Sumsets being squares

Alon, Angel, Benjamini and Lubetzky [1] recently studied an old problem of Euler on sumsets for which all elements of A+B are integer squares. Improving their result we prove: 1. There exists a set A of 3 positive integers and a corresponding set B?[0,N] with |B|?(logN)^15/17, such that all elements of A+B are perfect squares. 2. There exists a set A of 3 integers and a corresponding set B?[0,N] with |B|?(logN)^9/11, such that all elements of the sets A, B and A+B are perfect squares. The proofs make use of suitably constructed elliptic curves of high rank.     

On weakly central C-algebras

A unital C^∗-algebra A is weakly central if and only if for every x∈A there exists a sequence of elementary unital completely positive maps αn on A such that the sequence (αn(x)) converges to a central element.     

A characterization of Leonard pairs using the notion of a tail

Let V denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations A:V→V and A^∗:V→V that satisfy (i) and (ii) below:

(i)

There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A^∗ is diagonal.

(ii)

There exists a basis for V with respect to which the matrix representing A^∗ is irreducible tridiagonal and the matrix representing A is diagonal.

We call such a pair a Leonard pair on V. In this paper, we characterize the Leonard pairs using the notion of a tail. This notion is borrowed from algebraic graph theory.     

On single input controllability for infinite dimensional linear systems

Given a controllable linear system {A, B} where A is a Volterra operator, there exists a vector b in the range of B such that {A, b} is controllable. The case where A is a convolution operator on L²(0, ∞) is discussed and an example is given where a controllable system is not replaceable by a single input controllable system.     

Upper triangularization of matrices by permutations and lower triangular similarity transformations

Let A be an arbitary (square) matrix. As is well known, there exists an invertible matrix S such that S^-1AS is upper triangular. The present paper is concerned with the observation that S can always be chosen in the form S=∏L, where ∏ is a permutation matrix and L is lower triangular. Assuming that the eigenvalues of A are given, the matrices ∏, L, and U=L^-1∏^-1A∏L are constructed in an explicit way. The construction gives insight into the freedom one has in choosing the permutation matrix ∏. Two cases where ∏ can be chosen to be the identity matrix are discussed (A diagonable, A a low order Toeplitz matrix). There is a connection with systems theory.     

The scrambling index of symmetric primitive matrices

A nonnegative square matrix A is primitive if some power A^k>0 (that is, A^k is entrywise positive). The least such k is called the exponent of A. In [2], Akelbek and Kirkland defined the scrambling index of a primitive matrix A, which is the smallest positive integer k such that any two rows of A^k have at least one positive element in a coincident position. In this paper, we give a relation between the scrambling index and the exponent for symmetric primitive matrices, and determine the scrambling index set for the class of symmetric primitive matrices. We also characterize completely the symmetric primitive matrices in this class such that the scrambling index is equal to the maximum value.     

Representability and Specht problem for G-graded algebras

Let W be an associative PI-algebra over a field F of characteristic zero, graded by a finite group G. Let idG(W) denote the T-ideal of G-graded identities of W. We prove: 1. [G-graded PI-equivalence] There exists a field extension K of F and a finite-dimensional Z/2Z×G-graded algebra A over K such that idG(W)=idG(A^∗) where A^∗ is the Grassmann envelope of A. 2. [G-graded Specht problem] The T-ideal idG(W) is finitely generated as a T-ideal. 3. [G-graded PI-equivalence for affine algebras] Let W be a G-graded affine algebra over F. Then there exists a field extension K of F and a finite-dimensional algebra A over K such that idG(W)=idG(A).     

Sets of permutations that generate the symmetric group pairwise

The paper contains proofs of the following results. For all sufficiently large odd integers n, there exists a set of ²n−1 permutations that pairwise generate the symmetric group Sn. There is no set of ²n−1+1 permutations having this property. For all sufficiently large integers n with n≡2mod4, there exists a set of ²n−2 even permutations that pairwise generate the alternating group An. There is no set of ²n−2+1 permutations having this property.     

