));

since implies , unless is the empty sequence (in which case, the premise is false anyway), we can take

 T:=Λs′<


text:
# How do you find all rational roots for 8y^4 - 6y^3 + 17y^2 - 12y + 2 = 0?

Aug 4, 2016

The rational roots are $\frac{1}{4}$, $\frac{1}{2}$.

The remaining two roots are $\pm \sqrt{2} i$

#### Explanation:

$f \left(y\right) = 8 {y}^{4} - 6 {y}^{3} + 17 {y}^{2} - 12 y + 2$

By the rational root theorem, any rational zeros of $f \left(y\right)$ must be expressible in the form $\frac{p}{q}$ for integers $p , q$ with $p$ a divisor of the constant term $2$ and $q$ a divisor of the coefficient $8$ of the leading term.

That means that the only possible rational zeros are:

$\pm \frac{1}{8} , \pm \frac{1}{4} , \pm \frac{1}{2} , \pm 1 , \pm 2$

In addition, note that $f \left(- y\right) = 8 {y}^{4} + 6 {y}^{3} + 17 {y}^{2} + 12 y + 2$ has all positive coefficients. Hence $f \left(y\right)$ has no negative zeros.

So the only possible rational zeros of $f \left(y\right)$ are:

$\frac{1}{8} , \frac{1}{4} , \frac{1}{2} , 1 , 2$

We find:

$f \left(\frac{1}{4}\right) = 8 {\left(\frac{1}{4}\right)}^{4} - 6 {\left(\frac{1}{4}\right)}^{3} + 17 {\left(\frac{1}{4}\right)}^{2} - 12 \left(\frac{1}{4}\right) + 2$

$= \frac{1 - 3 + 34 - 96 + 64}{32} = 0$

$f \left(\frac{1}{2}\right) = 8 {\left(\frac{1}{2}\right)}^{4} - 6 {\left(\frac{1}{2}\right)}^{3} + 17 {\left(\frac{1}{2}\right)}^{2} - 12 \left(\frac{1}{2}\right) + 2$

$= \frac{2 - 3 + 17 - 24 + 8}{4} = 0$

So $y = \frac{1}{4}$ and $y = \frac{1}{2}$ are zeros and $\left(4 y - 1\right)$ and $\left(2 y - 1\right)$ are factors:

$8 {y}^{4} - 6 {y}^{3} + 17 {y}^{2} - 12 y + 2$

$= \left(4 y - 1\right) \left(2 {y}^{3} - {y}^{2} + 4 y - 2\right)$

$= \left(4 y - 1\right) \left(2 y - 1\right) \left({y}^{2} + 2\right)$

${y}^{2} + 2 \ge 2 > 0$ for all Real values of $y$, so there are no more Real, let alone rational, zeros.

The last two zeros are $\pm \sqrt{2} i$ since:

$\left(y - \sqrt{2} i\right) \left(y + \sqrt{2} i\right) = {y}^{2} - {\left(\sqrt{2} i\right)}^{2} = {y}^{2} + 2$

format: plain


text:
What happens when the a (amplitude) of a sine graph is negative -2 sin (1/4 x)?

1 Answer
Apr 29, 2015

It simply flips your graph upsidedown. Where it should have a positive amplitude, now gets negative and viceversa:
For example:
if you choose $x = \pi$ yo get $\sin \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$ but with minus 2 in front your amplitude becomes: $- 2 \frac{\sqrt{2}}{2} = - \sqrt{2}$:
Graphically you can see this comparing:
$y = 2 \sin \left(\frac{x}{4}\right)$
graph{2sin(x/4) [-11.25, 11.25, -5.625, 5.625]}
with:
$y = - 2 \sin \left(\frac{x}{4}\right)$
graph{-2sin(x/4) [-12.66, 12.65, -6.33, 6.33]}


text:
# memoir: book + articles

I have a book (memoir class) and few articles (memoir[article] class)

I want to put articles in the end of the book, so that they would appear as they were processed originally)

I would like to have a single tex document (since I have ctross-references between book and articles, and between articles)

I understand, that some editing will be required

Thank you very much in advance

%%%% Minimal example of standalone article

\documentclass[12pt,oneside,openany, article]{memoir}

\usepackage{blindtext}

\title{Some creative thinking about miscellaneous stupid things}

\author{Victor Ivrii}

\begin{document}

\maketitle

\chapter{Introduction}

\blindtext[5]

\bibliographystyle{amsplain}

\begin{thebibliography}{9}

\bibitem[AM]{crap}

John Doe \emph{Some crap}

\end{thebibliography}

\end{document}

%%%Embedding it into a book

\documentclass[12pt,oneside,openany]{memoir}

\usepackage{blindtext}

\begin{document}

\title{My Book}

\author{Victor}

\chapter{Some crap}

\blindtext

\renewcommand{\thesection}{\arabic{section}}

\title{Some creative thinking about miscellaneous stupid things}

\author{Victor}

\maketitle

\section{Introduction}

\blindtext[5]

\bibliographystyle{amsplain}

\begin{thebibliography}{9}

\bibitem[AM]{crap}

John Doe \emph{Some crap}

\end{thebibliography}

\end{document}

%%%%%%%%%%%

We see that

1. Running heads are displaying in the article CHAPTER (actually previous chapter of the book) while it should be SECTION inside the article

2. Bibliography in the embedded article starts from the new page (unlike in the article) and the word "Bibliography" is printed in larger fonts

PS Sure, book has its own bibliography which should look like bibliography in the book

\renewcommand{\bibsection}{\section*{\bibname}\prebibhook}

The result was that running header over Bibliography was the one from the previous section. I changed it to

\renewcommand{\bibsection}{\section{\bibname}\prebibhook}

which fixed the running header but added unwanted "section" number to Bibliography, which I fixed by preceding it by

\setcounter{secnumdepth}{0}

• Welcome, your question is a bit too broad. What kind of answer are you expecting? Should just work? Where is the problem right now? – Johannes_B Feb 25 '18 at 10:29
• Crosspost on texhax. – Johannes_B Feb 26 '18 at 7:51
• This should not have that hard to fix. But please provide a minimal example so we have something to work with. – daleif Feb 26 '18 at 9:46
• Please explain what you mean by "do a similar thing with the TOC"? The bibliography and the TOC are different constructions, and this does not share structure. – daleif Mar 2 '18 at 10:14
• Please ask a new question instead of completely change the existing question. – Johannes_B Mar 2 '18 at 10:28

Something like this might be a starting point.

\documentclass[12pt,oneside,openany]{memoir}

\usepackage{kantlipsum}

% need reactivation for \nouppercaseheads to have an effect

\setsecnumdepth{subsubsection}
\maxsecnumdepth{subsubsection}

% adjusted from memoir sources (look for \ifartopt)
\newcommand\InternalArticleOption{
% assumes oneside, use "both" instead of "right" in twosided case
\createmark{section}{right}{shownumber}{}{. \ }
\createmark{subsection}{right}{shownumber}{}{. \ }
\createmark{subsubsection}{right}{shownumber}{}{. \ }
}
% reactivation
\renewcommand{\maketitlehookb}{%
\vskip -1.5\topsep\vskip -1.5\partopsep}
\renewcommand{\maketitlehookc}{%
\vskip -1.5\topsep\vskip -1.5\partopsep}
% adjust bibliography (it is using \chapter* by deafault, under
% articl option, \chapter* behaves like \section*, we just foce it
% to use \section*)
\renewcommand{\bibsection}{\section*{\bibname}\prebibhook}
% remove chapter from section number
\renewcommand{\thesection}{\arabic{section}}
}

\begin{document}

\title{My Book}

\author{Victor}

\chapter{Some crap}

\kant

\InternalArticleOption

\title{Some creative thinking about miscellaneous stupid things}

\author{Victor}

\maketitle

\section{Introduction}

\kant[5]

\subsection{Test}

\kant[7]

\bibliographystyle{amsplain}

\begin{thebibliography}{9}

\bibitem[AM]{crap}

John Doe \emph{Some crap}

\end{thebibliography}

\end{document}

Addition: There is a bug in memoir in its handling of \maketitle, it does not reset the thanks data, which may result in accumulative thanks build up. Also, the counter value used by \thanks continues from where it left of. Easy fix:

\makeatletter
\renewcommand\maketitlehooka{
\setcounter{footnote}{0}
\global\let\@thanks\@empty
}
\makeatother

in the preamble. memoir probably never had repeated use of \maketitle in mind.

Normally when I make docs that lists articles (often seen in PhD theses), I tend to write my own interface that does not use \maketitle. Then the student can also include information like where this was published, abstract etc. From that we then build a suitable title page for the article (and add whatever we like to the general ToC).

EDIT: memoir not resitting thanks is not really a bug, it is even mentioned in the manual, with the suggestion of using

\emptythanks
\maketitle

if you need to use \maketitle many times, since \emptythanks is just \global\let\@thanks\@empty, the suggestion from above reduces to

\renewcommand\maketitlehooka{
\emptythanks
}

As multi using \maketitle is not that frequent, I'll just leave it at that, since \emptythanks is already mentioned in the manual (aka I do not see this a a bug, as I first wrote and as such will not fix it at present).

