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# Shimura-Taniyama conjecture

2010 Mathematics Subject Classification: Primary: 11G05 Secondary: 11D4111F1111F8011G18 [MSN][ZBL]

Shimura–Taniyama–Weil conjecture, Taniyama–Shimura conjecture, Taniyama–Weil conjecture, modularity conjecture

A conjecture that postulates a deep connection between elliptic curves (cf. Elliptic curve) over the rational numbers and modular forms (cf. Modular form). It has been completely proved thanks to the fundamental work of A. Wiles and R. Taylor [Wi], [TaWi], and its further refinements [Di], [CoDiTa].

Let $\def\G{\Gamma}\G_0(N)$ be the group of matrices in $\def\SL{\textrm{SL}}\SL_2(\Z)$ which are upper-triangular modulo a given positive integer $N$. It acts as a discrete group of Mobius transformations (cf. also Discrete group of transformations; Fractional-linear mapping) on the Poincaré upper half-plane $H=\{c\in\C | \textrm{Im}(z) > 0\}$ (cf. also Poincaré model). A cusp form of weight $2$ for $\G_0(N)$ is an analytic function $f$ on $H$ satisfying the relation

$$f\Big(\frac{az+b}{cz+d}\Big) = (zc+d)^2 f(z) \textrm{ for all } \begin{pmatrix}a&b\\c&d\end{pmatrix}\in\G_0(N),$$ together with suitable growth conditions on the boundary of $H$ (cf. also Modular form). The function $f$ is periodic of period $1$, and it can be written as a Fourier series in $q=e^{2\pi iz}$ with no constant term: $f(z)=\sum_{n=1}^\infty \def\l{\lambda} \l_n q^n$. The Dirichlet series $L(f,s) = \sum \l_n n^{-s}$ is called the $L$-function attached to $f$ (cf. also Fourier coefficients of automorphic forms; Dirichlet $L$-function). It is essentially the Mellin transform of $f$:

$$\Lambda(f,s):= \G(s)L(f,s) = (2\pi)^2 \int_0^\infty f(iy)y^{s-1}dy.$$ The space of cusp forms of weight $2$ on $\G_0(N)$ is a finite-dimensional vector space and is preserved by the involution $W_N$ defined by $W_N(f)(z) = Nz^2f(-1/(Nz))$. E. Hecke has shown that if $f$ lies in one of the two eigenspaces for this involution (with eigenvalue $w=\pm1$), then $L(f,s)$ satisfies the functional equation $\Lambda(f,s)=-w\Lambda(f,2-s)$, and that $L(f,s)$ has an analytic continuation to all of $\C$.

Let $E$ be an elliptic curve over the rational numbers, and let $L(E,s)$ denote its Hasse–Weil $L$-series. The curve $E$ is said to be modular if there exists a cusp form $f$ of weight $2$ on $\G_0(N)$, for some $N$, such that $L(E,s)=L(f,s)$. The Shimura–Taniyama conjecture asserts that every elliptic curve over $\Q$ is modular. Thus, it gives a framework for proving the analytic continuation and functional equation for $L(E,s)$. It is prototypical of a general relationship between the $L$-functions attached to arithmetic objects and those attached to automorphic forms (cf. also Automorphic form), as described in the far-reaching Langlands program.

A. Weil's refinement of the conjecture predicts that the integer $N$ is equal to the arithmetic conductor of $E$ (cf. also Elliptic curve). One now knows that every elliptic curve is modular. Wiles' method proceeds by viewing the Shimura–Taniyama conjecture in a wider framework which predicts the modularity of the (two-dimensional) Galois representations arising from the cohomology of varieties over $\Q$ (cf. also Galois theory).

The modularity of $E$ can also be formulated as the statement that $E$ is a quotient of the modular curve $X_0(N)$ over $\Q$; this curve represents the solution to the moduli problem of classifying pairs $(A,C)$ consisting of an elliptic curve $A$ with a distinguished cyclic subgroup $C$ of order $N$. Alternately, if $E$ is modular, then there is a (non-constant) complex-analytic uniformization $H/\G_0(N) \to E(\C)$.

The importance of the Shimura–Taniyama conjecture is manifold. Firstly, it gives the analytic continuation of $L(E,s)$ for a large class of elliptic curves. The $L$-function itself plays a key role in the study of $E$, most notably through the celebrated Birch–Swinnerton Dyer conjecture. Secondly, the modular curve $X_0(N)$ is endowed with a natural collection of algebraic points arising from the theory of complex multiplication (cf. also Elliptic curve), and the existence of a modular parametrization allows the construction of points on $E$ defined over Abelian extensions of certain imaginary quadratic fields. This fact was exploited by B.H.. Gross and D. Zagier and by V.A. Kolyvagin to give strong evidence for the Birch–Swinnerton Dyer conjecture for $E$, under the assumption that $E$ is modular.

The Shimura–Taniyama conjecture admits various generalizations. Replacing $\Q$ by an arbitrary number field $K$, it predicts that an elliptic curve $E$ over $K$ is associated to an automorphic form on $\textrm{GL}_2(K)$. When $K$ is totally real, such an $E$ is often uniformized by a Shimura curve attached to a suitable quaternion algebra (cf. also Quaternion) over $K$ with exactly one split place at infinity (when $K$ is of odd degree, or when $E$ has at least one prime of multiplicative reduction). In the context of function fields over finite fields, the Shimura–Taniyama conjecture admits an analogue which was established earlier by V.G. Drinfel'd using methods different from those of Wiles.

#### References

 [BrCoDiTa] C. Breuil, B. Conrad, F. Diamond, R. Taylor, "On the Modularity of elliptic curves" J. Amer. Math. Soc. 14 (2001), no. 4, 843–939 MR1839918 Zbl 0982.11033 [CoDiTa] B. Conrad, F. Diamond, R. Taylor, "Modularity of certain potentially Barsotti–Tate Galois representations" J. Amer. Math. Soc. J. Amer. Math. Soc. 12 (1999), no. 2, 521–567. MR639612 Zbl 0923.11085 [Di] F. Diamond, "On deformation rings and Hecke rings" Ann. of Math., 144 : 1–2 (1996) pp. 137–166 MR1405946 Zbl 0867.11032 [TaWi] R. Taylor, A. Wiles, "Ring-theoretic properties of certain Hecke algebras" Ann. of Math., 141 : 2–3 (1995) pp. 553–572 MR1333036 Zbl 0823.11030 [Wi] A. Wiles, "Modular elliptic curves and Fermat's last theorem" Ann. of Math., 141 : 2–3 (1995) pp. 443–551 MR1333035 Zbl 0864.11029 Zbl 0823.11029
How to Cite This Entry:
Shimura–Taniyama conjecture. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Shimura%E2%80%93Taniyama_conjecture&oldid=23018


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# How do I split a string into sentences while maintaining all the spaces in Java? [closed]

I tried the following regex:

sentences = sb.toString().split("(?<=[a-z])*\\.\\s*");

I am using a stringBuilder sb and converting it to string and then using a split function
The regex checks for 0 or more characters before '.' and 0 or more spaces after the '.'

However, it doesn't work for the following input

Hello World. Shipped to U.S on Friday.We are here .Good input

But I need to keep the space before We are here

Req Output

Hello World
Shipped to U.S on Friday
We are here
Good input

• why cant u jst split on the . – leoOrion Aug 8 '20 at 18:42
• Well, remove the \\s* part then? – 41686d6564 Aug 8 '20 at 18:46
• @leoOrion I can only assume that the OP doesn't want to split on the dot in "15.05", for example. – 41686d6564 Aug 8 '20 at 18:47
• 15.05?? uh?? what? – leoOrion Aug 8 '20 at 18:49
• @leoOrion Um, what's so weird about 15.05 (as an example)? – 41686d6564 Aug 8 '20 at 18:50

use this regex: ([^\.]+)(\.|$)*? you can read about group matchers and see the full matches here : https://regex101.com/r/yV9GES/5 edit: updated the link for answer in the comment. • my bad! the closest i can come to match the OP's req is this regex: /([^\.]+)(\.|$)*?/ It does break for Shipped to U.S on Friday though. – Khushboo Aug 8 '20 at 19:46

Split your string using \\. i.e. on .

Demo:

import java.util.Arrays;

public class Main {
public static void main(String[] args) {
System.out.println(Arrays.toString("Hello World. We are here .Good input.".split("\\.")));
}
}

Output:

[Hello World,  We are here , Good input]

• I think we need to add trim() for every generated statement. – hitesh bedre Aug 8 '20 at 18:47
• @hiteshbedre - No. OP wants to keep the space. Please check his expected output. – Arvind Kumar Avinash Aug 8 '20 at 18:48

Why do you have to use a RegEx?

You can simply use indexOf and substring

  public List<String> splitOnDot(String input) {
List<String> result = new ArrayList<>();
int idx;
while ((idx = input.indexOf('.')) != -1) {

@Test

format: plain


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## Files in this item

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## Description

 Title: Absorptions In The Visible Of Protonated Pyrene Collisionally Cooled To 15��k Author(s): Rice, C.A. Contributor(s): Maier, John P.; Gause, Oliver; Hardy, Francois Xavier Subject(s): Mini-symposium: Beyond the Mass-to-Charge Ratio: Spectroscopic Probes of the Structures of Ions Abstract: Protonated polycyclic hydrocarbons have been added to the list of suggested carriers of the diffuse interstellar absorptions. To test this proposition requires laboratory spectra measured under interstellar conditions, in particular with the rotational and vibrational degrees of freedom equilibrated to low temperatures. This has been achieved for protonated pyrene with absorption bands in the visible, using an ion trap and collisional cooling to $\approx$\,15\,K. A two-photon excitation-dissociation scheme was employed to record the (1)\,$^1$A$'~\leftarrow~X\,^1$A$'$ electronic spectrum on around 10$^5$ ions per duty cycle. The origin band of the absorption spectrum of this relatively large polycyclic aromatic species with 27 atoms is located at 4858.86\,\AA. Two further comparably intense spectral features are present at 4834.48 and 4809.32\,\AA. This is one of the largest protonated aromatics studied in the gas phase and compared to astonomical observations; however, it is not a carrier of known diffuse interstellar bands. Issue Date: 2014-06-18 Publisher: International Symposium on Molecular Spectroscopy Citation Info: Rice, C.A.; Maier, J.P.; Gause, O.; Hardy, F.X. ABSORPTIONS IN THE VISIBLE OF PROTONATED PYRENE COLLISIONALLY COOLED TO 15 K. Proceedings of the International Symposium on Molecular Spectroscopy, Urbana, IL, June 16-21, 2014. DOI: 10.15278/isms.2014.WG04 Genre: CONFERENCE PAPER/PRESENTATION Type: Text Language: English URI: http://hdl.handle.net/2142/50872 DOI: 10.15278/isms.2014.WG04 Rights Information: Copyright 2014 by the authors. Licensed under a Creative Commons Attribution 4.0 International License. http://creativecommons.org/licenses/by/4.0/ Date Available in IDEALS: 2014-09-172015-04-14


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# Variational message passing¶

This section briefly describes the variational message passing (VMP) framework, which is currently the only implemented inference engine in BayesPy. The variational Bayesian (VB) inference engine in BayesPy assumes that the posterior approximation factorizes with respect to nodes and plates. VMP is based on updating one node at a time (the plates in one node can be updated simultaneously) and iteratively updating all nodes in turns until convergence.

