Positive Cubature Formulas and Marcinkiewicz–Zygmund Inequalities on Spherical Caps

Let Π _n ^d denote the space of all spherical polynomials of degree at most n on the unit sphere $\mathbb{S}^{d}Let Π _n ^d denote the space of all spherical polynomials of degree at most n on the unit sphere \mathbbS^d\mathbb{S}^{d} of ℝ ^d+1, and let d(x,y) denote the geodesic distance arccos x⋅y between x,y ? \mathbbS^dx,y\in\mathbb{S}^{d} . Given a spherical cap

B(e,a)={x ? \mathbbSd:d(x,e) ￡ a}    (e ? \mathbbSd, a ? (0,p) is bounded awayfrom p),B(e,\alpha)=\big\{x\in\mathbb{S}^{d}:d(x,e)\leq\alpha\big\}\quad \bigl(e\in\mathbb{S}^{d},\ \alpha\in(0,\pi)\ \mbox{is bounded awayfrom}\ \pi\bigr),      2. Heat flows for a nonconvex Signorini type problem in $$ {\mathbb{R}^{N}} $$      A. Arkhipova 《Journal of Mathematical Sciences》2011,176(6):732-758 We prove the existence of a global heat flow u : Ω ×  \mathbbR+ ? \mathbbRN {\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 \mathbbRN {\mathbb{R}^{N}}) with boundary ∂ [`(W)] \bar{\Omega } such that φ(∂Ω) ⊂ \mathbbRN {\mathbb{R}^{N}} is given by a smooth noncompact hypersurface S. Bibliography: 30 titles.      3. Affine extractors over prime fields   总被引：1，自引：0，他引：1   Amir Yehudayoff 《Combinatorica》2011,31(2):245-256 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.      4. Hypersurfaces in the Lorentz-Minkowski space satisfying Lkψ?=?Aψ?+?b      Pascual Lucas  H. Fabián Ramírez-Ospina 《Geometriae Dedicata》2011,153(1):151-175 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 \mathbbSm1(r)×\mathbbRn-m{\mathbb{S}^m_1(r)\times\mathbb{R}^{n-m}}, \mathbbHm(-r)×\mathbbRn-m{\mathbb{H}^m(-r)\times\mathbb{R}^{n-m}}, with k + 1 ≤ m ≤ n − 1, or \mathbbLm×\mathbbSn-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).      5. Compact Surfaces with Constant Gaussian Curvature in Product Spaces      Juan A. Aledo  Victorino Lozano  José A. Pastor 《Mediterranean Journal of Mathematics》2010,7(3):263-270 We prove that the only compact surfaces of positive constant Gaussian curvature in \mathbbH2×\mathbbR{\mathbb{H}^{2}\times\mathbb{R}} (resp. positive constant Gaussian curvature greater than 1 in \mathbbS2×\mathbbR{\mathbb{S}^{2}\times\mathbb{R}}) whose boundary Γ is contained in a slice of the ambient space and such that the surface intersects this slice at a constant angle along Γ, are the pieces of a rotational complete surface. We also obtain some area estimates for surfaces of positive constant Gaussian curvature in \mathbbH2×\mathbbR{\mathbb{H}^{2}\times\mathbb{R}} and positive constant Gaussian curvature greater than 1 in \mathbbS2×\mathbbR{\mathbb{S}^{2}\times\mathbb{R}} whose boundary is contained in a slice of the ambient space. These estimates are optimal in the sense that if the bounds are attained, the surface is again a piece of a rotational complete surface.      6. <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-harmonic forms and stable hypersurfaces in space forms      Peng Zhu 《Archiv der Mathematik》2011,97(3):271-279 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}}.      7. Initial Enlargement of Filtrations and Entropy of Poisson Compensators      Stefan Ankirchner  Jakub Zwierz 《Journal of Theoretical Probability》2011,24(1):93-117 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}.      