Number Of Representations Research Articles

Sums of powers of integers and hyperharmonic numbers

In this paper, we obtain a new formula for the sums of k-th powers of the first n positive integers, Sk(n), that involves the hyperharmonic numbers and the Stirling numbers of the second kind. Then, using an explicit representation for the hyperharmonic numbers, we generalize this formula to the sums of powers of an arbitrary arithmetic progression. Furthermore, we express the Bernoulli polynomials in terms of hyperharmonic polynomials and Stirling numbers of the second kind. Finally, we extend the obtained formula for Sk(n) to negative values of n.

"Academization" of the saxophone: genesis, main milestones and current trends

The leading research topic in this article is the analysis of the specifics of the development of saxophone art in a symphony orchestra. The chosen problem requires the study of this direction from the perspective of a broader socio-historical context, within which it is possible to identify special signs of musical culture, both in the general context and in individual genres. In this vein, the study of saxophone music in a symphony orchestra allows us to outline, firstly, the origins of the origin of interest and the introduction of a special instrument into the cultural panorama, secondly, to trace the ways and impulses of its spread, and thirdly, to find out the problems of the saxophone as part of a symphony orchestra. In order to fully understand its special place in the cultural continuum of a certain place and epoch, music for the saxophone should be considered not just as a specific artifact that has been embodied in creativity, performance, education, but as a multi-level and three-dimensional concept in the musical art of the XIX-XXI centuries. The disclosure of the role of the saxophone as part of a symphony orchestra in the article is not just carried out through the prism of the representation of a certain number of facts, names, events, and can be traced as the correspondence of the development of the instrument to those historical, political, socio-economic and, above all, cultural and artistic processes that have formed a kind of creative reinterpretation of one of the most popular modern instruments. The purpose of the study is to substantiate the genesis, development trends and the current state of saxophone art in general, as well as, in particular, in the composition of a symphony orchestra. In the study, the author focuses on iconic performers and teachers, whose personal achievements in the field of concert, research and musical and social activities had a significant impact on the formation of the saxophone as part of a symphony orchestra.

The irony of an ‘international faculty’

Using the autoethnographic case study method, this article examines how my positionality as a foreign-born faculty member intersects with the institutional rhetoric of diversity and inclusion present in many predominantly White institutions. My vignettes show that foreign-born faculty, although contributing to the representation of diversity numbers, are positioned as knowledge providers in the discussions about the ‘global’, the ‘cultural’ and sometimes the ‘racial’, thus, ironically reinforce the embedded White institutional culture. This article argues that foreign-born faculty members could make use of their cultural positions to unpack the classed and racial culture on campus and to cultivate students’ anthropological sensibility. In other words, foreign-born faculty are in a unique position of recognising the limitations of the current diversity and inclusion rhetoric in predominantly White institutions (PWI), but also, they have the potential of decentring the White, middle-class cultural norms. This article concludes with some pedagogical implications.

Spatial Reasoning Profile of The Students with Good Number Sense Ability

Mathematics teacher must have a good basic mathematics comprehension such as number sense and spatial reasoning before teach it to the students. Those ability will help a person to solve mathematical problems effectively. The aim of this study is to describe how number sense and spatial reasoning was used to solve mathematical problems. This research is a qualitative-descriptive research with 27 of second year pre-service mathematics teachers as a participant. Subject were selected using purposive sampling by conducting number sense ability test. The one with good number senses abilities were given advanced test related to spatial reasoning. The result of this research shows that pre-service mathematics teacher with good number sense ability can solve the given problems correctly. Subjects can flexibly conduct mental calculation, mathematical computation, and create various number representation accurately and are able to create efficient problem-solving strategy by combining mental rotation and spatial visualization within the process.

On a class number formula of Hurwitz

In a little-known paper Hurwitz gave an infinite series representation of the class number for positive definite binary quadratic forms. In this paper we give a similar formula in the indefinite case. We also give a simple proof of Hurwitz’s formula and indicate some extensions.

