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From the viewpoint of atomic bond relaxation, an analytical approach was put forward to elucidate the physical origins of crystal size and cross-sectional shape dependency of piezoelectric potentials in GaN nanowires and nanotubes. It is demonstrated that (i) size-induced increase in piezoelectric potential is attributed to the coupling effect of the rising piezoelectric coefficient and both the reducing dielectric constant and elastic constant caused by the surface atomic coordination number loss, bond energy perturbation, and surface-to-volume ratio rising; (ii) as the number of sides for polygonal nanowires or nanotubes with the same equivalent radius decreases, the surface-to-volume ratio rises, and the piezoelectric potential increases; and (iii) the nanotubes can generate a piezoelectric potential higher than their nanowire counterparts due to their larger surface-to-volume ratios. The proposed formulation offers a scientific basis for the fabrication, optimization, and modulation of one-dimensional GaN-based piezoelectric nanometer devices. [ABSTRACT FROM AUTHOR]

Non-Darcy flow in pore media is a central challenge for subsurface energy extraction and geological conservation, and the effects of medium geometry features on the transition from Darcy to non-Darcy flow regimes in single-phase flow have not been comprehensively studied. This study quantitatively analyzed the impact of medium geometry features on the flow regime transition from the pore scale by constructing a series of two-dimensional models with different medium geometry features (particle disorder, particle radius, particle shape, and particle alignment angle). The results show that the geometric features influence the Darcy permeability and the flow regime transition by modulating the viscous dissipation rate of the system. With increasing disorder, a decrease in the particle radius, and a decrease in the number of polygonal edges, the system will enter the non-Darcy flow regime earlier. In addition, we analyze the intrinsic connection between vortices, the inertial core region, and medium geometrical features in the non-Darcy flow regime by introducing a typical pore–throat model. The non-Darcy flow regime transitions from a vortex-dominated state to an inertial core region-dominated state as the azimuthal angle of the medium pore increases. [ABSTRACT FROM AUTHOR]

The aim of the study is to investigate the effect of point reduction on the dimensions of a part produced by two different additive technologies, Stereolithography (SLA) and Fused Deposition Modelling (FDM). Also, the effect of polygonization methods on the number of points and file size. For the purpose of the experiment, a part was fabricated which was made up of geometric elements, which served to investigate the influence of the reverse geometric modelling on parameter variation. Subsequently, the model was made into two samples by additive manufacturing FDM and SLA on Stratasys F370 (FDM) and Form 3 (SLA). From the fabricated samples, 3D data was collected on a Metrotom industrial CT 1500, cloud of points was obtained and polygonised in various ways in CAQ software GOM Inspect and CARE software VG Studio MAX. The evaluation as well as the process reverse modelling was performed in Autodesk Powershape software. The results show that different polygonization methods affect the number of polygonal mesh points and the volume of data. It was also found that the reduction of the points by which the polygonal network is formed leads to variations in the dimensions of the geometric features. Another important factor is the method of additive manufacturing of the component, where we have shown point reduction has a different effect for a component manufactured by SLA and a component manufactured by FDM. [ABSTRACT FROM AUTHOR]

In this paper, we consider representations of integers as sums of at most four distinct polygonal numbers with a prescribed number of repeats of each distinct polygonal number. We compare such representations with classical polygonal numbers, and those representations with generalized polygonal numbers. Our main result is that representations with classical polygonal numbers are equidistributed in the sense that the number of representations in the nonnegative quadrant in four-dimensional space is asymptotically 1 16 of the representations in the entire space. [ABSTRACT FROM AUTHOR]

For the Poisson equation posed in a domain containing a large number of polygonal perforations, we propose a low-dimensional coarse approximation space based on a coarse polygonal partitioning of the domain. Similarly to other multiscale numerical methods, this coarse space is spanned by locally discrete harmonic basis functions. Along the subdomain boundaries, the basis functions are piecewise polynomial. The main contribution of this article is an error estimate regarding the H 1 -projection over the coarse space; this error estimate depends only on the regularity of the solution over the edges of the coarse partitioning. For a specific edge refinement procedure, the error analysis establishes superconvergence of the method even if the true solution has a low general regularity. Additionally, this contribution numerically explores the combination of the coarse space with domain decomposition (DD) methods. This combination leads to an efficient two-level iterative linear solver which reaches the fine-scale finite element error in few iterations. It also bodes well as a preconditioner for Krylov methods and provides scalability with respect to the number of subdomains. [ABSTRACT FROM AUTHOR]

