Smoothness of the moduli space of complexes of coherent sheaves on an abelian or a projective K3 surface

For an abelian or a projective K3 surface X over an algebraically closed field k, consider the moduli space of the objects E in Db(Coh(X)) satisfying and Hom(E,E)≅k. Then we can prove that is smooth and has a symplectic structure.     

Continuous extension operators and convexity

Given a nonempty closed subset A of a Hilbert space X, we denote by L(A) the space of all bounded Lipschitz mappings from A into X, equipped with the supremum norm. We show that there is a continuous mapping F_c:L(A)?L(X) such that for each g∈L(A), F_c(g)|_A=g, , and . We also prove that the corresponding set-valued extension operator is lower semicontinuous.     

Self-correspondences of the generic fibre of Lefschetz pencils and the Leray filtration

In this paper we give a detailed analysis of the interaction between homological self-correspondences of the general fibre Y/k(t) of the Lefschetz fibration of a Lefschetz pencil on a smooth projective variety X/k, and the Leray filtration of ρ. We derive the result that, if the standard conjecture B(Y) holds, then the operator is algebraic, where is defined as the inverse of L on LPn−1(X) and 0 on LkPj(X) for (1,n−1)≠(k,j); in the course of our proof we see that, under the above assumption, the Künneth projectors for i≠n−1,n,n+1 are algebraic.     

An Erdős–Ko–Rado Theorem for Direct Products

Letn_i, k_ibe positive integers,i=1, ..., d,satisfyingn_i≥2k_i.LetX₁, ..., X_dbe pairwise disjoint sets with |X_i| =n_i.Letbe the family of those (k₁+···+k_d)-element sets which have exactlyk_ielements inX_i, i=1,..., d.It is shown that if⊂is an intersecting family then ||/||≤max_ik_i/n_i,and this is best possible. The proof is algebraic, although in thed=2 case a combinatorial argument is presented as well.     

Likelihood ratio orderings of spacings of heterogeneous exponential random variables

Let X₁,X₂,…,X_n be independent exponential random variables such that X_i has failure rate λ for i=1,…,p and X_j has failure rate λ^* for j=p+1,…,n, where p≥1 and q=n-p≥1. Denote by D_i:n(p,q)=X_i:n-X_i-1:n the ith spacing of the order statistics , where X_0:n≡0. It is shown that D_i:n(p,q)?_lrD_i+1:n(p,q) for i=1,…,n-1, and that if λ?λ^* then , and for i=1,…,n, where ?_lr denotes the likelihood ratio order. The main results are used to establish the dispersive orderings between spacings.     

The index formula and the spectral shift function for relatively trace class perturbations

We compute the Fredholm index, index(DA), of the operator DA=(d/dt)+A on L²(R;H) associated with the operator path , where (Af)(t)=A(t)f(t) for a.e. t∈R, and appropriate f∈L²(R;H), via the spectral shift function ξ(⋅;A₊,A₋) associated with the pair (A₊,A₋) of asymptotic operators A_±=A(±∞) on the separable complex Hilbert space H in the case when A(t) is generally an unbounded (relatively trace class) perturbation of the unbounded self-adjoint operator A₋.We derive a formula (an extension of a formula due to Pushnitski) relating the spectral shift function ξ(⋅;A₊,A₋) for the pair (A₊,A₋), and the corresponding spectral shift function ξ(⋅;H₂,H₁) for the pair of operators in this relative trace class context,This formula is then used to identify the Fredholm index of DA with ξ(0;A₊,A₋). In addition, we prove that index(DA) coincides with the spectral flow of the family {A(t)}_t∈R and also relate it to the (Fredholm) perturbation determinant for the pair (A₊,A₋): with the choice of the branch of ln(detH(⋅)) on C₊ such thatWe also provide some applications in the context of supersymmetric quantum mechanics to zeta function and heat kernel regularized spectral asymmetries and the eta-invariant.     

