Maximal groups in βS can be trivial

We show that it is consistent that if S is the free semigroup or free group on countably many generators, there exist idempotents in βS whose maximal group is a singleton. The same conclusion holds if S is the semigroup of finite subsets of ω under the operation of union.     

Automorphism groups of certain simple 2-(q,3,λ) designs constructed from finite fields

Let F be a finite field of characteristic not 2, and SF a subset with three elements. Consider the collection

S={S·a+b | a,bF, a≠0}.

Then (F,S) is a simple 2-design and the parameter λ of (F,S) is 1, 2, 3 or 6. We find in this paper the full automorphism group of (F,S). Namely, if we put U={r | {0,1,r}S} and K the subfield of F generated by U, then the automorphisms of (F,S) are the maps of the form xg(α(x))+b, xF, where bF, α : F→F is a field automorphism fixing U, and g is a linear transformation of F considered as a vector space over K.     

Quot Functors for Deligne-Mumford Stacks

Abstract

Given a separated and locally finitely-presented Deligne-Mumford stack 𝒳 over an algebraic space S, and a locally finitely-presented 𝒪_𝒳-module ?, we prove that the Quot functor Quot(?/𝒳/S) is represented by a separated and locally finitely-presented algebraic space over S. Under additional hypotheses, we prove that the connected components of Quot(?/𝒳/S) are quasi-projective over S.     

A logarithmic Gauss curvature flow and the Minkowski problem

Let X₀ be a smooth uniformly convex hypersurface and f a postive smooth function in Sⁿ. We study the motion of convex hypersurfaces X(·,t) with initial X(·,0)=θX₀ along its inner normal at a rate equal to log(K/f) where K is the Gauss curvature of X(·,t). We show that the hypersurfaces remain smooth and uniformly convex, and there exists θ^*>0 such that if θ<θ^*, they shrink to a point in finite time and, if θ>θ^*, they expand to an asymptotic sphere. Finally, when θ=θ^*, they converge to a convex hypersurface of which Gauss curvature is given explicitly by a function depending on f(x).     

Laguerre Expansions of Gel′fand-Shilov Spaces

The subspaces G_α, G^β, and G^β_α (α, β ≥ 0)of Schwartz′ space S⁺ in (0, + ∞) are associated with the Hankel transform in the same way as the Gel′fand-Shilov spaces S_α, S^β, and S^β_α are associated with the Fourier transform. Indeed, if we consider the Hankel transform H_γ (γ < −1) defined by _γ(ƒ)(t) = ∫^∞₀ (xt)^−γ/2x^γJ_γ([formula]) ƒ(x) dx then _γ is an isomorphism from G_α, G^β, and G^β_α onto G^α, G_β, and G^α_β respectively. So. the spaces G^α_α are invariant for _γ. In this paper, we characterize the spaces G^α_α (α > 1) in terms of their Fourier-Laguerre coefficients. Also, we characterize the range of the Fourier-Laplace operator _D defined by _D(ƒ)(w) = ∫^∞₀ ƒ(t) e^{−(1/2)((1 + w)/(1 − w))t} for w D = {w : |w| ≤ 1} when it acts on the space G^α_α.     

Additivity of Lie multiplicative maps on triangular algebras

Let 𝒜 and ? be unital algebras over a commutative ring ?, and ? be a (𝒜,??)-bimodule, which is faithful as a left 𝒜-module and also as a right ?-module. Let 𝒰?=?Tri(𝒜,??,??) be the triangular algebra and 𝒱 any algebra over ?. Assume that Φ?:?𝒰?→?𝒱 is a Lie multiplicative isomorphism, that is, Φ satisfies Φ(ST???TS)?=?Φ(S)Φ(T)???Φ(T)Φ(S) for all S, T?∈?𝒰. Then Φ(S?+?T)?=?Φ(S)?+?Φ(T)?+?Z _S,T for all S, T?∈?𝒰, where Z _S,T is an element in the centre 𝒵(𝒱) of 𝒱 depending on S and T.     

Counting functions and expected values for the lattice profile at n

Recently, Dorfer and Winterhof introduced and analyzed a lattice test for sequences of length n over a finite field. We determine the number of sequences η of length n with given largest dimension S_n(η)=S for passing this test. From this result we derive an exact formula for the expected value of S_n(η). For the binary case we characterize the (infinite) sequences η with maximal possible S_n(η) for all n.     