In this paper, we study universal sums of triangular numbers and squares. Specifically, we prove that a sum of triangular numbers and squares is universal if and only if it represents 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 10 , 13 , 14 , 15 , 18 , 19 , 20 , 23 , 27 , 28 , 34 , 41 , 47 , and 48. [ABSTRACT FROM AUTHOR]

This article examines a complex passage of Aristotle's Physics in which a Pythagorean doctrine is explained by means of a mathematical example involving gnomons. The traditional interpretation of this passage (proposed by Milhaud and Burnet) has recently been challenged by Ugaglia and Acerbi, who have proposed a new one. The aim of this article is to analyse difficulties in their account and to advance a new interpretation. All attempts at interpreting the passage so far have assumed that 'gnomons' should indicate 'odd numbers'. In this article it is argued that the usage of 'gnomon' related to polygonal numbers, which is normally considered late, could be backdated to at least the fifth/fourth centuries b.c. ; in particular, it explains the link between the philosophical explanandum and the mathematical explanans in Aristotle's passage. [ABSTRACT FROM AUTHOR]

We determine extremal graphs with respect to the number of independent sets for polygonal chains joined by cutting edges. We show that for any class of polygonal chains, independently of the number of sides on each polygon, the zig-zag polygonal chain has the extremal minimum value. Meanwhile, the polygonal chain at distance 2 between any two consecutive cutting edges provides the extremal maximum value. These results generalize and formalize previous ideas about phenylene chains. To recognize those extremal topologies, extremal values for the product between two Fibonacci numbers with complementary indices are determined. The product between two Fibonacci numbers with complementary indices are also applied for recognizing extremal graphs when a path is extended with one more vertex, or when any independent subgraph is linked to the path. Our results are achieved by applying decomposition of the input graph through the vertex and edge division rules. [ABSTRACT FROM AUTHOR]

In this paper, we investigate arithmetic progressions in integer-valued quadratic polynomials. We prove two results for integer-valued quadratic polynomials. First, we prove that there is no three-term arithmetic progression with a perfect square as a common difference for a class of integer-valued quadratic polynomials. We then find a class of numbers depending on integer-valued quadratic polynomials, which cannot work as a common difference for three-term arithmetic progression. Further, we deduce some results for arithmetic progressions in the sequence of polygonal and centered polygonal numbers. [ABSTRACT FROM AUTHOR]

In real world applications, data are subject to ambiguity due to several factors; fuzzy sets and fuzzy numbers propose a great tool to model such ambiguity. In case of hesitation, the complement of a membership value in fuzzy numbers can be different from the non-membership value, in which case we can model using intuitionistic fuzzy numbers as they provide flexibility by defining both a membership and a non-membership functions. In this article, we consider the intuitionistic fuzzy linear programming problem with intuitionistic polygonal fuzzy numbers, which is a generalization of the previous polygonal fuzzy numbers found in the literature. We present a modification of the simplex method that can be used to solve any general intuitionistic fuzzy linear programming problem after approximating the problem by an intuitionistic polygonal fuzzy number with n edges. This method is given in a simple tableau formulation, and then applied on numerical examples for clarity. [ABSTRACT FROM AUTHOR]

Let, be the sequence of polygonal numbers where, is an integer. In this paper, we describe two wave networks in cellular automata that can generate the sequences, in real-time. The first is a wave network that yields a cellular automaton with 5/2 + 2 states that generates this sequence when s = 4 is an even integer. With the second wave network, we obtain a cellular automaton with s states when s = 5 is an odd integer, or a cellular automaton with four states when s = 3. [ABSTRACT FROM AUTHOR]

