Algebraic Hamiltonian actions

In this paper we deal with a Hamiltonian action of a reductive algebraic group G on an irreducible normal affine Poisson variety X. We study the quotient morphism \({\mu_{G,X}//G : X//G \rightarrow \mathfrak{g} //G}\) of the moment map \({\mu_{G,X} : X\rightarrow \mathfrak{g}}\) . We prove that for a wide class of Hamiltonian actions (including, for example, actions on generically symplectic varieties) all fibers of the morphism μ _G,X //G have the same dimension. We also study the “Stein factorization” of μ _G,X //G. Namely, let C _G,X denote the spectrum of the integral closure of \({\mu_{G,X}^{*}(\mathbb{K}[\mathfrak{g}]^G)}\) in \({\mathbb{K}(X)^G}\) . We investigate the structure of the \({\mathfrak{g} //G}\) -scheme C _G,X . Our results partially generalize those obtained by F. Knop for the actions on cotangent bundles and symplectic vector spaces.     

On powersmooth numbers

A natural number n is called y-smooth (y-powersmooth, respectively) for a positive number y if every prime (prime power) dividing n is bounded from above by y. Let ψ(x, y) and ψ*(x, y) denote the quantity of y-smooth and y-powersmooth integers restricted by x, respectively. In this paper we investigate function ψ*(x, y) in general. We derive formulas for finding exact calculation of ψ*(x, y) for large x and relatively small y and give theoretical estimates for this function and for a function of the greatest powersmooth integer. This results can be used in the cryptography and number theory to estimate the convergence of factorization algorithms.     

Projections of self-similar sets with no separation condition

Two natural symplectic constructions, the Lagrangian suspension and Seidel’s quantum representation of the fundamental group of the group of Hamiltonian diffeomorphisms, Ham(M), with (M, ω) a monotone symplectic manifold, admit categorifications as actions of the fundamental groupoid Π(Ham(M)) on a cobordism category recently introduced in [BC14] and, respectively, on a monotone variant of the derived Fukaya category. We show that the functor constructed in [BC14] that maps the cobordism category to the derived Fukaya category is equivariant with respect to these actions.     

New characterization of <Emphasis Type="Italic">S</Emphasis><Subscript>4</Subscript>(<Emphasis Type="Italic">q</Emphasis>) by its noncommuting graph

Let G be a nonabelian group, and associate the noncommuting graph ?(G) with G as follows: the vertex set of ?(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. Let S ₄(q) be the projective symplectic simple group, where q is a prime power. We prove that if G is a group with ?(G) ? ?(S ₄(q)) then G ? S ₄(q).     

Annihilating and power-commuting generalized skew derivations on lie ideals in prime rings

Let R be a prime ring of characteristic different from 2 and 3, Q_r its right Martindale quotient ring, C its extended centroid, L a non-central Lie ideal of R and n ≥ 1 a fixed positive integer. Let α be an automorphism of the ring R. An additive map D: R → R is called an α-derivation (or a skew derivation) on R if D(xy) = D(x)y + α(x)D(y) for all x, y ∈ R. An additive mapping F: R → R is called a generalized α-derivation (or a generalized skew derivation) on R if there exists a skew derivation D on R such that F(xy) = F(x)y + α(x)D(y) for all x, y ∈ R.     

Relative symplectic caps, 4-genus and fibered knots

We prove relative versions of the symplectic capping theorem and sufficiency of Giroux’s criterion for Stein fillability and use these to study the 4-genus of knots. More precisely, suppose we have a symplectic 4-manifold X with convex boundary and a symplectic surface Σ in X such that ?Σ is a transverse knot in ?X. In this paper, we prove that there is a closed symplectic 4-manifold Y with a closed symplectic surface S such that (X,Σ) embeds into (Y,S) symplectically. As a consequence we obtain a relative version of the symplectic Thom conjecture. We also prove a relative version of the sufficiency part of Giroux’s criterion for Stein fillability, namely, we show that a fibered knot whose mondoromy is a product of positive Dehn twists bounds a symplectic surface in a Stein filling. We use this to study 4-genus of fibered knots in \(\mathbb {S}^{3} \). Further, we give a criterion for quasipositive fibered knots to be strongly quasipositive.     

