Number Of Partitions Research Articles

On the number of partitions of a number into distinct divisors

Let $p_{\textrm{dsd}} (n)$ be the number of partitions of $n$ into distinct squarefree divisors of $n$. In this note, we find a lower bound for $p_{\textrm{dsd}} (n)$, as well as a sequence of $n$ for which $p_{\textrm{dsd}} (n)$ is unusually large.

Arithmetic properties of 5-regular partitions into distinct parts

A partition is said to be [Formula: see text]-regular if none of its parts is a multiple of [Formula: see text]. Let [Formula: see text] denote the number of 5-regular partitions into distinct parts (equivalently, into odd parts) of [Formula: see text]. This function has also close connections to representation theory and combinatorics. In this paper, we study arithmetic properties of [Formula: see text]. We provide full characterization of the parity of [Formula: see text], present several congruences modulo 4, and prove that the generating function of the sequence [Formula: see text] is lacunary modulo any arbitrary positive powers of 5.

Green upgrading of SPring-8 to produce stable, ultrabrilliant hard X-ray beams.

SPring-8-II is a major upgrade project of SPring-8 that was inaugurated in October 1997 as a third-generation synchrotron radiation light source. This upgrade project aims to achieve three goals simultaneously: achievement of excellent light source performance, refurbishment of aged systems, and significant reduction in power consumption for the entire facility. A small emittance of 50 pm rad will be achieved by (1) replacing the existing double-bend lattice structure with a five-bend achromat one, (2) lowering the stored beam energy from 8 to 6 GeV, (3) increasing the horizontal damping partition number from 1to 1.3, and (4) enhancing horizontal radiation damping by installing damping wigglers in long straight sections. The use of short-period in-vacuum undulators allows ultrabrilliant X-rays to be provided while keeping a high-energy spectral range even at the reduced electron-beam energy of 6 GeV. To reduce power consumption, the dedicated, aged injector system has been shut down and the high-performance linear accelerator of SACLA, a compact X-ray free-electron laser (XFEL) facility, is used as the injector of the ring in a time-shared manner. This allows the simultaneous operation of XFEL experiments at SACLA and full/top-up injection of the electron beam into the ring. This paper overviews the concept of the SPring-8-II project, the system design of the light source and the details of the accelerator component design.

Probabilities of transitions among endogenous regimes in asset returns and Environmental, Social and Governance scores

AbstractAssets' returns can be efficiently clustered in regimes, that are suitably defined non‐overlapping intervals creating a partition of the real numbers. This paper explores the relationship between the transition probabilities from one regime to another in assets' returns and the assets' MSCI Environmental, Social and Governance (ESG) scores. We apply the proposed methodology to the relevant empirical instance of the assets in the STOXX® Global 1800 Index. We consider three regimes—low, medium and high, on the basis of the variation range of the considered returns. Regimes are endogenous, in that their identification comes out from an entropy‐based optimization problem over the possible ranges of variation of the returns. We specifically investigate the possible linear relationship between transition probabilities among regimes and the ESG scores for different geographic regions, namely, America, Europe and Asia Pacific. The reference empirical period is the quadrennium 2018–2021. Results suggest that assets that are low ranked in ESG tend to remain in the low state of returns, if they are in the low state, while they tend to switch from higher to lower return states when the initial state is higher. On the other hand, assets that are highly ranked in the ESG dimensions, are likely to switch from a lower to a higher return state, when they are in a lower state or to remain in the same state when they are in a higher state. Results are more evident for America and Asia Pacific regions rather than Europe where regulation on ESG integration is at a more developed stage with respect to the other regions.

Simulating adiabatic quantum computing with parameterized quantum circuits

Abstract Adiabatic quantum computing is a universal model for quantum computing whose implementation using a gate-based quantum computer requires depths that are unreachable in the early fault-tolerant era. To mitigate the limitations of near-term devices, a number of hybrid approaches have been pursued in which a parameterized quantum circuit prepares and measures quantum states and a classical optimization algorithm minimizes an objective function that encompasses the solution to the problem of interest. In this work, we propose a different approach starting by analyzing how a small perturbation of a Hamiltonian affects the parameters that minimize the energy within a family of parameterized quantum states. We derive a set of equations that allow us to compute the new minimum by solving a constrained linear system of equations that is obtained from measuring a series of observables on the unperturbed system. We then propose a discrete version of adiabatic quantum computing that can be implemented in a near-term device while at the same time is insensitive to the initialization of the parameters and to other limitations hindered in the optimization part of variational quantum algorithms. We compare our proposed algorithm with the variational quantum eigensolver on two classical optimization problems, namely MaxCut and number partitioning, and on a quantum-spin configuration problem, the transverse-field ising chain model, and confirm that our approach demonstrates superior performance.