The polygonal interval-valued fuzzy number is constructed based on the polygonal fuzzy number and the interval-valued fuzzy number. Its main feature is that the linear operation of finite ordered points reduces the complexity of traditional interval-valued fuzzy number operations. This research presents a generalized distance formula between two polygonal interval-valued fuzzy numbers and explores topological properties under the distance of polygonal interval-valued fuzzy numbers. In addition, we adopt the TOPSIS (technique for order preference by similarity to an ideal solution) and prospect theory approach for the multi-attribute decision-making problem. The information of attributes describes with polygonal interval-valued fuzzy numbers, and we then implement optimized ranking on the alternatives according to the profit and loss ratio. Finally, we verify the effectiveness and practicability of the decision-making method and fuzzy numbers at polygonal interval-valued fuzzy numbers in e-commerce risk assessment. [ABSTRACT FROM AUTHOR]

In this research, we applied the idea stemming from the equation (x−y)(x 3+y 3 ) = 4(w 2−p 2 )T 6to generalize the homogeneous quartic equation (x + y)(x 3 + y 3 ) = kw 2 p 2, where k = 3 + n², n ∈ Z. Through the application of appropriate transformations, it becomes a simplified equation of the form u² + 3v 2 = kp 2, where varying equations are obtained by altering the values of n. We formulated equations using specific n values such as 4, 8, 11, 13, 15, 16, 17, 18, 19 and 20. Solving these equations yielded a range of patterns, the solutions of which were determined to be non-zero integral solutions. [ABSTRACT FROM AUTHOR]

This article introduces a new equation of the form 945(x2 + y2) − 1889xy + 10(x + y) + 100 = pzq and explores its solution by gradually changing the parameter q from 1 to 10 with generalized value q = m. A transformation x = u + v and y = u − v simplifies the equation which results (u + 10)2 + 3779v2 = pzq. We have employed multiple techniques to discover integral solutions for the equation and conducted a thorough analysis to examine the properties and characteristics of these solutions. [ABSTRACT FROM AUTHOR]

The paper deals with the exploration of subsequences of polygonal numbers of different sides derived through step-by-step elimination of terms of the original sequences. Eliminations are based on special rules similarly to how the classic sieve of Eratosthenes was developed through the elimination of multiples of primes. These elementary number theory activities, appropriate for technology-enhanced secondary mathematics education courses, are supported by a spreadsheet, Wolfram Alpha, Maple, and the Online Encyclopedia of Integer Sequences. General formulas for subsequences of polygonal numbers referred to in the paper as polygonal number sieves of order k, that include base-two exponential functions of k, have been developed. Different problem-solving approaches to the derivation of such and other sieves based on the technology-immune/technology-enabled framework have been used. The accuracy of computations and mathematical reasoning is confirmed through the technique of computational triangulation enabled by using more than one digital tool. A few relevant excerpts from the history of mathematics are briefly featured. [ABSTRACT FROM AUTHOR]

We study the historical development of the figurate numbers, from ancient Greece through Pierre de Fermat to the work of Blaise Pascal, examining key passages for the construction and computation of the figurate numbers as well as their use in enumerating combinations. To assist in a transition to algebraic equivalents of these verbal passages, we introduce tools from transformational grammar to rewrite the surface structure of certain sentences in terms of more streamlined sentences amenable to algebraic formation. These linguistic techniques allow for the identification of key nouns that become single algebraic symbols, while other linguistic rules show how these nouns are modified either by algebraic operations or by the inclusion of modern-day subscripts. The material can be adapted for a number of undergraduate mathematics classroom settings, e.g., general education courses, college algebra, discrete mathematics. [ABSTRACT FROM AUTHOR]

In this paper, we consider the solvability over non-negative integers of certain Diophantine equations coming from representations of integers as sums of pentagonal numbers (counting the number of dots in a regular pentagon). We study a general method to obtain generalized versions of Cauchy's lemma. Using this, we show the "pentagonal theorem of 63", which states that a sum of pentagonal numbers represents every non-negative integer if and only if it represents the integers 1 , 2 , 3 , 4 , 6 , 7 , 8 , 9 , 11 , 13 , 14 , 17 , 18 , 19 , 23 , 28 , 31 , 33 , 34 , 39 , 42 , 63. We further show that these integers form a unique minimal universality criterion set. [ABSTRACT FROM AUTHOR]