Sphericity test in a GMANOVA–MANOVA model with normal error

For the GMANOVA–MANOVA model with normal error: , , we study in this paper the sphericity hypothesis test problem with respect to covariance matrix: Σ=λI_q (λ is unknown). It is shown that, as a function of the likelihood ratio statistic Λ, the null distribution of Λ^2/n can be expressed by Meijer’s function, and the asymptotic null distribution of −2logΛ is (as n→∞). In addition, the Bartlett type correction −2ρlogΛ for logΛ is indicated to be asymptotically distributed as with order n⁻² for an appropriate Bartlett adjustment factor −2ρ under null hypothesis.     

On improved estimators of the generalized variance

Treated in this paper is the problem of estimating with squared error loss the generalized variance | Σ | from a Wishart random matrix S: p × p W_p(n, Σ) and an independent normal random matrix X: p × k N(ξ, Σ I_k) with ξ(p × k) unknown. Denote the columns of X by X₍₁₎ ,…, X_(k) and set ψ⁽⁰⁾(S, X) = {(n − p + 2)!/(n + 2)!} | S |, ψ⁽ⁱ⁾(X, X) = min[ψ⁽ⁱ⁻¹⁾(S, X), {(n − p + i + 2)!/(n + i + 2)!} | S + X₍₁₎ X′₍₁₎ + + X_(i) X′_(i) |] and Ψ⁽ⁱ⁾(S, X) = min[ψ⁽⁰⁾(S, X), {(n − p + i + 2)!/(n + i + 2)!}| S + X₍₁₎ X′₍₁₎ + + X_(i) X′_(i) |], i = 1,…,k. Our result is that the minimax, best affine equivariant estimator ψ⁽⁰⁾(S, X) is dominated by each of Ψ⁽ⁱ⁾(S, X), i = 1,…,k and for every i, ψ⁽ⁱ⁾(S, X) is better than ψ⁽ⁱ⁻¹⁾(S, X). In particular, ψ^(k)(S, X) = min[{(n − p + 2)!/(n + 2)!} | S |, {(n − p + 2)!/(n + 2)!} | S + X₍₁₎X′₍₁₎|,…,| {(n − p + k + 2)!/(n + k + 2)!} | S + X₍₁₎X′₍₁₎ + + X_(k)X′_(k)|] dominates all other ψ's. It is obtained by considering a multivariate extension of Stein's result (Ann. Inst. Statist. Math. 16, 155–160 (1964)) on the estimation of the normal variance.     

A solution to a problem of Cameron on sum-free complete sets

For any subsets A and B of an additive group G, define A + B = { a + b: a ε A and b ε B} and −A = {−a: a ε A}. A subset S of G is said to be sum-free, complete, and symmetric respectively if S + S S^c, S + S S^c, and S = −S. Cameron asked if for all sufficiently large moduli m there exists a sum-free complete set in Z/mZ that is not symmetric. We answer Cameron's question by showing there exists such a set for all moduli greater than or equal to 890626. We also show that every sum-free complete set in Z/mZ that is not symmetric can be used to construct a counter-example to a conjecture of Conway disproved by Marica. Conway conjectured that for any finite set S of integers, |S + S| |S --- S|.     

Path coupling without contraction

Path coupling is a useful technique for simplifying the analysis of a coupling of a Markov chain. Rather than defining and analysing the coupling on every pair in Ω×Ω, where Ω is the state space of the Markov chain, analysis is done on a smaller set SΩ×Ω. If the coefficient of contraction β is strictly less than one, no further analysis is needed in order to show rapid mixing. However, if β=1 then analysis (of the variance) is still required for all pairs in Ω×Ω. In this paper we present a new approach which shows rapid mixing in the case β=1 with a further condition which only needs to be checked for pairs in S, greatly simplifying the work involved. We also present a technique applicable when β=1 and our condition is not met.     

Optimal Recovery of the Derivative of Periodic Analytic Functions from Hardy Classes

LetS_β{z : |Im z|<β}. For 2π-periodic functions which are analytic inS_βwithp-integrable boundary values, we construct an optimal method of recovery off′(ξ), ξS_β, using information about the valuesf(x₁), mldr;, f(x_n), x_j[0, 2π).     

A generalization of Burnside's combinatorial lemma

Let G be a finite group which acts on a set S. We present a method of computing the entire distribution of G-orbits of S (the number of k-element G-orbits of S for all k) in terms of the number of s ε S fixed by every σ ε H for subgroups H of G, and the Möbius function μ(·, ·) defined on the subgroup lattice of G. We deduce Burnside's lemma as a consequence of our result.     