IMPLEMENTASI ALGORITMA I-SOS DALAM PENYELESAIAN TRAVELING SALESMAN PROBLEM (TSP)

The problem with TSP is an attempt to find the shortest distance traveled by a salesman in visiting each city without having to visit the same city more than once. The purpose of implementing the I-SOS algorithm in this case is to find the minimum distance traveled, a solution can be obtained after going through the calculation of the mutualism phase, commensalism phase, parasitism phase and predation phase. The resolution of TSP problems in the study begins with the process of identifying each city by providing a random value to represent each destination city. The random value used is between 0 and 1, the random results obtained will then be sorted with the provision that the smallest random value will be used as the initial for city A while the largest random value is used as the initial for city D. In the first random, the random value | 0.5 | 0.27 | 0.75 | 0.25 | the city representation of the random number is | C | B | D | A | or if the values are sorted, the city order will be obtained, namely A = 0.25, B = 0.27, C = 0.5 and D = 0.75, this process will continue until all the organisms defined in the ecosystem are formed

Assessment of Turbulence Models for Low Reynolds Number Flows and Their Computational Costs, Part 2: Square Prism in Cross Flow

AbstractThe accuracy of six turbulent flow modeling techniques in an unsteady solution is evaluated against experimental data for a square prism in cross flow. The selected models, shear stress transport (SST), SST-SAS, Reynolds stress model (RSM), partially averaged Navier–Stokes (PANS)-SST, detached eddy simulation (DES), and large eddy simulations (LES), models are the same as those presented in Part 1 of this study, which focused on flow in a staggered tube bank. For this geometry, the SST model proved to be effective at capturing the averaged Nusselt values per side of the square with relatively low computational costs. The SST model, however, showed poorer fidelity to the local Nusselt number profile compared to the experimental data. The LES approach provided a more accurate representation of the local Nusselt number but the computational cost was significantly higher. The PANS modification to the SST model did provide a noticeable improvement in accuracy at a reasonable cost while the SAS modification did not see the same improvement. These conclusions are generally consistent with those found for the staggered tube bank in Part 1 of this study. This study can be used as a guide for the industrial user to select a turbulence model for a similar problem with a low Reynolds number and significant flow separation.

On the derivation of exact eigenstates of the generalized squeezing operator

We construct the states that are invariant under the action of the generalized squeezing operator for arbitrary positive integer k. The states are given explicitly in the number representation. We find that for a given value of k there are k such states. We show that the states behave as n −k/4 when occupation number n → ∞ . This implies that for any k ≥ 3 the states are normalizable. For a given k, the expectation values of operators of the form are finite for positive integer j < (k/2 − 1) but diverge for integer j ≥ (k/2 − 1). For k = 3 we also give an explicit form of these states in the momentum representation in terms of Bessel functions.

Synthesis and Implementation of IIR Filter using VHDL Language

VHDL is a hardware description language of different hardware architectures at divers’ levels of abstraction. The language is used for design, modelling, and investigation of hardware. In this paper, it is presented hardware implementation architectures of IIR filter using synthesizable VHDL. The MATLAB Filter Design and Analysis Tool (FDA Tool) is usedto calculate filter coefficients. Then, the filter is simulated and implemented on the Xilinx xc3s500efg320-4. A fixed-point number representation is used in the implementation of the filter. The proposed design of the filter includes two architectures: parallel architecture and programmable multiplexing architecture, each type has its advantages and disadvantages. In the results, some figures depending on the size of design (filter length) are introduced in the synthesis report.

The Evolutionary History of Brains for Numbers.

Humans and other animals share a number sense', an intuitive understanding of countable quantities. Having evolved independent from one another for hundreds of millions of years, the brains of these diverse species, including monkeys, crows, zebrafishes, bees, and squids, differ radically. However, in all vertebrates investigated, the pallium of the telencephalon has been implicated in number processing. This suggests that properties of the telencephalon make it ideally suited to host number representations that evolved by convergent evolution as a result of common selection pressures. In addition, promising candidate regions in the brains of invertebrates, such as insects, spiders, and cephalopods, can be identified, opening the possibility of even deeper commonalities for number sense.

Intermediate symmetric construction of transformation between anyon and Gentile statistics

Gentile statistics describes fractional statistical systems in the occupation number representation. Anyon statistics researches those systems in the winding number representation. Both of them are intermediate statistics between Bose–Einstein and Fermi–Dirac statistics. The second quantization of Gentile statistics shows a lot of advantages. According to the symmetry requirement of the wave function and the property of braiding, we give the general construction of transformation between anyon and Gentile statistics. In other words, we introduce the second quantization form of anyons in an easier way. This construction is a correspondence between two fractional statistics and gives a new description of anyon. Basic relations of second quantization operators, the coherent state and Berry phase are also discussed.

