Green’s relations and regularity for semigroups of transformations that preserve order and a double direction equivalence

Let T _X be the full transformation semigroup on a set X,

Properties of Riesz fractional derivatives for Korteweg–de Vries solitons

Riesz fractional derivatives are defined as fractional powers of the Laplacian, D ^α  = (?Δ) ^α/2 for ${\alpha \in \mathbb{R}}Riesz fractional derivatives are defined as fractional powers of the Laplacian, D ^α  = (−Δ) ^α/2 for a ? \mathbbR{\alpha \in \mathbb{R}}. For the soliton solution of the Korteweg–de Vries equation, u ₀(X) with X = x − 4t, these derivatives, u _α (X) = D ^α u ₀(X), and their Hilbert transforms, v _α (X) = −HD ^α u ₀(X), can be expressed in terms of the full range Hurwitz Zeta functions ζ₊(s, a) and ζ₋(s, a), respectively. New properties are established for u _α (X) and v _α (X). It is proved that the functions w _α (X) = u _α (X) + iv _α (X) with α > −1 are solutions of the differential equation

-\fracddX(Pa(X)\fracdwdX)+Qa(X)w = lra(X)w,       X ? \mathbbR,-\frac{\rm d}{{\rm d}X}\left(P_{\alpha}(X)\frac{{\rm d}w}{{\rm d}X}\right)+Q_{\alpha}(X)w = \lambda\rho_{\alpha}(X)w,\qquad X \in \mathbb{R},      3. Operator-valued H ∞-calculus in Inter- and Extrapolation Spaces      Markus Haase 《Integral Equations and Operator Theory》2006,56(2):197-228 We generalize a Hilbert space result by Auscher, McIntosh and Nahmod to arbitrary Banach spaces X and to not densely defined injective sectorial operators A. A convenient tool proves to be a certain universal extrapolation space associated with A. We characterize the real interpolation space ( X,D( Aa ) ?R( Aa ) )q,p{\left( {X,\mathcal{D}{\left( {A^{\alpha } } \right)} \cap \mathcal{R}{\left( {A^{\alpha } } \right)}} \right)}_{{\theta ,p}} as { x  ?  X|t - q\textRea y1 ( tA )x, t - q\textRea y2 ( tA )x ? L*p ( ( 0,￥ );X ) } {\left\{ {x\, \in \,X|t^{{ - \theta {\text{Re}}\alpha }} \psi _{1} {\left( {tA} \right)}x,\,t^{{ - \theta {\text{Re}}\alpha }} \psi _{2} {\left( {tA} \right)}x \in L_{*}^{p} {\left( {{\left( {0,\infty } \right)};X} \right)}} \right\}}      4. Well-Posedness of Fractional Differential Equations on Vector-Valued Function Spaces      Shangquan Bu 《Integral Equations and Operator Theory》2011,71(2):259-274 We study the well-posedness of the fractional differential equations with infinite delay (P 2): Da u(t)=Au(t)+òt-￥a(t-s)Au(s)ds + f(t), (0 ￡ t ￡ 2p){D^\alpha u(t)=Au(t)+\int^{t}_{-\infty}a(t-s)Au(s)ds + f(t), (0\leq t \leq2\pi)}, where A is a closed operator in a Banach space ${X, \alpha > 0, a\in {L}^1(\mathbb{R}_+)}${X, \alpha > 0, a\in {L}^1(\mathbb{R}_+)} and f is an X-valued function. Under suitable assumptions on the parameter α and the Laplace transform of a, we completely characterize the well-posedness of (P 2) on Lebesgue-Bochner spaces Lp(\mathbbT, X){L^p(\mathbb{T}, X)} and periodic Besov spaces B p,qs(\mathbbT, X){{B} _{p,q}^s(\mathbb{T}, X)} .      5. The noncommutative Choquet boundary II: Hyperrigidity      William Arveson 《Israel Journal of Mathematics》2011,184(1):349-385 A (finite or countably infinite) set G of generators of an abstract C*-algebra A is called hyperrigid if for every faithful representation of A on a Hilbert space A ⊆ B(H) and every sequence of unital completely positive linear maps ϕ 1, ϕ 2,... from B(H) to itself, limn ? ￥ ||fn(g) - g|| = 0,"g ? G T limn ? ￥ fn(a) - a|| = 0,"a ? A.\mathop {\lim }\limits_{n \to \infty } ||{\phi _n}(g) - g|| = 0,{\forall _g} \in G \Rightarrow \mathop {\lim }\limits_{n \to \infty } {\phi _n}(a) - a|| = 0,{\forall _a} \in A.      6. Maps Between Uniform Algebras Preserving Norms of Rational Functions      Rumi Shindo 《Mediterranean Journal of Mathematics》2011,8(1):81-95 Let A, B be uniform algebras. Suppose that A 0, B 0 are subgroups of A −1, B −1 that contain exp A, exp B respectively. Let α be a non-zero complex number. Suppose that m, n are non-zero integers and d is the greatest common divisor of m and n. If T : A 0 → B 0 is a surjection with ||T(f)mT(g)n - a||￥ = ||fmgn - a||￥{\|T(f)^{m}T(g)^{n} - \alpha\|_{\infty} = \|f^{m}g^{n} - \alpha\|_{\infty}} for all f,g ? A0{f,g \in A_0}, then there exists a real-algebra isomorphism [(T)\tilde] : A ? B{\tilde{T} : A \rightarrow B} such that [(T)\tilde](f)d = (T(f)/T(1))d{\tilde{T}(f)^d = (T(f)/T(1))^d} for every f ? A0{f \in A_0}. This result leads to the following assertion: Suppose that S A , S B are subsets of A, B that contain A −1, B −1 respectively. If m, n > 0 and a surjection T : S A → S B satisfies ||T(f)mT(g)n - a||￥ = ||fmgn - a||￥{\|T(f)^{m}T(g)^{n} - \alpha\|_{\infty} = \|f^{m}g^{n} - \alpha\|_{\infty}} for all f, g ? SA{f, g \in S_A}, then there exists a real-algebra isomorphism [(T)\tilde] : A ? B{\tilde{T} : A \rightarrow B} such that [(T)\tilde](f)d = (T(f)/T(1))d{\tilde{T}(f)^d = (T(f)/T(1))^d} for every f ? SA{f \in S_A}. Note that in these results and elsewhere in this paper we do not assume that T(exp A) = exp B.      7. Numerical verification of condition for approximately midconvex functions      P. Spurek  Ja. Tabor 《Aequationes Mathematicae》2012,83(3):223-237 Let X be a normed space and V be a convex subset of X. Let a\colon \mathbbR+ ? \mathbbR+{\alpha \colon \mathbb{R}_+ \to \mathbb{R}_+}. A function f \colon V ? \mathbbR{f \colon V \to \mathbb{R}} is called α-midconvex if f (\fracx + y2)-\fracf(x) + f(y)2 ￡ a(||x - y||)    for  x, y ? V.f \left(\frac{x + y}{2}\right)-\frac{f(x) + f(y)}{2}\leq \alpha(\|x - y\|)\quad {\rm for} \, x, y \in V.      8. Bi-Parametric Potentials, Relevant Function Spaces and Wavelet-Like Transforms      Ilham A. Aliev 《Integral Equations and Operator Theory》2009,65(2):151-167 We introduce new potential type operators Jab = (E+(-D)b/2)-a/bJ^{\alpha}_{\beta} = (E+(-\Delta)^{\beta/2})^{-\alpha/\beta}, (α > 0, β > 0), and bi-parametric scale of function spaces Hab, p(\mathbbRn)H^{\alpha}_{\beta , p}({\mathbb{R}}^n) associated with Jαβ. These potentials generalize the classical Bessel potentials (for β = 2), and Flett potentials (for β = 1). A characterization of the spaces Hab, p(\mathbbRn)H^{\alpha}_{\beta, p}({\mathbb{R}}^n) is given with the aid of a special wavelet–like transform associated with a β-semigroup, which generalizes the well-known Gauss-Weierstrass semigroup (for β = 2) and the Poisson one (for β = 1).      