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Suppose that $Γ=(G, σ)$ is a connected signed graph with at least one cycle. The number of positive, negative and zero eigenvalues of the adjacency matrix of $Γ$ are called positive inertia index, negative inertia index and nullity of $Γ$, which are denoted by $i_+(Γ)$, $i_-(Γ)$ and $η(Γ)$, respectively. Denoted by $g$ the girth, which is the length of the shortest cycle of $Γ$. We study relationships between the girth and the negative inertia index of $Γ$ in this article. We prove $i_{-}(Γ)\geq \lceil\frac{g}{2}\rceil-1$ and extremal signed graphs corresponding to the lower bound are characterized. Furthermore, the signed graph $Γ$ with $i_{-}(Γ)=\lceil\frac{g}{2}\rceil$ for $g\geq 4$ are given. As a by-product, the connected signed graphs with given positive inertia index, nullity and given girth are also determined, respectively.
We study the problem of finding a subgroup of a given order in a finite group, where the group is represented by its Cayley table. We analyze the complexity of the problem in the special case of abelian groups and present an optimal algorithm for finding a subgroup of a given order when the input is given in the form of a Cayley table. To the best of our knowledge, no prior work has addressed the complexity of this problem under the Cayley table representation.
In this paper, we study the induced homological sequence and the induced merge tree of a discrete Morse function on a tree. A discrete Morse function on a tree gives rise to a sequence of Betti numbers that keep track of the number of components at each critical value. A discrete Morse function on a tree also gives rise to an induced merge tree which keeps track of component birth, death, and merging information. These topological indicators are similar but neither one contains the information of the other. We show that given a merge tree and a homological sequence along with some mild conditions on their relationship, there is a discrete Morse function on a tree that induces both the given merge tree and the given homological sequence.
Given a compact Riemann surface $C$, the line in $H^0(J_C,\, 2Θ)$ orthogonal to the sections vanishing at $0$ produces a natural projective structure on $C$. We investigate the properties of this projective structure.
In this paper, we introduce a way to measure the intelligence (or relevance) of an approximation of a given real number in a given model of approximation. Based on the notion of complexity of a number, defined as the number of its digits (in a given base), we introduce a function noted $μ$ (called a measure of intelligence) that associates to any approximation $\mathbf{app}$ of a given real number in a given model a positive number $μ(\mathbf{app})$, which measures the quality of that approximation. More precisely, an approximation $\mathbf{app}$ is deemed intelligent if and only if $μ(\mathbf{app}) \geq 1$. We illustrate the theory with several numerical examples and apply it to the rational model. In this case, we show that it is consistent with the classical theory of rational Diophantine approximation. We conclude by stating an open problem, namely whether any real number can be intelligently approximated in a given model for which it is a limit point.
We give an explicit upper bound for the number of equivalence classes of binary forms with rational integral coefficients of given degree and given discriminant, and with given splitting field. Further, we give an explicit upper bound for the number of irreducible binary forms with rational integral coefficients with given invariant order. Our bounds depend on as few parameters as possible. For instance, we show that the number of equivalence classes of irreducible binary forms with rational integral coefficients of degree r with given invariant order has an upper bound depending only on r. We have proved more general results for binary forms with coefficients in the ring of S-integers of a number field.
The first degree-based entropy of a graph is the Shannon entropy of its degree sequence normalized by the degree sum. Its correct interpretation as a measure of uniformity of the degree sequence requires the determination of its extremal values given natural constraints. In this paper, we prove that the graphs with given size that minimize the first degree-based entropy are the colex graphs.
As shown by Johannes Kepler in 1609, in the two-body problem, the shape of the orbit, a given ellipse, and a given non-vanishing constant angular momentum determines the motion of the planet completely. Even in the three-body problem, in some cases, the shape of the orbit, conservation of the centre of mass and a constant of motion (the angular momentum or the total energy) determines the motion of the three bodies. We show, by a geometrical method, that choreographic motions, in which equal mass three bodies chase each other around a same curve, will be uniquely determined for the following two cases. (i) Convex curves that have point symmetry and non-vanishing angular momentum are given. (ii) Eight-shaped curves which are similar to the curve for the figure-eight solution and the energy constant are given. The reality of the motion should be tested whether the motion satisfies an equation of motion or not. Extensions of the method for generic curves are shown. The extended methods are applicable to generic curves which does not have point symmetry. Each body may have its own curve and its own non-vanishing masses.
An individual's identity in a human society is specified by his or her name. Differently from family names, usually inherited from fathers, a given name for a child is often chosen at the parents' disposal. However, their decision cannot be made in a vacuum but affected by social conventions and trends. Furthermore, such social pressure changes in time, as new names gain popularity while some other names are gradually forgotten. In this paper, we investigate how popularity of given names has evolved over the last century by using datasets collected in Korea, the province of Quebec in Canada, and the United States. In each of these countries, the average popularity of given names exhibits typical patterns of rise and fall with a time scale of about one generation. We also observe that notable changes of diversity in given names signal major social changes.
In this paper, we give an algorithm to determine all local A-packets containing a given irreducible representation of a p-adic classical group. Especially, we can determine whether a given irreducible representation is of Arthur type or not.