Multi-valued neural networks I: a multi-valued associative memory

A new concept of a multi-valued associative memory is introduced, generalizing a similar one in fuzzy neural networks. We expand the results on fuzzy associative memory with thresholds, to the case of a multi-valued one: we introduce the novel concept of such a network without number representation of weights and data, investigate its properties, and give a learning algorithm in the multi-valued case. We discovered conditions under which it is possible to store given pairs of network variable patterns in such a multi-valued associative memory. In the multi-valued neural network, not all variables are numbers, but elements or subsets of a lattice, i.e., they are all only partially ordered. Lattice operations are used to build the network output by inputs. In this paper, the lattice is assumed to be Brouwer and determines the implication used, together with other lattice operations, to determine the neural network output. We give the example of the network use to classify aircraft/spacecraft trajectories.

A time-space tradeoff for Lehman’s deterministic integer factorization method

Fermat's well-known factorization algorithm is based on finding a representation of natural numbers $N$ as the difference of squares. In 1895, Lawrence generalized this idea and applied it to multiples $kN$ of the original number. A systematic approach to choose suitable values for $k$ was introduced by Lehman in 1974, which resulted in the first deterministic factorization algorithm considerably faster than trial division. In this paper, we construct a time-space tradeoff for Lawrence's generalization and apply it together with Lehman's result to obtain a deterministic integer factorization algorithm with runtime complexity $O(N^{2/9+o(1)})$. This is the first exponential improvement since the establishment of the $O(N^{1/4+o(1)})$ bound in 1977.

A Modified Method for Finding Initial Basic Feasible Solution for Fuzzy Transportation Problems Involving Generalized Trapezoidal Fuzzy Numbers

In this paper, we propose a modified method for finding an initial basic feasible solution(IBFS) for fuzzy transportation problem(FTP) involving generalized trapezoidal fuzzy numbers(GTFNS). Without disturbing the fuzzy nature, using the parametric representation of fuzzy numbers, the optimal solution of the FTPP is obtained. A numerical illustration is given to demonstrate the suitability and efficiency of the proposed procedure and the solution is compared with the existing methods.

Contribuições do Material Dourado para o ensino de números decimais numa turma do 6º ano do Ensino Fundamental II

Apresentamos neste trabalho, uma experiência com uso do Material Dourado e Quadro do Valor de Lugar (QVL) em aulas de Matemática na Escola Municipal de Ensino Fundamental Irmão Pio Barroso. O objetivo do estudo foi investigar as contribuições do uso do Material Dourado e QVL no ensino do sistema decimal de numeração, em especial a representação dos números decimais em uma turma de 6º ano. As atividades investigativas com material didático visaram a construção e fortalecimento dos conhecimentos prévios que os alunos adquiriram em anos anteriores dando sequência em conteúdos posteriores relacionados às operações fundamentais e números decimais. Desenvolveu-se um estudo qualitativo do tipo Pesquisa-ação a partir de uma problemática vivenciada na prática docente em relação ao ensino de números decimais. Os instrumentos de pesquisa foram o diário de campo da pesquisadora, questionário e material produzido pelos alunos. Destaca-se como resultados as contribuições vivenciadas pelos alunos relacionadas à participação na aula, interação com os colegas, socialização de dúvidas e verbalização de conceitos relacionados à representação de números decimais. O estudo contribui para o campo da educação matemática ao evidenciar o potencial do Material Dourado e QVL para o estudo de números decimais em turmas de 6º ano e sugere-se a realização de novas pesquisas.

Separable effects of the approximate number system, symbolic number knowledge, and number ordering ability on early arithmetic development

There is evidence that early variations in the development of an approximate number system (ANS) and symbolic number understanding are both influences on the later development of formal arithmetic skills. We report a large-scale (N = 552) longitudinal study of the predictors of arithmetic spanning a critical developmental period (the first 3 years of formal education). Variations in early knowledge of symbolic representations of number and the ordinal associations between them are direct predictors of later arithmetic skills. The development of number ordering ability is in turn predicted by earlier variations in arithmetic, the ANS (numerosity judgments), and rapid automatized naming (RAN). These findings have important implications for theories of numerical and arithmetical development and potentially for the teaching of these skills.

