An Estimate of the Enumeration Function of Multiplicative Partitions

Let f(n) denote the number of multiplicative partitions of a positive intger. We prove that if n is an odd number,then f(n)≤n/15+7/5.

Entropy of Partitions on Quantum Logic

Partition and entropy of partitions in quantum logic are introduced and their properties are investigated.The results are generalized to the general case of T-norm and T-conorm.

A New Formula for Partitions in a Set of Entities into Empty and Nonempty Subsets, and Its Application to Stochastic and Agent-Based Computational Models

In combinatorics, a Stirling number of the second kind S (n,k)? is the number of ways to partition a set of n objects into k nonempty subsets. The empty subsets are also added in the models presented in the article in order to describe properly the absence of the corresponding type i of state in the system, i.e. when its “share” Pi =0?. Accordingly, a new equation for partitions P (N, m)? in a set of entities into both empty and nonempty subsets was derived. The indistinguishableness of particles (N identical atoms or molecules) makes only sense within a cluster (subset) with the size?0≤ni ≥N. The first-order phase transition is indeed the case of transitions, for example in the simplest interpretation, from completely liquid state?typeL = {n1 =N, n2 = 0} to the completely crystalline state??typeC= {n1 =0, n2 = N }. These partitions are well distinguished from the physical point of view, so they are ‘typed’ differently in the model. Finally, the present developments in the physics of complex systems, in particular the structural relaxation of super-cooled liquids and glasses, are discussed by using such stochastic cluster-based models.

Root-Patterns to Algebrising Partitions

The study of the confluences of the roots of a given set of polynomials—root-pattern problem— does not appear to have been considered. We examine the situation, which leads us on to Young tableaux and tableaux representations. This in turn is found to be an aspect of multipartite partitions. We discover, and show, that partitions can be expressed algebraically and can be “differentiated” and “integrated”. We show a complete set of bipartite and tripartite partitions, indicating equivalences for the root-pattern problem, for select pairs and triples. Tables enumerating the number of bipartite and tripartite partitions, for small pairs and triples are given in an appendix.

A heuristic clustering algorithm based on high density-connected partitions

Clustering data with varying densities and complicated structures is important,while many existing clustering algorithms face difficulties for this problem. The reason is that varying densities and complicated structure make single algorithms perform badly for different parts of data. More intensive parts are assumed to have more information probably,an algorithm clustering from high density part is proposed,which begins from a tiny distance to find the highest density-connected partition and form corresponding super cores,then distance is iteratively increased by a global heuristic method to cluster parts with different densities. Mean of silhouette coefficient indicates the cluster performance. Denoising function is implemented to eliminate influence of noise and outliers. Many challenging experiments indicate that the algorithm has good performance on data with widely varying densities and extremely complex structures. It decides the optimal number of clusters automatically.Background knowledge is not needed and parameters tuning is easy. It is robust against noise and outliers.

Bijections between Lattice Paths and Plane Partitions

By using lattice paths in the three-dimensional space we obtain bijectively an interpretation for the overpartitions of a positive integer n in terms of a set of plane partitions of n . We also exhibit two bijections between unrestricted partitions of n and different subsets of plane partitions of n .

Cytotoxic Effect of Methanol Extracts and Partitions of Two Mexican Desert Plants against the Murine Lymphoma L5178Y-R

Pachycereus marginatus (DC.) Britton & Rose and Ibervillea sonorae (S. Watson) Greene have been used in the Mexican traditional medicine for the treatment of various diseases, including cancer. The present study aims to investigate the cytotoxic activity of these plants against a murine lymphoma. Soxhlet extraction of dried and powdered plant material was performed with methanol. Also, a further partitioning of these methanolic extracts with hexane and ethyl acetate was achieved. The in vitro cytotoxic activity against the murine lymphoma L5178Y-R cell line was assessed via the colorimetric MTT assay. The methanol extract from P. marginatus exhibited high cytotoxic activity (up to 94%) at concentrations ranging from 3.9 to 500 μg/mL;however, hexane and ethyl acetate partitions from this methanolic extract showed lower but significant (p < 0.05) concentration-dependent cytotoxicity (hexane partition up to 94% at 500 μg/mL;ethyl acetate partition up to 94% at 65.5 μg/mL). The methanolic extract and partitions derived from I. sonorae also showed significant (p < 0.05) and concentration-dependent cytotoxicity against L5178Y-R cells at concentrations ranging from 7.81 to 500 μg/mL (methanolic extract up to 63% at 500 μg/mL;hexane partition up to 76% at 250 μg/mL;ethyl acetate partition up to 73% at 500 μg/mL). These results demonstrate that the methanol extracts and partitions from P. marginatus and I. sonorae possess significant cytotoxic activity against the murine lymphoma L5178Y-R and validate the ethnobotanical use of these plants for the treatment of diseases consistent with cancer symptomatology. Previous scientific reports describe the isolation of isoquinoline alkaloids of P. marginatus as well as cucurbitacins from I. sonorae, phytochemicals that could be responsible for their observed cytotoxic activity in this research. The direct extraction with methanol of medicinal plants allows extracting of both high and low-polarity compounds, contrary to the simple extraction with water that only allows obtaining compounds of high polarity. The subsequent partition of the methanol extract with a solvent of low polarity (hexane) and another of medium polarity (ethyl acetate) allows making a preliminary fractionation of the bioactive molecules present in the plant that will facilitate the bioguided chromatographic isolation of the pure compounds responsible for the biological activity of the plant.

