Heat flows for a nonconvex Signorini type problem in $$ {\mathbb{R}^{N}} $$

We prove the existence of a global heat flow u : Ω ×  \mathbbR⁺ ? \mathbbR^N {\mathbb{R}^{+}} \to {\mathbb{R}^{N}}, N > 1, satisfying a Signorini type boundary condition u(∂Ω ×  \mathbbR⁺ {\mathbb{R}^{+}}) ⊂  \mathbbRⁿ {\mathbb{R}^{n}}), n \geqslant 2 n \geqslant 2 , and \mathbbR^N {\mathbb{R}^{N}}) with boundary ∂ [`(W)] \bar{\Omega } such that φ(∂Ω) ⊂ \mathbbR^N {\mathbb{R}^{N}} is given by a smooth noncompact hypersurface S. Bibliography: 30 titles.     

Sharp real-part theorems for high order derivatives

We obtain a representation for the sharp coefficient in an estimate of the modulus of the nth derivative of an analytic function in the upper half-plane \mathbbC₊ {\mathbb{C}_{+} } . It is assumed that the boundary value of the real part of the function on ?\mathbbC₊ \partial {\mathbb{C}_{+} } belongs to L ^p . This representation is specified for p = 1 and p = 2. For p = ∞ and for derivatives of odd order, an explicit formula for the sharp coefficient is found. A limit relation for the sharp coefficient in a pointwise estimate for the modulus of the n-th derivative of an analytic function in a disk is found as the point approaches the boundary circle. It is assumed that the boundary value of the real part of the function belongs to L ^p . The relation in question contains the sharp constant from the estimate of the modulus of the n-th derivative of an analytic function in \mathbbC₊ {\mathbb{C}_{+} } . As a corollary, a limit relation for the modulus of the n-th derivative of an analytic function with the bounded real part is obtained in a domain with smooth boundary. Bibliography: 8 titles.     

Classical operators on BLOCH spaces

Let \mathbbD \mathbb{D} ⁿ denote the unit polydisk and let B _n denote the unit ball in \mathbbC \mathbb{C} ⁿ , n ≥1. We study weighted composition operators on the α-Bloch spaces B^a {\mathcal{B}^\alpha } ( \mathbbD \mathbb{D} ⁿ ), α > 1. We also study Cesàro type operators on the α-Bloch spaces B^a {\mathcal{B}^\alpha } (B _n ), α > 0. Bibliography: 15 titles.     

On the dimension of invariant measures of endomorphisms of $${\mathbb{C}\mathbb{P}^k}$$

Let f be an endomorphism of \mathbbC\mathbbP^k{\mathbb{C}\mathbb{P}^k} and ν be an f-invariant measure with positive Lyapunov exponents (λ ₁, . . . , λ _k ). We prove a lower bound for the pointwise dimension of ν in terms of the degree of f, the exponents of ν and the entropy of ν. In particular our result can be applied for the maximal entropy measure μ. When k = 2, it implies that the Hausdorff dimension of μ is estimated by dim_Hm 3 [(log d)/(l₁)] + [(log d)/(l₂)]{{\rm dim}_\mathcal{H}\mu \geq {{\rm log} d \over \lambda_1} + {{\rm log} d \over \lambda_2}}, which is half of the conjectured formula. Our method for proving these results consists in studying the distribution of the ν-generic inverse branches of f ⁿ in \mathbbC\mathbbP^k{\mathbb{C}\mathbb{P}^k} . Our tools are a volume growth estimate for the bounded holomorphic polydiscs in \mathbbC\mathbbP^k{\mathbb{C}\mathbb{P}^k} and a normalization theorem for the ν-generic inverse branches of f ⁿ .     

Multipliers for logarithmic Cauchy integrals in the ball

Let B _n denote the unit ball in \mathbbC \mathbb{C} ⁿ , n ≥ 1. Let K \mathcal{K} ₀(n) denote the class of functions defined for z ∈ B _n as a constant plus the integral of the kernel log(1/(1 −〈z, ζ〉)) against a complex Borel measure on the sphere {ζ ∈ \mathbbC \mathbb{C} ⁿ ,: |ζ| = 1}. Properties of holomorphic functions g such that fg ∈ K \mathcal{K} ₀(n) for all f ∈ K \mathcal{K} ₀(n) are studied. The extended Cesàro operators are investigated on the spaces K \mathcal{K} ₀(n), n ≥ 1. Bibliography: 15 titles.     