9. Green’s relations and regularity for semigroups of transformations that preserve double direction equivalence   总被引：1，自引：0，他引：1   Lun-Zhi Deng  Ji-Wen Zeng  Bo Xu 《Semigroup Forum》2010,80(3):416-425 Let T X denote the full transformation semigroup on a set X. For an equivalence E on X, let $T_{E^*}(X)=\{\alpha\in T_X:\forall x,y\in X,(x,y)\in E\Leftrightarrow(x\alpha,y\alpha)\in E\}.$ Then $T_{E^{*}}(X)Let T X denote the full transformation semigroup on a set X. For an equivalence E on X, let TE*(X)={a ? TX:"x,y ? X,(x,y) ? E?(xa,ya) ? E}.T_{E^*}(X)=\{\alpha\in T_X:\forall x,y\in X,(x,y)\in E\Leftrightarrow(x\alpha,y\alpha)\in E\}.      10. Boxicity of Circular Arc Graphs      Diptendu Bhowmick  L. Sunil Chandran 《Graphs and Combinatorics》2011,27(6):769-783 A k-dimensional box is a Cartesian product R 1 × · · · × R k where each R i is a closed interval on the real line. The boxicity of a graph G, denoted as box(G), is the minimum integer k such that G can be represented as the intersection graph of a collection of k-dimensional boxes. That is, two vertices are adjacent if and only if their corresponding boxes intersect. A circular arc graph is a graph that can be represented as the intersection graph of arcs on a circle. We show that if G is a circular arc graph which admits a circular arc representation in which no arc has length at least p(\fraca-1a){\pi(\frac{\alpha-1}{\alpha})} for some a ? \mathbbN 3 2{\alpha\in\mathbb{N}_{\geq 2}}, then box(G) ≤ α (Here the arcs are considered with respect to a unit circle). From this result we show that if G has maximum degree D < ?\fracn(a-1)2a?{\Delta < \lfloor{\frac{n(\alpha-1)}{2\alpha}}\rfloor} for some a ? \mathbbN 3 2{\alpha \in \mathbb{N}_{\geq 2}}, then box(G) ≤ α. We also demonstrate a graph having box(G) > α but with D = n\frac(a-1)2a+ \fracn2a(a+1)+(a+2){\Delta=n\frac{(\alpha-1)}{2\alpha}+ \frac{n}{2\alpha(\alpha+1)}+(\alpha+2)}. For a proper circular arc graph G, we show that if D < ?\fracn(a-1)a?{\Delta < \lfloor{\frac{n(\alpha-1)}{\alpha}}\rfloor} for some a ? \mathbbN 3 2{\alpha\in \mathbb{N}_{\geq 2}}, then box(G) ≤ α. Let r be the cardinality of the minimum overlap set, i.e. the minimum number of arcs passing through any point on the circle, with respect to some circular arc representation of G. We show that for any circular arc graph G, box(G) ≤ r + 1 and this bound is tight. We show that if G admits a circular arc representation in which no family of k ≤ 3 arcs covers the circle, then box(G) ≤ 3 and if G admits a circular arc representation in which no family of k ≤ 4 arcs covers the circle, then box(G) ≤ 2. We also show that both these bounds are tight.      11. Rectifiability of metric flat chains and fractional masses      E. Stepanov 《Journal of Mathematical Sciences》2010,167(3):406-417 We prove that every real flat chain T of finite mass in a complete separable metric space E is rectifiable when \mathbbMa (T) < + ￥ {\mathbb{M}^\alpha }(T) < + \infty for some α ∈ [0, 1), where \mathbbMa (T) {\mathbb{M}^\alpha }(T) is the α-mass of T. Bibliography: 12 titles.      12. Generalized uncertainty inequalities      Alessio Martini 《Mathematische Zeitschrift》2010,265(4):831-848 The Heisenberg–Pauli–Weyl (HPW) uncertainty inequality on \mathbbRn{\mathbb{R}^n} says that || f ||2 ￡ Ca,b|| |x|a f||2\fracba+b|| (-D)b/2f||2\fracaa+b.\| f \|_2 \leq C_{\alpha,\beta}\| |x|^\alpha f\|_2^\frac{\beta}{\alpha+\beta}\| (-\Delta)^{\beta/2}f\|_2^\frac{\alpha}{\alpha+\beta}.      13. Actions of symbolic dynamical systems on C*-algebras. II. Simplicity of C*-symbolic crossed products and some examples      Kengo Matsumoto 《Mathematische Zeitschrift》2010,265(4):735-760 A C*-symbolic dynamical system ${(\mathcal{A}, \rho, \Sigma)}A C*-symbolic dynamical system (A, r, S){(\mathcal{A}, \rho, \Sigma)} consists of a unital C*-algebra A{\mathcal{A}} and a finite family { ra }a ? S{\{ \rho_\alpha \}_{\alpha \in \Sigma}} of endomorphisms ρ α of A{\mathcal{A}} indexed by symbols α of Σ satisfying some conditions. The endomorphisms ra, a ? S{\rho_\alpha, \alpha \in \Sigma } yield both a subshift Λ and a C*-algebra of a Hilbert C*-bimodule. The obtained C*-algebra is regarded as a crossed product of A{\mathcal{A}} by the subshift Λ. We will study simplicity condition of these C*-algebras. Some examples such as irrational rotation Cuntz–Krieger algebras will be studied.      