Affine \mathbb{T} -varieties of complexity one and locally nilpotent derivations

Let X = Spec A be a normal affine variety over an algebraically closed field k of characteristic 0 endowed with an effective action of a torus \mathbbT \mathbb{T} of dimension n. Let also ∂ be a homogeneous locally nilpotent derivation on the normal affine \mathbbZⁿ {\mathbb{Z}^n} -graded domain A, so that ∂ generates a k ₊-action on X that is normalized by the \mathbbT \mathbb{T} -action.     

Singular del Pezzo surfaces that are equivariant compactifications

We determine which singular del Pezzo surfaces are equivariant compactifications of \mathbbG_\texta² \mathbb{G}_{\text{a}}^2 , to assist with proofs of Manin’s conjecture for such surfaces. Additionally, we give an example of a singular quartic del Pezzo surface that is an equivariant compactification of \mathbbG_\texta {\mathbb{G}_{\text{a}}} ⋊ \mathbbG_\textm {\mathbb{G}_{\text{m}}} . Bibliography: 32 titles.     

Automorphisms of a finitary factor power of an infinite symmetric group

We consider a semigroup FP_\textfin⁺ ( \mathfrakS_\textfin( \mathbbN ) ) FP_{\text{fin}}^{+} \left( {{\mathfrak{S}_{\text{fin}}}\left( \mathbb{N} \right)} \right) defined as a finitary factor power of a finitary symmetric group of countable order. It is proved that all automorphisms of FP_\textfin⁺ ( \mathfrakS_\textfin( \mathbbN ) ) FP_{\text{fin}}^{+} \left( {{\mathfrak{S}_{\text{fin}}}\left( \mathbb{N} \right)} \right) are induced by permutations from \mathfrakS( \mathbbN ) \mathfrak{S}\left( \mathbb{N} \right) .     

Counting Ω-ideals

Let ${\mathbb{A}}Let \mathbbA{\mathbb{A}} be a universal algebra of signature Ω, and let I{\mathcal{I}} be an ideal in the Boolean algebra P_\mathbbA{\mathcal{P}_{\mathbb{A}}} of all subsets of \mathbbA{\mathbb{A}} . We say that I{\mathcal{I}} is an Ω-ideal if I{\mathcal{I}} contains all finite subsets of \mathbbA{\mathbb{A}} and f(Aⁿ) ? I{f(A^{n}) \in \mathcal{I}} for every n-ary operation f ? W{f \in \Omega} and every A ? I{A \in \mathcal{I}} . We prove that there are 2^2à₀{2^{2^{\aleph_0}}} Ω-ideals in P_\mathbbA{\mathcal{P}_{\mathbb{A}}} provided that \mathbbA{\mathbb{A}} is countably infinite and Ω is countable.     

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.     

Uniform rates of approximation by short asymptotic expansions in the CLT for quadratic forms

Let X, X ₁, X ₂,… be i.i.d. \mathbbR^d {\mathbb{R}^d} -valued real random vectors. Assume that E X = 0 and that X has a nondegenerate distribution. Let G be a mean zero Gaussian random vector with the same covariance operator as that of X. We study the distributions of nondegenerate quadratic forms \mathbbQ[ S_N ] \mathbb{Q}\left[ {{S_N}} \right] of the normalized sums S _N  = N ^−1/2 (X ₁ + ⋯ + X _N ) and show that, without any additional conditions,

Isoperimetric estimates for the first eigenfunction of a class of linear elliptic problems

We find some optimal estimates for the first eigenfunction of a class of elliptic equations whose prototype is - ( gu_xi )_xi = lgu \textin W ì \mathbbRⁿ - {\left( {\gamma u_{{x_{i} }} } \right)}_{{x_{i} }} = \lambda \gamma u\,{\text{in}}\,\Omega \subset \mathbb{R}^{n} with Dirichlet boundary condition, where γ is the normalized Gaussian function in \mathbbRⁿ \mathbb{R}^{n} . To this aim we make use of the Gaussian symmetrization which transforms a domain into an half-space with the same Gaussian measure. The main tools we use are the properties of the weighted rearrangements and in particular the isoperimetric inequality with respect to Gaussian measure.     

