Number Of Partitions Research Articles

Infinite Products of Binomials and Partitions of Numbers

Infinite Products of Binomials and Partitions of Numbers

Solving Combinational Optimization Problems With Evolutionary Single-Pixel Imaging

In this paper, an evolutionary single-pixel imaging (SPI) scheme is proposed to solve combinational optimization problems. SPI is a unique optical imaging technique by replacing the pixelated sensor array in a conventional camera with a single-pixel detector. SPI is conventionally employed for capturing object images or performing image processing tasks. It is a novel attempt to leverage SPI for processing other types of data in addition to images. An Ising machine model is implemented optically with SPI for solving two combinational optimization problems including number partition and graph maximum cut. The binary illumination patterns are encoded based on the spinning states of all the elements and the object image pixel values are encoded to be proportional to the pairwise weighting factors. As the mathematical inner product between object image and illumination pattern, the recorded single-pixel intensity values simulate the Hamiltonian function values. In each iteration, the feedback of single-pixel values is used to update the illumination patterns by selection, crossover and mutation. As the illumination patterns are evolving, an optimal solution of the Hamiltonian function can be finally obtained by our proposed optoelectronic Ising machine scheme.

Theoretical analysis of immunochromatographic assay and consideration of its operating parameters for efficient designing of high-sensitivity cardiac troponin I (hs-cTnI) detection

Troponin is the American College of Cardiology and American Heart Association preferred biomarker for diagnosing acute myocardial infarction (MI). We provide a modeling framework for high sensitivity cardiac Troponin I (hs-cTnI) detection in chromatographic immunoassays (flow displacement mode) with an analytical limit of detection, i.e., LOD < 10 ng/L. We show that each of the various control parameters exert a significant influence over the design requirements to reach the desired LOD. Additionally, the design implications in a multiplexed fluidic network, as in the case of Simple Plex™ Ella instrument, are significantly affected by the choice of the number of channels or partitions in the network. We also provide an upgrade on the existing LOD equation to evaluate the necessary minimum volume to detect a particular concentration by considering the effects of stochastics and directly incorporating the target number of copies in each of the partitions in case of multiplexed networks. Even though a special case of cTnI has been considered in this study, the model and analysis are analyte agnostic and may be applied to a wide class of chromatographic immunoassays. We believe that this contribution will lead to more efficient designing of the immunochromatographic assays.

A Multiclustering Evolutionary Hyperrectangle-Based Algorithm

Clustering is a grouping technique that has long been used to relate data homogeneously. With the huge growth of complex datasets from different sources in the last decade, new paradigms have emerged. Multiclustering is a new concept within clustering that attempts to simultaneously generate multiple clusters that are bound to be different from each other, allowing to analyze and discover hidden patterns in the dataset compared to single clustering methods. This paper presents a hybrid methodology based on an evolutionary approach with the concepts of hyperrectangle for multiclustering, called MultiCHCClust. The algorithm is applied in a post-processing stage and it improves the results obtained for a clustering algorithm with respect to the partitioning of the dataset and the optimization of the number of partitions, achieving a high degree of compactness and separation of the partitioned dataset as can be observed in a complete experimental study.

A modular idealizer chain and unrefinability of partitions with repeated parts

Recently Aragona et al. have introduced a chain of normalizers in a Sylow 2-subgroup of Sym(2n), starting from an elementary abelian regular subgroup. They have shown that the indices of consecutive groups in the chain depend on the number of partitions into distinct parts and have given a description, by means of rigid commutators, of the first n − 2 terms in the chain. Moreover, they proved that the (n − 1)-th term of the chain is described by means of rigid commutators corresponding to unrefinable partitions into distinct parts. Although the mentioned chain can be defined in a Sylow p-subgroup of Sym(pn), for p > 2 computing the chain of normalizers becomes a challenging task, in the absence of a suitable notion of rigid commutators. This problem is addressed here from an alternative point of view. We propose a more general framework for the normalizer chain, defining a chain of idealizers in a Lie ring over ℤm whose elements are represented by integer partitions. We show how the corresponding idealizers are generated by subsets of partitions into at most m − 1 parts and we conjecture that the idealizer chain grows as the normalizer chain in the symmetric group. As evidence of this, we establish a correspondence between the two constructions in the case m = 2.