Radially increasing minimizing surfaces or deformations under pointwise constraints on positions and gradients

In this paper, we prove existence of radially symmetric minimizersu_A(x)=U_A(|x|), having U_A(⋅)AC monotone and increasing, for the convex scalar multiple integral(∗ ) among those u(⋅) in the Sobolev space. Here, |∇u(x)| is the Euclidean norm of the gradient vector and B_R is the ball ; while A is the boundary data.Besides being e.g. superlinear (but no growth needed if (∗) is known to have minimum), our Lagrangian?^∗∗:R×R→[0,∞] is just convex lsc and and ?^∗∗(s,⋅) is even; while ρ₁(⋅) and ρ₂(⋅) are Borel bounded away from .Remarkably, (∗) may also be seen as the calculus of variations reformulation of a distributed-parameter scalar optimal control problem. Indeed, state and gradient pointwise constraints are, in a sense, built-in, since ?^∗∗(s,v)=∞ is freely allowed.     

Cells in hyperspaces

Let C(X) denote the hyperspace of subcontinua of a continuum X. For A∈C(X), define the hyperspace . Let k∈N, k?2. We prove that A is contained in the core of a k-od if and only if C(A,X) contains a k-cell.     

Stochastic matrices and a property of the infinite sequences of linear functionals

Our starting point is the proof of the following property of a particular class of matrices. Let T={T_i,j} be a n×m non-negative matrix such that ∑_jT_i,j=1 for each i. Suppose that for every pair of indices (i,j), there exists an index l such that T_i,l≠T_j,l. Then, there exists a real vector k=(k₁,k₂,…,k_m)^T,k_i≠k_j,i≠j;0<k_i?1, such that, if i≠j.Then, we apply that property of matrices to probability theory. Let us consider an infinite sequence of linear functionals , corresponding to an infinite sequence of probability measures {μ_(·)(i)}_i∈N, on the Borel σ-algebra such that, . The property of matrices described above allows us to construct a real bounded one-to-one piecewise continuous and continuous from the left function f such that

     

The higher-order Picone identity and comparison of half-linear differential equations of even order

An identity of the Picone type for higher-order half-linear ordinary differential operators of the form and where p_j and P_j, j=0,…,n, are continuous functions defined on [a,b] and , is derived and then the Sturmian comparison theory for the corresponding 2nth-order equations l_α[x]=0 and L_α[y]=0 based on this identity is developed.     

On almost prime k-tuples

Let k?2 and ai,bi(1?i?k) be integers such that ai>0 and _∏1?i<j?k(aibj−ajbi)≠0. Let Ω(m) denote the total number of prime factors of m. Suppose has no fixed prime divisors. Results of the form where rk is asymptotic to klogk have been obtained by using sieve methods, in particular weighted sieves. In this paper, we use another kind of weighted sieve due to Selberg to obtain improved admissible values for rk.     

Bogomolov restriction theorem for Higgs bundles

Let (E,θ) be a stable Higgs bundle of rank r on a smooth complex projective surface X equipped with a polarization H. Let C⊂X be a smooth complete curve with [C]=n⋅H. If where , then we prove that the restriction of (E,θ) to C is a stable Higgs bundle. This is a Higgs bundle analog of Bogomolov's restriction theorem for stable vector bundles.     

A short proof of a cross-intersection theorem of Hilton

Families A₁,…,A_k of sets are said to be cross-intersecting if for any A_i∈A_i and A_j∈A_j, i≠j. A nice result of Hilton that generalises the Erd?s-Ko-Rado (EKR) Theorem says that if r≤n/2 and A₁,…,A_k are cross-intersecting sub-families of , then

     

Schur-class multipliers on the Arveson space: De Branges-Rovnyak reproducing kernel spaces and commutative transfer-function realizations

An interesting and recently much studied generalization of the classical Schur class is the class of contractive operator-valued multipliers S(λ) for the reproducing kernel Hilbert space H(kd) on the unit ball Bd⊂Cd, where kd is the positive kernel kd(λ,ζ)=1/(1−〈λ,ζ〉) on Bd. The reproducing kernel space H(KS) associated with the positive kernel KS(λ,ζ)=(I−S(λ)S^∗(ζ))⋅kd(λ,ζ) is a natural multivariable generalization of the classical de Branges-Rovnyak canonical model space. A special feature appearing in the multivariable case is that the space H(KS) in general may not be invariant under the adjoints of the multiplication operators on H(kd). We show that invariance of H(KS) under for each j=1,…,d is equivalent to the existence of a realization for S(λ) of the form S(λ)=D+C⁻¹(I−λ₁A₁−?−λdAd)(λ₁B₁+?+λdBd) such that connecting operator has adjoint U^∗ which is isometric on a certain natural subspace (U is “weakly coisometric”) and has the additional property that the state operators A₁,…,Ad pairwise commute; in this case one can take the state space to be the functional-model space H(KS) and the state operators A₁,…,Ad to be given by (a de Branges-Rovnyak functional-model realization). We show that this special situation always occurs for the case of inner functions S (where the associated multiplication operator MS is a partial isometry), and that inner multipliers are characterized by the existence of such a realization such that the state operators A₁,…,Ad satisfy an additional stability property.     

