Positive Integer Research Articles

A Goldbach Theorem for Laurent Polynomials with Positive Integer Coefficients

We establish an analogue of the Goldbach conjecture for Laurent polynomials with positive integer coefficients.

Connections between two classes of generalized Fibonacci numbers squared and permanents of (0,1) Toeplitz matrices

By considering the tiling of an N-board (a linear array of N square cells of unit width) with new types of tile that we refer to as combs, we give a combinatorial interpretation of the product of two consecutive generalized Fibonacci numbers s n (where s n = ∑ i = 1 q v i s n − m i , s 0 = 1 , s n < 0 = 0 , where v i and m i are positive integers and m 1 < ⋯ < m q ) each raised to an arbitrary non-negative integer power. A ( w , g ; m ) -comb is a tile composed of m rectangular sub-tiles of dimensions w × 1 separated by gaps of width g. The interpretation is used to give combinatorial proof of new convolution-type identities relating s n 2 for the cases q = 2, v i = 1 , m 1 = M , m 2 = m + 1 for M = 0, m to the permanent of a (0,1) Toeplitz matrix with 3 nonzero diagonals which are − 2 , M−1, and m above the leading diagonal. When m = 1, these identities reduce to ones connecting the Padovan and Narayana's cows numbers.

On groups with BFC-covered word values

For a group G and a positive integer n write Bn(G)={x∈G:|xG|≤n}. If s≥1 and w is a group word, say that G satisfies the (n,s)-covering condition with respect to the word w if there exists a subset S⊆G such that |S|≤s and all w-values of G are contained in Bn(G)S. In a natural way, this condition emerged in the study of probabilistically nilpotent groups of class two. In this paper we obtain the following results.Let w be a multilinear commutator word on k variables and let G be a group satisfying the(n,s)-covering condition with respect to the word w. Then G has a soluble subgroup T such that[G:T]and the derived length of T are both(k,n,s)-bounded. (Theorem 1.1.)Letk≥1and G be a group satisfying the(n,s)-covering condition with respect to the wordγk. Then (1)γ2k−1(G)has a subgroup T such that[γ2k−1(G):T]and|T′|are both(k,n,s)-bounded; and (2) G has a nilpotent subgroup U such that[G:U]and the nilpotency class of U are both(k,n,s)-bounded. (Theorem 1.2.)

Implications of Ramsey Choice principles in ZF$\mathsf {ZF}$

AbstractThe Ramsey Choice principle for families of ‐element sets, denoted , states that every infinite set has an infinite subset with a choice function on . We investigate for which positive integers and the implication is provable in . It will turn out that beside the trivial implications , under the assumption that every odd integer is the sum of three primes (known as ternary Goldbach conjecture), the only non‐trivial implication which is provable in is .

P-matrix powers

A P-matrix is a matrix all of whose principal minors are positive. We demonstrate that the fractional powers of a P-matrix are also P-matrices. This insight allows us to affirmatively address a longstanding conjecture raised in Hershkowitz and Johnson (1986) [8]: It is shown that if Ak is a P-matrix for all positive integers k, then the eigenvalues of A are positive.

A Conjecture for the Clique Number of Graphs Associated with Symmetric Numerical Semigroups of Arbitrary Multiplicity and Embedding Dimension

A subset S of non-negative integers No is called a numerical semigroup if it is a submonoid of No and has a finite complement in No. An undirected graph G(S) associated with S is a graph having V(G(S))={vi:i∈No∖S} and E(G(S))={vivj⇔i+j∈S}. In this article, we propose a conjecture for the clique number of graphs associated with a symmetric family of numerical semigroups of arbitrary multiplicity and embedding dimension. Furthermore, we prove this conjecture for the case of arbitrary multiplicity and embedding dimension 7.

