On the variety generated by all nilpotent lattice-ordered groups

In 1974, J. Martinez introduced the variety of weakly Abelian lattice-ordered groups; it is defined by the identity

     

A New Proof of Spinks' Theorem

Skew lattices form a class of non-commutative lattices. Spinks' Theorem [Matthew Spinks, On middle distributivity for skew lattices, ] states that for symmetric skew lattices the two distributive identities and are equivalent. Up to now only computer proofs of this theorem have been known. In the present paper the author presents a direct proof of Spinks' Theorem. In addition, a new result is proved showing that the assumption of symmetry can be omitted for cancellative skew lattices.     

Varieties generated by countably compact Abelian groups

We prove under the assumption of Martin's Axiom that every precompact Abelian group of size belongs to the smallest class of groups that contains all Abelian countably compact groups and is closed under direct products, taking closed subgroups and continuous isomorphic images.

     

On the Diophantine equation

Let be an odd prime. In this paper, using some theorems of Adachi and the author, we prove that if and , then the equation , and the equation , have no integral solutions respectively. Here is th Bernoulli number.

     

The product of finitely based varieties of lattice-ordered groups

The question as to whether a product of two finitely based varieties of lattice-ordered groups is finitely based is considered. It is proved that varieties and are finitely based; here is a variety of lattice-ordered groups defined by identities [x ⁿ,y ⁿ] =e and [[x,y] z, [x ₁,y ₁] z ₁] =e; is a variety of lattice-ordered nilpotent groups of class s, defined by an identity [x ₁,x ₂,...,x _(s+1)] =e; V is an arbitrary finitely based variety of lattice-ordered groups. Translated fromAlgebra i Logika, Vol. 33, No. 3, pp. 255–263, May–June, 1994.Supported by the Russian Foundation for Fundamental Research, grant No. 93-011-1524.     

Invariant Hilbert schemes and desingularizations of symplectic reductions for classical groups

Let $G \subset GL(V)$ be a reductive algebraic subgroup acting on the symplectic vector space $W=(V \oplus V^*)^{\oplus m}$ , and let $\mu :\ W \rightarrow Lie(G)^*$ be the corresponding moment map. In this article, we use the theory of invariant Hilbert schemes to construct a canonical desingularization of the symplectic reduction $\mu ^{-1}(0)/\!/G$ for classes of examples where $G=GL(V)$ , $O(V)$ , or $Sp(V)$ . For these classes of examples, $\mu ^{-1}(0)/\!/G$ is isomorphic to the closure of a nilpotent orbit in a simple Lie algebra, and we compare the Hilbert–Chow morphism with the (well-known) symplectic desingularizations of $\mu ^{-1}(0)/\!/G$ .     

Hessian nilpotent polynomials and the Jacobian conjecture

Let and let be the Laplace operator. The main goal of the paper is to show that the well-known Jacobian conjecture without any additional conditions is equivalent to what we call the vanishing conjecture: for any homogeneous polynomial of degree , if for all , then when , or equivalently, when . It is also shown in this paper that the condition () above is equivalent to the condition that is Hessian nilpotent, i.e. the Hessian matrix is nilpotent. The goal is achieved by using the recent breakthrough work of M. de Bondt, A. van den Essen and various results obtained in this paper on Hessian nilpotent polynomials. Some further results on Hessian nilpotent polynomials and the vanishing conjecture above are also derived.

     

Norm of an Integral Operator Related to the Harmonic Bergman Projection

The norm of the operator

Ladders and canonical varieties of completely regular semigroups

A completely regular semigroup is a (disjoint) union of its (maximal) subgroups. We consider it here with the unary operation of inversion within its maximal subgroups. Their totality \(\mathcal {C}\mathcal {R}\) forms a variety whose lattice of subvarieties is denoted by \(\mathcal {L}(\mathcal {C}\mathcal {R})\). On it, one defines the relations \(\mathbf {B}^\wedge \) and \(\mathbf {B}^\vee \) by

