Duality and interpolation of anisotropic Triebel–Lizorkin spaces

We study properties of anisotropic Triebel–Lizorkin spaces associated with general expansive dilations and doubling measures on using wavelet transforms. This paper is a continuation of (Bownik in J Geom Anal 2007, to appear, Trans Am Math Soc 358:1469–1510, 2006), where we generalized the isotropic methods of dyadic -transforms of Frazier and Jawerth (J Funct Anal 93:34–170, 1990) to non-isotropic settings. By working at the level of sequence spaces, we identify the duals of anisotropic Triebel–Lizorkin spaces. We also obtain several real and complex interpolation results for these spaces. The author was partially supported by the NSF grants DMS-0441817 and DMS-0653881. The author wishes to thank Michael Frazier and Dachun Yang for valuable comments and discussions on this work.     

Duality of weighted anisotropic Besov and Triebel–Lizorkin spaces

Let A be an expansive dilation on ${{\mathbb R}^n}$ and w a Muckenhoupt ${\mathcal A_\infty(A)}$ weight. In this paper, for all parameters ${\alpha\in{\mathbb R} }$ and ${p,q\in(0,\infty)}$ , the authors identify the dual spaces of weighted anisotropic Besov spaces ${\dot B^\alpha_{p,q}(A;w)}$ and Triebel?CLizorkin spaces ${\dot F^\alpha_{p,q}(A;w)}$ with some new weighted Besov-type and Triebel?CLizorkin-type spaces. The corresponding results on anisotropic Besov spaces ${\dot B^\alpha_{p,q}(A; \mu)}$ and Triebel?CLizorkin spaces ${\dot F^\alpha_{p,q}(A; \mu)}$ associated with ${\rho_A}$ -doubling measure??? are also established. All results are new even for the classical weighted Besov and Triebel?CLizorkin spaces in the isotropic setting. In particular, the authors also obtain the ${\varphi}$ -transform characterization of the dual spaces of the classical weighted Hardy spaces on ${{\mathbb R}^n}$ .     

Product Systems of C*-correspondences and Baaj—Skandalis Duality

Acta Mathematica Sinica, English Series - Let X = {Xp}]p∈P be a product system over a lattice ordered group (G, P) with coefficients in a C*-algebra $${\cal A}$$ . In this paper, we study the...     

Duality of Unary Convex Functions Programming

Most nonliner programming problems consist of functions which are sums of unary,convexfunctions of linear fuctions.In this paper.we derive the duality forms of the unary oonvex optimization,and these technuqucs are applied to the geometric programming and minimum discrimination informationproblems.     

Pure Anderson motives and abelian ��-sheaves

Pure t-motives were introduced by G. Anderson as higher dimensional generalizations of Drinfeld modules, and as the appropriate analogs of abelian varieties in the arithmetic of function fields. In order to construct moduli spaces for pure t-motives the second author has previously introduced the concept of abelian ??-sheaf. In this article we clarify the relation between pure t-motives and abelian ??-sheaves. We obtain an equivalence of the respective quasi-isogeny categories. Furthermore, we develop the elementary theory of both structures regarding morphisms, isogenies, Tate modules, and local shtukas. The later are the analogs of p-divisible groups.     

Duality and the Poincaré–Hopf Inequalities

In this article we obtain a duality result for an n-manifold N with boundary ∂N = N + ⊔N ⁻ a disjoint union, where N ⁺ and N ⁻ are arbitrarily chosen parts in ∂N and need not be compact. This duality result is used to generalize the Poincaré–Hopf inequalities in a non-compact setting.     

Duality and well-posedness in convex interpolation ∗)

We consider the problem for convex interpolation with minimal L_p norm of the second derivative, 1 < p < +α. Convergence of a class of dual methods is established and numerical results are presented. It is proved that if p 2 then the solution of the problem is locally Lipschitz with respect to the data in the uniform metric.     

On Möbius Duality and Coarse-Graining

We study duality relations for zeta and Möbius matrices and monotone conditions on the kernels. We focus on the cases of families of sets and partitions. The conditions for positivity of the dual kernels are stated in terms of the positive Möbius cone of functions, which is described in terms of Sylvester formulae. We study duality under coarse-graining and show that an \(h\)-transform is needed to preserve stochasticity. We give conditions in order that zeta and Möbius matrices admit coarse-graining, and we prove they are satisfied for sets and partitions. This is a source of relevant examples in genetics on the haploid and multi-allelic Cannings models.     

