Number Of Representations Research Articles

A generalization of complex, dual and hyperbolic quaternions: Hybrid quaternions

Hybrid numbers are a new non-commutative number system which is a generalization of the complex (i2 = ?1), dual (?2 = 0), and hyperbolic numbers (h2 = 1). In this article, firstly we define a new quaternion system called hybrid quaternions by taking the coefficients of real quaternions as hybrid numbers. This new quaternion system is a combination of complex quaternions (biquaternions), hyperbolic (perplex) quaternions, and dual quaternions, and it can be viewed as a generalization of these quaternion systems. Then, we present the basic properties of hybrid quaternions including fundamental operations, conjugates, inner product, vector product, and norm. Finally, we give a schematic representation of numbers and quaternions.

Data and Knowledge Modelling as the Methodological Foundation of the Digital Humanities

Digital humanities is a multidisciplinary field that leverages digital technology and methodologies to explore and answer questions pertaining to the humanities. It is a dynamic intersection between the domains of computer science and the humanities, promoting innovation, collaboration, and research at the highest levels. However, as a relatively young field, the methodological foundations of the digital humanities are still being established. This paper seeks to explore the core methodologies that underpin digital humanities. The modelling of data, information, and knowledge can be considered one of the foundations of digital humanities. One of the arguments confirming this is that the development of digital humanities and the development of technologies in general are the development of ways to formalise and present data and knowledge. Science has come a long way from the modelling and computer representation of numbers to generating texts and art on the basis of prescribed inputs. With the advent of artificial intelligence, especially machine learning and deep learning techniques, the potential for more sophisticated and nuanced data modelling in the digital humanities has expanded significantly, linking computational capabilities with humanistic inquiries in unprecedented ways. The article considers the periodization, classification, and trends of approaches and methods for modelling data, information and knowledge in the humanities. The article provides an overview of existing examples and data models of different complexity from various humanities disciplines, including history, linguistics, literary criticism, and cultural studies.

Polynomial Expressions for Certain Arithmetic Functions

t: We exhibit polynomial expressions for the arithmetic functions and , the number of representations of n as a sum of k squares and k triangular numbers, respectively, and also for the color partitions

유리수의 순환마디에 대한 규칙성 분석

일반적으로, 유리수와 관련된 소수표현의 문제는 정수론에서 주로 다룬다. 이 논문에서는 잘 알려진 군론을 이 용하여 이러한 문제를 고찰해 보았다. 먼저, 유리수의 순환소수(repeating decimal)를 찾는 알고리즘을 고찰하여 이를 python으로 코딩을 해 보았다. 그리고 분모가 소수(prime)인 분수의 소수(decimal)표현을 군(group)의 관점에 서 고찰하였다. 구체적으로, 2와 5가아닌 소수 p에 대하여  의 순환마디 개수는 (Z , Xp)의 부분군 <10>의 위 수와 같다. 또한 (a=1,2,...,p-1)의 순환마디의 종류는 부분군 <10>의 잉여류의 개수만큼 나타난다.

Захист Осіріса. Формальні міркування щодо висловів 1076 та 1077 текстів саркофагів

One of the main episodes of the Osirian myth is the infamous assault of Seth on his brother Osiris. Their father, the god Geb, took all the precautions to protect the body of his murdered son from any future attack by Seth. A well-protected building housed the dead body, where Geb appointed powerful guards to protect the body of Osiris. Fortunately, few coffins from the Middle Kingdom (2055–1650 BC) preserved for us a strikingly interesting record of these apotropaic beings both in word and image. The standard publication of the Coffin Texts gathered the names of these guardians in two spells with numbers 1076 and 1077. Yet, lack of sufficient documentation and predominant textual bias in previous studies caused a great deal of confusion regarding these guardians as far as their number and visual representation are concerned. This contribution addresses these formal issues and analyzes the arrangement of the names and depictions of the guardians in these coffins. The close scrutiny revealed some hitherto undocumented features in the original sources.

Structural and functional connectivity of the inferior temporal numeral area.

A growing body of evidence suggests that in adults, there is a spatially consistent "inferior temporal numeral area" (ITNA) in the occipitotemporal cortex that appears to preferentially process Arabic digits relative to non-numerical symbols and objects. However, very little is known about why the ITNA is spatially segregated from regions that process other orthographic stimuli such as letters, and why it is spatially consistent across individuals. In the present study, we used diffusion-weighted imaging and functional magnetic resonance imaging to contrast structural and functional connectivity between left and right hemisphere ITNAs and a left hemisphere letter-preferring region. We found that the left ITNA had stronger structural and functional connectivity than the letter region to inferior parietal regions involved in numerical magnitude representation and arithmetic. Between hemispheres, the left ITNA showed stronger structural connectivity with the left inferior frontal gyrus (Broca's area), while the right ITNA showed stronger structural connectivity to the ipsilateral inferior parietal cortex and stronger functional coupling with the bilateral IPS. Based on their relative connectivity, our results suggest that the left ITNA may be more readily involved in mapping digits to verbal number representations, while the right ITNA may support the mapping of digits to quantity representations.