A construction for equidistant permutation arrays of index one

An equidistant permutation array (EPA) which we denote by A(r, λ; ν) is a ν × r array such that every row is a permutation of the integers 1, 2,…, r and such that every pair of distinct rows has precisely λ columns in common. R(r, λ) is the maximum ν such that there exists an A(r, λ; ν). In this paper we show that R(n² + n + 2, 1) ? 2n² + n where n is a prime power.     

Jordan zero-product preserving additive maps on operator algebras

Let Φ:A→B be an additive surjective map between some operator algebras such that AB+BA=0 implies Φ(A)Φ(B)+Φ(B)Φ(A)=0. We show that, under some mild conditions, Φ is a Jordan homomorphism multiplied by a central element. Such operator algebras include von Neumann algebras, C^∗-algebras and standard operator algebras, etc. Particularly, if H and K are infinite-dimensional (real or complex) Hilbert spaces and A=B(H) and B=B(K), then there exists a nonzero scalar c and an invertible linear or conjugate-linear operator U:H→K such that either Φ(A)=cUAU⁻¹ for all A∈B(H), or Φ(A)=cUA^∗U⁻¹ for all A∈B(H).     

Thin Hessenberg pairs

A square matrix is called Hessenberg whenever each entry below the subdiagonal is zero and each entry on the subdiagonal is nonzero. Let V denote a nonzero finite-dimensional vector space over a field K. We consider an ordered pair of linear transformations A:V→V and A^∗:V→V which satisfy both (i) and (ii) below.

(i)

There exists a basis for V with respect to which the matrix representing A is Hessenberg and the matrix representing A^∗ is diagonal.

(ii)

There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing A^∗ is Hessenberg.

We call such a pair a thin Hessenberg pair (or TH pair). This is a special case of a Hessenberg pair which was introduced by the author in an earlier paper. We investigate several bases for V with respect to which the matrices representing A and A^∗ are attractive. We display these matrices along with the transition matrices relating the bases. We introduce an “oriented” version of called a TH system. We classify the TH systems up to isomorphism.     

A just basis

An old problem of P. Erdös and P. Turán asks whether there is a basisA of order 2 for which the number of representationsn=a+a′, a,a′∈A is bounded. Erd?s conjectured that such a basis does not exist. We answer a related finite problem and find a basis for which the number of representations is bounded in the square mean. Writing σ (n)=|{(a, a ^t ) ∈A ²:a+a′=n}| we prove that there exists a setA of nonnegative integers that forms a basis of order 2 (that is,s(n)≥1 for alln), and satisfies ∑_{n ? N} σ(N)² = O(N).     

Steinitz classes for Galois extensions of Dedekind rings

Galois extensions of commutative rings have been studied by Chase, Harrison, Rosenberg and others. Suppose B|A is such an extension with (finite) group G where both A and B are Dedekind rings. Then the Steinitz class ^s B|A is an element in the class group Cl(A) which vanishes if and only if B is a free A-module. It is shown that ^s B|A = 1 except possibly when the characteristic char(A) ≠ 2 and G has a cyclic Sylow 2-subgroup ≠ 1. In the exceptional case there is a unique (normal) subgroup H of G with index 2 and ^s B|A = ^s C|A where C = B ^H is the fixed ring. The remaining quadratic case is known and easily treated.     

Toeplitz operators on Bergman spaces

Let ? be an element in \(H^\infty (D) + C(\overline D )\) such that ?^* is locally sectorial. In this paper it is shown that the Toeplitz operator defined on the Bergman spaceA ² (D) is Fredholm. Also, it is proved that ifS is an operator onA ²(D) which commutes with the Toeplitz operatorT _? whose symbol ? is a finite Blaschke product, thenS H ^∞ (D) is contained inH ^∞ (D). Moreover, some spectral properties of Toeplitz operators are discussed, and it is shown that the spectrum of a class of Toeplitz operators defined on the Bergman spaceA ² (D), is not connected.     

On simultaneous diagonalization of one Hermitian and one symmetric form

It is remarked that if A, B ? M_n(C), A = A^t, and B? = B^t, B positive definite, there exists a nonsingular matrix U such that (1) ū^tBU = I and (2) U^tAU is a diagonal matrix with nonnegative entries. Some related actions of the real orthogonal group and equations involving the unitary group are studied.     

