On the hypergroup structure of central Λ(p) sets

Considering multidimensional structure of the multi-exposure images, a new Tensor product and Tensor-singular value decomposition based Multi-Exposure image Fusion (TT-MEF) method is proposed. The main innovation of this work is to explore a new feature representation of multi-exposure images in the new tensor domain and design the fusion strategy on this basis. Specifically, the luminance and the chrominance channels are fused separately to maintain color consistency. For the luminance fusion, the luminance channel of multi-exposure images is divided into two parts, that is, de-mean term and mean term. The de-mean term is represented as a tensor to extract the feature. Then, the tensor product and tensor-singular value decomposition (T-SVD) are used to design a tensor feature extractor. Furthermore, a fusion strategy of the de-mean term is presented according to the visual saliency model, and a fusion strategy of the mean term is defined by the local and the global visual weights to control counterpoise between the local and global luminance. For the chrominance fusion, a new fusion strategy is also designed by the tensor product and T-SVD, similar to the luminance fusion. Finally, the fused image is obtained by combining the luminance and chrominance fusion. Experimental results show that the proposed TT-MEF method generally outperforms the existing state-of-the-art in terms of subjective visual quality and objective evaluation.

We give a general theory of matrix elements (ME’s) of the unitary irreducible representations (UIR’s) of linear semisimple Lie groups and of reductive Lie groups. This theory connects together the following things, (1) MEUIR’s of all the representation series of a noncompact Lie group, (2) MEUIR’s of compact and noncompact forms of the same complex Lie group. The theory presented is based on the results of the theory of the principal nonunitary series representations and on a theorem which states that ME’s of the principal nonunitary series representations are entire analytic functions of continuous representation parameters. The principle of analytic continuation of Clebsch–Gordan coefficients (CGC’s) of finite dimensional representations to CGC’s of the tensor product of a finite and an infinite dimensional representation and to CGC’s of the tensor product of two infinite dimensional representations is proved. ME’s for any UIR of the group U(n) and of the group U(n,1) are obtained. The explicit expression for all CGC’s summed over the multiplicity of the irreducible representation in the tensor product decomposition is derived.

On generalized perfect difference sets constructed from Sidon sets

(1) Bd(E) and B(E) are isometrically isomorphic for finite E ⊂ Γ. Bd(E) = `∞(E) characterizes I0 sets E and B(E) = `∞(E) characterizes Sidon sets E. [In general, Sidon sets are distinct from I0 sets. Within the group of integers Z, the set {2}n ⋃ {2+n}n is helsonian (hence Sidon) but not I0.] (2) Both are Fσ in 2 (as is also the class of finite unions of I0 sets). (3) There is an analogue for I0 sets of the sup-norm partition construction used with Sidon sets. (4) A set E is Sidon if and only if, there is some r ∈ R and positive integer N such that, for all finite F ⊂ E, there is some H ⊂ F with |H| ≥ r|F | and H is an I0 set of degree at most N . [Here |S| denotes the cardinality of S; two different but comparable definitions of degree for I0 sets are made below.] (5) IF all Sidon subsets of Z are finite unions of I0 sets, the number of I0 sets required is bounded by some function of the Sidon constant. This is also true in the category of all discrete abelian groups.

Some general connections between martingales and character ratios of finite groups are developed. As an application we sharpen the convergence rate in a central limit theorem for the character ratio of a random representation of the symmetric group on transpositions. A generalization of these results is given for Jack measure on partitions. We also give a probabilistic proof of a result of Burnside and Brauer on the decomposition of tensor products.

Si q≥2 est un nombre entier, on designe par Sq(n) la somme des chiffres en base q du nombre entier naturel n et par vq(n) sa valuation q-adique. L’objectif de cet article est d’etudier des sommes d’exponentielles de la forme ∑n≤xexp(2iπ(lmSq(n)+km′Sq(n+1)+θn)) afin d’en deduire certaines proprietes statistiques des nombres entiers n pour lesquels Sq(n) et Sq(n+1) appartiennent a des progressions arithmetiques donnees. Ceci permet d’etendre les resultats obtenus par Gelfond en 1968 et ceux obtenus par Mauduit-Sarkozy en 1996.

This is the first part of a two-part paper. Our paper was motivated by two classical papers: A paper of Sir Charles Wheatstone published in 1844 on representing certain powers of an integer as sums of arithmetic progressions and a paper of J. J. Sylvester published in 1882 for determining the number of ways a positive integer can be represented as the sum of a sequence of consecutive integers. There have been many attempts to extend Sylvester Theorem to the number of representations for an integer as the sums of different types of sequences, including sums of certain arithmetic progressions. In this part of the paper, we will make yet one more extension: We will describe a procedure for computing the number of ways a positive integer can be represented as the sums of all possible arithmetic progressions, together with an example to illustrate how this procedure can be carried out. In the process of doing this, we will also give an extension of Wheatstone’s work. In the second part of the paper, we will continue on the problems initiated by Wheatstone by studying certain relationships among the representations for different powers of an integer as sums of arithmetic progressions.