## Standard update equation¶

The general update equation for the factorized approximation of node is the following:

(1)

where and are the set of parents and children of , respectively. Thus, the posterior approximation of a node is updated by taking a sum of the expectations of all log densities in which the node variable appears. The expectations are over the approximate distribution of all other variables than . Actually, not all the variables are needed, because the non-constant part depends only on the Markov blanket of . This leads to a local optimization scheme, which uses messages from neighbouring nodes.

The messages are simple for conjugate exponential family models. An exponential family distribution has the following log probability density function:

(2)

where is the set of parents, is the sufficient statistic vector, is the natural parameter vector, is the negative log normalizer, and is the log base function. Note that the log density is linear with respect to the terms that are functions of : and . If a parent has a conjugate prior, (2) is also linear with respect to the parent’s sufficient statistic vector. Thus, (2) can be re-organized with respect to a parent as

where is the sufficient statistic vector of and the constant part is constant with respect to . Thus, the update equation (1) for can be written as

where the summation is over all the child nodes of . Because of the conjugacy, depends (multi)linearly on the parents’ sufficient statistic vector. Similarly, depends (multi)linearly on the expectations of the children’s and co-parents’ sufficient statistics. This gives the following update equation for the natural parameter vector of the posterior approximation :

(3)

## Variational messages¶

The update equation (3) leads to a message passing scheme: the term is a function of the parents’ sufficient statistic vector and the term can be interpreted as a message from the child node . Thus, the message from the child node to the parent node is

which can be computed as a function of the sufficient statistic vector of the co-parent nodes of and the sufficient statistic vector of the child node . The message from the parent node to the child node is simply the expectation of the sufficient statistic vector:

In order to compute the expectation of the sufficient statistic vector we need to write as

where is the natural parameter vector of . Now, the expectation of the sufficient statistic vector is defined as

(4)

We call this expectation of the sufficient statistic vector as the moments vector.

## Lower bound¶

Computing the VB lower bound is not necessary in order to find the posterior approximation, although it is extremely useful in monitoring convergence and possible bugs. The VB lower bound can be written as

where is the set of all observed variables and is the set of all latent variables. It can also be written as

which shows that observed and latent variables contribute differently to the lower bound. These contributions have simple forms for exponential family nodes. Observed exponential family nodes contribute to the lower bound as follows:

where is the observed data. On the other hand, latent exponential family nodes contribute to the lower bound as follows:

If a node is partially observed and partially unobserved, these formulas are applied plate-wise appropriately.

## Terms¶

To summarize, implementing VMP requires one to write for each stochastic exponential family node:

: the expectation of the prior natural parameter vector

Computed as a function of the messages from parents.

: natural parameter vector of the posterior approximation

Computed as a sum of and the messages from children.

: the posterior moments vector

Computed as a function of as defined in (4).

: the moments vector for given data

Computed as a function of of the observed data .

: the expectation of the negative log normalizer of the prior

Computed as a function of parent moments.

: the negative log normalizer of the posterior approximation

Computed as a function of .

: the log base measure for given data

Computed as a function of the observed data .

: the message to parent

Computed as a function of the moments of this node and the other parents.

Deterministic nodes require only the following terms:

: the posterior moments vector

Computed as a function of the messages from the parents.

: the message to a parent

Computed as a function of the messages from the other parents and all children.


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Date: Tue, 10 Jul 2007 05:07:09 -0700 Reply-To: Paige Miller Sender: "SAS(r) Discussion" From: Paige Miller Organization: http://groups.google.com Subject: Re: Can I do this multiplr regression? Comments: To: sas-l@uga.edu In-Reply-To: Content-Type: text/plain; charset="us-ascii" On Jul 10, 2:28 am, davidlcass...@MSN.COM (David L Cassell) wrote: > x...@LSU.EDU wrote: > > >A colleague asked me if he can do this multiple regression. He first > >calculated differences from X1 and X2 and used it as the dependent variable > >(Y). Then he wanted to run regression Y = X0 X1 X3 X4 where Y is the > >difference between X1 and X2, and he wanted to put X1 as one predictor. X0, > >X3, X4 are independent from Y. > > >Can this multiple regression be done? Which assumption is violated? > > This, in itself, does not violate any regression assumptions. Of course: > > [1] regression assumptions may be violated that have nothing to do with > using Y=X1-X2. > > and > > [2] there may be serious logical violations in doing this. > > #2 depends on the data, the meaning of the variables, the underlying > theory involved, the scope of the data, etc. It may be entirely feasible > to perform the regression, even though scientifically it could be a hideous > nightmare. I think this is a subject-matter problem more than a > statistical one. I would add that since Y is a function of X1 and other variables, then putting X1 in the right hand side as a predictor is very problematic. Specifically, Y may be highly correlated with X1. Now that in itself isn't a problem. The problem is that if the large regression doesn't improve the fit over the regression involving Y and only X1 and an intercept, then the large regression is useless. It simply is explaining the a priori correlation between Y and X1. -- Paige Miller paige\dot\miller \at\ kodak\dot\com

Back to: Top of message | Previous page | Main SAS-L page


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• ### Magnetism of monomer MnO and heterodimer FePt@MnO nanoparticles(1702.04312)

Feb. 14, 2017 cond-mat.mes-hall
We report about the magnetic properties of antiferromagnetic (AF) MnO nanoparticles (NPs) with different sizes (6-19nm). Using a combination of polarized neutron scattering and magnetometry we were able to resolve previously observed peculiarities. Magnetometry, on the one hand, reveals a peak in the zero field cooled (ZFC) magnetization curves at low temperatures (~25K) but no feature around the N\'eel temperature at 118K. On the other hand, polarized neutron scattering shows the expected behavior of the AF order parameter vanishing around 118K. Moreover, hysteresis curves measured at various temperatures reveal an exchange bias effect indicating a coupling of an AF core to a ferromagnetic (FM)-like shell. ZFC data measured at various fields exclude a purely superparamagnetic (SPM) scenario. We conclude that the magnetic behavior of MnO particles can be explained by a superposition of SPM-like thermal fluctuations of the AF-N\'eel vector inside the AF core and a strong magnetic coupling to a ferrimagnetic Mn$_2$O$_3$ or Mn$_3$O$_4$ shell. In addition, we have studied heterodimer ('Janus') particles, where a FM FePt particle is attached to the AF MnO particle. Via the exchange bias effect, the magnetic moment of the FePt subunit is stabilized by the MnO.
• ### Phase diagram of Eu magnetic ordering in Sn-flux-grown Eu(Fe$_{1-x}$Co$_{x}$)$_{2}$As$_{2}$ single crystals(1612.02238)

The magnetic ground state of the Eu$^{2+}$ moments in a series of Eu(Fe$_{1-x}$Co$_{x}$)$_{2}$As$_{2}$ single crystals grown from the Sn flux has been investigated in detail by neutron diffraction measurements. Combined with the results from the macroscopic properties (resistivity, magnetic susceptibility and specific heat) measurements, a phase diagram describing how the Eu magnetic order evolves with Co doping in Eu(Fe$_{1-x}$Co$_{x}$)$_{2}$As$_{2}$ is established. The ground-state magnetic structure of the Eu$^{2+}$ spins is found to develop from the A-type antiferromagnetic (AFM) order in the parent compound, via the A-type canted AFM structure with some net ferromagnetic (FM) moment component along the crystallographic $\mathit{c}$ direction at intermediate Co doping levels, finally to the pure FM order at relatively high Co doping levels. The ordering temperature of Eu declines linearly at first, reaches the minimum value of 16.5(2) K around $\mathit{x}$ = 0.100(4), and then reverses upwards with further Co doping. The doping-induced modification of the indirect Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between the Eu$^{2+}$ moments, which is mediated by the conduction $\mathit{d}$ electrons on the (Fe,Co)As layers, as well as the change of the strength of the direct interaction between the Eu$^{2+}$ and Fe$^{2+}$ moments, might be responsible for the change of the magnetic ground state and the ordering temperature of the Eu sublattice. In addition, for Eu(Fe$_{1-x}$Co$_{x}$)$_{2}$As$_{2}$ single crystals with 0.10 $\leqslant$ $\mathit{x}$ $\leqslant$ 0.18, strong ferromagnetism from the Eu sublattice is well developed in the superconducting state, where a spontaneous vortex state is expected to account for the compromise between the two competing phenomena.
• ### Magnetic polarization of Ir in underdoped, non-superconducting Eu(Fe$_{0.94}$Ir$_{0.06}$)$_{2}$As$_{2}$(1602.00228)

Using polarized neutron diffraction and x-ray resonant magnetic scattering (XRMS) techniques, multiple phase transitions were revealed in an underdoped, non-superconducting Eu(Fe$_{1-x}$Ir$_{x}$)$_{2}$As$_{2}$ ($\mathit{x}$ = 0.06) single crystal. Compared with the parent compound EuFe$_{2}$As$_{2}$, the tetragonal-to-orthorhombic structural phase transition and the antiferromagnetic order of the Fe$^{2+}$ moments are significantly suppressed to $\mathit{T_{S}}$ = 111 (2) K and $\mathit{T_{N,Fe}}$= 85 (2) K by 6% Ir doping, respectively. In addition, the Eu$^{2+}$ spins order within the $\mathit{ab}$ plane in the A-type antiferromagnetic structure similar to the parent compound. However, the order temperature is evidently suppressed to $\mathit{T_{N,Eu}}$= 16.0 (5) K by Ir doping. Most strikingly, the XRMS measurements at the Ir $\mathit{L_{3}}$ edge demonstrates that the Ir 5$\mathit{d}$ states are also magnetically polarized, with the same propagation vector as the magnetic order of Fe. With $\mathit{T_{N,Ir}}$ = 12.0 (5) K, they feature a much lower onset temperature compared with $\mathit{T_{N,Fe}}$. Our observation suggests that the magnetism of the Eu sublattice has a considerable effect on the magnetic nature of the 5$\mathit{d}$ Ir dopant atoms and there exists a possible interplay between the localized Eu$^{2+}$ moments and the conduction $\mathit{d}$-electrons on the FeAs layers.
• ### Magnetic ground state of superconducting Eu(Fe0.88Ir0.12)2As2: A combined neutron diffraction and first-principles calculation study(1502.01921)

