Positive Integer Research Articles

Is counting a bad idea? Complex relations among children's fraction knowledge, eye movements, and performance in visual fraction comparisons.

Understanding fraction magnitudes is crucial for mathematical development but is challenging for many children. Visualizations, such as tape diagrams, are thought to leverage children's early proportional reasoning skills. However, depending on children's prior knowledge, these visualizations may encourage various strategies. Children with lower fraction knowledge might rely on counting, leading to natural number bias and low performance, whereas those with higher knowledge might rely on more efficient strategies based on magnitude. This study explores the relationship between students' general fraction knowledge and their ability to visually compare fraction magnitudes represented with tape diagrams. A total of 67 children completed a fraction knowledge test and a set of comparison tasks with discretized and continuous tape diagrams while their eye movements, accuracy, and response times were recorded. Cluster analysis identified three groups. The first group, high-achieving and applying magnitude-based strategies, showed high accuracy and short response times, indicating efficiency. A second high-achieving group frequently used counting strategies, which was unexpected. This group achieved the highest accuracy but the longest response times, indicating less efficiency. The third group, low-achieving and rarely using counting strategies, had the lowest accuracy and short response times. These students tended to compare absolute sizes rather than relative sizes (i.e., showing a size bias). None of the groups exhibited a natural number bias. The study suggests that counting, although inefficient, does not necessarily lead to bias or low performance. Instead, biased reasoning with fraction visualizations can originate from reliance on absolute sizes.

The spectral ζ-function for quasi-regular Sturm–Liouville operators

In this work, we analyze the spectral ζ-function associated with the self-adjoint extensions, TA,B, of quasi-regular Sturm–Liouville operators that are bounded from below. By utilizing the Green’s function formalism, we find the characteristic function, which implicitly provides the eigenvalues associated with a given self-adjoint extension TA,B. The characteristic function is then employed to construct a contour integral representation for the spectral ζ-function of TA,B. By assuming a general form for the asymptotic expansion of the characteristic function, we describe the analytic continuation of the ζ-function to a larger region of the complex plane. We also present a method for computing the value of the spectral ζ-function of TA,B at all positive integers. We provide two examples to illustrate the methods developed in the paper: the generalized Bessel and Legendre operators. We show that in the case of the generalized Bessel operator, the spectral ζ-function develops a branch point at the origin, while in the case of the Legendre operator it presents, more remarkably, branch points at every nonpositive integer value of s.

Some New Constructions of q-ary Codes for Correcting a Burst of at Most t Deletions.

In this paper, we construct q-ary codes for correcting a burst of at most t deletions, where t,q≥2 are arbitrarily fixed positive integers. We consider two scenarios of error correction: the classical error correcting codes, which recover each codeword from one read (channel output), and the reconstruction codes, which allow to recover each codeword from multiple channel reads. For the first scenario, our construction has redundancy logn+8loglogn+o(loglogn) bits, encoding complexity O(q7tn(logn)3) and decoding complexity O(nlogn). For the reconstruction scenario, our construction can recover the codewords with two reads and has redundancy 8loglogn+o(loglogn) bits. The encoding complexity of this construction is Oq7tn(logn)3, and decoding complexity is Oq9t(nlogn)3. Both of our constructions have lower redundancy than the best known existing works. We also give explicit encoding functions for both constructions that are simpler than previous works.

Odometers, Backward Continued Fractions, and Counting Rationals

It has been more than twenty years since Moshe Newman, based on work by Neil Calkin and Herbert Wilf, introduced an explicit bijection between the rational and natural numbers. Interestingly, this bijection is dynamic in nature. Indeed, Newman’s map has the property that the orbit of zero provides the required bijection. Claudio Bonanno and Stefano Isola, using continued fractions expansions, described the dynamics of its first return time map T. They proved that it is topologically conjugated to the dyadic odometer. In this article, we prove that the correct numerical system needed to analyze this map is the backward continued fractions. Indeed, this approach has the advantage that it provides explicitly the action of T on the expansion. As a by-product, we naturally obtain an explicit formula for Minkowski’s question mark function in terms of backward continued fractions. The whole point of Newman was to provide an explicit bijection, our approach shares the same taste for the explicit.

