On the distribution of the overlaps at given disorder

We prove a rigorous version of the following heuristic statement: if, in a spin glass model, the extended Ghirlanda–Guerra identities are valid, at given disorder the distribution of the overlap of two configurations is discrete, and its support (the smallest closed set that carries this distribution) is non-random. To cite this article: G. Parisi, M. Talagrand, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Ghirlanda–Guerra identities and ultrametricity: An elementary proof in the discrete case

In this Note we give another proof of the fact that a random overlap array, which satisfies the Ghirlanda–Guerra identities and whose elements take values in a finite set, is ultrametric with probability one. The new proof bypasses random change of density invariance principles for directing measures of such arrays and, in addition to the Dovbysh–Sudakov representation, is based only on elementary algebraic consequences of the Ghirlanda–Guerra identities.     

Embedding of a pseudo-point residual design into a Möbius plane

Let $\text{U}$ be a class of subsets of a finite set X. Elements of $\text{U}$ are called blocks. Let υ, t, λ and k be nonnegative integers such that υ?k?t?0. A pair (X, $\text{U}$) is called a (υ, k, λ) t-design, denoted by S_λ(t, k, υ), if (1) |X| = υ, (2) every t-subset of X is contained in exactly λ blocks and (3) for every block A in $\text{U}$, |A| = k. A Möbius plane M is an S₁(3, q+1, q²+1) where q is a positive integer. Let ∞ be a fixed point in M. If ∞ is deleted from M, together with all the blocks containing ∞, then we obtain a point-residual design M^*. It can be easily checked that M^* is an S_q(2, q+1, q²). Any S_q(2, q+1, q²) is called a pseudo-point-residual design of order q, abbreviated by PPRD(q). Let A and B be two blocks in a PPRD(q)M^*. A and B are said to be tangent to each other at z if and only if A∩B={z}. M^* is said to have the Tangency Property if for any block A in M^*, and points x and y such that x?A and y?A, there exists at most one block containing y and tangent to A at x. This paper proves that any PPRD(q)M^* is uniquely embeddable into a Möbius plane if and only if M^* satisfies the Tangency Property.     

A deletion-invariance property for random measures satisfying the Ghirlanda–Guerra identities

We show that if a discrete random measure on the unit ball of a separable Hilbert space satisfies the Ghirlanda–Guerra identities then by randomly deleting half of the points and renormalizing the weights of the remaining points we obtain the same random measure in distribution up to rotations.     

Hochschild cohomology ring modulo nilpotence of a one point extension of a quiver algebra defined by two cycles and a quantum-like relation

We consider a one point extension algebra B of a quiver algebra A _q over a field k defined by two cycles and a quantum-like relation depending on a nonzero element q in k. We determine the Hochschild cohomology ring of B modulo nilpotence and show that if q is a root of unity, then B is a counterexample to Snashall-Solberg’s conjecture.     

Distance sets that are a shift of the integers and Fourier basis for planar convex sets

The aim of this paper is to prove that if a planar set A has a difference set Δ(A) satisfying Δ(A) ? ?⁺ + s for suitable s then A has at most 3 elements. This result is motivated by the conjecture that the disk has no more than 3 orthogonal exponentials. Further, we prove that if A is a set of exponentials mutually orthogonal with respect to any symmetric convex set K in the plane with a smooth boundary and everywhere non-vanishing curvature, then #(A ∩ [?q, q]²) ≦ C(K) _q where C(K) is a constant depending only on K. This extends and clarifies in the plane the result of Iosevich and Rudnev. As a corollary, we obtain the result from [8] and [9] that if K is a centrally symmetric convex body with a smooth boundary and non-vanishing curvature, then L ²(K) does not possess an orthogonal basis of exponentials.     

A Common Generalization to Theorems on Set Systems with <Emphasis Type="Italic">L</Emphasis>-intersections

In this paper, we provide a common generalization to the well-known Erdös–Ko–Rado Theorem, Frankl–Wilson Theorem, Alon–Babai–Suzuki Theorem, and Snevily Theorem on set systems with L-intersections. As a consequence, we derive a result which strengthens substantially the well-known theorem on set systems with k-wise L-intersections by Füredi and Sudakov [J. Combin. Theory, Ser. A, 105, 143–159 (2004)]. We will also derive similar results on L-intersecting families of subspaces of an n-dimensional vector space over a finite field F _q , where q is a prime power.     

On the maximum size of a k-zero-free set in an abelian group

A subset A of elements in an abelian group G is called k-zero-free if the equation x ₁ + x ₂ + ... + x _k = 0 has no solution in A. A k-zero-free set A in G is called maximal if A ∪ {x} is k-zero-free for no x ∈ G\A. Some bounds for the maximum size of a k-zero free set are obtained. In particular, we determine the maximum speed of a k-zero-free arithmetic progression in the cyclic group Z _n and find the upper and lower bounds for the maximum size of a k-zero-free set in an abelian group G. We describe the structure of a maximal k-zero-free set A in the cyclic group Z _n provided that gcd(n, k) = 1 and k|A| ≥ n + 1.     