Congruences modulo powers of 5 for the crank parity function

In 1988, Andrews and Garvan introduced the partition statistic “crank” in order to give combinatorial interpretation for Ramanujan’s celebrated partition congruence modulo 11. In 2009, Choi, Kang and Lovejoy established congruences modulo powers of 5 for the crank parity function pC (n) by utilizing the theory of modular forms, where pC (n) denotes the difference between the number of partitions of n with even crank and the number of partitions of n with odd crank. In this paper, we provide a completely elementary proof of this congruence family. Moreover, we also prove several new congruences modulo small powers of 5.

Distributed Optimization for Autonomous Restoration in DER-Rich Distribution System

Autonomous service restoration (ASR) of active distribution network (ADN) reduces service restoration time with the help of measurement devices, smart switches and distributed energy resources (DERs) considering the system's operational and radiality constraints. Restoration scheme can be deployed in both centralized and distributed manner. However, with increasing data points, the requirement for measurement availability at the control center for the decision support makes centralized optimization challenging. The decomposition and coordination scheme of distributed algorithms enhances its ability to solve large scale optimization problems. Moreover, distributed optimization enables robustness, scalability and resiliency compared to centralized optimization. In this work, a) the distributed ASR problem is solved using a novel penalty-driven distributed alternating direction method of multipliers algorithm (PD-ADMM), b) a switch level decomposition and coordination technique is proposed, c) load restoration, DER utilization are considered directly and radiality of ADN is enforced using commodity flow model. The applicability of the algorithm is validated using modified IEEE 33-bus, IEEE 123-bus and 1069-bus test systems for contingency cases, communication failure, scalability, accuracy and sensitivity to number of partitions, starting points, change in DER / solar generation.

Finding biclique partitions of co-chordal graphs

The biclique partition number (bp) of a graph G is referred to as the least number of complete bipartite (biclique) subgraphs that are required to cover the edges of the graph exactly once. In this paper, we show that the biclique partition number of a co-chordal (complementary graph of chordal) graph G=(V,E) is less than the number of maximal cliques (mc) of its complementary graph: A chordal graph Gc=(V,Ec). We first provide a general framework of the “divide and conquer” heuristic of finding minimum biclique partitions of co-chordal graphs based on clique trees. Furthermore, a heuristic of complexity O(|V|(|V|+|Ec|)) is proposed by applying a lexicographic breadth-first search to find structures called moplexes. Either heuristic gives us a biclique partition of G with a size of mc(Gc)−1. In addition, we prove that both of our heuristics can solve the minimum biclique partition problem on G exactly if its complement Gc is chordal and clique vertex irreducible. We also show that mc(Gc)−2≤bp(G)≤mc(Gc)−1 if G is a split graph.

Задача о беспорядках

The problem of characterizing symmetric necklaces as disorders is considered. Such an idea is an alternative approach to the one using Poya's enumeration theorem, including Burnside's lemma. The relationship between partitions of numbers and permutation types is shown. Disorder types are associated with necklace types considered as permutations in which no element remains in place, as well as with disorders with fixed elements. The distributions of disorders by types for necklaces up to the seventh degree are studied. The number and enumeration of disorders of each type is carried out, and their evenness is established. Particular attention is paid to the study of the symmetric necklaces properties. Classes of symmetry and asymmetry are enumerated for each necklace type considered. At the same time, the concepts of chiral and achiral symmetry are introduced as varieties of axial symmetry. Relationships between the power of symmetry classes and the order of symmetry of necklaces are revealed. The concept of symmetry diagrams for necklaces is introduced. Diagrams could be applied for determining the properties of invariance of symmetric and asymmetric necklaces of a given length. The concept of congruence is used as an equivalence relation between necklaces. It allows implementing a geometric approach to the study of necklaces and the visualization of the results obtained. For this purpose, necklaces are associated with multiconnected non-oriented graphs, the vertices of which are the vertices of regular polygons. In this case, the number of vertices in the graph corresponds to the necklace’s length, and the number of connected components coincides with the number of bead colors. As an application of the results obtained, the possibility of studying the accent dynamics of poetic stanzas vertically using necklaces is considered. Empirical data on the use of various types of seven-line stanzas in poetic practice are presented. It is noted that approximately a quarter of the existing symmetric types of necklaces for various reasons do not find practical application.