The incidence structure of subspaces with well-scaled frames

In this paper, the incidence structure of classes of subspaces that generalize the regular (unimodular) subspaces of rational coordinate spaces is studied. Let F the a field and S - F β {0}. A subspace, V, of a coordinate space over F is S-regular if every elementary vector of V can be scaled by an element of F β {0} so that all of its non-zero entries are elements of S. A subspace that is {−1, +1 }-regular over the rational field is regular.Associated with a subspace, V, over an arbitrary (respectively, ordered) field is a matroid (oriented matroid) having as circuits (signed circuits) the set of supports (signed supports) of elementary vectors of V. Fundamental representation properties are established for the matroids that arise from certain classes of subspaces. Matroids that are (minor) minimally non-representable by various classes of subspaces are identified. A unique representability results is established for the oriented matroids of subspaces that are dyadic (i.e., {±2⁰, ±2¹, ±2², …}-regular) over the rationals. A self-dual characterization is established for the matroids of S-regular subspaces which generalizes Minty's characterization of regular spaces as digraphoids.     

On Lp-Generalization of a Theorem of Adamyan, Arov, and Kreın

The paper deals with problems relating to the theory of Hankel operators. Let G be a bounded simple connected domain with the boundary Γ consisting of a closed analytic Jordan curve. Denote by _n,p(G), 1p<∞, the class of all meromorphic functions on G that can be represented in the form h=β/α, where β belongs to the Smirnov class E_p(G), α is a polynomial degree at most n, α0. We obtain estimates of s-numbers of the Hankel operator A_f constructed from fL_p(Γ), 1p<∞, in terms of the best approximation Δ_n,p of f in the space L_p(Γ) by functions belonging to the class _n,p(G).     

A new approach to Euler splines

Starting from the exponential Euler polynomials discussed by Euler in “Institutions Calculi Differentialis,” Vol. II, 1755, the author introduced in “Linear operators and approximation,” Vol. 20, 1972, the so-called exponential Euler splines. Here we describe a new approach to these splines. Let t be a constant such that t=|t|e^iα, −π<α<π,t≠0,t≠1.. Let S₁(x:t) be the cardinal linear spline such that S₁(v:t) = t^v for all v ε Z. Starting from S₁(x:t) it is shown that we obtain all higher degree exponential Euler splines recursively by the averaging operation . Here S_n(x:t) is a cardinal spline of degree n if n is odd, while is a cardinal spline if n is even. It is shown that they have the properties S_n(v:t) = t^v for v ε Z.     

On a problem of Duke–Erdős–Rödl on cycle-connected subgraphs

In this short note, we prove that for β<1/5 every graph G with n vertices and n^2−β edges contains a subgraph G^′ with at least cn^2−2β edges such that every pair of edges in G^′ lie together on a cycle of length at most 8. Moreover edges in G^′ which share a vertex lie together on a cycle of length at most 6. This result is best possible up to the constant factor and settles a conjecture of Duke, Erdős, and Rödl.     

Essential independent sets and long cycles

An independent set S of a graph G is said to be essential if S has a pair of vertices that are distance two apart in G. For SV(G) with S≠, let Δ(S)=max{d_G(x)|xS}. We prove the following theorem. Let k2 and let G be a k-connected graph. Suppose that Δ(S)d for every essential independent set S of order k. Then G has a cycle of length at least min{|G|,2d}. This generalizes a result of Fan.     

Approximation mit einer erweiterten Klasse von Exponentialsummen

Wir behandeln das Problem, eine stetige Funktion f im Intervall [0, 1] mit einer erweiterten Klasse von Exponentialsummen gleichmäβig zu approximieren. Die Klasse V_n^τ(S) besteht dabei aus allen reellwertigen Lösungen von homogenen, linearen Differentialgleichungen n-ter Ordnung mit konstanten Koeffizienten, bei denen das charakteristische Polynom nur Nullstellen in einer Menge S der komplexen Zahlen besitzt. Wir geben einen sehr kurzen Beweis dafür, daβ jede solche Summe n-ter Ordnung höchstens n − 1 Nullstellen in [0, 1] besitzt, wenn die Frequenzen im Streifen T={λC:|Imλ|<π} liegen. Bei Beschränkung auf T={λC:0<|Imλ|≤π} läβt sich eine Minimallösung notwendig und hinreichend charakterisieren durch eine Alternante der Länge n + k + 1 und die Minimallösung ist eindeutig bestimmt, falls die Frequenzen im Innern von T^* liegen.     

On optimal simultaneous rational approximation to with being some kind of cubic algebraic function

It is shown that each rational approximant to (ω,ω²)^τ given by the Jacobi–Perron algorithm (JPA) or modified Jacobi–Perron algorithm (MJPA) is optimal, where ω is an algebraic function (a formal Laurent series over a finite field) satisfying ω³+kω-1=0 or ω³+kdω-d=0. A result similar to the main result of Ito et al. [On simultaneous approximation to (α,α²) with α³+kα-1=0, J. Number Theory 99 (2003) 255–283] is obtained.