Ultrametricity indices for the Euclidean and Boolean hypercubes

Motivated by Murtagh’s experimental observation that sparse random samples of the hypercube become more and more ultrametric as the dimension increases, we consider a strict version of his ultrametricity coefficient, an index derived from Rammal’s degree of ultrametricity, and a topological ultrametricity index. First, we prove that the three ultrametricity indices converge in probability to one as dimension increases, if the sample size remains fixed. This is done for uniformly and normally distributed samples in the Euclidean hypercube, and for uniformly distributed samples in F₂ ^N with Hamming distance, as well as for very general probability distributions. Further, this holds true for random categorial data in complete disjunctive form. A second result is that the ultrametricity indices vanish in the limit for the full hypercube F₂ ^N as dimensionN increases,whereby Murtagh’s ultrametricity index is largest, and the topological ultrametricity index smallest, if N is large.     

Cross-intersecting sub-families of hereditary families

Families A₁,A₂,…,Ak of sets are said to be cross-intersecting if for any i and j in {1,2,…,k} with i≠j, any set in Ai intersects any set in Aj. For a finite set X, let X² denote the power set of X (the family of all subsets of X). A family H is said to be hereditary if all subsets of any set in H are in H; so H is hereditary if and only if it is a union of power sets. We conjecture that for any non-empty hereditary sub-family H≠{∅} of X² and any k?|X|+1, both the sum and the product of sizes of k cross-intersecting sub-families A₁,A₂,…,Ak (not necessarily distinct or non-empty) of H are maxima if A₁=A₂=?=Ak=S for some largest starSofH (a sub-family of H whose sets have a common element). We prove this for the case when H is compressed with respect to an element x of X, and for this purpose we establish new properties of the usual compression operation. As we will show, for the sum, the condition k?|X|+1 is sharp. However, for the product, we actually conjecture that the configuration A₁=A₂=?=Ak=S is optimal for any hereditary H and any k?2, and we prove this for a special case.     

Factorization of alternating sums of Virasoro characters

G. Andrews proved that if n is a prime number then the coefficients ak and ak+n of the product _∞(q,q)/_∞(qn,qn)=k_∑akqk have the same sign, see [G. Andrews, On a conjecture of Peter Borwein, J. Symbolic Comput. 20 (1995) 487-501]. We generalize this result in several directions. Our results are based on the observation that many products can be written as alternating sums of characters of Virasoro modules.     

Derived representation type and Gorenstein projective modules of an algebra under crossed product

Let H and its dual H* be finite dimensional semisimple Hopf algebras. In this paper, we firstly prove that the derived representation types of an algebra A and the crossed product algebra A#σH are coincident. This is an improvement of the conclusion about representation type of an algebra in Li and Zhang [Sci China Ser A, 2006, 50: 1-13]. Secondly, we give the relationship between Gorenstein projective modules over A and that over A#σH. Then, using this result, it is proven that A is a finite dimensional CM-finite Gorenstein algebra if and only if so is A#σH.     

A zero-one law for random subgroups of some totally disconnected groups

Let A be a locally compact group topologically generated by d elements and let k > d. Consider the action, by precomposition, of Γ = Aut(F _k ) on the set of marked, k-generated, dense subgroups $ {D_{k,A}}: = \left\{ {\eta \in {\text{Hom}}\left( {{F_k},A} \right)\left| {\overline {\left\langle {\phi \left( {{F_k}} \right)} \right\rangle } = A} \right.} \right\} Let A be a locally compact group topologically generated by d elements and let k > d. Consider the action, by precomposition, of Γ = Aut(F _k ) on the set of marked, k-generated, dense subgroups D_k,A: = { h ? \textHom( F_k,A )| [`( á f( F_k ) ñ )] = A } {D_{k,A}}: = \left\{ {\eta \in {\text{Hom}}\left( {{F_k},A} \right)\left| {\overline {\left\langle {\phi \left( {{F_k}} \right)} \right\rangle } = A} \right.} \right\} . We prove the ergodicity of this action for the following two families of simple, totally disconnected, locally compact groups:

•	A = PSL2(K) where K is a non-Archimedean local field (of characteristic ≠ 2);
•	A = Aut0(T q+1)—the group of orientation-preserving automorphisms of a q + 1 regular tree, for q \geqslant 2.q \geqslant 2.

In contrast, a recent result of Minsky’s shows that the same action fails to be ergodic for A = PSL₂(C) and, when k is even, also for A = PSL₂(R). Therefore, if k \geqslant 4 k \geqslant 4 is even and K is a local field (with char(K) ≠ 2), the action of Aut(F _k ) on D_{k,\textPS\textL2(K)} {D_{k,{\text{PS}}{{\text{L}}_2}(K)}} is ergodic if and only if K is non-Archimedean. Ergodicity implies that every “measurable property” either holds or fails to hold for almost every k-generated dense subgroup of A.     