• One more problem: accumulation of thanks: %%% Example\documentclass[12pt]{memoir} \begin{document} \title{About rhinos} \thanks{To Rhinos} \maketitle Rhinos are cool! \title{About raccoons} \thanks{To Raccoonss} \maketitle Raccoons are cool! \end{document} – Victor Feb 26 '18 at 12:04
• @Victor suggestion added, also added this to my todo list for memoir – daleif Feb 26 '18 at 13:34


text:
# C6H5Cl + Na + SbCl3 = NaCl + (C6H5)3Sb | Balanced | Chemical Reaction Details

## + sodium + = sodium chloride + |

### News Only 5% of POPULATION would know

Table of Content

 3C6H5Cl + 6Na + SbCl3 → 6NaCl + (C6H5)3Sb sodium sodium chloride 3 6 1 6 1 Hệ số Nguyên - Phân tử khối (g/mol) Số mol Khối lượng (g)

## Further information about equation 3C6H5Cl + 6Na + SbCl3 → 6NaCl + (C6H5)3Sb

### What is reaction condition of C6H5Cl () reacts with Na (sodium) reacts with SbCl3 () ?

No information found for this chemical equation

Explanation: The ideal environmental conditions for a reaction, such as temperature, pressure, catalysts, and solvent. Catalysts are substances that speed up the pace (velocity) of a chemical reaction without being consumed or becoming part of the end product. Catalysts have no effect on equilibrium situations.

### How reactions can happened and produce NaCl (sodium chloride) and (C6H5)3Sb () ?

In a full sentence, you can also say C6H5Cl () reacts with Na (sodium) reacts with SbCl3 () and produce NaCl (sodium chloride) and (C6H5)3Sb ()

### Phenomenon after C6H5Cl () reacts with Na (sodium) reacts with SbCl3 ()

This equation does not have any specific information about phenomenon.

In this case, you just need to observe to see if product substance (C6H5)3Sb, appearing at the end of the reaction.

Or if any of the following reactant substances SbCl3, disappearing

## Further questions related to chemical reactions 3C6H5Cl + 6Na + SbCl3 → 6NaCl + (C6H5)3Sb

### Rating

3C6H5Cl + 6Na + SbCl3 = 6NaCl + (C6H5)3Sb | Chemical Equation

Rating: 5 / 5 stars

sodium


text:
# Beams - structural beams#

This notebook shows how to use Beam elements model cantilever beams, how to get the results and how those results compare to theory.

A “Beam” in DAVE consists of a number of discrete segments and has a different formulation than the beams typically found in FEM packages.

from DAVE import *
s = Scene();

Equilibrium-core version = 2.48 from c:\python\miniconda3\envs\book\lib\site-packages\pyo3d.cp39-win_amd64.pyd
default resource folders:
c:\python\miniconda3\envs\book\lib\site-packages\DAVE\resources
C:\Users\beneden\DAVE_models
C:\data\Dave\Book\DAVE-book\DAVE-notebooks
Blender found at: C:\Program Files\Blender Foundation\Blender 3.3\blender.exe

Beams are elastic nodes that can be created between axis systems.

left = s.new_axis("left")
right = s.new_axis("right")
beam = s.new_beam("beam", nodeA=left, nodeB = right,
EA = 1000,
EIy = 1e5, EIz = 1e5, GIp = 1e3,
L=10,
mass = 0,
n_segments=20)
right.set_free()
s.solve_statics()

c:\python\miniconda3\envs\book\lib\site-packages\DAVE\scene.py:1668: UserWarning: new_axis is deprecated, use new_frame instead
warnings.warn("new_axis is deprecated, use new_frame instead")

True

from DAVE.jupyter import *;

show(s, camera_pos=(5,-10,6), lookat = (5,0,2), width=1000, height = 200)

 Warning! Please use "settings.use_parallel_projection" instead!

c:\python\miniconda3\envs\book\lib\site-packages\DAVE\jupyter\jupyter.py:194: UserWarning: VTK/Vedo issue: plotter is None
warnings.warn("VTK/Vedo issue: plotter is None")

A Beam node is rigidly fixed to the axis systems at its ends. It is fixed in the positive X-direction, so it departs from node A along the X-axis and arrives at ndoe B from the negative X-direction.

At the momement we have a massless beam with its left side fixed (axis system “left” is fixed) and its right size free (the right end of the beam is fixed to axis system “right”, but “right” is not connected to anything and is free to move).

This is one of those cases that frequently occur in textbooks and for which we can easily check the results.

First place a point on the right side so that we can apply a force:

s.new_point("point", parent = right)
force = s.new_force("force", parent = "point")

## Cantilever beam with a force#

force.force = (0,0,-100)

s.solve_statics()

True

show(s, camera_pos=(5,-10,0), lookat = (5,0,0), width=1000, height = 300)

c:\python\miniconda3\envs\book\lib\site-packages\DAVE\visual_helpers\vtkHelpers.py:533: RuntimeWarning: invalid value encountered in divide
axis = axis / length
c:\python\miniconda3\envs\book\lib\site-packages\DAVE\visual_helpers\vtkHelpers.py:594: RuntimeWarning: invalid value encountered in divide
axis = axis / length

 Warning! Please use "settings.use_parallel_projection" instead!

Theory tells us that the deflection and rotation of the right end should be:

$$\delta = -PL^3 / 3EI$$

and

$$\Theta = PL^2 / 2EI$$

Get the values from the model:

L = beam.L
EI = beam.EIy
P = -force.force[2]

And calculate their values

The deflection is the vertical displacement of the axis system on the right side of the beam:

right.z

-0.33604653085104813

-P*L**3 / (3*EI)

-0.3333333333333333

Theta is the slope. The slope can be calculated from the rotation of the axis system, but it is also available as “tilt”, which is a percentage.

right.tilt_y / 100

0.05014161613258149

P*L**2 / (2*EI)

0.05

It is also possible to obtain the forces in the beam.

The moments are available at the nodes.

import matplotlib.pyplot as plt

plt.plot(beam.X_nodes, beam.bending[:,1])
plt.xlabel('Distance along the beam [m]')

## Cantilever with moment#

We can do the same with a moment

force.force = (0,0,0)
force.moment = (0,1000,0)

s.solve_statics()

True

show(s, camera_pos=(5,-10,3), lookat = (5,0,-1), width=1000, height = 300)

c:\python\miniconda3\envs\book\lib\site-packages\DAVE\visual_helpers\vtkHelpers.py:533: RuntimeWarning: invalid value encountered in divide
axis = axis / length

 Warning! Please use "settings.use_parallel_projection" instead!

c:\python\miniconda3\envs\book\lib\site-packages\DAVE\jupyter\jupyter.py:194: UserWarning: VTK/Vedo issue: plotter is None
warnings.warn("VTK/Vedo issue: plotter is None")

Theory tells us that the deflection and rotation of the right end should be:

$$\delta = ML^2 / 2EI$$

and

$$\Theta = ML / EI$$

Get the values from the model:

M = force.moment[1]  # moment about Y-axis

M*L**2 / (2*EI)

0.5

-right.z

0.4995839925972531

M*L / EI

0.1

right.tilt_y / 100

0.09983341664682813

## Tension and torsion#

Beam can also take tension and torsion.

force.force = (100,0,0)
force.moment = (100,0,0)

s.solve_statics()

True

show(s, camera_pos=(5,-10,3), lookat = (5,0,-1), width=1000, height = 300)

 Warning! Please use "settings.use_parallel_projection" instead!

Tension and torsion forces in the beam are available not at the nodes but at the centers for the beam segments.

plt.plot(beam.X_midpoints, beam.torsion);

plt.plot(beam.X_midpoints, beam.tension);

Now this looks horrible!

Fortunately this is just the way matplotlib displays data which is almost constant.

Looking at the data directly it appears to be comfortingly constant:

beam.torsion

[99.99999999999781,
99.99999999999814,
99.99999999999709,
99.99999999999842,
99.99999999999537,
99.99999999999955,
99.9999999999967,
99.99999999999459,
100.00000000000048,
99.99999999999915,
99.99999999999892,
100.00000000000448,
100.00000000000048,
99.9999999999956,
100.00000000000381,
99.99999999998425,
100.00000000000159,
100.00000000000026,
100.00000000000136,
99.99999999998536]

import matplotlib
matplotlib.rcParams['axes.formatter.useoffset'] = False

plt.plot(beam.X_midpoints, beam.torsion);
plt.ylim((99.9, 100.1));

plt.plot(beam.X_midpoints, beam.tension);
plt.ylim((99.9, 100.1));

Modelling a distributed load is not possible in this way. Loads can only be added at points.

To model a distributed load we would have to manually discretise the model and add discrete loads at each node.

But these is a shortcut that we can take: The beams can have a mass. We can use the mass to model a distributed load as gravity.

s.delete('force')

q = 10 # kN/m

beam.mass = beam.L * q  / 9.81

s.solve_statics()

True

show(s, camera_pos=(5,-10,0), lookat = (5,0,0), width=1000, height = 300)

 Warning! Please use "settings.use_parallel_projection" instead!

Theory tells us that the deflection and rotation of the right end should be:

$$\delta = qL^4 / 8EI$$

and

$$\Theta = qL^3 / 6EI$$

-q*L**4 / (8*EI)

-0.125

right.z

-0.12539340701469195

q*L**3 / (6*EI)

0.016666666666666666

right.tilt_y / 100

0.01668937486888241

## Large deflections#

All the tests done so far were performed on small deflections. That is for a good reason. The Euler/Bernouilli beam theory is only applicable to small displacements. This is because in this theory the moment is derived from the deflection:

$$d^2/dx^2(EI d^2w / dx^2) = q$$

so the moment in the beam is proprtional to the change in slope. This is only valid for small changes.

