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# zbMATH — the first resource for mathematics

Intersections of Hirzebruch–Zagier divisors and CM cycles. (English) Zbl 1238.11069
Lecture Notes in Mathematics 2041. Berlin: Springer (ISBN 978-3-642-23978-6/pbk; 978-3-642-23979-3/ebook). viii, 140 p. (2012).
The celebrated theorem of Gross-Zagier [B. Gross and D. Zagier, “Heegner points and derivatives of $$L$$-series”, Invent. Math. 84, 225–320 (1986; Zbl 0608.14019)] gives a formula relating certain linear combinations of Heegner points on integral models of the modular curve $$X_0(N)$$ to Fourier coefficents of Eisenstein series that appear in the integral representation of an associated Rankin-Selberg $$L$$-function at its central value. A series of papers of S. S. Kudla, e.g. [“Central derivatives of Eisenstein series and height pairings”, Ann. Math. (2) 146, No. 3, 545–646 (1997; Zbl 0990.11032)] and [“Integrals of Borcherds forms”, Compos. Math. 137, No. 3, 293–349 (2003; Zbl 1046.11027)], propose to extend such a formula to the general setting of Shimura varieties and Eisenstein series on higher rank groups. The authors give some compelling evidence for a special case of this conjecture, namely in the setting of Hilbert modular surfaces, where a moduli interpretation is available. In particular, by developing some techniques in deformation theory (e.g. the theory of displays due to T. Zink [“The display of a formal $$p$$-divisible group”, Astérisque 278, 127–248 (2002; Zbl 1008.14008)]), the authors are able to perform the calculations required to prove some special cases of Kudla’s proposed conjecture, building on previous works of J. H. Bruinier and second author ([“CM values of Hilbert modular functions”, Invent. Math. 163, No. 2, 229–288 (2006; Zbl 1093.11041)], [T. Yang, “An arithmetic intersection formula on Hilbert modular surfaces”, Am. J. Math. 132, No. 5, 1275–1309 (2010; Zbl 1206.14049)], and [T. Yang, “Arithmetic intersection on a Hilbert modular surface and the Faltings height” (preprint)]).
Roughly, the authors prove the following type of result. Let $$F$$ be a real quadratic field, with $$\sigma \in \text{Gal}(F/{\mathbb{Q}})$$ the nontrivial automorphism of its Galois group over $${\mathbb{Q}}$$. Associated to $$F$$ is a Hilbert modular surface $$M$$, which can be defined as the moduli space of all $$\mathcal{O}_F$$-polarized RM abelian surfaces $$\mathcal{A}$$. Given a positive integer $$m$$, Hirzebruch and Zagier constructed a divisor on $$M({\mathbb{C}})$$, which was later given a moduli theoretic interpretation by Kudla-Rapoport [S. S. Kudla and M. Rapoport, “Arithmetic Hirzebruch-Zagier cycles”, J. Reine Angew. Math. 515, 155–244 (1999; Zbl 1048.11048)]. To describe this latter interpretation, let $$T(m)$$ denote the moduli space of pairs $$(\mathcal{A}, j)$$, where $$\mathcal{A}$$ is an $$\mathcal{O}_F$$-polarized RM abelian surface, and $$j$$ is a Rosati fixed endomorphism satisfying $$j \circ j = m$$ and $$j \circ x = x^{\sigma} \circ j$$ for all $$x \in \mathcal{O}_F$$. The “forget-$$j$$” morphism $$T(m) \rightarrow M$$ has image given by a codimension one cycle whose complex fibre is equal to the divisor constructed by Hirzebruch and Zagier. The moduli space $$T(m)$$ can then justifiably be referred to as a Hirzebrich-Zagier divisor. The main aim of the work is to relate the intersection multiplicities of three such divisors to the Fourier coefficients of the central derivative of a certain Siegel Eisenstein series of genus three.
To be somewhat more precise, the authors establish the following main result (Theorem E). Let $$E$$ be a quartic CM field containing $$F$$ as a real quadratic subfield. Let us suppose that $$E$$ is either Galois over $${\mathbb{Q}}$$, with Galois group $$\text{Gal}(E/{\mathbb{Q}}) \cong {\mathbb{Z}}/4{\mathbb{Z}}$$, or else that $$E$$ is not Galois over $${\mathbb{Q}}$$. Given a CM type $$\Sigma$$ of $$E$$, let us write $$E_{\Sigma}$$ to denote its reflex field, with $$\mathcal{O}_{\Sigma}$$ the ring of integers of $$E_{\Sigma}$$. One can associate to $$E$$ a reflex algebra $$E^{\sharp}$$ of $$E$$, which is the $${\mathbb{Q}}$$-algebra defined up to isomorphism by the existence of an $$\text{Aut}({\mathbb{C}}/{\mathbb{Q}})$$-equivariant bijection $\{ \text{CM types of $$E$$}\} \longrightarrow \text{Hom}(E^{\sharp}, {\mathbb{C}}), ~~~ \Sigma \longmapsto \phi_{\Sigma}$ for which $$\phi_{\Sigma}(E^{\sharp}) = E_{\Sigma}$$. Let $$F^{\sharp}$$ denote the maximal totally real subalgebra of $$E^{\sharp}$$. In the setting described above, $$E^{\sharp}$$ is a certain quartic CM field, with $$F^{\sharp}$$ its maximal totally real subfield. Anyhow, one can associate to the pair $$(E, \Sigma)$$ an algebraic stack $$CM_{\Sigma}$$, defined as the moduli space of principally polarized abelian varieties over $$\mathcal{O}_{\Sigma}$$-schemes with CM by $$\mathcal{O}_{\Sigma}$$, satisfying a certain technical condition (the so-called Kottwitz condition of $$\S 3.2$$). The forgetful morphism $$CM_{\Sigma} \longrightarrow M_{\mathcal{O}_{\Sigma}}$$ contains a codimension two cycle on $$M_{\mathcal{O}_{\Sigma}}$$. To a given pair $$T(m)$$ and $$CM_{\Sigma}$$, one can associate a finite intersection pairing $$\langle T(m): CM_{\Sigma} \rangle_{\text{fin}}$$, which is essentially the sum of lengths of all local rings in the intersection $$T(m) \bigcap CM_{\Sigma}$$ ($$\S 5.4$$). Following the constructions of Kudla [loc. cit.] and Bruinier [J. H. Bruinier, “Borcherds products and Chern classes of Hirzebruch-Zagier divisors”, Invent. Math. 138, No. 1, 51–83 (1999; Zbl 1011.11027)], one can also define for any integer $$m$$ an associated Green function $$G(m, v, \cdot)$$ on $$M({\mathbb{C}})$$, where $$v \in {\mathbb{R}}$$ is some fixed parameter. Taking the (finite) sum over all complex points of $$CM_{\Sigma}$$ then gives rise to a function $$G(m, v, CM_{\Sigma})$$ on $$M({\mathbb{C}})$$. One can then define the intersection of the formal pair $$\widehat{T}(m, v) := (T(m), G(m, v, \cdot))$$ and $$CM_{\Sigma}$$ to be the sum $\langle \widehat{T}(m, v): CM_{\Sigma} \rangle_{\text{fin}} = \langle T(m): CM_{\Sigma} \rangle_{\text{fin}} + \frac{1}{2}G(m, v, CM_{\Sigma}).$ Now, turning to the automorphic side, the authors define an $$\mathcal{O}_F$$-polarized CM module $${\mathbb{T}}$$ to be a projective $$\mathcal{O}_E$$-module of rank one equipped with a perfect $${\mathbb{Z}}$$-valued symplectic form (satisfying some compatibility condition with respect to the $$\mathcal{O}_E$$-action). To such a module $${\mathbb{T}}$$, there is an associated CM-type $$\Sigma$$. The set $$X_{\Sigma}$$ of isomorphism classes of such modules $${\mathbb{T}}$$ with associated CM type $$\Sigma$$ is finite. To each $${\mathbb{T}} \in X_{\Sigma}$$, one can attach a quadratic space $$\mathcal{C}({\mathbb{T}})$$ of rank two over the adele ring $${\mathbb{A}}_{F^{}\sharp}$$, which is incoherent in the sense that it does not arise from any quadratic space over $$F^{\sharp}$$. Using the Weil representation, one can associate to this quadratic space $$\mathcal{C}({\mathbb{T}})$$ a Hilbert modular Eisenstein series $$E(\tau, s, {\mathbb{T}})$$ of parallel weight one on $$\text{GL}_2({\mathbb{A}}_{F^{\sharp}})$$. Here, $$\tau = u+iv$$ denotes an element in the $$F^{\sharp}$$ upper half space $$\mathfrak{H}_{F^{\sharp}}$$, which can be identified with two copies on the usual upper-half space $$\mathfrak{H} = \{ z \in {\mathbb{C}}: \operatorname{Im}(z) > 0 \}$$. One can then define $E(s, \tau, \Sigma) = \sum_{ {\mathbb{T}} \in X_{\Sigma}} E(\tau, s, {\mathbb{T}})$ to denote the finite sum over modules $${\mathbb{T}} \in X_{\Sigma}$$ of such Eisenstein series, which in the setting described above turns out not to depend on the choice of $$\Sigma$$. The incoherence of the underlying quadratic space $$\mathcal{C}({\mathbb{T}})$$ implies that $$E(\tau, s, \Sigma)$$ vanishes at $$s=0$$, whence the derivative at $$s=0$$ is given by the sum $E'(\tau, 0, \Sigma) = \sum_{\alpha \in F^{\sharp}} c_{\Sigma}(\alpha, v) q^{\alpha}$ for some parameter $$v \in {\mathbb{R}}$$. Here, $$q^{\alpha} = e(\text{Tr}_{F^{\sharp}/{\mathbb{Q}}} (\alpha \tau))$$, where $$e(x) = \exp(2\pi i x )$$. Writing $$\iota: \mathfrak{H} \longrightarrow \mathfrak{H}_{F^{\sharp}}$$ to denote the diagonal embedding, the pullback of $$E'(\tau, 0, \Sigma)$$ to the complex upper-half space $$\mathfrak{H}$$ is a non-holomorphic modular form of weight two, with Fourier series expansion $E'(\iota(\tau), 0, \Sigma) = \sum_{m \in {\mathbb{Z}}} b_{\Sigma}(m, v) q^m,$ where the coefficients $$b_{\Sigma}(m, v)$$ are given by the formula $b_{\Sigma}(m,v) = \sum _{\substack{ \alpha \in F^{\sharp} \\ \text{Tr}_{ F^\sharp/{\mathbb{Q}}^{(\alpha)}} =m }} c_{\Sigma}(\alpha, v).$ The authors prove, under some mild technical hypothesis on the ramification of $$E$$ (Hypothesis B), that for any parameter $$v \in {\mathbb{R}}$$ and any nonzero integer $$m$$, $\langle \widehat{T}(m, v): CM_{\Sigma} \rangle = -\frac{1}{W_E} \cdot b_{\sigma}(m, v),$ where $$W_E$$ denotes the number of roots of unity in $$E^{\times}$$.
The first chapter of the monograph explains the main results. The second chapter introduces the notion of $$\mathfrak{c}$$-polarized RM and CM modules for $$\mathfrak{c} \supset \mathcal{O}_F$$ a fractional $$\mathcal{O}_F$$-ideal, then describes how to construct quadratic forms from some associated spaces of endomorphisms. The third chapter defines $$\mathfrak{c}$$-polarized RM and CM abelian surfaces. The fourth chapter gives a construction of the Eisenstein series $$E(\tau, s, {\mathbb{T}})$$. The fifth chapter describes the main technical results in more detail. The sixth and final chapter makes up the technical core of the work, developing techniques from the theory of Dieudonné modules, the Grothendieck-Messing deformation theory of $$p$$-divisible groups and Zink’s theory of displays to perform various local calculations required for the proofs.
Though formidably technical, the reviewer recommends this beautiful monograph to anyone interested in the circle of conjecture proposed by Kudla et al., particularly from the point of view of arithmetic geometry. The work contains many useful references and intricate proofs that do not appear elsewhere, and is likely to be extremely useful to future progress in the area.