Cyclic Arithmetic and Combinatorial Operations

Despite the ubiquity of the rings Z n as the finite, cyclic models of integer arithmetic, it is rather less well-known that cyclic models of (positive) arithmetic with exponentiation exist only for cycle length 1, 2, 6, 42, and 1806. We explore finite cyclic models of other arithmetical and combinatorial operations on positive arithmetic: fixed base exponentiation, factorial, and binomial coefficients. In each case we find whether infinitely or finitely many models are possible. The case of fixed base exponentiation is particularly interesting, where for base b > 2 we find that the compatible cycle sizes form a multiplicative monoid of positive integers, with infinitely many irreducible elements, and curiously sparse prime factors.

Division quaternion algebras over some cyclotomic fields

Let p 1, p 2 be two distinct prime integers, let n be a positive integer, n ≥ 3 and let ξn be a primitive root of order n of the unity. In the 3rd section of this paper we obtain a complete characterization for a quaternion algebra H ( p 1 , p 2 ) to be a division algebra over the nth cyclotomic field Q ( ξ n ) , when n ∈ { 3 , 4 , 6 , 7 , 8 , 9 , 11 , 12 } and we also obtain a characterization for a quaternion algebra H ( p 1 , p 2 ) to be a division algebra over the nth cyclotomic field Q ( ξ n ) , when n ∈ { 5 , 10 } . In the 4th section we obtain a complete characterization for a quaternion algebra H Q ( ξ n ) ( p 1 , p 2 ) to be a division algebra, when n = l k , with l a prime integer, l ≡ 3 (mod 4) and k a positive integer. In the last section of this article we obtain a complete characterization for a quaternion algebra H Q ( ξ l ) ( p 1 , p 2 ) to be a division algebra, when l is a Fermat prime number.

Ordered semigroups containing covered (m,n)-ideals

In this article, we extend the concept of covered (m, n)-ideals by defining covered (m, n)-ideals of ordered semigroups, where m, n are nonnegative integers. The results obtained generalize those of covered one-sided ideals and covered bi-ideals. We additionally investigate the conditions under which, in an (m, n)-regular ordered semigroup S, every covered (m, n)-ideal of an (m, n)-ideal I of S is also a covered (m, n)-ideal of S.

On distance k-domination number of graphs under product operations

For a positive integer k, a subset S ⊆ V(G) is said to be a k-dominating set of G if every vertex v ∈ V(G)\S is at a distance of at most k from some vertex of S. The cardinality of the minimum k-dominating set of G is the k-domination number, denoted by γk(G). In this article, we determine the distance k-domination number for two graphs under some product operations in terms of its factor graphs.

Primes as Sums of Fibonacci Numbers

The purpose of this paper is to discuss the relationship between prime numbers and sums of Fibonacci numbers. One of our main results says that for every sufficiently large integer k k there exists a prime number that can be represented as the sum of k k different and non-consecutive Fibonacci numbers. This property is closely related to, and based on, a prime number theorem for certain morphic sequences. In our case, these morphic sequences are based on the Zeckendorf expansion of a positive integer n n — we write n n as the sum of non-consecutive Fibonacci numbers. More precisely, we are concerned with the Zeckendorf sum-of-digits function z \mathsf {z} , which returns the minimal number of Fibonacci numbers needed to write a positive integer as their sum. The proof of such a prime number theorem for z \mathsf {z} , combined with a corresponding local result, constitutes the central contribution of this paper, from which the result stated in the beginning follows. Problems of this type have been discussed intensively in the context of the base- q q expansion of integers. The driving forces of this development were the Gelfond problems (1968/1969), more specifically the behavior of the sum-of-digits function in base q q along the sequence of primes and along integer-valued polynomials, and the Sarnak conjecture. Mauduit and Rivat resolved the question on the sum of digits of prime numbers (2010) and the sum of digits of squares (2009). Later the second author (2017) proved Sarnak’s conjecture for the class of automatic sequences, which are based on the q q -ary expansion of integers, and which generalize the sum-of-digits function in base q q considerably. For the (partial) solution of the Gelfond problems (1967/1968), Mauduit and Rivat have developed a powerful method that is based on techniques for “cutting off digits”, on sophisticated estimates for Fourier terms, and on estimates for exponential sums. These techniques — together with a new decomposition of finite automata — were also the basis for the second author’s result on automatic sequences. In order to obtain corresponding results for Fibonacci numbers, we have to extend Mauduit and Rivat’s method considerably. In fact, we are departing significantly from this method, proving the statement that exp ⁡ ( 2 π i ϑ z ( n ) ) \exp (2\pi i \vartheta \hspace {0.5pt}\mathsf {z}(n)) has level of distribution 1 1 . This latter result forms an essential part of our treatment of the occurring sums of type I \mathrm {I} and I I \mathrm {II} and uses Gowers norms related to the Zeckendorf sum-of-digits function as a central technical tool. Gowers norms are a higher order generalization of the above-mentioned Fourier terms, and their appearance in our method is intimately tied to the iterated application of a new generalization of van der Corput’s inequality.