$$\begin{aligned} \begin{array}{lll} \mathcal {U}\ \mathbf {B}^\wedge \ \mathcal {V}&{} \Longleftrightarrow &{} \mathcal {U}\cap \mathcal {B} =\mathcal {V}\cap \mathcal {B}, \\ \mathcal {U}\ \mathbf {B}^\vee \ \mathcal {V}&{} \Longleftrightarrow &{} \mathcal {U}\vee \mathcal {B} =\mathcal {V}\vee \mathcal {B} , \end{array} \end{aligned}$$

respectively, where \(\mathcal {B}\) denotes the variety of all bands. This is a study of the interplay between the \(\cap \)-subsemilatice \(\triangle \) of \(\mathcal {L}(\mathcal {C}\mathcal {R})\) of upper ends of \(\mathbf {B}^\wedge \)-classes and their \(\mathbf {B}^\vee \)-classes. The main tool is the concept of a ladder and their \(\mathbf {B}^\vee \)-classes, an indispensable part of the important Polák’s theorem providing a construction for the join of varieties of completely regular semigroups. The paper includes the tables of ladders of the upper ends of most \(\mathbf {B}^\wedge \)-classes. Canonical varieties consist of two ascending countably infinite chains which generate most of the upper ends of \(\mathbf {B}^\wedge \)-classes.

     

Hausdorff measures and dimension on

We consider the Hausdorff measures , , defined on with the topology induced by the metric

for all . We study its properties, their relation to the ``Lebesgue measure" defined on by R. Baker in 1991, and the associated Hausdorff dimension. Finally, we give some examples.

     

Asymptotic Behaviour of a Random Walk Killed on a Finite Set

We study asymptotic behavior, for large time n, of the transition probability of a two-dimensional random walk killed when entering into a non-empty finite subset A. We show that it behaves like \(4 \tilde u_{A}(x) \tilde u_{-A}(-y) (\lg n)^{-2} p^{n}(y- x)\) for large n, uniformly in the parabolic regime \(|x|\vee |y| =O(\sqrt n)\), where p ⁿ (y-x) is the transition kernel of the random walk (without killing) and \(\tilde u_{A}\) is the unique harmonic function in the ‘exterior of A’ satisfying the boundary condition \(\tilde u_{A}(x) \sim \lg |x|\) at infinity.     

A primitive trinomial of degree 6972593

The only primitive trinomials of degree over are and its reciprocal.

     

Symmetry and regularity of extremals of an integral equation related to the Hardy�CSobolev inequality

Let ?? be a real number satisfying 0?<????<?n, ${0\leq t<\alpha, \alpha{^\ast}(t)=\frac{2(n-t)}{n-\alpha}}$ . We consider the integral equation $$u(x)=\int\limits_{{\mathbb{R}^n}}\frac{u^{{\alpha{^\ast}(t)}-1}(y)}{|y|^t|x-y|^{n-\alpha}}\,dy,\quad\quad\quad\quad\quad\quad\quad(1)$$ which is closely related to the Hardy?CSobolev inequality. In this paper, we prove that every positive solution u(x) is radially symmetric and strictly decreasing about the origin by the method of moving plane in integral forms. Moreover, we obtain the regularity of solutions to the following integral equation $$u(x)=\int\limits_{{\mathbb{R}^n}}\frac{|u(y)|^{p}u(y)}{|y|^t|x-y|^{n-\alpha}}\, dy\quad\quad\quad\quad\quad\quad\quad(2)$$ that corresponds to a large class of PDEs by regularity lifting method.     

Asymptotics of Orthogonal Polynomials for a Weight with a Jump on [?1,1]

We consider the orthogonal polynomials on [−1,1] with respect to the weight

On the Intersection of the Normalizers of the \boldsymbol{\cal{F}}-Residuals of Subgroups of a Finite Group