The magnetic order of the localized Eu$^{2+}$ spins in optimally-doped Eu(Fe$_{1-x}$Ir$_{x}$)$_{2}$As$_{2}$ ($\mathit{x}$ = 0.12) with superconducting transition temperature $\mathit{T_{SC}}$ = 22 K was investigated by single-crystal neutron diffraction. The Eu$^{2+}$ moments were found to be ferromagnetically aligned along the $\mathit{c}$-direction with an ordered moment of 7.0(1) $\mu_{B}$ well below the magnetic phase transition temperature $\mathit{T_{C}}$ = 17 K. No evidence of the tetragonal-to-orthorhombic structural phase transition was found in this compound within the experimental uncertainty, in which the spin-density-wave (SDW) order of the Fe sublattice is supposed to be completely suppressed and the superconductivity gets fully developed. The ferromagnetic groud state of the Eu$^{2+}$ spins in Eu(Fe$_{0.88}$Ir$_{0.12}$)$_{2}$As$_{2}$ was supported by the first-principles density functional calculation. In addition, comparison of the electronic structure calculations between Eu(Fe$_{0.875}$Ir$_{0.125}$)$_{2}$As$_{2}$ and the parent compound EuFe$_{2}$As$_{2}$ indicates stronger hybridization and more expanded bandwith due to the Ir substitution, which together with the introduction of electrons might work against the Fe-SDW in favor of the superconductivity.
• ### Magnetic and structural transitions in La$_{0.4}$Na$_{0.6}$Fe$_2$As$_2$ single crystals(1412.7447)

La$_{0.4}$Na$_{0.6}$Fe$_2$As$_2$ single crystals have been grown out of an NaAs flux in an alumina crucible and characterized by measuring magnetic susceptibility, electrical resistivity, specific heat, as well as single crystal x-ray and neutron diffraction. La$_{0.4}$Na$_{0.6}$Fe$_2$As$_2$ single crystals show a structural phase transition from a high temperature tetragonal phase to a low-temperature orthorhombic phase at T$_s$\,=\,125\,K. This structural transition is accompanied by an anomaly in the temperature dependence of electrical resistivity, anisotropic magnetic susceptibility, and specific heat. Concomitant with the structural phase transition, the Fe moments order along the \emph{a} direction with an ordered moment of 0.7(1)\,$\mu_{\textup{B}}$ at \emph{T}\,=\,5 K. The low temperature stripe antiferromagnetic structure is the same as that in other \emph{A}Fe$_{2}$As$_{2}$ (\emph{A}\,=\,Ca, Sr, Ba) compounds. La$_{0.5-x}$Na$_{0.5+x}$Fe$_2$As$_2$ provides a new material platform for the study of iron-based superconductors where the electron-hole asymmetry could be studied by simply varying La/Na ratio.
• ### Magnetic structure of the Eu2+ moments in superconducting EuFe2(As1-xPx)2 with x = 0.19(1409.0432)

The magnetic structure of the Eu2+ moments in the superconducting EuFe2(As1-xPx)2 sample with x = 0.19 has been determined using neutron scattering. We conclude that the Eu2+ moments are aligned along the c direction below T_C = 19.0(1) K with an ordered moment of 6.6(2) mu_B in the superconducting state. An impurity phase similar to the underdoped phase exists within the bulk sample which orders antiferromagnetically below T_N = 17.0(2) K. We found no indication of iron magnetic order, nor any incommensurate magnetic order of the Eu2+ moments in the sample.
• ### Approaching the Ground State of Frustrated A-site Spinels: A Combined Magnetization and Polarized Neutron Scattering Study(1405.4694)

We re-investigate the magnetically frustrated, {\it diamond-lattice-antiferromagnet} spinels FeAl$_2$O$_4$ and MnAl$_2$O$_4$ using magnetization measurements and diffuse scattering of polarized neutrons. In FeAl$_2$O$_4$, macroscopic measurements evidence a "cusp" in zero field-cooled susceptibility around 13~K. Dynamic magnetic susceptibility and {\it memory effect} experiments provide results that do not conform with a canonical spin-glass scenario in this material. Through polarized neutron scattering studies, absence of long-range magnetic order down to 4~K is confirmed in FeAl$_2$O$_4$. By modeling the powder averaged differential magnetic neutron scattering cross-section, we estimate that the spin-spin correlations in this compound extend up to the third nearest-neighbour shell. The estimated value of the Land\'{e} $g$ factor points towards orbital contributions from Fe$^{2+}$. This is also supported by a Curie-Weiss analysis of the magnetic susceptibility. MnAl$_2$O$_4$, on the contrary, undergoes a magnetic phase transition into a long-range ordered state below $\approx$ 40~K, which is confirmed by macroscopic measurements and polarized neutron diffraction. However, the polarized neutron studies reveal the existence of prominent spin-fluctuations co-existing with long-range antiferromagnetic order. The magnetic diffuse intensity suggests a similar short range order as in FeAl$_2$O$_4$. Results of the present work supports the importance of spin-spin correlations in understanding magnetic response of frustrated magnets like $A$-site spinels which have predominant short-range spin correlations reminiscent of the "spin liquid" state.
• ### Coexistence of Superconductivity and Ferromagnetism in P-doped EuFe2As2(1401.5463)

Jan. 22, 2014 cond-mat.supr-con
The magnetic structure of the Eu2+ moments in the superconducting EuFe2(As1-xPx)2 sample with x = 0.15 has been determined using element specific x-ray resonant magnetic scattering. Combining magnetic, thermodynamic and scattering measurements, we conclude that the long range ferromagnetic order of the Eu2+ moments aligned primarily along the c axis coexists with the bulk superconductivity at zero field. At an applied magnetic field >= 0.6 T, superconductivity still coexists with the ferromagnetic Eu2+ moments which are polarized along the field direction. We propose a spontaneous vortex state for the coexistence of superconductivity and ferromagnetism in EuFe2(As0.85P0.15)2.
• ### Magnetization distribution and orbital moment in the non-Superconducting Chalcogenide Compound K0.8Fe1.6Se2(1311.0492)

Nov. 3, 2013 cond-mat.supr-con
We have used polarized and unpolarized neutron diffraction to determine the spatial distribution of the magnetization density induced by a magnetic field of 9 T in the tetragonal phase of K0.8Fe1.6Se2. The maximum entropy reconstruction shows clearly that most of the magnetization is confined to the region around the iron atoms whereas there is no significant magnetization associated with either Se or K atoms. The distribution of magnetization around the Fe atom is slightly nonspherical with a shape which is extended along the [0 0 1] direction in the projection. Multipolar refinement results show that the electrons which give rise to the paramagnetic susceptibility are confined to the Fe atoms and their distribution suggests that they occupy 3d t2g-type orbitals with around 66% in those of xz/yz symmetry. Detail modeling of the magnetic form factor indicates the presence of an orbital moment to the total paramagnetic moment of Fe2+
• ### Distinguishing $s^{\pm}$ and $s^{++}$ electron pairing symmetries by neutron spin resonance in superconducting NaFe$_{0.935}$Co$_{0.045}$As(1308.2453)

Aug. 12, 2013 cond-mat.supr-con
A determination of the superconducting (SC) electron pairing symmetry forms the basis for establishing a microscopic mechansim for superconductivity. For iron pnictide superconductors, the $s^\pm$-pairing symmetry theory predicts the presence of a sharp neutron spin resonance at an energy below the sum of hole and electron SC gap energies ($E\leq 2\Delta$) below $T_c$. On the other hand, the $s^{++}$-pairing symmetry expects a broad spin excitation enhancement at an energy above $2\Delta$ below $T_c$. Although the resonance has been observed in iron pnictide superconductors at an energy below $2\Delta$ consistent with the $s^\pm$-pairing symmetry, the mode has also be interpreted as arising from the $s^{++}$-pairing symmetry with $E\ge 2\Delta$ due to its broad energy width and the large uncertainty in determining the SC gaps. Here we use inelastic neutron scattering to reveal a sharp resonance at E=7 meV in SC NaFe$_{0.935}$Co$_{0.045}$As ($T_c = 18$ K). On warming towards $T_c$, the mode energy hardly softens while its energy width increases rapidly. By comparing with calculated spin-excitations spectra within the $s^{\pm}$ and $s^{++}$-pairing symmetries, we conclude that the ground-state resonance in NaFe$_{0.935}$Co$_{0.045}$As is only consistent with the $s^{\pm}$-pairing, and is inconsistent with the $s^{++}$-pairing symmetry.
• ### Magnetic anisotropy in hole-doped superconducting Ba 0.67K 0.33Fe 2As2 probed by polarized inelastic neutron scattering(1301.1397)

Jan. 8, 2013 cond-mat.supr-con
We use polarized inelastic neutron scattering (INS) to study spin excitations of optimally hole-doped superconductor Ba$_{0.67}$K$_{0.33}$Fe$_2$As$_{2}$ ($T_c=38$ K). In the normal state, the imaginary part of the dynamic susceptibility, $\chi^{\prime\prime}(Q,\omega)$, shows magnetic anisotropy for energies below $\sim$7 meV with c-axis polarized spin excitations larger than that of the in-plane component. Upon entering into the superconducting state, previous unpolarized INS experiments have shown that spin gaps at $\sim$5 and 0.75 meV open at wave vectors $Q=(0.5,0.5,0)$ and $(0.5,0.5,1)$, respectively, with a broad neutron spin resonance at $E_r=15$ meV. Our neutron polarization analysis reveals that the large difference in spin gaps is purely due to different spin gaps in the c-axis and in-plane polarized spin excitations, resulting resonance with different energy widths for the c-axis and in-plane spin excitations. The observation of spin anisotropy in both opitmally electron and hole-doped BaFe$_2$As$_2$ is due to their proximity to the AF ordered BaFe$_2$As$_2$ where spin anisotropy exists below $T_N$.
• ### Direct observation of low energy nuclear spin excitations in HoCrO$_3$ by high resolution neutron spectroscopy(1212.4338)