The implementation in DAVE is valid for large deflections, it is just that the analytical formula used above are not.

## Number of segments#

Beams use a discrete model (see theory https://davedocs.online/beams.html).

When a higher number of segments is used:

• The model become more accurate

• The solver becomes slower

• (The numerical errors build up)

So the number of nodes should be selected with some caution. In general 20 seems to be a good number for beams although in some cases (for example pure tension or torsion) a single segment is just as good.

It is easy to use the “plot_effect” function in scene to check the effect of the number of segments.

Here we calculate the effect of the number of segments on the vertical position of the second end of the beam:

s.plot_effect(evaluate="s['point'].gz",
change_property="n_segments",
change_node="beam",
start=1,
to=25.0,
steps=25);

setting 1.0 results in -0.25043539538010456
setting 2.0 results in -0.15639089359229383
setting 3.0 results in -0.13899103714473474

setting 4.0 results in -0.1329040421012639
setting 5.0 results in -0.1300872247358817

setting 6.0 results in -0.1285572544156294
setting 7.0 results in -0.12763482040356924

setting 8.0 results in -0.1270361175408637

setting 9.0 results in -0.12662568342623784

setting 10.0 results in -0.12633209478116098

setting 11.0 results in -0.12611496195310684

setting 12.0 results in -0.12594964178054247

setting 13.0 results in -0.12582112540100712

setting 14.0 results in -0.12571911178709774

setting 15.0 results in -0.12563692195711026

setting 16.0 results in -0.12556940415251053

setting 17.0 results in -0.12551358019856465

setting 18.0 results in -0.12546679931017593

setting 19.0 results in -0.12542720873744273

setting 20.0 results in -0.1253934070146907

setting 21.0 results in -0.12536431820383623

setting 22.0 results in -0.12533910520139316

setting 23.0 results in -0.12531710883253766

setting 24.0 results in -0.1252978043245916

setting 25.0 results in -0.12528076971477245

– end –


text:
Ex11 6

$\def\abs#1{|#1|}\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\def\tr{\mathord{\mbox{tr}}}\mathbf{Exercise\ 11.6}$

Show that all Hamming codes have distance $3$ so correct single bit flip errors.

Add a New Comment
page revision: 0, last edited: 18 Nov 2012 20:24


text:
# Reflection principle for intuitionistic Zermelo–Fraenkel?

The well-known reflection principle for classical Zermelo–Fraenkel states:

For any formula $$\varphi(x_1,\ldots,x_n)$$ of the language of ZFC with free variables $$x_1,\ldots,x_n$$, ZFC proves $$\forall M_0. \exists M \supseteq M_0. \forall x_1,\ldots,x_n \in M. (\varphi(x_1,\ldots,x_n) \Leftrightarrow \varphi^M(x_1,\ldots,x_n)),$$ where $$\varphi^M$$ is the $$M$$-relativization of $$\varphi$$.

Question. Does the analogous theorem hold with ZFC replaced by IZF, intuitionistic Zermelo–Fraenkel? All proofs of the reflection principle for ZFC that I know of don't obviously carry over to IZF, firstly because of the failure of Scott's trick and secondly because we cannot, unlike in classical texts, assume without loss of generality that formulas don't contain universal quantifiers.

Surely this has been studied before, but I wasn't able to track down a reference.

Motivation. With such a reflection principle in place, we could mimic the definition of ZFC/S, a conservative extension of ZFC useful for category theory, to create IZF/S. The reflection principle is what powers the automatic transfer from a given theorem formulated, for instance, for all small groups, to one formulated for all groups, and this leap shouldn't require non-intuitionistic techniques.

• I added the top-level arXiv tag lo.logic, and a link to the IZF axioms (are they the ones you are thinking of, Ingo?) – David Roberts Dec 20 '18 at 23:46
• For how large a fragment of logic can you prove the reflection principle for IZF? – Andrej Bauer Dec 21 '18 at 7:08
• @IngoBlechschmidt you can prove that using full separation and collection. Define $X := \{ x \in \{0\} | (\exists y)y \in C \}$, then by collection there exists a set $C'$ such that if $X$ is inhabited, then so is $C' \cap C$. Note that $C'$ is not definable, and in fact by a result due to Friedman and Scedrov it is impossible in general for such a $C'$ to be definable. Even with this trick it is difficult to deal with formulas of the form $(\exists y)\phi(x, y)$, with reflection as you stated it, because it is necessary to repeat transfinitely. – aws Dec 24 '18 at 15:25
• @aws Lubarsky provides an answer of the independence of Reflection over $\mathbf{CZF}$ in his article Independence Results around Constructive ZF. – Hanul Jeon Feb 19 '19 at 4:15
• @HanulJeon Thanks. I discussed this with Ingo a bit offlist and he noticed that reflection implies full separation, so in fact reflection is not even provable in $\mathbf{CZF}$ with powerset and $\mathbf{RDC}$. But Lubarsky's proof is interesting too - it shows reflection is not provable even with full separation (but without powerset). – aws Feb 19 '19 at 9:21


text:
# Illustration Wave function in the classically forbidden / allowed region

Get illustration

Share — copy and redistribute the material in any medium or format

Adapt — remix, transform, and build upon the material for any purpose, even commercially.

Sharing and adapting of the illustration is allowed with indication of the link to the illustration.

Example of a wave function that penetrates into a classically forbidden region (red). The wave function oscillates in the classically allowed region (blue) between $$x_1$$ and $$x_2$$. In classically forbidden region the wave function runs towards positive or negative infinity. For certain total energies $$W$$ of the particle, the wave function decreases exponentially. This property of the wave function enables the quantum tunneling.

In the classically allowed region between $$x_1$$ and $$x_2$$, the signs of the wave function $$\mathit{\Psi}(x)$$ and the curvature $$\frac{\partial^2 \mathit{\Psi}(x) }{\partial x^2}$$ (second spatial derivative of the wave function) are opposite at the location $$x$$. This behavior leads to the oscillation of the wave function in the classically allowed region. In the classically forbidden region, the wave function and its curvature always have the same sign (both positive or both negative at the location $$x$$).


text:
Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

# Charged domain boundaries stabilized by translational symmetry breaking in the hybrid improper ferroelectric Ca3–xSrxTi2O7

## Abstract

Charged domain walls and boundaries in ferroelectric materials display distinct phenomena, such as an increased conductivity due to the accumulation of bound charges. Here, we report the electron microscopy observations of atomic-scale arrangements at charged domain boundaries in the hybrid improper ferroelectric Ca2.46Sr0.54Ti2O7. Like in the prototype improper ferroelectric YMnO3, we find that charged domain boundaries in Ca2.46Sr0.54Ti2O7 correspond to out-of-phase boundaries, which separate adjacent domains with a fractional translational shift of the unit cell. In addition, our results show that strontium ions are located at charged domain boundaries. The out-of-phase boundary structure may decrease the polarization charge at the boundary because of the ferrielectric nature of Ca2.46Sr0.54Ti2O7, thereby promoting the stabilization of the charged state. By combining atomic-resolution imaging and density-functional theory calculations, this study proposes an unexplored stabilization mechanism of charged domain boundaries and structural defects accompanying out-of-phase translational shifts.

## Introduction

The physical properties of ferroelectric domain walls and boundaries are sometimes different from those of domains themselves1,2,3; examples include ferroelectricity at the antiphase boundary of an antiferroelectric4 and polar ferroelectric structures at ferroelastic boundaries5. Domain walls can have higher or lower conductivity than domains, a feature that could be used in high-density storage and switching devices2,6,7. In particular, charged domain walls, in which electric polarization vectors are opposed between adjacent domains, have attracted considerable attention because these walls could induce an enhanced conductivity in ferroelectric materials8. Charged domain walls are energetically unfavorable9; however, they are obtained by combining other structural parameters because ferroelectricity is not a primary order parameter in materials known as improper ferroelectrics10. For instance, improper ferroelectric hexagonal manganese oxides (RMnO3: R = Sc, Y, or Ho–Lu) exhibit peculiar cloverleaf domains with head-to-head or tail-to-tail domain walls involving polarization-dependent conductivity11,12,13, which serves as a bias switching for devices14.

Recently, perovskite oxides whose ferroelectricity is induced by more than two nonpolar octahedral tilts (hybrid improper ferroelectricity) have been extensively studied because of the possibility of achieving giant magnetoelectric effects15. Through the exploration of hybrid improper ferroelectrics, numerous peculiar charged domain walls were found in Ca3–xSrxTi2O7 (x = 0.54)16. This material has a layered perovskite structure with the A21am (No. 36) space group, which comprises a stack of perovskite and rock-salt layers, as shown in Fig. 1. The structure is classified as a Ruddlesden–Popper phase (Can+1TinO3n+1) with n = 2. The existence of two nonpolar TiO6 tilts leads to Ca/Sr displacements along the [100] axis, inducing ferroelectricity accompanying charged domain walls. Notably, the number of charged domain walls in the crystal is substantially increased through Sr substitution16. However, the microscopic structure of the charged domain walls remains elusive. In particular, the effect of Sr substitution has not been revealed, which motivated us to perform atomic-resolution scanning transmission electron microscopy (STEM) with energy-dispersive X-ray spectroscopy (EDS).