##### MSC:
 11G18 Arithmetic aspects of modular and Shimura varieties 11-02 Research exposition (monographs, survey articles) pertaining to number theory 11G15 Complex multiplication and moduli of abelian varieties 14G40 Arithmetic varieties and schemes; Arakelov theory; heights
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# Feit–Thompson theorem

In mathematics, the Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable. It was proved by Walter Feit and John Griggs Thompson (1962, 1963).

## History

The contrast that these results show between groups of odd and even order suggests inevitably that simple groups of odd order do not exist.

William Burnside (1911, p. 503 note M)

William Burnside (1911, p. 503 note M) conjectured that every nonabelian finite simple group has even order. Richard Brauer (1957) suggested using the centralizers of involutions of simple groups as the basis for the classification of finite simple groups, as the Brauer–Fowler theorem shows that there are only a finite number of finite simple groups with given centralizer of an involution. A group of odd order has no involutions, so to carry out Brauer's program it is first necessary to show that non-cyclic finite simple groups never have odd order. This is equivalent to showing that odd order groups are solvable, which is what Feit and Thompson proved.

The attack on Burnside's conjecture was started by Michio Suzuki (1957), who studied CA groups; these are groups such that the Centralizer of every non-trivial element is Abelian. In a pioneering paper he showed that all CA groups of odd order are solvable. (He later classified all the simple CA groups, and more generally all simple groups such that the centralizer of any involution has a normal 2-Sylow subgroup, finding an overlooked family of simple groups of Lie type in the process, that are now called Suzuki groups.)

Feit, Thompson, and Marshall Hall (1960) extended Suzuki's work to the family of CN groups; these are groups such that the Centralizer of every non-trivial element is Nilpotent. They showed that every CN group of odd order is solvable. Their proof is similar to Suzuki's proof. It was about 17 pages long, which at the time was thought to be very long for a proof in group theory.

The Feit–Thompson theorem can be thought of as the next step in this process: they show that there is no non-cyclic simple group of odd order such that every proper subgroup is solvable. This proves that every finite group of odd order is solvable, as a minimal counterexample must be a simple group such that every proper subgroup is solvable. Although the proof follows the same general outline as the CA theorem and the CN theorem, the details are vastly more complicated. The final paper is 255 pages long.

## Significance of the proof

The Feit–Thompson theorem showed that the classification of finite simple groups using centralizers of involutions might be possible, as every nonabelian simple group has an involution. Many of the techniques they introduced in their proof, especially the idea of local analysis, were developed further into tools used in the classification. Perhaps the most revolutionary aspect of the proof was its length: before the Feit–Thompson paper, few arguments in group theory were more than a few pages long and most could be read in a day. Once group theorists realized that such long arguments could work, a series of papers that were several hundred pages long started to appear. Some of these dwarfed even the Feit–Thompson paper; the paper by Michael Aschbacher and Stephen D. Smith on quasithin groups was 1,221 pages long.

## Revision of the proof

Many mathematicians have simplified parts of the original Feit–Thompson proof. However all of these improvements are in some sense local; the global structure of the argument is still the same, but some of the details of the arguments have been simplified.

The simplified proof has been published in two books: (Bender & Glauberman 1994), which covers everything except the character theory, and (Peterfalvi 2000, part I), which covers the character theory. This revised proof is still very hard, and is longer than the original proof, but is written in a more leisurely style.

A fully formal proof, checked with the Coq proof assistant, was announced in September 2012 by Georges Gonthier and fellow researchers at Microsoft Research and INRIA.[1]

## An outline of the proof

Instead of describing the Feit–Thompson theorem directly, it is easier to describe Suzuki's CA theorem and then comment on some of the extensions needed for the CN-theorem and the odd order theorem. The proof can be broken up into three steps. We let G be a non-abelian (minimal) simple group of odd order satisfying the CA condition. For a more detailed exposition of the odd order paper see Thompson (1963) or (Gorenstein 1980) or Glauberman (1999).

### Step 1. Local analysis of the structure of the group G

This is easy in the CA case because the relation "a commutes with b" is an equivalence relation on the non-identity elements. So the elements break up into equivalence classes, such that each equivalence class is the set of non-identity elements of a maximal abelian subgroup. The normalizers of these maximal abelian subgroups turn out to be exactly the maximal proper subgroups of G. These normalizers are Frobenius groups whose character theory is reasonably transparent, and well-suited to manipulations involving character induction. Also, the set of prime divisors of |G| is partitioned according to the primes which divide the orders of the distinct conjugacy classes of maximal abelian subgroups of |G|. This pattern of partitioning the prime divisors of |G| according to conjugacy classes of certain Hall subgroups (a Hall subgroup is one whose order and index are relatively prime) which correspond to the maximal subgroups of G (up to conjugacy) is repeated in both the proof of the Feit–Hall–Thompson CN-theorem and in the proof of the Feit–Thompson odd-order theorem. Each maximal subgroup M has a certain nilpotent Hall subgroup Mσ with normalizer contained in M, whose order is divisible by certain primes forming a set σ(M). Two maximal subgroups are conjugate if and only if the sets σ(M) are the same, and if they are not conjugate then the sets σ(M) are disjoint. Every prime dividing the order of G occurs in some set σ(M). So the primes dividing the order of G are partitioned into equivalence classes corresponding to the conjugacy classes of maximal subgroups. The proof of the CN-case is already considerably more difficult than the CA-case: the main extra problem is to prove that two different Sylow subgroups intersect in the identity. This part of the proof of the odd-order theorem takes over 100 journal pages. A key step is the proof of the Thompson uniqueness theorem, stating that abelian subgroups of normal rank at least 3 are contained in a unique maximal subgroup, which means that the primes p for which the Sylow p-subgroups have normal rank at most 2 need to be considered separately. Bender later simplified the proof of the uniqueness theorem using Bender's method. Whereas in the CN-case, the resulting maximal subgroups M are still Frobenius groups, the maximal subgroups that occur in the proof of the odd-order theorem need no longer have this structure, and the analysis of their structure and interplay produces 5 possible types of maximal subgroups, called types I, II, III, IV, V. Type I subgroups are of "Frobenius type", a slight generalization of Frobenius group, and in fact later on in the proof are shown to be Frobenius groups. They have the structure MFU where MF is the largest normal nilpotent Hall subgroup, and U has a subgroup U0 with the same exponent such that MFU0 is a Frobenius group with kernel MF. Types II, III, IV, V are all 3-step group with structure MFUW1, where MFU is the derived subgroup of M. The subdivision into types II, III, IV and V depends on the structure and embedding of the subgroup U as follows:

• Type II: U is nontrivial abelian and its normalizer is not contained in M.
• Type III: U is nontrivial abelian and its normalizer is contained in M.
• Type IV: U is nonabelian.
• Type V: U is trivial.

All but two classes of maximal subgroups are of type I, but there may also be two extra classes of maximal subgroups, one of type II, and one of type II, III, IV, or V.

### Step 2. Character theory of G

If X is an irreducible character of the normalizer H of the maximal abelian subgroup A of the CA group G, not containing A in its kernel, we can induce X to a character Y of G, which is not necessarily irreducible. Because of the known structure of G, it is easy to find the character values of Y on all but the identity element of G. This implies that if X1 and X2 are two such irreducible characters of H and Y1 and Y2 are the corresponding induced characters, then Y1 − Y2 is completely determined, and calculating its norm shows that it is the difference of two irreducible characters of G (these are sometimes known as exceptional characters of G with respect to H). A counting argument shows that each non-trivial irreducible character of G arises exactly once as an exceptional character associated to the normalizer of some maximal abelian subgroup of G. A similar argument (but replacing abelian Hall subgroups by nilpotent Hall subgroups) works in the proof of the CN-theorem. However, in the proof of the odd-order theorem, the arguments for constructing characters of G from characters of subgroups are far more delicate, and use the Dade isometry between character rings rather than character induction, since the maximal subgroups have a more complicated structure and are embedded in a less transparent way. The theory of exceptional characters is replaced by the theory of a coherent set of characters to extend the Dade isometry. Roughly speaking, this theory says that the Dade isometry can be extended unless the groups involved have a certain precise structure. Peterfalvi (2000) described a simplified version the character theory due to Dade, Sibley, and Peterfalvi.

### Step 3. The final contradiction

By step 2, we have a complete and precise description of the character table of the CA group G. From this, and using the fact that G has odd order, sufficient information is available to obtain estimates for |G| and arrive at a contradiction to the assumption that G is simple. This part of the argument works similarly in the CN-group case.

In the proof of the Feit–Thompson theorem, however, this step is (as usual) vastly more complicated. The character theory only eliminates some of the possible configurations left after step 1. First they show that the maximal subgroups of type I are all Frobenius groups. If all maximal subgroups are type I then an argument similar to the CN case shows that the group G cannot be an odd-order minimal simple group, so there are exactly two classes of maximal subgroups of types II, III, IV or V. Most of the rest of the proof now focuses on these two types of maximal subgroup S and T and the relation between them. More character-theoretic arguments show that they cannot be of types IV or V. The two subgroups have a precise structure: the subgroup S is of order pq×q×(pq–1)/(p–1) and consists of all automorphisms of the underlying set of the finite field of order pq of the form xaxσ+b where a has norm 1 and σ is an automorphism of the finite field, where p and q are distinct primes. The maximal subgroup T has a similar structure with p and q reversed. The subgroups S and T are closely linked. Taking p>q, one can show that the cyclic subgroup of S of order (pq–1)/(p–1) is conjugate to a subgroup of the cyclic subgroup of T of order (qp–1)/(q–1). (In particular, the first number divides the second, so if the Feit–Thompson conjecture is true, it would assert that this cannot happen, and this could be used to finish the proof at this point. The conjecture is still unproven, however.)