8. S<Superscript>1</Superscript>-Valued Sobolev maps      P. Mironescu 《Journal of Mathematical Sciences》2010,170(3):340-355 We describe the structure of the space Ws,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 Ws,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).      9. Normalizers and self-normalizing subgroups II      Boris Širola 《Central European Journal of Mathematics》2011,9(6):1317-1332 Let \mathbbK\mathbb{K} be a field, G a reductive algebraic \mathbbK\mathbb{K}-group, and G 1 ≤ G a reductive subgroup. For G 1 ≤ G, the corresponding groups of \mathbbK\mathbb{K}-points, we study the normalizer N = N G (G 1). In particular, for a standard embedding of the odd orthogonal group G 1 = SO(m, \mathbbK\mathbb{K}) in G = SL(m, \mathbbK\mathbb{K}) we have N ≅ G 1 ⋊ μ m ( \mathbbK\mathbb{K}), the semidirect product of G 1 by the group of m-th roots of unity in \mathbbK\mathbb{K}. The normalizers of the even orthogonal and symplectic subgroup of SL(2n, \mathbbK\mathbb{K}) were computed in [Širola B., Normalizers and self-normalizing subgroups, Glas. Mat. Ser. III (in press)], leaving the proof in the odd orthogonal case to be completed here. Also, for G = GL(m, \mathbbK\mathbb{K}) and G 1 = O(m, \mathbbK\mathbb{K}) we have N ≅ G 1 ⋊ \mathbbK\mathbb{K} ×. In both of these cases, N is a self-normalizing subgroup of G.      10. Quasiregularly elliptic link complements      Pekka Pankka  Kai Rajala 《Geometriae Dedicata》2011,151(1):1-8 We extend the theorem of B. Daniel about the existence and uniqueness of immersions into \mathbbSn × \mathbbR or \mathbbHn × \mathbbR{\mathbb{S}^{n}\,\times\,\mathbb{R}\, {\rm or}\, \mathbb{H}^{n}\,\times\,\mathbb{R}} to the Riemannian product of two space forms. More precisely, we prove the existence and uniqueness of an isometric immersion of a Riemannian manifold into the Riemannian product of two space forms.      11. Fujita-Liouville Type Theorem for Coupled Fourth-Order Parabolic Inequalities      Zhaoxin JIANG  Sining ZHENG 《数学年刊B辑(英文版)》2012,33(1):107-112 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.      12. Metric Flips with Calabi Ansatz      Jian Song  Yuan Yuan 《Geometric And Functional Analysis》2012,22(1):240-265 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.      13. Vietoris–Rips Complexes of Planar Point Sets      Erin W. Chambers  Vin de Silva  Jeff Erickson  Robert Ghrist 《Discrete and Computational Geometry》2010,44(1):75-90 Fix a finite set of points in Euclidean n-space \mathbbEn\mathbb{E}^{n} , thought of as a point-cloud sampling of a certain domain D ì \mathbbEnD\subset\mathbb{E}^{n} . The Vietoris–Rips complex is a combinatorial simplicial complex based on proximity of neighbors that serves as an easily-computed but high-dimensional approximation to the homotopy type of D. There is a natural “shadow” projection map from the Vietoris–Rips complex to \mathbbEn\mathbb{E}^{n} that has as its image a more accurate n-dimensional approximation to the homotopy type of D.      14. Density of the polynomials in Hardy and Bergman spaces of slit domains      John R. Akeroyd 《Arkiv f?r Matematik》2011,49(1):1-16 It is shown that for any t, 0<t<∞, there is a Jordan arc Γ with endpoints 0 and 1 such that G\{1} í \mathbbD:={z:|z| < 1}\Gamma\setminus\{1\}\subseteq\mathbb{D}:=\{z:|z|<1\} and with the property that the analytic polynomials are dense in the Bergman space \mathbbAt(\mathbbD\G)\mathbb{A}^{t}(\mathbb{D}\setminus\Gamma) . It is also shown that one can go further in the Hardy space setting and find such a Γ that is in fact the graph of a continuous real-valued function on [0,1], where the polynomials are dense in Ht(\mathbbD\G)H^{t}(\mathbb{D}\setminus\Gamma) ; improving upon a result in an earlier paper.      