Another motivation for this work is to reveal the role of crystallographic defects in Ca3–xSrxTi2O7. For RMnO3, recent extensive studies have revealed that antiphase boundaries, which separate adjacent domains with a translational shift, have played an important role in forming charged domain walls17,18,19,20. Translation domains (i.e., domains in which the translational symmetry is broken) are formed due to the trimerization of MnO5, which results in the stabilization of charged ferroelectric domain walls11. However, in the improper ferroelectric Ca3–xSrxTi2O7, the existence and effect of translation domains have not been observed at the atomic scale.

Here, we observed atomic-scale local structures to unveil the stabilization mechanism of the numerous charged defects in Ca2.46Sr0.54Ti2O7. The experimental results show that some charged boundaries are out-of-phase boundaries (translational boundaries) that contain Sr ions at the boundaries. Charged out-of-phase boundaries separate both the ferroelectric polarization directions and the crystallographic domains with phase shifts. Notably, an out-of-phase boundary is defined as a boundary between two regions of a crystal displaced by a fractional translation of a unit cell21,22,23. In particular, an out-of-phase boundary is called an antiphase boundary when the displacement phase is π (i.e., half of the lattice constant). Also note that, in this paper, a domain boundary represents a crystallographic defect that separates two ferroelectric domains with opposite polarization directions while a domain wall means a ferroelectric domain boundary without a compositional change. The out-of-phase displacement allows local electric polarization vectors of adjacent domains to align in the same direction at the boundaries, which stabilizes the charged domain boundaries. This study proves that charged domain boundaries are correlated with crystallographic defects in Ca2.46Sr0.54Ti2O7.

## Results and discussion

### Dark-field images of ferroelectric domains

To explore charged structural defects in the hybrid improper ferroelectric Ca2.46Sr0.54Ti2O7, its (1$$\bar{1}$$0) plane was observed. Observations from the [1$$\bar{1}$$0] direction are expected to be crucial because numerous charged defects run along this direction, which is at 45° to the macroscopic polarization directions16. Figure 2a, b shows dark-field images in the (1$$\bar{1}$$0) plane, which depict the characteristic defects and the ferroelectric domains with polarization pointing toward the left- and right-hand side of the image, as denoted by the blue and red arrows, respectively. The image was obtained under a two-beam condition by exciting a Bragg reflection, which causes the breakdown of Friedel’s law24,25,26. The imaging method identifies ferroelectric domains as bright and dark areas when the ferroelectric polarization is parallel and antiparallel to the Bragg reflection used in the dark-field imaging, respectively. Thus, the images demonstrate that the specimen has numerous neutral 180° and charged head-to-head and tail-to-tail ferroelectric domains. Curved charged structural defects exist, as reported in ref. 27. However, sharp straight charged domain boundaries are frequently formed, as indicated by the squares labeled 3a and 3b. Moreover, broad domain boundaries are also visualized (labeled S6a). Atomic-resolution observations were performed to reveal these characteristic defects.

### Atomic-resolution observations in charged domain boundaries

Figure 3a shows the high-angle annular dark-field STEM (HAADF-STEM) image of the corresponding area indicated in Fig. 2. The lattice structure can be described using five ions between the rock-salt layers, as depicted in Fig. 1: The Ca, Ti, Sr, Ti, and Ca ions are arranged along the [001] axis. The dark lines correspond to the rock-salt layer, since the rock-salt layer has less atomic density than the perovskite layer. Noticeably, the left region is misaligned compared with the right region. The boundary is pinched at an intergrowth with seven layers (yellow arrowhead, Fig. 3b), which agrees with the contrast of the region labeled 3b in Fig. 2. A two-layer displacement occurs at the boundary, as indicated by the dotted line in Fig. 3c. The displacement of the layers corresponds to 0.1889c along the [001] direction (where c is the lattice constant for the c axis), which demonstrates that the boundary is out of phase. Moreover, the boundary is sharp at the atomic scale. The EDS maps also reveal this boundary structure. In the matrix domains, the EDS maps show that Ca ions occupy the alkaline-earth-metal sites of the rock-salt layer (Fig. 3d), Sr ions occupy the alkaline-earth-metal sites of the perovskite layer (Fig. 3e), and Ti ions occupy the metal sites of the oxygen octahedron (Fig. 3f). The atomic arrangement shown in Fig. 3g agrees with the crystal structure illustrated in Fig. 1. However, at the out-of-phase boundary, the EDS maps of Fig. 3d–g show that the alkaline-earth-metal site of the perovskite layer coincides with that of the rock-salt site. As indicated by the dotted lines, the Sr layers (red dots) are located adjacent to the Ca layers (green dots) at the boundary. In the dark-field image of Fig. 2, the domain contrast indicates a tail-to-tail charged boundary, delineating a sharp straight line: The polarization is inverted across the boundary, as indicated by the Ps arrows, and the boundary lies at an angle of approximately 80° to the polarization in the two domains. These results demonstrate that the charged domain boundary coincides with the out-of-phase boundary. The charged boundary that resembles a domain wall in the dark-field image of Fig. 2 is technically a charged defect. Furthermore, the HAADF-STEM image shows brighter dots at the boundary, indicating the presence of heavier ions. The EDS maps show that Sr ions are segregated at the boundary. From these observations, it can be inferred that the formation of charged domain boundaries is influenced by the out-of-phase boundaries that contain Sr ions. The increase in the number of charged domain walls through Sr substitution16 can be explained by the presence of Sr ions at the charged boundaries because Sr ions are likely to form charged boundaries, as observed in this study. From the microscopy study, it was also found that another out-of-phase boundary with one-layer shift coincides with a head-to-head charged boundary involving Sr segregation (Supplementary Fig. 1), demonstrating that out-of-phase boundaries are important in the formation of charged domain boundaries.

It is important to explain why charged domain boundaries are formed at out-of-phase boundaries. Ferroelectric domains are formed as a consequence of the interplay between the wall energy and the depolarization fields induced by the bound charges due to the electric polarization28. Charged domain walls (boundaries) are energetically unfavorable compared with neutral domain walls because they create bound charges, which increase the electrostatic energy of the wall. Bulk Ca2.46Sr0.54Ti2O7 is a ferrielectric material whose polarization is caused by the combination of the two left displacements and one right displacement in perovskite and rock-salt layers, respectively (Fig. 1). When the structure is displaced by 0.1889c at the out-of-phase boundary, the oppositely polarized layers show the same displacement directions, as indicated by the dashed lines in Supplementary Fig. 2. The bound charge ρ is expressed as $$\rho =-\nabla \cdot {P}_{{{{{{\rm{s}}}}}}}$$, where Ps is the electric polarization. Since the bound charge is caused by the divergence of the polarization, parallel polarization directions reduce the local bound charge. Therefore, the configuration observed is energetically favorable, which possibly explains the origin of the charged out-of-phase boundary. Moreover, first-principles calculations show that the polarization vectors differ from layer to layer27,29. The ferrielectricity of Ca2.46Sr0.54Ti2O7 can be described by a stack of parallel- and antiparallel-layered polarizations as shown in Supplementary Fig. 3. Each layered polarization vector points toward opposite directions when the charged domain wall is formed. However, some layers have the same polarization directions if the domain is shifted by two layers (0.1889c) or one layer (0.122c), which are experimentally observed in Fig. 3 and Supplementary Fig. 1, respectively. Owing to the bound charge, these configurations should be more stable than boundaries without a translational shift. This mechanism is possible because of the ferrielectric nature of Ca2.46Sr0.54Ti2O7, which exhibits layered polarization vectors with opposite directions. Hence, this phenomenon should be characteristic of layered perovskite structures showing ferrielectricity.

Furthermore, the difference between Ca2.46Sr0.54Ti2O7 and YMnO3 can also be explained. In YMnO3, the trimerization of MnO5 causes cloverleaf charged ferroelectric domain walls due to the structural phase transition11. The cloverleaf domains comprise ferroelectric domain walls that coincide with antiphase boundaries because ferroelectric walls with antiphase boundaries have lower energy than only ferroelectric walls or antiphase boundaries11,17. Although YMnO3 and Ca2.46Sr0.54Ti2O7 share several properties, such as the formation of charged defects, translation domains, and improper ferroelectricity due to structural order parameters, the origin of the charged domain boundaries accompanying the translation domains in Ca2.46Sr0.54Ti2O7 differs from that in YMnO3. The out-of-phase boundaries in Ca2.46Sr0.54Ti2O7 are not induced by the phase transition from the I4/mmm to the A21am space group because the shear structure of the layer observed in Fig. 3 is not related to the symmetry of the phase transition. This is significantly different from the antiphase boundary due to the transition in YMnO3. The result obtained for Ca2.46Sr0.54Ti2O7 is also supported by the presence of Sr ions at the boundaries because the phase transition will not move Sr, which would take their positions after the material has become crystalline. Thus, the out-of-phase boundaries should exist in the I4/mmm paraelectric phase. Therefore, as the temperature is decreased, charged boundaries are selectively formed at the out-of-phase boundary positions because of the above-explained energy gain at the ferroelectric transition.