Since it will always be clear from the context, we will continue to use the symbol I H SB n sb i for Graph HT ht , when dealing with a specific sequence Graph HT ht . Then (15) is easily manipulated to become Graph HT ht (16) which is (14). [Extracted from the article]

In this paper, we consider representations of integers as sums of heptagonal numbers with a prescribed number of repeats of each heptagonal number appearing in the sum. In particular, we investigate the classification of such sums which are universal, i.e., those that represent every positive integer. We prove an explicit finite bound such that a given sum is universal if and only if it represents positive integer up to the given bound. [ABSTRACT FROM AUTHOR]

The aim of this study was to investigate the involvement of the mesencephalic superior colliculus (SC) in the pathogenetic mechanism of SIDS, a syndrome frequently ascribed to arousal failure from sleep. We analyzed the brains of 44 infants who died suddenly within the first 7 months of life, among which were 26 infants with SIDS and 18 controls. In-depth neuropathological investigations of serial sections of the midbrain showed the SC layered cytoarchitectural organization already well known in animals, as made up of seven distinct layers, but so far never highlighted in humans, albeit with some differences. In 69% of SIDS cases but never in the controls, we observed alterations of the laminar arrangement of the SC deep layers (precisely, an increased number of polygonal cells invading the superficial layers and an increased presence of intensely stained myelinated fibers). Since it has been demonstrated in experimental studies that the deep layers of the SC exert motor control including that of the head, their developmental disorder could lead to the failure of newborns who are in a prone position to resume regular breathing by moving their heads in the sleep-arousal phase. The SC anomalies highlighted here represent a new step in understanding the pathogenetic process that leads to SIDS. [ABSTRACT FROM AUTHOR]

Copyright of Chinese Journal of High Pressure Physics is the property of Chinese Journal of High Pressure Physics Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Summary: We present visual proofs illustrating certain identities relating hexagonal numbers to other figurate numbers. [ABSTRACT FROM AUTHOR]

Integrating multiple independent functions into a single optical component is one of the most important topics in research on photoelectric systems. In this paper, we propose a multifunctional all-dielectric metasurface that can achieve a variety of non-diffractive beams depending on the polarization state of the incident light. Using the anisotropic TiO2 rectangular column as the unit structure, the three functions of generating polygonal Bessel vortex beams under left-handed circularly polarized incidence, Airy vortex beams under right-handed circularly polarized incidence and polygonal Airy vortex-like beams under linearly polarized incidence are realized. In addition, the number of polygonal beam sides and the position of focal plane can be adjusted. The device could facilitate further developments in scaling complex integrated optical systems and fabricating efficient multifunctional components. [ABSTRACT FROM AUTHOR]

Figure 3 illustrates this equation for HT ht I . i The case k I = i 7, HT ht , for HT ht . Notation The I k i -gonal number with I n i sides HT ht can be defined with reference to triangular numbers HT ht via the equation HT ht for HT ht [[2]]. [Extracted from the article]

Minimal neighbour designs (NDs) are used when a response of a treatment (direct effect) is affected by the treatment(s) applied in the neighbouring units. Minimal generalised NDs are preferred when minimal NDs cannot be constructed. Through the method of cyclic shifts (Rule I), the conditions for the existence of minimal circular generalised NDs are discussed, in which v/2 unordered pairs do not appear as neighbours. Certain generators are also developed to obtain minimal circular generalised NDs in blocks of two different sizes, where k2 = 3, 4 and 5. All these designs are constructed using i sets of shifts for k1 and two for k2. [ABSTRACT FROM AUTHOR]

