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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.

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.

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.

We relate binary words with a given number of subsequences to continued fractions of rational numbers with a given denominator. We deduce that there are binary strings of length $O(\log n \log \log n)$ with exactly $n$ subsequences; this can be improved to $O(\log n)$ under assumption of Zaremba's conjecture.

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 express each Fréchet class of multivariate Bernoulli distributions with given margins as the convex hull of a set of densities, which belong to the same Fréchet class. This characterisation allows us to establish whether a given correlation matrix is compatible with the assigned margins and, if it is, to easily construct one of the corresponding joint densities. % Such %representation is based on a polynomial expression of the distributions of a Fréchet class. We reduce the problem of finding a density belonging to a Fréchet class and with given correlation matrix to the solution of a linear system of equations. Our methodology also provides the bounds that each correlation must satisfy to be compatible with the assigned margins. An algorithm and its use in some examples is shown.

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.

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.

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.

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 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.

We investigate the parameterized complexity of the graph editing problem called Editing to a Graph with a Given Degree Sequence, where the aim is to obtain a graph with a given degree sequence σby at most k vertex or edge deletions and edge additions. We show that the problem is W[1]-hard when parameterized by k for any combination of the allowed editing operations. From the positive side, we show that the problem can be solved in time 2^{O(k(Δ+k)^2)}n^2 log n for n-vertex graphs, where Δ=max σ, i.e., the problem is FPT when parameterized by k+Δ. We also show that Editing to a Graph with a Given Degree Sequence has a polynomial kernel when parameterized by k+Δif only edge additions are allowed, and there is no polynomial kernel unless NP\subseteq coNP/poly for all other combinations of allowed editing operations.

The aim of edge editing or modification problems is to change a given graph by adding and deleting of a small number of edges in order to satisfy a certain property. We consider the Edge Editing to a Connected Graph of Given Degrees problem that asks for a graph G, non-negative integers d,k and a function δ:V(G)->{1,...,d}, whether it is possible to obtain a connected graph G' from G such that the degree of v is δ(v) for any vertex v by at most k edge editing operations. As the problem is NP-complete even if δ(v)=2, we are interested in the parameterized complexity and show that Edge Editing to a Connected Graph of Given Degrees admits a polynomial kernel when parameterized by d+k. For the special case δ(v)=d, i.e., when the aim is to obtain a connected d-regular graph, the problem is shown to be fixed parameter tractable when parameterized by k only.

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 show that the orthogonal projection operator onto the range of the adjoint of a linear operator T can be represented as UT, where U is an invertible linear operator. Using this representation we obtain a decomposition of a multivariate Normal random variable Y as the sum of a linear transformation of Y that is independent of TY and an affine transformation of TY. We then use this decomposition to prove that the regular conditional distribution of a multivariate Normal random variable Y given a linear transformation TY is again a multivariate Normal distribution. This result is equivalent to the well-known result that given a k-dimensional component of a n-dimensional multivariate Normal random variable, where k < n, the regular conditional distribution of the remaining (n - k)-dimensional component is a (n - k)-dimensional multivariate Normal distribution.

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.

A variety of groups does not contain all metabelian groups if and only if there is an absolute bound for the nilpotency classes of powerful $p$-groups in the given variety. Similarly, a variety contains only finitely many finite $p$-groups of any given coclass if and only if not every group that is an extension of an abelian group by an elementary abelian $p$-group belongs to that variety.