Regular Partition Research Articles

Congruences for 9-regular partitions modulo 3

It is proved that the number of 9-regular partitions of n is divisible by 3 when n is congruent to 3 mod 4, and by 6 when n is congruent to 13 mod 16. An infinite family of congruences mod 3 holds in other progressions modulo powers of 4 and 5. A collection of conjectures includes two congruences modulo higher powers of 2 and a large family of “congruences with exceptions” for these and other regular partitions mod 3.

Microanatomy of the male and female reproductive tracts in the long‐tailed butterfly ray Gymnura poecilura, an elasmobranch with unusual characteristics

The anatomy of the male and female reproductive systems was investigated in the long-tailed butterfly ray Gymnura poecilura using gross observation and light microscopy. The testes are highly asymmetrical, to the extent that only the left testis is functional and the right testis is completely absent. Both of the male genital ducts are present and symmetrical, although spermatozoa only occur in the left duct. The genital ducts are straight and unconvoluted, with regular incomplete internal partitions throughout. Females do not possess a right ovary, nor do the oviducal glands exhibit distinct club and papillary zones, and the baffle zone lacks baffle plates. In all sections of the gland, the tubules display different secretory activities depending on the proximity to the gland lumen. The gland produces a thin egg membrane that encases each egg individually, while the endometrium is formed into trophonemata.

Efficient windows query processing with expanded grid cells on wireless spatial data broadcasting for pervasive computing

Smart mobile devices and advanced wireless networks make the vision of pervasive computing come into reality through context aware services. A wireless spatial data broadcasting is a natural way of pervasive information services by providing information related to mobile clients' location context to an arbitrary number of clients. To support location aware information services efficiently, we propose a spatial index based on expanded grid cells, named SIEC, aiming at spatial data broadcasting on air. The SIEC is organized with expanded cells of a regular grid partition for spatial data items, allows mobile clients to quickly access the queried data items by lowering the number of grid cells the clients have to access. Simulation studies show that the proposed scheme outperforms the existing schemes regarding the access time, tuning time, and energy consumed while processing given queries.

Urban Mobility Dynamics Based on Flexible Discrete Region Partition

Understanding the urban mobility patterns is essential for the planning and management of public infrastructure and transportation services. In this paper we focus on taxicab moving trajectory records and present a new approach to modeling and analyzing urban mobility dynamics. The proposed method comprises two phases. First, discrete space partition based on flexible grid is developed to divide urban environment into finite nonoverlapping subregions. By integrating mobility origin-destination points with covered region, the partitioned discrete subregions have better spatial semantics scalability. Then, we study mobility activity and its distribution randomness during given time periods among discrete subregions. Moreover, we also carry out the analysis of mobility linkage of mobility trips between different regions by O-D matrix. We present a case study with real dataset of taxicab mobility logs in Shenzhen, China, to demonstrate and evaluate the methodology. The experimental results show that the proposed method outperforms the clustering partition and regular partition methods.

Influence of modularity and regularity on disparity of atelostomata sea urchins.

A modularity approach is used to study disparity rates and evolvability of sea urchins belonging to the Atelostomata superorder. For this purpose, the pentameric sea urchin architecture is partitioned into modular spatial components and the interference between modules is quantified using areas and a measurement of the regularity of the spatial partitions. This information is used to account for the variability through time (disparity) and potential for morphological variation and evolution (evolvability) in holasteroid echinoids. We obtain that regular partitions of the space produce modules with high modular integrity, whereas irregular partitions produce low modular integrity; the former ones are related with high morphological disparity (facilitation hypothesis). Our analysis also suggests that a pentameric body plan with low regularity rates in Atelostomata reflects a stronger modular integration among modules than within modules, which could favors bilaterality against radial symmetry. Our approach constitutes a theoretical platform to define and quantify spatial organization in partitions of the space that can be related to modules in a morphological analysis.

GPU-Based Sphere Tracing for Radial Basis Function Implicits

Ray tracing of implicit surfaces based on radial basis functions can demand high computational cost in the presence of a large number of radial centers. Recently, it was presented the least squares hermite radial basis functions (LS-HRBF) Implicits, a method for implicit surface reconstruction from Hermitian data (points equipped with their normal vectors) which makes use of iterative center selection in order to reduce the number of centers. In the present work, we propose an antialiazed sphere tracing algorithm fully implemented in OpenGL Shader Language for ray tracing LS-HRBF Implicits, which exploits a regular partition of unity for strong parallelization. We show that interactive frame rates can be achieved for surfaces composed of thousands of centers even when rendering effects such as cube mapping, soft shadows and ambient occlusion are used.

