Functor category dualities for varieties of Heyting algebras

Let be a finitely generated variety of Heyting algebras and let be the class of subdirectly irreducible algebras in . We prove that is dually equivalent to a category of functors from into the category of Boolean spaces. The main tool is the theory of multisorted natural dualities.     

Pseudovarieties defining classes of sofic subshifts closed under taking shift equivalent subshifts

For a pseudovariety of ordered semigroups, let be the class of sofic subshifts whose syntactic semigroup lies in . It is proved that if contains then is closed under taking shift equivalent subshifts, and conversely, if is closed under taking conjugate subshifts then contains and . Almost finite type subshifts are characterized as the irreducible elements of , which gives a new proof that the class of almost finite type subshifts is closed under taking shift equivalent subshifts.     

Monoids with sub-log-exponential free spectra

A classic result from the 1960s states that the asymptotic growth of the free spectrum of a finite group is sub-log-exponential if and only if is nilpotent. Thus a monoid is sub-log-exponential implies , the pseudovariety of semigroups with nilpotent subgroups. Unfortunately, little more is known about the boundary between the sub-log-exponential and log-exponential monoids.The pseudovariety consists of those finite semigroups satisfying (x^ωy^ω)^ω(y^ωx^ω)^ω(x^ωy^ω)^ω≈(x^ωy^ω)^ω. Here it is shown that a monoid is sub-log-exponential implies . A quick application: a regular sub-log-exponential monoid is orthodox. It is conjectured that a finite monoid is sub-log-exponential if and only if it is , the finite monoids in having nilpotent subgroups. The forward direction of the conjecture is proved; moreover, the conjecture is proved for when is completely (0)-simple. In particular, the six-element Brandt monoid (the Perkins semigroup) is sub-log-exponential.     

On the Ricci and Einstein equations on the pseudo-euclidean and hyperbolic spaces

We consider tensors T=fg on the pseudo-euclidean space Rn and on the hyperbolic space Hn, where n?3, g is the standard metric and f is a differentiable function. For such tensors, we consider, in both spaces, the problems of existence of a Riemannian metric , conformal to g, such that , and the existence of such a metric which satisfies , where is the scalar curvature of . We find the restrictions on the Ricci candidate for solvability and we construct the solutions when they exist. We show that these metrics are unique up to homothety, we characterize those globally defined and we determine the singularities for those which are not globally defined. None of the non-homothetic metrics , defined on Rn or Hn, are complete. As a consequence of these results, we get positive solutions for the equation , where g is the pseudo-euclidean metric.     

Invariant algebraic curves for the cubic Liénard system with linear damping

We show that the system , with f,g polynomials of degree 1 and 3 respectively cannot have simultaneously an algebraic invariant curve and a limit cycle.     

Moduli of metaplectic bundles on curves and Theta-sheaves

We give a geometric interpretation of the Weil representation of the metaplectic group, placing it in the framework of the geometric Langlands program.For a smooth projective curve X we introduce an algebraic stack of metaplectic bundles on X. It also has a local version , which is a gerbe over the affine Grassmanian of G. We define a categorical version of the (nonramified) Hecke algebra of the metaplectic group. This is a category of certain perverse sheaves on , which act on by Hecke operators. A version of the Satake equivalence is proved describing as a tensor category. Further, we construct a perverse sheaf on corresponding to the Weil representation and show that it is a Hecke eigen-sheaf with respect to .     

A formula for inserting point masses

Let be a probability measure on the unit circle and be the measure formed by adding a pure point to . We give a formula for the Verblunsky coefficients of , based on a result of Simon.     

Estimates of weighted Hardy-Littlewood averages on the p-adic vector space

In the p-adic vector space , we characterize those non-negative functions ψ defined on for which the weighted Hardy-Littlewood average is bounded on (1?r?∞), and on . Also, in each case, we find the corresponding operator norm ‖Uψ‖.     

Additive maps on standard operator algebras preserving parts of the spectrum

Let and be standard operator algebras on infinite dimensional complex Banach spaces X and Y, respectively, and let Φ be a unital additive surjection from  onto . We introduce thirteen parts of the spectrum for elements in  and , and prove that if Φ preserves any one of these parts of the spectrum, then it is either an isomorphism or an anti-isomorphism.     

A one-to-one correspondence between colorings and stable sets

Given a graph G, we construct an auxiliary graph with vertices such that the set of all stable sets of is in one-to-one correspondence with the set of all colorings of G. Then, we show that the Max-Coloring problem in G reduces to the Maximum Weighted Stable set problem in .     

Minimum deviation, quasi-LU factorization of nonsingular matrices

Not all matrices enjoy the existence of an LU factorization. For those that do not, a number of “repairs” are possible. For nonsingular matrices we offer here a permutation-free repair in which the matrix is factored , with and collectively as near as possible to lower and upper triangular (in a natural sense defined herein). Such factorization is not generally unique in any sense. In the process, we investigate further the structure of matrices without LU factorization and permutations that produce an LU factorization.