14. Packing dimension and Ahlfors regularity of porous sets in metric spaces      Esa Järvenpää  Maarit Järvenpää  Antti Käenmäki  Tapio Rajala  Sari Rogovin  Ville Suomala 《Mathematische Zeitschrift》2010,266(1):83-105 Let X be a metric measure space with an s-regular measure μ. We prove that if A ì X{A\subset X} is r{\varrho} -porous, then dimp(A) ￡ s-crs{{\rm {dim}_p}(A)\le s-c\varrho^s} where dimp is the packing dimension and c is a positive constant which depends on s and the structure constants of μ. This is an analogue of a well known asymptotically sharp result in Euclidean spaces. We illustrate by an example that the corresponding result is not valid if μ is a doubling measure. However, in the doubling case we find a fixed N ì X{N\subset X} with μ(N) = 0 such that dimp(A) ￡ dimp(X)-c(log\tfrac1r)-1rt{{\rm {dim}_p}(A)\le{\rm {dim}_p}(X)-c(\log \tfrac1\varrho)^{-1}\varrho^t} for all r{\varrho} -porous sets A ì X\ N{A \subset X{\setminus} N} . Here c and t are constants which depend on the structure constant of μ. Finally, we characterize uniformly porous sets in complete s-regular metric spaces in terms of regular sets by verifying that A is uniformly porous if and only if there is t < s and a t-regular set F such that A ì F{A\subset F} .      15. Implicit standard Jacobi gives high relative accuracy      Froilán M. Dopico  Plamen Koev  Juan M. Molera 《Numerische Mathematik》2009,113(4):519-553 We prove that the Jacobi algorithm applied implicitly on a decomposition A = XDX T of the symmetric matrix A, where D is diagonal, and X is well conditioned, computes all eigenvalues of A to high relative accuracy. The relative error in every eigenvalue is bounded by O(ek(X)){O(\epsilon \kappa (X))} , where e{\epsilon} is the machine precision and k(X) o ||X||2·||X-1||2{\kappa(X)\equiv\|X\|_2\cdot\|X^{-1}\|_2} is the spectral condition number of X. The eigenvectors are also computed accurately in the appropriate sense. We believe that this is the first algorithm to compute accurate eigenvalues of symmetric (indefinite) matrices that respects and preserves the symmetry of the problem and uses only orthogonal transformations.      16. A Generalized Radon Transform on the Plane      Zhongkai Li  Futao Song 《Constructive Approximation》2011,33(1):93-123 A new generalized Radon transform R α, β on the plane for functions even in each variable is defined which has natural connections with the bivariate Hankel transform, the generalized biaxially symmetric potential operator Δ α, β , and the Jacobi polynomials Pk(b, a)(t)P_{k}^{(\beta,\,\alpha)}(t). The transform R α, β and its dual Ra, b*R_{\alpha,\,\beta}^{\ast} are studied in a systematic way, and in particular, the generalized Fuglede formula and some inversion formulas for R α, β for functions in La, bp(\mathbbR2+)L_{\alpha,\,\beta}^{p}(\mathbb{R}^{2}_{+}) are obtained in terms of the bivariate Hankel–Riesz potential. Moreover, the transform R α, β is used to represent the solutions of the partial differential equations Lu:=?j=1majDa, bju=fLu:=\sum_{j=1}^{m}a_{j}\Delta_{\alpha,\,\beta}^{j}u=f with constant coefficients a j and the Cauchy problem for the generalized wave equation associated with the operator Δ α, β . Another application is that, by an invariant property of R α, β , a new product formula for the Jacobi polynomials of the type Pk(b, a)(s)C2ka+b+1(t)=còòPk(b, a)P_{k}^{(\beta,\,\alpha)}(s)C_{2k}^{\alpha+\beta+1}(t)=c\int\!\!\int P_{k}^{(\beta,\,\alpha)} is obtained.      17. Growth of permutational extensions      Laurent?BartholdiEmail author  Anna?Erschler 《Inventiones Mathematicae》2012,189(2):431-455 We study the geometry of a class of group extensions, containing permutational wreath products, which we call “permutational extensions”. We construct for all k∈ℕ a torsion group K k with growth function vKk(n) ~ exp(n1-(1-a)k),       23-3/a+22-2/a+21-1/a=2,v_{K_k}(n)\sim\exp(n^{1-(1-\alpha)^k}),\qquad 2^{3-3/\alpha}+2^{2-2/\alpha}+2^{1-1/\alpha}=2,      18. Uniform rates of approximation by short asymptotic expansions in the CLT for quadratic forms      F. Götze  A. Yu. Zaitsev 《Journal of Mathematical Sciences》2011,176(2):162-189 Let X, X 1, X 2,… be i.i.d. \mathbbRd {\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[ SN ] \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, DN(a) = supx | \textP{ \mathbbQ[ SN - a ] \leqslant x } - \textP{ \mathbbQ[ G - a ] \leqslant x } - Ea(x) | = O( N - 1 ) \Delta_N^{(a)} = \mathop {{\sup }}\limits_x \left| {{\text{P}}\left\{ {\mathbb{Q}\left[ {{S_N} - a} \right] \leqslant x} \right\} - {\text{P}}\left\{ {\mathbb{Q}\left[ {G - a} \right] \leqslant x} \right\} - {E_a}(x)} \right| = \mathcal{O}\left( {{N^{ - 1}}} \right)      19. A categorification of Morelli’s theorem      Bohan Fang  Chiu-Chu Melissa Liu  David Treumann  Eric Zaslow 《Inventiones Mathematicae》2011,186(1):79-114       20. Clifford Analysis Techniques for Spherical PDE      Hong Liu  John Ryan 《Journal of Fourier Analysis and Applications》2002,8(6):535-564 For a ? R\alpha \in \mathbf{R}, the class of a-\alpha -order spherical harmonic functions in an open set W í\Omega \subseteq Sn-1\mathbf{S}^{n-1}, Ha(W)H^{\alpha }(\Omega ) is defined as the C2-C^{2}-solutions of Dau=0\Delta _{\alpha }u=0; where Da=Ds+a(n+a-2)\Delta _{\alpha }=\Delta _{s}+\alpha (n+\alpha -2) is the spherical Laplace--Beltrami operator of order a\alpha and Ds\Delta _{s} is the radially independent part of the Laplace operator. We obtain a Green's integral formula for the functions in Ha(W)H^{\alpha }(\Omega ) with kernel expressed as a Gegenbauer function. As generalizations, higher order spherical iterated Dirac operators are defined in a polynomial form. Integral representations of the null solutions to these operators and an intertwining formula relating these operators on the sphere and their analogues in Euclidean space are presented.      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Operator-valued H ∞-calculus in Inter- and Extrapolation Spaces