(k,k-1)-双正则图的平衡Judicious Partitions(英文)

Bollobás和Scott提出猜想:任意一个边数为m且最小度大于1的图存在顶点集的平衡二部划分使得每一部分点集的导出子图包含的边数不超过m/3.Bollobás和Scott证明了绝大部分正则图存在顶点集的平衡二部划分使得每一部分点集的导出子图包含的边数比m/4小.这里讨论(k,k-1)-双正则图的平衡二部划分,证明了每一个(k,k-1)-双正则图存在平衡二部划分使得每一部分点集的导出子图包含的边数是m/4左右.

Singly Covered Minimal Elements of Linked Partitions and Cycles of Permutations

Linked partitions were introduced by Dykema(Dykema K J. Multilinear function series and transforms in free probability theory. Adv. Math., 2005, 208(1):351–407) in the study of the unsymmetrized T-transform in free probability theory.Permutation is one of the most classical combinatorial structures. According to the linear representation of linked partitions, Chen et al.(Chen W Y C, Wu S Y J, Yan C H. Linked partitions and linked cycles. European J. Combin., 2008, 29(6): 1408–1426) de?ned the concept of singly covered minimal elements. Let L(n, k) denote the set of linked partitions of [n] with k singly covered minimal elements and let P(n, k) denote the set of permutations of [n] with k cycles. In this paper, we mainly establish two bijections between L(n, k) and P(n, k). The two bijections from a different perspective show the one-to-one correspondence between the singly covered minimal elements in L(n, k) and the cycles in P(n, k).

SYNCHRONIZATION OF CHAOTIC SYSTEMS VIA LINEAR PARTITIONS

In this paper, a new method for synchronization of chaotic systems is proposed. The method is based on the linear partitions of the chaotic system under consideration. A chaotic system is partitioned into a linear part and a nonlinear part. Under some assumptions, another chaotic system which can synchronize with the original one can be contructed. The proposed method is easier to verify and more general than the conventional one in some sense.

The Fractal and the Recurrence Equations Concerning the Integer Partitions

This paper introduced a way of fractal to solve the problem of taking count of the integer partitions, furthermore, using the method in this paper some recurrence equations concerning the integer partitions can be deduced, including the pentagonal number theorem.

A Study on Associated Rules and Fuzzy Partitions for Classification

The amount of data for decision making has increased tremendously in the age of the digital economy. Decision makers who fail to proficiently manipulate the data produced may make incorrect decisions and therefore harm their business. Thus, the task of extracting and classifying the useful information efficiently and effectively from huge amounts of computational data is of special importance. In this paper, we consider that the attributes of data could be both crisp and fuzzy. By examining the suitable partial data, segments with different classes are formed, then a multithreaded computation is performed to generate crisp rules (if possible), and finally, the fuzzy partition technique is employed to deal with the fuzzy attributes for classification. The rules generated in classifying the overall data can be used to gain more knowledge from the data collected.

The Goldbach Conjecture Is True

Let 2m>2, m∈ℤ, be the given even number of the Strong Goldbach Conjecture Problem. Then, m can be called the median of the problem. So, all Goldbach partitions (p,q)exist a relationship, p=m−dand q=m+d, where p≤qand d is the distance from m to either p or q. Now we denote the finite feasible solutions of the problem as S(2m)={ (2,2m−2),(3,2m−3),⋅⋅⋅,(m,m) }. If we utilize the Eratosthenes sieve principle to efface those false objects from set S(2m)in pistages, where pi∈P, pi≤2m, then all optimal solutions should be found. The Strong Goldbach Conjecture is true since we proved that at least one optimal solution must exist to the problem. The Weak Goldbach Conjecture is true since it is a special case of the Strong Goldbach Conjecture. Therefore, the Goldbach Conjecture is true.

Effects of projectile parameters on the momentum transfer and projectile melting during hypervelocity impact

The effects of projectile/target impedance matching and projectile shape on energy,momentum transfer and projectile melting during collisions are investigated by numerical simulation.By comparing the computation results with the experimental results,the correctness of the calculation and the statistical method of momentum transfer coefficient is verified.Different shapes of aluminum,copper and heavy tungsten alloy projectiles striking aluminum,basalt,and pumice target for impacts up to 10 km/s are simulated.The influence mechanism of the shape of the projectile and projectile/target density on the momentum transfer was obtained.With an increase in projectile density and length-diameter ratio,the energy transfer time between the projectile and targets is prolonged.The projectile decelerates slowly,resulting in a larger cratering depth.The energy consumed by the projectile in the excavation stage increased,resulting in lower mass-velocity of ejecta and momentum transfer coefficient.The numerical simulation results demonstrated that for different projectile/target combinations,the higher the wave impedance of the projectile,the higher the initial phase transition velocity and the smaller the mass of phase transition.The results can provide theoretical guidance for kinetic impactor design and material selection.

