On a Regev-Seeman conjecture about ℤ2-graded tensor products

If A,B are superalgebras then, besides A?B, a ?₂-graded tensor product A $ \bar \otimes $ B arises. Kemer proved that if A,B are T-prime algebras then A? B is multi-linear equivalent to a suitable T-prime algebra C. Regev and Seeman conjectured that this holds for A $ \bar \otimes $ B as well. In this paper we prove their conjecture is true indeed, by means of G-graded polynomial identities. The results obtained are valid over any infinite field of characteristic ≠ 2.     

Factorization of Analytic Functions and Operator Inequalities

If A and B are contraction operators on a Hilbert space ${\mathcal{H}}$ that commute with a shift operator S, it is shown that A = BC for some contraction operator C on ${\mathcal{H}}$ that commutes with S if and only if ${AA^{*} \leq BB^{*}}$ .     

Immunity properties and strong positive reducibilities

We use certain strong Q-reducibilities, and their corresponding strong positive reducibilities, to characterize the hyperimmune sets and the hyperhyperimmune sets: if A is any infinite set then A is hyperimmune (respectively, hyperhyperimmune) if and only if for every infinite subset B of A, one has ${\overline{K}\not\le_{\rm ss} B}$ (respectively, ${\overline{K}\not\le_{\overline{\rm s}} B}$ ): here ${\le_{\overline{\rm s}}}$ is the finite-branch version of s-reducibility, ??_ss is the computably bounded version of ${\le_{\overline{\rm s}}}$ , and ${\overline{K}}$ is the complement of the halting set. Restriction to ${\Sigma^0_2}$ sets provides a similar characterization of the ${\Sigma^0_2}$ hyperhyperimmune sets in terms of s-reducibility. We also show that no ${A \geq_{\overline{\rm s}}\overline{K}}$ is hyperhyperimmune. As a consequence, ${\deg_{\rm s}(\overline{K})}$ is hyperhyperimmune-free, showing that the hyperhyperimmune s-degrees are not upwards closed.     

Suitable families of boxes and kernels of staircase starshaped sets in {\mathbb{R}^d}

Let S be an orthogonal polytope in ${\mathbb{R}^d}$ . There exists a suitable family ${\mathcal{C}}$ of boxes with ${S = \cup \{C : C {\rm in} \mathcal{C}\}}$ such that the following properties hold:

The staircase kernel Ker S is a union of boxes in ${\mathcal{C}}$ . Let ${\mathcal{V}}$ be the family of vertices of boxes in ${\mathcal{C}}$ , and let ${v_o\, \epsilon \mathcal{V}}$ . Point v _o belongs to Ker S if and only if v _o sees via staircase paths in S every point w in ${\mathcal{V}}$ . Moreover, these staircase paths may be selected to consist of edges of boxes in ${\mathcal{C}}$ . Let B be a box in ${\mathcal{C}}$ with vertices of B in Ker S. Box B lies in Ker S if and only if, for some b in rel int B and for every translate H of a coordinate hyperplane at ${b, b \epsilon}$ Ker (H ∩ S). For point p in S, p belongs to Ker S if and only if, for every x in S, there exist some p ? x geodesic λ (p, x) and some corresponding ${\mathcal{C}}$ - chain D containing λ (p, x) such that D is staircase starshaped at p.

     

Projective Hausdorff gaps

Todor?evi? (Fund Math 150(1):55–66, 1996) shows that there is no Hausdorff gap (A, B) if A is analytic. In this note we extend the result by showing that the assertion “there is no Hausdorff gap (A, B) if A is coanalytic” is equivalent to “there is no Hausdorff gap (A, B) if A is ${{\bf \it{\Sigma}}^{1}_{2}}$ ”, and equivalent to ${\forall r \; (\aleph_1^{L[r]}\,< \aleph_1)}$ . We also consider real-valued games corresponding to Hausdorff gaps, and show that ${\mathsf{AD}_\mathbb{R}}$ for pointclasses Γ implies that there are no Hausdorff gaps (A, B) if ${{\it{A}} \in {\bf \it{\Gamma}}}$ .     

