On rational invariants of a set of symmetric matrices

Some identities resulting from the Cayley-Hamilton theorem are derived. Some applications include: (a) for k = 1,2,…,n ? 1 a condition is found for a pair (A,B) of symmetric operators acting in Euclidean n-space to have common invariant k-subspace (provided that A does not have multiple eigenvalues); (b) it is shown that the field of rational invariants of (A,B) is isomorphic to a subfield of a rational function field with n(n+3)/2 generators consisting of elements symmetric with respect to the permutaion group P_n; (c) it is shown that any rational invariant of (g+2) symmetric operators A,B,C₁,C₂,…, C_g can be expressed as a rational function of invariants of one or two operators that are taken for pairs (A,B), (A,C₂),…, (A,C_g, (A,B+C₁), (A,B+C₂),…,(A,B+C_g).     

On the Gröbner complexity of matrices

In this paper we show that if for an integer matrix A the universal Gröbner basis of the associated toric ideal I_A coincides with the Graver basis of A, then the Gröbner complexity u(A) and the Graver complexity g(A) of its higher Lawrence liftings agree, too. In fact, if the universal Gröbner basis of I_A coincides with the Graver basis of A, then also the more general complexities u(A,B) and g(A,B) agree for arbitrary B. We conclude that for the matrices A_3×3 and A_3×4, defining the 3×3 and 3×4 transportation problems, we have u(A_3×3)=g(A_3×3)=9 and u(A_3×4)=g(A_3×4)≥27. Moreover, we prove that u(A_a,b)=g(A_a,b)=2(a+b)/gcd(a,b) for positive integers a,b and .     

Generalizations of weakly peripherally multiplicative maps between uniform algebras

Let A and B be uniform algebras on first-countable, compact Hausdorff spaces X and Y, respectively. For f∈A, the peripheral spectrum of f, denoted by σπ(f)={λ∈σ(f):|λ|=‖f‖}, is the set of spectral values of maximum modulus. A map T:A→B is weakly peripherally multiplicative if σπ(T(f)T(g))∩σπ(fg)≠∅ for all f,g∈A. We show that if T is a surjective, weakly peripherally multiplicative map, then T is a weighted composition operator, extending earlier results. Furthermore, if T₁,T₂:A→B are surjective mappings that satisfy σπ(T₁(f)T₂(g))∩σπ(fg)≠∅ for all f,g∈A, then T₁(f)T₂(1)=T₁(1)T₂(f) for all f∈A, and the map f?T₁(f)T₂(1) is an isometric algebra isomorphism.     

Optimally small sumsets in finite abelian groups

Let G be a finite abelian group of order g. We determine, for all 1?r,s?g, the minimal size μ_G(r,s)=min|A+B| of sumsets A+B, where A and B range over all subsets of G of cardinality r and s, respectively. We do so by explicit construction. Our formula for μ_G(r,s) shows that this function only depends on the cardinality of G, not on its specific group structure. Earlier results on μ_G are recalled in the Introduction.     

Simultaneous quasidiagonalization of complex matrices

Let

be the complex algebra generated by a pair of n × n Hermitian matrices A, B. A recent result of Watters states that A, B are simultaneously unitarily quasidiagonalizable [i.e., A and B are simultaneously unitarily similar to direct sums C₁⊕…⊕C_t,D₁⊕…⊕D_t for some t, where C_i, D_i are k_i × k_i and k_i?2(1?i?t)] if and only if [p(A, B), A]² and [p(A, B), B]² belong to the center of

for all polynomials p(x, y) in the noncommuting variables x, y. In this paper, we obtain a finite set of conditions which works. In particular we show that if A, B are positive semidefinite, then A, B are simultaneously quasidiagonalizable if (and only if) [A, B]², [A², B]² and [A, B²]² commute with A, B.     

Generalized Bowen-Franks groups of integral matrices with the same zeta function

In this paper the relation between the zeta function of an integral matrix and its generalized Bowen-Franks groups is studied. Suppose that A and B are nonnegative integral matrices whose invertible part is diagonalizable over the field of complex numbers and A and B have the same zeta function. Then there is an integer m, which depends only on the zeta function, such that, for any prime q such that gcd(q,m)=1, for any g(x)∈Z[x] with g(0)=1, the q-Sylow subgroup of the generalized Bowen-Franks group BF_g(x)(A) and BF_g(x)(B) are the same. In particular, if m=1, then zeta function determines generalized Bowen-Franks groups.     

On sum sets of sidon sets,II

LetG be a finite abelian group,G?{Z _n, Z₂?Z_2n}. Then every sequenceA={g ₁,...,g_t} of $t = \frac{{4\left| G \right|}}{3} + 1$ elements fromG contains a subsequenceB?A, |G|=|G| such that $\sum\nolimits_{g_i \in B^{g_i } } { = 0 (in G)} $ . This bound, which is best possible, extends recent results of [1] and [22] concerning the celebrated theorem of Erdös-Ginzburg-Ziv [21].     