The conclusion from applying character theory to the group G is that G has the following structure: there are primes p>q such that (pq–1)/(p–1) is coprime to p–1 and G has a subgroup given by the semidirect product PU where P is the additive group of a finite field of order pq and U its elements of norm 1. Moreover G has an abelian subgroup Q of order prime to p containing an element y such that P0 normalizes Q and (P0)y normalizes U, where P0 is the additive group of the finite field of order p. (For p=2 a similar configuration occurs in the group SL2(2q), with PU a Borel subgroup of upper triangular matrices and Q the subgroup of order 3 generated by ${\displaystyle \scriptstyle y=\left({\begin{smallmatrix}0&1\\1&1\end{smallmatrix}}\right)}$.) To eliminate this final case, Thompson used some fearsomely complicated manipulations with generators and relations, which were later simplified by Peterfalvi (1984), whose argument is reproduced in (Bender & Glauberman 1994). The proof examines the set of elements a in the finite field of order pq such that a and 2–a both have norm 1. One first checks that this set has at least one element other than 1. Then a rather difficult argument using generators and relations in the group G shows that the set is closed under taking inverses. If a is in the set and not equal to 1 then the polynomial N((1–a)x+1)–1 has degree q and has at least p distinct roots given by the elements x in Fp, using the fact that x→1/(2–x) maps the set to itself, so pq, contradicting the assumption p>q.

## Use of oddness

The fact that the order of the group G is odd is used in several places in the proof, as follows (Thompson 1963).

• The Hall–Higman theorem is sharper for groups of odd order.
• For groups of odd order, all non-principal characters occur in complex conjugate pairs.
• Several results about p-groups only hold for odd primes p.
• If a group of odd order has no elementary abelian subgroups of rank 3, then its derived group is nilpotent. (This fails for the symmetric group S4 of even order.)
• Several arguments involving character theory fail for small primes, especially for the prime 2.

## References

1. ^ "Feit–Thompson theorem has been totally checked in Coq". Msr-inria.inria.fr. 2012-09-20. Archived from the original on 2016-11-19. Retrieved 2012-09-25.

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So we'll first find the initial X and why components of the velocity. This would be equaling essentially the initial velocity co sign if ADA and for the wide component the initial velocity sign of data. And so this is gonna be equaling 3.50 meters per second again for both of them. And then this will be co sign of 50 degrees sign of 50 degrees for the X component. We then have 2.25 meters per second and for the wide component, we then have 2.68 meters per second. Now we can find the time of flight. Um, however, we need to first find the final velocity in the direction the final velocity in the UAE direction because we don't have time is gonna be equaling weaken. Say negative because we know that it's gonna be traveling downwards. This would be a negative square root of V. Why initial squared plus two times the acceleration in the why Direction times, delta y. And so from here we can then, of course, substitute. This would be equaling negative square root and this would be 2.68 meters per second quantity squared plus two multiplied by negative 9.80 meters per second squared multiplied by negative 0.500 meters and we have a final. Why velocity equaling negative 4.12 meters per second. Now we can solve. For the time of flight t, this would be equaling essentially v Y final minus the why initial divided by G and this would be equaling or better yet divided by the acceleration, the war direction. And so this would be equaling negative 4.12 meters per second minus 2.68 meters per second, divided by negative 9.80 meters per second squared and we find that t the time of flight is equaling 0.694 seconds. And so the horizontal distance traveled. During that time, Delta X would simply be equaling the ex initial times t, this is gonna be equaling 2.25 meters per second multiplied by 0.694 seconds and me find that Then the horizontal distance traveled Delta axes equaling 1.56 meters. This would be our final answer. That is the end of the solution. Thank you for watching

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# upper bound on infinite series by exponentials

I'm interested in upper bounding the following infinite series by exponential functions for x>1 (I would be fine with x>2 as well).

\begin{eqnarray*} \sum_{n=1}^{\infty}e^{-\frac{x^2n^2}{2}}\\ \sum_{n=1}^{\infty}x^2ne^{-\frac{x^2n^2}{2}}\\ \sum_{n=1}^{\infty}x^2(x^2n^2-1)e^{-\frac{x^2n^2}{2}}\\ \sum_{n=1}^{\infty}nx^2(x^2n^2-3)e^{-\frac{x^2n^2}{2}} \end{eqnarray*}

By plotting them it seems that something like the following holds for $x>\sqrt{6}$. Essentially first term multiplied by 1.05

\begin{eqnarray*} \sum_{n=1}^{\infty}e^{-\frac{x^2n^2}{2}}\le 1.05e^{-\frac{x^2}{2}}\\ \sum_{n=1}^{\infty}x^2ne^{-\frac{x^2n^2}{2}}\le 1.05x^2e^{-\frac{x^2}{2}}\\ \sum_{n=1}^{\infty}x^2(x^2n^2-1)e^{-\frac{x^2n^2}{2}}\le 1.05x^2(x^2-1)e^{-\frac{x^2}{2}}\\ \sum_{n=1}^{\infty}nx^2(x^2n^2-3)e^{-\frac{x^2n^2}{2}}\le 1.05x^2(x^2-3)e^{-\frac{x^2}{2}} \end{eqnarray*} I was wondering if there was a way of formalizing the above. For example is it possible to show that the right-hand side of the above inequalities minus the left-hand side has only one zero. Then verifying its location by plotting and deduce the inequalities above hold after that value.

Also in case it might be helpful the first function is essentially the theta function \begin{equation*} \vartheta_3(q)=\sum_{n=1}^\infty q^{n^2} \end{equation*}

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The first inequality you can get from comparison to $\sum_{n=1}^{\infty}e^{-\frac{x^2n}{2}}$ and then looking at it as a geometric series. On the others you need to prove essentially (after some reduction) that $\sum_{n=1}^{\infty}n^2e^{-\frac{x^2n^2}{2}}$ can be bounded by the same thing or something similar. In each case your bound for $\sum_{n=1}^{\infty} a_nb_n^n$ is just $\frac{a_1b_1}{1-b_1}$ which looks a lot like the geometric series.

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I suggest comparing $e^{-x^2n^2/2+2\ln(n)}$ and $e^{-x^2n/2}$. I just worked it out and it shows that you can use a strict comparison for $x$ sufficiently large. –  Stephen Sturgeon Jul 29 '13 at 18:16


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# Selling to Cournot oligopolists: pricing under uncertainty & generalized mean residual life

We study a classic Cournot market, which we extend to a two-stage game with endogenous cost formation: the retailers' marginal cost represents purchases from a price-setting, revenue-maximizing supplier. Any demand uncertainty falls to the supplier, who acts first and sets the wholesale price under incomplete information concerning the retailers' willingness to pay. We introduce the generalized mean residual life (GMRL) function of the supplier's belief distribution F and show that his revenue function is unimodal, if the GMRL function is decreasing - (DGMRL) property - and F has finite second moment. In this case, we characterize the supplier's optimal price as a fixed point of his MRL function and provide a bound on the market-inefficiency due to demand uncertainty, which is tight over the class of DMRL distributions. We then turn to the class of DGMRL random variables and study their moments, limiting behavior, and closure properties. Under the additional assumption that F has a density, we establish its relationship to the widely used class of increasing generalized failure rate (IGFR) random variables. If a random variable is IGFR, then it is DGMRL. We provide a sufficient condition for the converse to be true and show that the limiting behavior of the GMRL and GFR functions is closely linked under a reciprocal relationship.

## Authors

• 14 publications
• 3 publications
• ### Measuring Market Performance with Stochastic Demand: Price of Anarchy and Price of Uncertainty

Globally operating suppliers face the rising challenge of wholesale pric...
08/11/2018 ∙ by Costis Melolidakis, et al. ∙ 0

• ### Upstream Demand Uncertainty in a Two-Echelon Supply Chain: Comparative Statics via Stochastic Orderings

We revisit the classic Cournot model and extend it to a two-echelon supp...
03/09/2018 ∙ by Constandina Koki, et al. ∙ 0

• ### Third-degree Price Discrimination Versus Uniform Pricing

We compare the revenue of the optimal third-degree price discrimination ...
12/11/2019 ∙ by Dirk Bergemann, et al. ∙ 0

• ### Simultaneous 2nd Price Item Auctions with No-Underbidding

We study the price of anarchy (PoA) of simultaneous 2nd price auctions (...
03/26/2020 ∙ by Michal Feldman, et al. ∙ 0

• ### An optimal mechanism charging for priority in a queue

We derive a revenue-maximizing scheme that charges customers who are hom...
02/16/2020 ∙ by Moshe Haviv, et al. ∙ 0

• ### Weighted mean inactivity time function with applications

The concept of mean inactivity time plays a crucial role in reliability,...
03/13/2021 ∙ by Antonio Di Crescenzo, et al. ∙ 0

• ### Latent Dirichlet Allocation for Internet Price War

Internet market makers are always facing intense competitive environment...
08/23/2018 ∙ by Chenchen Li, et al. ∙ 0

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## 2 The Model

We consider a vertical market with a monopolistic upstream manufacturer, distributor or service provider, henceforth supplier selling a homogeneous product (or resource) to downstream symmetric retailers. The supplier acts first (Stackelberg leader) and chooses a unique wholesale price for all retailers (linear pricing scheme without price differentiation), see also Tyagi (1999). The supplier produces at a constant marginal cost which we normalize to zero. This corresponds to the situation in which the supplier has ample quantity to cover any possible demand by the retailers and his lone decision variable is his wholesale price, or equivalently his profit margin, .

To study the market in which the supplier is less informed than the retailers about the retail demand level , we assume that after the supplier’s pricing decision but prior to the retailers’ order decisions, a value for is realized from a continuous cummulative distribution (cdf) , with finite mean and nonnegative values, i.e., . Equivalently, can be thought of as the supplier’s belief about the demand level and, hence, about the retailers’ willingness-to-pay his price. We will write for the tail distribution of ,

for its probability density function (pdf), whenever such a pdf exists, and

for its support, with , and . The case is not excluded111Formally, this case contradicts the assumption that is continuous or non-atomic. However, it is allowed so that unnecessary notation is avoided. It should cause no confusion. and corresponds to the situation where the supplier is completely informed about the retail demand level.

Given the demand realization , the total quantity that the retailers will order from the supplier for any possible wholesale price will be denoted by . Using this notation, and under risk neutrality, the supplier aims to maximize his expected profit function , which is equal to

 Πs(r)=r⋅Eαq(r∣α) (1)

In general, depending on the form of second stage competition, the quantity may vary. In this paper, we focus on markets with linear demand, see Mills (1959), Petruzzi and Dada (1999) and Huang et al. (2013) among others, and allow for a wide range of competition structures between the retailers, see Table 1. All structures presented in Table 1 give rise – in equilibrium – to essentially the same and hence to the same objective function for the supplier. More importantly, in all these structures, the second-stage equilibrium between the retailers is unique and hence, is uniquely determined under the assumption that the retailers follow their equilibrium strategies in the game induced by each wholesale price (subgame perfect equilibrium concept).To ease the exposition, we restrict to retailers and generalize our results to arbitrary number of symmetric retailers in Subsection 3.3. Specifically, we assume that each retailer faces the inverse demand function

 pi=α−βqi−γqj, (2)

for and . Here, denotes the potential market size (primary demand), the store-level factor and the degree of product differentiation or substitutability between the retailers, see Singh and Vives (1984), Wu et al. (2012). As usual, we assume that . Each retailer’s only cost is the wholesale price that she pays to the supplier. Hence, each retailer aims to maximize her profit function , which is equal to

 Πi(qi,qj)=qi(pi−r) (3)

Given the wholesale price set by the supplier and the realized value of the demand parameter , our model allows the retailers to engage in various forms of competition, the most common of which are summarized in Table 1. Here, denotes the positive part, i.e., and is a function of which denotes the equilibrium quantity that retailer orders from the supplier for realized demand and posted wholesale price . Since, there is no uncertainty on the side of retailers, is also the quantity that each retailer will sell to the market (in equilibrium).