A dedekind finite borel set

In this paper we prove three theorems about the theory of Borel sets in models of ZF without any form of the axiom of choice. We prove that if ${B\subseteq 2^\omega}$ is a G _??? -set then either B is countable or B contains a perfect subset. Second, we prove that if 2 ^?? is the countable union of countable sets, then there exists an F _??? set ${C\subseteq 2^\omega}$ such that C is uncountable but contains no perfect subset. Finally, we construct a model of ZF in which we have an infinite Dedekind finite ${D\subseteq 2^\omega}$ which is F _??? .     

Permanent Versus Determinant over a Finite Field

Let $ \mathbb{F} $ be a finite field of characteristic different from 2. We study the cardinality of sets of matrices with a given determinant or a given permanent for the set of Hermitian matrices $ {{\mathcal{H}}_n}\left( \mathbb{F} \right) $ and for the whole matrix space M _n ( $ \mathbb{F} $ ). It is known that for n = 2, there are bijective linear maps Φ on $ {{\mathcal{H}}_n}\left( \mathbb{F} \right) $ and M _n ( $ \mathbb{F} $ ) satisfying the condition per A = det Φ(A). As an application of the obtained results, we show that if n ≥ 3, then the situation is completely different and already for n = 3, there is no pair of maps (Φ, ?), where Φ is an arbitrary bijective map on matrices and ? : $ \mathbb{F} $ → $ \mathbb{F} $ is an arbitrary map such that per A = ?(det Φ(A)) for all matrices A from the spaces $ {{\mathcal{H}}_n}\left( \mathbb{F} \right) $ and M _n ( $ \mathbb{F} $ ), respectively. Moreover, for the space M _n ( $ \mathbb{F} $ ), we show that such a pair of transformations does not exist also for an arbitrary n > 3 if the field $ \mathbb{F} $ contains sufficiently many elements (depending on n). Our results are illustrated by a number of examples.     

Uniqueness of diffusion operators and capacity estimates

Let Ω be a connected open subset of R ^d . We analyse L ₁-uniqueness of real second-order partial differential operators ${H = - \sum^d_{k,l=1} \partial_k c_{kl} \partial_l}$ and ${K = H + \sum^d_{k=1}c_k \partial_k + c_0}$ on Ω where ${c_{kl} = c_{lk} \in W^{1,\infty}_{\rm loc}(\Omega), c_k \in L_{\infty,{\rm loc}}(\Omega), c_0 \in L_{2,{\rm loc}}(\Omega)}$ and C(x) = (c _kl (x)) > 0 for all ${x \in \Omega}$ . Boundedness properties of the coefficients are expressed indirectly in terms of the balls B(r) associated with the Riemannian metric C ^?1 and their Lebesgue measure |B(r)|. First, we establish that if the balls B(r) are bounded, the Täcklind condition ${\int^\infty_R dr r({\rm log}|B(r)|)^{-1} = \infty}$ is satisfied for all large R and H is Markov unique then H is L ₁-unique. If, in addition, ${C(x) \geq \kappa (c^{T} \otimes c)(x)}$ for some ${\kappa > 0}$ and almost all ${x \in \Omega}$ , ${{\rm div} c \in L_{\infty,{\rm loc}}(\Omega)}$ is upper semi-bounded and c ₀ is lower semi-bounded, then K is also L ₁-unique. Secondly, if the c _kl extend continuously to functions which are locally bounded on ?Ω and if the balls B(r) are bounded, we characterize Markov uniqueness of H in terms of local capacity estimates and boundary capacity estimates. For example, H is Markov unique if and only if for each bounded subset A of ${\overline\Omega}$ there exist ${\eta_n \in C_c^\infty(\Omega)}$ satisfying , where ${\Gamma(\eta_n) = \sum^d_{k,l=1}c_{kl} (\partial_k \eta_n) (\partial_l \eta_n)}$ , and for each ${\varphi \in L_2(\Omega)}$ or if and only if cap(?Ω) = 0.     