Projective Freeness of Algebras of Real Symmetric Functions

Let \mathbb Dⁿ:={z=(z₁,?, z_n) ? \mathbb Cⁿ:|z_j| < 1,   j=1,?, n}{\mathbb {D}^n:=\{z=(z_1,\ldots, z_n)\in \mathbb {C}^n:|z_j| < 1, \;j=1,\ldots, n\}}, and let [`(\mathbbD)]ⁿ{\overline{\mathbb{D}}^n} denote its closure in \mathbb Cⁿ{\mathbb {C}^n}. Consider the ring

Hypersurfaces in the Lorentz-Minkowski space satisfying Lkψ?=?Aψ?+?b

We study hypersurfaces in the Lorentz-Minkowski space \mathbbLⁿ⁺¹{\mathbb{L}^{n+1}} whose position vector ψ satisfies the condition L _k ψ = Aψ + b, where L _k is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed k = 0, . . . , n − 1, A ? \mathbbR^(n+1)×(n+1){A\in\mathbb{R}^{(n+1)\times(n+1)}} is a constant matrix and b ? \mathbbLⁿ⁺¹{b\in\mathbb{L}^{n+1}} is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)th mean curvature, open pieces of totally umbilical hypersurfaces \mathbbSⁿ₁(r){\mathbb{S}^n_1(r)} or \mathbbHⁿ(-r){\mathbb{H}^n(-r)}, and open pieces of generalized cylinders \mathbbS^m₁(r)×\mathbbR^n-m{\mathbb{S}^m_1(r)\times\mathbb{R}^{n-m}}, \mathbbH^m(-r)×\mathbbR^n-m{\mathbb{H}^m(-r)\times\mathbb{R}^{n-m}}, with k + 1 ≤ m ≤ n − 1, or \mathbbL^m×\mathbbS^n-m(r){\mathbb{L}^m\times\mathbb{S}^{n-m}(r)}, with k + 1 ≤ n − m ≤ n − 1. This completely extends to the Lorentz-Minkowski space a previous classification for hypersurfaces in \mathbbRⁿ⁺¹{\mathbb{R}^{n+1}} given by Alías and Gürbüz (Geom. Dedicata 121:113–127, 2006).     

Biharmonic submanifolds of {\mathbb{C}P^n}

We give some general results on proper-biharmonic submanifolds of a complex space form and, in particular, of the complex projective space. These results are mainly concerned with submanifolds with constant mean curvature or parallel mean curvature vector field. We find the relation between the bitension field of the inclusion of a submanifold [`(M)]{\bar{M}} in \mathbbCPⁿ{\mathbb{C}P^n} and the bitension field of the inclusion of the corresponding Hopf-tube in \mathbbS²ⁿ⁺¹{\mathbb{S}^{2n+1}}. Using this relation we produce new families of proper-biharmonic submanifolds of \mathbbCPⁿ{\mathbb{C}P^n}. We study the geometry of biharmonic curves of \mathbbCPⁿ{\mathbb{C}P^n} and we characterize the proper-biharmonic curves in terms of their curvatures and complex torsions.     

Initial Enlargement of Filtrations and Entropy of Poisson Compensators

Let μ be a Poisson random measure, let \mathbbF\mathbb{F} be the smallest filtration satisfying the usual conditions and containing the one generated by μ, and let \mathbbG\mathbb{G} be the initial enlargement of \mathbbF\mathbb{F} with the σ-field generated by a random variable G. In this paper, we first show that the mutual information between the enlarging random variable G and the σ-algebra generated by the Poisson random measure μ is equal to the expected relative entropy of the \mathbbG\mathbb{G}-compensator relative to the \mathbbF\mathbb{F}-compensator of the random measure μ. We then use this link to gain some insight into the changes of Doob–Meyer decompositions of stochastic processes when the filtration is enlarged from  \mathbbF\mathbb{F} to  \mathbbG\mathbb{G}. In particular, we show that if the mutual information between G and the σ-algebra generated by the Poisson random measure μ is finite, then every square-integrable \mathbbF\mathbb{F}-martingale is a \mathbbG\mathbb{G}-semimartingale that belongs to the normed space S¹\mathcal{S}^{1} relative to  \mathbbG\mathbb{G}.     