Retail market structure Retailer ’s order
Singh and Vives (1984)
Cournot competition – product differentiation
Bertrand competition – product differentiation
Single retailer no/full returns
Competing retailers (orders/price) – no returns
Competing retailers (orders/price) – full returns
Yang and Zhou (2006)
Collusion between retailers – product differentiation

The standard Cournot and Betrand outcomes arise as special cases of the above. In particular, for , the goods are independent and we have the monopoly solution . For , the goods are perfect substitutes with in Bertrand competition (at zero price) and in Cournot competition. All of the above are assumed to be common knowledge among the participants in the market (the supplier and the retailers).

## 3 Equilibrium analysis: supplier’s optimal wholesale price

We restrict attention to subgame perfect equlibria222Technically, these are perfect Bayes-Nash equilibria, since the supplier has a belief about the retailers’ types, i.e. their willingness-to-pay his price, that depends on the value of the stochastic demand parameter . of the extensive form, two-stage game, in which for any wholesale price, , set by the supplier, the retailers order their equilibrium quantities, , with in the game induced by .

Under subgame perfection, uniqueness of the equlibrium strategies in all the retail market competition-structures of Table 1 implies that, conditional on , the supplier correctly predicts the retailers’ orders (optimal responses) for choosing wholesale price . Hence, taking the expectation over all possible values of , (1) becomes . Based on the second column of Table 1, the aggregate quantity that the retailers will order from the supplier in equilibrium has the general form , for all competition-structures, where is a suitable model-specific constant. Thus, the supplier’s expected profit maximization problem becomes

 maxr≥0Π∗s(r)=λM⋅maxr≥0rE(α−r)+ (4)

From the supplier’s perspective, we are interested in finding conditions such that the maximization problem in (4) admits a unique and finite optimal wholesale price, .

### 3.1 Deterministic Market

First, we treat the case in which the supplier knows the primary demand (deterministic market). According to the notation introduced in Section 2, this corresponds to the case . In this case and it is straightforward that . Hence, the complete information two-stage game has a unique subgame perfect Nash equilibrium, under which the supplier sells with optimal price and each retailer orders quantity as determined by Table 1.

### 3.2 Stochastic Market

The equilibrium behavior of the market in which the supplier does not know the demand level (stochastic market) is less straightforward. Now, and the supplier is interested in finding an that maximizes his expected profit in (4). For an arbitrary demand distribution , may not be concave (nor quasi-concave) and, hence, not unimodal, in which case the solution to the supplier’s optimization problem is not immediate. To obtain a general unimodality condition, we proceed by differentiating the supplier’s revenue function , see also Leonardos and Melolidakis (2018). First, since is nonnegative, we write , for . Since, and is non-atomic by assumption, we have that

for any . With this formulation, both the supplier’s revenue function and its first derivative can be expressed in terms of the mean residual life (MRL) function of . In general, the MRL function of a nonnegative random variable

with cumulative distribution function (cdf)

and finite expectation, , is defined as

 m(r):=E(α−r∣α>r)=1¯F(r)∫∞r¯F(u)du,for r

and , otherwise, see, e.g., Shaked and Shanthikumar (2007); Lai and Xie (2006) or Belzunce et al. (2016). Using this notation, we obtain that and

 dΠ∗sdr(r)=λM(m(r)−r)¯F(r)=λMr(m(r)r−1)¯F(r) (5)

for . Based on (5), the first order condition (FOC) for the supplier’s revenue function is that or equivalently that . We call the expression

 \e(r):=m(r)r,0

the generalized mean residual life (GMRL) function, see Leonardos and Melolidakis (2018), due to its connection to the generalized failure rate (GFR) function , defined and studied by Lariviere (1999) and Lariviere and Porteus (2001). Its meaning is straightforward: while the MRL function at point measures the expected additional demand, given the current demand , the GMRL function measures the expected additional demand as a percentage of the given current demand. Similarly to the GFR function, the GMRL function has an appealing interpreation from an economic perspective, as it is related to the price elasticity of expected or mean demand, , see also Xu et al. (2010). Specifically,

 (7)

which implies that corresponds to the inverse of the price elasticity of expected demand. Hence, the FOC asserts that the supplier’s payoff must be maximized at the point(s) of unitary elasticity. Also, realistic problems must have a price elasticity that eventually becomes greater than , see Lariviere (2006). Consequently, for an economically meaningful analysis, we focus on distributions for which eventually becomes less than , i.e., distributions for which is finite. Observe that for a nonnegative random demand with continuous distribution and finite expectation , and hence .

Based on these considerations, it remains to derive conditions that guarantee the existence and uniqueness of an that satisfies the FOC and to show that this indeed corresponds to a maximum of the supplier’s revenue function. This is established in the following Theorem which is the main result of the present Section.

###### Theorem 3.1 (Equilibrium wholesale prices in the stochastic market).

Assume that the nonnegative demand parameter, , follows a continuous (non-atomic) distribution , with support .

• If an optimal price for the supplier exists, then satisfies the fixed point equation

 r∗=m(r∗) (8)
• If the generalized mean residual life (GMRL) function, , of is strictly decreasing and is finite, then in equilibrium, the supplier’s optimal price exists and is the unique solution of (8). In this case, , if , and , otherwise.

###### Proof.

Since for , the sign of the derivative is determined by the term and any critical point satisfies . Hence, the necessary part of the theorem is obvious from (5) and the continuity of . For the sufficiency part, it remains to check that such a critical point exists and corresponds to a maximum under the assumptions that is strictly decreasing and . Clearly, is continuous and . Hence, starts increasing on . However, the limiting behavior of and hence of as approaches from the left, may vary depending on whether is finite or not. If is finite, i.e. if the support of is bounded, then . Hence, eventually becomes less than 1 and a critical point that corresponds to a maximum exists without any further assumptions. Strict monotonicity of implies that this is unique. If , then an optimal solution may not exist because the limiting behavior of as may vary, see Example 3.3 or Bradley and Gupta (2003). In this case, the condition of finite second moment ensures that . In particular, as we show in Leonardos and Melolidakis (2018), if the GMRL function of a random variable with unbounded support is decreasing, then if and only if is finite. This establishes existence. Uniqueness follows again from strict monotonicity of which precludes intervals of the form that give rise to multiple optimal solutions.

To prove the second claim of the sufficiency part, note that is equivalent to . Then, the DGMRL property implies that for all , hence . In this case, and hence is given explicitly by , which may be compared with the optimal of the complete information case. On the other hand, if , then for all , which implies that must be in . ∎

The economic interpretation of the sufficiency conditions in part (b) of Theorem 3.1 is immediate. By (7), demand distributions with the DGMRL property are precisely distributions that exhibit increasing price elasticity of expected demand. By Leonardos and Melolidakis (2018), Theorem 3.2, finiteness of the second moment is required to ensure that the expended demand will eventually become elastic, even in the case of unbounded support. In sum, part (b) characterizes in terms of their mathematical properties, demand distributions that model linear markets with monotone and eventually elastic expected demand. These conditions are derived in a broad probabilistic context and apply to distributions that may neither be absolutely continuous (do not possess a density) nor have a connected support.

###### Remark 3.2.

Strict monotonocity can be relaxed to weak monotonicity in the statement of Theorem 3.1 without significant loss of generality. This relies on the explicit characterization of distributions with MRL functions that contain linear segments which is given in Proposition 10 of Hall and Wellner (1981). Namely, on some interval if and only if for all . If is unbounded, this implies that has the Pareto distribution on with scale parameter . In this case, , see Example 3.3, which is precluded by the requirement that . Hence, to replace strict by weak monotonicity – but still retain equilibrium uniqueness – it suffices to exclude distributions that contain intervals with in their support, for which for all .

###### Example 3.3 (Pareto distribution).

The Pareto distribution is the unique distribution with constant GMRL and GFR functions over its support. Let be Pareto distributed with pdf , and parameters and (for we get , which contradicts the basic assumptions of our model). To simplify, let , so that , , and . The mean residual life of is given by and, hence, is decreasing on and increasing on . However, the GMRL function is decreasing for and is constant thereafter, hence, is DGMRL. Similarly, for the failure (hazard) rate is decreasing, but the generalized failure rate is constant and, hence, is IGFR. The payoff function of the supplier is

 Π∗s(r)=λMrm(r)¯F(r)=λM(k−1){r(r−rk+k)if 0≤r<1r2−kif r≥1,

which diverges as , for and remains constant for . In particular, for , the second moment of is infinite, i.e. , which shows that for DGMRL distributions, we may not drop the assumption that the second moment of is finite, for part (b) of Theorem 3.1 to hold. Contrary, for , we get as the unique optimal wholesale price, which is indeed the unique fixed point of .

### 3.3 General case with n identical retailers

To ease the exposition, we restricted our presentation to identical retailers. However, the present analysis applies to arbitrary number of symmetric retailers for most competition-structures that are given in Table 1. This relies on the fact, that these structures give rise to a unique second-stage equilibrium, and hence, to a well defined expected profit function for the supplier, as in (4), under the subgame perfect equilibrium concept, for any . Moreover, the total quantity that is ordered from the supplier depends on only up to a scaling constant. Thus, the approach to the supplier’s expected profit maximization in the first-stage remains the same independently of the number of second-stage retailers.

To avoid unnecessary notation, we restrict attention to classic Cournot competition. Formally, let , with denote the set of symmetric retailers. A strategy profile (retailers’ orders from the supplier) is denoted by with and . Assuming linear inverse demand function , the payoff function of retailer , for , is given by . Under these assumptions, the second stage corresponds to a linear Cournot oligopoly with constant marginal cost, . Hence, each retailer’s equilibrium strategy, , is given by , for . Accordingly, in the first stage, the supplier’s expected revenue function on the equilibrium path is given by . Hence, it is maximized again at if the supplier knows or at if the supplier only knows the distribution of . Based on the above, the number of second-stage retailers affects the supplier’s revenue function only up to a scaling constant and Theorem 3.1 is stated unaltered for any . The other forms of competition admit similar generalizations provided that the retailers remain symmetric.

