Cohomology and deformations of the infinite-dimensional filiform Lie algebra

Denote the infinite-dimensional -graded Lie algebra defined by basis e_i, i1, and relations [e₁,e_i]=e_i+1 for all i2. We compute in this article the bracket structure on , and in relation to this, we establish that there are only finitely many true deformations of in each non-positive weight by constructing them explicitly. It turns out that in weight 0 one gets exactly the other two filiform Lie algebras.     

Plancherel–Polya-type inequalities for entire functions of exponential type in

The goal of the paper is to prove generalizations of the classical Plancherel–Polya inequalities in which point-wise sampling of functions (δ-distributions) is replaced by more general compactly supported distributions on . As an application it is shown that a function , 1p∞, which is an entire function of exponential type is uniquely determined by a set of numbers {Ψ_j(f)}, , where {Ψ_j}, , is a countable sequence of compactly supported distributions. In the case p=2 a reconstruction method of a Paley–Wiener function f from a sequence of samples {Ψ_j(f)}, , is given. This method is a generalization of the classical result of Duffin–Schaeffer about exponential frames on intervals.     

Simultaneous similarity and triangularization of sets of 2 by 2 matrices

Let be a finite or infinite sequence of 2×2 matrices with entries in an integral domain. We show that, except in a very special case, is (simultaneously) triangularizable if and only if all pairs (A_j,A_k) are triangularizable, for 1j,k∞. We also provide a simple numerical criterion for triangularization.Using constructive methods in invariant theory, we define a map (with the minimal number of invariants) that distinguishes simultaneous similarity classes for non-commutative sequences over a field of characteristic ≠2. We also describe canonical forms for sequences of 2×2 matrices over algebraically closed fields, and give a method for finding sequences with a given set of invariants.     

Ore-type and Dirac-type theorems for matroids

Let M be a connected binary matroid having no -minor. Let be a collection of cocircuits of M. We prove there is a circuit intersecting all cocircuits of if either one of two things hold:

(i) For any two disjoint cocircuits and in it holds that .

(ii) For any two disjoint cocircuits and in it holds that .

Part (ii) implies Ore's Theorem, a well-known theorem giving sufficient conditions for the existence of a hamilton cycle in a graph. As an application of part (i), it is shown that if M is a k-connected regular matroid and has cocircumference c^*2k, then there is a circuit which intersects each cocircuit of size c^*−k+2 or greater.We also extend a theorem of Dirac for graphs by showing that for any k-connected binary matroid M having no -minor, it holds that for any k cocircuits of M there is a circuit which intersects them.     

Theorems about the attractor for incompressible non-Newtonian flow driven by external forces that are rapidly oscillating in time but have a smooth average

This paper discusses the incompressible non-Newtonian fluid with rapidly oscillating external forces g(x,t)=g(x,t,t/) possessing the average g⁰(x,t) as →0⁺, where 0<₀<1. Firstly, with assumptions (A₁)–(A₅) on the functions g(x,t,ξ) and g⁰(x,t), we prove that the Hausdorff distance between the uniform attractors and in space H, corresponding to the oscillating equations and the averaged equation, respectively, is less than O() as →0⁺. Then we establish that the Hausdorff distance between the uniform attractors and in space V is also less than O() as →0⁺. Finally, we show for each [0,₀].     

Approximate range searching using binary space partitions

We show how any BSP tree for the endpoints of a set of n disjoint segments in the plane can be used to obtain a BSP tree of size for the segments themselves, such that the range-searching efficiency remains almost the same. We apply this technique to obtain a BSP tree of size O(nlogn) such that -approximate range searching queries with any constant-complexity convex query range can be answered in O(min_>0{(1/)+k}logn) time, where k is the number of segments intersecting the -extended range. The same result can be obtained for disjoint constant-complexity curves, if we allow the BSP to use splitting curves along the given curves.We also describe how to construct a linear-size BSP tree for low-density scenes consisting of n objects in such that -approximate range searching with any constant-complexity convex query range can be done in O(logn+min_>0{(1/^d−1)+k}) time.     

Semifields of order with left nucleus and center

In [G. Marino, O. Polverino, R. Trombetti, On -linear sets of PG(3,q³) and semifields, J. Combin. Theory Ser. A 114 (5) (2007) 769–788] it has been proven that there exist six non-isotopic families (i=0,…,5) of semifields of order q⁶ with left nucleus and center , according to the different geometric configurations of the associated -linear sets. In this paper we first prove that any semifield of order q⁶ with left nucleus , right and middle nuclei and center is isotopic to a cyclic semifield. Then, we focus on the family by proving that it can be partitioned into three further non-isotopic families: , , and we show that any semifield of order q⁶ with left nucleus , right and middle nuclei and center belongs to the family .     