Quantifying the uncertainty of partitions for infinite mixture models

Bayesian clustering models, such as Dirichlet process mixture models (DPMMs), are sophisticated flexible models. They induce a posterior distribution on the set of all partitions of a set of observations. Analysing this posterior distribution is of great interest, but it comes with several challenges. First of all, the number of partitions is overwhelmingly large even for moderate values of the number of observations. Consequently the sample space of the posterior distribution of the partitions is not explored well by MCMC samplers. Second, due to the complexity of representing the uncertainty of partitions, usually only maximum a posteriori estimates of the posterior distribution of partitions are provided and discussed in the literature. In this paper we propose a numerical and graphical method for quantifying the uncertainty of the clusters of a given partition of the data and we suggest how this tool can be used to learn about the partition uncertainty.

A Symmetrical Fuzzy Neural Network Regression Method Coordinating Structure and Parameter Identifications for Regression

Fuzzy neural networks have both the interpretability of fuzzy systems and the self-learning ability of neural networks, but they will face the challenge of “rule explosion” when dealing with high-dimensional data. Moreover, the structure and parameter identifications of models are generally performed in two stages, and this always attends to one thing and loses another in terms of interpretability and predictive performance. In this paper, a fuzzy neural network regression method (FNNR) that coordinates structure identification and parameter identification is proposed. To alleviate the problem of rule explosion, the structure identification and parameter identification are coordinated in the training process, and the numbers of fuzzy rules and fuzzy partitions are effectively limited, while the parameters of fuzzy rules are optimized. The symmetrical architecture of the FNNR is designed for automatic structure identification. An alternate training strategy is adopted by treating discrete and continuous parameters differently, and thus the convergence efficiency of the algorithm is improved. To enhance interpretability, regularized terms are designed from fuzzy rule level and fuzzy partition level to guide the model to learn fuzzy rules with simple structures and clear semantics. The experimental results show that the proposed method has both a compact structure and high precision.

Universal Quantum Optimization with Cold Atoms in an Optical Cavity.

Cold atoms in an optical cavity have been widely used for quantum simulations of many-body physics, where the quantum control capability has been advancing rapidly in recent years. Here, we show the atom cavity system is universal for quantum optimization with arbitrary connectivity. We consider a single-mode cavity and develop a Raman coupling scheme by which the engineered quantum Hamiltonian for atoms directly encodes number partition problems. The programmability is introduced by placing the atoms at different positions in the cavity with optical tweezers. The number partition problem solution is encoded in the ground state of atomic qubits coupled through a photonic cavity mode, which can be reached by adiabatic quantum computing. We construct an explicit mapping for the 3-SAT and vertex cover problems to be efficiently encoded by the cavity system, which costs linear overhead in the number of atomic qubits. The atom cavity encoding is further extended to quadratic unconstrained binary optimization problems. The encoding protocol is optimal in the cost of atom number scaling with the number of binary degrees of freedom of the computation problem. Our theory implies the atom cavity system is a promising quantum optimization platform searching for practical quantum advantage.

Enumerating numerical sets associated to a numerical semigroup

A numerical set T is a subset of N0 that contains 0 and has finite complement. The atom monoid of T is the set of x∈N0 such that x+T⊆T. Marzuola and Miller introduced the anti-atom problem: how many numerical sets have a given atom monoid? This is equivalent to asking for the number of integer partitions with a given set of hook lengths. We introduce the void poset of a numerical semigroup S and show that numerical sets with atom monoid S are in bijection with certain order ideals of this poset. We use this characterization to answer the anti-atom problem when S has small type.

