On the spectrum of critical sets in latin squares of order 2n

Suppose that L is a latin square of order m and P ? L is a partial latin square. If L is the only latin square of order m which contains P, and no proper subset of P has this property, then P is a critical set of L. The critical set spectrum problem is to determine, for a given m, the set of integers t for which there exists a latin square of order m with a critical set of size t. We outline a partial solution to the critical set spectrum problem for latin squares of order 2ⁿ. The back circulant latin square of even order m has a well‐known critical set of size m²/4, and this is the smallest known critical set for a latin square of order m. The abelian 2‐group of order 2ⁿ has a critical set of size 4ⁿ‐3ⁿ, and this is the largest known critical set for a latin square of order 2ⁿ. We construct a set of latin squares with associated critical sets which are intermediate between the back circulant latin square of order 2ⁿ and the abelian 2‐group of order 2ⁿ. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 25–43, 2008     

Singular Set of a Convex Potential in Two Dimensions

Let u be a convex potential of the optimal transfer map from a convex open set X to a nonconvex open set Y in the plane. If u only has singularities whose sets of supports are one dimension, then under some mild assumptions on Y, we show that the singular set of u are disjoint union of countably many C ¹ curves. Or we can say that the singular set is a C ¹ manifold.     

On some sets of dictionaries whose ω ‐powers have a given

A dictionary is a set of finite words over some finite alphabet X. The ω ‐power of a dictionary V is the set of infinite words obtained by infinite concatenation of words in V. Lecomte studied in [10] the complexity of the set of dictionaries whose associated ω ‐powers have a given complexity. In particular, he considered the sets ??( Σ ⁰k) (respectively, ??( Π ⁰_k), ??( Δ ¹₁)) of dictionaries V ? 2* whose ω ‐powers are Σ ⁰_k‐sets (respectively, Π ⁰_k‐sets, Borel sets). In this paper we first establish a new relation between the sets ??( Σ ⁰₂) and ??( Δ ¹₁), showing that the set ??( Δ ¹₁) is “more complex” than the set ??( Σ ⁰₂). As an application we improve the lower bound on the complexity of ??( Δ ¹₁) given by Lecomte, showing that ??( Δ ¹₁) is in Σ ¹ ₂(2^2*)\ Π ⁰₂. Then we prove that, for every integer k ≥ 2 (respectively, k ≥ 3), the set of dictionaries ??( Π ⁰_k+1) (respectively, ??( Σ ⁰_{k +1})) is “more complex” than the set of dictionaries ??( Π ⁰_k) (respectively, ??( Σ ⁰_k)) (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Construction of Difference Sets in High Exponent 2-Groups Using Representation Theory

Nontrivial difference sets in groups of order a power of 2 are part of the family of difference sets called Menon difference sets (or Hadamard), and they have parameters (2^2d+2, 2^2d+1±2 ^d , 2^2d ±2 ^d ). In the abelian case, the group has a difference set if and only if the exponent of the group is less than or equal to 2 ^d+2. In [14], the authors construct a difference set in a nonabelian group of order 64 and exponent 32. This paper generalizes that result to show that there is a difference set in a nonabelian group of order 2^2d+2 with exponent 2 ^d+3. We use representation theory to prove that the group has a difference set, and this shows that representation theory can be used to verify a construction similar to the use of character theory in the abelian case.     

Geometrical solutions of some quadratic equations with non-real roots

This note gives geometrical/graphical methods of finding solutions of the quadratic equation ax ² + bx + c = 0, a p 0, with non-real roots. Three different cases which give rise to non-real roots of the quadratic equation have been discussed. In case I a geometrical construction and its proof for finding the solutions of the quadratic equation ax ² + bx + c = 0, a p 0, when a,b,c ] R, the set of real numbers, are presented. Case II deals with the geometrical solutions of the quadratic equation ax ² + bx + c = 0, a p 0, when b ] R, the set of real numbers; and a,c ] C, the set of complex numbers. Finally, the solutions of the quadratic equation ax ² + bx + c = 0, a p 0, when a,c ] R, the set of real numbers, and b ] C, the set of complex numbers, are presented in case III.     

Diffeomorphisms with <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-shadowing property

We give a characterization of structurally stable diffeomorphisms by making use of the notion of L ^p -shadowing property. More precisely, we prove that the set of structurally stable diffeomorphisms coincides with the C ¹-interior of the set of diffeomorphisms having L ^p -shadowing property.     

The orders of graphs with prescribed degree sets

The degree set ??^G of a graph G is the set of degrees of the vertices of G. For a finite nonempty set S of positive integers, all positive integers p are determined for which there exists a graph G of order p such that ??^G = S.     

Toral algebraic sets and function theory on polydisks

A toral algebraic set A is an algebraic set in ℂ ⁿ whose intersection with T ⁿ is sufficiently large to determine the holomorphic functions on A. We develop the theory of these sets, and give a number of applications to function theory in several variables and operator theoretic model theory. In particular, we show that the uniqueness set for an extremal Pick problem on the bidisk is a toral algebraic set, that rational inner functions have zero sets whose irreducible components are not toral, and that the model theory for a commuting pair of contractions with finite defect lives naturally on a toral algebraic set.     

-regularity of the Aronsson equation in

For a nonnegative, uniformly convex HC²(R²) with H(0)=0, if uC(Ω), ΩR², is a viscosity solution of the Aronsson equation (1.7), then uC¹(Ω). This generalizes the C¹-regularity theorem on infinity harmonic functions in R² by Savin [O. Savin, C¹-regularity for infinity harmonic functions in dimensions two, Arch. Ration. Mech. Anal. 176 (3) (2005) 351–361] to the Aronsson equation.     

