On Some Approximation Properties for a Sequence of λ-Bernstein Type Operators

In 2010, Long and Zeng introduced a new generalization of the Bernstein polynomials that depends on a parameter and called -Bernstein polynomials. After that, in 2018, Lain and Zhou studied the uniform convergence for these -polynomials and obtained a Voronovaskaja-type asymptotic formula in ordinary approximation. This paper studies the convergence theorem and gives two Voronovaskaja-type asymptotic formulas of the sequence of -Bernstein polynomials in both ordinary and simultaneous approximations. For this purpose, we discuss the possibility of finding the recurrence relations of the -th order moment for these polynomials and evaluate the values of -Bernstein for the functions , is a non-negative integer

Algorithms for computing sums of powers of arithmetic progressions by using Eulerian numbers

The sums of powers of arithmetic progressions is of the form a^p+(a+d)^p +(a+2d)^p+\cdots+(a+(n-1)d)^p, where n\geq 1, p is a non-negative integer, and a and d are complex numbers with d\neq 0. This sum can be computed using classical Eulerian numbers \cite{worpitzky1883studien} and general Eulerian numbers \cite{xiong2013general}. This paper gives a new method using classical Eulerian numbers to compute this sum. The existing formula that uses general Eulerian numbers are more algorithmically complex due to more numbers to compute. This paper presents and focuses on two novel algorithms involving both types of Eulerian numbers. This paper gives a comparison to Xiong \textit{et al.}’s result with general Eulerian numbers \cite{xiong2013general}. Moreover, an analysis of theoretical time complexities is presented to show our algorithm is less complex. Various values of p are analyzed in the proposed algorithms to add significance to the results. The experimental results show the proposed algorithm remains around 70\% faster as p increases.

On the Inverse Problem for Some Topological Indices

The study of the inverse problem (IP) based on the topological indices (TIs) deals with the numerical relations to TIs. Mathematically, the IP can be expressed as follows: given a graph parameter/TI that assigns a non-negative integer value g to every graph within a given family G of graphs, find some G ∈ G for which TI G = g . It was initiated by the Zefirov group in Moscow and later Gutman et al. proposed it. In this paper, we have established the IP only for the Y -index, Gourava indices, second hyper-Zagreb index, reformulated first Zagreb index, and reformulated F -index since they are closely related to each other. We have also studied the same which is true for the molecular, tree, unicyclic, and bicyclic graphs.

Limit theorems for numbers of multiple returns in non-conventional arrays

For a $\psi $ -mixing process $\xi _0,\xi _1,\xi _2,\ldots $ we consider the number ${\mathcal N}_N$ of multiple returns $\{\xi _{q_{i,N}(n)}\in {\Gamma }_N,\, i=1,\ldots ,\ell \}$ to a set ${\Gamma }_N$ for n until either a fixed number N or until the moment $\tau _N$ when another multiple return $\{\xi _{q_{i,N}(n)}\in {\Delta }_N,\, i=1,\ldots ,\ell \}$ , takes place for the first time where ${\Gamma }_N\cap {\Delta }_N=\emptyset $ and $q_{i,N}$ , $i=1,\ldots ,\ell $ are certain functions of n taking on non-negative integer values when n runs from 0 to N. The dependence of $q_{i,N}(n)$ on both n and N is the main novelty of the paper. Under some restrictions on the functions $q_{i,N}$ we obtain Poisson distributions limits of ${\mathcal N}_N$ when counting is until N as $N\to \infty $ and geometric distributions limits when counting is until $\tau _N$ as $N\to \infty $ . We obtain also similar results in the dynamical systems setup considering a $\psi $ -mixing shift T on a sequence space ${\Omega }$ and studying the number of multiple returns $\{ T^{q_{i,N}(n)}{\omega }\in A^a_n,\, i=1,\ldots ,\ell \}$ until the first occurrence of another multiple return $\{ T^{q_{i,N}(n)}{\omega }\in A^b_m,\, i=1,\ldots ,\ell \}$ where $A^a_n,\, A_m^b$ are cylinder sets of length n and m constructed by sequences $a,b\in {\Omega }$ , respectively, and chosen so that their probabilities have the same order.

