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Within the context of Indian religions, Jainism has long been recognized for its extensive use of permutations and combinations. However, the application of these principles within Buddhist scriptures has received relatively little scholarly attention. This paper introduces a new example of the specific application of permutations and combinations in Buddhist scriptures. In this paper, we focus on the first saṅghādisesa rule in the Theravāda-vinaya, which lists a series of element sets and arranges these elements according to a certain pattern known as "ten-roots" (mūla), and we discover that these arrangements form a regular numerical sequence, called "oblong numbers". Moreover, similar patterns with different quantities are also found in the fourth Pārājika and the fifth saṅghādisesa rules. This indicates that the compilers of the Theravāda-vinaya did not use this mathematical knowledge without basis. Interestingly, we also found the use of this sequence in the Bakhshālī manuscript. Therefore, in this article, after summarizing and verifying the arrangement rules of the Theravāda-vinaya, we discuss whether the oblong numbers were influenced by Greek mathematics. [ABSTRACT FROM AUTHOR]

How do task experiences support proximal and longer‐term desires to reengage? Central to this question is the interplay between perceptions of task difficulty and the situational interest. This interplay is increasingly researched but not yet well understood. Micro‐analytic, latent studies are important tools for replicating past findings and extending our understanding in this area. This study aimed to extend our understanding of the interplay between readiness (prior knowledge and individual interest), a sequence of mathematics task experiences (situational interest and perceived difficulty) and reengagement beyond the set of tasks (i.e. optional follow‐up engagement). Participants (ages 20–49; n = 301) completed a series of maths tasks online, self‐reporting their task experiences after each task. Participants were offered an optional opportunity to follow‐up with the task content as a measure of their continued interest. Both readiness components assessed were important predictors of future situational interest (positive) and perceptions of task difficulty (negative). Persistent interconnections between the task experience components across time supported past evidence regarding the important role of situational interest and the challenges of perceived difficulty for sustained engagement. Modelling indicated that task success (β = 0.36), situational interest (β = 0.16) and perceptions of task difficulty (β = 0.21) can converge on choices to reengage with an object going forward. [ABSTRACT FROM AUTHOR]

This work is a segment of an ongoing doctoral research in Brazil. The Leonardo numbers and the Leonardo sequence have gained attention from mathematicians and the academic community. Despite being a relatively new sequence within mathematical literature, its discussion has intensified over the past five years, giving rise to other branches, with contributions and associations to other topics in mathematics. Thus, the aim of this study was to construct and present the state of the art of the Leonardo sequence, considering its historical aspects and highlighting works on its evolutionary process in the epistemic-mathematical field, regarding its generalization, complexification, hyper complexification, and combinatorial model during the last five years (2019-2023). The methodology used was a bibliographic study, where the state of the art was carried out through the mapping of publications on the subject. Twenty-four research works related to the key descriptors "Leonardo sequence", "Leonardo numbers", "complexification", "generalization", "hybrids", and "combinatorial model" were found, cataloged, and discussed. From the analysis of these studies, it is noted that its development in pure mathematics has advanced to other branches and discoveries, and that, albeit timidly, research on the subject has emerged directed towards the field of education, especially in the initial teacher training and, particularly, in Brazil. [ABSTRACT FROM AUTHOR]

The exploration of children's drawings as mathematical representations is a current focus in early years mathematics education research. This paper presents a qualitative analysis of 72 kindergarten to Grade 3 (5 to 8 years old) children's drawings produced during problem-solving tasks centred on multiplicative strategies. Existing frameworks for the developmental sequence of mathematical drawings and the progression of children's strategies for multiplicative situations were an interpretive lens through which to analyse the drawings. Children used pictographic and iconic drawing types to represent the "story" in the problem and the multiplicative strategies employed to solve the tasks. Exploration of the children's drawings suggested that as children's drawings become more structural, schematic in nature, it may be easier for children to show their understanding of the structural elements of multiplicative relationships. Results revealed that structural elements of multiplicative relationships were more easily seen in iconic representations; however, both pictographic and iconic drawings were useful to observe counting, additive, and multiplicative strategies when mathematical elements of the problem were visible. Additional representations attached to the drawings (e.g. numerical) were needed to confirm children's strategies when their drawings lacked structure. These findings have implications for how young children's drawings are interpreted by classroom teachers. The interpretation of these drawings suggested that some children may not yet realise how their drawings in mathematics need to shift from illustrations of the problem's story context to representing mathematical ideas and processes — which requires intentional teaching of the purpose of drawings for mathematical contexts. [ABSTRACT FROM AUTHOR]