Not all quadrative norms are strongly stable

A norm N on an algebra A is called quadrative if N(x²) ≤ N(x)² for all x A, and strongly stable if N(x^k) ≤ N(x)^k for all x A and all k = 2, 3, 4…. Our main purpose in this note is to show that not all quadrative norms are strongly stable.     

New examples of Cantor sets in <Emphasis Type="Italic">S</Emphasis><Superscript>1</Superscript> that are not <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript>-minimal

Although every Cantor subset of the circle (S¹) is the minimal set of some homeomorphism of S¹, not every such set is minimal for a C¹ diffeomorphism of S¹. In this work, we construct new examples of Cantor sets in S¹ that are not minimal for any C¹-diffeomorphim of S¹.     

On the performance of peeling algorithms

The peeling of a d-dimensional set of points is usually performed with successive calls to a convex hull algorithm; the optimal worst-case convex hull algorithm, known to have an O(n^˙ Log (n)) execution time, may give an O(n^˙n^˙ Log (n)) to peel all the set; an O(n^˙n) convex hull algorithm, m being the number of extremal points, is shown to peel every set with an O(n-n) time, and proved to be optimal; an implementation of this algorithm is given for planar sets and spatial sets, but the latter give only an approximate O(n^˙n) performance.     

An Improvement to “A Note on Euclidean Ramsey Theory”

We characterize those subsets Y⊆ℝ ⁿ such that for every infinite X⊆ℝ ⁿ , there is a red/blue coloring of ℝ ⁿ having no monochromatic red set similar to X and no monochramtic blue set similar to Y.     

Global periodicity and openness of the set of solutions for discrete dynamical systems

In this paper, we find additional conditions to be satisfied by a globally periodic discrete dynamical system, so that its good set (the set of initial conditions providing well-defined solutions) is an open set of ? ^k or ? ^k . We will pay especial attention to the rational case and several examples will be given.     

Building Symmetric Designs With Building Sets

We introduce a uniform technique for constructing a family of symmetric designs with parameters (v(q ^m+1-1)/(q-1), kq ^m ,q ^m), where m is any positive integer, (v, k, ) are parameters of an abelian difference set, and q = k ²/(k - ) is a prime power. We utilize the Davis and Jedwab approach to constructing difference sets to show that our construction works whenever (v, k, ) are parameters of a McFarland difference set or its complement, a Spence difference set or its complement, a Davis–Jedwab difference set or its complement, or a Hadamard difference set of order 9 · 4 ^d , thus obtaining seven infinite families of symmetric designs.     

Recognition of the Group O+ 10(2) from Its Spectrum

We prove that if the set of orders of elements of a finite group G coincides with the set of orders of elements of the group D=O₁₀ ⁺(2), then G is isomorphic to D. In other words, O ₁₀ ⁺(2) is recognizable from its spectrum.     

On the Local Uniqueness of Solutions of Variational Inequalities Under H-Differentiability

In this paper, we give some sufficient conditions for the local uniqueness of solutions to nonsmooth variational inequalities where the underlying functions are H-differentiable and the underlying set is a closed convex set/polyhedral set/box/polyhedral cone. We show how the solution of a linearized variational inequality is related to the solution of the variational inequality. These results extend/unify various similar results proved for C ¹ and locally Lipschitzian variational inequality problems. When specialized to the nonlinear complementarity problem, our results extend/unify those of C ² and C ¹ nonlinear complementarity problems.     

Certaines formes linéaires pathologiques sur un espace de Banach dual

We construct a Banach spaceE such thatE′ isw ^*-separable, andf∈E″/E, which isw ^*-continuous on every set ofE′ which is thew ^*-closure of a countablebounded set ofE′.      

Geometry of free cyclic submodules over ternions

Given the algebra T of ternions (upper triangular 2×2 matrices) over a commutative field F we consider as set of points of a projective line over T the set of all free cyclic submodules of T ². This set of points can be represented as a set of planes in the projective space over F ⁶. We exhibit this model, its adjacency relation, and its automorphic collineations. Despite the fact that T admits an F-linear antiautomorphism, the plane model of our projective line does not admit any duality.     

Universal Spectra and Tijdeman's Conjecture on Factorization of Cyclic Groups

The purpose of this article is to study the Hilbert space W²\mathcal{ W}^2 A spectral set W\Omega in \RRⁿ\RR^n is a set of finite Lebesgue measure such that L² ( W)L^2 ( \Omega ) has an orthogonal basis of exponentials { e^{2 pi á\la, x ?} : \la ? \La }\{ e^{2 \pi i \langle \la, x \rangle} : \la \in \La \} restricted to W\Omega. Any such set \La\La is called a spectrum for W\Omega. It is conjectured that every spectral set W\Omega tiles \RRⁿ\RR^n by translations. A tiling set \sT\sT of translations has a \textit{ universal spectrum} \La\La if every set W\Omega that tiles \RRⁿ\RR^n by \sT\sT is a spectral set with spectrum \La\La. Recently Lagarias and Wang showed that many periodic tiling sets \sT\sT have universal spectra. Their proofs used properties of factorizations of abelian groups, and were valid for all groups for which a strong form of a conjecture of Tijdeman is valid. However, Tijdeman's original conjecture is not true in general, as follows from a construction of Szabó [17], and here we give a counterexample to Tijdeman's conjecture for the cyclic group of order 900. This article formulates a new sufficient condition for a periodic tiling set to have a universal spectrum, and applies it to show that the tiling sets in the given counterexample do possess universal spectra.