K-Regular Matroids

<p>The class of matroids representable over all fields is the class of regular matroids. The class of matroids representable over all fields except perhaps GF(2) is the class of near-regular matroids. Let k be a non-negative integer. This thesis considers the class of k-regular matroids, a generalization of the last two classes. Indeed, the classes of regular and near-regular matroids coincide with the classes of 0-regular and 1-regular matroids, respectively. This thesis extends many results for regular and near-regular matroids. In particular, for all k, the class of k-regular matroids is precisely the class of matroids representable over a particular partial field. Every 3-connected member of the classes of either regular or near-regular matroids has a unique representability property. This thesis extends this property to the 3-connected members of the class of k-regular matroids for all k. A matroid is [omega] -regular if it is k-regular for some k. It is shown that, for all k [greater than or equal to] 0, every 3-connected k-regular matroid is uniquely representable over the partial field canonically associated with the class of [omega] -regular matroids. To prove this result, the excluded-minor characterization of the class of k-regular matroids within the class of [omega] -regular matroids is first proved. It turns out that, for all k, there are a finite number of [omega] -regular excluded minors for the class of k-regular matroids. The proofs of the last two results on k-regular matroids are closely related. The result referred to next is quite different in this regard. The thesis determines, for all r and all k, the maximum number of points that a simple rank-r k-regular matroid can have and identifies all such matroids having this number. This last result generalizes the corresponding results for regular and near-regular matroids. Some of the main results for k-regular matroids are obtained via a matroid operation that is a generalization of the operation of [Delta] - Y exchange. This operation is called segment-cosegment exchange and, like the operation of [Delta] - Y exchange, has a dual operation. This thesis defines the generalized operation and its dual, and identifies many of their attractive properties. One property in particular, is that, for a partial field P, the set of excluded minors for representability over P is closed under the operations of segment-cosegment exchange and its dual. This result generalizes the corresponding result for [Delta] - Y and Y - [Delta] exchanges. Moreover, a consequence of it is that, for a prime power q, the number of excluded minors for GF(q)-representability is at least 2q-4.</p>

K-Regular Matroids

<p>The class of matroids representable over all fields is the class of regular matroids. The class of matroids representable over all fields except perhaps GF(2) is the class of near-regular matroids. Let k be a non-negative integer. This thesis considers the class of k-regular matroids, a generalization of the last two classes. Indeed, the classes of regular and near-regular matroids coincide with the classes of 0-regular and 1-regular matroids, respectively. This thesis extends many results for regular and near-regular matroids. In particular, for all k, the class of k-regular matroids is precisely the class of matroids representable over a particular partial field. Every 3-connected member of the classes of either regular or near-regular matroids has a unique representability property. This thesis extends this property to the 3-connected members of the class of k-regular matroids for all k. A matroid is [omega] -regular if it is k-regular for some k. It is shown that, for all k [greater than or equal to] 0, every 3-connected k-regular matroid is uniquely representable over the partial field canonically associated with the class of [omega] -regular matroids. To prove this result, the excluded-minor characterization of the class of k-regular matroids within the class of [omega] -regular matroids is first proved. It turns out that, for all k, there are a finite number of [omega] -regular excluded minors for the class of k-regular matroids. The proofs of the last two results on k-regular matroids are closely related. The result referred to next is quite different in this regard. The thesis determines, for all r and all k, the maximum number of points that a simple rank-r k-regular matroid can have and identifies all such matroids having this number. This last result generalizes the corresponding results for regular and near-regular matroids. Some of the main results for k-regular matroids are obtained via a matroid operation that is a generalization of the operation of [Delta] - Y exchange. This operation is called segment-cosegment exchange and, like the operation of [Delta] - Y exchange, has a dual operation. This thesis defines the generalized operation and its dual, and identifies many of their attractive properties. One property in particular, is that, for a partial field P, the set of excluded minors for representability over P is closed under the operations of segment-cosegment exchange and its dual. This result generalizes the corresponding result for [Delta] - Y and Y - [Delta] exchanges. Moreover, a consequence of it is that, for a prime power q, the number of excluded minors for GF(q)-representability is at least 2q-4.</p>

On exponential Diophantine equation 17x +83y = z2 and 29x +71y = z2

This Diophantine is an equation that many researchers are interested in and studied in many form such 3x +5y · 7z = u2, (x+1)k + (x+2)k + … + (2x)k = yn and kax + lby = cz. The extensively studied form is ax + by = cz. In this paper we show that the Diophantine equations 17x +83y = z2 and 29x +71y = z2 has a unique non – negative integer solution (x, y, z) = (1,1,10)

Infinite sequences of almost symmetric non-Weierstrass numerical semigroups

Let u be any positive integer. We construct infinite sequences of almost symmetric non-Weierstrass numerical semigroups whose conductors are the genera double minus $$2u-1$$ 2 u - 1 . Moreover, let v be any non-negative integer. We give an infinite number of non-Weierstrass numerical semigroups whose conductors are the genera double minus 2v.

On deficiency problems for graphs

Motivated by analogous questions in the setting of Steiner triple systems and Latin squares, Nenadov, Sudakov and Wagner [Completion and deficiency problems, Journal of Combinatorial Theory Series B, 2020] recently introduced the notion of graph deficiency. Given a global spanning property $\mathcal P$ and a graph $G$ , the deficiency $\text{def}(G)$ of the graph $G$ with respect to the property $\mathcal P$ is the smallest non-negative integer t such that the join $G*K_t$ has property $\mathcal P$ . In particular, Nenadov, Sudakov and Wagner raised the question of determining how many edges an n-vertex graph $G$ needs to ensure $G*K_t$ contains a $K_r$ -factor (for any fixed $r\geq 3$ ). In this paper, we resolve their problem fully. We also give an analogous result that forces $G*K_t$ to contain any fixed bipartite $(n+t)$ -vertex graph of bounded degree and small bandwidth.