Figurate numbers are numbers that can be represented by a regular and discrete geometric pattern of evenly-spaced points. Their study has attracted the attention of many mathematicians and scientists since the dawn of mathematical history, including Pythagoras of Samos (582 BC-507 BC), Diophantus of Alexandria (200/214-284/298), Fibonacci (1170-1250), Pierre de Fermat (1601-1665), Leonhard Euler (1707-1783), Waclaw Franciszek Sierpińshi (1882-1969) [1]. Although it is not one of the basic topics, it serves many fields of mathematics, such as Number Theory and Geometry. Many special numbers are related to figurate numbers. Polynomial values, some theorems and solutions of Diophantine equations are expressed in figurate numbers and studied [2, 3, 4]. Two-dimensional figurate numbers are known as polygonal numbers. In the early 1990's, polygonal numbers were expressed and visualised with the help of computers [5]. Richard K. Guy asked the question 'Every number is expressible as the sum of how many polygonal numbers?' [6]. As can be understood from his study, it is seen that most of the studies conducted are about ordinary polygonal numbers. But Euler proved the 'generalized pentagonal number theorem' about partitions [7]. [ABSTRACT FROM AUTHOR]

The paper deals with the exploration of subsequences of polygonal numbers of different sides derived through step-by-step elimination of terms of the original sequences. Eliminations are based on special rules similarly to how the classic sieve of Eratosthenes was developed through the elimination of multiples of primes. These elementary number theory activities, appropriate for technology-enhanced secondary mathematics education courses, are supported by a spreadsheet, Wolfram Alpha, Maple, and the Online Encyclopedia of Integer Sequences. General formulas for subsequences of polygonal numbers referred to in the paper as polygonal number sieves of order k, that include base-two exponential functions of k, have been developed. Different problem-solving approaches to the derivation of such and other sieves based on the technology-immune/technology-enabled framework have been used. The accuracy of computations and mathematical reasoning is confirmed through the technique of computational triangulation enabled by using more than one digital tool. A few relevant excerpts from the history of mathematics are briefly featured.

Copyright of Baghdad Science Journal is the property of Republic of Iraq Ministry of Higher Education & Scientific Research (MOHESR) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

'Slide scanners' are rapid optical microscopes equipped with automated and accurate x-y travel stages with virtual z-motion that cannot be rotated. In biomedical microscopic imaging, they are widely deployed to generate whole-slide images (WSI) of tissue samples in various modes of illumination. The availability of WSI has motivated the development of instrument-agnostic advanced image analysis software, helping drug development, pathology, and many other areas of research. Slide scanners are now being modified to enable polarised petrographic microscopy by simulating stage rotation with the acquisition of multiple rotation angles of the polariser–analyser pair for observing randomly oriented anisotropic materials. Here we report on the calibration strategy of one repurposed slide scanner and describe a pilot image analysis pipeline designed to introduce the wider audience to the complexity of performing computer-assisted feature recognition on mineral groups. The repurposed biological scanner produces transmitted light plane- and cross-polarised (TL-PPL and XPL) and unpolarised reflected light (RL) WSI from polished thin sections or slim epoxy mounts at various magnifications, yielding pixel dimensions from ca. 2.7 × 2.7 to 0.14 × 0.14 µm. A data tree of 14 WSI is regularly obtained, containing two RL and six of each PPL and XPL WSI (at 18° rotation increments). This pyramidal image stack is stitched and built into a local server database simultaneously with acquisition. The pyramids (multi-resolution 'cubes') can be viewed with freeware locally deployed for teaching petrography and collaborative research. The main progress reported here concerns image analysis with a pilot open-source software pipeline enabling semantic segmentation on petrographic imagery. For this purpose, all WSI are post-processed and aligned to a 'fixed' reflective surface (RL), and the PPL and XPL stacks are then summarised in one image, each with ray tracing that describes visible light reflection, absorption, and O- and E-wave interference phenomena. The maximum red-green-blue values were found to best overcome the limitation of refractive index anisotropy for segmentation based on pixel-neighbouring feature maps. This strongly reduces the variation in dichroism in PPL and interference colour in XPL. The synthetic ray trace WSI is then combined with one RL to estimate modal mineralogy with multi-scale algorithms originally designed for object-based cell segmentation in pathological tissues. This requires generating a small number of polygonal expert annotations that inform a training dataset, enabling on-the-fly machine learning classification into mineral classes. The accuracy of the approach was tested by comparison with modal mineralogy obtained by energy-dispersive spectroscopy scanning electron microscopy (SEM-EDX) for a suite of rocks of simple mineralogy (granulites and peridotite). The strengths and limitations of the pixel-based classification approach are described, and phenomena from sample preparation imperfections to semantic segmentation artefacts around fine-grained minerals and/or of indiscriminate optical properties are discussed. Finally, we provide an outlook on image analysis strategies that will improve the status quo by using the first-pass mineralogy identification from optical WSI to generate a location grid to obtain targeted chemical data (e.g., by SEM-EDX) and by considering the rock texture. [ABSTRACT FROM AUTHOR]