## 4 Market Efficiency

We now turn to study the effect of upstream demand uncertainty on the efficiency of the vertical market. Unless stated otherwise, and to avoid unnecessary notation, we consider classic Cournot competition with linear demand and arbitrary number of competing retailers in the second stage, see also Subsection 3.3. After scaling to , this implies that the equilibrium order quantities are for each and any wholesale price . The supplier’s optimal wholesale price, , is given by Theorem 3.1.

Markets with incomplete information are usually inefficient in the sense that trades that are profitable for all market participants may actually not take place. In the current model, such inefficiencies appear as values of for which a transaction does not occur in equilibrium under incomplete information, although such a transaction would have been beneficial for all parties involved, i.e., supplier, retailers and consumers.

If , then the retailers buy units and there is an immediate stockout. Hence, for a continuous distribution of , the probabilitiy of no-trade in equilibrium under incomplete information is equal to . To study this probability as a measure of market inefficiency, we restrict attention to the family of DMRL distributions, i.e., distributions for which is non-increasing. In this case, we have

###### Theorem 4.1.

For any demand distribution with the DMRL property, the probability of no-trade in the equilibrium of the stochastic market cannot exceed the bound . This bound is tight over all DMRL distributions.

###### Proof.

Expressing the distribution function in terms of the MRL function, e.g. see Guess and Proschan (1988), we get . Hence, by the DMRL property and the monotonicity of the exponential function, it follows that . Since , we conclude that

. If the MRL function is constant, as is the case for the exponential distribution, see

Example 4.2, then all inequalities above hold as equalities, which establishes the second claim of the Theorem. ∎

Examples 4.3 and 4.2 highlight the tightness of the no-trade probability bound that is derived in Theorem 4.1. Example 4.4 shows that this bound cannot be extended to the class of DGMRL distributions. The conclusions are summarized in Figure 1.

###### Example 4.2 (Exponential distribution).

Let , with , and pdf . Since , for , the MRL function is constant over its support and, hence, is both DMRL and IMRL but strictly DGMRL, as , for . By Theorem 3.1, the optimal strategy of the supplier is . The probability of no transaction is equal to , confirming that the bound derived in Theorem 4.1 is tight. Thus, the exponential distribution is the least favorable, over the class of DMRL distributions, in terms of efficiency at equilibrium.

###### Example 4.3 (Beta distribution).

This example refers to a special case of the Beta distribution, also known as the Kumaraswamy distribution, see

Jones (2009). Let with , and pdf . Then, and for . Since the MRL function is decreasing, Theorem 3.1 applies and the optimal price of the supplier is . Hence, as . This shows that the upper bound of in Theorem 4.1 is still tight over distributions with strictly decreasing MRL, i.e., it is not the flatness of the exponential MRL that generated the large inefficiency.

###### Example 4.4 (Generalized Pareto or Pareto II distribution).

This example shows that the bound of Theorem 4.1 does not extend to the class of DGMRL distributions. Let , with pdf and cdf , with . For the parametrization and , with , the cdf becomes . Moreover, , since for any . Hence, by a standard calculation, , which shows that is DGMRL but not DMRL. In this case, and , which shows that the probability of a stockout may become arbitrarily large for values of close to . The “pathology” of this example relies on the fact that as .

### 4.2 Division of realized market profits

If the realized value of is larger than , then a transaction between the supplier and the retailers takes place. In this case, we measure market efficiency in terms of the realized market profits. Specifically, we fix a demand distribution (which satisfies the sufficiency conditions of Theorem 3.1) with support and a realized demand level and compare the individual realized profits of the supplier and each retailer between the deterministic and the stochastic markets. For clarity, we summarize the necessary quantities in Table 2.

Upstream Demand for the Supplier
Uncertain Deterministic
 Equilibrium Wholesale Price
 r∗=mF(r∗)
 r∗=α/2
Realized Profits in Equilibrium
Supplier
Retailer
Aggregate

We are interested in addressing the following questions: First, how do the supplier’s (retailers’) realized profits compare between the stochastic and the deterministic market? Second, how does retail competition and demand uncertainty affect the supplier’s (retailers’) share of realized market profits? Third, how does the level or retail competition – number of retailers – affect supplier’s profits in both markets? The answers are summarized in the following Theorem which follows rather immediately from Table 2. To avoid technicalities, we assume throughout that is large enough, so that (e.g. ).

###### Theorem 4.5.

Let denote a demand distribution with support , the respective optimal wholesale price in the stochastic market, and , with , a realized demand level, for which trading between supplier and retailers takes place in both the stochastic and the deterministic market. Let, also, and denote the supplier’s share of realized profits in the stochastic and deterministic markets respectively. Then,

1. , with equality only for . In particular, for any .

2. decreases in the realized demand level .

3. is independent of the demand level .

4. is higher than for values of , equal for , and lower otherwise.

5. and both increase in the level of retail competition.

Finally, each retailer’s profit in the stochastic market, , is strictly higher than her profit in the determinstic market for all demand levels and less otherwise, with equality only for .

###### Proof.

By Table 2, we have that: (i) if and only if which holds with strict inequality for all values of , except for for which the quantities are equal. The second part of statement (i) is immediate. For (ii) , and for (iii) . Now, (iv) and (v) directly follow from the previous calculations. Finally, if and only if which holds with strict inequality for all values of and with equality for . ∎

The statements of Theorem 4.5 are rather intuitive. The supplier is always better off if he is informed about the retail demand level, (i). Moreover, he captures a larger share of the realized market profits for lower values of realized demand (but not lower than the no-trade threshold of ) in the stochastic market, (ii), whereas his share of profits is constant (with respect to the demand level) in the deterministic market, (iii). Yet, there exists an interval of demand realizations, namely , for which the supplier’s profits, albeit less than in the deterministic market, represent a larger share of the aggregate market profits in the stochastic market, (iv). In any case, retail competition benefits the supplier, (v). Finally, and abiding to intuition, each retailer makes a larger profit for higher realized demand values in the case that the supplier prices under uncertainty. These observations confirm the existence of conflicting incentives regarding demand-information disclosure between the retailers and the supplier.

#### 4.2.1 Market performance: aggregate profits

We next turn to the comparison of the aggregate market profits between the deterministic and the stochastic market. As before, we fix a demand distribution (which is again assumed to satisfy the sufficiency conditions of Theorem 3.1) with support , and evaluate the ratio of the aggregate realized market profits in the stochastic market to the aggregate market profits in the deterministic market. To study market performance under the two scenarios, we need to evaluate the combined effect of two competing forces: demand uncertainty and retail competition. For a realized demand , there is a stockout and the realized aggregated profits are equal to . In this case, the stochastic market performs arbitrarily worse than the deterministic market and the ratio is equal to for any number of competing retailers. Hence, for a non-trivial analysis, we study the question of how good can the stochastic market perform when compared to the deterministic market. We have the following333Again, to avoid technicalities, and to ensure that for any , we assume that is large enough, e.g. ..

###### Theorem 4.6.

Let denote a demand distribution with support , the respective optimal wholesale price in the stochastic market, and , with , a realized demand level, for which trading between supplier and retailers takes place in both the stochastic and the deterministic market. Let, also, denote the ratio of the aggregate realized profits in the stochastic market to the aggregate profits in the deterministic market. Then,

1. for , if , if and if .

2. is maximized for for , for which it is equal to . Moreover, converges to as for any .

3. increases in the level of competition for demand levels and decreases thereafter.

###### Proof.

By Table 2, a direct substitution yields that for . Taking the derivative of the ratio with respect to yields which shows that the ratio is increasing , and decreasing thereafter. In particular, the values of for which the ratio is larger than or equal to are , for , for and for . This establishes (i) and after some trivial algebra, also (ii). To obtain (iii) we take the partial derivative of with respect to . ∎

Theorem 4.6 asserts that there exists an interval of realized demand values, whose upper bound depends on the number of competing retailers, for which the stochastic market outperforms (in terms of aggregate profits) the deterministic marke, (i). The effect of increasing retail competition on the aggregate profits of the stochastic market is twofold. First, the range (interval) of demand values for which the ratio of aggregate profits exceeds is reduced to a single point as competition increases (). Second, for larger values of realized demand, the ratio converges to as . This shows that uncertainty on the side of the supplier is less detrimental for the aggregate market profits when levels of retail cometition are low. In particular, for , the aggregate profits of the stochastic market remain strictly higher than the profits of the deterministic market for all large enough realized demand levels. As competition increases this remains true only for lower (but still above the no-trade threshold) demand levels. However, as competition increases, the ratio becomes arbitrarily low in the limit, i.e., for higher demand realizations.

The statements of Theorem 4.6 are illustrated in Figure 2. Here but the picture is essentially the same for any choice of demand distribution that satisfies the sufficiency conditions of Theorem 3.1 and for which is large enough, i.e., .

## 5 Comparative Statics

The closed form expression of (8) facilitates a comparative statics and sensitivity analysis on the demand distribution’s paremeters via the rich theory of stochastic orders, see Shaked and Shanthikumar (2007), Lai and Xie (2006) and Belzunce et al. (2016). Because in equilibrium, both the total quantity that will be sold to the market and the retail price are monotone in the wholesale price , we restrict henceforth attention to the response of in varying distribution characteristics.

To obtain a meaningful comparison between different markets, we assume throughout equilibrium uniqueness and hence, unless stated otherwise, we consider only distributions for which Theorem 3.1 applies444Since the DGMRL property is satisfied by a very broad class of distributions, see Banciu and Mirchandani (2013), Kocabıyıkoğlu and Popescu (2011) and Leonardos and Melolidakis (2018), we do not consider this as a significant restriction. Still, since it is sufficient (together with finitenes of the second moment) but not necessary for the existence of a unique optimal price, the analysis naturally applies to any other distribution that guarantees equilibrium existence and uniqueness.. First, we introduce some notation. Let be two nonnegative random variables with MRL functions and , respectively, such that for every . Then, we say that is less than in the mean residual life order, denoted by . This ordering plays a key role in the present model. Specifically, by (7), we have that for any if and only if for any , i.e., if and only if the price elasticity of expected demand in market is less than the price elasticity of expected demand in market for any wholesale price . This motivates the following definition: we will say that market is less elastic than market , denoted by , if for every . Based on the above, if and only if . Using this notation, the following Lemma captures the importance of the characterization in (8).

###### Lemma 5.1.

Let be two nonnegative, continuous and strictly DGMRL demand distributions with finite second moments. If is less elastic than , then the supplier’s optimal wholesale price is lower in market than in market . In symbols, if , then .

###### Proof.

By definition, implies that for every which by (7) is equivalent to for all . Hence, by (8), for all , where the second inequality follows from the assumption that is strictly DGMRL. Since is the unique solution of , it follows that . ∎

In short, Lemma 5.1 states that the supplier charges a lower price in a more elastic market. Although trivial to prove once Theorem 3.1 has been established, it is the key to the comparative statics analysis in the present model. Indeed, combining the above, the task of comparing the optimal wholesale price for varying demand distribution parameters – such as market size or demand variability – essentially reduces to comparing demand distributions (markets) according to their elasticities or equivalently according to their MRL functions. Such conditions can be found in Shaked and Shanthikumar (2007) and provide the framework for the subsequent analysis.