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Delays in meeting math requirements can impede the progress among community college students who aspire to earn a baccalaureate degree. To investigate this issue, we used state administrative data from Texas to examine the prevalence and predictors of math course repetition and how math course repetition predicts transfer students' outcomes. More than a third of community college transfer students take additional introductory mathematics coursework despite having fulfilled the requirement—a phenomenon we referred to as "horizontal repetition"—and one sixth of community college students take redundant coursework within a given mathematics course sequence, referred to as "vertical repetition." Using regression models controlling for student backgrounds, academic experiences, and institutional fixed effects, we found that horizontal repetition was linked to lower GPA and, among degree recipients, increased time to degree and excess credits. Vertical repetition was negatively associated with GPA and degree completion and positively linked to increased time to degree and excess credits. Location of course repetition shaped student outcomes, where math course repetitions occurring at the university appear to drive many of the negative associations between both horizontal and vertical repetition and student outcomes. As community colleges and universities across the country consider the efficacy of course sequences and transfer pathways, our research offers insights into patterns and implications of course repetition in core math courses. [ABSTRACT FROM AUTHOR]

One essential goal of mathematics teaching is to develop the habit of mind and the ability to look for and recognize structures, to probe into and act upon structures, and to reason and justify in terms of general structures. Framed by the five practices for orchestrating productive mathematics discussions (Smith & Stein, 2011), this paper uses the Horse Rider problem as an example to illustrate how teachers can nurture student structural thinking through careful sequencing of mathematical tasks. [ABSTRACT FROM AUTHOR]

This study proposed an evaluation model of mathematics teaching quality under recurrent neural network for the sake of making the evaluation model of mathematics teaching quality have good fault tolerance. This model decomposes the initial data sequence of mathematics teaching quality evaluation into high- and low-frequency sequence by wavelet analysis and reconstructs it by using phase space. After introducing the recurrent neural network model, the data is reconstructed after model training, and the data mining is carried out for the evaluation of mathematics teaching quality. In the process of constructing the evaluation model, the evaluation index system should be constructed based on three dimensions firstly, and the evaluation index of association rules should be defined, so as to realize deep dig of data and obtain the phase space distribution of data and then carry out the constraint test of parameters to evaluate the mathematics teaching quality scientifically and accurately. After verification, it is known that the average values of training error and test error of the model proposed in this paper are 3.02% and 2.61%, and the average values of absolute error and relative error are 0.58 and 3.82%. This model can retain the valid data information in the initial sequence, and the evaluation results of mathematics teaching quality are relatively ideal, which greatly improves the efficiency and level of mathematics teaching. [ABSTRACT FROM AUTHOR]

Many students who experience difficulty with mathematics may also require behavioral support. We suggest educators create an appropriate scope and sequence for mathematics intervention by focusing on the critical mathematics content. Educators can then design the intervention with a common session structure. Finally, educators can embed and intensify supports for a student by increasing the dosage, focusing on the alignment of the intervention, and attending to transfer. [ABSTRACT FROM AUTHOR]

The concrete–representational–abstract (CRA) sequence of instruction is an explicit methodology for teaching mathematics using multiple representations of concepts. Learning concepts through multiple representations fosters conceptual understanding and mathematical thinking. This article describes how a special education teacher used explicit CRA instruction with two elementary students with emotional and behavioral disorders. Its aims are to describe and provide a rationale for explicit CRA instruction. We describe lesson activities, methods, materials, and procedures. Finally, we offer suggestions for effective implementation. [ABSTRACT FROM AUTHOR]

Socioeconomic status (SES) has been shown to be associated with children's arithmetic knowledge as early as kindergarten, which is an important issue, given that early numeracy knowledge and skills usually correlate to later academic achievement in arithmetic. In line with recent research, it is argued that the regular practice of exercises involving cardinality, ordinality and arithmetic transformations during class time could accelerate the progress of children, particularly among low-SES children. Participants were 118 kindergarteners (Mage = 3.87). Teachers were randomly assigned to the intervention group (immediate implementation of the intervention; N = 64) or the control group (intervention delayed, N = 54). This intervention consisted of playful exercises involving quantities and numerical transformations. The results indicated that the intervention tended to accelerate progress on the verbal numerical sequence, particularly among lower-SES children. In addition, the intervention tended to accelerate progress on cardinality and arithmetic. These results provide preliminary evidence of the effectiveness of the intervention to help children acquire basic mathematical skills. [ABSTRACT FROM AUTHOR]