Dec. 18, 2012 cond-mat.str-el
We have investigated low energy nuclear spin excitations in strongly correlated electron compound HoCrO$_3$. We observe clear inelastic peaks at $E = 22.18 \pm 0.04$ $\mu eV$ in both energy loss and gain sides. The energy of the inelastic peaks remains constant in the temperature range 1.5 - 40 K at which they are observed. The intensity of the inelastic peak increases at first with increasing temperature and then decreases at higher temperatures. The temperature dependence of the energy and intensity of the inelastic peaks is very unusual compared to that observed in other Nd, Co and V compounds. Huge quasielastic scattering appears at higher temperatures presumably due to the fluctuating electronic moments of the Ho ions that get increasingly disordered at higher temperatures.
• ### Strong coupling of Sm and Fe magnetism in SmFeAsO as revealed by magnetic x-ray scattering(1107.1778)

The magnetic structures adopted by the Fe and Sm sublattices in SmFeAsO have been investigated using element specific x-ray resonant and non-resonant magnetic scattering techniques. Between 110 and 5 K, the Sm and Fe moments are aligned along the c and a directions, respectively according to the same magnetic representation $\Gamma_{5}$ and the same propagation vector (1, 0, 0.5). Below 5 K, magnetic order of both sublattices change to a different magnetic structure and the Sm moments reorder in a magnetic unit cell equal to the chemical unit cell. Modeling of the temperature dependence for the Sm sublattice as well as a change in the magnetic structure below 5 K provide a clear evidence of a surprisingly strong coupling between the two sublattices, and indicate the need to include anisotropic exchange interactions in models of SmFeAsO and related compounds.
• ### Soft X-ray resonant scattering study of single-crystal LaSr$_2$Mn$_2$O$_7$(1003.2229)

March 10, 2010 cond-mat.str-el
Soft X-ray resonant scattering studies at the Mn $L_{\texttt{II, III}}$- and the La $M_{\texttt{IV, V}}$- edges of single-crystal LaSr$_2$Mn$_2$O$_7$ are reported. At low temperatures, below $T_\texttt{N} \approx 160$ K, energy scans with a fixed momentum transfer at the \emph{A}-type antiferromagnetic (0 0 1) reflection around the Mn $L_{\texttt{II, III}}$-edges with incident linear $\sigma$ and $\pi$ polarizations show strong resonant enhancements. The splitting of the energy spectra around the Mn $L_{\texttt{II, III}}$-edges may indicate the presence of a mixed valence state, e.g., Mn$^{3+}$/Mn$^{4+}$. The relative intensities of the resonance and the clear shoulder-feature as well as the strong incident $\sigma$ and $\pi$ polarization dependences strongly indicate its complex electronic origin. Unexpected enhancement of the charge Bragg (0 0 2) reflection at the La $M_{\texttt{IV, V}}$-edges with $\sigma$ polarization has been observed up to 300 K, with an anomaly appearing around the orbital-ordering transition temperature, $T_{\texttt{OO}} \approx 220$ K, suggesting a strong coupling (competition) between them.
• ### EuFe_2As_2 magnetic structure determination(0906.1508)

Sept. 23, 2009 cond-mat.str-el
Magnetic resonant X ray scattering experiments have been performed on a single crystal of EuFe_2As_2 at the Eu L_3 absorption edge. This has permitted us to determine for the first time the orientation of Eu magnetic moments in the antiferromagnetic ordered phase below T_N=19 K as parallel to the a-axis. In addition, non-resonant magnetic X ray measurements indicate that Fe magnetic moments align along the same direction in the spin density wave phase (T<T_S~190 K). Temperature dependences indicate that the onset of the Eu AFM phase barely affects the Fe magnetic sublattice.


text:
### Times

Which times on a digital clock have a line of symmetry? Which look the same upside-down? You might like to try this investigation and find out!

### Clock Hands

This investigation explores using different shapes as the hands of the clock. What things occur as the the hands move.

### Ten Green Bottles

Do you know the rhyme about ten green bottles hanging on a wall? If the first bottle fell at ten past five and the others fell down at 5 minute intervals, what time would the last bottle fall down?

# Millennium Man

### Why do this problem?

This problem is a challenging one for children who are familiar with the measurement of time and calendars.

### Key questions

How often do leap years occur?
What day of the week was the 1st January 2001 ... 1st January 2002 etc.?
What happens after $28$ years?


text:
# What are the critical values, if any, of  f(x)= x cos x −3 sin x +2x in [0,2pi]?

Nov 19, 2017

$0$, $2 \pi$ and a value near $4$ Please see below.

#### Explanation:

$f ' \left(x\right) = x \sin x - 2 \cos x + 2$

By inspection $x = 0$ and $x = 2 \pi$ are solutions.

$f ' \left(\pi\right) > 0$ and $f ' \left(\frac{3 \pi}{2}\right) < 0$ so there is another critical number in $\left[\pi , \frac{3 \pi}{2}\right]$.

Numerical or technological techniques find the solution at about $4$.


text:
# Gauge Field Tensor from Wilson Loop

+ 4 like - 0 dislike
5878 views

It is possible to introduce the gauge field in a QFT purely on geometric arguments. For simplicity, consider QED, only starting with fermions, and seeing how the gauge field naturally emerges. The observation is that the derivative of the Dirac field doesn't have a well-defined transformation, because: $$n^\mu \partial_\mu \,\psi = \lim\limits_{\epsilon\rightarrow 0}\big[\psi(x+\epsilon n)-\psi(x)\big],$$ i.e. the derivative combines two fields at different spacetime points (having different transformation rules). We need to introduce a parallel transporter $U(y,x)$ that transforms as $$U(y,x) \rightarrow e^{ig\alpha(y)}U(y,x) e^{-ig \alpha(x)},$$ such that we can adapt the defintion of the derivative into a covariant derivative, that transforms in a well-defined way: $$n^\mu D_\mu \,\psi = \lim\limits_{\epsilon\rightarrow 0}\big[\psi(x+\epsilon n)- U(x+\epsilon n,x)\psi(x)\big].$$ From geometric arguments, it is straightforward to show that the parallel transporter is a Wilson line: $$U(y,x) = \mathcal{P}\,e^{ig\int\limits_x^y dz^\mu\, A_\mu(z)},$$ which introduces a new field, namely the gauge field $A_\mu$. See e.g. Peskin & Schroeder Chapter 15 for more detail.

However.. Where the interaction term $\bar{\psi} A_\mu \psi$ emerged in a natural way, I totally don't see how the kinetic terms emerge. The standard way to proceed, is to consider a Wilson loop (a Wilson line on a closed path), and use Stokes' theorem: $$\text{exp}\left\{ig\oint_\mathcal{C}dx^\mu\, A_\mu \right\} = \text{exp}\left\{ig\int_\Sigma dx^\mu \wedge dx^\nu\,\left(\partial_\mu A_\nu - \partial_\nu A_\mu \right)\right\},$$ where of course $\partial_\mu A_\nu - \partial_\nu A_\mu \equiv F_{\mu\nu}$. In Peskin & Schroeder, they then consider a small rectangular loop, and see that in the limit $\epsilon\rightarrow 0$, $F_{\mu\nu}$ is invariant. But what's the point? I mean, the transformation law for $A_\mu$ is easily calculated from the definition of the Wilson loop: $$A_\mu\rightarrow A_\mu+\partial_\mu \alpha,$$ making $F_{\mu\nu}$ invariant by definition: $$F_{\mu\nu}\rightarrow \partial_\mu A_\nu - \partial_\nu A_\mu +\square \alpha-\square \alpha.$$

I would have liked to see a calculation, starting from a particular loop parameterisation, that naturally leads to the correct kinetic terms in the Lagrangian, as was the case for the interaction term. In other words $$\text{exp}\left\{ig\oint_\mathcal{C}dx^\mu\, A_\mu \right\} \leadsto -\frac{1}{4}\left(F_{\mu\nu}\right)^4,$$ but I have no idea how to do it.

Or is the idea simply 'look I have found some quadratic derivative terms that are invariant, now let me fiddle a bit and put its square in $\mathcal{L}$'? If yes, then why do Peskin & Schroeder bother calculating a loop parameterisation (p484), if using Stokes' theorem would have been enough to find $F_{\mu\nu}$ somewhere?

This post imported from StackExchange Physics at 2014-08-15 09:33 (UCT), posted by SE-user freddieknets

look I have found some quadratic derivative terms that are invariant, now let me fiddle a bit and put its square in L?

I think that's it, I have not heard of any deeper insight than just that.

OP's unease comes from

then why do Peskin & Schroeder bother calculating a loop parameterisation (p484), if using Stokes' theorem would have been enough to find $F_{\mu\nu}$ somewhere?

But, calculating from a small loop isn't really more complicated than using Stoke's theorem, remember Stokes theorem is proved exactly by this kind of small loop calculation.

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# optimality of the Carnot cycle

It is well-known and is stated always and everywhere that the Carnot cycle is the most efficient thermodynamic cycle. Although stated as such it is rarely proven. One proof is in the article "Engine Efficiencies and the Second Law of Thermodynamics", (AMERICAN JOURNAL OF PHYSICS VOLUME 37, NUMBER 11 NOVEMBER 1969), Tobin proves that among all engines performing a cycle, so that the engine returns to its starting state, the Carnot cycle is the most efficient where efficiency is defined as $\epsilon= 1-\frac{Q_{rejected}}{Q_{absorbed}}$, and the Carnot cycle is defined as one consisting of an isothermal-adiabatic-isothermal-adiabatic segment. The proof assumes the Clausius equality, that is in a reversible cycle $\oint \frac{\delta Q}{T} = 0$.
Then using the geometry of the picture below a simple argument over the areas enclosed in the cycles IABCDJI vs. IAEFGDJI (heat absorbed) or IJDAI vs. GHEAIJDG (heat rejected) will show that any other cycle working between the same temperature limits such as EFGHE will have lower efficiency than the Carnot cycle ABCDA.