### 5.1 Market Size

We start with the response of the equilibrium wholesale price to transformations that intuitively correspond to a larger market. Unless otherwise stated, we assume that the random demand is such that it satisfies the sufficiency conditions of Theorem 3.1 and hence that the supplier’s optimal wholesale price exists and is unique.

#### 5.1.1 Reestimating Demand

Let denote the random demand in an instance of the market under consideration. Let denote a positive constant and an additional random source of demand that is independent of . Moreover, let denote the equilibrium wholesale price in the initial market and the equilibrium wholesale prices in the markets with random demand and respectively. How does compare to and ?

While the answer for is rather straightforward, see Theorem 5.2 below, the case of is more complicated. Specifically, since DGMRL random variables are not closed under convolution, see Leonardos and Melolidakis (2018), the random variable may not be DGMRL. This may lead to multiple equilibrium wholesale prices in the market, irrespectively of whether is DGMRL or not. To deal with the possible multiplicity of equilibria, we will write to denote the set of all possible equilibrium wholesale prices. Here, denotes the MRL function of a demand distribution, e.g., . To ease the notation, we will also write , when all elements of the set are less or equal than all elements of the set .

Theorem 5.2 largely confirms the intuition that wholesale prices are higher in the larger and markets. However, it also reveals that this is not always the case, and particularly for the market, that it hinges on additional, more restrictive assumptions for and .

###### Theorem 5.2.

Let be a nonnegative and continuous demand distribution with finite second moment.

1. If is DGMRL and is a positive constant, then .

2. If is DMRL and

is a nonnegative, continuous random variable with finite second moment and independent of

, then , i.e., for any equilibrium wholesale price of the market.

###### Proof.

The proof of part (i) follows directly from the preservation property of the -order that is stated in Theorem 2.A.11 of Shaked and Shanthikumar (2007). Specifically, since is the MRL function of , we have that , for all , with the inequality following from the assumption that is DGMRL. Hence, which by Lemma 5.1 implies that .

Part (ii) follows from Theorem 2.A.11 of Shaked and Shanthikumar (2007). The proof necessitates that is DMRL and hence requiring that is merely DGMRL is not enough. Since, is DMRL, we know that for all . Together with , this implies that , for all . Hence, , which implies that in this case, is a lower bound to the set of all possible wholesale equilibrium prices in the market. ∎

#### 5.1.2 Preservation of Market Size & Wholesale Price

Next, we turn attention to operations on demand distributions that preserve the -order and hence the order of wholesale prices. Specifically, let denote two different demand distributions, such that . In this case, we know by Lemma 5.1 that . We are interested in determining transformations of that preserve the -order and hence the ordering . Again, to avoid technicalities, we assume that satisfy the sufficiency conditions of Theorem 3.1.

###### Theorem 5.3.

Let denote two nonnegative, continuous and strictly DGMRL demand distributions, with finite second moments, such that .

1. If is a nonnegative, IFR distribution, independent of and , then .

2. If is an increasing, convex function, then .

3. If for some , then .

###### Proof.

Part (i) follows from Lemma 2.A.8 of Shaked and Shanthikumar (2007). Since the resulting distributions may not be DGMRL nor DMRL, the setwise notation is necessary. Part (ii) follows from Theorem 2.A.19. Equilibrium uniqueness is retained in the transformed markets, , since the DGMRL class of distributions is closed under increasing, convex transformations, see Leonardos and Melolidakis (2018). Finally, part (iii) follows from Theorem 2.A.19. However, the DGMRL class is not closed under mixtures and hence, in this case, the market may have multiple equilibria, which necessitates, as in part (i), the setwise statement for the wholesale equilibrium prices of the market. ∎

If instead of , and are ordered in the stronger hazard rate, -order, i.e., if for all , denoted by , then part (i) of Theorem 5.3 remains true by Lemma 2.A.10 of Shaked and Shanthikumar (2007), even if is merely DMRL (instead of IFR).

Although Theorems 5.3 and 5.2 are immediate once


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4 Replies Latest reply on Sep 13, 2005 11:45 AM by Dustin Hunter

# Compile Warnings

When I compile using the latest JBoss I link against \${jboss.home}/lib and \${jboss.home}/server/all/lib and I get the warnings below. These jar files certainly do not exist in the directories, and I think they are being generated from the classpath's defined in some of the manifest files. I'm I just using jboss wrong or are these warnings expected?

[javac] warning: [path] bad path element "C:\foo\tools\jboss-4.0.3RC2\lib\log4j.jar": no such file or directory
[javac] warning: [path] bad path element "C:\foo\tools\jboss-4.0.3RC2\lib\log4j-core.jar": no such file or directory
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[javac] warning: [path] bad path element "C:\foo\tools\jboss-4.0.3RC2\server\all\lib\antlr-2.7.2.jar": no such file or directory
[javac] warning: [path] bad path element "C:\foo\tools\jboss-4.0.3RC2\server\all\deploy\ejb3.deployer\log4j.jar": no such file or directory
[javac] warning: [path] bad path element "C:\foo\tools\jboss-4.0.3RC2\server\all\deploy\ejb3.deployer\log4j-core.jar": no such file or directory
[javac] warning: [path] bad path element "C:\foo\lib\jaxws-2.0_20050825\lib\jsr173_1.0_api.jar": no such file or directory
[javac] 10 warnings

• ###### 1. Re: Compile Warnings

Can you please show how you are attempting to copile your code. What command are you using? Are you using Ant? If you are using Ant what does the script look like?

• ###### 2. Re: Compile Warnings

I'm sorry I assumed that others were having this problem as well, and the build code was not needed. I assumed this because the manifest in the lib/commons-logging.jar specifically reference "log4j.jar log4j-core.jar" on the classpath (which do not exist).

Yes, we are using Ant. Code below. Thanks for looking.

```<path id="jboss.compile.classpath">
<fileset dir="\${jboss.dir}/lib">
<include name="**/*.jar" />
</fileset>
<fileset dir="\${jboss.dir}/server/all/lib">
<include name="**/*.jar" />
</fileset>
<fileset dir="\${jboss.dir}/server/all/deploy/ejb3.deployer">
<include name="*.jar" />
</fileset>
<fileset dir="\${jboss.dir}/server/all/deploy/jboss-aop-jdk50.deployer">
<include name="*.jar" />
</fileset>
</path>

<path id="compile.classpath">
<path refid="jboss.compile.classpath"/>
<path refid="testng.compile.classpath"/>
</path>

<javac destdir="\${classes.dir}"
source="\${jcompiler.version}"
target="\${jcompiler.version}"
deprecation="on"
debug="\${jcompiler.debug}"
optimize="\${jcompiler.optimize}">

<classpath>
<path refid="compile.classpath"/>
</classpath>
<src path="\${src.dir}/java/"/>
<compilerarg value="-Xlint" compiler="javac1.5"/>
</javac>```

• ###### 3. Re: Compile Warnings

What is your component that you are trying to compile? Generally the only jar you need to reference is jbossall-client.jar in the folder {jboss.home}/client

• ###### 4. Re: Compile Warnings

We are building server components, beans, web services, etc. I thought the client libs were for client applications accessing jboss as opposed to running within (but I might very well be confused... still learning).

It works if I use :

```<path id="jboss.compile.classpath">
<fileset dir="\${jboss.dir}/client" includes="**/jbossall-client.jar" />
<!--fileset dir="\${jboss.dir}/lib">
<include name="**/*.jar" />
</fileset>
<fileset dir="\${jboss.dir}/server/all/lib">
<include name="**/*.jar" />
</fileset-->
<fileset dir="\${jboss.dir}/server/all/deploy/ejb3.deployer">
<include name="*.jar" />
</fileset>
<fileset dir="\${jboss.dir}/server/all/deploy/jboss-aop-jdk50.deployer">
<include name="*.jar" />
</fileset>
</path>```

Guess the jbossall-client does not contain the ejb3.0 stuff yet.

Thanks


text:
Browse Questions

# H-atoms in ground state (13.6eV) are excited by monochromatic radiations of photon of energy 12.1eV. Find the number of spectral lines emitted in H atom.

$(a)\;1\qquad(b)\;2\qquad(c)\;3\qquad(d)\;4$

$E_n=-13.6+12.1=1.5eV$
$\therefore n^2=\large\frac{E_1}{E_n}$
$\Rightarrow \large\frac{13.6}{1.5}$$=9$
$\Rightarrow n=3$
Thus transition occurs from $3^{rd}$ to $1^{st}$ orbit.
$\therefore$ spectral lines emitted $=\sum \Delta n$
$\Rightarrow \sum(3-1)$
$\Rightarrow \sum(2)$
$\Rightarrow 1+2=3$
Hence (c) is the correct answer.


text:
# On the distributivity of arithmetic and boolean operations in 2ⁿ

✍ → Written on 2021-08-24 in 4843 words. Part of cs IT-Security pqcrypto

## Motivation

In the field of hardware security, you often need to convert between boolean and arithmetic masking (e.g. Coron et al, CHES 2014). As a result, you often need to verify an algorithm mixing arithmetic and boolean operations. One specific usecase I had, was verifying whether $a \oplus (b \land c) == (a \oplus b) \land (a \oplus c)$ holds true in $\mathbb{Z}/{16}\mathbb Z$.

Mixing such operations essentially means you don’t know these laws by heart and resort to generating truthtables to verify your claims. In this blogpost, we do it systematically. It answers questions like:

• Is XOR distributive over AND?

• Is AND distributive over XOR?

• Is OR distributive over XOR?

## The law of distributivity

The law of distributivity is easy to state: Let x and y be any two elements of a given set S. Let $\odot$ and $\oplus$ be two binary operations. We call $\odot$ left-distributive over $\oplus$ if and only if …

$x \odot (y \oplus z) = (x \odot y) \oplus (x \odot z)$

We call $\odot$ right-distributive over $\oplus$ if and only if …

$(y \oplus z) \odot x = (y \odot x) \oplus (z \odot x)$

We call $\odot$ distributive over $\oplus$ if it is left-distributive and right-distributive.

Usually, you know this property from rings which establish distributivity between addition and multiplication.