Electric–Magnetic Duality and WDVV Equations

We consider the associativity (or WDVV) equations as they appear in the Seiberg–Witten theory and prove that they are covariant under general electric–magnetic duality transformations. We discuss the consequences of this covariance from various perspectives.     

A Duality Theorem Associated with the Smash Product of Graded Ring and G-set

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Lefschetz Property,Schur–Weyl Duality and a q-Deformation of Specht polynomial

We describe the Schur–Weyl duality for a polynomial representation of the quantum group and the Hecke algebra of type A from a viewpoint of a q-analog of the strong Lefschetz property. A q-deformation of the Specht polynomial appears as a constituent of bases for irreducible components.     

Duality of Besov,Triebel–Lizorkin and Herz spaces with variable exponents

Our aim is to prove duality and reflexivity of Besov spaces, Triebel–Lizorkin spaces and Herz spaces with variable exponents.     

Picard’s theorem and the Rickman construction

Nevanlinna theory (value-distribution theory) has its genesis in Picard’s discovery that a function analytic in the plane which omits two values is constant. Nearly a century later, attention turned to the analogous situation in Rn, n≥3, where entire functions are necesarily replaced by entire quasiregular mappings. This expository article centers on one of Seppo Rickman’s main contributions to this issue, including an outline of his famous example showing that the omitted set in R3, while finite, can be much larger than possible in the plane.     

A comparison of the alpha‐ and beta‐characteristic functions in cooperative non‐sidepayment n‐person games 1

This article reports a test of theories of payoff allocation in n‐person game‐theoretic systems. An experimental study was conducted to test the relative predictive accuracy of three solution concepts (imputation set, stable set, core) in the context of 4‐person, 2‐strategy non‐sidepayment games. Predictions from each of the three solution concepts were computed on the basis of both α‐effectiveness (von Neumann‐Morgenstern) and β‐effectiveness (Aumann), making a total of six predictive theories under test. Two important results emerged. First, the data show that the g‐imputation set was more accurate than the a‐imputation set, the β‐stable set was more accurate than the α‐stable set, and the (3‐core was more accurate than the α‐core; in other words, for each of the solutions tested, the prediction from any solution concept based on (β‐effectiveness was more accurate than the prediction from the same solution based on a‐effectiveness. Second, the β‐core was the most accurate of the six theories tested. Results are interpreted as showing that β‐effectiveness is superior to a‐effectiveness as a basis for payoff predictions in cooperative non‐sidepayment games.     

Symmetric Duality and Self-duality for Nonsmooth Mathematical Programming

Ｓｙｍｍｅｔｒｉｃ　Ｄｕａｌｉｔｙ　ａｎｄ　Ｓｅｌｆ－ｄｕａｌｉｔｙ　ｆｏｒ　Ｎｏｎｓｍｏｏｔｈ　Ｍａｔｈｅｍａｔｉｃａｌ　ＰｒｏｇｒａｍｍｉｎｇＤｏｎｇＪｉａｌｉ（董加礼）（ＤｅｐａｒｔｍｅｎｔｏｆａｐｐｌｉｅｄＭａｔｈｅｍａｔｉｃｓａ，Ｊｉｌｉｎ...     

New applications of Picard?s successive approximations

The iterative method of successive approximations, originally introduced by Émile Picard in 1890, is a basic tool for proving the existence of solutions of initial value problems regarding ordinary first order differential equations. In the present paper, it is shown that this method can be modified to get estimates for the growth of solutions of linear differential equations of the typef(k)+Ak−1(z)f(k−1)+?+A₁(z)f^′+A₀(z)f=0 with analytic coefficients. A short comparison to the growth results in the literature, obtained by means of different methods, is also given. It turns out that many known results can be proved by applying Picard?s successive approximations in an effective way. Self-contained considerations are carried out in the complex plane and in the unit disc, and some remarks about solutions of real linear differential equations are made.     

Warfield’s Duality and Malcev’s Matrix

In this work, we investigate relations between Malcev’s matrices of a torsion-free group G of finite rank and Malcev’s matrices of groups Hom(R,G) and Hom(G,R), where G is a locally free group and R is a torsion-free group of rank 1.