Notably, in planar defects with dislocations, one domain is shifted with respect to the adjacent domain, and this shift is characterized by a displacement vector30,31. If the displacement vector is a translation vector of the disordered structure, the boundary is known as an antiphase boundary. By contrast, if the displacement vector is not a lattice translation vector, the boundary is known as a stacking fault. As shown in Fig. 3, the domains are shifted by fractions of the unit cell, demonstrating that the displacement vector is not a lattice translation vector. Thus, the out-of-phase boundary observed in Fig. 3 is also defined as a stacking fault. This study thus reveals charged stacking faults in ferroelectrics, which have not been reported in layered perovskite structures. The displacement vector of the stacking fault could not be determined here because of the diffraction contrast due to the ferroelectricity and the low symmetry of the structure via dark-field imaging. However, Fig. 3 shows that the displacement in the (1$$\bar{1}$$0) plane occurs only along the [001] direction. Besides, the HAADF-STEM images shown in Supplementary Fig. 4 indicate that no displacement can be seen in boundaries for which the dark-field image and the STEM images display boundary contrast when the (001) plane is investigated. These images also show that this type of boundary runs along the [1$$\bar{1}$$0] direction, and the boundary is a planar defect.

### Charged domain boundaries with rotated out-of-phase domains

Another structural defect is observed in Fig. 4a, in which a head-to-head boundary is formed. Figure 4b–f shows magnified HAADF-STEM images and corresponding EDS maps around the defect. Similar to Fig. 3, the rock-salt layers are misaligned by two layers, indicating the formation of an out-of-phase boundary. However, the boundary comprises four ions in width. Besides, the presence of Sr and Ti ions indicates that these regions form a perovskite layer (SrTiO3) because Sr ions preferentially occupy the sites of the perovskite layer, as indicated by the density-functional theory calculations (Supplementary Note 5), and the rock-salt layer is not formed. At the edges of the boundary, dark lines can be observed parallel to the [001] axis in the HAADF-STEM image, whose contrast is characteristic of the rock-salt layer. The elemental maps in Fig. 4c demonstrate that these edges comprise Ca ions parallel to the [001] axis. Since this region has Ca layers that correspond to the rock-salt layer along the [001] direction, the region is considered a 90°-rotated domain whose [001] axis is perpendicular to the crystal [001] axis. The dark-field image in Fig. 4 indicates that this boundary has a head-to-head charged configuration, as illustrated by the arrows in Fig. 4a. These SrTiO3 regions seemingly stabilize the charged domain boundary. Contrary to Ca3Ti2O7, the perovskite structure of simple cubic SrTiO3 is nonpolar. Thus, it is energetically more favorable to form the SrTiO3 perovskite structure at the charged domain boundary. The presence of a nonpolar region increases the stability of the charged domain boundary because these regions separate oppositely polarized domains. Thus, these results reveal that the local structures and origins of the charged domain boundaries in this case are different from those illustrated in Fig. 3, although they are formed in the same crystal.

Finally, the effects of Sr substitution in Ca2.46Sr0.54Ti2O7 are explained. Previous polarization–hysteresis measurements have revealed that the spontaneous polarization magnitude decreases as the Sr content increases16. Such a reduction occurs because the structure of Sr-substituted Ca3–xSrxTi2O7 is nonpolar (space group P42/mnm for 0.915 < x < 1 and I4/mmm for x > 1)32. Accordingly, the increased Sr content reduces the octahedral tilt in the crystal structure, which decreases the polarization. Additionally, Sr substitution results in a decrease in the polarization through the formation of SrTiO3 regions in a crystal. Our observations show out-of-phase boundaries that comprise the SrTiO3 perovskite structure. Furthermore, a large perovskite region with higher Sr content than the matrix region was observed as shown in Fig. 2a and Supplementary Fig. 6. The areas with the SrTiO3 perovskite structure have no polarization because of the nonpolar simple cubic structure of SrTiO3. Hence, the extended perovskite region reduces the spontaneous polarization, which is defined as the electric polarization per volume.

## Conclusions

The dark-field and HAADF-STEM images prove that the crystal has distinct structural boundaries and defects. The elemental maps show the presence of out-of-phase boundaries, which contain Sr ions. Furthermore, it was demonstrated that charged domain boundaries correspond to out-of-phase boundaries because the ferrielectric structure compensates for the polarization charge at the boundaries by means of translational shifts. Notably, this study reveals that some charged boundaries considered as charged domain walls in a previous work16 could actually be charged defects. The structural differences between charged walls and defects are evidenced in their macroscopic behaviors. For example, charged domains walls can be moved via an external electric field, whereas domains separated by charged defects remain stationary under an electric field higher than the coercive field because they are pinned by Sr atoms. Besides, in charged defects of ferroelectrics, the valence states of cations are found to be different from those of domains33. Thus, the charged out-of-phase boundaries presented in this study may have different valence states of Sr and Ti ions at the boundaries. These properties will be explored in future works. Our results provide mechanistic insights into the microscopic structure of boundaries and the Sr substitution effect on the properties of the layered perovskite oxide Ca3–xSrxTi2O7. Moreover, this study shows that atomic-resolution elemental mapping is important for understanding structural defects and macroscopic properties in ferroelectric materials.

## Methods

### Transmission electron microscopy

Atomic-resolution HAADF-STEM and EDS-STEM mapping were performed using a transmission electron microscope (JEM-ARM200F, JEOL Co., Ltd., Japan) equipped with a spherical aberration corrector and double silicon drift detectors. The acceleration voltage, probe semiangle, and current were 200 kV, 22 mrad, and 60 pA, respectively. The solid angle for the whole collection system was ~1.96 Sr. in the EDS mapping. The angular detection range of the HAADF detector for scattered electrons was 90–170 mrad. EDS-STEM images were processed using Fourier filtering for noise reduction34. The single crystal was grown via the floating zone method. Thin specimens along the (1$$\bar{1}$$0) plane were prepared via focused ion beam at 30 kV up to a thickness <100 nm. Subsequent Ar-ion milling at 4 kV and an incident angle of 8° was used to remove the damage from the specimen and reduce the specimen thickness.

The polarization direction vector Ps of each ferroelectric domain was determined from the contrast of the dark-field images, which were captured under two-beam conditions24,25. In these conditions, the intensity of a Bragg reflection $${{{{{{\boldsymbol{g}}}}}}}_{{{{{{\boldsymbol{hkl}}}}}}}$$ is different from that of a reflection with space inversion $${{{{{{\boldsymbol{g}}}}}}}_{\bar{{{{{{\boldsymbol{h}}}}}}}\bar{{{{{{\boldsymbol{k}}}}}}}\bar{{{{{{\boldsymbol{l}}}}}}}}$$, and the breakdown of Friedel’s law occurs. Consequently, the contrast in the ferroelectric domain can be expressed as Ps$$\cdot {{{{{{\boldsymbol{g}}}}}}}_{{{{{{\boldsymbol{hkl}}}}}}}$$: The dark-field image indicates ferroelectric domains as bright areas when the Bragg reflection is parallel to the polarization. Conversely, ferroelectric domains are depicted as dark areas when the reflection is antiparallel to the polarization. The contrast disappears when the polarization is perpendicular to the reflection. The disappearance of the contrast was confirmed by using the 008 reflection, as shown in Supplementary Fig. 7.

### Density-functional theory calculations

All calculations were conducted based on the spin-polarized density-functional theory using the Vienna ab initio simulation package (VASP 6.1.0)35. The core–valence electron interaction was described using the projector augmented wave method36, and the exchange-correlation part was treated within the Perdew–Burke–Ernzerhof generalized gradient approximation37. The on-site Coulomb interaction (Ueff) of the Ti ions was set to 3 eV38. The cutoff energy for the plane-wave basis was 550 eV. The atomic positions shown in Fig. 1 were relaxed to obtain the most stable structure: The Hellman–Feynman forces on every atom converged to be <0.01 eV Å−1. Besides, convergence concerning self-consistent iterations was assumed when the total energy difference between cycles was <1.0  × 10−7 eV. The basis calculations were performed in the unit cell of Ca3Ti2O7 with 48 atoms. Monkhorst–Pack k-point meshes39 (9 × 9 × 3) with a Gaussian searing of 0.05 eV were used to sample the Brillouin zone.

## Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## References

1. Nataf, G. F. et al. Domain-wall engineering and topological defects in ferroelectric and ferroelastic materials. Nat. Rev. Phys. 2, 634–648 (2020).

2. Catalan, G., Seidel, J., Ramesh, R. & Scott, J. F. Domain wall nanoelectronics. Rev. Mod. Phys. 84, 119 (2012).

3. Vasudevan, R. K. et al. Domain wall conduction and polarization-mediated transport in ferroelectrics. Adv. Funct. Mater. 23, 2592–2616 (2013).

4. Wei, X.-K. et al. Ferroelectric translational antiphase boundaries in nonpolar materials. Nat. Commun. 5, 1–8 (2014).

5. Yokota, H. et al. Direct evidence of polar nature of ferroelastic twin boundaries in CaTiO3 obtained by second harmonic generation microscope. Phys. Rev. B 89, 144109 (2014).