Estimation of the number of iterations in integer programming algorithms using the regular partitions method

We review the results of studying integer linear programming algorithms which exploit properties of problem relaxation sets. The main attention is paid to the estimation of the number of iterations of these algorithms by means of the regular partitions method and other approaches. We present such estimates for some cutting plane, branch and bound (Land and Doig scheme), and L-class enumeration algorithms and consider questions of their stability. We establish the upper bounds for the average number of iterations of the mentioned algorithms as applied to the knapsack problem and the set packing one.

On calculation of the interweight distribution of an equitable partition

We derive recursive and direct formulas for the interweight distribution of an equitable partition of a hypercube. The formulas involve a three-variable generalization of the Krawtchouk polynomials. Keywords: equitable partition; regular partition; partition design; strong distance invariance; interweight distribution; distance distribution; Krawtchouk polynomial

An approximate blow-up lemma for sparse pseudorandom graphs

We state a sparse approximate version of the blow-up lemma, showing that regular partitions in sufficiently pseudorandom graphs behave almost like complete partite graphs for embedding graphs with maximum degree Δ. We show that (p,γ)-jumbled graphs, with γ=o(pmax(2Δ,Δ+3/2)n), are “sufficiently pseudorandom”.The approach extends to random graphs Gn,p with p≫(lognn)1/Δ.

Functional quantizer design for source localization in sensor networks

In this paper, we address the problem of quantizer design optimized for a source localization application in acoustic sensor networks where physically separated sensors make measurements of acoustic signal energy, quantize them, and transmit the quantized data to a fusion node, which then produces an estimate of the source location. We propose an iterative regular quantizer design algorithm that minimizes the localization error. To construct regular quantization partitions, we suggest the average distance error as a metric in the functional quantization since the distance is monotonic in each sensor reading. Furthermore, to guarantee minimization of the localization error, we propose a new technique to update the codewords and prove that the localization error can be reduced at each iteration while the average distance error remains nonincreasing by applying our update technique. Our experiments show that our proposed algorithm yields significantly improved performance as compared with traditional quantizer designs.

Image encryption based on the finite field cosine transform

In this paper, a novel image encryption scheme is proposed. The technique involves two steps, where the finite field cosine transform is recursively applied to blocks of a given image. In the first step, the image blocks to be transformed result from the regular partition of subimages of the original image. The transformed subimages are regrouped and an intermediate image is constructed. In the second step, a secret-key determines the positions of the intermediate image blocks to be transformed. Besides complying with the main properties required by image encryption methods, the proposed scheme provides benefits related to computational complexity and encoding of the ciphered-images.

Numerical integration based on trivariate C 2 quartic spline quasi-interpolants

In this paper we consider the space generated by the scaled translates of the trivariate C 2 quartic box spline B defined by a set X of seven directions, that forms a regular partition of the space into tetrahedra. Then, we construct new cubature rules for 3D integrals, based on spline quasi-interpolants expressed as linear combinations of scaled translates of B and local linear functionals.We give weights and nodes of the above rules and we analyse their properties.Finally, some numerical tests and comparisons with other known integration formulas are presented.

Coarse-grained spin density-functional theory: Infinite-volume limit via the hyperfinite

Coarse-grained spin density functional theory (SDFT) is a version of SDFT which works with number/spin densities specified to a limited resolution — averages over cells of a regular spatial partition — and external potentials constant on the cells. This coarse-grained setting facilitates a rigorous investigation of the mathematical foundations which goes well beyond what is currently possible in the conventional formulation. Problems of existence, uniqueness, and regularity of representing potentials in the coarse-grained SDFT setting are here studied using techniques of (Robinsonian) nonstandard analysis. Every density which is nowhere spin-saturated is V-representable, and the set of representing potentials is the functional derivative, in an appropriate generalized sense, of the Lieb internal energy functional. Quasi-continuity and closure properties of the set-valued representing potentials map are also established. The extent of possible non-uniqueness is similar to that found in non-rigorous studies of the conventional theory, namely non-uniqueness can occur for states of collinear magnetization which are eigenstates of Sz.