Early visual deprivation disrupts the mental representation of numbers in visually impaired children

Several shreds of evidence indicate that visual deprivation does not alter numerical competence neither in adults nor in children. However, studies reporting non-impaired numerical abilities in the visually impaired population present some limitations: (a) they mainly assessed the ability to process numbers (e.g. mathematical competence) rather than represent numbers (e.g. mental number line); (b) they principally focused on positive rather than negative number estimates; (c) they investigated numerical abilities in adult individuals except one focusing on children (Crollen et al. in Cognition 210:104586, 2021). Overall, this could limit a comprehensive explanation of the role exerted by vision on numerical processing when vision is compromised. Here we investigated how congenital visual deprivation affects the ability to represent positive and negative numbers in horizontal and sagittal planes in visually impaired children (thirteen children with low vision, eight children with complete blindness, age range 6–15 years old). We adapted the number-to-position paradigm adopted by Crollen et al. (Cognition 210:104586, 2021), asking children to indicate the spatial position of positive and negative numbers on a graduated rule positioned horizontally or sagittally in the frontal plane. Results suggest that long-term visual deprivation alters the ability to identify the spatial position of numbers independently of the spatial plane and the number polarity. Moreover, results indicate that relying on poor visual acuity is detrimental for low vision children when asked to localize both positive and negative numbers in space, suggesting that visual experience might have a differential role in numerical processing depending on number polarity. Such findings add knowledge related to the impact of visual experience on numerical processing. Since both positive and negative numbers are fundamental aspects of learning mathematical principles, the outcomes of the present study inform about the need to implement early rehabilitation strategies to prevent the risk of numerical difficulties in visually impaired children.

Dual-Gaussian Pell and Pell-Lucas numbers

In this study, we define a new type of Pell and Pell-Lucas numbers which are called dual-Gaussian Pell and dual-Gaussian Pell-Lucas numbers. We also give the relationship between negadual-Gaussian Pell and Pell-Lucas numbers and dual-complex Pell and Pell-Lucas numbers. Also, some sum ve product properties of Pell and Pell-Lucas numbers are given. Moreover, we obtain the Binet’s formula, generating function, d’Ocagne’s identity, Catalan’s identity, Cassini’s identity and some sum formulas for these new type numbers. Some algebraic proporties of dual-Gaussian Pell and Pell-Lucas numbers are investigated. Futhermore, we give the matrix representation of dual-Gaussian Pell and Pell-Lucas numbers.

On the Degree Distribution of Haros Graphs

Haros graphs are a graph-theoretical representation of real numbers in the unit interval. The degree distribution of the Haros graphs provides information regarding the topological structure and the associated real number. This article provides a comprehensive demonstration of a conjecture concerning the analytical formulation of the degree distribution. Specifically, a theorem outlines the relationship between Haros graphs, the corresponding continued fraction of its associated real number, and the subsequent symbolic paths in the Farey binary tree. Moreover, an expression that is continuous and piecewise linear in subintervals defined by Farey fractions can be derived from an additional conclusion for the degree distribution of Haros graphs.

Frequency set selection for multi-frequency steady-state visual evoked potential-based brain-computer interfaces.

Multi-frequency steady-state visual evoked potential (SSVEP) stimulation and decoding methods enable the representation of a large number of visual targets in brain-computer interfaces (BCIs). However, unlike traditional single-frequency SSVEP, multi-frequency SSVEP is not yet widely used. One of the key reasons is that the redundancy in the input options requires an additional selection process to define an effective set of frequencies for the interface. This study investigates systematic frequency set selection methods. An optimization strategy based on the analysis of the frequency components in the resulting multi-frequency SSVEP is proposed, investigated and compared to existing methods, which are constructed based on the analysis of the stimulation (input) signals. We hypothesized that minimizing the occurrence of common sums in the multi-frequency SSVEP improves the performance of the interface, and that selection by pairs further increases the accuracy compared to selection by frequencies. An experiment with 12 participants was conducted to validate the hypotheses. Our results demonstrated a statistically significant improvement in decoding accuracy with the proposed optimization strategy based on multi-frequency SSVEP features compared to conventional techniques. Both hypotheses were validated by the experiments. Performing selection by pairs and minimizing the number of common sums in selection by pairs are effective ways to select suitable frequency sets that improve multi-frequency SSVEP-based BCI accuracies. This study provides guidance on frequency set selection in multi-frequency SSVEP. The proposed method in this study shows significant improvement in BCI performance (decoding accuracy) compared to existing methods in the literature.