But this proof assumes tacitly that pieces EFG and GHE are such that $dS>0$ and $dS<0$, respectively.
Here I am thinking of a nasty cycle such as shown in the next figure in red where the cycle cannot be neatly separated into two and only two monotonic parts having $dS>0$ or $dS<0$. There is no obvious reason why that restriction should hold in general. Clearly this monotonicity should hold if the engine has only temperature as its independent "force" but if the engine's state needs more than one independent force variable than the situation is more complicated. Instead it is reasonable to assume that any physical cycle should have finite number of monotonic pieces, kind of a "thermodynamic Dirichlet condition" but it could also have self intersections, little loops, and other ugly features when projected on the TS plane.

How would then one prove the optimality of a Carnot cycle? Or ugly cycles on the ST plane are non-physical even if there are more than one thermodynamic variables?

[] []

• I guess my expectation of a proof is different from yours. While I find Carnot's argument (I think the wikipedia reference is essentially the original argument of optimality) is absolutely brilliant but I would not call it a proof in a mathematical sense. I expect that optimality should follow from geometry and from the Clausius integral $\oint \frac {\delta Q}{T}=0$ as a mathematical theorem somewhere along Tobin's reasoning, I just do not know how. – hyportnex Dec 22 '16 at 22:58


text:
# Artifacts in TikZ picture when using scale option [closed]

When compiling the example code below with pdflatex Version 3.1415926-2.4-1.40.13 (TeX Live 2012/W32TeX), I observe an artifact (horizontal line starting from the start point of the objects) in the scaled-down version of the tikzpicture. Any help on how to fix this would be appreciated.

\documentclass{article}
\usepackage{tikz}

\begin{document}

\def\myFig{
+(0,0) to [out=90,in=270]  +(0,1)
to [out=90,in=270]  +(1,3)
to [out=90,in=270]  +(-0.3,1)
to [out=90,in=180]  +(1,1)
to [out=0,in=90]    +(1,-1)
to [out=270,in=90]  +(-0.3,-1)
to [out=270,in=90]  +(1,-3)
to [out=270,in=90]  +(0,-1)
to [out=270,in=0]   +(-0.3,-0.3)
to [out=180,in=0]   +(-2.8,0)
to [out=180,in=270] +(-0.3,0.3)
-- cycle;
}

scale=1: figure looks ok

\begin{tikzpicture}[scale=1]
\draw[step=1,lightgray] (-5,0) grid (5,10);
\draw[gray,thick,fill=gray!10] (-4.2,2.5) \myFig;
\draw[thick,fill=gray!40,rotate=-18] (0.7,1.1) \myFig;
\end{tikzpicture}

\vspace{1cm}
scale smaller than 1:  figure contains additional lines

\begin{tikzpicture}[scale=0.4]
\draw[step=1,lightgray] (-5,0) grid (5,10);
\draw[gray,thick,fill=gray!10] (-4.2,2.5) \myFig;
\draw[thick,fill=gray!40,rotate=-18] (0.7,1.1) \myFig;
\end{tikzpicture}

\end{document}

-

## closed as off topic by zeroth, Stephen, Werner, Kurt, mafpMar 18 '13 at 21:39

Questions on TeX - LaTeX Stack Exchange are expected to relate to TeX, LaTeX or related typesetting systems within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here. If this question can be reworded to fit the rules in the help center, please edit the question.

I can't reproduce this. Which TikZ version are you using? –  Jake Mar 18 '13 at 20:07
Could you please provide pictures, I get the correct image (perfectly scaled, and nothing more). –  zeroth Mar 18 '13 at 20:08
No additional lines here (TeX Live2012 + PGF version 2.10). What version of PGF/TikZ do you use? –  Gonzalo Medina Mar 18 '13 at 20:08
@Mathias could you provide a link to the PDF, just to rule out any discrepancies between versions, etc.? –  zeroth Mar 18 '13 at 20:42
Sure, here's the link: dl.dropbox.com/u/1280560/document.pdf –  Mathias Mar 19 '13 at 6:19


text:
## Main

The MINimal fluorescence photon FLUXes (MINFLUX) imaging concept separates individual fluorophores at subdiffraction distances by switching them randomly ‘on’ and ‘off’, while establishing their position with an excitation light pattern featuring one or more intensity zeros, such as a donut1,2. Probing the fluorophore position with the central zero of a donut-shaped excitation beam substantially increases the localization precision for a given number of detected fluorescence photons. Previous studies showed that around 2,500 photons suffice to obtain precisions <1 nm (standard deviation) in the focal plane. Likewise, roughly 2 nm precisions were attained in three dimensions, demonstrating the capability of MINFLUX nanoscopy to resolve the spatial distribution of fluorophores at molecular scales3,4.

All fluorescence nanoscopy concepts distinguish neighboring fluorophores by consecutively transferring them from a dark ‘off’ to a detectable ‘on’ state and back5. To this end, MINFLUX nanoscopy has so far relied exclusively on switchable or activatable fluorophores, that is molecules where the ‘on/off’ switching is afforded by state transitions within the fluorophore itself. However, the use of intrinsic state transitions places several constraints on the fluorophore, including on the brightness and the switching kinetics2,3,4,6,7,8,9,10,11. This is especially disadvantageous in multicolor recordings since the brightness and switching kinetics of different fluorophores have to be matched within a narrow range, often by applying specific buffers. Initial MINFLUX implementations were limited to two-color recordings.

In DNA-based point accumulation for imaging in nanoscale topography (DNA-PAINT) nanoscopy the ‘on/off’ modulation is implemented differently, namely by transient binding of diffusing fluorescently labeled oligonucleotides (denoted as imager strands) to complementary docking strands that are conjugated to a target protein such as an antibody12,13,14. While diffusing fluorophores contribute less detectable fluorescence and hence are largely ‘off’, bound fluorophores are ‘on’ because they deliver a stream of fluorescence photons from a fixed coordinate until bleaching or dissociation of the imager strand (which equals to going back ‘off’). Since the fluorophore does not need to be intrinsically switchable or activatable, bright and stable fluorophores can be employed.

As it typically uses widefield illumination and recording with a camera, establishing the ‘on/off’ state contrast in DNA-PAINT is challenged by the ‘background’ fluorescence from diffusing (‘off’ state) imager strands. This is particularly true when the desire to increase imaging speed calls for high concentrations of diffusing imager strands. As a result, most cellular DNA-PAINT recordings are performed in the total internal reflection fluorescence or highly inclined and laminated optical mode15,16.

We reasoned that by combining DNA-PAINT with MINFLUX recording, we could synergistically benefit from the advantages of both methods. As in the current MINFLUX nanoscopy implementations, the ‘background’ fluorescence stemming from diffusing imager strands is suppressed by the confocal pinhole, DNA-PAINT MINFLUX nanoscopy can be used in the far-field mode. DNA-PAINT MINFLUX nanoscopy is expected to provide the same single-digit nanometer resolution as conventional MINFLUX nanoscopy. As the state-switching kinetics are determined by the binding of an imager strand to a docking strand, no dedicated buffer systems are required and the kinetics can be adapted to the density of the targets by tuning the concentration of the imager strand. As in conventional MINFLUX nanoscopy using photoswitchable dyes, also in DNA-PAINT MINFLUX nanoscopy the individual localizations are recorded one by one. Thus, the imaging time scales with the number of targets, making single-beam scanning MINFLUX particularly suited for recording small regions of interest. An intrinsic benefit of using PAINT is the fact that when densely packed molecules are imaged, successive fluorophore docking avoids the interaction of fluorophores belonging to neighboring target molecules. Hence coactivation and mutual fluorophore quenching is largely avoided. Finally, as multiple orthogonal imager strands can be applied sequentially, each binding to a different docking strand (Exchange DNA-PAINT)17, addressing multiple targets should also be straightforward. For an overview of synergies, see Extended Data Fig. 1.

To establish DNA-PAINT MINFLUX nanoscopy, we first systematically explored the influence of the experimental key parameters laser power, confocal pinhole diameter and imager strand concentration on the image quality and the recording speed. Specifically, we determined the influence of these three parameters on (1) the time between valid events (tbtw), (2) the background emission frequency (fbg), which is determined by the microscope, (3) the center-frequency ratio (CFR), a filter parameter for localizations during image acquisition4 and (4) the localization precision (σr). These parameters together provide a measure of the image quality, the average success of the localization process and the time for recording a DNA-PAINT MINFLUX image. For a description of the detailed analysis, see Supplementary Notes. In brief, we found that in DNA-PAINT MINFLUX imaging a sufficiently low imager strand concentration is a key determinant of the localization precision; a too low imager strand concentration, however, increases tbtw, and thereby the overall recording time. Reducing background by decreasing the pinhole diameter improves the localization precision. Conversely, a smaller pinhole increases tbtw, ultimately requiring the identification of an optimal pinhole size. The localization precision scales with increasing laser intensity. At the available laser intensities, we did not observe a relevant effect on tbtw. The CFR proved to be an easily accessible and reliable indicator for the expectable localization precision within an experimental series. For measurements with Atto 655 bound to the imager strand, the analysis suggested as a good starting point an excitation laser power (at 640 nm) of roughly 62 µW in the sample for the first MINFLUX iteration, a pinhole diameter of 0.4 Airy units (AU) and, for proteins with an overall density similar to nuclear pore proteins, an imager strand concentration of 2 nM.

Using these parameters, we first recorded two-dimensional (2D) DNA-PAINT MINFLUX images of various cellular structures exhibiting different densities of the target proteins (Fig. 1). To this end, chemically fixed genome-edited U2OS and HeLa cell lines, endogenously expressing fusions of a host protein and a fluorescent protein, were used. The fluorescent proteins were decorated with a nanobody featuring a docking strand. Confocal overview images recording the fluorescent protein fluorescence were used to identify smaller regions that were subsequently imaged by MINFLUX nanoscopy. As the cell lines expressing TOM70-Dreiklang (mitochondrial outer membrane), Nup96-GFP (subunit of the nuclear pore complex) and Vimentin-rsEGFP2 (vimentin cytoskeleton) exhibited a moderate target protein density, we chose an imager strand concentration of 2 nM. For the Zyxin-rsEGFP2 expressing cells that exhibited a slightly less dense distribution of target proteins, we chose 2.5 nM imager strand. For all images, the localization precision (σr) of an individual localization event was in the range of 2.4 to 2.7 nm (Supplementary Table 1 and Extended Data Fig. 2); note that the images display all recorded valid localization events (Fig. 1). As previously demonstrated, the individual localizations of single binding events can also be combined7, resulting in higher nominal localization precisions of 0.8 to 1.1 nm (σrc) (Supplementary Table 1). The localization precisions achieved with DNA-PAINT MINFLUX were comparable to the localization precision achieved when using the photo-switching of Alexa Fluor 647 for MINFLUX nanoscopy (σr=3.0 nm; σrc=1.4 nm) (Fig. 1e and Supplementary Table 1).