We generalize a Hilbert space result by Auscher, McIntosh and Nahmod to arbitrary Banach spaces X and to not densely defined injective sectorial operators A. A convenient tool proves to be a certain universal extrapolation space associated with A. We characterize the real interpolation space ( X,D( A^a ) ?R( A^a ) )_q,p{\left( {X,\mathcal{D}{\left( {A^{\alpha } } \right)} \cap \mathcal{R}{\left( {A^{\alpha } } \right)}} \right)}_{{\theta ,p}} as

Well-Posedness of Fractional Differential Equations on Vector-Valued Function Spaces

We study the well-posedness of the fractional differential equations with infinite delay (P ₂): D^a u(t)=Au(t)+ò^t_-￥a(t-s)Au(s)ds + f(t), (0 ￡ t ￡ 2p){D^\alpha u(t)=Au(t)+\int^{t}_{-\infty}a(t-s)Au(s)ds + f(t), (0\leq t \leq2\pi)}, where A is a closed operator in a Banach space ${X, \alpha > 0, a\in {L}^1(\mathbb{R}_+)}${X, \alpha > 0, a\in {L}^1(\mathbb{R}_+)} and f is an X-valued function. Under suitable assumptions on the parameter α and the Laplace transform of a, we completely characterize the well-posedness of (P ₂) on Lebesgue-Bochner spaces L^p(\mathbbT, X){L^p(\mathbb{T}, X)} and periodic Besov spaces B _p,q^s(\mathbbT, X){{B} _{p,q}^s(\mathbb{T}, X)} .     

The noncommutative Choquet boundary II: Hyperrigidity

A (finite or countably infinite) set G of generators of an abstract C*-algebra A is called hyperrigid if for every faithful representation of A on a Hilbert space A ⊆ B(H) and every sequence of unital completely positive linear maps ϕ ₁, ϕ ₂,... from B(H) to itself,