Limited empirical research has investigated the impact of interdisciplinary STEM approaches on students' learning with a particular dearth in the area of assessment. A lack of alignment between interdisciplinary approaches in STEM teaching and its assessment strategies is notable. In this study, we adopt an inquiry approach to the ways in which interdisciplinary learning of science and mathematics impacted on student learning through the use of a STEM challenge that involved a real-world task. Two secondary school teachers, one who teaches science and the other mathematics, participated in the study and jointly planned and taught a sequence of science and mathematics lessons using an interdisciplinary approach to their Year 8 class of 24 students over several weeks. Data of students' project reports, video of their project presentation, and teacher interviews were analysed to identify the intersection of disciplinary specific and generic skills in assessment practice of an interdisciplinary task. Our findings highlight the importance of having an assessment rubric with criteria that addressed both disciplinary specific skills as well as generic STEM skills. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study (λ, μ) -uniformly distributed and (λ, μ)-well distributed double sequences. We also define the concept of A-uniformly distributed double sequence. [ABSTRACT FROM AUTHOR]

In this work, we show the equivalence of ideal Wijsman convergence and Wijsman convergence and the equivalence of ideal Hausdorff convergence and Hausdorff convergence for the nested sequences of sets. Then we prove that the ideal Wijsman limit and the ideal Hausdorff limit of the nested sequence of sets are equivalent each other for every admissible ideal. [ABSTRACT FROM AUTHOR]

We consider a multi-variable generalization of the zeta function defined by Arakawa and Kaneko. We establish its analytic continuation to an entire function, whence it follows that values at non-positive integers are expressed as multi-indexed poly-Bernoulli numbers of C-type. We also prove that its values at positive integers are in the space spanned by multiple zeta values. [ABSTRACT FROM AUTHOR]

This study presents a teaching experiment in which second-grade primary school students compared a city and a town according to population estimates. The activities were presented to them as Fermi problems and required an analysis of the reality to identify sub-problems that students could deal with. The students' products were analyzed from the perspective of mathematical modeling and it was observed that the students developed their own mathematical methods to make the required estimates. Learning opportunities were identified when students needed concepts and procedures that they had not yet worked with. The results show that estimation acts as a facilitator for the generation of mathematical models that enable students to connect their real-world knowledge to new mathematical learning. In the conclusions, there is a discussion of the implications of the use of Fermi problem sequences to initiate mathematical modeling and link up mathematical and extra-mathematical knowledge about social problems in the first years of primary education. [ABSTRACT FROM AUTHOR]

Learning about space is part of the teaching regimen in various elementary-level disciplines, including science and mathematics, in France and Quebec alike. Understanding space plays a central role in the ability to grasp a number of mathematical and scientific concepts. However, until now, few studies have focused on how to valorize it in class or on the milestones that can be used to guide the development of teaching sequences that promote students' understanding of space. This article presents an empirical study of an activity-generating structure (AGS)—initially developed to help with the design and analysis of teaching sequences in mathematics—as a mechanism for designing and analyzing a science teaching sequence that aims to foster a better apprehension of space. Our experiment shows how the AGS allows us, first, to adapt a classic teaching sequence on the discovery of the compass in a way that promotes students' acquisition of spatial understanding, and second, to analyze the impact of the adapted sequence on what students learn about space. [ABSTRACT FROM AUTHOR]

The sequence of balancing numbers (Bn) is defined by the recurrence relation Bn = 6Bn-1 -Bn-2 for n ≥ 2 with initial conditions B0 = 0 and B1 = 1. Bn is called the nth balancing number. In this paper, we find all repdigits in the base b, which are sums of four balancing numbers. As a result of our theorem, we state that if Bn is repdigit in the base b and has at least two digits, then (n, b) = (2, 5), (3, 6). Namely, B2 = 6 = (11)5 and B3 = 35 = (55)6. [ABSTRACT FROM AUTHOR]

Benford's law is an empirical "law" governing the frequency of leading digits in numerical data sets. Surprisingly, for mathematical sequences the predictions derived from it can be uncannily accurate. For example, among the first billion powers of 2, exactly 301029995 begin with digit 1, while the Benford prediction for this count is 10 9 log 10 2 = 301029995.66 ... . Similar "perfect hits" can be observed in other instances, such as the digit 1 and 2 counts for the first billion powers of 3. We prove results that explain many, but not all, of these surprising accuracies, and we relate the observed behavior to classical results in Diophantine approximation as well as recent deep conjectures in this area. [ABSTRACT FROM AUTHOR]