No decreasing sequence of cardinals

In set theory without the Axiom of Choice (AC), we investigate the set-theoretic strength of the principle NDS which states that there is no function f on the set ω of natural numbers such that for everyn ∈ ω, f (n + 1) ? f (n), where for sets x and y, x ? y means that there is a one-to-one map g : x → y, but no one-to-one map h : y → x. It is a long standing open problem whether NDS implies AC. In this paper, among other results, we show that NDS is a strong axiom by establishing that AC^LO (AC restricted to linearly ordered sets of non-empty sets, and also equivalent to AC in ZF, the Zermelo–Fraenkel set theory minus AC) ? NDS in ZFA set theory (ZF with the Axiom of Extensionality weakened in order to allow the existence of atoms). The latter result provides a strongly negative answer to the question of whether “every Dedekind-finite set is finite” implies NDS addressed in G. H. Moore “Zermelo’s Axiom of Choice. Its Origins, Development, and Influence” and in P. Howard–J. E. Rubin “Consequences of the Axiom of Choice”. We also prove that AC^WO (AC restricted to well-ordered sets of non-empty sets) ? NDS in ZF (hence, “every Dedekind-finite set is finite” ? NDS in ZF, either) and that “for all infinite cardinals m, m + m = m” ? NDS in ZFA.     

Closable Multipliers

Let (X, μ) and (Y, ν) be standard measure spaces. A function \({\varphi\in L^\infty(X\times Y,\mu\times\nu)}\) is called a (measurable) Schur multiplier if the map S _φ , defined on the space of Hilbert-Schmidt operators from L ₂(X, μ) to L ₂(Y, ν) by multiplying their integral kernels by φ, is bounded in the operator norm. The paper studies measurable functions φ for which S _φ is closable in the norm topology or in the weak* topology. We obtain a characterisation of w*-closable multipliers and relate the question about norm closability to the theory of operator synthesis. We also study multipliers of two special types: if φ is of Toeplitz type, that is, if φ(x, y) = f(x ? y), \({x,y\in G}\), where G is a locally compact abelian group, then the closability of φ is related to the local inclusion of f in the Fourier algebra A(G) of G. If φ is a divided difference, that is, a function of the form (f(x) ? f(y))/(x ? y), then its closability is related to the “operator smoothness” of the function f. A number of examples of non-closable, norm closable and w*-closable multipliers are presented.     

A sharp estimate of positive integral points in 6-dimensional polyhedra and a sharp estimate of smooth numbers

Inspired by Durfee Conjecture in singularity theory, Yau formulated the Yau number theoretic conjecture(see Conjecture 1.3) which gives a sharp polynomial upper bound of the number of positive integral points in an n-dimensional(n≥3) polyhedron. It is well known that getting the estimate of integral points in the polyhedron is equivalent to getting the estimate of the de Bruijn function ψ(x, y), which is important and has a number of applications to analytic number theory and cryptography. We prove the Yau number theoretic conjecture for n = 6. As an application, we give a sharper estimate of function ψ(x, y) for 5≤y 17, compared with the result obtained by Ennola.     

Quasiplanar Diagrams and Slim Semimodular Lattices

For elements x and y in the (Hasse) diagram D of a finite bounded poset P, x is on the left of y, written as x λ y, if x and y are incomparable and x is on the left of all maximal chains through y. Being on the right, written as x ? y, is defined analogously. The diagram D is quasiplanar if λ and ? are transitive and for any pair (x,y) of incomparable elements, if x is on the left of some maximal chain through y, then x λ y. A planar diagram is quasiplanar, and P has a quasiplanar diagram iff its order dimension is at most 2. We are interested in diagrams only up to similarity. A finite lattice is slim if it is join-generated by the union of two chains. The main result gives a bijection between the set of (the similarity classes of) finite quasiplanar diagrams and that of (the similarity classes of) planar diagrams of finite slim semimodular lattices. This bijection allows one to describe finite posets of order dimension at most 2 by finite slim semimodular lattices, and conversely. As a corollary, we obtain that there are exactly (n?2)! quasiplanar diagrams of size n.     

A Generalized Block Replacement Policy with Minimal Repair and General Random Repair Costs for a Multi-unit System

A generalization of the block replacement policy (BRP) is proposed and analysed for a multi-unit system which has the specific multivariate distribution. Under such a policy an operating system is preventively replaced at times kT (k = 1, 2, 3,...), as in the ordinary BRP, and the replacement of the failed system at failure is not mandatory; instead, a minimal repair to the component of the system can be made. The choice of these two possible actions is based on some random mechanism which is age-dependent. The cost of the ith minimal repair of the component at age y depends on the random part C(y) and the deterministic part C_i(y). The aim of the paper is to find the optimal block interval T which minimizes the long-run expected cost per unit time of the policy.     