Maps Between Uniform Algebras Preserving Norms of Rational Functions

Let A, B be uniform algebras. Suppose that A ₀, B ₀ are subgroups of A ⁻¹, B ⁻¹ that contain exp A, exp B respectively. Let α be a non-zero complex number. Suppose that m, n are non-zero integers and d is the greatest common divisor of m and n. If T : A ₀ → B ₀ is a surjection with ||T(f)^mT(g)ⁿ - a||_￥ = ||f^mgⁿ - a||_￥{\|T(f)^{m}T(g)^{n} - \alpha\|_{\infty} = \|f^{m}g^{n} - \alpha\|_{\infty}} for all f,g ? A₀{f,g \in A_0}, then there exists a real-algebra isomorphism [(T)\tilde] : A ? B{\tilde{T} : A \rightarrow B} such that [(T)\tilde](f)^d = (T(f)/T(1))^d{\tilde{T}(f)^d = (T(f)/T(1))^d} for every f ? A₀{f \in A_0}. This result leads to the following assertion: Suppose that S _A , S _B are subsets of A, B that contain A ⁻¹, B ⁻¹ respectively. If m, n > 0 and a surjection T : S _A → S _B satisfies ||T(f)^mT(g)ⁿ - a||_￥ = ||f^mgⁿ - a||_￥{\|T(f)^{m}T(g)^{n} - \alpha\|_{\infty} = \|f^{m}g^{n} - \alpha\|_{\infty}} for all f, g ? S_A{f, g \in S_A}, then there exists a real-algebra isomorphism [(T)\tilde] : A ? B{\tilde{T} : A \rightarrow B} such that [(T)\tilde](f)^d = (T(f)/T(1))^d{\tilde{T}(f)^d = (T(f)/T(1))^d} for every f ? S_A{f \in S_A}. Note that in these results and elsewhere in this paper we do not assume that T(exp A) = exp B.     

Numerical verification of condition for approximately midconvex functions

Let X be a normed space and V be a convex subset of X. Let a\colon \mathbbR₊ ? \mathbbR₊{\alpha \colon \mathbb{R}_+ \to \mathbb{R}_+}. A function f \colon V ? \mathbbR{f \colon V \to \mathbb{R}} is called α-midconvex if

Bi-Parametric Potentials, Relevant Function Spaces and Wavelet-Like Transforms

We introduce new potential type operators J^a_b = (E+(-D)^b/2)^-a/bJ^{\alpha}_{\beta} = (E+(-\Delta)^{\beta/2})^{-\alpha/\beta}, (α > 0, β > 0), and bi-parametric scale of function spaces H^a_{b, p}(\mathbbRⁿ)H^{\alpha}_{\beta , p}({\mathbb{R}}^n) associated with J^α_β. These potentials generalize the classical Bessel potentials (for β = 2), and Flett potentials (for β = 1). A characterization of the spaces H^a_{b, p}(\mathbbRⁿ)H^{\alpha}_{\beta, p}({\mathbb{R}}^n) is given with the aid of a special wavelet–like transform associated with a β-semigroup, which generalizes the well-known Gauss-Weierstrass semigroup (for β = 2) and the Poisson one (for β = 1).     

Green’s relations and regularity for semigroups of transformations that preserve double direction equivalence

Let T _X denote the full transformation semigroup on a set X. For an equivalence E on X, let $T_{E^*}(X)=\{\alpha\in T_X:\forall x,y\in X,(x,y)\in E\Leftrightarrow(x\alpha,y\alpha)\in E\}.$ Then $T_{E^{*}}(X)Let T _X denote the full transformation semigroup on a set X. For an equivalence E on X, let

