Bounds for ratios of eigenvalues using traces

Let A be an n × n matrix with real eigenvalues λ₁ ? … ? λ_n, and let 1 ? k < l ? n. Bounds involving trA and trA² are introduced for λ_k/λ_l, (λ_k ? λ_l)/(λ_k + λ_l), and {kλ_k + (n ? l + 1)λ_l}²/{kλ²_k + (n ? l + 1)λ²_l}. Also included are conditions for λ_l >; 0 and for λ_k + λ_l > 0.     

Semidefinite descriptions of low-dimensional separable matrix cones

Let K⊂E, K^′⊂E^′ be convex cones residing in finite-dimensional real vector spaces. An element y in the tensor product E⊗E^′ is K⊗K^′-separable if it can be represented as finite sum , where x_l∈K and for all l. Let S(n), H(n), Q(n) be the spaces of n×n real symmetric, complex Hermitian and quaternionic Hermitian matrices, respectively. Let further S₊(n), H₊(n), Q₊(n) be the cones of positive semidefinite matrices in these spaces. If a matrix A∈H(mn)=H(m)⊗H(n) is H₊(m)⊗H₊(n)-separable, then it fulfills also the so-called PPT condition, i.e. it is positive semidefinite and has a positive semidefinite partial transpose. The same implication holds for matrices in the spaces S(m)⊗S(n), H(m)⊗S(n), and for m?2 in the space Q(m)⊗S(n). We provide a complete enumeration of all pairs (n,m) when the inverse implication is also true for each of the above spaces, i.e. the PPT condition is sufficient for separability. We also show that a matrix in Q(n)⊗S(2) is Q₊(n)⊗S₊(2)- separable if and only if it is positive semidefinite.     

Rademacher averages on noncommutative symmetric spaces

Let E be a separable (or the dual of a separable) symmetric function space, let M be a semifinite von Neumann algebra and let E(M) be the associated noncommutative function space. Let (εk)_k?1 be a Rademacher sequence, on some probability space Ω. For finite sequences (xk)_k?1 of E(M), we consider the Rademacher averages k_∑εk⊗xk as elements of the noncommutative function space and study estimates for their norms ‖k_∑εk⊗xkE_‖ calculated in that space. We establish general Khintchine type inequalities in this context. Then we show that if E is 2-concave, ‖k_∑εk⊗xkE_‖ is equivalent to the infimum of over all yk, zk in E(M) such that xk=yk+zk for any k?1. Dual estimates are given when E is 2-convex and has a nontrivial upper Boyd index. In this case, ‖k_∑εk⊗xkE_‖ is equivalent to . We also study Rademacher averages _∑i,jεi⊗εj⊗xij for doubly indexed families (xij)_i,j of E(M).     

Legendrian tori and the semi-classical limit

Let X be CPn or a compact smooth quotient of the n-dimensional complex hyperbolic space, n>1. Let L be a hermitian holomorphic line bundle (with hermitian connection) on X chosen as follows: if X=CPn then L is the hyperplane bundle, and in the second case L is chosen so that L⊗(n+1)=KX⊗E, where KX is the canonical line bundle and E is a flat line bundle. The unit circle bundle P in L^∗ is a contact manifold. Let k^′ be a fixed positive integer. We construct certain Legendrian tori in P (the construction depends, in particular, on the choice of k^′) and sequences {uk}, k=k^′m, , of holomorphic sections of L⊗k associated to these tori. We study asymptotics of the norms ‖ukk_‖ as m→+∞ and, in particular, apply this result to construct explicitly certain non-trivial holomorphic automorphic forms on the n-dimensional complex hyperbolic space. We obtain an n>1 analogue of the classical period formula (this is a well-known statement for automorphic forms on the upper half plane, n=1).     

Chains of prime ideals in tensor products of algebras

This paper investigates the length of particular chains of prime ideals in tensor products of algebras over a field k. As an application, we compute dim(A⊗_kA) for a new family of domains A that are k-algebras.     

Chen’s Conjecture and Its Generalization

Let l1, l2, ..., lg be even integers and x be a sufficiently large number. In this paper, the authors prove that the number of positive odd integers k ≤ x such that （k ＋l1）^2, （k ＋l2）^2, ..., （k ＋lg）^2 can not be expressed as 2^n＋p^α is at least c（g）x, where p is an odd prime and the constant c（g） depends only on g.     