Axiomatisability problems for S-acts, revisited

This paper discusses necessary and sufficient conditions on a monoid S, such that a class of left S-acts is first order axiomatisable. Such questions have previously been considered by Bulman-Fleming, Gould, Stepanova and others. Let $\mathcal{C}$ be a class of embeddings of right S-acts. A left S-act B is $\mathcal{C}$ -flat if tensoring with B preserves the embeddings in $\mathcal{C}$ . We find two sets (depending on a property of $\mathcal{C}$ ) of necessary and sufficient conditions on S such that the class of all $\mathcal{C}$ -flat left S-acts is axiomatisable. These results are similar to the ??replacement tossings?? results of Gould and Shaheen for S-posets. Further, we show how to axiomatise some classes using both replacement tossings and interpolation conditions, thus throwing some light on the former technique.     

Mixed-mean inequality for submatrix

For an m × n matrix B = (b _ij ) _m×n with nonnegative entries b _ij , let B(k, l) denote the set of all k × l submatrices of B. For each A ∈ B(k, l), let a _A and g _A denote the arithmetic mean and geometric mean of elements of A respectively. It is proved that if k is an integer in ( $\tfrac{m} {2}$ ,m] and l is an integer in ( $\tfrac{n} {2}$ , n] respectively, then $$\left( {\prod\limits_{A \in B\left( {k,l} \right)} {a_A } } \right)^{\tfrac{1} {{\left( {_k^m } \right)\left( {_l^n } \right)}}} \geqslant \frac{1} {{\left( {_k^m } \right)\left( {_l^n } \right)}}\left( {\sum\limits_{A \in B\left( {k,l} \right)} {g_A } } \right),$$ with equality if and only if b _ij is a constant for every i, j.     

Proof of the BMV conjecture

We prove the BMV (Bessis, Moussa, Villani, [1]) conjecture, which states that the function ${t \mapsto \mathop{\rm Tr}\exp(A-tB)}$ , ${t \geqslant 0}$ , is the Laplace transform of a positive measure on [0,∞) if A and B are ${n \times n}$ Hermitian matrices and B is positive semidefinite. A semi-explicit representation for this measure is given.     

Numerical Radii for Tensor Products of Operators

For bounded linear operators A and B on Hilbert spaces H and K, respectively, it is known that the numerical radii of A, B and ${A\otimes B}$ are related by the inequalities ${w(A)w(B)\le w(A\otimes B)\le {\rm min}\{\|A\|w(B), w(A)\|B\|\}}$ . In this paper, we show that (1) if ${w(A\otimes B) = w(A)w(B)}$ , then w(A) = ρ(A) or w(B) = ρ(B), where ρ(·) denotes the spectral radius of an operator, and (2) if A is hyponormal, then ${w(A\otimes B) = w(A)w(B) = \|A\|w(B)}$ . Here (2) confirms a conjecture of Shiu’s and is proven via dilating the hyponormal A to a normal operator N with the spectrum of N contained in that of A. The latter is obtained from the Sz.-Nagy–Foia? dilation theory.     

A sharp equivalence between H ∞ functional calculus and square function estimates

Let (T _t ) _{t?≥ 0} be a bounded analytic semigroup on L ^p (Ω), with 1?<?p?<?∞. Let ?A denote its infinitesimal generator. It is known that if A and A ^* both satisfy square function estimates ${\bigl\|\bigl(\int_{0}^{\infty} \vert A^{\frac{1}{2}} T_t(x)\vert^2 {\rm d}t \bigr)^{\frac{1}{2}}\bigr\|_{L^p} \lesssim \|x\|_{L^p}}$ and ${\bigl\|\bigl(\int_{0}^{\infty} \vert A^{*\frac{1}{2}} T_t^*(y) \vert^2 {\rm d}t \bigr)^{\frac{1}{2}}\bigr\|_{L^{p^\prime}} \lesssim \|y\|_{L^{p^\prime}}}$ for ${x\in L^p(\Omega)}$ and ${y\in L^{p^\prime}(\Omega)}$ , then A admits a bounded ${H^{\infty}(\Sigma_\theta)}$ functional calculus for any ${\theta>\frac{\pi}{2}}$ . We show that this actually holds true for some ${\theta<\frac{\pi}{2}}$ .     