For a positive integer n , let $\mathcal T(n)$ denote the set of all integers greater than or equal to n. A sum of generalised m -gonal numbers g is called tight $\mathcal T(n)$ -universal if the set of all nonzero integers represented by g is equal to $\mathcal T(n)$. We prove the existence of a minimal tight $\mathcal T(n)$ -universality criterion set for a sum of generalised m -gonal numbers for any pair $(m,n)$. To achieve this, we introduce an algorithm giving all candidates for tight $\mathcal T(n)$ -universal sums of generalised m -gonal numbers. Furthermore, we provide some experimental results on the classification of tight $\mathcal T(n)$ -universal sums of generalised m -gonal numbers. [ABSTRACT FROM AUTHOR]

Recently, Jha has found identities that connect certain sums over the divisors of n to the number of representations of n as a sum of squares and triangular numbers. In this note, we state a generalized result that gives such relations for s-gonal numbers for any integer s ≥ 3. [ABSTRACT FROM AUTHOR]

We determine all natural numbers that can be expressed as the difference of two m-gonal numbers. For each such number, we determine the number of possible expression as the difference of two m-gonal numbers. [ABSTRACT FROM AUTHOR]

Given a set S of n points, a weight function w to associate a non-negative weight to each point in S , a positive integer k ≥ 1 , and a real number > 0 , we present algorithms for computing a spanner network G (S , E) for the metric space (S , d w) induced by the weighted points in S. The weighted distance function d w on the set S of points is defined as follows: for any p , q ∈ S , d w (p , q) is equal to w (p) + d π (p , q) + w (q) if p ≠ q , otherwise, d w (p , q) is 0. Here, d π (p , q) is the Euclidean distance between p and q if points in S are in ℝ d , otherwise, it is the geodesic (Euclidean) distance between p and q. The following are our results: (1) When the weighted points in S are located in ℝ d , we compute a k -vertex fault-tolerant (4 +) -spanner network of size O (k n 2 ). (2) When the weighted points in S are located in the relative interior of the free space of a polygonal domain , we detail an algorithm to compute a k -vertex fault-tolerant (4 +) -spanner network with O (k n h + 1 2 lg n) edges. Here, h is the number of simple polygonal holes in . (3) When the weighted points in S are located on a polyhedral terrain , we propose an algorithm to compute a k -vertex fault-tolerant (4 +) -spanner network, and the number of edges in this network is O (k n 2 lg n). [ABSTRACT FROM AUTHOR]

Copyright of Journal of Zhejiang University (Science Edition) is the property of Journal of Zhejiang University (Science Edition) Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Some convolution-type identities involving polygonal numbers and Horadam numbers are derived. The method of proof is to properly relate the generating functions to each other. Additionally, we prove a general non-convolutional result involving these number families and discuss some of the consequences.

In this article we obtain an approximation of a given fuzzy number by a unique polygonal fuzzy number preserving its main characteristics such as core, support, and expected interval. For this, we consider the well-known weighted L 2 metric. We show that this weighted polygonal approximation is equivalent to a strict convex quadratic optimization problem finite dimensional with linear inequality and equality constraint. Then, we present efficient and practice solution methods illustrated with several examples. Some properties of invariance of the approximation operator are also obtained. The results presented here extend and improve the methods obtained for weighted trapezoidal approximation and piecewise linear approximation of fuzzy numbers. [ABSTRACT FROM AUTHOR]

In this paper, basic information about centered polygonal numbers and continuous fractions are given. By using the continuous fraction algorithm, the relations between centered polygonal numbers and continued fractions have been studied. A general form for continued fractions of ratios of centered polygonal numbers of consecutive order is explained. [ABSTRACT FROM AUTHOR]