The inversion formulae for automorphisms of polynomial algebras and rings of differential operators in prime characteristic

Let K be an arbitrary field of characteristic p>0. We find an explicit formula for the inverse of any algebra automorphism of any of the following algebras: the polynomial algebra P_n?K[x₁,…,x_n], the ring of differential operators D(P_n) on P_n, D(P_n)⊗P_m, the n’th Weyl algebra A_n or A_n⊗P_m, the power series algebra K[[x₁,…,x_n]], the subalgebra T_k1,…,kn of D(P_n) generated by P_n and the higher derivations , 0≤j<p^k_i, i=1,…,n (where k₁,…,k_n∈N), T_k1,…,kn⊗P_m or an arbitrary central simple (countably generated) algebra over an arbitrary field.     

Monochromatic Vs multicolored paths

Letl andk be positive integers, and letX={0,1,...,l ^k?1}. Is it true that for every coloring δ:X×X→{0,1,...} there either exist elementsx ₀<x ₁<...<x _l ofX with δ(x ₀,x ₁)=δ(x ₁,x ₂)=...=δ(x _l?1,x _l), or else there exist elementsy ₀<y ₁<...<y _k ofX with δ(y _i?1,y _i) ∈ δ(y _j?1,y _j) for all 1<-i<j≤k? We prove here that this is the case if eitherl≤2, ork≤4, orl≥(3k)^2k . The general question remains open.     

On continuous maps between Grassmann manifolds

LetG _n,k denote the Grassmann manifold ofk-planes in ?ⁿ. We show that for any continuous mapf: G _n,k→G_n,l the induced map inZ/2-cohomology is either zero in positive dimensions or has image in the subring generated by w₁(γ_{n, k}), provided 1≤l<k≤[n/2] andn≥k+2l-1. Our main application is to obtain negative results on the existence of equivariant maps between oriented Grassmann manifolds. We also obtain positive results in many cases on the existence of equivariant maps between oriented Grassmann manifolds.     

Spectral decomposition and Gelfand’s theorem

We consider the spectral decomposition of A, the generator of a polynomially bounded n-times integrated group whose spectrum set $\sigma(A)=\{i\lambda_{k};k\in\mathbb{\mathbb{Z}}^{*}\}We consider the spectral decomposition of A, the generator of a polynomially bounded n-times integrated group whose spectrum set s(A)={il_k;k ? \mathbb\mathbbZ^*}\sigma(A)=\{i\lambda_{k};k\in\mathbb{\mathbb{Z}}^{*}\} is discrete and satisfies ?\frac1|l_k|^ld_kⁿ < ￥\sum \frac{1}{|\lambda_{k}|^{\ell}\delta_{k}^{n}}<\infty , where ℓ is a nonnegative integer and d_k=min(\frac|l_k+1-l_k|2,\frac|l_k-1-l_k|2)\delta _{k}=\min(\frac{|\lambda_{k+1}-\lambda _{k}|}{2},\frac{|\lambda _{k-1}-\lambda _{k}|}{2}) . In this case, Theorem 3, we show by using Gelfand’s Theorem that there exists a family of projectors (P_k)_{k ? \mathbb\mathbbZ^*}(P_{k})_{k\in\mathbb{\mathbb{Z}}^{*}} such that, for any x∈D(A ^n+ℓ ), the decomposition ∑P _k x=x holds.     

A note on the probabilistic approach to Turán's problem

The Turán number T(n, l, k) is the smallest possible number of edges in a k-graph on n vertices such that every l-set of vertices contains an edge. Given a k-graph H = (V(H), E(H)), we let X_s(S) equal the number of edges contained in S, for any s-set S?V(H). Turán's problem is equivalent to estimating the expectation E(X_l), given that min(X_l) ≥ 1. The following lower bound on the variance of X_s is proved:

$\text{Var(X}{}_{\mathrm{s}}\text{)?mm}\text{n?2k}\text{s?k}\text{n}\text{s}{}^{\mathrm{?1}}\text{n}\text{k}{}^1$

, where m = |E(H)| and $\text{m}\text{= (}{}_{\mathrm{k}}{}^{\mathrm{n}}\text{) ? m}$. This implies the following: putting t(k, l) = lim_n→∞T(n, l, k)(_kⁿ)^?1 then t(k, l) ≥ T(s, l, k)((_k^s) ? 1)^?1, whenever s ≥ l > k ≥ 2. A connection of these results with the existence of certain t-designs is mentioned.     