## A set of operations

Let us denote all numbers in binary, e.g. $10_2$ for decimal 2. Let $n$ define $2^n$ as the number of elements in the set. Now, we consider various operations:

$a + b$ is the common addition in $\mathbb Z/2^n \mathbb Z$, e.g. $2 + 3$ is $5$ in $\mathbb Z/2^3 \mathbb Z$

SUB

$a - b$ is subtraction in $\mathbb Z/2^n \mathbb Z$, e.g. $2 - 3$ is $7$ in $\mathbb Z/2^3 \mathbb Z$

MUL

$a \cdot b$ is the common multiplication in $\mathbb Z/2^n \mathbb Z$, e.g. $2 \cdot 3$ is $6$ in $\mathbb Z/2^3 \mathbb Z$

DIV

$a / b$ is division in $\mathbb Z/2^n \mathbb Z$ with the result $0$ in special case $b = 0$, e.g. $6 / 3$ is $2$ in $\mathbb Z/2^3 \mathbb Z$

XOR

$a \oplus b$ is a bitwise XOR, e.g. $5 \cdot 1$ is $4$

AND

$a \land b$ is a bitwise AND, e.g. $5 \land 1$ is $1$

OR

$a \lor b$ is a bitwise OR, e.g. $6 \lor 1$ is $7$

IF

$a \rightarrow b$ is an implication between two bits, e.g. if a then (b) else (false)

XNOR

$\neg (a \oplus b)$ is the negation of bitwise XOR, e.g. 1 and 1 gives 1

NAND

$\neg (a \land b)$ is the negation of bitwise AND, e.g. 1 and 1 gives 0

You can think of {ADD, SUB, MUL, DIV} as arithmetic operations and {XOR, AND, OR, IF, XNOR, NAND} as boolean operations.

## Evaluation

Now, we generate a table for $n = 1$ where l stands for left-distributivity and r stand for right-distributivity. We always consider the row operation to be distributive over the column operation.

↓ is distributive over → ADD SUB MUL DIV XOR AND OR IF XNOR NAND

SUB

MUL

l,r

l,r

l,r

l,r

l,r

l,r

l,r

DIV

l,r

l,r

l,r

l,r

l,r

l,r

l,r

XOR

AND

l,r

l,r

l,r

l,r

l,r

l,r

l,r

OR

l,r

l,r

l,r

l,r

l,r

l,r

IF

l

l

l

l

l

l

XNOR

NAND

Apparently, …

• {ADD, SUB, XOR, XNOR, NAND} is not distributive over any other operation.
For example, ADD is not distributive over ADD because $x + (y + z) \neq (x + y) + (x + z)$.
For example, ADD is not distributive over XOR because $x 1 + (1 \oplus 1) = 1 \neq 0 = (1 + 1) \oplus (1 + 1)$

• MUL is distributive over MUL because $x \cdot (y \cdot z) = (x \cdot y) \cdot (x \cdot z)$ and $(y \cdot z) \cdot x = (y \cdot x) \cdot (z \cdot x)$ in $\mathbb Z/2\mathbb Z$. Why? Because either all values are one (then both sides are one) or one value is zero (then both sides are zero).

• I call IF “left-biased” because IF returns 0 whenever the left argument is 0. This the reason why IF is left-distributive, but never right-distributive.

… in $\mathbb Z/2\mathbb Z$.