Energies of an Electron in a One-Dimensional Lattice Using the Dirac Equation: The Coulomb Potential

The energies of an electron in a one-dimensional crystal are studied with both the Schrödinger and Dirac equations using the plane wave expansion method. The crystalline potential sensed by the electron in a cell was calculated by accounting for the Coulombic (electrostatic) interaction between the electron and the surrounding cores (immobile positive ions at the center of the crystal cells). The energies and wave functions of the electron were calculated as a function of four parameters: the period ap of the lattice, the dimension ndim of the matrix in the momentum space, the partition number lpa in which the unit cell is divided to calculate the potential and the number of cores nco that affect the electron. It was found that 8000 cores (surrounding the electron) were needed to reach our convergence criterion. An analytical equation that accurately describes the behavior of the energies in function of the cores that affect the electron was also found. As case studies, the energies for pseudo-lithium and pseudo-graphene were obtained as a first approximation for one-dimensional lattices. Subsequently, the energies of an isolated dimer nanoparticle were also calculated using the supercell method.

Experimental study of the design and performance optimisation of a self-heating methanol on-board reforming reactor for hydrogen production

There is an increasing research on efficient and compact on-board reformer, based on which a serpentine plate type self-heating reformer integrating preheating, reforming and combustion has been designed and fabricated, which can be disassembled and assembled flexibly. In this paper, experiments on the self-heating reformer have been carried out to optimise the differential structure of different chambers. It can be found that: the combustion chamber needs to be partitioned and the resistance loss is small and stable for a long time when the number of partitions is 3. The combustion chamber with O2/CH3OH = 1.5 has the highest wall temperature of 240 °C. Hydrogen yield can be increased to more than two times after optimising the evaporation chamber. The catalyst in the reforming chamber must be uniformly distributed, and the methanol conversion rate is stable at 75 % with a mass of 110 g and Weight Hourly Space Velocity (WHSV) = 8 h−1. The optimum operating conditions of the reactor is WHSV = 8 h−1 with air flow rate of 25 L/min, at which the reactor starts up within 20 min, the hydrogen yield stabilises within 40 min, and the hydrogen yield is 9.8 L/min and remains stable for 120 min.

Study on the influence of geometric characteristics of micro-periodic structures with slits on sound absorption properties

In recent years, there has been considerable interest in acoustic metamaterial technology that leverages sub-wavelength scale microperiodic structures to control the behavior of sound waves for sound absorption material development. This technology offers high design flexibility in unit shapes and holds promise for the development of sound absorption materials with diverse characteristics. However, the field remains in its foundational exploration phase, with the detailed relationship between geometric shapes and sound absorption properties yet to be fully elucidated. To address this gap, we conducted a thorough investigation focusing on their relationship by examining the transport parameters of the Johnson-Champoux-Allard-Lafarge-Pride (JCALP) model, which are closely associated with sound absorption properties. We designed a cubic lattice with slits on the incident surface of the unit shape as a microperiodic structure and examined the normal incident sound absorption characteristics using the finite element method. Our results indicate that manipulation of slit partition numbers and slit widths allows for control of transport parameters, suggesting the potential to achieve various sound absorption properties.