On Galois Algebras Satisfying the Fundamental Theorem

Let B be a Galois algebra over a commutative ring R with Galois group G such that B ^H is a separable subalgebra of B for each subgroup H of G. Then it is shown that B satisfies the fundamental theorem if and only if B is one of the following three types: (1) B is an indecomposable commutative Galois algebra, (2) B = Re ⊕ R(1 ? e) where e and 1 ? e are minimal central idempotents in B, and (3) B is an indecomposable Galois algebra such that for each separable subalgebra A, V _B (A) = ?∑ _g∈G(A) J _g , and the centers of A and B ^G(A) are the same where V _B (A) is the commutator subring of A in B, J _g  = {b ∈ B | bx = g(x)b for each x ∈ B} for a g ∈ G, and G(A) = {g ∈ G | g(a) = a for all a ∈ A}.     

A Uniqueness Theorem in a Finitely Accessible Additive Category

It is proved that if A ₁,A ₂,..., A _m and B ₁,B ₂,..., B _n are objects in a finitely accessible additive category \(\mathcal{A}\) such that their pure injective envelopes are indecomposable, and there are pure monomorphisms μ:A ₁?⊕?A ₂?⊕?...?⊕?A _m →B ₁?⊕?B ₂?⊕?...?⊕?B _n and ν:B ₁?⊕?B ₂?⊕?...?⊕?B _n →A ₁?⊕?A ₂?⊕?...?⊕?A _m , then m?=?n and there are a permutation σ and pure monomorphisms A _i →B _σ(i) and B _σ(i)→A _i for every i?=?1, 2, ..., n.     

Relationship between solutions of families of two-point boundary value problems and Cauchy problems

We carry out a qualitative analysis and suggest a method for the solution of the two-point boundary value problems A _l υ = g, x ? [0, l], l ? (0, L) = B ? R ₊; $$ \alpha v'_x \left| {_{x = + 0} = F_1 (v,l)} \right|_{x = 0} , \beta v'_x \left| {_{x = l - 0} = F_2 (v,l)} \right|_{x = l} , $$ , where α, β ? R, υ is the unknown function, g = g(x) is a given real function, F ₁(y, l) and F ₂(y, l) are known real functions defined on the sets Y ₁ × B and Y ₂ × B, respectively, where Y ₁ ∪ Y ₂ ? R, and A _l is the restriction of A corresponding to the embedding parameter l. (Here A is an operator taking an arbitrary function in the set of function classes defined in the paper to C(B ₁), where B ₁ = [0, L).) The study takes into account the dependence of solutions of various versions of these two-point boundary value problems on the parameter l. We construct algorithms for the reduction of these families of two-point boundary value problems to Cauchy problems for ordinary differential equations and integro-differential equations that contain only first derivatives of the unknown functions with respect to the parameter l.     

The characteristic polynomial of a principal subpencil of a Hermitian matrix pencil

Given n-square Hermitian matrices A,B, let A_i,B_i denote the principal (n?1)- square submatrices of A,B, respectively, obtained by deleting row i and column i. Let μ, λ be independent indeterminates. The first main result of this paper is the characterization (for fixed i) of the polynomials representable as det(μA_i?λB_i) in terms of the polynomial det(μA?λB) and the elementary divisors, minimal indices, and inertial signatures of the pencil μA?λB. This result contains, as a special case, the classical interlacing relationship governing the eigenvalues of a principal sub- matrix of a Hermitian matrix. The second main result is the determination of the number of different values of i to which the characterization just described can be simultaneously applied.     

Schatten p-norm inequalities related to an extended operator parallelogram law

Let C_p be the Schatten p-class for p>0. Generalizations of the parallelogram law for the Schatten 2-norms have been given in the following form: if A={A₁,A₂,…,A_n} and B={B₁,B₂,…,B_n} are two sets of operators in, then C₂

     

Constructive Axiomatization of Plane Hyperbolic Geometry

We provide a universal axiom system for plane hyperbolic geometry in a firstorder language with two sorts of individual variables, ‘points’ (upper‐case) and ‘lines’ (lowercase), containing three individual constants, A₀, A₁, A₂, standing for three non‐collinear points, two binary operation symbols, φ and ι, with φ(A, B) = l to be interpreted as ‘𝓁 is the line joining A and B’ (provided that A ≠ B, an arbitrary line, otherwise), and ι(g, h) = P to be interpreted as 𝓁P is the point of intersection of g and h (provided that g and h are distinct and have a point of intersection, an arbitrary point, otherwise), and two binary operation symbols, π₁(P, 𝓁) and —₂(P, 𝓁), with π_i(P, 𝓁) = g (for i = 1, 2) to be interpreted as ‘g is one of the two limiting paralle lines from P to 𝓁 (provided that P is not on 𝓁, an arbitrary line, otherwise).     