Microplastic discharge from a wastewater treatment plant: long term monitoring to compare two analytical techniques, LDIR and optical microscopy while also assessing the removal efficiency of a bubble curtain.

In this study we compare two parallel analytical methods while also testing a microplastics mitigation method. We assess the effectiveness of a bubble curtain to reduce microplastics in a wastewater treatment plant (WWTP)-effluent canal during the course of six months (>70 samples) using two analytical techniques: laser direct infrared (LDIR) and optical microscopy (OM) covering a size range of 0.02 to 5 mm. Comparison of the two analytical strategies shows similar trends, fluctuations, and correlating particle and fibre numbers. However, absolute values of particles differ, and the strategies provide different levels of information: LDIR is capable of identifying the plastic type as well as shape, while OM cannot determine the plastic type. Furthermore LIDR has a lower size limit (10-20 μm) than OM (50 μm). While information obtained by OM in general is far less detailed it is more affordable. This research also shows that the bubble curtain pilot does not have a measurable effect on the particle concentration. Possible effects of the curtain are hidden in the temporal variations. This research also reveals that individual samples show a large variation in particle numbers, illustrating that single measurements might give a poor representation of environmental particle number.

English as the Lingua Franca in Europe

Introduction. Throughout several decades, English has been strengthening its position as lingua franca in international communication. The relevance of the given study is justified by the examination of the English language from the perspective of contact variantology and in the context of gradually emerging new version of English – Euro-English. The aim of the presented work is to consider the socio-political prerequisites for the strengthening of the English language role for intra-European communication, as well as to try to explain the significance of Brussels for this process. Methodology and sources. Scientific publications of domestic and foreign linguists, sociolinguists (D.S. Borodina, Z.G. Proshina, B. Kachru, D. Crystal, E. Edwards), as well as interviews with European politicians and public figures (Michel Guerin, Pascal Smet, Sven Gatz) all comprise the research material. The method of synthesis and analysis, descriptive method, comparative language analysis is used in the process of work on this article. Results and discussion. The widespread usage of English in Europe has led to the creation of several mixed language variants and intra-European communication in English gives the latter the status of a supranational communication language. Belgium is seen as the center for the formation of this new language variant, which is explained by the representation of a significant number of international organizations and corporations in Brussels. In the context of the competition between French and Dutch, the difficult linguistic situation in Belgium finds a peculiar resolution in the regular use of English which some activists propose to make the third official language of the Brussels-Capital Region. Euro-English has already acquired several characteristic lexical, morphosyntactic and phonetical features, whilst the withdrawal of Great Britain from the European Union has provided serious possibilities for Euro-English to continue its formation as an independent language variant free from “care” and control of English native speakers. Conclusion. The existence of the English language variants involves the preservation of uniqueness for each nation where these are used, as well as the enrichment of the language from which these variants are derived from. Gradual formation of Euro-English confirms the presence of high number of professional and everyday communications under which the representatives of various European nations find common language – and that is English.

An Efficient CRT-Base Power-of-Two Scaling in Minimally Redundant Residue Number System.

In this paper, we consider one of the key problems in modular arithmetic. It is known that scaling in the residue number system (RNS) is a rather complicated non-modular procedure, which requires expensive and complex operations at each iteration. Hence, it is time consuming and needs too much hardware for implementation. We propose a novel approach to power-of-two scaling based on the Chinese Remainder Theorem (CRT) and rank form of the number representation in RNS. By using minimal redundancy of residue code, we optimize and speed up the rank calculation and parity determination of divisible integers in each iteration. The proposed enhancements make the power-of-two scaling simpler and faster than the currently known methods. After calculating the rank of the initial number, each iteration of modular scaling by two is performed in one modular clock cycle. The computational complexity of the proposed method of scaling by a constant Sl=2l associated with both required modular addition operations and lookup tables is estimeted as k and 2k+1, respectively, where k equals the number of primary non-redundant RNS moduli. The time complexity is log2k+l modular clock cycles.