New limits of the Lie product formula type in Banach algebras

We find new limits of the Lie product formula type in Banach algebras with unit. Some sample results: Let X, Y, Z be Banach algebras with unit, xn,ynn∈N⊂X×Y convergent sequences with limn→∞xn=x, limn→∞yn=y and T:X×Y→Z a continuous bilinear operator with T1,1=1. Then for all sequences of natural numbers ann∈N with limn→∞an=∞ we have limn→∞T∏k=1nexkank+nk+2n,∏k=1neykank+2nk+3nan=eln43Tx,1+ln2T1,y;limn→∞T∏k=1ncosxkannn+k,∏k=1ncoskyknann2+k2nan2=e-ln2Tx2,1+1-π4T1,y22.

Wannabe Bounded Treewidth Graphs Admit a Polynomial Kernel for Directed Feedback Vertex Set

In the Directed Feedback Vertex Set (DFVS) problem, given a digraph D and a positive integer k , the goal is to check if there exists a set of at most k vertices whose deletion from D results in a directed acyclic graph. The existence of a polynomial kernel for DFVS , parameterized by the solution size k , is a central open problem in Kernelization. In this paper, we give a polynomial kernel for DFVS parameterized by k plus the size of a treewidth- η modulator (of the underlying undirected graph), where η is any fixed positive integer. Since the status of the existence of a polynomial kernel for DFVS (parameterized by the solution size) is open for a very long time now, and it is known to not admit a polynomial kernel when the parameter is the size of a treewidth-2 modulator, solution size plus the size of the treewidth- η modulator makes for an interesting choice of parameter to study. In fact, the polynomial kernelization complexity of DFVS parameterized by the size of the undirected feedback vertex set (treewidth-1 modulator) in the underlying undirected graph, has already been studied in literature. Our choice of parameter strictly encompasses previous positive kernelization results on DFVS . Our result is based on a novel application of the tool of important separators embedded in state-of-the-art machinery such as protrusion decompositions.

On algebraic degrees of certain exponential sums over finite fields

Abstract From a complex perspective, Birch and Bombieri (1985) proposed the study of an important class of n-dimensional exponential sums for n = 4 {n=4} , which was then generalized by Katz (1987) to general positive integer n. Our previous paper (2022) enhanced the preceding work from a complex point of view and expanded the subject to a p-adic point of view for any positive integer n. In this paper, we investigate the algebraic degree of the above exponential sum as an algebraic integer.

On the number of subdirect products involving semigroups of integers and natural numbers

On the number of subdirect products involving semigroups of integers and natural numbers

Multi-Dimensional Markov Chains of M/G/1 Type

We consider an irreducible discrete-time Markov process with states represented as (k, i) where k is an M-dimensional vector with non-negative integer entries, and i indicates the state (phase) of the external environment. The number n of phases may be either finite or infinite. One-step transitions of the process from a state (k, i) are limited to states (n, j) such that n ≥ k−1, where 1 represents the vector of all 1s. We assume that for a vector k ≥ 1, the one-step transition probability from a state (k, i) to a state (n, j) may depend on i, j, and n − k, but not on the specific values of k and n. This process can be classified as a Markov chain of M/G/1 type, where the minimum entry of the vector n defines the level of a state (n, j). It is shown that the first passage distribution matrix of such a process, also known as the matrix G, can be expressed through a family of nonnegative square matrices of order n, which is a solution to a system of nonlinear matrix equations.

