Erdos-Ko-Rado Theorems of Labeled Sets

For k = (k1, ··· , kn) ∈ Nn, 1 ≤ k1 ≤···≤ kn, let Lkr be the family of labeled r-sets on k given by Lkr := {{(a1, la1), ··· , (ar, lar)} : {a1, ··· , ar} ■[n],lai ∈ [kai],i = 1, ··· , r}. A family A of labeled r-sets is intersecting if any two sets in A intersect. In this paper we give the sizes and structures of intersecting families of labeled r-sets.     

Higher trace forms and essential dimension in central simple algebras

We show that the essential dimension of a finite-dimensional central simple algebra coincides with the essential dimension of its r-linear trace form, for any r ≥ 3. Received: 15 March 2006     

On the Problem of Parameter Estimation in Exponential Sums

Let I≥1 be an integer, ω ₀=0<ω ₁<⋯<ω _I ≤π, and for j=0,…,I, a _j ∈ℂ, a_-j=[`(a_j)]a_{-j}={\overline{{a_{j}}}}, ω _−j =−ω _j , and a_j 1 0a_{j}\not=0 if j 1 0j\not=0. We consider the following problem: Given finitely many noisy samples of an exponential sum of the form

On zero sum subsequences of restricted size

Let G be a finite abelian group with exp(G) = e. Let s(G) be the minimal integer t with the property that any sequence of t elements in G contains an e-term subsequence with sum zero. Let n, mand r be positive integers and m ≥ 3. Furthermore, η(C _m ^r ) = a _r (m − 1) + 1, for some constant a _r depending on r and n is a fixed positive integer such that

On some extremal problems of different metrics for differentiable functions on the axis

For an arbitrary fixed segment [α, β] ⊂ R and given r ∈ N, A _r , A ₀, and p > 0, we solve the extremal problem

Poisson Numbers and Poisson Distributions in Subset Surprisology

Given a family of k + 1 real-valued functions f₀ , ?,f_kf_0 , \ldots ,f_k defined on the set { 1, ?,n}\{ 1, \ldots ,n\} and measuring the intensity of certain signals, we want to investigate whether these functions are T₀ , ?,T_k ,T_0 , \ldots ,T_k , the size a of the collection of numbers j ? { 1, ?,n}j \in \{ 1, \ldots ,n\} whose signals f₀ (j), ?,f_k (j)f_0 (j), \ldots ,f_k (j) exceed the corresponding threshold values T₀ , ?,T_kT_0 , \ldots ,T_k simultaneously for all 0, ?,k0, \ldots ,k is surprisingly large (or small) in comparison to the family of cardinalities

Simplicial complexes and Macaulay’s inverse systems

Let Δ be a simplicial complex on V = {x ₁, . . . , x _n }, with Stanley–Reisner ideal ${I_{\Delta}\subseteq R=k[x_1,\ldots, x_n]}Let Δ be a simplicial complex on V = {x ₁, . . . , x _n }, with Stanley–Reisner ideal I_D í R=k[x₁,?, x_n]{I_{\Delta}\subseteq R=k[x_1,\ldots, x_n]} . The goal of this paper is to investigate the class of artinian algebras A=A(D,a₁,?,a_n) = R/(I_D,x₁^a₁,?,x_n^a_n){A=A(\Delta,a_1,\ldots,a_n)= R/(I_{\Delta},x_1^{a_1},\ldots,x_n^{a_n})} , where each a _i ≥ 2. By utilizing the technique of Macaulay’s inverse systems, we can explicitly describe the socle of A in terms of Δ. As a consequence, we determine the simplicial complexes, that we will call levelable, for which there exists a tuple (a ₁, . . . , a _n ) such that A(Δ, a ₁, . . . , a _n ) is a level algebra.     

Graphic sequences with a realization containing intersecting cliques

Let r≥ 1, k≥ 2 and Fm1 ,...,mki;r denote the most general definition of a friendship graph, that is, the graph of Kr+m1 , . . . , Kr+mk meeting in a common r set, where Kr+mi is the complete graph on r + mi vertices. Clearly, | Fm1 ,...,mki;r | = m1+ ··· + mk + r. Let σ(Fm1 ,...,mki;r , n) be the smallest even integer such that every n-term graphic sequence π = (d1, d2, . . . , dn) with term sum σ(π) = d1 + d2 + ··· + dn ≥σ(Fm1 ,...,mki;r,n) has a realization G containing Fm1 ,...,mki;r as a subgraph. In this paper, we determine σ(Fm1 ,...,mki;r,n) for n sufficiently large.     