6. Seidel, J. et al. Conduction at domain walls in oxide multiferroics. Nat. Mater. 8, 229–234 (2009).

7. Sharma, P. et al. Nonvolatile ferroelectric domain wall memory. Sci. Adv. 3, e1700512 (2017).

8. Sluka, T., Tagantsev, A. K., Bednyakov, P. & Setter, N. Free-electron gas at charged domain walls in insulating BaTiO3. Nat. Commun. 4, 1–6 (2013).

9. Bednyakov, P. S., Sturman, B. I., Sluka, T., Tagantsev, A. K. & Yudin, P. V. Physics and applications of charged domain walls. NPJ Comput. Mater. 4, 65 (2018).

10. Toledano, P. & Toledano, J. Landau Theory Of Phase Transitions, The: Application To Structural, Incommensurate, Magnetic And Liquid Crystal Systems. vol. 3 (World Scientific Publishing Company, 1987).

11. Choi, T. et al. Insulating interlocked ferroelectric and structural antiphase domain walls in multiferroic YMnO3. Nat. Mater. 9, 253 (2010).

12. Wu, W., Horibe, Y., Lee, N., Cheong, S.-W. & Guest, J. R. Conduction of topologically protected charged ferroelectric domain walls. Phys. Rev. Lett. 108, 77203 (2012).

13. Meier, D. et al. Anisotropic conductance at improper ferroelectric domain walls. Nat. Mater. 11, 284 (2012).

14. Mundy, J. A. et al. Functional electronic inversion layers at ferroelectric domain walls. Nat. Mater. 16, 622 (2017).

15. Benedek, N. A. & Fennie, C. J. Hybrid improper ferroelectricity: a mechanism for controllable polarization-magnetization coupling. Phys. Rev. Lett. 106, 107204 (2011).

16. Oh, Y. S., Luo, X., Huang, F.-T., Wang, Y. & Cheong, S.-W. Experimental demonstration of hybrid improper ferroelectricity and the presence of abundant charged walls in (Ca, Sr)3Ti2O7 crystals. Nat. Mater. 14, 407 (2015).

17. Kumagai, Y. & Spaldin, N. A. Structural domain walls in polar hexagonal manganites. Nat. Commun. 4, 1–8 (2013).

18. Jungk, T., Hoffmann, Á., Fiebig, M. & Soergel, E. Electrostatic topology of ferroelectric domains in YMnO3. Appl. Phys. Lett. 97, 12904 (2010).

19. Chae, S. C. et al. Evolution of the domain topology in a ferroelectric. Phys. Rev. Lett. 110, 167601 (2013).

20. Zhang, Q.-H. et al. Topology breaking of the vortex in multiferroicY0.67Lu0.33MnO3. Appl. Phys. Lett. 105, 12902 (2014).

21. Zurbuchen, M. A. et al. Morphology, structure, and nucleation of out-of-phase boundaries (OPBs) in epitaxial films of layered oxides. J. Mater. Res. 22, 1439–1471 (2007).

22. Zurbuchen, M. A. et al. Bismuth volatility effects on the perfection of SrBi2Nb2O9 and SrBi2Ta2O9 films. Appl. Phys. Lett. 82, 4711–4713 (2003).

23. Pan, X. Q., Jiang, J. C., Theis, C. D. & Schlom, D. G. Domain structure of epitaxial Bi4Ti3O12 thin films grown on (001) SrTiO3 substrates. Appl. Phys. Lett. 83, 2315–2317 (2003).

24. Tanaka, M. & Honjo, G. Electron optical studies of barium titanate single crystal films. J. Phys. Soc. Japan 19, 954–970 (1964).

25. Fujimoto, F. Dynamical theory of electron diffraction in Laue-case, I. General theory. J. Phys. Soc. Japan 14, 1558–1568 (1959).

26. Huang, F.-T. et al. Domain topology and domain switching kinetics in a hybrid improper ferroelectric. Nat. Commun. 7, 11602 (2016).

27. Lee, M. H. et al. Hidden antipolar order parameter and entangled Néel-type charged domain walls in hybrid improper ferroelectrics. Phys. Rev. Lett. 119, 157601 (2017).

28. Mitsui, T. & Furuichi, J. Domain structure of rochelle salt and KH2PO4. Phys. Rev. 90, 193 (1953).

29. Benedek, N. A., Mulder, A. T. & Fennie, C. J. Polar octahedral rotations: a path to new multifunctional materials. J. Solid State Chem. 195, 11–20 (2012).

30. Amelinckx, S. & Van Landuyt, J. Transmission electron microscopy. (Academic Press, 2003).

31. De Graef, M. Introduction to Conventional Transmission Electron Microscopy. (Cambridge University Press, 2003).

32. Huang, F.-T. et al. Topological defects at octahedral tilting plethora in bi-layered perovskites. NPJ Quant. Mater. 1, 16017 (2016).

33. Rojac, T. et al. Domain-wall conduction in ferroelectric BiFeO3 controlled by accumulation of charged defects. Nat. Mater. 16, 322–327 (2017).

34. Ishizuka, K., Eilers, P. H. C. & Kogure, T. Optimal noise filters in high-resolution electron microscopy. Micros. Today 15, 16–21 (2007).

35. Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 54, 11169 (1996).

36. Blöchl, P. E. Projector augmented-wave method. Phys. Rev. B 50, 17953 (1994).

37. Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865 (1996).

38. Li, C. F. et al. Structural transitions in hybrid improper ferroelectric Ca3Ti2O7 tuned by site-selective isovalent substitutions: A first-principles study. Phys. Rev. B 97, 184105 (2018).

39. Monkhorst, H. J. & Pack, J. D. Special points for Brillouin-zone integrations. Phys. Rev. B 13, 5188 (1976).

40. Li, G. J., Liu, X. Q., Lu, J. J., Zhu, H. Y. & Chen, X. M. Crystal structural evolution and hybrid improper ferroelectricity in Ruddlesden-Popper Ca3-xSrxTi2O7 ceramics. J. Appl. Phys. 123, 014101 (2018).

41. Momma, K. & Izumi, F. VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data. J. Appl. Crystallogr. 44, 1272–1276 (2011).

## Acknowledgements

This work was supported in part by JSPS KAKENHI Grant Numbers JP19H05814, JP19H05625, JP21K14538, and JP21H04625. The crystal growth at Rutgers was supported by the center for Quantum Materials Synthesis (cQMS), funded by the Gordon and Betty Moore Foundation’s EPiQS initiative through grant GBMF10104, and by Rutgers University.

## Author information

Authors

### Contributions

H.N., K.K., H.T. and S. Mori conceived the research. K.K. and H.N. performed the STEM observations and analyzed the data. S. Mine and M.M. performed the density-functional theory calculations. G.B. and S.-W.C. synthesized the single crystal. S. Mori initiated and supervised the research project. H.N. wrote the manuscript. All authors discussed the results and contributed to the manuscript.

### Corresponding author

Correspondence to Hiroshi Nakajima.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Communications Materials thanks Zijian Hong and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Primary Handling Editor: Aldo Isidori.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Nakajima, H., Kurushima, K., Mine, S. et al. Charged domain boundaries stabilized by translational symmetry breaking in the hybrid improper ferroelectric Ca3–xSrxTi2O7. Commun Mater 2, 109 (2021). https://doi.org/10.1038/s43246-021-00215-1

• Accepted:

• Published:

• DOI: https://doi.org/10.1038/s43246-021-00215-1


text:
# AELIP-34: Laminar Allocation for NFT Gated Deals

Author SeaGolem Approved TBD TBD 2022-09-08

## Simple Summary

This AELIP proposes to create a new allocation mechanism, commonly referred to as a "Laminar allocation", for investors in NFT gated deals when there is excess interest in a capped deal. In the proposed allocation, smaller investors are deallocated less than larger investors to ensure a wider distribution of deal tokens.

## Abstract

In contrast to a proportional allocation which always provides the highest pool share to the largest investor, this system proposes to calculate a maximum allocation (cutoff) based on the number of NFTs used and the investment values they each represent.

In this system, investment under the cutoff value will not be diluted, and all investments above would be limited to this cutoff value. For NFT gated deals, an investor may still spread the value of their investment across multiple NFTs held in the same wallet. Instead of doing the calculation per investor wallet, the calculation will be done per NFT used to invest in the pool.

## Motivation

NFTs are scarce by nature and therefore alternative allocation mechanisms in NFT gated pools are immune to Sybil attacks on the pool through multiple wallets creation. Thanks to this property, new ways to allocate investments can be created.

NFTs are very often associated with communities composed of people with common interests that already established strong social bounds. In a sense, investing with your NFT community is more akin to investing with a group of friends rather than competing against unknown market actors.

Laminar allocation aims at sponsors wanting more decentralization for their pool outcome and prioritizing equity among community members.

## Specification

### Overview

Allocation/Deallocation option: When creating an uncapped NFT Gated deal, a sponsor or protocol will have the option to use the "Laminar allocation" instead of the currently only option "Proportional Deallocation". Investors will then associate their investment with either one or multiple NFTs. Hence, each NFT always represents a non-zero investment value.

When the time of allocating pool tokens to the investors comes, if the raised capital is above the desired capital target, a laminar allocation will happen. It consists of calculating the maximum allocation (cut-off) allowed for each NFT, in order for the deal to distribute, without dilution, to the maximum number of NFTs whose values are under this cut-off.