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

Metric Flips with Calabi Ansatz

We study the limiting behavior of the K?hler–Ricci flow on \mathbbP(O_\mathbbPⁿ ?O_\mathbbPⁿ(-1)^?(m+1)){{\mathbb{P}(\mathcal{O}_{\mathbb{P}^n} \oplus \mathcal{O}_{\mathbb{P}^n}(-1)^{\oplus(m+1)})}} for m, n ≥ 1, assuming the initial metric satisfies the Calabi symmetry. We show that the flow either shrinks to a point, collapses to \mathbbPⁿ{{\mathbb{P}^n}} or contracts a subvariety of codimension m + 1 in the Gromov–Hausdorff sense. We also show that the K?hler–Ricci flow resolves a certain type of cone singularities in the Gromov–Hausdorff sense.     

Spectral decomposition and Gelfand’s theorem

We consider the spectral decomposition of A, the generator of a polynomially bounded n-times integrated group whose spectrum set $\sigma(A)=\{i\lambda_{k};k\in\mathbb{\mathbb{Z}}^{*}\}We consider the spectral decomposition of A, the generator of a polynomially bounded n-times integrated group whose spectrum set s(A)={il_k;k ? \mathbb\mathbbZ^*}\sigma(A)=\{i\lambda_{k};k\in\mathbb{\mathbb{Z}}^{*}\} is discrete and satisfies ?\frac1|l_k|^ld_kⁿ < ￥\sum \frac{1}{|\lambda_{k}|^{\ell}\delta_{k}^{n}}<\infty , where ℓ is a nonnegative integer and d_k=min(\frac|l_k+1-l_k|2,\frac|l_k-1-l_k|2)\delta _{k}=\min(\frac{|\lambda_{k+1}-\lambda _{k}|}{2},\frac{|\lambda _{k-1}-\lambda _{k}|}{2}) . In this case, Theorem 3, we show by using Gelfand’s Theorem that there exists a family of projectors (P_k)_{k ? \mathbb\mathbbZ^*}(P_{k})_{k\in\mathbb{\mathbb{Z}}^{*}} such that, for any x∈D(A ^n+ℓ ), the decomposition ∑P _k x=x holds.     

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.     

Minimal Cubic Cones via Clifford Algebras

In this paper, we construct two infinite families of algebraic minimal cones in ^n{\mathbb{R}^{n}}. The first family consists of minimal cubics given explicitly in terms of the Clifford systems. We show that the classes of congruent minimal cubics are in one to one correspondence with those of geometrically equivalent Clifford systems. As a byproduct, we prove that for any n ≥ 4, n ≠ 16k + 1, there is at least one minimal cone in \mathbbRⁿ{\mathbb{R}^{n}} given by an irreducible homogeneous cubic polynomial. The second family consists of minimal cones in \mathbbR^m2{\mathbb{R}^{m^2}}, m ≥ 2, defined by an irreducible homogeneous polynomial of degree m. These examples provide particular answers to the questions on algebraic minimal cones in \mathbbRⁿ{\mathbb{R}^{n}} posed by Wu-Yi Hsiang in the 1960s.     

Weakly Singular and Microscopically Hypersingular Load Perturbation for a Nonlinear Traction Boundary Value Problem: a Functional Analytic Approach

Let Ω ⁱ and Ω ^o be two bounded open subsets of \mathbbRⁿ{{\mathbb{R}}^{n}} containing 0. Let G ⁱ be a (nonlinear) map from ?Wⁱ×\mathbbRⁿ{\partial\Omega^{i}\times {\mathbb{R}}^{n}} to \mathbbRⁿ{{\mathbb{R}}^{n}} . Let a ^o be a map from ∂Ω ^o to the set M_n(\mathbbR){M_{n}({\mathbb{R}})} of n × n matrices with real entries. Let g be a function from ∂Ω ^o to \mathbbRⁿ{{\mathbb{R}}^{n}} . Let γ be a positive valued function defined on a right neighborhood of 0 in the real line. Let T be a map from ]1-(2/n),+￥[×M_n(\mathbbR){]1-(2/n),+\infty[\times M_{n}({\mathbb{R}})} to M_n(\mathbbR){M_{n}({\mathbb{R}})} . Then we consider the problem

Absolutely convergent fourier series. An improvement of the Beurling-Helson theorem