The covariety of numerical semigroups with fixed Frobenius number

Denote by m(S)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathrm m}(S)$$\\end{document} the multiplicity of a numerical semigroup S. A covariety is a nonempty family C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr {C}$$\\end{document} of numerical semigroups that fulfils the following conditions: there is the minimum of C,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr {C},$$\\end{document} the intersection of two elements of C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr {C}$$\\end{document} is again an element of C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr {C}$$\\end{document} and S\\{m(S)}∈C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S\\backslash \\{{\\mathrm m}(S)\\}\\in \\mathscr {C}$$\\end{document} for all S∈C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S\\in \\mathscr {C}$$\\end{document} such that S≠min(C).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S\ e \\min (\\mathscr {C}).$$\\end{document} In this work we describe an algorithmic procedure to compute all the elements of C.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr {C}.$$\\end{document} We prove that there exists the smallest element of C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr {C}$$\\end{document} containing a set of positive integers. We show that A(F)={S∣Sis a numerical semigroup with Frobenius numberF}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr {A}(F)=\\{S\\mid S \\hbox { is a numerical semigroup with Frobenius number }F\\}$$\\end{document} is a covariety, and we particularize the previous results in this covariety. Finally, we will see that there is the smallest covariety containing a finite set of numerical semigroups.

Frobenius Numbers Associated with Diophantine Triples of x2-y2=zr

We give an explicit formula for the p-Frobenius number of triples associated with Diophantine Equations x2−y2=zr (r≥2), that is, the largest positive integer that can only be represented in p ways by combining the three integers of the solutions of Diophantine equations x2−y2=zr. This result is also a generalization because if r=2 and p=0, the (0-)Frobenius number of the Pythagorean triple has already been given. To find p-Frobenius numbers, we use geometrically easy to understand figures of the elements of the p-Apéry set, which exhibits symmetric appearances.

Maldistributed sequences in metric spaces

Let a kind of measure μ be given on the set of all positive integers N. By a μ-maldistributed sequence in a metric space (X,d) we mean any sequence possessing μ-extremely many points in each nonempty open subspace of (X,d). Let s(X) denote the metric space of all sequences in (X,d) endowed with the Fréchet metric. In this paper we study topological properties of the set of all maldistributed sequences in X as a subspace s(X). We show that if (X,d) is separable then this set is residual for a large class of commonly used measures μ on N.

P-286 Comparative study of frozen-thawed embryo transfer with modified natural cycle versus HRT cycle in previously cancelled true natural cycle patients

Abstract Study question Is there any difference in outcome of frozen embryo transfer(FET) between hormone replacement therapy(HRT) and modified natural cycle(MNC) in previously cancelled true natural cycle patients Summary answer In patients with previous history of cancelled true natural cycle, MNC results in higher clinical pregnancy rates after FET, when compared to HRT cycle What is known already Preparation of endometrium lays down the foundation for conception and subsequent successful pregnancy.There has been a plethora of endometrial preparation protocols described in the literature. The established protocols are true natural cycle,MNC with or without luteal phase support(LPS) and HRT cycle with or without gonadotrophin releasing hormone agonist(GnRha) suppression. Nevertheless,there is no consensus on protocol which may provide the best outcome in patients who have had cancelled true natural cycle.In such situations,HRT cycle may be considered as an easy option as the cancellation rate is low.However,HRT treatment is known for higher early pregnancy and obstetric complications,compared to MNC Study design, size, duration Prospective observational study,where 155 patients undergoing FET who had minimum one true natural cycle cancellation were recruited.They were divided into group A undergoing HRT cycle(n = 77) and group B for MNC cycle(n = 78).The study was conducted in a fertility unit over six months from June-November 2023.Primary outcome measures were serum betaHCG estimation and ultrasonographic documentation of intrauterine pregnancy.Results were analysed using statistical software SPSS version25 and p value of less than 0.05 was considered statistically significant Participants/materials, setting, methods HRT patients received injectable GnRha 3.75 mg on day21 of previous cycle.Oral estradiol valerate 6mg daily started on day2 of menses. When ET was minimum 7mm, luteal phase support(LPS) was started with daily oral and vaginal progesterone(30mg and 600mcg respectively),followed by FET. MNC patients received Letrozole 5mg daily from day 2-6 of menstruation.rhCG 250mcg was given subcutaneously,when ET was at least 7mm,dominant follicle 16-18mm and progesterone(P4) less than 1ng/ml.LPS(vaginal progesterone600mcg) was started 36hours after rHCG,followed by FET Main results and the role of chance Mean ET in MNC group was 8.92mm ± SD1.38, which was higher than HRT group (8.31 mm ± SD0.89). This difference was statistically significant with a p value of 0.014.Serum beta HCG estimation was done 2weeks after FET. In HRT group beta HCG was positive in 41 out of 77(53.25%) and in MNC group positivity was higher being 48 out of 78(61.54%). However, this difference was not statistically significant. Transvaginal ultrasonography(TVS) was done to confirm intrauterine pregnancy 2weeks after beta HCG assay. Clinical pregnancy rate(CPR) was higher in MNC group with 32 out of 78(41.03%),compared to 18 out of 77(23.38%) in HRT group. This difference is statistically significant with p value of 0.019 Limitations, reasons for caution Relatively small population in this study may be the reason for caution to interpret the statistical data. Study with higher numbers may bring about more robust conclusions Wider implications of the findings Patients with failed true natural cycles may be considered for modified natural cycle(MNC) stimulation for FET, without attempt of another true natural cycle stimulation Trial registration number NOT APPLICABLE