Equivariant cohomology distinguishes the geometric structures of toric hyperkähler manifolds

Toric hyperkähler manifolds are the hyperkähler analogue of symplectic toric manifolds. The theory of Bielawski and Dancer tells us that, while a symplectic toric manifold is determined by a Delzant polytope, a toric hyperkähler manifold is determined by a smooth hyperplane arrangement. The purpose of this paper is to show that a toric hyperkähler manifold up to weak hyperhamiltonian T -isometry is determined not only by a smooth hyperplane arrangement up to weak linear equivalence but also by its equivariant cohomology H* _T (M; ?) with a point â in H ²(M;?) \ {0} up to weak H*(BT; ?)-algebra isomorphism preserving â.     

Gaiotto’s Lagrangian Subvarieties via Derived Symplectic Geometry

Let Bun_G be the moduli space of G-bundles on a smooth complex projective curve. Motivated by a study of boundary conditions in mirror symmetry, Gaiotto (2016) associated to any symplectic representation of G a Lagrangian subvariety of T^?Bun_G. We give a simple interpretation of (a generalization of) Gaiotto’s construction in terms of derived symplectic geometry. This allows to consider a more general setting where symplectic G-representations are replaced by arbitrary symplectic manifolds equipped with a Hamiltonian G-action and with an action of the multiplicative group that rescales the symplectic form with positive weight.     

Stability of Cubic and Quartic Functional Equations in Non-Archimedean Spaces

We prove generalized Hyers-Ulam–Rassias stability of the cubic functional equation f(kx+y)+f(kx?y)=k[f(x+y)+f(x?y)]+2(k ³?k)f(x) for all \(k\in \Bbb{N}\) and the quartic functional equation f(kx+y)+f(kx?y)=k ²[f(x+y)+f(x?y)]+2k ²(k ²?1)f(x)?2(k ²?1)f(y) for all \(k\in \Bbb{N}\) in non-Archimedean normed spaces.     

Ergodic properties of skew products in infinite measure

Let (Ω, µ) be a shift of finite type with a Markov probability, and (Y, ν) a non-atomic standard measure space. For each symbol i of the symbolic space, let Φ_i be a non-singular automorphism of (Y, ν). We study skew products of the form (ω, y) ? (σω, Φ_ω0 (y)), where σ is the shift map on (Ω, µ). We prove that, when the skew product is recurrent, it is ergodic if and only if the Φ_i’s have no common non-trivial invariant set.     

Planar Graded Lattices and the <Emphasis Type="Bold">c</Emphasis><Subscript>1</Subscript>-Median Property

Let L be a lattice of finite length, ξ = (x ₁,…, x _k )∈L ^k , and y ∈ L. The remoteness r(y, ξ) of y from ξ is d(y, x ₁)+?+d(y, x _k ), where d stands for the minimum path length distance in the covering graph of L. Assume, in addition, that L is a graded planar lattice. We prove that whenever r(y, ξ) ≤ r(z, ξ) for all z ∈ L, then y ≤ x ₁∨?∨x _k . In other words, L satisfies the so-called c ₁ -median property.     

Heat Kernels for Isotropic-Like Markov Generators on Ultrametric Spaces: a Survey

Let (X, d) be a locally compact separable ultrametric space. Let D be the set of all locally constant functions having compact support. Given a measure m and a symmetric function J(x, y) we consider the linear operator L^Jf(x) = ∫(f(x) ? f(y)) J(x, y)dm(y) defined on the set D. When J(x, y) is isotropic and satisfies certain conditions, the operator (?L^J, D) acts in L²(X,m), is essentially self-adjoint and extends as a self-adjoint Markov generator, its Markov semigroup admits a continuous heat kernel p^J (t, x, y). When J(x, y) is not isotropic but uniformly in x, y is comparable to isotropic function J(x, y) as above the operator (?L^J, D) extends in L²(X,m) as a self-adjointMarkov generator, its Markov semigroup admits a continuous heat kernel p^J(t, x, y), and the function p^J(t, x, y) is uniformly comparable in t, x, y to the function p^J(t, x, y), the heat kernel related to the operator (?L^J,D).     

Propagation of sound pulses in a homogeneous layer under an inhomogeneous half space

We consider the propagation of two-dimensional sound pulses in a homogeneous layer ?y ₁?y?0. It is bounded by a plane stratified inhomogeneous half spacey?0 on one side and a perfectly reflecting boundary on the other. A line source is situated in the layer. The boundary condition isφ=0 or?φ/?y=0 aty=?y ₁, whereφ is the acoustic velocity potential. We suppose that the velocity of wave propagationc is given byc ^?2=p?qe ^?αy iny>0, wherep, q, α are real and positive andp>q. It is equal to C′ in the layer where C′ is a constant. The method of dual integral transformation is used and the velocity potentialφ is obtained after using asymptotic expressions for some of the functions which are in the integrand. We obtain the incident, reflected, multiply-reflected and diffracted pulses in the layer.