Enhanced performance of cyclone separator through coupling of centrifugal force, electrostatic force and particle pre-charge

The electrostatic enhancement can effectively improve the performance of cyclone separator at low load or high-temperature, but its application is limited owing to the reduction of spark voltage in high-temperature. Aiming at this question, a new technology coupling centrifugal force, electrostatic force and particle pre-charge was proposed. The characteristics of particles charging and agglomeration were analyzed, and the influence of different factors on the change of particle partition number concentration and total removal efficiency of cyclone separator were studied. The results showed that, increasing the charge generator voltage and decreasing the main pipe gas velocity can increase the particles charge quantity and improve agglomeration, which is benefit for the efficiency improvement of either cyclone separator or electrostatic cyclone separator, especially at low inlet gas velocity. Higher charge generator voltage and lower inlet gas velocity, more obvious efficiency improvement. The total removal efficiency can be increased by up to 10 % and 9 % respectively for the cyclone separator and electrostatic cyclone separator because of particle pre-charge at 10 m/s of inlet gas velocity, and the removal efficiency of fine particles can also be effectively improved. Comparing with the electrostatic cyclone separator, achieving the same total removal efficiency, the internal voltage can be reduced 16 kV through particle pre-charge. The electrostatic force and centrifugal force were the main factor of the small particles and big particles removal, respectively.

Billiard partitions, Fibonacci sequences, SIP classes, and quivers

Starting from billiard partitions which arose recently in the description of periodic trajectories of ellipsoidal billiards in d d -dimensional Euclidean space, we introduce a new type of separable integer partition classes, called type B. We study the numbers of basis partitions with d d parts and relate them to the Fibonacci sequence and its natural generalizations. Remarkably, the generating series of basis partitions can be related to the quiver generating series of symmetric quivers corresponding to the framed unknot via knots-quivers correspondence, and to the count of Schröder paths.

On the Number of Partitions of $n$ into Exactly $m$ Parts Whose Even Parts are Distinct

On the Number of Partitions of $n$ into Exactly $m$ Parts Whose Even Parts are Distinct

Effect of partition arrangement of metal hydrides and phase change materials on hydrogen absorption performance in the metal hydride reactor

Metal hydride (MH) suffers from strong thermal effects during hydrogen absorption. To fully utilize the reaction heat during hydrogen absorption, phase change material (PCM) can be used to achieve efficient thermal management in the reactor. A novel MH-PCM reactor with the partition arrangement of MH and PCM was proposed in this paper. Then, the two-dimensional multiphysics coupled model of the reactor was developed, and the effects of the number of partitions n, the mass of PCM and the hydrogen absorption pressure Pa on the hydrogen absorption performance were discussed detailedly. The results indicated that the novel MH-PCM reactor not only increases the heat transfer area and boots the hydrogen absorption rate, but also improves the uniformity of temperature distribution. As the n increases from 2 to 5, the hydrogen absorption time gradually shortens. Compared to the conventional MH-PCM reactor, the time for 90% hydrogen absorption shortens by 67.13%. Furthermore, the hydrogen absorption rate can be enhanced with the increasing mass of PCM and Pa. However, the larger these values are, the weaker the enhancement effect becomes. The results of this paper may provide a new direction for the optimized design and engineering application of MH-PCM reactors.

Partition theorems and the Chinese Remainder Theorem

The famous partition theorem of Euler states that partitions of n into distinct parts are equinumerous with partitions of n into odd parts. Another famous partition theorem due to MacMahon states that the number of partitions of n with all parts repeated at least once equals the number of partitions of n where all parts must be even or congruent to 3(mod6). These partition theorems were further extended by Glaisher, Andrews, Subbarao, Nyirenda and Mugwangwavari. In this paper, we utilize the Chinese Remainder Theorem to prove a comprehensive partition theorem that encompasses all existing partition theorems. We also give a natural generalization of Euler's theorem based on a special complete residue system. Furthermore, we establish interesting congruence connections between the partition function p(n) and related partition functions.

Colour Class Domination equivalence partition and Colourful resolving sets in graphs