Application of (σ,δ)-cyclic codes in DNA codes over a non-chain ring

For a prime [Formula: see text] and a positive integer [Formula: see text], let [Formula: see text] be the finite field of characteristic [Formula: see text], and [Formula: see text] be a non-chain ring for a positive integer [Formula: see text]. In this paper, we study the [Formula: see text]-cyclic codes over [Formula: see text]. Further, we study the application of these codes in finding DNA codes. Toward this, we first define a Gray map to find classical codes over [Formula: see text] using codes over the ring [Formula: see text]. Later, we find the conditions for a [Formula: see text]-cyclic code to be reversible. Finally, we provide DNA codes using this algebraic method with some classical codes of better parameters.

Wronskian Representation of the Solutions to the KdV Equation

A method to construct solutions to the Korteweg-de-Vries (KdV) equation in terms of wronskians is given. For this, a particular type of polynomials is considered and we obtain for each positive integer n, rational solutions in terms of determinants of order n. Explicit solutions can be easily constructed and rational solutions from order 1 until order 10 are given.

On two variants of split graphs: 2-unipolar graph and k-probe-split graph

A graph is called split if its vertex set can be partitioned into a stable set and a clique. In this article, we studied two variants of split graphs. A graph G is polar if its vertex set can be partitioned into two sets A and B such that G[A] is a complete multipartite graph and G[B] is a disjoint union of complete graphs. A 2-unipolar graph is a polar graph G such that G[A] is a clique and G[B] is the disjoint union of complete graphs with at most two vertices. We present a minimal forbidden induced subgraph characterization for 2-unipolar graphs. In addition, we show that they can be represented as an intersection of substars of special cacti. Let G be a graph class, the G-width of a graph G is the minimum positive integer k such that there exist k independent sets N1, … , Nk such that a set F of nonedges of G, whose endpoints belong to some Ni with i = 1, … , k, can be added so that the resulting graph G′ belongs to G. We say that a graph G is k-probe-G if it has G-width at most k and when G is the class of split graphs it is denominated k-probe-split. We prove that deciding, given a graph G and a positive integer k, whether G is a h-probe-split graph for some h ≤ k is NP-complete. Besides, a characterization by minimal forbidden induced subgraphs for 2-probe-split cographs is presented.