Point spectra of partially power-bounded operators

Let T be an operator on a separable Banach space, and denote by σ_p(T) its point spectrum. We answer a question left open in (Israel J. Math. 146 (2005) 93–110) by showing that it is possible that be uncountable, yet Tⁿ∞. We further investigate the relationship between the growth of sequences (n_k) such that sup_kT^n_k<∞ and the possible size of .Analogous results are also derived for continuous operator semigroups (T_t)_t0.     

Sharp tridiagonal pairs

Let denote a field and let V denote a vector space over with finite positive dimension. We consider a pair of -linear transformations A:V→V and A^*:V→V that satisfies the following conditions: (i) each of A,A^* is diagonalizable; (ii) there exists an ordering of the eigenspaces of A such that A^*V_iV_i-1+V_i+V_i+1 for 0id, where V_-1=0 and V_d+1=0; (iii) there exists an ordering of the eigenspaces of A^* such that for 0iδ, where and ; (iv) there is no subspace W of V such that AWW, A^*WW, W≠0, W≠V. We call such a pair a tridiagonal pair on V. It is known that d=δ and for 0id the dimensions of coincide. We say the pair A,A^* is sharp whenever dimV₀=1. A conjecture of Tatsuro Ito and the second author states that if is algebraically closed then A,A^* is sharp. In order to better understand and eventually prove the conjecture, in this paper we begin a systematic study of the sharp tridiagonal pairs. Our results are summarized as follows. Assuming A,A^* is sharp and using the data we define a finite sequence of scalars called the parameter array. We display some equations that show the geometric significance of the parameter array. We show how the parameter array is affected if Φ is replaced by or or . We prove that if the isomorphism class of Φ is determined by the parameter array then there exists a nondegenerate symmetric bilinear form , on V such that Au,v=u,Av and A^*u,v=u,A^*v for all u,vV.     

Properties of multilevel block -circulants

In a previous paper we characterized unilevel block α-circulants , , 0mn-1, in terms of the discrete Fourier transform of , defined by . We showed that most theoretical and computational problems concerning A can be conveniently studied in terms of corresponding problems concerning the Fourier coefficients F₀,F₁,…,F_n-1 individually. In this paper we show that analogous results hold for (k+1)-level matrices, where the first k levels have block circulant structure and the entries at the (k+1)-st level are unstructured rectangular matrices.     

disjoint cycles containing specified independent vertices

Let k1 be an integer and G be a graph of order n3k satisfying the condition that σ₂(G)n+k-1. Let v₁,…,v_k be k independent vertices of G, and suppose that G has k vertex-disjoint triangles C₁,…,C_k with v_iV(C_i) for all 1ik.Then G has k vertex-disjoint cycles such that

(i) for all 1ik.

(ii) , and

(iii) At least k-1 of the k cycles are triangles.

The condition of degree sum σ₂(G)n+k-1 is sharp.

A characterization of the natural embedding of the split Cayley hexagon in by intersection numbers

In this paper, we prove that a set of q⁵+q⁴+q³+q²+q+1 lines of with the properties that (1) every point of is incident with either 0 or q+1 elements of , (2) every plane of is incident with either 0, 1 or q+1 elements of , (3) every solid of is incident with either 0, 1, q+1 or 2q+1 elements of , and (4) every hyperplane of is incident with at most q³+3q²+3q members of , is necessarily the set of lines of a regularly embedded split Cayley generalized hexagon in .     

Hilbert's Tenth Problem for function fields of varieties over number fields and p-adic fields

Let k be a subfield of a p-adic field of odd residue characteristic, and let be the function field of a variety of dimension n1 over k. Then Hilbert's Tenth Problem for is undecidable. In particular, Hilbert's Tenth Problem for function fields of varieties over number fields of dimension 1 is undecidable.     

Sobolev–Hardy space with general weight

In this paper, it is defined the kth order Sobolev–Hardy space with norm

Then the corresponding Poincaré inequality in this space is obtained, and the results are given that this space is embedded in with weight and in with weight ^q/2 for 1q<2. Moreover, we prove that the constant of k-improved Hardy–Sobolev inequality with general weight is optimal. These inequalities turn to be some known versions of Hardy–Sobolev inequalities in the literature by some particular choice of weights.     

Turán type reverse Markov inequalities for compact convex sets

For a compact convex set the well-known general Markov inequality holds asserting that a polynomial p of degree n must have p^′c(K)n²p. On the other hand for polynomials in general, p^′ can be arbitrarily small as compared to p.The situation changes when we assume that the polynomials in question have all their zeroes in the convex set K. This was first investigated by Turán, who showed the lower bounds p^′(n/2)p for the unit disk D and for the unit interval I[-1,1]. Although partial results provided general lower estimates of order , as well as certain classes of domains with lower bounds of order n, it was not clear what order of magnitude the general convex domains may admit here.Here we show that for all bounded and convex domains K with nonempty interior and polynomials p with all their zeroes lying in K p^′c(K)np holds true, while p^′C(K)np occurs for any K. Actually, we determine c(K) and C(K) within a factor of absolute numerical constant.     

