Nonvanishing of Kronecker coefficients for rectangular shapes

We prove that for any partition (λ₁,…,λd²) of size ?d there exists k?1 such that the tensor square of the irreducible representation of the symmetric group Sk?d with respect to the rectangular partition (k?,…,k?) contains the irreducible representation corresponding to the stretched partition (kλ₁,…,kλd²). We also prove a related approximate version of this statement in which the stretching factor k is effectively bounded in terms of d. We further discuss the consequences for geometric complexity theory which provided the motivation for this work.     

Almost perfect powers in consecutive integers (II)

Let k ≥ 4 be an integer. We find all integers of the form by^l where l ≥ 2 and the greatest prime factor of b is at most k (i.e. nearly a perfect power) such that they are also products of k consecutive integers with two terms omitted.     

New applications of Picard?s successive approximations

The iterative method of successive approximations, originally introduced by Émile Picard in 1890, is a basic tool for proving the existence of solutions of initial value problems regarding ordinary first order differential equations. In the present paper, it is shown that this method can be modified to get estimates for the growth of solutions of linear differential equations of the typef(k)+Ak−1(z)f(k−1)+?+A₁(z)f^′+A₀(z)f=0 with analytic coefficients. A short comparison to the growth results in the literature, obtained by means of different methods, is also given. It turns out that many known results can be proved by applying Picard?s successive approximations in an effective way. Self-contained considerations are carried out in the complex plane and in the unit disc, and some remarks about solutions of real linear differential equations are made.     

Reverse inequalities for zonoids and their application

We prove inequalities for mixed volumes of zonoids with isotropic generating measures. A special case is an inequality for zonoids that is reverse to the generalized Urysohn inequality, between mean width and another intrinsic volume; here the equality case characterizes parallelepipeds. We apply this to a question from stochastic geometry. It is known that among the stationary Poisson hyperplane processes of given positive intensity in n-dimensional Euclidean space, the ones with rotation invariant distribution are characterized by the fact that they yield, for k∈{2,…,n}, the maximal intensity of the intersection process of order k. We show that, if the kth intersection density is measured in an affine-invariant way, the processes of hyperplanes with only n fixed directions are characterized by a corresponding minimum property.     

The Weyl-type asymptotic formula for biharmonic Steklov eigenvalues on Riemannian manifolds

Let Ω be a bounded domain with C²-smooth boundary in an n-dimensional oriented Riemannian manifold. It is well known that for the biharmonic equation Δ²u=0 in Ω with the condition u=0 on ∂Ω, there exists an infinite set {uk} of biharmonic functions in Ω with positive eigenvalues {λk} satisfying on ∂Ω. In this paper, by a new method we establish the Weyl-type asymptotic formula for the counting function of the biharmonic Steklov eigenvalues λk.     

On fine fractal properties of generalized infinite Bernoulli convolutions

The paper is devoted to the investigation of generalized infinite Bernoulli convolutions, i.e., the distributions μξ of the following random variables: where ak are terms of a given positive convergent series; ξk are independent random variables taking values 0 and 1 with probabilities p0k and p1k correspondingly.We give (without any restriction on {an}) necessary and sufficient conditions for the topological support of ξ to be a nowhere dense set. Fractal properties of the topological support of ξ and fine fractal properties of the corresponding probability measure μξ itself are studied in details for the case where ak?rk:=ak+1+ak+2+? (i.e., rk−1?2rk) for all sufficiently large k. The family of minimal dimensional (in the sense of the Hausdorff–Besicovitch dimension) supports of μξ for the above mentioned case is also studied in details. We describe a series of sets (with additional structural properties) which play the role of minimal dimensional supports of generalized Bernoulli convolutions. We also show how a generalization of M. Cooper's dimensional results on symmetric Bernoulli convolutions can easily be derived from our results.     

Powerful arithmetic progressions

We give a complete characterization of so-called powerful arithmetic progressions, i.e. of progressions whose kth term is a kth power for all k. We also prove that the length of any primitive arithmetic progression of powers can be bounded both by any term of the progression different from 0 and ±1, and by its common difference. In particular, such a progression can have only finite length.     

Categorical data clustering with automatic selection of cluster number

In this paper, we investigate the problem of determining the number of clusters in the k-modes based categorical data clustering process. We propose a new categorical data clustering algorithm with automatic selection of k. The new algorithm extends the k-modes clustering algorithm by introducing a penalty term to the objective function to make more clusters compete for objects. In the new objective function, we employ a regularization parameter to control the number of clusters in a clustering process. Instead of finding k directly, we choose a suitable value of regularization parameter such that the corresponding clustering result is the most stable one among all the generated clustering results. Experimental results on synthetic data sets and the real data sets are used to demonstrate the effectiveness of the proposed algorithm.     

On analytic equivalence of functions at infinity

In this paper we define the relation of analytic equivalence of functions at infinity. We prove that if the ?ojasiewicz exponent at infinity of the gradient of a polynomial f∈R[x₁,…,xn] is greater or equal to k−1, then there exists ε>0 such that for every polynomial P∈R[x₁,…,xn] of degree less or equal to k, whose coefficients of monomials of degree k are less or equal ε, the polynomials f and f+P are analytically equivalent at infinity.     