To investigate whether DNA-PAINT MINFLUX nanoscopy is indeed suitable for addressing densely packed protein distributions, we imaged U2OS cells in which the abundant nonhistone chromatin protein HMG-I/Y was endogenously tagged with the fluorescent protein rsEGFP2. An imager strand concentration of 0.5 nM enabled recordings of the distribution of HMG-I/Y (Fig. 1c) with a localization precision (σr) of 2.3 nm (Supplementary Table 1 and Extended Data Fig. 2).

Cryo-electron tomography of in vitro reconstituted vimentin filaments suggested the assembly of four-stranded protofibrils with a right-handed supertwist18. In parts of the DNA-PAINT MINFLUX recorded filaments we indeed identified patterns that were highly suggestive of a twist, whereas in other parts this was not obvious (Fig. 1f). To determine whether insufficient sampling of localization events was causing these differences in the visibility of twists, we analyzed the accumulated localizations at different time points during a prolonged DNA-PAINT MINFLUX recording. Visual inspection suggested that in the first 4–5 hours of the DNA-PAINT MINFLUX recording new localizations continuously enhanced the vimentin imaging, while after 6–7 hours, new localizations did not add to the vimentin structure (Extended Data Fig. 3). This impression was fully in line with a Fourier ring correlation (FRC) analysis19 of the images recorded at the different time points. After 6–7 hours, the FRC resolution value reached a plateau (Extended Data Fig. 3). We conclude that most of the accessible binding sites had been captured, and that a prolongation of the recoding time would not have improved the recording further. We also note that the progression of the FRC resolution values could be used as an abort criterion to determine the endpoint of DNA-PAINT MINFLUX recordings.

So far, all MINFLUX recordings were restricted to at most two-color channels. To demonstrate three-channel DNA-PAINT MINFLUX nanoscopy, we decorated TOM70-Dreiklang expressing cells with an anti-GFP nanobody conjugated to a docking strand, whereas the inner membrane proteins Mic60 and the subunit beta of the F1FO-ATP synthase (ATP5B) were labeled using specific antibodies. The three-dimensional (3D) MINFLUX imaging of the three channels was performed sequentially by adding and washing out the respective imager strands. The obtained localizations were rendered in three dimensions and overlayed (Fig. 2). The resulting 3D localization precision for all three target proteins was roughly 5.4 nm (σr) and 3.1 nm (σz) (Supplementary Table 1). When the individual localizations of single binding events were combined7, we achieved localization precisions of roughly 2.0 nm (σrc) and 0.8 nm (σzc) (Supplementary Table 1).

Altogether, this study quantified the influence of the laser power, pinhole size and imager strand concentration on the image quality, thereby narrowing down the parameter space for future DNA-PAINT MINFLUX applications (Supplementary Notes). Since fluorescence microscopes render nothing but the fluorophores in the sample, the concept of optical resolution can only be applied to the fluorophores. To be able to draw meaningful conclusions about the target molecules at the <5 nm scale, the size and mobility of the linker between the molecule and the fluorophore have to be taken into account. To fully harness the nanometer optical resolution potential of MINFLUX nanoscopy, these sample parameters deserve further attention and improvement. In addition to the size of the label, the completeness of the labeling and the fraction of fluorophores that can be successfully localized must also be taken into account. DNA-PAINT MINFLUX makes it possible to localize each binding site several times. Therefore, missing localizations due to premature bleaching of the fluorophore are avoided with this technique.

Compared to initial MINFLUX implementations that relied on the popular STORM dyes Alexa Fluor 647, CF660C and CF680 as switchable fluorophores, (multicolor) DNA-PAINT MINFLUX nanoscopy is simpler to apply, as it avoids the use of complex buffer systems and the need to adjust to different blinking kinetics of the fluorophores in use (for a detailed comparison of the concepts, see Extended Data Fig. 1). The MINFLUX localization process remains unchanged compared to previous implementations. Therefore, DNA-PAINT MINFLUX nanoscopy provides the same unbiased, high-precision localization demonstrated in previous studies2,3. Finally, we note that by multiplexing DNA-PAINT labeling through the application of multiple orthogonal strands, our study paves the way for 3D MINFLUX imaging with nanometer resolution within cells with n > 3 channels.

## Methods

### Cell lines

The genome-edited U2OS cell lines HMGA1-rsEGFP2 (homozygous), Zyxin-rsEGFP2 (homozygous) and Vimentin-rsEGFP2 (heterozygous) were described in ref. 20. The heterozygous TOMM70A-Dreiklang U2OS cell line was generated as described in ref. 20. The homozygous NUP96-mEGFP cell line U2OS-CRISPR-NUP96-mEGFP clone no. 195 (300174)21 and the NUP107-mEGFP cell line HK-2xZFN-mEGFP-Nup107 (300676)22 were purchased from CLS GmbH (CLS Cell Lines Service GmbH).

### Cell culture

U2OS cells were cultivated in McCoy’s 5a medium (Thermo Fisher Scientific), supplemented with 100 U ml−1 penicillin, 100 μg ml−1 streptomycin, 1 mM Na-pyruvate and 10% (v/v) FBS (Invitrogen) at 37 °C, 5% CO2. HeLa Kyoto cells (HK-2xZFN-mEGFP-Nup107) were cultivated in DMEM, high glucose, GlutaMAX Supplement, pyruvate (Thermo Fisher Scientific), supplemented with 100 U ml−1 penicillin, 100 μg ml−1 streptomycin and 10% (v/v) FBS (Invitrogen) at 37 °C, 5% CO2.

### Sample preparation

The cells were cultured for 1 day on cover slips (Marienfeld) or in eight-well chambered cover slips (ibidi) and fixed in prewarmed 8% formaldehyde in PBS for 10 min. Fixed cells were permeabilized with 0.5% (v/v) Triton X-100 in PBS for 5 min. NUP107-mEGFP cells were fixed in 2.4% formaldehyde in PBS for 30 min at room temperature and after fixation incubated with 0.1 M NH4Cl in PBS for 5 min. Then, NUP107-mEGFP cells were permeabilized with 0.25% (v/v) Triton X-100. Afterward, all cells were blocked in antibody incubation buffer (Massive Photonics) for roughly 30 min. The cells were incubated for 1 h with the MASSIVE-TAG-Q anti-GFP single domain antibody (Massive Photonics) or with the FluoTag-Q anti-GFP single domain antibody (conjugated with Alexa Fluor 647) (NanoTag Biotechnologies) in antibody incubation buffer (Massive Photonics) at a dilution of 1:100. The cells were then washed three times with 1× washing buffer (Massive Photonics). For multiplexing, the cells were fixed, permeabilized and blocked as described above. Afterward, the cells were incubated for 1 h at room temperature with primary antibodies against Mic60 (Proteintech) at a concentration of 1.235 µg ml−1 and ATP synthase subunit beta (Abcam) at a concentration of 5 µg ml−1 in antibody incubation buffer (Massive Photonics). After three washing steps with PBS, the cells were incubated with polyclonal secondary antibodies coupled to DNA-PAINT docking sites, targeting mouse and rabbit IgGs (Massive Photonics) at a dilution of 1:400 each and with MASSIVE-TAG-Q anti-GFP single domain antibody (Massive Photonics) at a dilution of 1:100. The cells were then washed three times with 1× washing buffer (Massive Photonics).

### Sample mounting and imaging buffer

For the stabilization of the samples during MINFLUX imaging, the samples were incubated with 100 µl of gold nanorod dispersion (A12-40-980-CTAB-DIH-1-25, Nanopartz Inc.) for 7 min, as described before2,4. To remove unbound nanorods, the samples were rinsed with PBS several times. For single-color DNA-PAINT imaging, aliquots (5 µM) of the DNA-PAINT imager strand conjugated to Atto 655 (Massive Photonics) were diluted in imaging buffer (Massive Photonics) (final concentrations indicated in Supplementary Table 1). Alternatively, for MINFLUX imaging of Alexa Fluor 647, standard STORM buffer containing 10 mM MEA (Sigma-Aldrich), 64 µg ml−1 catalase from bovine liver (Sigma-Aldrich), 0.4 mg ml−1 glucose oxidase from Aspergillus niger (Sigma-Aldrich), 50 mM Tris/HCl, 10 mM NaCl and 10% (w/v) glucose, pH 8.0 was used23. Cover slips were sealed with picodent twinsil (picodent) on cavity slides (Brand GmbH & CO KG). For multiplexing, eight-well chambered cover slips (ibidi) were used. After incubation with gold nanorod dispersion and washing as described above, aliquots (5 µM) of the DNA-PAINT imager strand (conjugated to Atto 655) (Massive Photonics) transiently binding to MASSIVE-TAG-Q anti-GFP single domain antibody were diluted in imaging buffer (final concentration 2 nM) (Massive Photonics) and added to the cells. After DNA-PAINT MINFLUX imaging, the cells were washed on the microscope stage five times with PBS and one time with imaging buffer (Massive Photonics). Subsequently, DNA-PAINT imager (conjugated to Atto 655) (Massive Photonics) transiently binding to the anti-rabbit IgG was diluted (final concentration 1 nM) and added. After recording of the second DNA-PAINT MINFLUX dataset this process was repeated and imager transiently binding to the anti-mouse IgG (final concentration 1 nM) was added.

### MINFLUX measurements

The data were acquired on an Abberior MINFLUX microscope (Abberior Instruments)4 using Imspector Software (v.16.3.11647M-devel-win64-MINFLUX, Abberior Instruments). For MINFLUX measurements, the Imspector MINFLUX sequence templates seqIIF (2D) and DefaultIIF3D (3D) provided and optimized by the manufacturer for samples with the dye Alexa Fluor 647 were used (MINFLUX sequences).

Cells were identified and placed in the focus using the 488 nm confocal scan of the microscope. If necessary, the persistent binding–unbinding activity of imager strands was verified in the 642-nm confocal scan. Before starting a MINFLUX measurement, the stabilization system of the microscope was activated. Measurements were conducted with a stabilization precision of typically below 1 nm. A region of interest was selected in the confocal scan image and laser power and pinhole size were adjusted in the software (indicated pinhole sizes in AU refer to the emission maximum of Atto 655 at 680 nm). For MINFLUX measurements of Alexa Fluor 647 (Fig. 1e) a laser power of 12 µW in the first iteration and a pinhole diameter of 0.6 AU were used. Finally, the MINFLUX measurement was started in the region of interest.