On generalized Stanley sequences

Let \({\mathbb{N}}\) denote the set of all nonnegative integers. Let \({k \ge 3}\) be an integer and \({A_{0} = \{a_{1}, \dots, a_{t}\} (a_{1} < \cdots < a_{t})}\) be a nonnegative set which does not contain an arithmetic progression of length k. We denote \({A = \{a_{1}, a_{2}, \ldots{}\}}\) defined by the following greedy algorithm: if \({l \ge t}\) and \({a_{1}, \dots{}, a_{l}}\) have already been defined, then \({a_{l+1}}\) is the smallest integer \({a > a_{l}}\) such that \({\{a_{1}, \dots, a_{l}\} \cup \{a\}}\) also does not contain a k-term arithmetic progression. This sequence A is called the Stanley sequence of order k generated by A₀. We prove some results about various generalizations of the Stanley sequence.     

Actions of symbolic dynamical systems on C*-algebras. II. Simplicity of C*-symbolic crossed products and some examples

A C*-symbolic dynamical system ${(\mathcal{A}, \rho, \Sigma)}A C*-symbolic dynamical system (A, r, S){(\mathcal{A}, \rho, \Sigma)} consists of a unital C*-algebra A{\mathcal{A}} and a finite family { r_a }_{a ? S}{\{ \rho_\alpha \}_{\alpha \in \Sigma}} of endomorphisms ρ _α of A{\mathcal{A}} indexed by symbols α of Σ satisfying some conditions. The endomorphisms r_a, a ? S{\rho_\alpha, \alpha \in \Sigma } yield both a subshift Λ and a C*-algebra of a Hilbert C*-bimodule. The obtained C*-algebra is regarded as a crossed product of A{\mathcal{A}} by the subshift Λ. We will study simplicity condition of these C*-algebras. Some examples such as irrational rotation Cuntz–Krieger algebras will be studied.     

A nice group structure on the orbit space of unimodular rows

If A is an affine algebra of dimension d ≥ 2, over a perfect field k, where char k ≠ 2 and c.d.₂ k ≤ 1, or if A = R[X], where R is a local, noetherian ring of dimension d ≥ 2, in which 2R = R, then the group structure of W. van der Kallen on the orbit space Um _d+1(A)/E _d+1(A) is given by coordinatewise multiplication via the product formula

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.     

Roots of algebraic equations and Clifford algebra

If an algebraic equation onIR admits real roots it admits hyperbolic roots which are elements of the set (x ₀ + εy ₀) where ε is a Clifford number having a square equal to 1. The equation can have hyperbolic or real roots

The Hermite Property of a Causal Wiener Algebra Used in Control Theory

Let \mathbbC₊ : = {s ? \mathbbC    |     Re(s) 3 0}{{\mathbb{C}}}_{+} := \{s \in {{\mathbb{C}}}\quad | \quad {\rm Re}(s) \geq 0\} and let A\mathcal{A} denote the Banach algebra

Generalized (<Emphasis Type="Italic">n</Emphasis>, <Emphasis Type="Italic">d</Emphasis>)-ray systems of points and inequalities for nonoverlapping domains and open sets

We solve the extremal problem of finding the maximum of the functional

Hausdorff–Besicovitch dimension of the graph of one continuous nowhere-differentiable function

We investigate fractal properties of the graph of the function

Extended equivariant Picard complexes and homogeneous spaces

Let k be a field of characteristic 0 and let [`(k)] \bar{k} be a fixed algebraic closure of k. Let X be a smooth geometrically integral k-variety; we set [`(X)] = X ×_k[`(k)] \bar{X} = X{ \times_k}\bar{k} and denote by [`(X)] \bar{X} . In [BvH2] we defined the extended Picard complex of X as the complex of Gal( [`(k)]

An anisotropic integral operator in high temperature superconductivity

A simplified model in superconductivity theory studied by P. Krotkov and A. Chubukov [KC1, KC2] led to an integral operator K — see (1), (2). They guessed that the equation E ₀(a, T) = 1, where E ₀ is the largest eigenvalue of the operator K, has a solution

Total Variation Distance for Poisson Subset Numbers

Let n be an integer and A₀,..., A_k random subsets of {1,..., n} of fixed sizes a₀,..., a_k, respectively chosen independently and uniformly. We provide an explicit and easily computable total variation bound between the distance from the random variable , the size of the intersection of the random sets, to a Poisson random variable Z with intensity λ = EW. In particular, the bound tends to zero when λ converges and for all j = 0,..., k, showing that W has an asymptotic Poisson distribution in this regime. Received February 24, 2005