A characterization of Burniat surfaces with K2=4 and of non nodal type

Let S be a minimal surface of general type with pg(S) = 0 and K_S~2= 4. Assume the bicanonical map ψ of S is a morphism of degree 4 such that the image of ψ is smooth. Then we prove that the surface S is a Burniat surface with K~2= 4 and of non nodal type.     

Nonmatricial version of the Arveson-Wittstock extension principle,and its generalization

We consider the algebra ? = ?(H) of bounded operators in a Hilbert space H, ?-bimodules, and morphisms of these bimodules into the algebra ?(L ? H), where L is a Hilbert space. We study the problem of extension of a morphism defined on a sub-?-bimodule Y ? Z to Z. This problem is solved for Ruan bimodules.     

The Levi problem for Riemann domains over Stein spaces with isolated singularities

We consider the following question: Let \({p:Y \rightarrow X}\) be an unbranched Riemann domain and assume that X is a Stein space and p is a Stein morphism. Does it follow that Y is Stein ? We show that the answer is affirmative if X has isolated singularities. This generalizes a result of Andreotti and Narasimhan.     

Secondary Hochschild Cohomology

For a B-algebra A we introduce a Hochschild-like cohomology and use it to describe simultaneous deformations of the product and of the B-algebra structure on A[[t]]. These deformations have the property that the natural projection A[[t]]→A is a morphism of B-algebras.     

Abelianization of the <Emphasis Type="Italic">F</Emphasis>-divided fundamental group scheme

Let (X , x ₀) be a pointed smooth proper variety defined over an algebraically closed field. The Albanese morphism for (X , x ₀) produces a homomorphism from the abelianization of the F-divided fundamental group scheme of X to the F-divided fundamental group of the Albanese variety of X. We prove that this homomorphism is surjective with finite kernel. The kernel is also described.     

Representing some families of monotone maps by principal lattice congruences

For a lattice L with 0 and 1, let Princ(L) denote the set of principal congruences of L. Ordered by set inclusion, it is a bounded ordered set. In 2013, G. Grätzer proved that every bounded ordered set is representable as Princ(L); in fact, he constructed L as a lattice of length 5. For {0, 1}-sublattices \({A \subseteq B}\) of L, congruence generation defines a natural map Princ(A) \({\longrightarrow}\) Princ(B). In this way, every family of {0, 1}-sublattices of L yields a small category of bounded ordered sets as objects and certain 0-separating {0, 1}-preserving monotone maps as morphisms such that every hom-set consists of at most one morphism. We prove the converse: every small category of bounded ordered sets with these properties is representable by principal congruences of selfdual lattices of length 5 in the above sense. As a corollary, we can construct a selfdual lattice L in G. Grätzer's above-mentioned result.     

Factorization of the reaction-diffusion equation,the wave equation,and other equations

We investigate equations of the form D _t u = Δu + ξ? _u for an unknown function u(t, x), t ∈ ?, x ∈ X, where D _t u = a ₀(u, t) + Σ _k=1 ^r a _k (t, u)? _t ^k u, Δ is the Laplace-Beltrami operator on a Riemannian manifold X, and ξ is a smooth vector field on X. More exactly, we study morphisms from this equation within the category PDE of partial differential equations, which was introduced by the author earlier. We restrict ourselves to morphisms of a special form—the so-called geometric morphisms, which are given by maps of X to other smooth manifolds (of the same or smaller dimension). It is shown that a map f: X → Y defines a morphism from the equation D _t u = Δu + ξ? _u if and only if, for some vector field Ξ and a metric on Y, the equality (Δ + ξ?)f*v = f*(Δ + Ξ?)v holds for any smooth function v: Y → ?. In this case, the quotient equation is D _t v = Δv + Ξ?v for an unknown function v(t, y), y ∈ Y. It is also shown that, if a map f: X → Y is a locally trivial bundle, then f defines a morphism from the equation D _t u = Δu if and only if fibers of f are parallel and, for any path γ on Y, the expansion factor of a fiber translated along the horizontal lift γ to X depends on γ only.     

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.     

Chowla’s cosine problem

A continuous linear map T from a Banach algebra A into another B approximately preserves the zero products if ‖T(a)T(b)‖ ≤ α‖a‖‖b‖ (a,b ∈ A, ab = 0) for some small positive α. This paper is mainly concerned with the question of whether any continuous linear surjective map T: A → B that approximately preserves the zero products is close to a continuous homomorphism from A onto B with respect to the operator norm. We show that this is indeed the case for amenable group algebras.     