Canonical form with respect to semiscalar equivalence for a matrix pencil with nonsingular first matrix

Polynomial n × n matrices A(x) and B(x) over a field \mathbbF \mathbb{F} are called semiscalar equivalent if there exist a nonsingular n × n matrix P over \mathbbF \mathbb{F} and an invertible n × n matrix Q(x) over \mathbbF \mathbb{F} [x] such that A(x) = PB(x)Q(x). We give a canonical form with respect to semiscalar equivalence for a matrix pencil A(x) = A _0x - A ₁, where A ₀ and A ₁ are n × n matrices over \mathbbF \mathbb{F} , and A ₀ is nonsingular.     

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-harmonic forms and stable hypersurfaces in space forms

We prove that a complete noncompact orientable stable minimal hypersurface in \mathbbSⁿ⁺¹{\mathbb{S}^{n+1}} (n ≤ 4) admits no nontrivial L ²-harmonic forms. We also obtain that a complete noncompact strongly stable hypersurface with constant mean curvature in \mathbbRⁿ⁺¹{\mathbb{R}^{n+1}} or \mathbbSⁿ⁺¹{\mathbb{S}^{n+1}} (n ≤ 4) admits no nontrivial L ²-harmonic forms. These results are generalized versions of Tanno’s result on stable minimal hypersurfaces in \mathbbRⁿ⁺¹{\mathbb{R}^{n+1}}.     

Affine extractors over prime fields

An affine extractor is a map that is balanced on every affine subspace of large enough dimension. We construct an explicit affine extractor AE from \mathbbFⁿ \mathbb{F}^n to \mathbbF\mathbb{F}, \mathbbF\mathbb{F} a prime field, so that AE(x) is exponentially close to uniform when x is chosen uniformly at random from an arbitrary affine subspace of \mathbbFⁿ \mathbb{F}^n of dimension at least δn, 0<δ≤1 a constant. Previously, Bourgain constructed such affine extractors when the size of \mathbbF\mathbb{F} is two. Our construction is in the spirit of but different than Bourgain’s construction. This allows for simpler analysis and better quantitative results.     

Numerical integration with polynomial exactness over a spherical cap

This paper presents rules for numerical integration over spherical caps and discusses their properties. For a spherical cap on the unit sphere \mathbbS²\mathbb{S}^2, we discuss tensor product rules with n ²/2 + O(n) nodes in the cap, positive weights, which are exact for all spherical polynomials of degree ≤ n, and can be easily and inexpensively implemented. Numerical tests illustrate the performance of these rules. A similar derivation establishes the existence of equal weight rules with degree of polynomial exactness n and O(n ³) nodes for numerical integration over spherical caps on \mathbbS²\mathbb{S}^2. For arbitrary d ≥ 2, this strategy is extended to provide rules for numerical integration over spherical caps on \mathbbS^d\mathbb{S}^d that have O(n ^d ) nodes in the cap, positive weights, and are exact for all spherical polynomials of degree ≤ n. We also show that positive weight rules for numerical integration over spherical caps on \mathbbS^d\mathbb{S}^d that are exact for all spherical polynomials of degree ≤ n have at least O(n ^d ) nodes and possess a certain regularity property.     

On the Second Cohomology of K?hler Groups

Carlson and Toledo conjectured that if an infinite group Γ is the fundamental group of a compact K?hler manifold, then virtually H²(G, \mathbb R) 1 0{H^{2}(\Gamma, {\mathbb R}) \ne 0} . We assume that Γ admits an unbounded reductive rigid linear representation. This representation necessarily comes from a complex variation of Hodge structure ( \mathbbC{\mathbb{C}} -VHS) on the K?hler manifold. We prove the conjecture under some assumption on the \mathbbC{\mathbb{C}} -VHS. We also study some related geometric/topological properties of period domains associated to such a \mathbbC{\mathbb{C}} -VHS.     