Distance-two labelings of digraphs

For positive integers j?k, an L(j,k)-labeling of a digraph D is a function f from V(D) into the set of nonnegative integers such that |f(x)-f(y)|?j if x is adjacent to y in D and |f(x)-f(y)|?k if x is of distance two to y in D. Elements of the image of f are called labels. The L(j,k)-labeling problem is to determine the -number of a digraph D, which is the minimum of the maximum label used in an L(j,k)-labeling of D. This paper studies -numbers of digraphs. In particular, we determine -numbers of digraphs whose longest dipath is of length at most 2, and -numbers of ditrees having dipaths of length 4. We also give bounds for -numbers of bipartite digraphs whose longest dipath is of length 3. Finally, we present a linear-time algorithm for determining -numbers of ditrees whose longest dipath is of length 3.     

On holomorphic domination, I

Let X be a separable Banach space and u:X→R locally upper bounded. We show that there are a Banach space Z and a holomorphic function h:X→Z with u(x)<‖h(x)‖ for x∈X. As a consequence we find that the sheaf cohomology group Hq(X,O) vanishes if X has the bounded approximation property (i.e., X is a direct summand of a Banach space with a Schauder basis), O is the sheaf of germs of holomorphic functions on X, and q?1. As another consequence we prove that if f is a C¹-smooth -closed (0,1)-form on the space X=L₁[0,1] of summable functions, then there is a C¹-smooth function u on X with on X.     

Reduced class groups grafting relative invariants

Let (X,T) be a regular stable conical action of an algebraic torus on an affine normal conical variety X defined over an algebraically closed field of characteristic zero. We define a certain subgroup of Cl(X//T) and characterize its finiteness in terms of a finite T-equivariant Galois descent of X. Consequently we show that the action (X,T) is equidimensional if and only if there exists a T-equivariant finite Galois covering such that is cofree. Moreover the order of is controlled by a certain subgroup of Cl(X). The present result extends thoroughly the equivalence of equidimensionality and cofreeness of (X,T) for a factorial X. The purpose of this paper is to evaluate orders of divisor classes associated to modules of relative invariants for a Krull domain with a group action. This is useful in studying on equidimensional torus actions as above. The generalization of R.P. Stanley?s criterion for freeness of modules of relative invariants plays an important role in showing key assertions.     

On conjugate adjacency matrices of a graph

Let G be a simple graph of order n. Let and , where a and b are two nonzero integers and m is a positive integer such that m is not a perfect square. We say that A^c=[c_ij] is the conjugate adjacency matrix of the graph G if c_ij=c for any two adjacent vertices i and j, for any two nonadjacent vertices i and j, and c_ij=0 if i=j. Let P_G(λ)=|λI-A| and denote the characteristic polynomial and the conjugate characteristic polynomial of G, respectively. In this work we show that if then , where denotes the complement of G. In particular, we prove that if and only if P_G(λ)=P_H(λ) and . Further, let P^c(G) be the collection of conjugate characteristic polynomials of vertex-deleted subgraphs G_i=G?i(i=1,2,…,n). If P^c(G)=P^c(H) we prove that , provided that the order of G is greater than 2.     

Continuity of the spectrum on a class of upper triangular operator matrices

Let B(H) denote the algebra of operators on an infinite dimensional complex Hilbert space H, and let A^○∈B(K) denote the Berberian extension of an operator A∈B(H). It is proved that the set theoretic function σ, the spectrum, is continuous on the set C(i)⊂B(Hi) of operators A for which σ(A)={0} implies A is nilpotent (possibly, the 0 operator) and at every non-zero λ∈σp(A^○) for some operators X and B such that λ∉σp(B) and σ(A^○)={λ}∪σ(B). If CS(m) denotes the set of upper triangular operator matrices , where Aii∈C(i) and Aii has SVEP for all 1?i?m, then σ is continuous on CS(m). It is observed that a considerably large number of the more commonly considered classes of Hilbert space operators constitute sets C(i) and have SVEP.