Partitions with rounded occurrences and attached parts

We introduce the number of (k,i)-rounded occurrences of a part in a partition and use q-difference equations to interpret a certain q-series S _k,i (a;x;q) as the generating function for partitions with bounded (k,i)-rounded occurrences and attached parts. When a=0 these partitions are the same as those studied by Bressoud in his extension of the Rogers-Ramanujan-Gordon identities to even moduli. When a=1/q we obtain a new family of partition identities.     

Higher rank numerical ranges and low rank perturbations of quantum channels

For a positive integer k, the rank-k numerical range Λk(A) of an operator A acting on a Hilbert space H of dimension at least k is the set of scalars λ such that PAP=λP for some rank k orthogonal projection P. In this paper, a close connection between low rank perturbation of an operator A and Λk(A) is established. In particular, for 1?r<k it is shown that Λk(A)⊆Λk−r(A+F) for any operator F with rank(F)?r. In quantum computing, this result implies that a quantum channel with a k-dimensional error correcting code under a perturbation of rank at most r will still have a (k−r)-dimensional error correcting code. Moreover, it is shown that if A is normal or if the dimension of A is finite, then Λk(A) can be obtained as the intersection of Λk−r(A+F) for a collection of rank r operators F. Examples are given to show that the result fails if A is a general operator. The closure and the interior of the convex set Λk(A) are completely determined. Analogous results are obtained for Λ_∞(A) defined as the set of scalars λ such that PAP=λP for an infinite rank orthogonal projection P. It is shown that Λ_∞(A) is the intersection of all Λk(A) for k=1,2,…. If A−μI is not compact for all μ∈C, then the closure and the interior of Λ_∞(A) coincide with those of the essential numerical range of A. The situation for the special case when A−μI is compact for some μ∈C is also studied.     

On the Wronskian combinants of binary forms

Given a sequence A=(A₁,…,A_r) of binary d-ics, we construct a set of combinants C={C_q:0≤q≤r,q≠1}, to be called the Wronskian combinants of A. We show that the span of A can be recovered from C as the solution space of an SL(2)-invariant differential equation. The Wronskian combinants define a projective imbedding of the Grassmannian G(r,S_d), and, as a corollary, any other combinant of A is expressible as a compound transvectant in C.Our main result characterises those sequences of binary forms that can arise as Wronskian combinants; namely, they are the ones such that the associated differential equation has the maximal number of linearly independent polynomial solutions. Along the way we deduce some identities which relate Wronskians to transvectants. We also calculate compound transvectant formulae for C in the case r=3.     

Approximating classes of sequences: The Hermitian case

Given a sequence {A_n} of matrices A_n of increasing dimension d_n with d_k>d_q for k>q, k,q∈N, we recently introduced the concept of approximating class of sequences (a.c.s.) in order to define a basic approximation theory for matrix sequences. We have shown that such a notion is stable under inversion, linear combinations, and product, whenever natural and mild conditions are satisfied. In this note we focus our attention on the Hermitian case and we show that is an a.c.s. for {f(A_n)}, if is an a.c.s. for {A_n}, {A_n} is sparsely unbounded, and f is a suitable continuous function defined on R. We also discuss the potential impact and future developments of such a result.     

On deep holes of standard Reed-Solomon codes

Determining deep holes is an important open problem in decoding Reed-Solomon codes. It is well known that the received word is trivially a deep hole if the degree of its Lagrange interpolation polynomial equals the dimension of the Reed-Solomon code. For the standard Reed-Solomon codes [p-1, k]p with p a prime, Cheng and Murray conjectured in 2007 that there is no other deep holes except the trivial ones. In this paper, we show that this conjecture is not true. In fact, we find a new class of deep holes for standard Reed-Solomon codes [q-1, k]q with q a power of the prime p. Let q≥4 and 2≤k≤q-2. We show that the received word u is a deep hole if its Lagrange interpolation polynomial is the sum of monomial of degree q-2 and a polynomial of degree at most k-1. So there are at least 2(q-1)qk deep holes if k q-3.     

Double k-sets in symplectic generalized quadrangles

In classical projective geometry, a double six of lines consists of 12 lines ? ₁, ? ₂, . . . , ? ₆, m ₁, m ₂, . . . , m ₆ such that the ? _i are pairwise skew, the m _i are pairwise skew, and ? _i meets m _j if and only if i ≠ j. In the 1960s Hirschfeld studied this configuration in finite projective spaces PG(3, q) showing they exist for almost all values of q, with a couple of exceptions when q is too small. We will be considering double-k sets in the symplectic geometry W(q), which is constructed from PG(3, q) using an alternating bilinear form. This geometry is an example of a generalized quadrangle, which means it has the nice property that if we take any line ? and any point P not on ?, then there is exactly one line through P meeting ?. We will discuss all of this in detail, including all of the basic definitions needed to understand the problem, and give a result classifying which values of k and q allow us to construct a double k-set of lines in W(q).