TE*(X)={a ? TX:"x,y ? X,(x,y) ? E?(xa,ya) ? E}.T_{E^*}(X)=\{\alpha\in T_X:\forall x,y\in X,(x,y)\in E\Leftrightarrow(x\alpha,y\alpha)\in E\}.      10. Boxicity of Circular Arc Graphs      Diptendu Bhowmick  L. Sunil Chandran 《Graphs and Combinatorics》2011,27(6):769-783 A k-dimensional box is a Cartesian product R 1 × · · · × R k where each R i is a closed interval on the real line. The boxicity of a graph G, denoted as box(G), is the minimum integer k such that G can be represented as the intersection graph of a collection of k-dimensional boxes. That is, two vertices are adjacent if and only if their corresponding boxes intersect. A circular arc graph is a graph that can be represented as the intersection graph of arcs on a circle. We show that if G is a circular arc graph which admits a circular arc representation in which no arc has length at least p(\fraca-1a){\pi(\frac{\alpha-1}{\alpha})} for some a ? \mathbbN 3 2{\alpha\in\mathbb{N}_{\geq 2}}, then box(G) ≤ α (Here the arcs are considered with respect to a unit circle). From this result we show that if G has maximum degree D < ?\fracn(a-1)2a?{\Delta < \lfloor{\frac{n(\alpha-1)}{2\alpha}}\rfloor} for some a ? \mathbbN 3 2{\alpha \in \mathbb{N}_{\geq 2}}, then box(G) ≤ α. We also demonstrate a graph having box(G) > α but with D = n\frac(a-1)2a+ \fracn2a(a+1)+(a+2){\Delta=n\frac{(\alpha-1)}{2\alpha}+ \frac{n}{2\alpha(\alpha+1)}+(\alpha+2)}. For a proper circular arc graph G, we show that if D < ?\fracn(a-1)a?{\Delta < \lfloor{\frac{n(\alpha-1)}{\alpha}}\rfloor} for some a ? \mathbbN 3 2{\alpha\in \mathbb{N}_{\geq 2}}, then box(G) ≤ α. Let r be the cardinality of the minimum overlap set, i.e. the minimum number of arcs passing through any point on the circle, with respect to some circular arc representation of G. We show that for any circular arc graph G, box(G) ≤ r + 1 and this bound is tight. We show that if G admits a circular arc representation in which no family of k ≤ 3 arcs covers the circle, then box(G) ≤ 3 and if G admits a circular arc representation in which no family of k ≤ 4 arcs covers the circle, then box(G) ≤ 2. We also show that both these bounds are tight.      11. Rectifiability of metric flat chains and fractional masses      E. Stepanov 《Journal of Mathematical Sciences》2010,167(3):406-417 We prove that every real flat chain T of finite mass in a complete separable metric space E is rectifiable when \mathbbMa (T) < + ￥ {\mathbb{M}^\alpha }(T) < + \infty for some α ∈ [0, 1), where \mathbbMa (T) {\mathbb{M}^\alpha }(T) is the α-mass of T. Bibliography: 12 titles.      12. Generalized uncertainty inequalities      Alessio Martini 《Mathematische Zeitschrift》2010,265(4):831-848 The Heisenberg–Pauli–Weyl (HPW) uncertainty inequality on \mathbbRn{\mathbb{R}^n} says that || f ||2 ￡ Ca,b|| |x|a f||2\fracba+b|| (-D)b/2f||2\fracaa+b.\| f \|_2 \leq C_{\alpha,\beta}\| |x|^\alpha f\|_2^\frac{\beta}{\alpha+\beta}\| (-\Delta)^{\beta/2}f\|_2^\frac{\alpha}{\alpha+\beta}.      13. Actions of symbolic dynamical systems on C*-algebras. II. Simplicity of C*-symbolic crossed products and some examples      Kengo Matsumoto 《Mathematische Zeitschrift》2010,265(4):735-760 A C*-symbolic dynamical system ${(\mathcal{A}, \rho, \Sigma)}A C*-symbolic dynamical system (A, r, S){(\mathcal{A}, \rho, \Sigma)} consists of a unital C*-algebra A{\mathcal{A}} and a finite family { ra }a ? S{\{ \rho_\alpha \}_{\alpha \in \Sigma}} of endomorphisms ρ α of A{\mathcal{A}} indexed by symbols α of Σ satisfying some conditions. The endomorphisms ra, a ? S{\rho_\alpha, \alpha \in \Sigma } yield both a subshift Λ and a C*-algebra of a Hilbert C*-bimodule. The obtained C*-algebra is regarded as a crossed product of A{\mathcal{A}} by the subshift Λ. We will study simplicity condition of these C*-algebras. Some examples such as irrational rotation Cuntz–Krieger algebras will be studied.      14. Packing dimension and Ahlfors regularity of porous sets in metric spaces      Esa Järvenpää  Maarit Järvenpää  Antti Käenmäki  Tapio Rajala  Sari Rogovin  Ville Suomala 《Mathematische Zeitschrift》2010,266(1):83-105 Let X be a metric measure space with an s-regular measure μ. We prove that if A ì X{A\subset X} is r{\varrho} -porous, then dimp(A) ￡ s-crs{{\rm {dim}_p}(A)\le s-c\varrho^s} where dimp is the packing dimension and c is a positive constant which depends on s and the structure constants of μ. This is an analogue of a well known asymptotically sharp result in Euclidean spaces. We illustrate by an example that the corresponding result is not valid if μ is a doubling measure. However, in the doubling case we find a fixed N ì X{N\subset X} with μ(N) = 0 such that dimp(A) ￡ dimp(X)-c(log\tfrac1r)-1rt{{\rm {dim}_p}(A)\le{\rm {dim}_p}(X)-c(\log \tfrac1\varrho)^{-1}\varrho^t} for all r{\varrho} -porous sets A ì X\ N{A \subset X{\setminus} N} . Here c and t are constants which depend on the structure constant of μ. Finally, we characterize uniformly porous sets in complete s-regular metric spaces in terms of regular sets by verifying that A is uniformly porous if and only if there is t < s and a t-regular set F such that A ì F{A\subset F} .      15. Implicit standard Jacobi gives high relative accuracy      Froilán M. Dopico  Plamen Koev  Juan M. Molera 《Numerische Mathematik》2009,113(4):519-553 We prove that the Jacobi algorithm applied implicitly on a decomposition A = XDX T of the symmetric matrix A, where D is diagonal, and X is well conditioned, computes all eigenvalues of A to high relative accuracy. The relative error in every eigenvalue is bounded by O(ek(X)){O(\epsilon \kappa (X))} , where e{\epsilon} is the machine precision and k(X) o ||X||2·||X-1||2{\kappa(X)\equiv\|X\|_2\cdot\|X^{-1}\|_2} is the spectral condition number of X. The eigenvectors are also computed accurately in the appropriate sense. We believe that this is the first algorithm to compute accurate eigenvalues of symmetric (indefinite) matrices that respects and preserves the symmetry of the problem and uses only orthogonal transformations.      16. A Generalized Radon Transform on the Plane      Zhongkai Li  Futao Song 《Constructive Approximation》2011,33(1):93-123 A new generalized Radon transform R α, β on the plane for functions even in each variable is defined which has natural connections with the bivariate Hankel transform, the generalized biaxially symmetric potential operator Δ α, β , and the Jacobi polynomials Pk(b, a)(t)P_{k}^{(\beta,\,\alpha)}(t). The transform R α, β and its dual Ra, b*R_{\alpha,\,\beta}^{\ast} are studied in a systematic way, and in particular, the generalized Fuglede formula and some inversion formulas for R α, β for functions in La, bp(\mathbbR2+)L_{\alpha,\,\beta}^{p}(\mathbb{R}^{2}_{+}) are obtained in terms of the bivariate Hankel–Riesz potential. Moreover, the transform R α, β is used to represent the solutions of the partial differential equations Lu:=?j=1majDa, bju=fLu:=\sum_{j=1}^{m}a_{j}\Delta_{\alpha,\,\beta}^{j}u=f with constant coefficients a j and the Cauchy problem for the generalized wave equation associated with the operator Δ α, β . Another application is that, by an invariant property of R α, β , a new product formula for the Jacobi polynomials of the type Pk(b, a)(s)C2ka+b+1(t)=còòPk(b, a)P_{k}^{(\beta,\,\alpha)}(s)C_{2k}^{\alpha+\beta+1}(t)=c\int\!\!\int P_{k}^{(\beta,\,\alpha)} is obtained.      17. Growth of permutational extensions      Laurent?BartholdiEmail author  Anna?Erschler 《Inventiones Mathematicae》2012,189(2):431-455 We study the geometry of a class of group extensions, containing permutational wreath products, which we call “permutational extensions”. We construct for all k∈ℕ a torsion group K k with growth function vKk(n) ~ exp(n1-(1-a)k),       23-3/a+22-2/a+21-1/a=2,v_{K_k}(n)\sim\exp(n^{1-(1-\alpha)^k}),\qquad 2^{3-3/\alpha}+2^{2-2/\alpha}+2^{1-1/\alpha}=2,      18. Uniform rates of approximation by short asymptotic expansions in the CLT for quadratic forms      F. Götze  A. Yu. Zaitsev 《Journal of Mathematical Sciences》2011,176(2):162-189 Let X, X 1, X 2,… be i.i.d. \mathbbRd {\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[ SN ] \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, DN(a) = supx | \textP{ \mathbbQ[ SN - a ] \leqslant x } - \textP{ \mathbbQ[ G - a ] \leqslant x } - Ea(x) | = O( N - 1 ) \Delta_N^{(a)} = \mathop {{\sup }}\limits_x \left| {{\text{P}}\left\{ {\mathbb{Q}\left[ {{S_N} - a} \right] \leqslant x} \right\} - {\text{P}}\left\{ {\mathbb{Q}\left[ {G - a} \right] \leqslant x} \right\} - {E_a}(x)} \right| = \mathcal{O}\left( {{N^{ - 1}}} \right)      19. A categorification of Morelli’s theorem      Bohan Fang  Chiu-Chu Melissa Liu  David Treumann  Eric Zaslow 《Inventiones Mathematicae》2011,186(1):79-114       20. Clifford Analysis Techniques for Spherical PDE      Hong Liu  John Ryan 《Journal of Fourier Analysis and Applications》2002,8(6):535-564 For a ? R\alpha \in \mathbf{R}, the class of a-\alpha -order spherical harmonic functions in an open set W í\Omega \subseteq Sn-1\mathbf{S}^{n-1}, Ha(W)H^{\alpha }(\Omega ) is defined as the C2-C^{2}-solutions of Dau=0\Delta _{\alpha }u=0; where Da=Ds+a(n+a-2)\Delta _{\alpha }=\Delta _{s}+\alpha (n+\alpha -2) is the spherical Laplace--Beltrami operator of order a\alpha and Ds\Delta _{s} is the radially independent part of the Laplace operator. We obtain a Green's integral formula for the functions in Ha(W)H^{\alpha }(\Omega ) with kernel expressed as a Gegenbauer function. As generalizations, higher order spherical iterated Dirac operators are defined in a polynomial form. Integral representations of the null solutions to these operators and an intertwining formula relating these operators on the sphere and their analogues in Euclidean space are presented.      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Boxicity of Circular Arc Graphs