A review of the metal stud stiffness on the sound insulation of gypsum-board partitions

For the sound insulation of a double-panel partition,the stud between two leaves creates a vibration transmission path,which can often be more critical and more important in the mid-frequency range than the airborne path through the cavity.Owing to the fact that partitions with light-weight steel studs are commonly used in building construction,studies on the sound insulation effect of such studs have been conducted.Especially,a model,initiated by Gu and Wang(1983),has been widely studied during the past decades.In the model,the steel stud is considered as an elastic spring with its cross-section stiffness in the sound insulation index prediction of such a partition.Experimental results of different stud profiles have been reported from different testing laboratories and more information has been gained to understand the stiffness effect of the stud on the sound insulation of the double-leaf partitions.In this paper,the authors have given this subject a thorough review and have concluded that a critical problem needs further investigation on the determination of the stiffness of the connecting elements in the double-leaf partitions.

Optimized Local Trigonometric Bases with Nonuniform Partitions

The authors provide optimized local trigonometric bases with nonuniform partitions which efficiently compress trigonometric functions. Numerical examples demonstrate that in many cases the proposed bases provide better compression than the optimized bases with uniform partitions obtained by Matviyenko.

Long Duration Coverage Control of Multiple Robotic Surface Vehicles Under Battery Energy Constraints

Dear Editor,This letter addresses long duration coverage problem of multiple robotic surface vehicles(RSVs) subject to battery energy constraints,in addition to uncertainties and disturbances. An anti-disturbance energy-aware control method is proposed for performing coverage task of RSVs. Firstly, a centroidal Voronoi tessellation(CVT) is used to optimize the partition of the given coverage area.

Experimental Investigation of Lee–Yang Criticality Using Non-Hermitian Quantum System

Lee–Yang theory clearly demonstrates where the phase transition of many-body systems occurs and the asymptotic behavior near the phase transition using the partition function under complex parameters. The complex parameters make the direct investigation of Lee–Yang theory in practical systems challenging. Here we construct a non-Hermitian quantum system that can correspond to the one-dimensional Ising model with imaginary parameters through the equality of partition functions. By adjusting the non-Hermitian parameter,we successfully obtain the partition function under different imaginary magnetic fields and observe the Lee–Yang zeros. We also observe the critical behavior of free energy in vicinity of Lee–Yang zero that is consistent with theoretical prediction. Our work provides a protocol to study Lee–Yang zeros of the one-dimensional Ising model using a single-qubit non-Hermitian system.

Restraint effect of partition wall on the tunnel floor heave in layered rock mass

The presence of horizontal layered rocks in tunnel engineering significantly impacts the stability and strength of the surrounding rock mass,leading to floor heave in the tunnel.This study focused on preparing layered specimens of rock-like material with varying thickness to investigate the failure behaviors of tunnel floors.The results indicate that thin-layered rock mass exhibits weak interlayer bonding,causing rock layers near the surface to buckle and break upwards when subjected to horizontal squeezing.With an increase in the layer thickness,a transition in failure mode occurs from upward buckling to shear failure along the plane,leading to a noticeable reduction in floor heave deformation.The primary cause of significant deformation in floor heave is upward buckling failure.To address this issue,the study proposes the installation of a partition wall in the middle of the floor to mitigate heave deformation of the rock layers.The results demonstrate that the partition wall has a considerable stabilizing effect on the floor,reducing the zone of buckling failure and minimizing floor heave deformation.It is crucial for the partition wall to be sufficiently high to prevent buckling failure and ensure stability.Through simulation calculations on an engineering example,it is confirmed that implementing a partition wall can effectively reduce floor heave and enhance the stability of tunnel floor.

Dependence of impact regime boundaries on the initial temperatures of projectiles and targets

Towards higher impact velocities,ballistic events are increasingly determined by the material temperatures.Related effects might range from moderate thermal softening to full phase transition.In particular,it is of great interest to quantify the conditions for incipient or full melting of metals during impact interactions,which result in a transition from still strength-affected to hydrodynamic material behavior.In this work,we investigate to which extent the respective melting thresholds are also dependent on the initial,and generally elevated,temperatures of projectiles and targets before impact.This is studied through the application of a model developed recently by the authors to characterize the transition regime between high-velocity and hypervelocity impact,for which the melting thresholds of materials were used as the defining quantities.The obtained results are expected to be of general interest for ballistic application cases where projectiles or targets are preheated.Such conditions might result,for example,from aerodynamic forces acting onto a projectile during atmospheric flight,explosive shapedcharge-jet formation or armor exposure to environmental conditions.The performed analyses also broaden the scientific understanding of the relevance of temperature in penetration events,generally known since the 1960s,but often not considered thoroughly in impact studies.