Relative asymptotic of multiple orthogonal polynomials for Nikishin systems

We prove the relative asymptotic behavior for the ratio of two sequences of multiple orthogonal polynomials with respect to the Nikishin systems of measures. The first Nikishin system is such that for each k, σ_k has a constant sign on its compact support consisting of an interval , on which almost everywhere, and a discrete set without accumulation points in . If denotes the smallest interval containing , we assume that Δ_k∩Δ_k+1=0/, k=1,…,m−1. The second Nikishin system is a perturbation of the first by means of rational functions r_k, k=1,…,m, whose zeros and poles lie in .     

Configurations in abelian categories. II. Ringel–Hall algebras

This is the second in a series on configurations in an abelian category . Given a finite poset (I,), an (I,)-configuration (σ,ι,π) is a finite collection of objects σ(J) and morphisms ι(J,K) or in satisfying some axioms, where J,KI. Configurations describe how an object X in decomposes into subobjects.The first paper defined configurations and studied moduli spaces of (I,)-configurations in , using the theory of Artin stacks. It showed well-behaved moduli stacks of objects and configurations in exist when is the abelian category coh(P) of coherent sheaves on a projective scheme P, or mod- of representations of a quiver Q.Write for the vector space of -valued constructible functions on the stack . Motivated by the idea of Ringel–Hall algebras, we define an associative multiplication * on using pushforwards and pullbacks along 1-morphisms between configuration moduli stacks, so that is a -algebra. We also study representations of , the Lie subalgebra of functions supported on indecomposables, and other algebraic structures on .Then we generalize all these ideas to stack functions , a universal generalization of constructible functions, containing more information. When Extⁱ(X,Y)=0 for all and i>1, or when for P a Calabi–Yau 3-fold, we construct (Lie) algebra morphisms from stack algebras to explicit algebras, which will be important in the sequels on invariants counting τ-semistable objects in .     

Inverse eigenproblem for -symmetric matrices and their approximation

Let be a nontrivial involution, i.e., R=R⁻¹≠±I_n. We say that is R-symmetric if RGR=G. The set of all -symmetric matrices is denoted by . In this paper, we first give the solvability condition for the following inverse eigenproblem (IEP): given a set of vectors in and a set of complex numbers , find a matrix such that and are, respectively, the eigenvalues and eigenvectors of A. We then consider the following approximation problem: Given an n×n matrix , find such that , where is the solution set of IEP and is the Frobenius norm. We provide an explicit formula for the best approximation solution by means of the canonical correlation decomposition.     

Ratewise-optimal non-sequential search strategies under constraints on the tests

Already in his Lectures on Search [A. Rényi, Lectures on the theory of search, University of North Carolina, Chapel Hill, Institute of Statistics, Mimeo Series No. 6007, 1969. [11]] Renyi suggested to consider a search problem, where an unknown is to be found by asking for containment in a minimal number m(n,k) of subsets A₁,…,A_m with the restrictions |A_i|k<n/2 for i=1,2,…,m.Katona gave in 1966 the lower bound m(n,k)logn/h(k/n) in terms of binary entropy and the upper bound m(n,k)(logn+1)/logn/k·n/k, which was improved by Wegener in 1979 to m(n,k)logn/logn/k(n/k-1).We prove here for k=pn that m(n,k)=logn+o(logn)/h(p), that is, ratewise optimality of the entropy bound: .Actually this work was motivated by a more recent study of Karpovsky, Chakrabarty, Levitin and Avresky of a problem on fault diagnosis in hypercubes, which amounts to finding the minimal number M(n,r) of Hamming balls of radius r=ρn with in the Hamming space , which separate the vertices. Their bounds on M(n,r) are far from being optimal. We establish bounds implying

However, it must be emphasized that the methods of prove for our two upper bounds are quite different.     

Equivalence relation groupoids associated with certain linearly ordered dimension groups

We consider linearly ordered, Archimedean dimension groups (G,G⁺,u) for which the group G/u is torsion-free. It will be shown that if, in addition, G/u is generated by a single element (i.e., ), then (G,G⁺,u) is isomorphic to for some irrational number τ(0,1). This amounts to an extension of related results where dimension groups for which G/u is torsion were considered. We will prove, in the case of the Fibonacci dimension group, that these results can be used to directly construct an equivalence relation groupoid whose C^*-algebra is the Fibonacci C^*-algebra.