15. Pluripotential Energy      Thomas Bloom  Norman Levenberg 《Potential Analysis》2012,36(1):155-176 We discuss a notion of the energy of a compactly supported measure in \mathbbCn \mathbb{C}^n for n > 1 which we show is equivalent to that defined by Berman, Boucksom, Guedj and Zeriahi. This generalizes the classical notion of logarithmic energy of a measure in the complex plane \mathbbC \mathbb{C} ; i.e., the case n = 1.      16. Absolutely convergent fourier series. An improvement of the Beurling-Helson theorem      V. V. Lebedev 《Functional Analysis and Its Applications》2012,46(2):121-132 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 || einf ||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 || einf ||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.      17. On the dimension of invariant measures of endomorphisms of $${\mathbb{C}\mathbb{P}^k}$$      Christophe Dupont 《Mathematische Annalen》2011,349(3):509-528 Let f be an endomorphism of \mathbbC\mathbbPk{\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 dimHm 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\mathbbPk{\mathbb{C}\mathbb{P}^k} . Our tools are a volume growth estimate for the bounded holomorphic polydiscs in \mathbbC\mathbbPk{\mathbb{C}\mathbb{P}^k} and a normalization theorem for the ν-generic inverse branches of f n .      18. Some presentations of the real number field      A. S. Morozov 《Algebra and Logic》2012,51(1):66-88 It is proved that every two Σ-presentations of an ordered field \mathbbR \mathbb{R} of reals over \mathbbH\mathbbF ( \mathbbR ) \mathbb{H}\mathbb{F}\,\left( \mathbb{R} \right) , whose universes are subsets of \mathbbR \mathbb{R} , are mutually Σ-isomorphic. As a consequence, for a series of functions f:\mathbbR ? \mathbbR f:\mathbb{R} \to \mathbb{R} (e.g., exp, sin, cos, ln), it is stated that the structure \mathbbR \mathbb{R} = 〈R, +, ×, <, 0, 1, f〉 lacks such Σ-presentations over \mathbbH\mathbbF ( \mathbbR ) \mathbb{H}\mathbb{F}\,\left( \mathbb{R} \right) .      19. Exotic smooth structures on small 4-manifolds with odd signatures      Anar Akhmedov  B. Doug Park 《Inventiones Mathematicae》2010,181(3):577-603 Let M be (2n-1)\mathbbCP2#2n[`(\mathbbCP)]2(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 \mathbbCP2#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.      20. Dvoretzky Type Theorems for Multivariate Polynomials and Sections of Convex Bodies      Vladimir L. Dol��nikov  Roman N. Karasev 《Geometric And Functional Analysis》2011,21(2):301-318 In this paper we prove the Gromov–Milman conjecture (the Dvoretzky type theorem) for homogeneous polynomials on \mathbbRn{\mathbb{R}^{n}}, and improve bounds on the number n(d, k) in the analogous conjecture for odd degrees d (this case is known as the Birch theorem) and complex polynomials.