The main result is a characterization of the generating sequences of the length of words in a regular language on κ symbols. We say that a sequence s of integers is regular if there is a finite graph G with two vertices i, t such that s[SUBn] is the number of path of length n from i to t in G. Thus the generating sequence of a regular language is regular. We prove that a sequence s is the generating sequence of a regular language on κ symbols if and if both sequence s = (s[SUBn])[SUBn]≥0 and t=(κ[SUPn]-s[SUBn])[SUBn]≥0 are regular. [ABSTRACT FROM AUTHOR]

We study the sixth-order recursive sequences of the form xn+1 =xn-5xnxn-4(an + bnxn-5xn) ,where an and bn are sequences of real numbers, via the technique of Lie group analysis. Symmetry generators associated with the group of transformations that map solutions onto themselves are obtained and exact solutions derived. The 'final constraint' when finding the symmetries, is used to split the solution into different categories. The result of this work generalizes a recent work by Elsayed et al. [ABSTRACT FROM AUTHOR]

In this paper we study that the second kind q-Euler numbers En,q and q-Euler Euler polynomials En,q(x) are analytic continued to Eq(s) and Eq(s,w). We investigate the new concept of dynamics of the zeros of analytic continued polynomials. Finally, we observe an interesting phenomenon of 'scattering' of the zeros of Eq(s,w). [ABSTRACT FROM AUTHOR]

The article offers information related to the deep content knowledge in mathematics. It examines the dialogic approach combined with hands on learning most appropriate for students learning deep content knowledge in mathematics; and several concepts including concepts of infinity; concepts of the infinitesimal; and concepts of mathematical sequences in the living world.

Over the ring of algebraic integers 𝒪 of a number field K , the quadratic congruence a 1 x 1 2 + ⋯ + a k x k 2 ≡ c modulo a nonzero ideal ℐ is considered. We prove explicit formulas for N k (a 1 , ... , a k , c , ℐ) and N k ∗ (a 1 , ... , a k , c , ℐ) , the number of incongruent solutions (x 1 , ... , x k) ∈ 𝒪 k and the number of incongruent solutions (x 1 , ... , x k) ∈ 𝒪 k with x 1 x 2 ⋯ x k coprime to ℐ , respectively. If ℐ is contained in a prime ideal 𝒫 containing the rational prime p , it is assumed that 𝒫 is unramified over p. Moreover, some interesting identities for exponential sums are proved. [ABSTRACT FROM AUTHOR]

The article focuses on the ins and outs of mathematical explanation. Topics include the crucial result here is that a connected graph has an Eulerian cycle iff it has no vertex with an odd degree, another phenomenon whose explanation involves mathematics appears in the life cycle of cicadas, and a periodic organism trying to avoid a periodic predator needs to minimise the number of years of overlap between itself and the predator.

Since 1997 a considerable effort has been spent on the study of the swap (switch) Markov chains on graphic degree sequences. All of these results assume some kind of regularity in the corresponding degree sequences. Recently, Greenhill and Sfragara published a breakthrough paper about irregular normal and directed degree sequences for which rapid mixing of the swap Markov chain is proved. In this paper we present two groups of results. An example from the first group is the following theorem: let be a directed degree sequence on n vertices. Denote by Δ the maximum value among all in- and out-degrees and denote by the number of edges in the realization. Assume furthermore that . Then the swap Markov chain on the realizations of is rapidly mixing. This result is a slight improvement on one of the results of Greenhill and Sfragara. An example from the second group is the following: let d be a bipartite degree sequence on the vertex set U ⊎ V, and let 0 < c1 ≤ c2 < |U| and 0 < d1 ≤ d2 < |V| be integers, where c1 ≤ d(v) ≤ c2: ∀v ∈ V and d1 ≤ d(u) ≤ d2: ∀u ∈ U. Furthermore assume that (c2 − c1 − 1)(d2 − d1 − 1) < max{c1(|V| − d2), d1(|U| − c2)}. Then the swap Markov chain on the realizations of d is rapidly mixing. A straightforward application of this latter result shows that when a random bipartite or directed graph is generated under the Erdős—Rényi G(n, p) model with mild assumptions on n and p then the degree sequence of the generated graph has, with high probability, a rapidly mixing swap Markov chain on its realizations. [ABSTRACT FROM AUTHOR]