Frobenius morphisms and stable module categories of repetitive algebras

Let k be the algebraic closure of a finite field F_q and A be a finite dimensional k-algebra with a Frobenius morphism F.In the present paper we establish a relation between the stable module category of the repetitive algebra  of A and that of the repetitive algebra of the fixed-point algebra A~F.As an application,it is shown that the derived category of A~F is equivalent to the subcategory of F-stable objects in the derived category of A when A has a finite global dimension.     

Theta characteristics of hyperelliptic graphs

We study theta characteristics of hyperelliptic metric graphs of genus g with no bridge edges. These graphs have a harmonic morphism of degree two to a metric tree that can be lifted to a morphism of degree two of a hyperelliptic curve X over K to the projective line, with K an algebraically closed field of char\({(K) \not =2}\), complete with respect to a non-Archimedean valuation, with residue field k of char\({(k)\not=2}\). The hyperelliptic curve has \({2^{2g}}\) theta characteristics. We show that for each effective theta characteristic on the graph, \({2^{g-1}}\) even and \({2^{g-1}}\) odd theta characteristics on the curve specialize to it; and \({2^g}\) even theta characteristics on the curve specialize to the unique not effective theta characteristics on the graph.     

Detecting fixed points of nonexpansive maps by illuminating the unit ball

We give necessary and sufficient conditions for a nonexpansive map on a finite-dimensional normed space to have a nonempty, bounded set of fixed points. Among other results we show that if f: V → V is a nonexpansive map on a finite-dimensional normed space V, then the fixed point set of f is nonempty and bounded if and only if there exist w₁,..., w _m in V such that {f(w _i ) ? w _i : i = 1,..., m} illuminates the unit ball. This yields a numerical procedure for detecting fixed points of nonexpansive maps on finite-dimensional spaces. We also discuss applications of this procedure to certain nonlinear eigenvalue problems arising in game theory and mathematical biology.     

Approximation of maps into spheres by regulous maps

Let X be a compact real algebraic set of dimension n. We prove that every Euclidean continuous map from X into the unit n-sphere can be approximated by a regulous map. This strengthens and generalizes previously known results.     

Moduli of Continuity of Harmonic Quasiregular Mappings in $\mathbb{B}^n$

We prove that ω _u (δ)?≤?Cω _f (δ), where \(u : \overline{\mathbb{B}^n} \rightarrow \mathbb{R}^n\) is the harmonic extension of a continuous map \(f:\mathbb{S}^{n-1}\rightarrow\mathbb{R}^n\), if u is a K-quasiregular map. Here C is a constant depending only on n, ω _f and K and ω _h denotes the modulus of continuity of h.     

Short cycles in repeated exponentiation modulo a prime

Given a prime p, we consider the dynamical system generated by repeated exponentiations modulo p, that is, by the map \({u \mapsto f_g(u)}\), where f _g (u) ≡ g ^u (mod p) and 0 ≤ f _g (u) ≤ p ? 1. This map is in particular used in a number of constructions of cryptographically secure pseudorandom generators. We obtain nontrivial upper bounds on the number of fixed points and short cycles in the above dynamical system.     

Compact Manifolds Covered by a Torus

Let X be a compact complex manifold which is the image of a complex torus by a holomorphic surjective map A→X. We prove that X is Kähler and that up to a finite étale cover, X is a product of projective spaces by a torus.     

Arnold diffusion for a complete family of perturbations

In this work we illustrate the Arnold diffusion in a concrete example — the a priori unstable Hamiltonian system of 2 + 1/2 degrees of freedom H(p, q, I, φ, s) = p ²/2+ cos q ? 1 + I ²/2 + h(q, φ, s; ε) — proving that for any small periodic perturbation of the form h(q, φ, s; ε) = ε cos q (a ₀₀ + a ₁₀ cosφ + a ₀₁ cos s) (a ₁₀ a ₀₁ ≠ 0) there is global instability for the action. For the proof we apply a geometrical mechanism based on the so-called scattering map. This work has the following structure: In the first stage, for a more restricted case (I* ~ π/2μ, μ = a ₁₀/a ₀₁), we use only one scattering map, with a special property: the existence of simple paths of diffusion called highways. Later, in the general case we combine a scattering map with the inner map (inner dynamics) to prove the more general result (the existence of instability for any μ). The bifurcations of the scattering map are also studied as a function of μ. Finally, we give an estimate for the time of diffusion, and we show that this time is primarily the time spent under the scattering map.     