The application of wire EDM to the machining of cylindrical parts is an emerging research topic. The current research is mainly focused on the processing of microcylindrical parts, and the processing removal of microparts is usually small. In the complete machining process, especially during rough machining, the removal amount of the workpiece is usually large. In recent years, there have been few studies in this area. This paper proposes the application of WEDM with contour approximation method (CAM-WEDM) to the machining of cylindrical parts with large removal volume. First, the basic principle of the contour approximation method is introduced, and the machining process is divided into three parts: roughing, semi-finishing and finishing. Roughing corresponds to polygon cutting. Second, the various indicators of polygon cutting are analyzed. The influence of the number of polygonal sides on the residual height and residual area is analyzed. Two methods are proposed for polygon cutting: sequential cutting and multiple cutting. The calculation formulas for the machining volume of the two methods are deduced. By comparing the two methods, it is found that different methods are applicable to different part sizes. At last, machining experiments were carried out. The selection of residual variables and the determination of polygon machining methods are detailed. The process parameters and machining time of each stage are listed. In addition, the measurement results of each stage of the machining are also analyzed. At the same time, by comparing with the general WEDT, it is concluded that CAM-WEDM can effectively process cylindrical parts with large removal volume. [ABSTRACT FROM AUTHOR]

For a subset S of nonnegative integers and a vector $\mathbf {a}=(a_1,\ldots ,a_k)$ of positive integers, define the set $V^{\prime }_S(\mathbf {a})=\{ a_1s_1+\cdots +a_ks_k : s_i\in S\}-\{0\}$. For a positive integer n, let $\mathcal T(n)$ be the set of integers greater than or equal to n. We consider the problem of finding all vectors $\mathbf {a}$ satisfying $V^{\prime }_S(\mathbf {a})=\mathcal T(n)$ when S is the set of (generalised) m-gonal numbers and n is a positive integer. In particular, we completely resolve the case when S is the set of triangular numbers. [ABSTRACT FROM AUTHOR]

In preschool, numbers and shapes typically appear as separate topics. This study explores how a game, designed as a guided play activity with figurate numbers, functions in a preschool context. The guided play involved parking Lego cars in a rectangular shape, and to find out for which number of Lego cars this is possible. Thirteen preschool children in three separate age groups, aged from four to six years, together with their teacher, participated in the study. Their communications through words and actions were recorded. The results exemplify how this guided play provides a rich context for engaging young children with mathematical activities such as counting, sorting, shaping, asking, justifying, and inferring, as well as emotional engagement with the activity. [ABSTRACT FROM AUTHOR]

As a special case of general fuzzy numbers, the polygonal fuzzy number can describe a fuzzy object by means of an ordered representation of finite real numbers. Different from general fuzzy numbers, the polygonal fuzzy numbers overcome the shortcoming of complex operations based on Zadeh's traditional expansion principle, and can maintain the closeness of arithmetic operation. Hence, it is feasible to use a polygonal fuzzy number to approximate a general fuzzy number. First, an extension theorem of continuous functions on a real compact set is given according to open set construction theorem. Then using Weierstrass approximation theorem and ordered representation of the polygonal fuzzy numbers, the convergence of a single hidden layer feedforward polygonal fuzzy neural network is proved. Secondly, the gradient vector of the approximation error function and the optimization parameter vector of the network are given by using the ordered representation of polygonal fuzzy numbers, and then the gradient descent algorithm is used to train the optimal parameters of the polygonal fuzzy neural network iteratively. Finally, two simulation examples are given to verify the approximation ability of the network. Simulation result shows that the proposed network and the gradient descent algorithm are effective, and the single hidden layer feedforward network have good abilities in learning and generalization. [ABSTRACT FROM AUTHOR]

Merca ['Congruence identities involving sums of odd divisors function', Proc. Rom. Acad. Ser. A Math. Phys. Tech. Sci. Inf. Sci.22(2) (2021), 119–125] posed three conjectures on congruences for specific convolutions of a sum of odd divisor functions with a generating function for generalised m-gonal numbers. Extending Merca's work, we complete the proof of these conjectures. [ABSTRACT FROM AUTHOR]