Linear recurring sequences over semirings were introduced recently. In this paper, after investigating some properties on zero-sum strongly k-noetherian halfrings (briefly zero-sum SKN-halfrings), we study some properties of families of linear recurring sequences over zero-sum SKN-halfrings and relationships between families of linear recurring sequences and theirs annihilators. We also characterize the finitely generated subsemimodules of the S-semimodule of all linear recurring sequences over S. [ABSTRACT FROM AUTHOR]

The paper introduces the notions of boundedness and convergence with speed for difference sequences, and characterizes certain matrix classes associating the sets of such classes of sequences involving the operator Δ and two speeds λ = (λk) and µ = (µk) ... The results obtained in this paper should easily extendible to difference sequences of higher orders, and even, in combination with multipliers. [ABSTRACT FROM AUTHOR]

In this paper we investigate the distribution properties of hybrid sequences which are made by combining Halton sequences in the ring of polynomials and digital Kronecker sequences. We give a full criterion for the uniform distribution and prove results on the discrepancy of such hybrid sequences. [ABSTRACT FROM AUTHOR]

When people consider a series of random binary events, such as tossing an unbiased coin and recording the sequence of heads (H) and tails (T), they tend to erroneously rate sequences with less internal structure or order (such as HTTHT) as more probable than sequences containing more structure or order (such as HHHHH). This is traditionally explained as a local representativeness effect: Participants assume that the properties of long sequences of random outcomes—such as an equal proportion of heads and tails, and little internal structure—should also apply to short sequences. However, recent theoretical work has noted that the probability of a particular sequence of say, heads and tails of length n , occurring within a larger (> n ) sequence of coin flips actually differs by sequence, so P (HHHHH) < P (HTTHT). In this alternative account, people apply rational norms based on limited experience. We test these accounts. Participants in Experiment 1 rated the likelihood of occurrence for all possible strings of 4, 5, and 6 observations in a sequence of coin flips. Judgments were better explained by representativeness in alternation rate, relative proportion of heads and tails, and sequence complexity, than by objective probabilities. Experiments 2 and 3 gave similar results using incentivized binary choice procedures. Overall the evidence suggests that participants are not sensitive to variation in objective probabilities of a sub-sequence occurring; they appear to use heuristics based on several distinct forms of representativeness. [ABSTRACT FROM AUTHOR]

In this paper, we show that there is no solution of the diophantine equation uk1+ uk2 +...+ukn-1 = uln+1+uln+2+...+uln+rr for special cases of k and l where the elements of sequence fung satisfy the relation un = Aun-1 - un-2 with u0 = 0, u1= 1 and A ≥ 3 is a positive integer. [ABSTRACT FROM AUTHOR]

The focus of this work is on two new techniques for imposing maximum length scale in topology optimization. Restrictions on the maximum length scale provide designers with full control over the optimized structure and open possibilities to tailor the optimized design for broader range of manufacturing processes by fulfilling the associated technological constraints. One of the proposed methods is based on combination of several filters and builds on top of the classical density filtering which can be viewed as a low pass filter applied to the design parametrization. The main idea is to construct band pass filter which restricts the appearance of very thin and very thick elements in the design. In combination with the robust design optimization formulation the methodology results in manufacturable designs without the need of any post processing. The second technique provides more strict control on the maximum design features and is developed with the help of morphological operators. The formulation relies on a small number of additional constraints. Both approaches are demonstrated on optimization problems in linear elasticity. [ABSTRACT FROM AUTHOR]

Ordinality is a fundamental aspect of numerical cognition. However, preverbal infants' ability to represent numerical order is poorly understood. In the present study we extended the evidence provided by Macchi Cassia, Picozzi, Girelli, and de Hevia (2012), showing that 4-month-old infants detect ordinal relationships within size-based sequences, to numerical sequences. In three experiments, we showed that at 4months of age infants fail to represent increasing and decreasing numerical order when numerosities differ by a 1:2 ratio (Experiment 1), but they succeed when numerosities differ by a 1:3 ratio (Experiments 2 and 3). Critically, infants showed the same behavioral signature (i.e., asymmetry) described by Macchi Cassia et al. for discrimination of ordinal changes in area: they succeed at detecting increasing but not decreasing order (Experiments 2 and 3). These results support the idea of a common (or at least parallel) development of ordinal representation for the two quantitative dimensions of size and number. Moreover, the finding that the asymmetry signature, previously reported for size-based sequences, extends to numerosity, points to the existence of a common constraint in ordinal magnitude processing in the first months of life. The present findings are discussed in the context of possible evolutionary and developmental sources of the ordinal asymmetry, as well as their implication for other related cognitive abilities. [ABSTRACT FROM AUTHOR]