Exotic smooth structures on small 4-manifolds with odd signatures

Let M be (2n-1)\mathbbCP²#2n[`(\mathbbCP)]<sup>2</sup>(2n-1)\mathbb{CP}^{2}\#2n\overline{\mathbb{CP}}{}^{2} for any integer n≥1. We construct an irreducible symplectic 4-manifold homeomorphic to M and also an infinite family of pairwise non-diffeomorphic irreducible non-symplectic 4-manifolds homeomorphic to M. We also construct such exotic smooth structures when M is \mathbbCP<sup>2</sup>#4[`(\mathbbCP)]²\mathbb{CP}{}^{2}\#4\overline {\mathbb{CP}}{}^{2} or 3\mathbbCP²#k[`(\mathbbCP)]²3\mathbb{CP}{}^{2}\#k\overline{\mathbb{CP}}{}^{2} for k=6,8,10.     

A criterion for the splitting of a principal bundle over a projective space

Let E_G be an algebraic principal G-bundle over \mathbbC\mathbbPⁿ ,\mathbb{C}\mathbb{P}^n , n  \mathbbC.\mathbb{C}. We prove that E_G admits a reduction of structure group to a one-parameter subgroup of G if and only if

On branch points of three-dimensional mappings with unbounded characteristic of quasiconformality

For open discrete mappings f:D\{ b } ? \mathbbR³ f:D\backslash \left\{ b \right\} \to {\mathbb{R}^3} of a domain D ì \mathbbR³ D \subset {\mathbb{R}^3} satisfying relatively general geometric conditions in D \ {b} and having an essential singularity at a point b ? \mathbbR³ b \in {\mathbb{R}^3} , we prove the following statement: Let a point y ₀ belong to [`(\mathbbR³)] \f( D\{ b } ) \overline {{\mathbb{R}^3}} \backslash f\left( {D\backslash \left\{ b \right\}} \right) and let the inner dilatation K _I (x, f) and outer dilatation K _O (x, f) of the mapping f at the point x satisfy certain conditions. Let B _f denote the set of branch points of the mapping f. Then, for an arbitrary neighborhood V of the point y ₀, the set V ∩ f(B _f ) cannot be contained in a set A such that g(A) = I, where I = { t ? \mathbbR:| t | < 1 } I = \left\{ {t \in \mathbb{R}:\left| t \right| < 1} \right\} and g:U ? \mathbbRⁿ g:U \to {\mathbb{R}^n} is a quasiconformal mapping of a domain U ì \mathbbRⁿ U \subset {\mathbb{R}^n} such that A ⊂ U.     

S<Superscript>1</Superscript>-Valued Sobolev maps

We describe the structure of the space W^s,p( \mathbbSⁿ;\mathbbS¹ ) {W^{s,p}}\left( {{\mathbb{S}^n};{\mathbb{S}^1}} \right) , where 0 < s < ∞ and 1 ≤ p < ∞. According to the values of s, p, and n, maps in W^s,p( \mathbbSⁿ;\mathbbS¹ ) {W^{s,p}}\left( {{\mathbb{S}^n};{\mathbb{S}^1}} \right) can either be characterised by their phases or by a couple (singular set, phase).     

Congruence-simple subsemirings of ℚ+

Commutative congruence-simple semirings have already been characterized with the exception of the subsemirings of ℝ⁺. Even the class CongSimp(\mathbb Q⁺)\mathit{\mathcal{C}ong\mathcal{S}imp}(\mathbb {Q}^{+}) of all congruence-simple subsemirings of ℚ⁺ has not been classified yet. We introduce a new large class of the congruence-simple saturated subsemirings of ℚ⁺. We classify all the maximal elements of CongSimp(\mathbbQ⁺)\mathit{\mathcal{C}ong\mathcal {S}imp}(\mathbb{Q}^{+}) and show that every element of CongSimp(\mathbbQ⁺)\{\mathbbQ⁺}\mathit{\mathcal{C}ong\mathcal{S}imp}(\mathbb{Q}^{+})\setminus\{\mathbb{Q}^{+}\} is contained in at least one of them.     

Upper bound for the Lempert function of smooth domains

An upper estimate for the Lempert function of any C ^1+ε -smooth bounded domain in \mathbbCⁿ{\mathbb{C}^n} is found in terms of the boundary distance.