## n == 2

↓ is distributive over → ADD SUB MUL DIV XOR AND OR IF XNOR NAND

SUB

MUL

l,r

l,r

l,r

DIV

r

XOR

AND

l,r

l,r

l,r

OR

l,r

l,r

IF

l

r

r

r

XNOR

NAND

l,r

l,r

l,r

## n == 3

↓ is distributive over → ADD SUB MUL DIV XOR AND OR IF XNOR NAND

SUB

MUL

l,r

l,r

DIV

XOR

AND

l,r

l,r

l,r

OR

l,r

l,r

IF

l

r

r

r

XNOR

NAND

l,r

l,r

l,r

## n == 4

↓ is distributive over → ADD SUB MUL DIV XOR AND OR IF XNOR NAND

SUB

MUL

l,r

l,r

DIV

XOR

AND

l,r

l,r

l,r

OR

l,r

l,r

IF

l

r

r

r

XNOR

NAND

l,r

l,r

l,r

## n == 5

↓ is distributive over → ADD SUB MUL DIV XOR AND OR IF XNOR NAND

SUB

MUL

l,r

l,r

DIV

XOR

AND

l,r

l,r

l,r

OR

l,r

l,r

IF

l

r

r

r

XNOR

NAND

l,r

l,r

l,r

## python3 source code

The following source code was used to generate these results:

``````#!/usr/bin/env python3

import itertools
import collections

# convenience utilities
bit = lambda bits, i: (bits >> i) & 1

def op_per_bit(op, n):
"""Apply a binary operation to the i-th bits ∀i in i-bits vectors"""
def per_bit(a, b):
acc = 0
for i in range(n):
acc |= op(bit(a, i), bit(b, i))
return acc
return per_bit

# definition of operations
OP = collections.namedtuple('OP', 'name func')
OPS = lambda n: [
OP('ADD ', lambda a, b: (a + b) % 2**n),
OP('SUB ', lambda a, b: (a - b) % 2**n),
OP('MUL ', lambda a, b: (a * b) % 2**n),
OP('DIV ', lambda a, b: (a // b) % 2**n if b != 0 else 0),
OP('XOR ', lambda a, b: a ^ b),
OP('AND ', lambda a, b: a & b),
OP('OR  ', lambda a, b: a | b),
OP('IF  ', op_per_bit(lambda bit1, bit2: int(not bit1 or bit2), n)),
OP('XNOR', op_per_bit(lambda bit1, bit2: int(not (bit1 ^ bit2)), n)),
OP('NAND', op_per_bit(lambda bit1, bit2: int(not (bit1 & bit2)), n))
]

def result_repr(left, right):
if left and right:
return "l,r  "
elif left:
return "l    "
elif right:
return "r    "
return "     "

ops = OPS(n)

print(" " * 5 + "│", end='')
for op in ops:
print(op[0] + "│", end='')
print()
print("–" * 56)

for op_1 in ops:
print(op_1[0] + " │", end='')
for op_2 in ops:
print(result_repr(tbl_left[op_1[0] + op_2[0]], tbl_right[op_1[0] + op_2[0]]), end='')
print()

if __name__ == "__main__":
for n in range(1, 6):
print("\n## n == {}\n".format(n))

# evaluation data
left_distributivity = {}
right_distributivity = {}

# evaluation
for op1, op2 in itertools.product(OPS(n), repeat=2):
left_dist, right_dist = True, True

for arg1, arg2, arg3 in itertools.product(range(2**n), repeat=3):
lhs = op1.func(arg1, op2.func(arg2, arg3))
rhs = op2.func(op1.func(arg1, arg2), op1.func(arg1, arg3))
left_dist = left_dist and (lhs == rhs)

lhs = op1.func(op2.func(arg2, arg3), arg1)
rhs = op2.func(op1.func(arg2, arg1), op1.func(arg3, arg1))
right_dist = right_dist and (lhs == rhs)

left_distributivity[op1.name + op2.name] = left_dist
right_distributivity[op1.name + op2.name] = right_dist

## Conclusion

It is interesting to see how the operation change with increasing n.

1. The tables don’t seem to change beyond n>2.

2. If n=2 …

1. {DIV, MUL, AND} lose most distributivity properties

2. MUL is not distributive over MUL anymore, because it does not only matter whether one argument is zero, but x occurs two times on the right-hand side. Thus for values like 2, we get a first-order expression on the left-hand side and a second-order expression on the right-hand side.

3. IF was sometimes left-distributive in the case of n=1. In n=2, it is only right-distributive over IF and XNOR. Right-distributivity over NAND is a new property.

4. The table n=2 is not a subset of n=1 because the properties of IF change a lot.

3. If n>2 …

1. DIV loses right-distributivity over AND

2. MUL is not distributive over XOR anymore

Update 2021-08-26: I fixed formatting issues and a bug in the python script.


text:
# Why is V_x of a projectile constant?

1. Jul 2, 2015

### NooDota

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

Why is the horizontal velocity for a projectile constant?

I understand it physically, since the vertical and horizontal velocities aren't related and don't affect each other, g will only affect V_y and neglect V_x, so it's constant.

But maths-wise, isn't V_x = V*Cos(theta)? How can it be constant if the angle is changing?

2. Relevant equations

V_x = V*Cos(theta)

3. The attempt at a solution

2. Jul 2, 2015

### rude man

V is also changing, such that Vcosθ is constant.
Good question!

3. Jul 2, 2015

### PeroK

I
$\theta$ refers to the initial angle at launch, not the changing angle at which the projectile moves thereafter.

Last edited: Jul 2, 2015
4. Jul 2, 2015

### NooDota

But the law for the horizontal velocity is V_x = V*Cos(theta) where theta is the current angle the velocity vector makes with the x+ axis, right?

5. Jul 2, 2015

### PeroK

No, that's wrong. $v$ is the magnitude of the initial velocity and $\theta$ is the initial angle. That's why $v_x$ is constant.

6. Jul 2, 2015

### SammyS

Staff Emeritus
That's only true if θ is defined that way. Clearly OP considers θ to be changing along the trajectory.

In this case we would likely θ0 as the initial value of θ,

As usual, rude man has answered well. - - short and complete.

7. Jul 2, 2015

### PeroK

Apologies, my mistake. I see what the OP was asking.

8. Jul 2, 2015

### rude man

Thanks Sammy. Owe you one!

9. Jul 2, 2015

### rude man

You are correct, and vcosθ is constant unless you try to include air friction. Which you don't want to do, trust me!

10. Jul 2, 2015

### NooDota

Okay, thanks.

Another question, it's not worth making a thread for.

If the velocity of a projectile is given by V = 25i -4.9j (y is directed upwards), has the projectile reached its highest point yet? The book says yes.

Since it's moving downwards, that means yes. But what if I initially threw it down, and it bounced back up higher than its starting point?

11. Jul 2, 2015

### SammyS

Staff Emeritus
That's no longer simple projectile motion.

12. Jul 2, 2015

### rude man

Then its V would no longer be i 25 - j 4.9.


text:
$$\renewcommand{\AA}{\text{Å}}$$

# comm_modify command¶

## Syntax¶

comm_modify keyword value ...

• one or more keyword/value pairs may be appended

• keyword = mode or cutoff or cutoff/multi or group or reduce/multi or vel

mode value = single, multi, or multi/old = communicate atoms within a single or multiple distances
cutoff value = Rcut (distance units) = communicate atoms from this far away
cutoff/multi collection value
collection = atom collection or collection range (supports asterisk notation)
value = Rcut (distance units) = communicate atoms for selected types from this far away
reduce/multi arg = none = reduce number of communicated ghost atoms for multi style
cutoff/multi/old type value
type = atom type or type range (supports asterisk notation)
value = Rcut (distance units) = communicate atoms for selected types from this far away
group value = group-ID = only communicate atoms in the group
vel value = yes or no = do or do not communicate velocity info with ghost atoms

## Examples¶

comm_modify mode multi reduce/multi
comm_modify mode multi group solvent
comm_modify mode multi cutoff/multi 1 10.0 cutoff/multi 2*4 15.0
comm_modify vel yes
comm_modify mode single cutoff 5.0 vel yes
comm_modify cutoff/multi * 0.0

## Description¶

This command sets parameters that affect the inter-processor communication of atom information that occurs each timestep as coordinates and other properties are exchanged between neighboring processors and stored as properties of ghost atoms.

Note

These options apply to the currently defined comm style. When you specify a comm_style or read_restart command, all communication settings are restored to their default or stored values, including those previously reset by a comm_modify command. Thus if your input script specifies a comm_style or read_restart command, you should use the comm_modify command after it.

The mode keyword determines whether a single or multiple cutoff distances are used to determine which atoms to communicate.

The default mode is single which means each processor acquires information for ghost atoms that are within a single distance from its sub-domain. The distance is by default the maximum of the neighbor cutoff across all atom type pairs.

For many systems this is an efficient algorithm, but for systems with widely varying cutoffs for different type pairs, the multi or multi/old mode can be faster. In multi, each atom is assigned to a collection which should correspond to a set of atoms with similar interaction cutoffs. In this case, each atom collection is assigned its own distance cutoff for communication purposes, and fewer atoms will be communicated. in multi/old, a similar technique is used but atoms are grouped by atom type. See the neighbor multi and neighbor multi/old commands for neighbor list construction options that may also be beneficial for simulations of this kind. The multi communication mode is only compatible with the multi neighbor style. The multi/old communication mode is comparable with both the multi and multi/old neighbor styles.

The cutoff keyword allows you to extend the ghost cutoff distance for communication mode single, which is the distance from the borders of a processor’s sub-domain at which ghost atoms are acquired from other processors. By default the ghost cutoff = neighbor cutoff = pairwise force cutoff + neighbor skin. See the neighbor command for more information about the skin distance. If the specified Rcut is greater than the neighbor cutoff, then extra ghost atoms will be acquired. If the provided cutoff is smaller, the provided value will be ignored, the ghost cutoff is set to the neighbor cutoff and a warning will be printed. Specifying a cutoff value of 0.0 will reset any previous value to the default. If bonded interactions exist and equilibrium bond length information is available, then also a heuristic based on that bond length is computed. It is used as communication cutoff, if there is no pair style present and no comm_modify cutoff command used. Otherwise a warning is printed, if this bond based estimate is larger than the communication cutoff used.

The cutoff/multi option is equivalent to cutoff, but applies to communication mode multi instead. Since the communication cutoffs are determined per atom collections, a collection specifier is needed and cutoff for one or multiple collections can be extended. Also ranges of collections using the usual asterisk notation can be given. Collections are indexed from 1 to N where N is the total number of collections. Note that the arguments for cutoff/multi are parsed right before each simulation to account for potential changes in the number of collections. Custom cutoffs are preserved between runs but if collections are redefined, one may want to re-specify the communication cutoffs. For granular pair styles,the default cutoff is set to the sum of the current maximum atomic radii for each collection. The cutoff/multi/old option is similar to cutoff/multi except it operates on atom types as opposed to collections.

The reduce/multi option applies to multi and sets the communication cutoff for a particle equal to the maximum interaction distance between particles in the same collection. This reduces the number of ghost atoms that need to be communicated. This method is only compatible with the multi neighbor style and requires a half neighbor list and Newton on. See the neighbor multi command for more information.

These are simulation scenarios in which it may be useful or even necessary to set a ghost cutoff > neighbor cutoff:

• a single polymer chain with bond interactions, but no pairwise interactions

• bonded interactions (e.g. dihedrals) extend further than the pairwise cutoff

• ghost atoms beyond the pairwise cutoff are needed for some computation

In the first scenario, a pairwise potential is not defined. Thus the pairwise neighbor cutoff will be 0.0. But ghost atoms are still needed for computing bond, angle, etc interactions between atoms on different processors, or when the interaction straddles a periodic boundary.

The appropriate ghost cutoff depends on the newton bond setting. For newton bond off, the distance needs to be the furthest distance between any two atoms in the bond, angle, etc. E.g. the distance between 1-4 atoms in a dihedral. For newton bond on, the distance between the central atom in the bond, angle, etc and any other atom is sufficient. E.g. the distance between 2-4 atoms in a dihedral.

In the second scenario, a pairwise potential is defined, but its neighbor cutoff is not sufficiently long enough to enable bond, angle, etc terms to be computed. As in the previous scenario, an appropriate ghost cutoff should be set.

In the last scenario, a fix or compute or pairwise potential needs to calculate with ghost atoms beyond the normal pairwise cutoff for some computation it performs (e.g. locate neighbors of ghost atoms in a manybody pair potential). Setting the ghost cutoff appropriately can insure it will find the needed atoms.

Note

In these scenarios, if you do not set the ghost cutoff long enough, and if there is only one processor in a periodic dimension (e.g. you are running in serial), then LAMMPS may “find” the atom it is looking for (e.g. the partner atom in a bond), that is on the far side of the simulation box, across a periodic boundary. This will typically lead to bad dynamics (i.e. the bond length is now the simulation box length). To detect if this is happening, see the neigh_modify cluster command.

The group keyword will limit communication to atoms in the specified group. This can be useful for models where no ghost atoms are needed for some kinds of particles. All atoms (not just those in the specified group) will still migrate to new processors as they move. The group specified with this option must also be specified via the atom_modify first command.

The vel keyword enables velocity information to be communicated with ghost particles. Depending on the atom_style, velocity info includes the translational velocity, angular velocity, and angular momentum of a particle. If the vel option is set to yes, then ghost atoms store these quantities; if no then they do not. The yes setting is needed by some pair styles which require the velocity state of both the I and J particles to compute a pairwise I,J interaction, as well as by some compute and fix commands.

Note that if the fix deform command is being used with its “remap v” option enabled, then the velocities for ghost atoms (in the fix deform group) mirrored across a periodic boundary will also include components due to any velocity shift that occurs across that boundary (e.g. due to dilation or shear).

## Restrictions¶

Communication mode multi is currently only available for comm_style brick.

## Default¶

The option defaults are mode = single, group = all, cutoff = 0.0, vel = no. The cutoff default of 0.0 means that ghost cutoff = neighbor cutoff = pairwise force cutoff + neighbor skin.


text:
# Limit point of basic sequence in Banach space

This question is about Schauder bases in Banach spaces. Suppose $(x_n)$ is a basic sequence in a Banach space $E$. Prove that $0$ is the only possible weak limit point of $\{x_n \in E \;\vert\; n \in \mathbb{N}\}$.

-
What is a "basic sequence"? A sequence of basis elements? –  Jonas Teuwen Dec 24 '10 at 14:10
It is a sequence that is a Schauder basis for its closed linear span. –  TCL Dec 24 '10 at 15:42

Edit: I was initially assuming that $E$ is the closed linear span of the basic sequence, but then I realized that from your question there is no justification for me having done so. Since my initial argument applied only to that case and I didn't know how to extend it, I did a Google search to find that one way to reduce to this case is to apply Mazur's theorem, and that a nice and succinct answer to your question is given as Lemma 1.6.1 in Topics in Banach space theory by Albiac and Kalton. (This is something I should have known, but I had apparently not thought enough about weak topologies.)
For what it's worth, here's my original argument that applies after reducing to the case where $E$ is the closed linear span of $\{x_n\}$.
Thus every element $x$ of $E$ has a unique representation as an infinite sum $x=\sum_n a_n(x)x_n$, and the coefficient functionals $x\mapsto a_n(x)$ are bounded. Let $B=\{x_n\}$ denote the basis, and suppose that $x$ is in $\overline{B}\setminus B$. You want to show that $x=0$, which is equivalent to showing that $a_n(x)=0$ for all $n$.
For each $n$ and each $\varepsilon\gt0$, consider the subbasic neigborhood $U_{n,\varepsilon}$ of $x$ defined by $U_{n,\varepsilon}= \{y:|a_n(y)-a_n(x)|\lt \epsilon \}$. Suppose to get a contradiction that $x\neq 0$, so there exists a $k$ such that $a_k(x)\neq 0$. Then $U_{k,\varepsilon}$ is disjoint from $B\setminus\{x_k\}$ for sufficiently small $\epsilon$, and since $x$ is in $\overline B$ you can deduce that $a_k(x)=1$. Since $x\neq x_k$, there exists $m\neq k$ such that $a_m(x)\neq0$, and the same argument indicated above shows that $a_m(x)$ must be $1$. Then $U_{k,1}\cap U_{m,1}$ is a neighborhood of $x$ that doesn't intersect $B$, a contradiction.


text:
## Saturday, October 2, 2010

### Great Math Homework Help Website

Math Help Forum

Hey guys, this website is an AWESOME homework tool that has been pretty much invaluable to me throughout some of my advanced math classes.  Just ask these guys anything and they'll usually tell you exactly what you need to know in order for you to grasp a concept that you've been struggling to wrap your mind around.

There's one thing you should know about this website in that, since it sucks deciphering stuff like "a^3 / square root of 21 + 7sin(theta)", they use a thing called LaTex (pronounced: lay-tech).  This is just a very simple, code-oriented way of clearly displaying Math symbols and equations exactly as you would write them down on paper.

Here's the basics of the LaTex you'll need to know for the forum:

To start off, just know that you'll have to wrap your LaTex code in tags like so: $"your code here"$.  The "post message" form on the forum even has a shortcut button where you just highlight all your code, then click the button, and it will automatically wrap everything in these tags.

Here's a nice, basic reference sheet to familiarize yourself with:

Alright, so it's basically telling you all the stuff you'll need to know for most basic arithmetic and trig.

Stuff like a^2 will give you a to the 2nd power.. \sqrt{4} will give you the square root of four.. \sin will give you the symbol for sin.. \frac{2}{3} to give 2 over 3.. etc.  Pretty basic stuff.

So using our example from earlier...  "(a^3 / square root of 21) + 7sin(theta)" would be coded as \frac{a^3}{\sqrt{21}} + \sin\theta"  I also just learned you could do other roots like 3 to the 4th root: \sqrt[4]{3}.  You learn something new everyday.

While this might look intimidating at first, it's seriously not.  Just learn by trial and error, click on other people's LaTex to see how they coded it, and you'll be using LaTex without even thinking about it soon.  (And honestly, you don't NEED to use LaTex to ask a question.  I just find it so much clearer and easier.  Ask a few questions just the normal way, then after you see how useful this site is, maybe start picking up LaTex.)

So now that you're well on your way to mastering the nuances of LaTex, you're ready to start getting some free, comprehensive math help.  Actually, even if you don't help, I suggest you to lurk the forum every once in a while anyway.  I might need you to explain some Calc to me!


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For the 7 bit hamming code ,calculate the probability of decoding to a wrong codeword if the code is used for a channel with bit error probability 0.2 ?

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+1 vote
Hamming code can correct 1 bit error. So if more than one bit gets flipped decoding will result in wrong codeword.

So P(more than 1 bit gets flipped) = 1 - P(0 or 1 bit gets flipped) = 1 - P(no bit gets flipped) - P(1 bit gets flipped)

P(particular bit gets flipped) = 0.2, hence P(particular bit is error free) = 0.8

Assuming error in each bit is independent of error in other bits,
P(0 bit gets flipped) = P(all bits are error free) = $(0.8)^7$

P(1 bit gets flipped) =  $\binom{7}{1}*(0.2)*(0.8)^6$

Ans = 0.4233