We consider the set $\mathcal{M}_n(\mathbb Z; H)$ of $n\times n$-matrices with integer elements of size at most $H$ and obtain a new upper bound on the number of matrices from $\mathcal{M}_n(\mathbb Z; H)$ with a given characteristic polynomial $f \in \mathbb Z[X]$, which is uniform with respect to $f$. This complements the asymptotic formula of A. Eskin, S. Mozes and N. Shah (1996) in which $f$ has to be fixed and irreducible. Using this result, among others, we obtain upper and lower bounds on the number of $s$-tuples of matrices from $\mathcal{M}_n(\mathbb Z; H)$, satisfying various multiplicative relations, including multiplicative dependence and bounded generation of a subgroup of $\mathrm{GL}_n(\mathbb Q)$. These problems generalise those studied in the scalar case $n=1$ by F. Pappalardi, M. Sha, I. E. Shparlinski and C. L. Stewart (2018) with an obvious distinction due to the non-commutativity of matrices. Motivated by these problems, we also prove various properties of the variety of complex matrices with fixed characteristic polynomial, including computing the degree of this variety.
In [S. Effler, F. Ruskey, A CAT algorithm for listing permutations with a given number of inversions, {\it I.P.L.}, 86/2 (2003)] the authors give an algorithm, which appears to be CAT, for generating permutations with a given major index. In the present paper we give a new algorithm for generating a Gray code for subexcedant sequences. We show that this algorithm is CAT and derive it into a CAT generating algorithm for permutations with a given major index.
There is a well-known connection between hypergraphs and bipartite graphs, obtained by treating the incidence matrix of the hypergraph as the biadjacency matrix of a bipartite graph. We use this connection to describe and analyse a rejection sampling algorithm for sampling simple uniform hypergraphs with a given degree sequence. Our algorithm uses, as a black box, an algorithm $\mathcal{A}$ for sampling bipartite graphs with given degrees, uniformly or nearly uniformly, in (expected) polynomial time. The expected runtime of the hypergraph sampling algorithm depends on the (expected) runtime of the bipartite graph sampling algorithm $\mathcal{A}$, and the probability that a uniformly random bipartite graph with given degrees corresponds to a simple hypergraph. We give some conditions on the hypergraph degree sequence which guarantee that this probability is bounded below by a positive constant.
We give a brief re-exposition of the theory due to Pauli and Sinclair of ramification polygons of Eisenstein polynomials over p-adic fields, their associated residual polynomials and an algorithm to produce all extensions for a given ramification polygon. We supplement this with an algorithm to produce all ramification polygons of a given degree, and hence we can produce all totally ramified extensions of a given degree.
This paper addresses the enumeration of rooted and unrooted hypermaps of a given genus. For rooted hypermaps the enumeration method consists of considering the more general family of multirooted hypermaps, in which darts other than the root dart are distinguished. We give functional equations for the generating series counting multirooted hypermaps of a given genus by number of darts, vertices, edges, faces and the degrees of the vertices containing the distinguished darts. We solve these equations to get parametric expressions of the generating functions of rooted hypermaps of low genus. We also count unrooted hypermaps of given genus by number of darts, vertices, hyperedges and faces.
All over the world, future parents are facing the task of finding a suitable given name for their child. This choice is influenced by different factors, such as the social context, language, cultural background and especially personal taste. Although this task is omnipresent, little research has been conducted on the analysis and application of interrelations among given names from a data mining perspective. The present work tackles the problem of recommending given names, by firstly mining for inter-name relatedness in data from the Social Web. Based on these results, the name search engine "Nameling" was built, which attracted more than 35,000 users within less than six months, underpinning the relevance of the underlying recommendation task. The accruing usage data is then used for evaluating different state-of-the-art recommendation systems, as well our new NameRank algorithm which we adopted from our previous work on folksonomies and which yields the best results, considering the trade-off between prediction accuracy and runtime performance as well as its ability to generate personalized recommendations. We also show, how the gathered inter-name relationships can be used for m
A subgroup $H$ of a group $G$ is said to be an $ICΦ$-subgroup of $G$ if $H \cap [H,G] \le Φ(H)$. We analyze the structure of a finite group $G$ under the assumption that some given subgroups of $G$ are $ICΦ$-subgroups of $G$. A new characterization of finite abelian groups and some new criteria for $2$-nilpotence and nilpotence of finite groups will be obtained. Moreover, we will obtain two criteria for a finite group to lie in a given solvably saturated formation containing the class of finite supersolvable groups.
Consider the following framework of universal decoding suggested in [MerhavUniversal]. Given a family of decoding metrics and random coding distribution (prior), a single, universal, decoder is optimal if for any possible channel the average error probability when using this decoder is better than the error probability attained by the best decoder in the family up to a subexponential multiplicative factor. We describe a general universal decoder in this framework. The penalty for using this universal decoder is computed. The universal metric is constructed as follows. For each metric, a canonical metric is defined and conditions for the given prior to be normal are given. A sub-exponential set of canonical metrics of normal prior can be merged to a single universal optimal metric. We provide an example where this decoder is optimal while the decoder of [MerhavUniversal] is not.
We have classified, upto isoclinism, certain groups with a given central factor. As an application, we classify, upto isoclinism, groups having at the most nine element centralizers. Among other results of independent interest, we have classified, upto isoclinism, groups having a central factor of order $p^3$, $p$ a prime. All these improves some previous results.
Working in a variant of the intersection type assignment system of Coppo, Dezani-Ciancaglini and Venneri [1981], we prove several facts about sets of terms having a given intersection type. Our main result is that every strongly normalizing term M admits a *uniqueness typing*, which is a pair $(Γ,A)$ such that 1) $Γ\vdash M : A$ 2) $Γ\vdash N : A \Longrightarrow M =_{βη} N$ We also discuss several presentations of intersection type algebras, and the corresponding choices of type assignment rules. Moreover, we show that the set of closed terms with a given type is uniformly separable, and, if infinite, forms an adequate numeral system. The proof of this fact uses an internal version of the Böhm-out technique, adapted to terms of a given intersection type.