Vectorizing and distributing number‐theoretic transform to count Goldbach partitions on Arm‐based supercomputers

SummaryIn this article, we explore the usage of scalable vector extension (SVE) to vectorize number‐theoretic transforms (NTTs). In particular, we show that 64‐bit modular arithmetic operations, including modular multiplication, can be efficiently implemented with SVE instructions. The vectorization of NTT loops and kernels involving 64‐bit modular operations was not possible in previous Arm‐based single instruction multiple data architectures since these architectures lacked crucial instructions to efficiently implement modular multiplication. We test and evaluate our SVE implementation on the A64FX processor in an HPE Apollo 80 system. Furthermore, we implement a distributed NTT for the computation of large‐scale exact integer convolutions. We evaluate this transform on HPE Apollo 70, Cray XC50, HPE Apollo 80, and HPE Cray EX systems, where we demonstrate good scalability to thousands of cores. Finally, we describe how these methods can be utilized to count the number of Goldbach partitions of all even numbers to large limits. We present some preliminary results concerning this problem, in particular a histogram of the number of Goldbach partitions of the even numbers up to 240.

Adaptive Bayesian learning for making risk-aware decisions: A case of trauma survival prediction

Decision tree (DT) models provide a transparent approach to prediction of patient’s outcomes within a probabilistic framework. Averaging over DT models under certain conditions can deliver reliable estimates of predictive posterior probability distributions, which is of critical importance in the case of predicting an individual patient’s outcome. Reliable estimations of the distribution can be achieved within the Bayesian framework using Markov chain Monte Carlo (MCMC) and its Reversible Jump extension enabling DT models to grow to a reasonable size. Existing MCMC strategies however have limited ability to control DT structures and tend to sample overgrown DT models, making unreasonably small partitions, thus deteriorating the uncertainty calibration. This happens because the MCMC explores a DT model parameter space within a limited knowledge of the distribution of data partitions. We propose a new adaptive strategy which overcomes this limitation, and show that in the case of predicting trauma outcomes the number of data partitions can be significantly reduced, so that the unnecessary uncertainty of estimating the predictive posterior density is avoided. The proposed and existing strategies are compared in terms of entropy which, being calculated for predicted posterior distributions, represents the uncertainty in decisions. In this framework, the proposed method has outperformed the existing sampling strategies, so that the unnecessary uncertainty in decisions is efficiently avoided.

Design optimization and analysis of a multi-temperature partition and multi-configuration integrated organic Rankine cycle for low temperature heat recovery

With the development of society and economy, more and more attention has been paid to energy saving where low temperature heat plays an important role. As an effective means to utilize low temperature heat, the organic Rankine cycle attracts great attention, where interval partition number, the optimization of interval temperature, system configuration optimization, working fluid selection and matching with heat source have significant influence on system performance. It is challenging to optimize organic Rankine cycle system considering these issues simultaneously. This study proposes a multi-temperature partition and multi-configuration integrated organic Rankine cycle system model to improve low temperature heat utilization and a comprehensive comparison with basic organic Rankine cycle, recuperative organic Rankine cycle and regenerative organic Rankine cycle is conducted to assess thermal and economic performance with ten working fluids. A numerical optimization analysis compared with other literature and the result of traversal, is performed to ensure the accuracy of mathematical models under the same settings. A series of thermal analysis for organic Rankine cycle systems with multiple system configurations, working fluids selection, condition of heat source and so on are implemented considering internal temperature optimization with evaporation temperature and regenerative ratio. A solution strategy framework is established for the proposed system models. In order to verify the practicability of the proposed model, a case study is built with refinery diesel as the heat source. The result shows that thermal efficiency of the proposed system model has an increase of 4.95–14.01 % compared with the basic organic Rankine cycle and the one with single interval temperature has better performance with 4.95 % higher thermal efficiency and 1.45 % higher total annual cost. Thus, the proposed system model and its corresponding solution strategy can be applied for low temperature heat recovery.

