The degree sequences of a graph with restrictions

<p>Two finite sequences <em>s</em>₁and <em>s</em>₂ of nonnegative integers are called bigraphical if there exists a bipartite graph <em>G</em> with partite sets <em>V</em>₁ and <em>V</em>₂ such that <em>s</em>₁ and <em>s</em>₂ are the degrees in <em>G </em>of the vertices in <em>V</em>₁ and <em>V</em>₂, respectively. In this paper, we introduce the concept of <em>1</em>-graphical sequences and present a necessary and sufficient condition for a sequence to be <em>1</em>-graphical in terms of bigraphical sequences.</p>

Real Vector Space and Related NotionsThis study was supported in part by JSPS KAKENHI Grant Numbers 17K00182 and 20K19863.

Summary. In this paper, we discuss the properties that hold in finite dimensional vector spaces and related spaces. In the Mizar language [1], [2], variables are strictly typed, and their type conversion requires a complicated process. Our purpose is to formalize that some properties of finite dimensional vector spaces are preserved in type transformations, and to contain the complexity of type transformations into this paper. Specifically, we show that properties such as algebraic structure, subsets, finite sequences and their sums, linear combination, linear independence, and affine independence are preserved in type conversions among TOP-REAL(n), REAL-NS(n), and n-VectSp over F Real. We referred to [4], [9], and [8] in the formalization.

Unified Vector Multiplication Approach for Calculating Convolution and Correlation

This paper is a theoretical analysis of discrete time convolution and correlation and to introduce a unified vector multiplication approach for calculating discrete convolution and correlation, both of which are important concepts in the design and analysis of signals and systems and are usually encountered in the first course in signals and systems analysis. There are software tools for calculating them, however, it is important to learn now to compute them by hand. Several methods have been proposed to compute them by hand, most of which can be very involving. However, a closer look at the concepts reveal that the convolution and correlation sums are actually vector multiplication with diagonalwise addition and for finite sequences, can be computed by hand the same way. The method is also extended to N-point circular convolution. The method also makes it clearer to see the similarities and differences between convolution and correlation.

Certain Distance-Based Topological Indices of Parikh Word Representable Graphs

Relating graph structures with words which are finite sequences of symbols, Parikh word representable graphs (PWRGs) were introduced. On the other hand, in chemical graph theory, graphs have been associated with molecular structures. Also, several topological indices have been defined in terms of graph parameters and studied for different classes of graphs. In this study, we derive expressions for computing certain topological indices of PWRGs of binary core words, thereby enriching the study of PWRGs.

Minkowski–Clarkson’s type inequalities

In this paper, we give some Minkowski–Clarkson’s type inequalities related to two finite sequences of real nonnegative numbers. In particular, we prove two inequalities which in some sense can be regarded as inverse Minkowski’s inequalities concerning the cases p ≥ 2 {p\geq 2} and 0 < p ≤ 1 {0<p\leq 1} . Moreover, for 1 < p < 2 {1<p<2} we prove another Minkowski–Clarkson’s type inequality.

On the pseudorandom properties of $ k $-ary Sidel'nikov sequences

<p style='text-indent:20px;'>In 2002 Mauduit and Sárközy started to study finite sequences of <inline-formula><tex-math id="M2">\begin{document}$ k $\end{document}</tex-math></inline-formula> symbols</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ E_{N} = \left(e_{1},e_{2},\cdots,e_{N}\right)\in \mathcal{A}^{N}, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M3">\begin{document}$ \mathcal{A} = \left\{a_{1},a_{2},\cdots,a_{k}\right\}(k\in \mathbb{N},k\geq 2) $\end{document}</tex-math></inline-formula> is a finite set of <inline-formula><tex-math id="M4">\begin{document}$ k $\end{document}</tex-math></inline-formula> symbols. Later many pseudorandom sequences of <inline-formula><tex-math id="M5">\begin{document}$ k $\end{document}</tex-math></inline-formula> symbols have been given and studied by using number theoretic methods. In this paper we study the pseudorandom properties of the <inline-formula><tex-math id="M6">\begin{document}$ k $\end{document}</tex-math></inline-formula>-ary Sidel'nikov sequences with length <inline-formula><tex-math id="M7">\begin{document}$ q-1 $\end{document}</tex-math></inline-formula> by using the estimates for certain character sums with exponential function, where <inline-formula><tex-math id="M8">\begin{document}$ q $\end{document}</tex-math></inline-formula> is a prime power. Our results show that Sidel'nikov sequences enjoy good well-distribution measure and correlation measure. Furthermore, we prove that the set of size <inline-formula><tex-math id="M9">\begin{document}$ \phi(q-1) $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M10">\begin{document}$ k $\end{document}</tex-math></inline-formula>-ary Sidel'nikov sequences is collision free and possesses the strict avalanche effect property provided that <inline-formula><tex-math id="M11">\begin{document}$ k = o(q^{\frac{1}{4}}) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M12">\begin{document}$ \phi $\end{document}</tex-math></inline-formula> denotes Euler's totient function.</p>