Narrow Results

Emerging applications such as optimal budget allocation and sensor placement impose problems of maximizing variants of submodular functions with constraints under a streaming setting. In this paper, we first devise a streaming algorithm based on Sieve-Streaming for maximizing a monotone diminishing return submodular (DR-submodular) function with a cardinality constraint on the integer lattice and show it is a one-pass algorithm with approximation ratio $\frac{1}{2}-𝜖$. The key idea to ensure one pass for the algorithm is to combine binary search for determining the level of an element with the exponential-growth method for estimating the OPT. Inspired by Sieve-Streaming++, we then improve the memory of the algorithm to $O(\frac{k}{𝜖})$ and the query complexity to $O(\frac{k{log}^{2}k}{𝜖})$.

The problem of maximization submodular functions with a cardinality constraint has been extensively researched in recent years. Balkanski and Singer were the first to study this class problem. Subsequently, Chekuri and Kent recently extended these results to more general constraints, that is, $k$-cardinality constraint, partition and laminar matroids, matching, knapsack constraints, and including their intersection. They proposed a $(1-1/e-𝜀)$ approximation randomized-parallel-greedy algorithm which are poly-logarithmic adaptivity. However, these existing approaches are hardly extended to the nonsubmodular case. In this paper, we investigate the problem of maximization on a nonsubmodular function subject to a cardinality constraint, provided the objective function is specified by a generic submodularity ratio $\gamma$. We design a $(1-e^{-\gamma }-𝜀)$-approximation Greedy algorithm by using the technical aspects to maximize the multilinear relaxation of the object function under the $k$-cardinality constraints. The adaptive of multilinear relaxation is $O(logn/{𝜀}^2)$; the number of oracle is $Õ(n/{𝜀}^4)$.

Since fractal functions are widely applied in dynamic systems and physics such as fractal growth and fractal antennas, this paper concerns fundamental problems of fractal continuous functions like cardinality of collection of fractal functions, box dimension of summation of fractal functions, and fractal linear space. After verifying that the cardinality of fractal continuous functions is the second category by Baire theory, we investigate the box dimension of sum of fractal continuous functions so as to discuss fractal linear space under fractal dimension. It is proved that the collection of 1-dimensional fractal continuous functions is a fractal linear space under usual addition and scale multiplication of functions. Particularly, it is revealed that the fractal function with the largest box dimension in the summation represents a fractal dimensional character whenever the other box dimension of functions exist or not. Simply speaking, the fractal function with the largest box dimension can absorb the other fractal features of functions in the summation.

It seems that a suitably constructed fuzzy set of natural numbers does form the most complete and adequate description of cardinality of a finite fuzzy set. Nevertheless, in many applications, one needs a simple scalar approximation (evaluation) of that cardinality. Usually, the well-known, but very imperfect concept of sigma-count of a fuzzy set is then used. The aim of this paper is to find the best approximation of cardinality of a finite fuzzy set by a single natural number.

Let R be an associative ring with identity, and let I be an (left, right, two-sided) ideal of R. Say that I is small if |I| < |R| and large if |R/I| < |R|. In this paper, we present results on small and large ideals. In particular, we study their interdependence and how they influence the structure of R. Conversely, we investigate how the ideal structure of R determines the existence of small and large ideals.

Given a genus 2 curve $\mathrm{\text{C}}$ defined over a finite field ${𝔽}_q$ of odd characteristic such that $2|\#\mathrm{\text{Jac}}(\mathrm{\text{C}})({𝔽}_q)$, we study the growth of the 2-adic valuation of the cardinality of the Jacobian over a tower of quadratic extensions of ${𝔽}_q$. In the cases of simpler regularity, we determine the exponents of the 2-Sylow subgroup of $\mathrm{\text{Jac}}(\mathrm{\text{C}})({𝔽}_{q^{2^k}})$.

It is well-known that the different kinds of multiresolution analysis (MRA) structures generate different wavelets. In this paper, we give two uniform formulas on the number of mother functions for various wavelets associated with MRA structures. These formulas show that the number of mother functions of wavelets is determined by the support of the Fourier transform of the scaling function in MRA structure.

We present an optimization approach to wavelet architecture that hinges on the Zak transform to formulate the construction as a minimization problem. The transform warrants parametrization of the quadrature mirror filter in terms of the possible integer sample values of the scaling function and the associated wavelet. The parameters are predicated to satisfy constraints derived from the conditions of regularity, compact support and orthonormality. This approach allows for the construction of nearly cardinal scaling functions when an objective function that measures deviation from cardinality is minimized. A similar objective function based on a measure of symmetry is also proposed to facilitate the construction of nearly symmetric scaling functions on the line.

The correct inference of gene regulatory networks for the understanding of the intricacies of the complex biological regulations remains an intriguing task for researchers. With the availability of large dimensional microarray data, relationships among thousands of genes can be simultaneously extracted. Among the prevalent models of reverse engineering genetic networks, S-system is considered to be an efficient mathematical tool. In this paper, Bat algorithm, based on the echolocation of bats, has been used to optimize the S-system model parameters. A decoupled S-system has been implemented to reduce the complexity of the algorithm. Initially, the proposed method has been successfully tested on an artificial network with and without the presence of noise. Based on the fact that a real-life genetic network is sparsely connected, a novel Accumulative Cardinality based decoupled S-system has been proposed. The cardinality has been varied from zero up to a maximum value, and this model has been implemented for the reconstruction of the DNA SOS repair network of Escherichia coli. The obtained results have shown significant improvements in the detection of a greater number of true regulations, and in the minimization of false detections compared to other existing methods.

Three necessary and sufficient conditions for a matroid in the sense of Betten and Wenzel to be disconnected are given in this paper.

It is known that the value of the exponential sum can be derived from the estimate of the cardinality |V|, the number of elements contained in the set where is the partial derivatives of with respect to . The cardinality of V in turn can be derived from the p-adic sizes of common zeros of the partial derivatives . This paper presents a method of determining the p-adic sizes of the components of (ξ,η) a common root of partial derivative polynomials of f(x,y) in Z_p[x,y] of degree five based on the p-adic Newton polyhedron technique associated with the polynomial. The degree five polynomial is of the form f(x,y) = ax⁵ + bx⁴y + cx³y² + sx + ty + k. The estimate obtained is in terms of the p-adic sizes of the coefficients of the dominant terms in f.