In this paper we give criteria for a finite group to belong to a formation. As applications, recent theorems of Li, Shen, Shi and Qian are generalized. Let G  be a finite group, $\cal F$ a formation and p  a prime. Let $D_{\mathcal {F}}(G)$ be the intersection of the normalizers of the $\cal F$ -residuals of all subgroups of G, and let $D_{\mathcal {F}}^{p}(G)$ be the intersection of the normalizers of $(H^{\cal F}O_{p'}(G))$ for all subgroups H of G. We then define $D_{\mathcal F}^{0}(G)=D_{\mathcal F, p}^{~0}(G)=1$ and $D_{\mathcal F}^{i+1}(G)/D_{\mathcal F}^{i}(G)=D_{\mathcal F}(G/D_{\mathcal F}^{i}(G))$ , $D_{\mathcal F, p}^{i+1}(G)/D_{\mathcal F, p}^{~i}(G)=D_{\mathcal F, p}(G/D_{\mathcal F, p}^{~i}(G))$ . Let $D_{\mathcal {F}}^{\infty}(G)$ and $D_{\mathcal {F}, p}^{~\infty}(G)$ denote the terminal member of the ascending series of $D_{\mathcal F}^{i}(G)$ and $D_{\mathcal F, p}^{~i}(G)$ respectively. In this paper we prove that under certain hypotheses, the the $\cal F$ -residual $G^{\cal F}$ is nilpotent (respectively,p-nilpotent) if and only if $G=D_{\mathcal {F}}^{\infty}(G)$ (respectively, $G=D_{\mathcal {F}, p}^{~\infty}(G)$ ). Further more, if the formation $\cal F$ is either the class of all nilpotent groups or the class of all abelian groups, then $G^{\cal F}$ is p-nilpotent if and only if and only if every cyclic subgroup of G order p and 4 (if p?=?2) is contained in $D_{\mathcal {F}, p}^{~\infty}(G)$ .     

Classification of positive solutions to an integral system with the poly-harmonic extension operator

In this paper, we investigate the positive solutions to the following integral system with a polyharmonic extension operator on R~+_n:{u(x)=c_n,a∫_?R_+~n(x_n~(1-a_v)(y)/|x-y|~(n-a))dy,x∈R_+~n,v(y)=c_n,a∫_R_+~n(x_n~(1-a_uθ)(x)/|x-y|~(n-a))dx,y∈ ?R_+~n,where n 2, 2-n a 1, κ, θ 0. This integral system arises from the Euler-Lagrange equation corresponding to an integral inequality on the upper half space established by Chen(2014). The explicit formulations of positive solutions are obtained by the method of moving spheres for the critical case κ =n-2+a/n-a,θ =n+2-a/ n-2+a. Moreover,we also give the nonexistence of positive solutions in the subcritical case for the above system.     

Integrals of polylogarithmic functions, recurrence relations, and associated Euler sums

We show that integrals of the form

and

satisfy certain recurrence relations which allow us to write them in terms of Euler sums. From this we prove that, in the first case for all and in the second case when is even, these integrals are reducible to zeta values. In the case of odd , we combine the known results for Euler sums with the information obtained from the problem in this form to give an estimate on the number of new constants which are needed to express the above integrals for a given weight .

The proofs are constructive, giving a method for the evaluation of these and other similar integrals, and we present a selection of explicit evaluations in the last section.

     

Polynomial structures for nilpotent groups

If a polycyclic-by-finite rank- group admits a faithful affine representation making it acting on properly discontinuously and with compact quotient, we say that admits an affine structure. In 1977, John Milnor questioned the existence of affine structures for such groups . Very recently examples have been obtained showing that, even for torsion-free, finitely generated nilpotent groups , affine structures do not always exist. It looks natural to consider affine structures as examples of polynomial structures of degree one. We introduce the concept of a canonical type polynomial structure for polycyclic-by-finite groups. Using the algebraic framework of the Seifert Fiber Space construction and a nice cohomology vanishing theorem, we prove the existence and uniqueness (up to conjugation) of canonical type polynomial structures for virtually finitely generated nilpotent groups. Applying this uniqueness to a result going back to Malcev, it follows that, for torsion-free, finitely generated nilpotent groups, each canonical polynomial structure is expressed in polynomials of limited degree. The minimal degree needed for obtaining a polynomial structure will determine the ``affine defect number'. We prove that the known counterexamples to Milnor's question have the smallest possible affine defect, i.e. affine defect number equal to one.