A k-dimensional box is a Cartesian product R ₁ × · · · × R _k where each R _i is a closed interval on the real line. The boxicity of a graph G, denoted as box(G), is the minimum integer k such that G can be represented as the intersection graph of a collection of k-dimensional boxes. That is, two vertices are adjacent if and only if their corresponding boxes intersect. A circular arc graph is a graph that can be represented as the intersection graph of arcs on a circle. We show that if G is a circular arc graph which admits a circular arc representation in which no arc has length at least p(\fraca-1a){\pi(\frac{\alpha-1}{\alpha})} for some a ? \mathbbN _{3 2}{\alpha\in\mathbb{N}_{\geq 2}}, then box(G) ≤ α (Here the arcs are considered with respect to a unit circle). From this result we show that if G has maximum degree D < ?\fracn(a-1)2a?{\Delta < \lfloor{\frac{n(\alpha-1)}{2\alpha}}\rfloor} for some a ? \mathbbN _{3 2}{\alpha \in \mathbb{N}_{\geq 2}}, then box(G) ≤ α. We also demonstrate a graph having box(G) > α but with D = n\frac(a-1)2a+ \fracn2a(a+1)+(a+2){\Delta=n\frac{(\alpha-1)}{2\alpha}+ \frac{n}{2\alpha(\alpha+1)}+(\alpha+2)}. For a proper circular arc graph G, we show that if D < ?\fracn(a-1)a?{\Delta < \lfloor{\frac{n(\alpha-1)}{\alpha}}\rfloor} for some a ? \mathbbN _{3 2}{\alpha\in \mathbb{N}_{\geq 2}}, then box(G) ≤ α. Let r be the cardinality of the minimum overlap set, i.e. the minimum number of arcs passing through any point on the circle, with respect to some circular arc representation of G. We show that for any circular arc graph G, box(G) ≤ r + 1 and this bound is tight. We show that if G admits a circular arc representation in which no family of k ≤ 3 arcs covers the circle, then box(G) ≤ 3 and if G admits a circular arc representation in which no family of k ≤ 4 arcs covers the circle, then box(G) ≤ 2. We also show that both these bounds are tight.     

Rectifiability of metric flat chains and fractional masses

We prove that every real flat chain T of finite mass in a complete separable metric space E is rectifiable when \mathbbM^a (T) < + ￥ {\mathbb{M}^\alpha }(T) < + \infty for some α ∈ [0, 1), where \mathbbM^a (T) {\mathbb{M}^\alpha }(T) is the α-mass of T. Bibliography: 12 titles.     