Rigidity and stability of the Leibniz and the chain rule

We study rigidity and stability properties of the Leibniz and chain rule operator equations. We describe which non-degenerate operators V, T ₁, T ₂,A: C ^k (?) → C(?) satisfy equations of the generalized Leibniz and chain rule type for f, g ∈ C ^k (?), namely, V (f · g) = (T ₁ f) · g + f · (T ₂ g) for k = 1, V (f · g) = (T ₁ f) · g + f · (T ₂ g) + (Af) · (Ag) for k = 2, and V (f ○ g) = (T ₁ f) ○ g · (T ₂ g) for k = 1. Moreover, for multiplicative maps A, we consider a more general version of the first equation, V (f · g) = (T ₁ f) · (Ag) + (Af) · (T ₂ g) for k = 1. In all these cases, we completely determine all solutions. It turns out that, in any of the equations, the operators V, T ₁ and T ₂ must be essentially equal. We also consider perturbations of the chain and the Leibniz rule, T (f ○ g) = Tf ○ g · Tg + B(f ○ g, g) and T (f · g) = Tf · g + f · Tg + B(f, g), and show under suitable conditions on B in the first case that B = 0 and in the second case that the solution is a perturbation of the solution of the standard Leibniz rule equation.     

Non-commutative operator Bohr inequality

It is shown that if A, B, X are Hilbert space operators such that X?γI, for the positive real number γ, and p,q>1 with 1/p+1/q=1, then |A−B|²?p|A|²+q|B|² with equality if and only if (1−p)A=B and γ||||A−B|²|||?|||p|A|²X+qX|B|²||| for every unitarily invariant norm. Moreover, if in addition A, B are normal and X is any Hilbert-Schmidt operator, then ‖δ_A,B²(X)‖₂?‖p|A|²X+qX|B|²‖₂ with equality if and only if (1−p)AX=XB.     

ℓ-group homomorphisms between reduced archimedean f-rings

Let A and B be reduced archimedean f-rings, A with identity e; let $A\,\mathop \to \limits^\gamma\,BLet A and B be reduced archimedean f-rings, A with identity e; let A \mathop ? ^g BA\,\mathop \to \limits^\gamma\,B be an ℓ-group homomorphism, and set w = γ (e). We show (with some vagaries of phrasing here) (1) γ = w·ρ for a canonical ℓ-ring homomorphism A \mathop ? ^r B (w)A\,\mathop \to \limits^\rho\,B (w), where B (w) is an extension of B in which w is a von Neumann regular element, and (2) for X _A ,X _B canonical representation spaces for A, B, γ is realized via composition with a unique partially defined continuous function from X _B to X _A .     

Artin's L-functions and one-dimensional characters

Let K/Q be a finite Galois extension with the Galois group G, let χ₁,…,χr be the irreducible non-trivial characters of G, and let A be the C-algebra generated by the Artin L-functions L(s,χ₁),…,L(s,χr). Let B be the subalgebra of A generated by the L-functions corresponding to induced characters of non-trivial one-dimensional characters of subgroups of G. We prove: (1) B is of Krull dimension r and has the same quotient field as A; (2) B=A iff G is M-group; (3) the integral closure of B in A equals A iff G is quasi-M-group.     

Lifting commuting pairs of C1 algebras

Given a commuting pair $\text{A}$₁, $\text{A}$₂ of abelian $\text{C}{}^1$ subalgebras of the Calkin algebra, we look for a commuting pair $\text{B}$₁,$\text{B}$₂ of $\text{C}{}^1$ subalgebras of $\text{B(H)}$ which project onto $\text{A}$₁ and $\text{A}$₂. We do not insist that $\text{B}$_i, be abelian, so $\text{B}$_i, may contain nontrivial compact operators. If X is the joint spectrum σ($\text{A}$₁, $\text{A}$₂), it is shown that the existence of a pair $\text{B}$₁, $\text{B}$₂ depends only on the element τ in Ext(X) determined by $\text{A}$₁, $\text{A}$₂. The set L(X) of those τ in Ext(X) which “lift” in this sense is shown to be a subgroup of Ext(X) when Ext(X) is Hausdorff, and also when $\text{A}$_i are singly generated. In this latter case, L(X) can be explicitly calculated for large classes of joint spectra. These results are applied to lift certain pairs of commuting elements of the Calkin algebra to pairs of commuting operators.     

On the exponential sum—product problem

Let g be an element of order T over a finite field Fp of p elements, where p is a prime. We show that for a very wide class of sets A, B ∈ {1, . . . , T} at least one of the sets

{gab:a∈A,b∈B}and{ga+gb:a∈A,b∈B}     

A relationship between stochastic matrices and eigenvalues of products

Let {B₁,…,B_n} be a set of n rank one n×n row stochastic matrices. The next two statements are equivalent: (1) If A is an n×n nonnegative matrix, then 1 is an eigenvalue ofB_kA for each k=1,…,n if and only if A is row stochastic. (2) The n×n row stochastic matrix S whose kth row is a row of B_k for k=1,…,n is nonsingular. For any set {B₁, B₂,…, B_p} of fewer than n row stochastic matrices of order n×n and of any rank, there exists a nonnegative n×n matrix A which is not row stochastic such that 1 is eigenvalue of every B_kA, k=1,…,p.