Subrecursive incomparability of the graphs of standard and dual Baire sequences

Our main question of interest is the existence or the non-existence of a subrecursive reduction between different representations of the irrational numbers. For any representation, considered as a total function, we consider the characteristic function of its graph. The graph is computably equivalent to the function itself, but not subrecursively equivalent. In some cases, the graph of a representation is subrecursively equivalent to an already known representation, but in other cases it is a new representation. In the present paper we undertake a systematic study of the graphs of standard and dual Baire sequences. By combining our new results with the previously known results on the graph of the continued fraction, we obtain a total of eight new subrecursive degrees, which lie strictly between the Dedekind cut and the continued fraction.

Representations of Real Numbers Induced by Probability Distributions on ℕ

Abstract We observe that a probability distribution supported by ℕ, induces a representation of real numbers in [0, 1) with digits in ℕ. We first study the Hausdorff dimension of sets with prescribed digits with respect to these representations. Then, we determine the prevalent frequency of digits and the Hausdorff dimension of sets with prescribed frequencies of digits. As examples, we consider the geometric distribution, the Poisson distribution and the zeta distribution.

Gendered Epithets in Two Prominent Romanian Dictionaries

Abstract The present paper seeks to investigate the representation of a relevant number of gendered epithets in two prominent Romanian dictionaries, the most recent edition of DOOM (2021, Third Edition) and the current edition of DEX (2016, Revised Edition based on the first and second editions). Taking into account the prescriptive dimension which both these dictionaries share and the fact that they both retain their status as significant reference books in present-day Romania, the paper examines the way in which these current dictionaries choose to describe these words, arguing for the significance of the “metadata” (Nunberg, 2018; Pullum, 2018) component in the lexicographical representation of lexical items which can function as gendered insults.

Resolving the Singularity by Looking at the Dot and Demonstrating the Undecidability of the Continuum Hypothesis

Einsteinian gravity, of which Newtonian gravity is a part, is fraught with the problem of singularity that has been established as a theorem by Hawking and Penrose. The hypothesis that founds the basis of both Einsteinian and Newtonian theories of gravity is that bodies with unequal magnitudes of masses fall with the same acceleration under the gravity of a source object. Since, the Einstein’s equations is one of the assumptions that underlies the proof of the singularity theorem, therefore, the above hypothesis is implicitly one of the founding pillars of the same. In this work, I demonstrate how one can possibly write a non-singular theory of gravity which manifests that the above mentioned hypothesis is only valid in an approximate sense in the “large distance” scenario. To mention a specific instance, under the gravity of the earth, a 5 kg and a 500 kg fall with accelerations which differ by approximately \(113.148\times 10^{-32}\) meter/sec\(^2\) and the more massive object falls with less acceleration. Further, I demonstrate why the concept of gravitational field is not definable in the “small distance” regime which automatically justifies why the Einstein’s and Newton’s theories fail to provide any “small distance” analysis. In course of writing down this theory, I demonstrate why the continuum hypothesis as spelled out by Goedel, is undecidable. The theory has several aspects which provide the following realizations: (i) Descartes’ self-skepticism concerning exact representation of numbers by drawing lines (ii) Born’s wish of taking into account “natural uncertainty in all observations” while describing “a physical situation” by means of “real numbers” (iii) Klein’s vision of having “a fusion of arithmetic and geometry” where “a point is replaced by a small spot” (iv) Goedel’s assertion about “non-standard analysis, in some version” being “the analysis of the future”.

Complexity Theory

Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including interactive proof systems, quantum information and computation, algorithmic coding theory, arithmetic complexity, expansion of hypergraphs and simplicial complexes, Markov chain Monte Carlo, and pseudorandomness. Many of the developments are related to diverse mathematical fields such as algebraic geometry, extremal combinatorics, combinatorial number theory, probability theory, representation theory, and operator algebras.

Additive representations of natural numbers

Additive representations of natural numbers

Right Fusiform Gray Matter Volume in Children with Long-Term Abacus Training Positively Correlates with Arithmetic Ability

Abacus-based mental calculation (AMC) training has a positive effect on number-related cognitive abilities. While visuospatial strategy may distinguish AMC from conventional calculation method, the underlying neural mechanism is still elusive. The current study aimed to address this question by examining the plasticity of fusiform induced by AMC training and whether this training affects the association between the volume of fusiform and behavioral performance in numerical cognitive tasks using voxel-based morphometry analysis. The results showed that gray matter volumes of bilateral fusiform were significantly smaller in the AMC group relative to the control group. In addition, the volume of right fusiform was positively correlated with digit memory span and negatively correlated with reaction time of an arithmetic operation task only within the AMC group. These results indicate that bilateral fusiform may be the essential neural substrate for AMC experts to recognize and reconstruct abacus-based representations for numbers. These results may advance our understanding of the neural mechanisms of AMC and shield some lights to potential interactions between brain development and cognitive training in children.