An Analogue of Beurling’s Theorem for NA Groups

In this paper, we prove Beurling’s theorem for NA groups, from which we derive some other versions of uncertainty principles.     

An Analogue of Gromov’s Waist Theorem for Coloring the Cube

It is proved that if we partition a d-dimensional cube into \(n^d\) small cubes and color the small cubes in \(m+1\) colors then there exists a monochromatic connected component consisting of at least \(f(d, m) n^{d-m}\) small cubes. Another proof of this result is given in Matdinov’s preprint (Size of components of a cube coloring, arXiv:1111.3911, 2011)     

An Extension of Mercer’s Theorem via Pontryagin Spaces

We extend Mercer’s theorem to a composition of the form RS, in which R and S are integral operators acting on a space L ²(X) generated by a locally finite measure space (X, ν). The operator R is compact and positive while S is continuous and having spectral decomposition based on well distributed eigenvalues. The proof is based on a Pontryagin space structure for L ²(X) constructed via the operators R and S themselves.     

Generalized Bochner’s Theorem for radial function

A radial function Φ(x) can be expressed by its generator ?(·) through Φ(x)=?(‖x‖). The positive de finite of the function Φ plays an important role in the radial basis interpolation. We can naturally use Bochner’s Theorem to check if Φ is positive de finite. This requires however a n-dimensional Fourier transformation and it is not very easy to calculate. Furthermore in a lot of cases we will use ? for spaces of various dimensions too, then for every fixed n we need do the Fourier transformation once to check if the function is positive definite in the n-dimensional space. The completely monotone function, which is discussed in [4], is positive definite for arbitrary space dimensions. With this technique we can very easily characterize the positive definite of a radial function through its generator. Unfortunately there is only a very small subset of radial function which is completely monotone. Thus this criterion excluded a lot of interesting functions such as compactly supported radial function, which are very use ful in application. Can we find some conditions (as the completely monotone function) only for the 1-dimensional Fourier transform of the generator ? to characterize a radial function Φ, which is positive definite in n-dimensional (fixed n) space? In this paper we defined a kind of incompletely monotone function of order α, for α=0, 1/2, 1, 3/2, 2, … (we denote the function class by ICM), in this sence a normal positvie function is in ICM₀; a positive monotone decreasing function is in ICM₁ and a positive monotone decreasing and convex function is in ICM₂. Based on this definition we get a generalized Bochner’s Theorem for radial function: If1-dimensional Fourier transform of the generator of a radial function can be written as $F{}_1\varphi (t) = \tilde F(\frac{{t^2 }}{2})$ , then corresponding radial function Φ(x) is positive definite as a n-variate function iff $\tilde F$ is an incompletely monotone function of order α=(n-1)/2 (or simply $\tilde F \in ICM_{\frac{{n - 1}}{2}} $ .     

A Complex Analogue of Toda’s Theorem

Toda (SIAM J. Comput. 20(5):865–877, 1991) proved in 1989 that the (discrete) polynomial time hierarchy, PH, is contained in the class P ^#P , namely the class of languages that can be decided by a Turing machine in polynomial time given access to an oracle with the power to compute a function in the counting complexity class #P. This result, which illustrates the power of counting, is considered to be a seminal result in computational complexity theory. An analogous result (with a compactness hypothesis) in the complexity theory over the reals (in the sense of Blum–Shub–Smale real machines (Blum et al. in Bull. Am. Math. Soc. 21(1):1–46, 1989) was proved in Basu and Zell (Found. Comput. Math. 10(4):429–454, 2010). Unlike Toda’s proof in the discrete case, which relied on sophisticated combinatorial arguments, the proof in Basu and Zell (Found. Comput. Math. 10(4):429–454, 2010) is topological in nature; the properties of the topological join are used in a fundamental way. However, the constructions used in Basu and Zell (Found. Comput. Math. 10(4):429–454, 2010) were semi-algebraic—they used real inequalities in an essential way and as such do not extend to the complex case. In this paper, we extend the techniques developed in Basu and Zell (Found. Comput. Math. 10(4):429–454, 2010) to the complex projective case. A key role is played by the complex join of quasi-projective complex varieties. As a consequence, we obtain a complex analogue of Toda’s theorem. The results of this paper, combined with those in Basu and Zell (Found. Comput. Math. 10(4):429–454, 2010), illustrate the central role of the Poincaré polynomial in algorithmic algebraic geometry, as well as in computational complexity theory over the complex and real numbers: the ability to compute it efficiently enables one to decide in polynomial time all languages in the (compact) polynomial hierarchy over the appropriate field.     

A Bipartite Analogue of Dilworth’s Theorem

Let m(n) be the maximum integer such that every partially ordered set P with n elements contains two disjoint subsets A and B, each with cardinality m(n), such that either every element of A is greater than every element of B or every element of A is incomparable with every element of B. We prove that . Moreover, for fixed ε ∈ (0,1) and n sufficiently large, we construct a partially ordered set P with n elements such that no element of P is comparable with other elements of P and for every two disjoint subsets A and B of P each with cardinality at least , there is an element of A that is comparable with an element of B.     