Let H = (V,E) be a simple graph. A subset S of V(H) is an equivalence set if the subgraph induced by S is component-wise complete. Let P = {V1,V2, , Vk} be a partition of V (H) where each Vi,1 ≤ i ≤ k is an equivalence set of H. The partition P is called a colourful domination equivalence partition if the set D = {u1,u2, , uk} where ui,1 ≤ i ≤ k are suitably chosen vertices one each from Vi, 1 ≤ i ≤ k is a dominating set. The trivial partition where every partite set is a singleton is an example of a colourful domination equivalence partition. The minimum cardinality of such a partition is called colourful domination equivalence partition number of G and is denoted by χγeq (H). Asubset S = {u1,u2, ,ur}ofV(H)iscalledaresolvingset of H if forany v in V (H), the codeofv withrespecttoS namely(d(v,u1), , d(v,ur)) (denotedbycS(v)) is distinct for distinct v where d(v,ui) denotes the distance between v and ui . Since u1, , ur are evidently resolved by S, it is enough if vertices in V − S are resolved by S. Given a proper colour partition P = {V1,V2, , Vk}, a resolving set S = {u1,u2, , uk} is said to be colourful with respect to P if ui ∈ Vi for every i, 1 ≤ i ≤ k. A proper colour partition P is said to be a colour resolving colour partition if some set of vertices, one each from each colour class is a revolving set. The minimum cardinality of a resolving set of H is called the metric dimension of G [8] and is denoted by Dim(H). The trivial partition consisting of n singletons where n is the order of H is a colourful resolving colour partition of H. The minimum cardinality of a colourful resolving colour partition of H is denoted by χre(H). Astudy of these two new parameters is initiated in this paper.

Entropy‐driven Progressive Compression of 3D Point Clouds

Abstract3D point clouds stand as one of the prevalent representations for 3D data, offering the advantage of closely aligning with sensing technologies and providing an unbiased representation of a measured physical scene. Progressive compression is required for real‐world applications operating on networked infrastructures with restricted or variable bandwidth. We contribute a novel approach that leverages a recursive binary space partition, where the partitioning planes are not necessarily axis‐aligned and optimized via an entropy criterion. The planes are encoded via a novel adaptive quantization method combined with prediction. The input 3D point cloud is encoded as an interlaced stream of partitioning planes and number of points in the cells of the partition. Compared to previous work, the added value is an improved rate‐distortion performance, especially for very low bitrates. The latter are critical for interactive navigation of large 3D point clouds on heterogeneous networked infrastructures.

A method to shorten the time of high-dimensional matrix operations and convolution operations

Serverless computing, or Function-as-a-Service (FaaS), is a cloud computing execution model where the cloud provider dynamically manages the allocation and provisioning of servers which is a new way to solve the problem that the need of client.By serverless computing clients can easily solve their problems anytime,anywhere.However, the complex convolution operations and high-dimensional matrix operations can still consume a lot of resources such as computation time and memory occupancy in serverless computing. In this paper, firstly, the author selects two frequently used models in serverless computing, and then divide them into different numbers of partitions respectively to study the impact of partitioning methods on the computation time and memory occupancy. The author show that different partitioning methods can make an obviously impact on the computation time and memory occupancy,whitch shows that the proper partitioning of high-dimensional matrices and tensors can make a significantly reduction of calculation time and memory occupancy.

Partitioned least squares

Linear least squares is one of the most widely used regression methods in many fields. The simplicity of the model allows this method to be used when data is scarce and allows practitioners to gather some insight into the problem by inspecting the values of the learnt parameters. In this paper we propose a variant of the linear least squares model allowing practitioners to partition the input features into groups of variables that they require to contribute similarly to the final result. We show that the new formulation is not convex and provide two alternative methods to deal with the problem: one non-exact method based on an alternating least squares approach; and one exact method based on a reformulation of the problem. We show the correctness of the exact method and compare the two solutions showing that the exact solution provides better results in a fraction of the time required by the alternating least squares solution (when the number of partitions is small). We also provide a branch and bound algorithm that can be used in place of the exact method when the number of partitions is too large as well as a proof of NP-completeness of the optimization problem.

Influence of an inclined thick partition on free convection heat transfer performance under surface radiation in a hollow block

In the present work, a computational analysis of the combined energy transfer by free convection, radiation, and heat conduction in 2D hollow block with an inclined partition was carried out. The inclined partition was made of a heat-conducting material and had a finite thickness. The closed space inside the brick was filled with air (Pr = 0.71) and the airflow is laminar. The external surfaces of the vertical borders are considered to be isothermal, and the remaining horizontal surfaces are supposed to be adiabatic. The finite difference technique is employed to work out numerically the differential equations. The in-house computational code was verified in detail using various problems. The major characteristics that determine the considered phenomena are surface emissivity of internal walls, Ra number, and material of solid walls and partition. As a result of the research, the distributions of streamlines and isotherms combined with the average Nusselt numbers were obtained. Depending on the material of the partition, the flow structure in the cavity changes significantly, which reflects the distribution of streamlines, namely, for the materials with low thermal conductivity the convective cell in the upper part of the cavity is elongated, while in the lower part it is shifted to the lower and left borders. The isotherms reflect the more intensive heating in the left part of the area. With an increase in the thermal conductivity coefficient, the streamlines reflecting the nature of the flow are restructured. It was shown that the presence of an inclined partition significantly reduces the intensity of convective heat transfer.