On Finite Analogues of Euler’s Constant

Abstract We introduce and study finite analogues of Euler’s constant in the same setting as finite multiple zeta values. We define a couple of candidate values from the perspectives of a “regularized value of $\zeta (1)$” and of Mascheroni’s and Kluyver’s series expressions of Euler’s constant using Gregory coefficients. Moreover, we reveal that the differences between them always lie in the ${\mathbb{Q}}$-vector space spanned by 1 and values of a finite analogue of logarithm at positive integers.

A Self-Formed Ag Nanostructure Based Neuromorphic Device Performing Arithmetic Computation and Area Integration: Influence of Presynaptic Pulsing Scheme on Mathematical Precision.

A material equivalent of a biosynapse is the key to neuromorphic architecture. Here we report a self-forming labyrinthine Ag nanostructure activated with a few pulses of 0.5 V, width and interval set at 50 ms, at current compliance (ICC) of 400 nA, serving as the active material for a highly stable device with programmable volatility. Both the conductance (G) and its retention time (tr) in the potentiated state are found to vary linearly with the pulse number for pulses of positive and negative polarities, with the nonlinearity factors being noticeably small, ∼0.03 for G during potentiation and ∼0.08 during depression. This was tested for over 200 days, and the results were highly reproducible. Relying on the high linearity, arithmetic operations involving counting of positive and negative integers were realized using pulses of both polarities, often by mixing them in the feeding sequence. The observed outcomes based on G and independently from tr are highly accurate, with deviations being typically less than ∼1.5% from the expected results. Notably, the way the pulse polarities are mixed is found to have an influence, with random sequences producing relatively larger deviations in integer estimation. However, deviations decreased with higher ICC values, which promoted stronger filament formation in the percolation networks. Besides, the G and tr values were found to vary with the pulse amplitude as well, which enabled the calculation of the area under a curve. Further, the device exhibited a simulation-based image classification accuracy of 94.95%, close to the ideal value (96.05%). Simulations utilizing the finite element method have showcased the uniqueness of the labyrinthine morphology, giving rise to field intensification along potential percolative paths.

Decomposition of hypercube graphs into paths and cycles having k edges

For an even graph G and positive integer p, q, and k, the pair (p, q) is an admissible pair if ( p + q ) k = | E ( G ) | . If a graph G admits a decomposition into p copies of P k + 1 , the path of length k, and q copies of Ck , the cycle of length k, for every admissible pair (p, q), then G has a { P k + 1 , C k } { p , q } -decomposition. In this paper, we give necessary and sufficient conditions for the existence of a { P k + 1 , C k } { p , q } -decomposition of n-dimensional hypercube graphs Qn when n is even, k ≥ 4 , and n ≡ 0 ( mod k ) .

On the page-number of a circulant graph

The page-number of a graph G, denoted by p(G), is the minimum k such that G has an embedding in a k-book. The circulant graph C(m, n) is such a graph that its vertex set is { v 0 , v 1 , … , v m − 1 } and edge set is { v i v i + 1 , v i v i + n | i = 0 , 1 , … , m − 1 } , where m, n are positive integers satisfying 2 ≤ n ≤ ⌊ m 2 ⌋ and indices is read modulo m. In the paper we show that the page-number of a circulant graph C(m, n) is three except for C ( 2 k , 2 ) , where the page-number of C ( 2 k , 2 ) is two.

A Discussion on the Solution(s) of the Diophantine Equation 3<sup>x</sup> + 15<sup>y</sup> =Z<sup>2</sup>

The absence of a generalized method for solving Diophantine equations having more unknowns than a number of equations is a challenge for researchers in different fields. The presence of the Diophantine equation is reported in the study of the Hydrogen spectrum, quantum Hall effect, chemistry, cryptography, etc. Some special types of Diophantine equations could be addressed with the help of Catalan’s conjecture and Congruence theory. The Diophantine equation 3x + 15y =z2 is addressed in this paper to find the solution(s) in positive integers. It is found that the equation has only two solutions of (x,y,z) as (1,0,2) and (0,1,4) in non-negative integers.

On vertices of outdegree [formula omitted] in minimally [formula omitted]-arc-connected digraphs

Let k be a positive integer, and D=(V(D),E(D)) be a minimally k-arc-connected simple digraph. Mader conjectured (Combinatorics 2 (1996) 423-449) that there are at least k+1 vertices of outdegree k in D. In this paper we prove that there are at least four vertices of outdegree k for k≥3.