Generalized balancing numbers

The positive integer x is a (k, l) -balancing number for y(x ≤ y — 2) if 1^k + 2^k + … + (x — 1)^k = (x + 1)^l + … + (y — 1)^l for fixed positive integers k and l. In this paper, we prove some effective and ineffective finiteness statements for the balancing numbers, using certain Baker-type Diophantine results and Bilu—Tichy theorem, respectively.     

Note on the number of integral ideals in Galois extensions

Let K be an algebraic number field of finite degree over the rational filed Q.Let ak be the number of integral ideals in K with norm k.In this paper we study the l-th integral power sum of ak,i.e.,∑k≤ x akl(l = 2,3,...).We are able to improve the classical result of Chandrasekharan and Good.As an application we consider the number of solutions of polynomial congruences.     

Lower bound inequalities for norms of symmetrized tensor powers

Let V be a complex inner product space of positive dimension m with inner product 〈·,·〉, and let T_n(V) denote the set of all n-linear complex-valued functions defined on V×V×?×V (n-copies). By S_n(V) we mean the set of all symmetric members of T_n(V). We extend the inner product, 〈·,·〉, on V to T_n(V) in the usual way, and we define multiple tensor products A₁⊗A₂⊗?⊗A_n and symmetric products A₁·A₂?A_n, where q₁,q₂,…,q_n are positive integers and A_i∈T_qi(V) for each i, as expected. If A∈S_n(V), then A^k denotes the symmetric product A·A?A where there are k copies of A. We are concerned with producing the best lower bounds for ‖A^k‖², particularly when n=2. In this case we are able to show that ‖A^k‖² is a symmetric polynomial in the eigenvalues of a positive semi-definite Hermitian matrix, M_A, that is closely related to A. From this we are able to obtain many lower bounds for ‖A^k‖². In particular, we are able to show that if ω denotes 1/r where r is the rank of M_A, and , then

     

Schur–Weyl duality over finite fields

We prove a version of the Schur–Weyl duality over finite fields. We prove that for any field k, if k has at least r + 1 elements, the Schur–Weyl duality holds for the rth tensor power of a finite dimensional vector space V. Moreover, if the dimension of V is at least r + 1, the natural map ${{k\mathfrak{S}_r \to \mathsf{End}_{{\rm GL}(V)}(V^{\otimes r})}}We prove a version of the Schur–Weyl duality over finite fields. We prove that for any field k, if k has at least r + 1 elements, the Schur–Weyl duality holds for the rth tensor power of a finite dimensional vector space V. Moreover, if the dimension of V is at least r + 1, the natural map k\mathfrakS_r ? End_GL(V)(V^?r){{k\mathfrak{S}_r \to \mathsf{End}_{{\rm GL}(V)}(V^{\otimes r})}} is an isomorphism. This isomorphism may fail if dim _k V is not strictly larger than r.     

Some inverse results for Hill's equation

We consider Hill's equation y″+(λ−q)y=0 where q∈L¹[0,π]. We show that if l_n—the length of the n-th instability interval—is of order O(n^−(k+2)) then the real Fourier coefficients a_n^k,b_n^k of q^(k)—k-th derivative of q—are of order O(n⁻²), which implies that q^(k) is absolutely continuous almost everywhere for k=0,1,2,….     

Forbidding just one intersection

Following a conjecture of P. Erdös, we show that if $\text{F}$ is a family of k-subsets of and n-set no two of which intersect in exactly l elements then for k ? 2l + 2 and n sufficiently large |$\text{F}$| ? (_{k ? l ? 1}^{n ? l ? 1}) with equality holding if and only if $\text{F}$ consists of all the k-sets containing a fixed (l + 1)-set. In general we show |$\text{F}$| ? d_kn^{max;{;l,k ? l ? 1};}, where d_k is a constant depending only on k. These results are special cases of more general theorems (Theorem 2.1–2.3).