The rank and generating set for inverse limits of wreath products of Abelian groups

We derive an exact formula for the topological rank d(W) of the inverse limit ${W = \ldots \wr A_2 \wr A_1}$ of iterated wreath products of arbitrary nontrivial finite Abelian groups. By using the language of automorphisms of a spherically homogeneous rooted tree, we construct and study a topological generating set for W with cardinality ${d(A_1) + \rho'}$ , where ${\rho'}$ is the topological rank of the profinite Abelian group ${A_2 \times A_3 \times \cdots}$ . In particular, if the group A ₁ is cyclic, this approach gives a minimal generating set for W.     

A remark on Nesterenko’s theorem for Ramanujan functions

We study the algebraic independence of values of the Ramanujan q-series $A_{2j+1}(q)=\sum_{n=1}^{\infty}n^{2j+1}q^{2n}/(1-q^{2n})$ or S _2j+1(q) (j≥0). It is proved that, for any distinct positive integers i, j satisfying $(i,j)\not=(1,3)$ and for any $q\in \overline{ \mathbb{Q}}$ with 0<|q|<1, the numbers A ₁(q), A _2i+1(q), A _2j+1(q) are algebraically independent over $\overline{ \mathbb{Q}}$ . Furthermore, the q-series A _2i+1(q) and A _2j+1(q) are algebraically dependent over $\overline{ \mathbb{Q}}(q)$ if and only if (i,j)=(1,3).     

Unitary Equivalence, Similarity and Calculation of K0-Group

In this paper, we are concerned with the classification of operators on complex separable Hilbert spaces, in the unitary equivalence sense and the similarity sense, respectively. We show that two strongly irreducible operators A and B are unitary equivalent if and only if W＊（A＋B）′≈M2（C）, and two operators A and B in B1（Ω） are similar if and only if A′（AGB）/J≈M2（C）. Moreover, we obtain V（H^∞（Ω,μ）≈N and Ko（H^∞（Ω,μ）≈Z by the technique of complex geometry, where Ω is a bounded connected open set in C, and μ is a completely non-reducing measure on Ω.     

On Zero-Sum {\mathbb{Z}_k} -Magic Labelings of 3-Regular Graphs

Let G =  (V, E) be a finite loopless graph and let (A, +) be an abelian group with identity 0. Then an A-magic labeling of G is a function ${\phi}$ from E into A ? {0} such that for some ${a \in A, \sum_{e \in E(v)} \phi(e) = a}$ for every ${v \in V}$ , where E(v) is the set of edges incident to v. If ${\phi}$ exists such that a =  0, then G is zero-sum A-magic. Let zim(G) denote the subset of ${\mathbb{N}}$ (the positive integers) such that ${1 \in zim(G)}$ if and only if G is zero-sum ${\mathbb{Z}}$ -magic and ${k \geq 2 \in zim(G)}$ if and only if G is zero-sum ${\mathbb{Z}_k}$ -magic. We establish that if G is 3-regular, then ${zim(G) = \mathbb{N} - \{2\}}$ or ${\mathbb{N} - \{2,4\}.}$      

Γ-inverses of Bounded Linear Operators

Let B(H) be the algebra of all the bounded linear operators on a Hilbert space H.For A,P and Q in B(H),if there exists an operator X∈ B(H) such thatAP X QA=A,X QAP X=X,(QAP X)*=QAP X and(X QAP)*=X QAP,then X is said to be the Γ-inverse of A associated with P and Q,and denoted by AP,Q+.In this note,we present some necessary and su?cient conditions for which A+P,Qexists,and give an explicit representation of AP,Q+(if AP,Q+exists).