An Analogue of Nakayama’s Conjecture for Johnson Schemes

Nakayamas Conjecture is one of the most famous theorems for representation theory of symmetric groups. Two general irreducible characters of a symmetric group belong to the same p-block if and only if the p-cores of the young diagrams corresponding to them are the same. The conjecture was first proven in 1947 by Brauer and Robinson. We consider an analogue of Nakayamas Conjecture for Johnson schemes.Received January 28, 2004     

Widder’s Representation Theorem for Symmetric Local Dirichlet Spaces

In classical PDE theory, Widder’s theorem gives a representation for non-negative solutions of the heat equation on \(\mathbb{R }^n\) . We show that an analogous theorem holds for local weak solutions of the canonical “heat equation” on a symmetric local Dirichlet space satisfying a local parabolic Harnack inequality.     

An Analogue of Beurling–Hörmander’s Theorem Associated with Partial Differential Operators

In this paper for a positive real number α we consider two partial differential operators D and D_α on the half–plane We define a generalized Fourier transform associated with the operators D and D_α. We establish an analogue of Beurling–H?rmander’s Theorem for this transform and we give some applications of this theorem.     

On positive positive-definite functions and Bochner’s Theorem

We recall an open problem on the error of quadrature formulas for the integration of functions from some finite dimensional spaces of trigonometric functions posed by Novak (1999) in [8] ten years ago and summarised recently in Novak and Wo?niakowski (2008) [9]. It is relatively easy to prove an error formula for the best quadrature rules with positive weights which shows intractability of the tensor product problem for such rules. In contrast to that, the conjecture that also quadrature formulas with arbitrary weights cannot decrease the error is still open.We generalise Novak’s conjecture to a statement about positive positive-definite functions and provide several equivalent reformulations, which show the connections to Bochner’s Theorem and Toeplitz matrices.     

A Generalization of Bochner’s Extension Theorem to Rough Tubes

This paper generalizes Bochner’s extension theorem to tubes X+iℝ ^m where the set X⊂ℝ ^m is not necessarily a manifold.     

An Analog of Titchmarsh’s Theorem for the Fourier–Walsh Transform

The space clos(X) of all nonempty closed subsets of an unbounded metric space X is considered. The space clos(X) is endowed with a metric in which a sequence of closed sets converges if and only if the distances from these sets to a fixed point θ are bounded and, for any r, the sequence of the unions of the given sets with the exterior balls of radius r centered at θ converges in the Hausdorff metric. The metric on clos(X) thus defined is not equivalent to the Hausdorff metric, whatever the initial metric space X. Conditions for a set to be closed, totally bounded, or compact in clos(X) are obtained; criteria for the bounded compactness and separability of clos(X) are given. The space of continuous maps from a compact space to clos(X) is considered; conditions for a set to be totally bounded in this space are found.     

An extension of Barta’s Theorem and geometric applications

We prove an extension of a theorem of Barta and we give some geometric applications. We extend Cheng’s lower eigenvalue estimates of normal geodesic balls. We generalize Cheng-Li-Yau eigenvalue estimates of minimal submanifolds of the space forms. We show that the spectrum of the Nadirashvili bounded minimal surfaces in have positive lower bounds. We prove a stability theorem for minimal hypersurfaces of the Euclidean space, giving a converse statement of a result of Schoen. Finally we prove generalization of a result of Kazdan–Kramer about existence of solutions of certain quasi-linear elliptic equations. Bessa and Montenegro were partially supported by CNPq Grant.     

On Weyl’s Theorem for Functions of Operators

Let H be a complex separable infinite dimensional Hilbert space. In this paper, a variant of the Weyl spectrum is discussed. Using the new spectrum, we characterize the necessary and sufficient conditions for both T and f(T) satisfying Weyl's theorem, where f ∈ Hol(σ(T)) and Hol(σ(T)) is defined by the set of all functions f which are analytic on a neighbourhood of σ(T) and are not constant on any component of σ(T). Also we consider the perturbations of Weyl's theorem for f(T).     

An analogue of Edmonds’ Branching Theorem for infinite digraphs

We extend Edmonds’ Branching Theorem to locally finite infinite digraphs. As examples of Oxley or Aharoni and Thomassen show, this cannot be done using ordinary arborescences, whose underlying graphs are trees. Instead we introduce the notion of pseudo-arborescences and prove a corresponding packing result. Finally, we verify some tree-like properties for these objects, but give also an example that their underlying graphs do in general not correspond to topological trees in the Freudenthal compactification of the underlying multigraph of the digraph.     

An Analogue of Hajós' Theorem for the Circular Chromatic Number (II)

This paper designs a set of graph operations, and proves that for 2k/d<3, starting from K_k/_d, by repeatedly applying these operations, one can construct all graphs G with _c(G)k/d. Together with the result proved in [20], where a set of graph operations were designed to construct graphs G with _c(G)k/d for k/d3, we have a complete analogue of Hajós' Theorem for the circular chromatic number. This research was partially supported by the National Science Council under grant NSC 89-2115-M-110-003