#### Quantification measurement series

In a measurement series (Supplementary Notes) one of the experimental parameters, namely laser power, pinhole size or imager concentration, was varied, while the other parameters were kept constant. Within one measurement series, we recorded 2D MINFLUX images of labeled nuclear pores close to the cover slip and kept the image size and the recording time (1 h) constant. All images were taken with the same MINFLUX iteration sequence. Multiple regions (1 × 1 µm) of the lower envelope of one nucleus were measured. Each region was imaged with a different experimental parameter. Each measurement series was repeated three times on different days with fresh samples.

### Daily alignment of the MINFLUX nanoscope

The shape of the intensity pattern ('donut') for fluorescence excitation was evaluated using immobilized fluorescent beads (GATTA-BEAD R, Gattaquant GmbH) and if necessary optimized by changing the spatial light modulator parameters. Additionally, the position of the pinhole was adjusted so that the confocal detection matched the excitation volume. If during measurement series more than one pinhole size was used, all pinhole positions were determined before starting the measurement series. The pinhole position was then adjusted before each measurement.

### MINFLUX data analysis

#### Data export

Each MINFLUX measurement was exported using Imspector Software (Abberior Instruments). The exported files contained a collection of recorded parameters for all valid localizations and also included discarded nonvalid localization attempts. Additional information of the measurement (laser power and so on) was stored manually. Both were imported in a custom analysis script written in MATLAB (R2018b) to calculate the following quantification parameters in an automated manner.

#### Quantification

For all calculations, only data of the last MINFLUX iterations (in two dimensions fourth, in three dimensions nineth, after one prelocalization iteration), which were also identified as valid (exported parameter VLD = 1), were used.

The first quantification parameter to be calculated was the time that passed between the localization of two valid events, in short, the time between events or tbtw. An emitting molecule is usually localized by the microscope several times in direct succession by repeating the last two MINFLUX iterations. These successive localizations are assigned to the same event via the same trace ID (exported parameter TID). Moreover, for each individual localization the time at which its localization process started was saved (exported parameter TIM). This allowed the determination of the start and end time of each molecule binding event. Each event (TID) was terminated after a predefined number of nonvalid localization attempts. The time of the first final nonvalid localization attempt was defined as the end time of the molecule binding event. Finally, tbtw was calculated as the time difference between two consecutive valid events by subtracting the end time of the first molecule from the start time of the second molecule. For each measurement, the median of the first 100 events was determined as tbtw.

### Time between molecule binding events tbtw calculated from the exported measurement parameters

Saved localization attempts are depicted as colored rectangles, arranged in order of their appearance. Valid localization attempts were saved with the exported parameter VLD = 1 and are shown as green, while the nonvalid localization attempts were saved with VLD = 0 and are shown in yellow. The beginning of a localization attempt is saved as a time stamp (exported parameter TIM), here shown simplified as dimensionless values from 1 to 10. Localization events belonging to the same molecule have the same trace ID (exported parameter TID). Here, the time between the two consecutive valid molecules is calculated as the time difference between the start of molecule 4 (TIM = 7) and the end of molecule 2 (TIM = 5).

The second quantification parameter was the background emission frequency (fbg). The fbg is continuously estimated by the MINFLUX microscope between valid events and is used by the system to identify emission events and to correct emission frequencies of localization events.

The third quantification parameter was the CFR. The CFR is the ratio of the effective emission frequency at the central position of the MINFLUX excitation pattern over the mean effective emission frequency over all outer positions and defined as $${{{\mathrm{CFR}}}} = f_{{{{\mathrm{eff}}}}}({{{\mathrm{central}}}}\,{{{\mathrm{position}}}})/f_{{{{\mathrm{eff}}}}}({{{\mathrm{outer}}}}\,{{{\mathrm{positions}}}})$$. The effective frequencies feff are the measured emission frequencies above a background automatically determined by the system. The value of the CFR is regarded as a quality measure for the localization process. For each measurement, the median CFR of all valid localizations in the last iteration was determined. The CFR is calculated directly by the microscope software and is also used for filtering in early iterations (exported parameter CFR). It therefore directly influences the measurement4.

To estimate the localization precision of a measurement as the third quantification parameter, the standard deviation σr was calculated for each molecule (at least five localizations with the same exported parameter TID) as $$\sigma _r = \sqrt {\left( {\sigma _x^2 + \sigma _y^2} \right)/2}$$ with the standard deviations of the x and y coordinates as determined by the microscope (exported parameter POS). The median σr represents the stated localization precision. The combined localization precision was estimated as $$\sigma _{r\mathrm{c}} = \left\langle {\left\langle {\sigma _r} \right\rangle /\surd n} \right\rangle _n$$, that is the weighted average of the average single localization precision σr divided by √n and weighted by the occurrence of n being the number of single localizations with the same TID. The precision in the z direction is often different from x and y, therefore we separately computed the combined localization precision in z: $$\sigma _{zc} = \left\langle {\left\langle {\sigma _z} \right\rangle /\surd n} \right\rangle _n$$.

#### CFR simulation

The CFR is a parameter that is directly calculated during image acquisition by the MINFLUX software. To understand and judge the CFR values from the experimental results we simulated the CFR dependency on pinhole size and imager strand concentration for a molecule that is located at the center of the MINFLUX targeted coordinate pattern (TCP) with background contributions included (Supplementary Notes and Supplementary Fig. 3). The excitation PSF $$h_{{{{\mathrm{exc}}}}}(x,y,z)$$ in shape of a 2D donut was determined via fast focus field calculations24 for high numerical apertures and using realistic values for the objective lens properties as well as an excitation wavelength λexc = 642 nm. The confocal detection PSF hdet(x,y,z) was calculated25 for a detection wavelength of λexc = 680 nm. We then calculated the resulting effective PSF $$h_{{{{\mathrm{eff}}}},i} = h_{{{{\mathrm{exc}}}},i} h_{{{{\mathrm{det}}}}}$$ for each exposure i by shifting hexc to the according exposure position in the MINFLUX TCP while keeping the confocal detection hdet centered. The background contribution due to diffusing imager strand was calculated in two steps. The resulting background intensity Bi in the effective excitation volume was calculated as $$B_i\approx {\int}_{\!x,y,z} {h_{{{{\mathrm{eff}}}},i}} \left( {x,y,z} \right) \times c_{{{{\mathrm{imager}}}}}dxdydz$$ for each exposure. For the CFR calculation, we assumed that the central donut exposure of the MINFLUX TCP is placed directly on the molecule, chosen here as the origin. In the case of a perfect donut zero, this leads to a detected emitter intensity of Icenter = 0 for this exposure. The signal intensity detected at different exposures is calculated as $$I_i\approx h_{{{{\mathrm{eff}}}},i}(0,0,0)$$. Correcting for the different total time spent in the inner and outer exposures, the mean background intensity $$\bar B_{{{{\mathrm{outer}}}}}$$ and mean signal intensity $$\bar I_{{{{\mathrm{outer}}}}}$$ was calculated for the outer exposures ($$i \ne 1$$). Therefore, we were able to calculate the CFR as $${{{\mathrm{CFR}}}} = \frac{{B_{{\mathrm{center}}} + I_{{\mathrm{center}}}}}{{\left\langle {\bar B_{{\mathrm{outer}}} + \bar I_{{\mathrm{outer}}}} \right\rangle }}$$ for different scenarios. We repeated the calculations for different concentrations c, adapted the pinhole size when determining hdet and used different values for the TCP diameter L.

#### Sample drift correction

Sample drift was corrected from the extracted molecule event position and time pairs by dividing the events into overlapping time windows of approximately 2,000 events per window, and generating a 2D or 3D rendered MINFLUX image (placing a Gaussian peak with standard deviation sigma of 2 nm at each estimated molecule position) and calculating 2D or 3D cross-correlations between images from different time windows. The center of the cross-correlation peak was fitted with a Gaussian function and its offset relative to the center of the cross-correlation presented the spatial sample shift between the corresponding time points. The drift curve that fulfilled all possible sample drift estimations for all possible time window pairs was estimated in a least squares sense. A smooth (cubic spline) interpolation of the estimated drift curve for all time points of all events was then subtracted from the molecule coordinates.

#### FRCxy calculations

For the determination of the FRC shown in Supplementary Table 1 and Extended Data Fig. 3 we implemented the algorithm described in ref. 19. In brief, a dataset of combined localizations (only x and y positions) was divided into two statistically independent subsets resulting in two subimages, each containing 50% of the combined localizations of the original dataset. Then, the average correlation of the Fourier transform of these subimages was calculated on rings of constant spatial frequency. The inverse of the spatial frequency at which the FRC drops below one-seventh was taken as a measure of the FRC resolution. We used combined localizations instead of single localizations for the estimation of the FRC resolution, because for single localizations the FRC is dominated by the large number of repeated localizations during one binding event and the calculated FRC resolution is then strictly proportional to the single localization precision. To obtain a more robust result, the random division into subsets was repeated several times and the obtained FRC resolutions for each division were averaged.

### Image rendering in two dimensions

All valid localization events were rendered using Imspector Software and displayed as 2D histograms with the bin size 4 nm (Fig. 1a–f) and 1 nm (Fig. 1f, close-up).

### Image rendering in three dimensions

Each MINFLUX measurement was exported with Imspector Software. The data were drift corrected (Sample drift correction) and the z position was scaled with the scaling factor 0.7 (ref. 3). A rendering of the resulting localizations where each localization was replaced by a Gaussian peak with sigma 5 nm was imported into the Imaris Software (Imaris x64, v.9.7.2, Bitplane AG). The data were displayed as a blend volume rendition.

### MINFLUX sequences

The MINFLUX microscope’s data acquisition is controlled by a set of parameters that are specified within a text file (see seqIIF.json and seqDefaultIIF3d.json in the Supplementary Data). The set of parameters defines a sequence that controls the iterative zooming in on single molecule events and was provided and optimized by the manufacturer for samples with the dye Alexa Fluor 647. The MINFLUX iteration process is described in ref. 4. In two dimensions, four iterations plus one prelocalization iteration were performed. In three dimensions, nine iterations plus one prelocalization iteration were performed. In the last iteration an L of 40 nm was used. Key parameters of the 2D iteration sequence include:

TCP parameter L (nm) Photon limit (minimal photon count) Dwell time (ms) CFR limit Laser power factor
Prelocalization   160 ≥1 Off 1
Iteration 1 288 150 ≥1 0.5 1
Iteration 2 151 100 ≥1 Off 2
Iteration 3 76 100 ≥1 0.8 4
Iteration 4 40 150 ≥1 Off 6

### Supplementary software and data

An additional software package is provided with the manuscript (https://doi.org/10.5281/zenodo.6563100) to facilitate reanalysis of the MINFLUX localization data. The package is written in MATLAB and contains experimental localization data of all recorded DNA-PAINT MINFLUX datasets, which are shown in this publication. The software applies analysis steps such as drift correction, precision estimation, as well as CFR and FRC calculations on each dataset.