Example: Let’s imagine the following scenario. A NFT gated uncapped pool is opened by a sponsor, trying to raise 50k sUSD. 10 unique holders each holding 1 NFT invest in the pool. The below table shows the outcome of a laminar allocation.

NFT Sprinkling: Laminar Allocation requires us to add a third allocation mechanism to NFTs in an Aelin deal: “NFT Sprinkling”. The 2 current ways to use NFTs at the moment are: You need at least 1 to participate. It is then used as an entrance ticket and has an unlimited investment amount. A cap per NFT is set. Having multiple NFTs becomes an advantage allowing for a greater investment.

The only way for an investor to skew the laminar allocation in its favor is to divide its investment across multiple NFTs, incentivizing them to buy more. To take advantage of that property this AELIP proposes to create a third option: “NFT Sprinkling”.

When selected by a sponsor at the creation of the pool, the investors will be able to spread their uncapped investment across multiple NFTs being held in the same wallet. Example: an investor wants to invest 40k sUSD and holds 4 NFTs. By selecting them, each NFT will be attributed the value of 10k sUSD.

In this case, a NFT does not just have a cap per investment or not, it also helps to reduce how much an investor is deallocated.

### Rationale

Proportional allocation partially rises from the difficulty to find an alternative not subjected to wallet gaming (Sybil attack). As wallets are free, unlimited and extremely easy to create, one malicious actor can take advantage of any solution based on a per wallet distribution. NFTs, by their nature, allow new approaches to emerge.

Visualizing the difference: The distribution problem is equivalent to the following one: a tube, with a constant section, filled with water, represents each NFT. The height of water in each tube represents the investment value held by each NFT. The capital target of the pool is represented by an empty tank below the tubes, of a volume corresponding to the investment target. How to fill the tank with the tubes' content?

Proportional: In this setup, each tube has a faucet at the bottom. You have to calculate the ratio (Total volume in the tubes/Capital Target).Then mark all the tubes using this ratio. Finally, use the faucets to empty each tube until the marking

Laminar: Here, you link all tubes together with 1 faucet at the bottom. Simply open the faucet until the tank is filled; gravity will take care of the distribution. Hence the term "Laminar"

Proportional/Laminar output comparison:

NFT Sprinkling: The only way to skew the allocation of a fixed investment amount is to spread it across multiple NFTs. NFT Sprinkling uses and formalizes this practice. In our use case, it can be considered a positive thing as it provides more utility to the NFTs from a collection and drives value into accumulating more of them. This has the potential to create buying pressure on a collection, increasing its floor value and thus be beneficial for all holders.

Impact of NFT sprinkling on the laminar allocation example: An uncapped NFT gated pool has been set, trying to raise $100,000 250 unique holders of 1 NFT have each invested$1,000 in a pool On top of this 1 unique holder used their NFTs to invest $50,000 Below is the impact of the number of NFTs this holder uses with NFT sprinkling on the pool and his own allocation. For reference, if proportional allocation were used the holders allocation would be$16,667. This holder would need to hold 50 NFTs to reach the same allocation he would have had using proportional allocation.

### Technical Specification

Below is an un-optimized python code calculating the laminar distribution that can be used as reference for the developments of the solidity code

### Test Cases

NA

Copyright and related rights waived via CC0.


text:
Open Access

# Limited and Dunford-Pettis operators on Banach lattices

### Abstract

This paper is devoted to investigation of conditions on a pair of Banach lattices $E; F$ under which every positive Dunford-Pettis operator $T:E\rightarrow F$ is limited. Mainly, it is proved that if every positive Dunford-Pettis operator $T:E\rightarrow F$ is limited, then the norm on $E'$ is order continuous or $F$ is finite dimensional. Also, it is proved that every positive Dunford-Pettis operator $T:E\rightarrow F$ is limited, if one of the following statements is valid:
1) The norm on $E^{\prime }$ is order continuous, and $F^{\prime }$ has weak$^{\ast }$ sequentially continuous lattice operations.
2) The topological dual $E^{\prime }$is discrete and its norm is order continuous.
3) The norm of $E^{\prime }$ is order continuous and the lattice operations in $E^{^{\prime }}$ are weak$^{\ast }$ sequentially continuous.
4) The norms of $E$ and of $E^{\prime }$ are order continuous.

Key words: Limited operator, Dunford-Pettis operator, Banach lattice, order continuous norm.

### Article Information

 Title Limited and Dunford-Pettis operators on Banach lattices Source Methods Funct. Anal. Topology, Vol. 25 (2019), no. 3, 205-210 MathSciNet MR4016207 Milestones Received 12/02/2019; Revised 04/04/2019 Copyright The Author(s) 2019 (CC BY-SA)

### Authors Information

Abdennabi EL Aloui
Department of Mathematics, Faculty Polydisciplinary of Larache, Abdelmalek Essaadi University, P.O. Box 745, Larache 92004, Morocco

Khalid Bouras
Department of Mathematics, Faculty Polydisciplinary of Larache, Abdelmalek Essaadi University, P.O. Box 745, Larache 92004, Morocco

Aziz Elbour
Department of Mathematics, Faculty of Science and Technology, Moulay Ismail University, P.O. Box 509, Boutalamine 52000, Errachidia, Morocco

### Citation Example

Khalid Bouras, Abdennabi EL Aloui, and Aziz Elbour, Limited and Dunford-Pettis operators on Banach lattices, Methods Funct. Anal. Topology 25 (2019), no. 3, 205-210.

### BibTex

@article {MFAT1205,
AUTHOR = {Khalid Bouras and Abdennabi EL Aloui and Aziz Elbour},
TITLE = {Limited and Dunford-Pettis operators on Banach lattices},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {25},
YEAR = {2019},
NUMBER = {3},
PAGES = {205-210},
ISSN = {1029-3531},
MRNUMBER = {MR4016207},
URL = {http://mfat.imath.kiev.ua/article/?id=1205},
}

Coming Soon.


text:
# Numerical solution of a locally Lipschitz differential equation. (Gross-Pitaevskii)

A (very simplified) version of the GP-equation, can be written as

$$\frac{d\alpha}{dt}=-Ui\alpha^*\alpha^2=-Ui|\alpha|^2\alpha$$,

Where $$i$$ is the imaginary unit. In contrast to what I first thought, this problem seems not one-sided Lipschitz, so that an Implicit Euler method is generally not sufficient to tame numerical instability...; it is locally Lipschitz however.

Are there other implicit or explicit methods that are guaranteed to be numerically stable requiring only local Lipschitzness?


text:
# There is a fire sensor system in a building. In long series of experim

There is a fire sensor system in a building. In long series of experiments, it was found that the accuracy of the fire sensor system can be described as follows:
- In the case of a fire, the sensor system will be activated with probability 0.97.
- In a night without fire, the sensor system may be activated with probability 0.01. The probability of a fire in this specific area is founded $$\displaystyle{\frac{{{2}}}{{{397}}}}$$ from the past data.
The fire sensor system is activated. Find the probability that there was actually a fire.

• Questions are typically answered in as fast as 30 minutes

### Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

vrangett

Step 1
Given Information :
In the case of a fire, the sensor system will be activated with probability 0.97
In a night without fire, the sensor system may be activated with probability 0.01.
The probability of a fire in this specific area is founded $$\displaystyle={\frac{{{2}}}{{{397}}}}={0.005}$$
First we will draw a probability contingency table with all the information provided:
$$\begin{array}{|c|c|} \hline &\text{Sensor system is activate}&\text{Sensor system is not activate}&Total\\ \hline \text{There is fire}&0.97 \cdot 0.005=0.0049&0.005-0.0049=0.0001&\frac{2}{397}=0.005\\ \hline \text{There is no fire}&0.01 \cdot (1-0.005)=0.00995&0.995-0.00995=0.98505&1-0.005=0.995\\ \hline Total&0.01485&0.98515&1\\ \hline \end{array}$$
Step 2
Now, if The fire sensor system is activated. Find the probability that there was actually a fire
We will use the principle of conditional probability:
$\begin{array}{|c|c|} \hline &\text{Sensor system is activated}\\ \hline \text{There is fire}&0.97 \cdot 0.005=0.0049\\ \hline \text{There is no fire}&0.01 \cdot (1-0.005)=0.00995\\ \hline Total&0.01485\\ \hline \end{array}$
$$\displaystyle{P}{\left(\text{There is fire}{\mid}\text{sensor system is activated}\right)}=$$
$$\displaystyle{\frac{{{P}{\left(\text{There is fire}\cap\text{sensor system is activated}\right)}}}{{\text{P(sensor system is activated)}}}}$$
$$\displaystyle{P}{\left(\text{There is fire}{\mid}\text{sensor system is activated}\right)}={\frac{{{0.0049}}}{{{0.01485}}}}={0.329}$$
probability that there was actually a fire $$\displaystyle={0.329}$$


text:
# Jon Dehdari

## How do I do squiggly lines in LaTeX?

\usepackage[normalem]{ulem}	% Part of the standard distribution
...
\uwave{text goes here}

## How do I do strikethrough lines in LaTeX?

For normal text,
\usepackage[normalem]{ulem}	% Part of the standard distribution
...
\sout{text goes here}

For math mode,

\usepackage{cancel}		% Part of the standard distribution
...
\cancel{math text goes here}

## How do I use my favorite TrueType font (TTF) in LaTeX?

Use the command lualatex or xelatex   – both of which are included in TeXLive, MikTeX, and MacTeX – and in the preamble add the following:
\usepackage{fontspec}
\setmainfont{Arial Unicode}	% For example

This also works for OpenType fonts (OTF), which is a superior format.

