Let V be a smooth equidimensional quasi-affine variety of dimension r over C and let F be a (p× s)-matrix of coordinate functions of C[V ], where s ≥ p+ r. The pair (V, F ) determines a vector bundle E of rank s− p over W := {x ∈ V | rkF (x) = p}. We associate with (V, F ) a descending chain of degeneracy loci of E (the generic polar varieties of V represent a typical example of this situation)...

We study the refined and unrefined crystal/BPS partition functions of D6-D2-D0 brane bound states for all toric Calabi-Yau threefolds without compact 4-cycles some non-toric examples. They can be written as products (generalized) MacMahon functions. check our expressions use them vacuum characters to gluings. then consider wall crossings discuss possible crystal descriptions different chambers....

Let V be a smooth equidimensional quasi-affine variety of dimension r over C and let F be a (p× s)-matrix of coordinate functions of C[V ], where s ≥ p+ r. The pair (V, F ) determines a vector bundle E of rank s− p over W := {x ∈ V | rkF (x) = p}. We associate with (V, F ) a descending chain of degeneracy loci of E (the generic polar varieties of V represent a typical example of this situation)...

Given continuous functions M and N of two variables, it is shown that if in a continuous iteration semigroup with only (M,N)-convex or (M,N)-concave elements there are two (M,N)-affine elements, then M = N and every element of the semigroup is M -affine. Moreover, all functions in the semigroup either are M -convex or M -concave.

We propose integral representations for wave functions of Bn, Cn, and Dn open Toda chains at zero eigenvalues of the Hamiltonian operators thus generalizing Givental representation for An. We also construct Baxter Q-operators for closed Toda chains corresponding to Lie algebras B∞, C∞, D∞, affine Lie algebras B (1) n , C (1) n , D (1) n and twisted affine Lie algebras A (2) 2n−1 and A (2) 2n . ...

Let V be a smooth equidimensional quasi-affine variety of dimension r over C and let F be a (p× s)-matrix of coordinate functions of C[V ], where s ≥ p+ r. The pair (V, F ) determines a vector bundle E of rank s− p over W := {x ∈ V | rkF (x) = p}. We associate with (V, F ) a descending chain of degeneracy loci of E (the generic polar varieties of V represent a typical example of this situation)...

It should be noted that the requirement of f to be convex in the definition of f -divergence is essential. In Euclidean spaces any convex function can be represented as the pointwise supremum of a family of affine functions and vice versa, every supremum of a family of affine functions produces a convex function. Take f(x) = 12 |x− 1| as an example. We see that it can be written as a pointwise ...

In this paper, we show that the property of tight affine frame decomposition of functions in L can be extended in a stable way to functions in Sobolev spaces when the generators of the tight affine frames satisfy certain mild regularity and vanishing moment conditions. Applying the affine frame operators Qj on j-th levels to any function f in a Sobolev space reveals the detailed information Qjf...

It is well-known that affine equivalence relations keep nonlineaerity invariant for all Boolean functions. The set of all Boolean functions, Fn, over IF n 2 , is naturally regarded as the 2 n dimensional vector space, IF n 2 . Thus, while analyzing the transformations acting on Fn, S22n , the group of all bijective mappings, defined from IF 2 2 onto itself should be considered. As it is shown i...

It is shown here that if (Y, ‖ · ‖Y ) is a Banach space in which martingale differences are unconditional (a UMD Banach space) then there exists c = c(Y ) ∈ (0,∞) with the following property. For every n ∈ N and ε ∈ (0, 1/2], if (X, ‖·‖X) is an n-dimensional normed space with unit ball BX and f : BX → Y is a 1-Lipschitz function then there exists an affine mapping Λ : X → Y and a sub-ball B∗ = ...