Generalized uncertainty inequalities

The Heisenberg–Pauli–Weyl (HPW) uncertainty inequality on \mathbbRⁿ{\mathbb{R}^n} says that

Actions of symbolic dynamical systems on C*-algebras. II. Simplicity of C*-symbolic crossed products and some examples

A C*-symbolic dynamical system ${(\mathcal{A}, \rho, \Sigma)}A C*-symbolic dynamical system (A, r, S){(\mathcal{A}, \rho, \Sigma)} consists of a unital C*-algebra A{\mathcal{A}} and a finite family { r_a }_{a ? S}{\{ \rho_\alpha \}_{\alpha \in \Sigma}} of endomorphisms ρ _α of A{\mathcal{A}} indexed by symbols α of Σ satisfying some conditions. The endomorphisms r_a, a ? S{\rho_\alpha, \alpha \in \Sigma } yield both a subshift Λ and a C*-algebra of a Hilbert C*-bimodule. The obtained C*-algebra is regarded as a crossed product of A{\mathcal{A}} by the subshift Λ. We will study simplicity condition of these C*-algebras. Some examples such as irrational rotation Cuntz–Krieger algebras will be studied.     

Packing dimension and Ahlfors regularity of porous sets in metric spaces

Let X be a metric measure space with an s-regular measure μ. We prove that if A ì X{A\subset X} is r{\varrho} -porous, then dim_p(A) ￡ s-cr^s{{\rm {dim}_p}(A)\le s-c\varrho^s} where dim_p is the packing dimension and c is a positive constant which depends on s and the structure constants of μ. This is an analogue of a well known asymptotically sharp result in Euclidean spaces. We illustrate by an example that the corresponding result is not valid if μ is a doubling measure. However, in the doubling case we find a fixed N ì X{N\subset X} with μ(N) = 0 such that dim_p(A) ￡ dim_p(X)-c(log\tfrac1r)^-1r^t{{\rm {dim}_p}(A)\le{\rm {dim}_p}(X)-c(\log \tfrac1\varrho)^{-1}\varrho^t} for all r{\varrho} -porous sets A ì X\ N{A \subset X{\setminus} N} . Here c and t are constants which depend on the structure constant of μ. Finally, we characterize uniformly porous sets in complete s-regular metric spaces in terms of regular sets by verifying that A is uniformly porous if and only if there is t < s and a t-regular set F such that A ì F{A\subset F} .     

Implicit standard Jacobi gives high relative accuracy

We prove that the Jacobi algorithm applied implicitly on a decomposition A = XDX ^T of the symmetric matrix A, where D is diagonal, and X is well conditioned, computes all eigenvalues of A to high relative accuracy. The relative error in every eigenvalue is bounded by O(ek(X)){O(\epsilon \kappa (X))} , where e{\epsilon} is the machine precision and k(X) o ||X||₂·||X^-1||₂{\kappa(X)\equiv\|X\|_2\cdot\|X^{-1}\|_2} is the spectral condition number of X. The eigenvectors are also computed accurately in the appropriate sense. We believe that this is the first algorithm to compute accurate eigenvalues of symmetric (indefinite) matrices that respects and preserves the symmetry of the problem and uses only orthogonal transformations.     

A Generalized Radon Transform on the Plane

A new generalized Radon transform R _α, β on the plane for functions even in each variable is defined which has natural connections with the bivariate Hankel transform, the generalized biaxially symmetric potential operator Δ _α, β , and the Jacobi polynomials P_k^(b, a)(t)P_{k}^{(\beta,\,\alpha)}(t). The transform R _α, β and its dual R_a, b^*R_{\alpha,\,\beta}^{\ast} are studied in a systematic way, and in particular, the generalized Fuglede formula and some inversion formulas for R _α, β for functions in L_a, b^p(\mathbbR²₊)L_{\alpha,\,\beta}^{p}(\mathbb{R}^{2}_{+}) are obtained in terms of the bivariate Hankel–Riesz potential. Moreover, the transform R _α, β is used to represent the solutions of the partial differential equations Lu:=?_j=1^ma_jD_a, b^ju=fLu:=\sum_{j=1}^{m}a_{j}\Delta_{\alpha,\,\beta}^{j}u=f with constant coefficients a _j and the Cauchy problem for the generalized wave equation associated with the operator Δ _α, β . Another application is that, by an invariant property of R _α, β , a new product formula for the Jacobi polynomials of the type P_k^(b, a)(s)C_2k^a+b+1(t)=còòP_k^(b, a)P_{k}^{(\beta,\,\alpha)}(s)C_{2k}^{\alpha+\beta+1}(t)=c\int\!\!\int P_{k}^{(\beta,\,\alpha)} is obtained.     

Growth of permutational extensions

We study the geometry of a class of group extensions, containing permutational wreath products, which we call “permutational extensions”. We construct for all k∈ℕ a torsion group K _k with growth function v_Kk(n) ~ exp(n^1-(1-a)k),       2^3-3/a+2^2-2/a+2^1-1/a=2,v_{K_k}(n)\sim\exp(n^{1-(1-\alpha)^k}),\qquad 2^{3-3/\alpha}+2^{2-2/\alpha}+2^{1-1/\alpha}=2,     

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,

Clifford Analysis Techniques for Spherical PDE

For a ? R\alpha \in \mathbf{R}, the class of a-\alpha -order spherical harmonic functions in an open set W í\Omega \subseteq S^n-1\mathbf{S}^{n-1}, H^a(W)H^{\alpha }(\Omega ) is defined as the C²-C^{2}-solutions of D_au=0\Delta _{\alpha }u=0; where D_a=D_s+a(n+a-2)\Delta _{\alpha }=\Delta _{s}+\alpha (n+\alpha -2) is the spherical Laplace--Beltrami operator of order a\alpha and D_s\Delta _{s} is the radially independent part of the Laplace operator. We obtain a Green's integral formula for the functions in H^a(W)H^{\alpha }(\Omega ) with kernel expressed as a Gegenbauer function. As generalizations, higher order spherical iterated Dirac operators are defined in a polynomial form. Integral representations of the null solutions to these operators and an intertwining formula relating these operators on the sphere and their analogues in Euclidean space are presented.