The above command sets that font for the entire document. If you want to use a special font for just a snippet of text, do the following:

\usepackage{fontspec}
\newfontfamily\myfont{Arial Unicode}
...
Foo \myfont{bar} baz

## How do I do hanging/old-style numbers in LaTeX?

You can get them by using Palatino, one of the finest and most mature fonts available. However, avoid the palatino package itself.
\usepackage[osf]{mathpazo}	% Mathpazo package includes accompanying math fonts

## How do I draw dependency arrows in LaTeX?

One way is to use the tree-dvips package, part of the texlive-humanities package in Debian/Ubuntu. Unfortunately you'll need to run the old-fashioned latex, dvipdf sequence since it depends on PostScript code.
\usepackage{tree-dvips}
...
\node{the}{the} \node{dog}{dog} \node{eats}{eats} \node{food}{food}

\anodecurve[t]{the}{dog}{1.1em}
\anodecurve[t]{eats}{the}{1.8em}
\anodecurve[t]{eats}{food}{1.2em}

## How do I get argmax in LaTeX formulas?

\DeclareMathOperator*{\argmax}{arg\,max}   % In your preamble
...
\argmax_{...}                              % In your formula

## How do I get really really small fonts in LaTeX?

For a font smaller than \tiny (or larger than \Huge), add the following in your preample:
\def\Tiny{ \font\Tinyfont = cmr10 at 3pt \relax  \Tinyfont}

Now you can use {\Tiny ... }. Replace the 3pt with any positive real-numbered value as you see fit. Unfortunately this only uses Computer Modern fonts with the roman typeface.

A less robust, but more elegant solution is to put the following command in your preamble:

\def\Tiny{\fontsize{5pt}{5pt} \selectfont}

The usage is the same as before, but this only scales down to 5pt. It works with most fonts. You can use the type1cm package for smaller sizes.

A really robust, but very different way is to use the \scalebox command from the graphicx package:

\usepackage{graphicx}                        % In your preamble.
...
\scalebox{0.20}{Really small text here}

You could of course define a macro in your preamble using this:
\usepackage{graphicx}                        % In your preamble.
\newcommand{\teeny}[1]{\scalebox{0.20}{#1}}  % In your preamble
...
\teeny{Really small text here, with math $\alpha_i$}

## How do I get bold Greek letters in math mode?

This is one of the lamest quirks about LaTeX 2ε. Anyways, here's the sad answer:
\usepackage{amsmath}	% In your preamble.  or \usepackage{amsbsy}
...
$\boldsymbol{\lambda}$

If   \boldsymbol{}   still doesn't work (like for ∑, ∏, ∫, etc.), then use the "poor man's bold", a bunch of jittered copies:
$\pmb{\sum}$

## How do I prevent a linguistic gloss from getting split across pages?

Use the samepage environment around the whole block.
\begin{samepage}
\exg. Dies ist eine erste Glosse \\
This is a first gloss

\end{samepage}

The above example uses the Linguex package, but the samepage environment should work for others as well.

## How do I do multi-line comments or verbatims?

\usepackage{verbatim}	% Part of the standard distribution
...
\begin{comment}
This is a comment
\end{verbatim}
\end{frame}

## How do I do hanging indents?

The easiest way is to use the description environment, with argumentless items:
\begin{description}
\item Text goes here
\item More text goes here
\end{description}

## How do I do force math-mode subscripted items underneath, in running text or in fractions?

Instead of the underscore _ use \limits_ :
... so that we have $\sum\limits_{i \in C} g(i)$ and then ...

$$r = \frac{ \sum\limits_{j=1} f(i) }{Z}$$

Use judiciously.

## How do I put a big "DRAFT" stamp across the first page of my document?

\usepackage[firstpage]{draftwatermark}  % Not part of the standard distribution.

Download the package at CTAN . If you want the message on every page, remove the optional argument, so just \usepackage{draftwatermark} . You can customize or change the message, such as \SetWatermarkText{Hi!} . See the documentation for more details.

## How do I produce an A4 (or Letter) document?

Current LaTeX distributions do not honor the page dimensions option (eg. a4paper or letterpaper) to your \documentclass command very well. You can put the following TeX commands in your preamble:
\pdfpagewidth=210 true mm
\pdfpageheight=297 true mm

Another solution is to use the geometry package, as:
\usepackage[a4paper]{geometry}   % Part of the standard distribution

This is a great package, but it will change your default margins as well. Manually adjusting the margins with this package is pretty easy; see the documentation for details.

## How do I do inline comments?

\newcommand{\comment}[1]{}
...
Hello \comment{My Comment} world

## How do I extract a vector image from a PDF, to use in my LaTeX document?

One relatively easy way is to use Inkscape, an excellent Free vector graphics program.
1. Open the PDF in Inkscape, selecting the page number that the image is found on.
2. Select the entire range of the image, from one corner of a rectangle that would enclose it, to the opposite corner.
3. Copy the selection, using the usual Control-C or clicking Edit->Copy.
4. Create a new document, either by typing Control-N or clicking the top-left icon.
5. Paste the saved image selection into the new document, using the usual Control-V or clicking Edit->Paste.
6. Go to the Document Properties dialog box, either by typing Shift-Control-D or clicking File->Document Properties.
7. Click on "Resize Page to Content", then on the button "Resize Page to Drawing or Selection"
8. Now save the new document, either by the usual Control-S or clicking File->Save.
9. In the bottom-right corner, choose the file type Portable Document Format (PDF).
10. You can leave most of the defaults, although I recommend making sure that the button "Convert texts to paths" is unchecked. This will allow text in the image to be searchable in your LaTeX document.
11. Now you can use this image PDF as a normal image in your LaTeX file, using the usual \includegraphics{} command (from the graphicx package).
12. Enjoy your awesome document! Don't forget to cite the original authors in your caption.

 '' –LaTeX


text:
Canonical commutation relation, from QM to QFT.

1. Feb 18, 2017

annihilatorM

1. The problem statement, all variables and given/known data

This is a system of n coupled harmonic oscillators in 1 dimension.

Since the distance between neighboring oscillators is $\Delta x$ one can characterize the oscillators equally well by $q(x,t)$ instead of $q_j(t)$. Then $q_{j \pm 1}$ should be replaced by $q(x \pm \Delta x , t)$. Show that the quantization condition

$$[\dot{q}_j(t), q_k (t)] = - \frac{i}{\mu} \delta _{jk}$$

can be expressed as

$$[ \dot{q} (x,t), q (x',t)] = - \frac{i}{\mu} \int_{ \frac{- \Delta x}{2}}^{\frac{ \Delta x}{2}} dy \delta (y-x'+x)$$

Hint: Distinguish the two cases $x=x'$ and $| x-x' | = j \Delta x$

2. Relevant equations

3. The attempt at a solution

Making the substitution of $q(x,t)$ in place of $q_j (t)$ I obtain

$$[ \dot{q} (x,t), q (x',t)] = \delta (x-x')$$

which is the expected commutation relation i've seen from other sources, however, I don't understand why the integration with respect to y is introduced or where the integration bounds come from. What is wrong with simply writing down what I have above?

2. Feb 23, 2017

Greg Bernhardt

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.


text:
residuals.mppm

0th

Percentile

Residuals for Point Process Model Fitted to Multiple Point Patterns

Given a point process model fitted to multiple point patterns, compute residuals for each pattern.

Keywords
models, spatial
Usage
# S3 method for mppm
residuals(object, type = "raw", ...,
fittedvalues = fitted.mppm(object))
Arguments
object

Fitted point process model (object of class "mppm").

Ignored.

type

Type of residuals: either "raw", "pearson" or "inverse". Partially matched.

fittedvalues

Advanced use only. Fitted values of the model to be used in the calculation.

Details

Baddeley et al (2005) defined residuals for the fit of a point process model to spatial point pattern data. For an explanation of these residuals, see the help file for residuals.ppm.

This function computes the residuals for a point process model fitted to multiple point patterns. The object should be an object of class "mppm" obtained from mppm.

The return value is a list. The number of entries in the list equals the number of point patterns in the original data. Each entry in the list has the same format as the output of residuals.ppm. That is, each entry in the list is a signed measure (object of class "msr") giving the residual measure for the corresponding point pattern.

Value

A list of signed measures (objects of class "msr") giving the residual measure for each of the original point patterns. See Details.

References

Baddeley, A., Turner, R., Moller, J. and Hazelton, M. (2005) Residual analysis for spatial point processes. Journal of the Royal Statistical Society, Series B 67, 617--666.

Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. London: Chapman and Hall/CRC Press.

mppm, residuals.mppm

Aliases
• residuals.mppm
Examples
# NOT RUN {
fit <- mppm(Bugs ~ x, hyperframe(Bugs=waterstriders))
r <- residuals(fit)
# compute total residual for each point pattern
rtot <- sapply(r, integral.msr)
# standardise the total residuals
areas <- sapply(windows.mppm(fit), area.owin)
rtot/sqrt(areas)
# }

Documentation reproduced from package spatstat, version 1.61-0, License: GPL (>= 2)

Community examples

Looks like there are no examples yet.