We consider the space A(\mathbbT)A(\mathbb{T}) of all continuous functions f on the circle \mathbbT\mathbb{T} such that the sequence of Fourier coefficients [^(f)] = { [^(f)]( k ), k ? \mathbbZ }\hat f = \left\{ {\hat f\left( k \right), k \in \mathbb{Z}} \right\} belongs to l ¹(ℤ). The norm on A(\mathbbT)A(\mathbb{T}) is defined by || f ||_A(\mathbbT) = || [^(f)] ||_{l¹ (\mathbbZ)}\left\| f \right\|_{A(\mathbb{T})} = \left\| {\hat f} \right\|_{l^1 (\mathbb{Z})}. According to the well-known Beurling-Helson theorem, if f:\mathbbT ? \mathbbT\phi :\mathbb{T} \to \mathbb{T} is a continuous mapping such that || e^inf ||_A(\mathbbT) = O(1)\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = O(1), n ∈ ℤ then φ is linear. It was conjectured by Kahane that the same conclusion about φ is true under the assumption that || e^inf ||_A(\mathbbT) = o( log| n | )\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\log \left| n \right|} \right). We show that if $\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\left( {{{\log \log \left| n \right|} \mathord{\left/ {\vphantom {{\log \log \left| n \right|} {\log \log \log \left| n \right|}}} \right. \kern-\nulldelimiterspace} {\log \log \log \left| n \right|}}} \right)^{1/12} } \right)$\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\left( {{{\log \log \left| n \right|} \mathord{\left/ {\vphantom {{\log \log \left| n \right|} {\log \log \log \left| n \right|}}} \right. \kern-\nulldelimiterspace} {\log \log \log \left| n \right|}}} \right)^{1/12} } \right), then φ is linear.     

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).     

Simultaneous inhomogeneous diophantine approximation on manifolds

In 1998, Kleinbock and Margulis proved Sprindzuk’s conjecture pertaining to metrical Diophantine approximation (and indeed the stronger Baker–Sprindzuk conjecture). In essence, the conjecture stated that the simultaneous homogeneous Diophantine exponent w ₀(x) = 1/n for almost every point x on a nondegenerate submanifold M \mathcal{M} of \mathbbRⁿ {\mathbb{R}^n} . In this paper, the simultaneous inhomogeneous analogue of Sprindzuk’s conjecture is established. More precisely, for any “inhomogeneous” vector θ ∈ \mathbbRⁿ {\mathbb{R}^n} we prove that the simultaneous inhomogeneous Diophantine exponent w ₀(x , θ) is 1/n for almost every point x on M \mathcal{M} . The key result is an inhomogeneous transference principle which enables us to deduce that the homogeneous exponent w ₀(x) is 1/n for almost all x ∈ M \mathcal{M} if and only if, for any θ ∈ \mathbbRⁿ {\mathbb{R}^n} , the inhomogeneous exponent w ₀(x , θ) = 1/n for almost all x ∈ M \mathcal{M} . The inhomogeneous transference principle introduced in this paper is an extremely simplified version of that recently discovered by us. Nevertheless, it should be emphasised that the simplified version has the great advantage of bringing to the forefront the main ideas while omitting the abstract and technical notions that come with describing the inhomogeneous transference principle in all its glory.     

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).     

Fujita-Liouville Type Theorem for Coupled Fourth-Order Parabolic Inequalities

This paper deals with a coupled system of fourth-order parabolic inequalities |u|t ≥ 2u + |v|q,|v|t ≥ 2v + |u|p in S = Rn × R+ with p,q > 1,n ≥ 1.A FujitaLiouville type theorem is established that the inequality system does not admit nontrivial nonnegative global solutions on S whenever n4 ≤ max(ppq+11,pqq+11).Since the general maximum-comparison principle does not hold for the fourth-order problem,the authors use the test function method to get the global non-existence of nontrivial solutions.     

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.     

On constructing total orders and solving vector optimization problems with total orders

In this paper, we introduce a construction method of total ordering cone on \mathbbRⁿ{\mathbb{R}^n} . It is shown that any total ordering cone on \mathbbRⁿ{\mathbb{R}^n} is isomorphic to the cone \mathbbRⁿ_lex{\mathbb{R}^n_{lex}} . Existence of a total ordering cone that contain given cone with a compact base is shown. By using this cone, a solving method of vector and set valued optimization problems is presented.