Jacobi Maaß forms

In this paper, we give a new definition for the space of non-holomorphic Jacobi Maaß forms (denoted by J _k,m ^nh ) of weight k∈? and index m∈? as eigenfunctions of a degree three differential operator \(\mathcal{C}^{k,m}\). We show that the three main examples of Jacobi forms known in the literature: holomorphic, skew-holomorphic and real-analytic Eisenstein series, are contained in J _k,m ^nh . We construct new examples of cuspidal Jacobi Maaß forms F _f of weight k∈2? and index 1 from weight k?1/2 Maaß forms f with respect to Γ₀(4) and show that the map f ? F _f is Hecke equivariant. We also show that the above map is compatible with the well-known representation theory of the Jacobi group. In addition, we show that all of J _k,m ^nh can be “essentially” obtained from scalar or vector valued half integer weight Maaß forms.     

Infinitesimal Hopf Algebras and the cd-Index of Polytopes

Infinitesimal bialgebras were introduced by Joni and Rota [JR]. The basic theory of these objects was developed in [Aff1] and [Aff2]. In this paper we present a simple proof of the existence of the cd-index of polytopes, based on the theory of infinitesimal Hopf algebras.For the purpose of this work, the main examples of infinitesimal Hopf algebras are provided by the algebra \ppp of all posets and the algebra k &;lt;ab&;gt; of noncommutative polynomials. We show that k &;lt;ab&;gt; satisfies the following universal property: given a graded infinitesimal bialgebra A and a morphism of algebras ζ _A \colon A→ k , there exists a unique morphism of graded infinitesimal bialgebras ψ\colon A → k&;lt;ab&;gt; such that ζ_{1,0}ψ=ζ_A, where ζ_{1,0} is evaluation at (1,0). When the universal property is applied to the algebra of posets and the usual zeta function ζ_{\ppp}(P)=1, one obtains the \abindex of posets ψ\colon \ppp→k &;lt;ab&;gt;.The notion of antipode is used to define an analog of the Möbius function of posets for more general infinitesimal Hopf algebras than \ppp , and this in turn is used to define a canonical infinitesimal Hopf subalgebra, called the eulerian subalgebra. All eulerian posets belong to the eulerian subalgebra of \ppp . The eulerian subalgebra of k &;lt;ab&;gt; is precisely the algebra spanned by c=a+b and d=ab+ba. The existence of the cd-index of eulerian posets is then an immediate consequence of the simple fact that eulerian subalgebras are preserved under morphisms of infinitesimal Hopf algebras.The theory also provides a version of the generalized Dehn—Sommerville equations for more general infinitesimal Hopf algebras than k &;lt;ab&;gt;.     

Generalizing the Krein–Rutman theorem,measures of noncompactness and the fixed point index

If L : Y → Y is a bounded linear map on a Banach space Y, the “radius of the essential spectrum” or “essential spectral radius” ρ(L) of L is well-defined and there are well-known formulas for ρ(L) in terms of measures of noncompactness. Now let \({C \subset D}\) be complete cones in a normed linear space (X, || · ||) and f : C → C a continuous map which is homogeneous of degree one and preserves the partial ordering induced by D. We prove (see Section 2) that various obvious analogs of the formulas for the essential spectral radius for the case f : C → C have serious defects, even when f is linear on C. We propose (see (3.5)) a definition for ρ _C (f), the “cone essential spectral radius of f,” which avoids these difficulties. If \({{\tilde r}_{C}(f)}\) denotes the (Bonsall) cone spectral radius of f, we conjecture (see Conjecture 4.1) that if \({\rho_{C}(f) < {\tilde r}_{C}(f)}\), then there exists \({u \in C {\backslash} \, \{0\}}\) with f(u) = ru where r ? r _C (f). If f satisfies certain additional conditions (for example, if f is a compact perturbation of a map which is linear on C), we obtain the conclusion of the conjecture; but in general we observe (Remark 4.7) that the conjecture is intimately related to old and difficult conjectures in asymptotic fixed point theory. In Section 5 we briefly discuss extensions of generalized max-plus operators which were our original motivation and for which Conjecture 4.1 is already nontrivial.