Higher integrality conditions, volumes and Ehrhart polynomials

A polytope is integral if all of its vertices are lattice points. The constant term of the Ehrhart polynomial of an integral polytope is known to be 1. In previous work, we showed that the coefficients of the Ehrhart polynomial of a lattice-face polytope are volumes of projections of the polytope. We generalize both results by introducing a notion of k-integral polytopes, where 0-integral is equivalent to integral. We show that the Ehrhart polynomial of a k-integral polytope P has the properties that the coefficients in degrees less than or equal to k are determined by a projection of P, and the coefficients in higher degrees are determined by slices of P. A key step of the proof is that under certain generality conditions, the volume of a polytope is equal to the sum of volumes of slices of the polytope.     

Dubrovin's duality for F-manifolds with eventual identities

A vector field E on an F-manifold (M,°,e) is an eventual identity if it is invertible and the multiplication X?Y:=X°Y°E−1 defines a new F-manifold structure on M. We give a characterization of such eventual identities, this being a problem raised by Manin (2005) [12]. We develop a duality between F-manifolds with eventual identities and we show that this duality is compatible with the local irreducible decomposition of F-manifolds and preserves the class of Riemannian F-manifolds. We find necessary and sufficient conditions on the eventual identity which ensure that the classes of harmonic Higgs bundles, DChk-structures and weak CV-structures are preserved by our duality. Examples of such structures are given in the case of a semi-simple multiplication. We use eventual identities to construct compatible pairs of metrics.     

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.     

Lyubeznik numbers of projective schemes

Let X be a projective scheme over a field k and let A be the local ring at the vertex of the affine cone of X under some embedding . We prove that, when char(k)>0, the Lyubeznik numbers λi,j(A) are intrinsic numerical invariants of X, i.e., λi,j(A) depend only on X, but not on the embedding.     

On extensions of a symplectic class

Let F be a fibration on a simply-connected base with symplectic fiber (M,ω). Assume that the fiber is nilpotent and T2k-separable for some integer k or a nilmanifold. Then our main theorem, Theorem 1.8, gives a necessary and sufficient condition for the cohomology class [ω] to extend to a cohomology class of the total space of F. This allows us to describe Thurston?s criterion for a symplectic fibration to admit a compatible symplectic form in terms of the classifying map for the underlying fibration. The obstruction due to Lalond and McDuff for a symplectic bundle to be Hamiltonian is also rephrased in the same vein. Furthermore, with the aid of the main theorem, we discuss a global nature of the set of the homotopy equivalence classes of fibrations with symplectic fiber in which the class [ω] is extendable.     

EXISTENCE OF GLOBAL ATTRACTORS FOR A NONLINEAR EVOLUTION EQUATION IN SOBOLEV SPACE Hk

In this paper we prove that the initial-boundary value problem for the nonlinear evolution equation ut = △u ＋ λu - u^3 possesses a global attractor in Sobolev space H^k for all k≥0, which attracts any bounded domain of H^k（Ω） in the H^k-norm. This result is established by using an iteration technique and regularity estimates for linear semigroup of operator, which extends the classical result from the case k ∈ [0, 1] to the case k∈ [0, ∞）.     

Deviations of discrete distributions and a question of Móri

In this note, we consider a question of Móri regarding estimating the deviation of the kth terms of two discrete probability distributions in terms of the supremum distance between their generating functions over the interval [0,1]. An optimal bound for distributions on finite support is obtained. Properties of Chebyshev polynomials are employed.     

k-Plane Clustering

A finite new algorithm is proposed for clustering m given points in n-dimensional real space into k clusters by generating k planes that constitute a local solution to the nonconvex problem of minimizing the sum of squares of the 2-norm distances between each point and a nearest plane. The key to the algorithm lies in a formulation that generates a plane in n-dimensional space that minimizes the sum of the squares of the 2-norm distances to each of m₁ given points in the space. The plane is generated by an eigenvector corresponding to a smallest eigenvalue of an n × n simple matrix derived from the m₁ points. The algorithm was tested on the publicly available Wisconsin Breast Prognosis Cancer database to generate well separated patient survival curves. In contrast, the k-mean algorithm did not generate such well-separated survival curves.     

The quaternionic evolution operator

In the recent years, the notion of slice regular functions has allowed the introduction of a quaternionic functional calculus. In this paper, motivated also by the applications in quaternionic quantum mechanics, see Adler (1995) [1], we study the quaternionic semigroups and groups generated by a quaternionic (bounded or unbounded) linear operator T=T₀+iT₁+jT₂+kT₃. It is crucial to note that we consider operators with components T?(?=0,1,2,3) that do not necessarily commute. Among other results, we prove the quaternionic version of the classical Hille–Phillips–Yosida theorem. This result is based on the fact that the Laplace transform of the quaternionic semigroup etT is the S-resolvent operator , the quaternionic analogue of the classical resolvent operator. The noncommutative setting entails that the results we obtain are somewhat different from their analogues in the complex setting. In particular, we have four possible formulations according to the use of left or right slice regular functions for left or right linear operators.