Explicit constructions of infinite families of MSTD sets

Text

We explicitly construct infinite families of MSTD (more sums than differences) sets, i.e., sets where |A+A|>|A−A|. There are enough of these sets to prove that there exists a constant C such that at least C/r⁴ of the r² subsets of {1,…,r} are MSTD sets; thus our family is significantly denser than previous constructions (whose densities are at most f(r)/²r/2 for some polynomial f(r)). We conclude by generalizing our method to compare linear forms ?₁A+?+?nA with ?i∈{−1,1}.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=vIDDa1R2.     

The Connes-Higson construction is an isomorphism

Let A be a separable C∗-algebra and B a stable C∗-algebra containing a strictly positive element. We show that the group Ext^−1/2(SA,B) of unitary equivalence classes of extensions of SA by B, modulo the extensions which are asymptotically split, coincides with the group of homotopy classes of such extensions. This is done by proving that the Connes-Higson construction gives rise to an isomorphism between Ext^−1/2(SA,B) and the E-theory group E(A,B) of homotopy classes of asymptotic homomorphisms from S²A to B.     

On sets with small doubling property

Suppose that G is an arbitrary Abelian group and A is any finite subset G. A set A is called a set with small sumset if, for some number K, we have |A + A| ≤ K|A|. The structural properties of such sets were studied in the papers of Freiman, Bilu, Ruzsa, Chang, Green, and Tao. In the present paper, we prove that, under certain constraints on K, for any set with small sumset, there exists a set Λ, Λ ? _? K log |A|, such that |A ν Λ| ? |A|/K ^1/2+? , where ? > 0. In contrast to the results of the previous authors, our theorem is nontrivial even for a sufficiently large K. For example, for K we can take |A| ^η , where η > 0. The method of proof used by us is quite elementary.     

Zero divisors and prime elements of bounded semirings

A semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. A bounded semiring is a semiring equipped with a compatible bounded partial order. In this paper, properties of zero divisors and prime elements of a bounded semiring are studied. In particular, it is proved that under some mild assumption, the set Z（A） of nonzero zero divisors of A is A ／ {0, 1}, and each prime element of A is a maximal element. For a bounded semiring A with Z（A） = A ／ {0, 1}, it is proved that A has finitely many maximal elements if ACC holds either for elements of A or for principal annihilating ideals of A. As an application of prime elements, we show that the structure of a bounded semiring A is completely determined by the structure of integral bounded semirings if either ｜Z（A）｜ = 1 or ｜Z（A）｜ -- 2 and Z（A）2 ≠ 0. Applications to the ideal structure of commutative rings are also considered. In particular, when R has a finite number of ideals, it is shown that the chain complex of the poset I（R） is pure and shellable, where I（R） consists of all ideals of R.     

On the existence of an arrow and a Bergson-Samuelson social welfare function

Let Σ be the set of all possible preferences over a given set of alternatives A. Let Ω be a proper subset of Σ and let P?Ωⁿ be a fixed profile of preferences. P is heterogeneous in Ω if for all a,b,c?A and Q?Ωⁿ, there exist three alternatives x,y,z?A such that Q(a,b,c)=P(x,y,z) where Q(B) denotes the subprofile over a set of alternatives B?A. An Arrow SWF ? is a function ?:Ωⁿ→Σ satisfying the conditions Pareto and IIA. A Bergson-Samuelson SWF is a function ?:P→Σ satisfying Pareto and Independence+Neutrality. The paper shows that (a) there exist a neutral nondictatorial Arrow SWF on Ω if and only if there exist a neutral nondictatorial Bergson-Samuelson SWF on P. (b) There exist a nondictatorial n person Bergson-Samuelson SWF on P if and only if there exists a 3 person Bergson-Samuelson SWF on P. (c) There exists a nondictatorial Arrow SWF on Ω if and only if there exists a nondictatorial Bergson-Samuelson SWF on P.