The brain constantly infers the causes of the inputs it receives and uses these inferences to generate statistical expectations about future observations. Experimental evidence for these expectations and their violations include explicit reports, sequential effects on reaction times, and mismatch or surprise signals recorded in electrophysiology and functional MRI. Here, we explore the hypothesis that the brain acts as a near-optimal inference device that constantly attempts to infer the time-varying matrix of transition probabilities between the stimuli it receives, even when those stimuli are in fact fully unpredictable. This parsimonious Bayesian model, with a single free parameter, accounts for a broad range of findings on surprise signals, sequential effects and the perception of randomness. Notably, it explains the pervasive asymmetry between repetitions and alternations encountered in those studies. Our analysis suggests that a neural machinery for inferring transition probabilities lies at the core of human sequence knowledge. [ABSTRACT FROM AUTHOR]

In this article, we introduce the Zweier Paranorm I-convergent double sequence spaces 2ZI(q), 2Z0I(q) and 2Z∞I(q) for q = (qij), a sequence of positive real numbers. We study some algebraic and topological properties on these spaces. [ABSTRACT FROM AUTHOR]

We prove that a general condition introduced by Colombo and Gobbino to study limits of curves of maximal slope allows us to characterize also minimizing movements along a sequence of functionals as curves of maximal slope of a limit functional. [ABSTRACT FROM AUTHOR]

In this paper, we show that the infinite log-monotonicity of sequences is preserved under the componentwise product. As applications, we can obtain that the central column sequences of some classical combinatorial triangles, such as the Catalan triangles and the Narayana triangle, are infinitely log-monotonic respectively. [ABSTRACT FROM AUTHOR]

Under some weak conditions, we use the moment inequality for-mixing sequence to investigate the Bahadur representation of sample quantiles based on this stochastic sequence. For all, we obtain a rate of Bahadur representation in the form of, which is close to the rate. [ABSTRACT FROM PUBLISHER]

We explore the classical pattern avoidance question in the case of irreducible permutations, i.e., those in which there is no index i such that σ(i + 1) - σ(i) = 1. The problem is addressed completely in the case of avoiding one or two patterns of length three, and several well known sequences are encountered in the process, such as Catalan, Motzkin, Fibonacci, Tribonacci, Padovan and Binary numbers. Also, we present constructive bijections between the set of Motzkin paths of length n - 1 and the sets of irreducible permutations of length n (respectively fixed point free irreducible involutions of length 2n) avoiding a pattern α for α ∈ {132, 213, 321}. This induces two new bijections between the set of Dyck paths and some restricted sets of permutations. [ABSTRACT FROM AUTHOR]

The period of balancing numbers modulo m, denoted by π(m), is the least positive integer t such that {Bt,Bt+1}≡{0, 1} (mod m), where Bt denotes the t-th balancing number. In this article, the periods of balancing numbers modulo product of consecutive Pell and Pell-Lucas numbers are examined. MSC 2010. 11B39 [ABSTRACT FROM AUTHOR]

First it is shown that the Nörlund matrix associated with the sequence of positive integers is a coposinormal operator on l² . This fact then turns out to be useful for showing that this operator is also posinormal and hyponormal. In contrast with the analogous weighted mean matrix result [6], the proof of hyponormality is accomplished without resorting to determinants or Sylvester's criterion. [ABSTRACT FROM AUTHOR]

A new three-parameter family of integer sequences Somos-4 is constructed. [ABSTRACT FROM AUTHOR]

The article talks about finding quotients of triangular numbers which are obtained by adding consecutive natural numbers and for the mathematical derivation, the author considers a series of triangular numbers. Methods such as diophantine equations, sequences and other mathematical theorems are used for finding out the result.

The article talks about finding out the maximal strictly composite runs which means deriving the maximum number of consecutive composite number that can come in a sequence. It adds that composite numbers means any number greater than zero which is neither prime nor the power of any prime number and theorems on composite numbers and prime numbers are used to find the result.

Given a set of t ≥ κ +2 words of length n over a κ-letter alphabet, it is proved that there exists a common subsequence among two of them of length at least n/κ + cn1-1/(t-k-2) for some c > 0 depending on κ and t. This is sharp up to the value of c. [ABSTRACT FROM AUTHOR]