Optical experimental solution for the multiway number partitioning problem and its application to computing power scheduling

Optical experimental solution for the multiway number partitioning problem and its application to computing power scheduling

A new variant of the fem for evaluation the strenght of structures of complex geometry with heterogeneous material structure

Structures are known to feature a complex geometry and a complex material structure which varies depending on their functional purpose, technological capabilities and design. The finite difference method, collocation method, variational method, theoretical and experimental method, finite element method, among other methods, are employed to calculate the stress-strain state (SST) of structures of complex geometry. The finite element method (FEM) has become the most common one of them all. The spline version of the finite element method (SV FEM) has proven to be an effective FEM method used to calculate the SST of structures of complex geometry. Unlike the conventional FEM method, a preliminary parametrization of the entire area of complex geometry using the parameters outlined by the canonical domain and a cubic approximation of the desired variables within each element yields consistent finite elements. The unbroken geometric continuity maintained between the elements along with the continuity of the desired quantities and their first derivatives along the line of contact of the elements makes it possible to obtain reliable results with just a small number of partitions into finite elements. One should note the application of the basics of the spline version of the finite element method in two-dimensional SV FEM-2 and three-dimensional SV FEM-3 arrangements. One should also make mention of the idea of producing a synthesized version of SV FEM. Two-dimensional and three-dimensional elements being synthesized into one element would usher in a new phase in the development of the SV FEM approach which would describe the complex structures of construction materials in addition to complex geometry. SV FEM has made it possible to evaluate the SST of structural elements of complex geometry along with the thin layers that penetrate it in all three directions. The versions of SV FEM that are available now serve to assess the SST of a broad range of various thin-walled and non-thin-walled structures of complex geometry, including multilayer structures with thin layers in all three directions, with hard and soft coatings along the edges and with local defects on the surfaces of structures. This paper presents a number of SST calculations of specific structures that have been carried out via the FEM versions described above.

The Deranged Bell Numbers

ABSTRACT Based on the combinatorial interpretation of the ordered Bell numbers, which count all the ordered partitions of the set [n] = {1, 2, … n}, this paper introduces the deranged partition as a free-fixed-block permutation of its blocks, then defines the deranged Bell numbers that count the total number of the deranged partitions of [n]. At first, we study the classical properties of these numbers (generating function, explicit formula, convolutions, etc.), we then present an asymptotic behavior of the deranged Bell numbers. Finally, we briefly review some results regarding the r-extension of these numbers by considering that the r first elements must be in distinct blocks.

Scalable and Secure Virtualization of HSM With ScaleTrust

Hardware security modules (HSMs) have been utilized as a trustworthy foundation for cloud services. Unfortunately, existing systems using HSMs fail to meet multi-tenant scalability arising from the emerging trends such as microservices, which utilize frequent cryptographic operations. As an alternative, cloud vendors provide HSMs as a service. However, such cloud-managed HSM usage models raise security concerns due to their untrusted and shared operating environment. We propose ScaleTrust, a scalable and secure system for key management. ScaleTrust allows us to scale the number of virtual HSM partitions, each of which is isolated with respect to each other and is robust against cloud insider attacks, while preserving physical isolation of the root of trust. To enable this, ScaleTrust uses Intel SGX and multiple HSM features, such as restricting key usage by controlling key attributes of in-HSM keys and establishing a secure channel using only HSM commands. Finally, we apply ScaleTrust to four real-world systems: Keyless SSL for TLS private key offloading, JSON Web Token authentication for microservices, key provisioning, and encryption in database systems. Our evaluation shows that ScaleTrust achieves multi-tenancy in a scalable way by providing multiple virtual HSMs with legacy HSM devices that are designed to support a single tenant. ScaleTrust provides security against insider threats while incurring 11.9% and 39.0% of end-to-end throughput and latency overhead for Keyless SSL compared to stand-alone HSMs.