Consider all the arithmetic progressions of odd numbers, no term of which is of the form 2 k + p, where k is a positive integer and p is an odd prime. Erdős ever asked whether all these progressions can be obtained from covering congruences. In this paper, we characterize all arithmetic progressions in which there are positive proportion natural numbers that can be expressed in the form 2 k +p, and give a quantitative form of Romanoff’s theorem on arithmetic progressions. As a corollary, we prove that the answer to the above Erdős problem is affirmative.

This work introduces and analyzes B-spline approximation spaces defined on general geo- metric domains obtained through a mapping from a parameter domain. These spaces are constructed as sparse-grid tensor products of univariate spaces in the parameter domain and are mapped to the physical domain via a geometric parametrization. Both the univariate approximation spaces and the geometric mapping are built using maximally smooth B-splines. We construct two such spaces, em- ploying either the sparse-grid combination technique or the hierarchical subspace decomposition of sparse-grid tensor products, and we prove their mathematical equivalence. Furthermore, we derive ap- proximation error estimates and inverse inequalities that highlight the advantages of sparse-grid tensor products. Specifically, under suitable regularity assumptions on the solution, these spaces achieve the same approximation order as standard tensor product spaces while using significantly fewer degrees of freedom. Additionally, our estimates indicate that, in the case of non-tensor-product domains, stronger regularity assumptions on the solution—particularly concerning isotropic (non-mixed) derivatives—are required to achieve optimal convergence rates compared to sparse-grid methods defined on tensor- product domains.

The decomposition of certain tensor products of representations in the principal series with representations in the supplementary series of the group SL (n, C) into irreducible components

Tensor product decompositions and open orbits in multiple flag varieties

Introduction 1. Primes in arithmetic progressions 2. The main theorems II.Oscillation theorems for sifting functions 3. The statements 4. Proof of Theorem B2 5. Proof of Proposition 3.1 6. Proof of Proposition 3.2 7. Proof of Theorem C: preliminaries 8. Proof of Theorem C: completion 9. Proof of Theorem Bl 10.Proof of Theorem B3 III.Oscillation theorems for primes 11.Proof of Theorem Al 12. Proof of Theorem A2 13.Proofs of Theorem A3 and Proposition 2.1 I. INTRODUCTION 1. PRIMES IN ARITHMETIC PROGRESSIONSFor x ~ 2 real, q a positive integer, and a an integer coprime to q, let 0 and ~ be defined by (1.1) O(x; q, a) = E logp = qJZq) (1 +~(x; q, a)), p~x p=amodq

We introduce a new, elementary method for studying random differences in arithmetic progressions and convergence phenomena along random sequences of integers. We apply our method to obtain significant improvements on two results. The first improvement is the following. Let l be a positive integer and {u 1 ≥ u 2 ≥ · · · } be a decreasing sequence of probabilities satisfying u n · n 1/(l+1)→∞. Let R = R ω be the random sequence obtained by selecting the natural number n with probability u n . Then every set A of natural numbers with positive upper density contains an arithmetic progression a, a+r, a+2r,..., a+lr of length l + 1 with difference r ∈ R ω . The best previous result (by M. Christ and us) was the condition u n ·n 2−l+1 → ∞ with a logarithmic rate. The new bound is better when l ≥ 4. Our second improvement concerns almost everywhere convergence of double ergodic averages. We construct a (random) sequence {r 1 < r 2 < · · · } of positive integers such that r n /n 2−ϵ → ∞ for all ϵ > 0 and, for any measure preserving transformation T of a probability space, the averages $$\frac{1}{N}\sum\limits_{n < N} {{T^n}{F_1}\left( x \right)} {T^{{r_n}}}{F_2}\left( x \right)$$ converge for almost every x. Our best previous result was the growth rate r n /n (1+1/14)−ϵ → ∞ of the sequence {r n }.

On square-free arithmetic progressions in infinite words

In their valuable book Board games around the world . Bell and Cornelius discuss possible investigations arising from the rather complicated medieval game of Rithmomachia . One such investigation gives rise to the following problem: Find all sets of four positive integers in which three of the four are in arithmetic progression, another three are in geometric progression, and another three are in harmonic progression (that is to say, their reciprocals are in arithmetic progression). The authors give two such sets, and state without proof that they are the only ones, apart from multiplication throughout by a common factor. The proof that this is so is an interesting investigation in itself, and opens up some further mathematics.