### Availability of materials

U2OS cells lines HMGA1-rsEGFP2, Zyxin-rsEGFP2, Vimentin-rsEGFP2 and TOMM70A-Dreiklang are available from the corresponding author upon reasonable request. All other materials are commercially available.

### Statistics and reproducibility

All experiments in this paper were performed independently at least three times and yielded similar results.

### Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this article.


text:
# Measure defined on Semi-Algebra and on Algebra

Let $\Omega$ be a set, $\mathcal B$ is a semi-algebra that contains $\Omega$, and let $\mu \colon \mathcal B\rightarrow [0,\infty]$ be a measure defined on $(\Omega,B)$. Now define the algebra $\mathcal A$ which contains all the finite (disjoint) unions of sets from $\mathcal B$: $$\mathcal A=\left\{E=E_1 \cup E_2 \cup\cdots\cup E_n: E_1,E_2,\cdots,E_n\in \mathcal B, E_i\cap E_j=\varnothing \right\}.$$ Now define $\nu\colon\mathcal A\rightarrow [0,\infty]$ as $\nu (E)=\sum_{k=1}^n \mu (E_k)$. I shall prove that $\nu$ is a measure on $\mathcal A$.

I'm having problems with showing the $\sigma$-additivity property. How do I show that if $Q_1,Q_2,\ldots\in\mathcal A$ and $\bigcup_{n=1}^\infty Q_n \in \mathcal A$ then $\nu\left(\bigcup_{n=1}^\infty Q_n\right)=\sum_n \nu(Q_n)$?

-
You assume that the sets $Q_n$ are pairwise disjoint. If $\bigcup_{n=1}^{+\infty}Q_n\in A$, write it as $E_1\cup\cdots\cup E_N$ where $E_i\in\mathcal A$ and are pairwise disjoint. Write $Q_n:=\bigcup_{j=1}^{N_n}E_{j,n}$. The sets $E_i\cap E_{j,n}$ are pairwise disjoint. –  Davide Giraudo Nov 27 '11 at 21:41
Please don't include things like "HW" in the subject line; that's what the [homework] tag is for. Thank you. –  Arturo Magidin Nov 27 '11 at 22:00
@Davide Giraudo: but what from here? I write that $\nu (\cup_n Q_n)=\sum_{k=1}^N \mu (E_k) = \sum_{k=1}^N \mu (\cup_{n=1}^\infty \cup_{j=1}^{N_n} E_k\cap E_{j,n} )$ but I cant continue further because I'm not sure that $\cup_{j=1}^{N_n} E_k\cap E_{j,n} \in B$ –  Max Nov 27 '11 at 22:22
@Max: Isn't a semialgebra necessarily closed under binary intersections? This is true for semirings, and a semialgebra is just a semiring that includes the whole set. –  Arturo Magidin Nov 27 '11 at 23:40
@Arturo Magidin: I'm sorry, but I dont know what are binary intersections. Can I prove the theorem without them (or without using the fact that its closed under these intersections)? –  Max Nov 27 '11 at 23:56


text:
## Algebra 1

$32 \space L$,$18 \space L$
As per the given statement let us consider $a$ be the amount of solution with quantity of acid 80% and $b$ be the amount of solution with quantity of acid 30%. Thus, $a+b=50$ ...(1) and $0.8a+0.3b=0.62(50)$ ...(2) From equation (1), we have $b=50-a$ Plug $b=50-a$ in $0.8a+0.3b=0.62(50)$ $0.8a+0.3(50-a)=0.62(50)$ $a=32 \space L$ and $b=50-32=18$ L Hence, the answers are: is $32 \space L$,$18 \space L$


text:
1
JEE Main 2016 (Online) 10th April Morning Slot
+4
-1
Concrete mixture is made by mixing cement, stone and sand in a rotating cylindrical drum. If the drum rotates too fast, the ingredients remain stuck to the wall of the drum and proper mixing of ingredients does not take place. The maximum rotational speed of the drum in revolutions per minute(rpm) to ensure proper mixing is close to :

(Take the radius of the drum to be 1.25 m and its axle to be horizontal) :
A
0.4
B
1.3
C
8.0
D
27.0
2
JEE Main 2016 (Online) 10th April Morning Slot
+4
-1
In the figure shown ABC is a uniform wire. If centre of mass of wire lies vertically below point A, then $${{BC} \over {AB}}$$ is close to :

A
1.85
B
1.37
C
1.5
D
3
3
JEE Main 2016 (Online) 9th April Morning Slot
+4
-1
A cubical block of side 30 cm is moving with velocity 2 ms−1 on a smooth horizontal surface. The surface has a bump at a point O as shown in figure. The angular velocity (in rad/s) of the block immediately after it hits the bump, is :

A
5.0
B
6.7
C
9.4
D
13.3
4
JEE Main 2016 (Offline)
+4
-1
A roller is made by joining together two cones at their vertices $$0$$. It is kept on two rails $$AB$$ and $$CD$$, which are placed asymmetrically (see figure), with its axis perpendicular to $$CD$$ and its center $$O$$ at the center of line joining $$AB$$ and $$CD$$ (see figure). It is given a light push so that it starts rolling with its center $$O$$ moving parallel to $$CD$$ in the direction shown. As it moves, the roller will tend to :
A
go straight
B
turn left and right alternately
C
turn left
D
turn right
JEE Main Subjects
Physics
Mechanics
Electricity
Optics
Modern Physics
Chemistry
Physical Chemistry
Inorganic Chemistry
Organic Chemistry
Mathematics
Algebra
Trigonometry
Coordinate Geometry
Calculus
EXAM MAP
Joint Entrance Examination


text:
# Chlorsulphonic Acid

1. Nov 22, 2004

### dextercioby

Hello,

I have a question.Can one (theoretically) obtain Chlorsulphonic acid by disolving chlorhydric acid into sulphuric acid like follows:
$$H_{2}SO_{4} +HCl=HSO_{3}Cl+H_{2}O$$.

If not,how can one obtain the Chlorsulphonic acid?

2. Nov 22, 2004

### chem_tr

It is not wrong, as hydrochloric acid is stronger than sulfuric acid, but dehydration spoils the logic behind this; so I recommend that you try thionyl chloride-mediated chlorination of sulfuric acid:

$$SOCl_2+HO-SO_2-OH \longrightarrow Cl-SO_2-OH + SO_2 + HCl$$

There are some alternatives to thionyl chloride, but this is the best. You may also use phosphorus trichloride (which gives phosphite acid, $H_3PO_3$) or phosphorus oxytrichloride (which gives phosphate acid, $H_3PO_4$), or even phosphorus pentachloride (giving phosphorus oxytrichloride and then phosphate acid), etc.

3. Nov 24, 2004

### dextercioby

Okay,thenx for the info provided.But yet,i have another question:
I remember that sulphuric acid's first acid constant is larger than the chlorhydric's one,and smaller than the ones by iodhydric acid,perchloric and permanganic acids.
So,that,but not only that,should make sulphuric acid stronger than the chlorhydric one,right??

4. Nov 24, 2004

### chem_tr

It is not correct. I am clearly sure that HCl is more powerful than H2SO4[\sub]. Oxygen does not readily give proton away; even if it is attached to an ametal like phosphorus or boron.

5. Nov 24, 2004

### dextercioby

You're right,i've searched through many textbooks upon the various theories for acids and bases,but the numbers,awkwardly,do not match.For instance,the numbers in "Lehrbuch der Chemie" (7 authors,VEB Deutscher Verlag fur Grundstoffindustrie,Leipzig,1971) are $$pK_{S} \sim -3$$ for the sulphuric acid for the reaction
$$H_{2}SO_{4}+H_{2}O\rightleftharpoons HSO_{4}^{-}+H_{3}O^{+}$$
,while for the hydrochloric one $$pK_{S} \sim -7$$,in between which there is a huge gap.In other books (French ones and Pauling's one) the gap between these numbers was smaller,but either way,they stated that the hydrocloric acid is stronger than the sulphuric one.The question ithat arises is this one;
If the hydrochloric acid is stronger than the sulphuric one,why does the chemical equilibrium in the reaction:
$$2NaCl+H_{2}SO_{4} \rightleftharpoons 2HCl+Na_{2}SO_{4}$$ shift leftwards,so that u cannot obtain sulphuric acid from hydrochloric acid and the sulphuric acid's salts???

Last edited: Nov 24, 2004
6. Nov 24, 2004

### chem_tr

The powerful the acid, the eager to be involved in reaction... It is just that simple, I think.

7. Nov 24, 2004

### dextercioby

It ain't that simple.Not only hydrochloric acid's stronger than the sulphuric one.Yet one cannot obtain sulphuric acid from its salts and a stronger acid.There has to be a deeper explanation...I am sure of it.
It's just that i don't see it...

8. Nov 24, 2004

### GCT

In response to the first post,

See if this makes sense to you, assuming that your proposed reaction takes place to a significant yield (relative to the dissociation reaction of each of the strong acids).......we have an acid solution, furthermore by adding sufuric acid (or vice versa) to the solution decreases its pH. It is essentially what your saying, I believe this will not happen. Remember that the acids in consideration have a very strong tendendcy to dissociate in water, increasing pH.


text:
# American Institute of Mathematical Sciences

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## A Gronwall lemma for functions of two variables and its application to partial differential equations of fractional order

 Faculty of Mathematics and Computer Science, University of Lodz, Banacha 22, 90-238 Lodz, Poland

* Corresponding author: Dariusz Idczak

Received  May 2020 Revised  January 2021 Early access March 2021

In the paper, a new Gronwall lemma for functions of two variables with singular integrals is proved. An application to weak relative compactness of the set of solutions to a fractional partial differential equation is given.

Citation: Dariusz Idczak. A Gronwall lemma for functions of two variables and its application to partial differential equations of fractional order. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2021019
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