ИССЛЕДОВАНИЕ ЗАДАЧ ДИСКРЕТНОЙ ОПТИМИЗАЦИИ С ЛОГИЧЕСКИМИ ОГРАНИЧЕНИЯМИ НА ОСНОВЕ МЕТОДА РЕГУЛЯРНЫХ РАЗБИЕНИЙ

ИССЛЕДОВАНИЕ ЗАДАЧ ДИСКРЕТНОЙ ОПТИМИЗАЦИИ С ЛОГИЧЕСКИМИ ОГРАНИЧЕНИЯМИ НА ОСНОВЕ МЕТОДА РЕГУЛЯРНЫХ РАЗБИЕНИЙ

Superconvergence of [formula omitted] finite element approximations for the Stokes problem by [formula omitted]-projection methods

A general superconvergence result of H(div) finite element approximations for the Stokes equations is established by using L2-projection method. Regularity assumptions for the Stokes problem with regular partitions is required. Numerical experiments are given to verify the theoretical results.

Superconvergence for discontinuous Galerkin finite element methods by [formula omitted]-projection methods

A general superconvergence of discontinuous Galerkin (DG) finite element method for the elliptic problem is established by using L2-projection method. Regularity assumptions for the elliptic problem with regular partitions are required. Numerical experiments are given to verify the theoretical results.

A Wowzer-type lower bound for the strong regularity lemma

The regularity lemma of Szemeredi asserts that one can partition every graph into a bounded number of quasi-random bipartite graphs. In some applications however, one would like to have a strong control on how quasi-random these bipartite graphs are. Alon, Fischer, Krivelevich and Szegedy obtained a powerful variant of the regularity lemma, which allows one to have an arbitrary control on this measure of quasi-randomness. However, their proof only guaranteed to produce a partition where the number of parts is given by the Wowzer function, which is the iterated version of the Tower function. We show here that a bound of this type is unavoidable by constructing a graph H, with the property that even if one wants a very mild control on the quasi-randomness of a regular partition, then any such partition of H must have a number of parts given by a Wowzer-type function.

Approximation of explicit model predictive control using regular piecewise affine functions: an input-to-state stability approach

Piecewise affine (PWA) feedback control laws defined on general polytopic partitions, as for instance obtained by explicit model predictive control, will often be prohibitively complex for fast systems. In this work the authors study the problem of approximating these high-complexity controllers by low-complexity PWA control laws defined on more regular partitions, facilitating faster on-line evaluation. The approach is based on the concept of input-to-state stability (ISS). In particular, the existence of an ISS Lyapunov function (LF) is exploited to obtain a priori conditions that guarantee asymptotic stability and constraint satisfaction of the approximate low-complexity controller. These conditions can be expressed as local semidefinite programs or linear programs, in case of 2-norm or 1, ∞-norm-based ISS, respectively, and apply to PWA plants. In addition, as ISS is a prerequisite for our approximation method, the authors provide two tractable computational methods for deriving the necessary ISS inequalities from nominal stability. A numerical example is provided that illustrates the main results.

Direct Design of a Polynomial Model Predictive Controlle

Suboptimal design of a model predictive controller is considered with a piecewise polynomial defined on a given partition of the state space. The motivation is that the optimal controller is piecewise affine and its computation requires computational geometric methods, which are difficult to apply when the plant is of high order and/or when its piecewise structure is nearly degenerate. The proposed approach can avoid geometric computation by using a regular partition and the sum-of-squares method. It is more flexible than the existing suboptimal approach with a regularly partitioned piecewise affine controller. It is also direct in that it does not need the computation of the optimal controller and is carried out with minimization of the distance to optimality. The asymptotic optimality of the proposed approach is discussed together with the stability of the resulting closed-loop system.

Approximate Explicit MPC on Simplicial Partitions with Guaranteed Stability for Constrained Linear Systems

This paper proposes an approximate explicit model predictive control design approach for regulating linear time-invariant systems subject to both state and control constraints. The proposed control law is implemented as a piecewise-affine function defined on a regular simplicial partition, and has two main positive features. First, the regularity of the simplicial partition allows a very efficient implementation of the control law on digital circuits, with computation performed in tens of nanoseconds. Second, the asymptotic stability of the closed-loop system is enforced a priori by design.