A method to enhance Apache Spark performance based on data segmentation and configuration parameters settings

When using modern big data processing tools, there is a problem of increasing the productivity of using modern frameworks in the context of effective setting of various configuration parameters. The object of the research is computational processes of processing big data with the use of technologies of high-performance frameworks. The subject is methods and approaches to the effective setting of configuration parameters of frameworks in the conditions of limitations of virtualization environments and local resources. The purpose of the study is to improve the performance of Apache Spark and Apache Hadoop deployment modes based on a combined approach that includes preprocess segmentation of input data and setting of basic and additional configuration parameters that take into account the limitations of the virtual environment and local resources. Achieving the set goal involves the following tasks: create a synthesized set of WordCount test data for using input data segmentation methods. Determine the composition of general and specific Apache Spark and Apache Hadoop configuration parameters that most affect the performance of frameworks in Spark Standalone and Hadoop Yarn (FIFO) deployment modes. Justify changes in the values of the configuration parameters (accepted by default) by setting the level of parallelism, the number of partitions of the input file according to the number of processor cores, the number of tasks assigned to each core and the system executor. Conduct experimental research to substantiate theoretical results and prove their use in practice. Methods. The research used the following methods: statistical analysis; a method of generating test data based on defined segmentation characteristics with arbitrary volumes of data; a systematic approach for comprehensive evaluation and analysis of performance of frameworks based on selected configuration parameters. The results. On the basis of the developed system of parameters for evaluating the performance of the studied frameworks, experiments were carried out, which include: the application of the method of segmentation of input data based on the division of the input file into paragraphs (lines) for different values of the ranges of the number of words and the number of letters in each word; setting the main parameters and specific ones, in particular, partitioning and parallelism, taking into account the characteristics of the virtual environment and the local resource. According to the obtained results, a detailed analysis of the use of the proposed methods to improve the performance of the studied frameworks with recommendations for choosing the optimal values of data segmentation parameters and configuration parameters was carried out. You are snowmen. The obtained results of the experiments allow us to conclude that the use of the proposed methods of setting the configuration parameters of Spark and Hadoop will increase the processing productivity: for small files (0.5–1 GB) on average up to 25–30%, for large ones (1.5–2.5 GB ) – up to 10–20% on average. At the same time, the average value of the execution time of one task decreased by 10-15% for files of different sizes and with different number of words in a line.

An algebraic global linelet preconditioner for incompressible flow solvers

Given the wide range of scales present in fluid dynamics, highly anisotropic meshes are often employed to resolve boundary layer flow at high Reynolds numbers. However, such meshes result in distorted elements, which can deteriorate the convergence of the preconditioned conjugate gradient solver (PCG) used to solve the discrete pressure equations. To address this issue, the Linelet Preconditioner is commonly used to accelerate the convergence rate of the PCG by building line segments (linelets) along the direction of the strongest couplings and applying a specific operation to each linelet. Under the current state of the art, linelets are constrained to operate independently within each domain partition. In these scenarios, PCG convergence was observed to deteriorate when the number of domain partitions is increased. This work proposes a communication step to be integrated into the preconditioning step, enabling different linelets to exchange information. The proposed method is observed to improve the convergence rate of linear systems with high stiffness due to highly stretched elements. Also, an algorithm is developed to generate the preconditioning matrix by purely algebraic considerations, in contrast to the usual approach of building linelets based on geometrical considerations. Furthermore, since the current approach is agnostic to the domain partition, it does not impose any constraint on the domain decomposition. Hence, the proposed method is a robust preconditioner with a numerical performance almost independent of the quality of the domain decomposition, providing a way for a more balanced load distribution.