Pythagorean triples using the relativistic velocity addition formula

In the special theory of relativity, there is an unusual formula for addition of velocities. We will use this formula to generate Pythagorean triples. We define a Pythagorean triple in the usual manner as a triple of positive integers (a, b, c) such that a2 + b2 = c2. The numbers a and b are called legs, and c is called the hypotenuse. Later, we will allow b to take on the value of any integer. A primitive Pythagorean triple is a Pythagorean triple for which a, b and c are relatively prime.

An Open Problem in Embedding

We propose and study the problem of finding the smallest nonnegative integer s such that a GDD ( m , n , 3 ; 0 , λ ) can be embedded into a BIBD ( m n + s , 3 , λ ) . We find the values of s for all cases except for the case where n ≡ 5 ( mod 6 ) and m ≡ 1 , 3 ( mod 6 ) and m ≥ 3 , which remains as an open problem.

Graphs whose \(l_p\)-optimal rankings are \(l_{\infty}\) Optimal

A ranking on a graph G is a function f : V ( G ) → { 1 , 2 , … , k } with the following restriction: if f ( u ) = f ( v ) for any u , v ∈ V ( G ) , then on every u v path in G , there exists a vertex w with f ( w ) &gt; f ( u ) . The optimality of a ranking is conventionally measured in terms of the l ∞ norm of the sequence of labels produced by the ranking. In \cite{jacob2017lp} we compared this conventional notion of optimality with the l p norm of the sequence of labels in the ranking for any p ∈ [ 0 , ∞ ) , showing that for any non-negative integer c and any non-negative real number p , we can find a graph such that the sets of l p -optimal and l ∞ -optimal rankings are disjoint. In this paper we identify some graphs whose set of l p -optimal rankings and set of l ∞ -optimal rankings overlap. In particular, we establish that for paths and cycles, if p &gt; 0 then l p optimality implies l ∞ optimality but not the other way around, while for any complete multipartite graph, l p optimality and l ∞ optimality are equivalent.

Exposure Of Positive Integer Solutions to an Equation Comprising Kaprekar Numbers

Exposure Of Positive Integer Solutions to an Equation Comprising Kaprekar Numbers

A Study On Maximal Embedding Dimension Numerical Semigroups

In this article, it is examine some the numerical semigroups W and W/2 such that W= &lt; p, q &gt; and W/2= &lt; p, p+q/2, q &gt; where p &lt; q and p, q are odd natural numbers.

Zero-Power Valued Mappings with Dependent Element of Associative Rings

Let R` be a semi prime rings. An additive mapping L: R`→R` is called a zero-power valued on V over L is preserving V and if for each p belongs to V, that’s meaning, there exists a positive integer n(p) &gt; 1 such that L^(n(p))(p) = 0. The main purpose of this article is to study the notion of an additive mapping which is named zero-power valued mapping with dependent element of a semi prime ring. As a factual information, we deliver some results concerning that.

Topology Meets Number Theory

Liouville proved the existence of a set L of transcendental real numbers now known as Liouville numbers. Erdős proved that while L is a small set in that its Lebesgue measure is zero, and even its s-dimensional Hausdorff measure, for each s > 0, equals zero, it has the Erdős property, that is, every real number is the sum of two numbers in L . He proved L is a dense G δ -subset of R and every dense G δ -subset of R has the Erdős property. While being a dense G δ -subset of R is a purely topological property, all such sets contain c Liouville numbers. Each dense G δ -subset of R , including L , is homeomorphic to the product N ℵ 0 of copies of the discrete space N of all natural numbers. Also this product space is homeomorphic to the space P of all irrational real numbers and the space T of all transcendental real numbers. Hence every dense G δ -subset of R has cardinality c . Indeed, any dense G δ -subset of R has a chain Xm , m ∈ ( 0 , ∞ ) of homeomorphic dense G δ -subsets such that X m ⊂ X n , for n < m, and X n ∖ X m has cardinality c . Finally, every real number r ≠ 1 is equal to ab , for some a , b ∈ L .