Non-symbolic and symbolic number lines are dissociated.

People use mental number lines for both symbolic numerals and numerosity, but little is known about how these two mental number lines are related. The current study investigated the association in effect size, directionality of the mental number line, and development between symbolic and non-symbolic mental number lines to determine if they were related to or independent from each other. We collected data from numerosity- and digit-matching tasks that used the following numbers: 11, 14, 17, 20, 23, 26, and 29. Tasks were performed by college undergraduates and the fifth-grade primary school students. The results showed that none of the effects for non-symbolic numbers was related to any of the effects for symbolic numbers, and vice versa, in both adults and children. Another notable finding was that the correlation between the SNARC (spatial-numerical association of response code) effect size and mathematical ability was negative in the adult group. These results are consistent with the dissociated processes hypothesis and suggest that mental number lines are notation-dependent. As shown by the SNARC effect, the mental number line might result in interference in the current task by an automatically activated spatial notation-dependent representation of numbers.

UGEMM: Unary Computing for GEMM Applications

General matrix multiplication (GEMM) is pervasive in various domains, such as signal processing, computer vision, and machine learning. Conventional binary architectures for GEMM exhibit poor scalability in area and energy efficiency, due to the spatial nature of number representation and computing. On the contrary, unary computing processes data in temporal domain with extremely simple logic. However, to date, there rarely exist efficient architectures for unary GEMM. In this work, we first present uGEMM, a hardware-efficient unary GEMM architecture enabled by universally compatible arithmetic units, which simultaneously achieves input-insensitivity and high output accuracy. Next, we demonstrate that the proposed uGEMM can reliably early terminate the computation and offers dynamic energy-accuracy scaling for real-world applications via an accuracy-aware metric. Finally, to propel the future research for unary computing, we open source our unary computing simulator, UnarySim.

What is to be learnt? Critical aspects of elementary arithmetic skills

In this paper, we present a way of describing variation in young children’s learning of elementary arithmetic within the number range 1–10. Our aim is to reveal what is to be learnt and how it might be learnt by means of discerning particular aspects of numbers. The Variation theory of learning informs the analysis of 2184 observations of 4- to 7-year-olds solving arithmetic tasks, placing the focus on what constitutes the ways of experiencing numbers that were observed among these children. The aspects found to be necessary to discern in order to develop powerful arithmetic skills were as follows: modes of number representations, ordinality, cardinality, and part-whole relation (the latter has four subcategories: differentiating parts and whole, decomposing numbers, commutativity, and inverse relationship between addition and subtraction). In the paper, we discuss particularly how the discernment of the aspects opens up for more powerful ways of perceiving numbers. Our way of describing arithmetic skills, in terms of discerned aspects of numbers, makes it possible to explain why children cannot use certain strategies and how they learn to solve tasks they could not previously solve, which has significant implications for the teaching of elementary arithmetic.

Neuroethology of number sense across the animal kingdom.

Many species from diverse and often distantly related animal groups (e.g. monkeys, crows, fish and bees) have a sense of number. This means that they can assess the number of items in a set - its 'numerosity'. The brains of these phylogenetically distant species are markedly diverse. This Review examines the fundamentally different types of brains and neural mechanisms that give rise to numerical competence across the animal tree of life. Neural correlates of the number sense so far exist only for specific vertebrate species: the richest data concerning explicit and abstract number representations have been collected from the cerebral cortex of mammals, most notably human and nonhuman primates, but also from the pallium of corvid songbirds, which evolved independently of the mammalian cortex. In contrast, the neural data relating to implicit and reflexive numerical representations in amphibians and fish is limited. The neural basis of a number sense has not been explored in any protostome so far. However, promising candidate regions in the brains of insects, spiders and cephalopods - all of which are known to have number skills - are identified in this Review. A comparative neuroscientific approach will be indispensable for identifying evolutionarily stable neuronal circuits and deciphering codes that give rise to a sense of number across phylogeny.