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.     

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.     

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).     

Compact Surfaces with Constant Gaussian Curvature in Product Spaces

We prove that the only compact surfaces of positive constant Gaussian curvature in \mathbbH²×\mathbbR{\mathbb{H}^{2}\times\mathbb{R}} (resp. positive constant Gaussian curvature greater than 1 in \mathbbS²×\mathbbR{\mathbb{S}^{2}\times\mathbb{R}}) whose boundary Γ is contained in a slice of the ambient space and such that the surface intersects this slice at a constant angle along Γ, are the pieces of a rotational complete surface. We also obtain some area estimates for surfaces of positive constant Gaussian curvature in \mathbbH²×\mathbbR{\mathbb{H}^{2}\times\mathbb{R}} and positive constant Gaussian curvature greater than 1 in \mathbbS²×\mathbbR{\mathbb{S}^{2}\times\mathbb{R}} whose boundary is contained in a slice of the ambient space. These estimates are optimal in the sense that if the bounds are attained, the surface is again a piece of a rotational complete surface.     

<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}}.     

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}.     

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).     

Normalizers and self-normalizing subgroups II

Let \mathbbK\mathbb{K} be a field, G a reductive algebraic \mathbbK\mathbb{K}-group, and G ₁ ≤ G a reductive subgroup. For G ₁ ≤ G, the corresponding groups of \mathbbK\mathbb{K}-points, we study the normalizer N = N _G (G ₁). In particular, for a standard embedding of the odd orthogonal group G ₁ = SO(m, \mathbbK\mathbb{K}) in G = SL(m, \mathbbK\mathbb{K}) we have N ≅ G ₁ ⋊ μ _m ( \mathbbK\mathbb{K}), the semidirect product of G ₁ by the group of m-th roots of unity in \mathbbK\mathbb{K}. The normalizers of the even orthogonal and symplectic subgroup of SL(2n, \mathbbK\mathbb{K}) were computed in [Širola B., Normalizers and self-normalizing subgroups, Glas. Mat. Ser. III (in press)], leaving the proof in the odd orthogonal case to be completed here. Also, for G = GL(m, \mathbbK\mathbb{K}) and G ₁ = O(m, \mathbbK\mathbb{K}) we have N ≅ G ₁ ⋊ \mathbbK\mathbb{K} ^×. In both of these cases, N is a self-normalizing subgroup of G.     

Quasiregularly elliptic link complements

We extend the theorem of B. Daniel about the existence and uniqueness of immersions into \mathbbSⁿ × \mathbbR or \mathbbHⁿ × \mathbbR{\mathbb{S}^{n}\,\times\,\mathbb{R}\, {\rm or}\, \mathbb{H}^{n}\,\times\,\mathbb{R}} to the Riemannian product of two space forms. More precisely, we prove the existence and uniqueness of an isometric immersion of a Riemannian manifold into the Riemannian product of two space forms.     

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.     

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.     

Vietoris–Rips Complexes of Planar Point Sets

Fix a finite set of points in Euclidean n-space \mathbbEⁿ\mathbb{E}^{n} , thought of as a point-cloud sampling of a certain domain D ì \mathbbEⁿD\subset\mathbb{E}^{n} . The Vietoris–Rips complex is a combinatorial simplicial complex based on proximity of neighbors that serves as an easily-computed but high-dimensional approximation to the homotopy type of D. There is a natural “shadow” projection map from the Vietoris–Rips complex to \mathbbEⁿ\mathbb{E}^{n} that has as its image a more accurate n-dimensional approximation to the homotopy type of D.     

Density of the polynomials in Hardy and Bergman spaces of slit domains

It is shown that for any t, 0<t<∞, there is a Jordan arc Γ with endpoints 0 and 1 such that G\{1} í \mathbbD:={z:|z| < 1}\Gamma\setminus\{1\}\subseteq\mathbb{D}:=\{z:|z|<1\} and with the property that the analytic polynomials are dense in the Bergman space \mathbbA^t(\mathbbD\G)\mathbb{A}^{t}(\mathbb{D}\setminus\Gamma) . It is also shown that one can go further in the Hardy space setting and find such a Γ that is in fact the graph of a continuous real-valued function on [0,1], where the polynomials are dense in H^t(\mathbbD\G)H^{t}(\mathbb{D}\setminus\Gamma) ; improving upon a result in an earlier paper.     

Pluripotential Energy

We discuss a notion of the energy of a compactly supported measure in \mathbbCⁿ \mathbb{C}^n for n > 1 which we show is equivalent to that defined by Berman, Boucksom, Guedj and Zeriahi. This generalizes the classical notion of logarithmic energy of a measure in the complex plane \mathbbC \mathbb{C} ; i.e., the case n = 1.     

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.     

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 .     

Some presentations of the real number field

It is proved that every two Σ-presentations of an ordered field \mathbbR \mathbb{R} of reals over \mathbbH\mathbbF ( \mathbbR ) \mathbb{H}\mathbb{F}\,\left( \mathbb{R} \right) , whose universes are subsets of \mathbbR \mathbb{R} , are mutually Σ-isomorphic. As a consequence, for a series of functions f:\mathbbR ? \mathbbR f:\mathbb{R} \to \mathbb{R} (e.g., exp, sin, cos, ln), it is stated that the structure \mathbbR \mathbb{R} = 〈R, +, ×, <, 0, 1, f〉 lacks such Σ-presentations over \mathbbH\mathbbF ( \mathbbR ) \mathbb{H}\mathbb{F}\,\left( \mathbb{R} \right) .     

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.     

Dvoretzky Type Theorems for Multivariate Polynomials and Sections of Convex Bodies

In this paper we prove the Gromov–Milman conjecture (the Dvoretzky type theorem) for homogeneous polynomials on \mathbbRⁿ{\mathbb{R}^{n}}, and improve bounds on the number n(d, k) in the analogous conjecture for odd degrees d (this case is known as the Birch theorem) and complex polynomials.