We establish a decomposition of the Jensen-Shannon divergence into a linear combination of a scaled Jeffreys' divergence and a reversed Jensen-Shannon divergence. Upper and lower bounds for the Jensen-Shannon divergence are then found in terms of the squared (total) variation distance. The derivations rely upon the Pinsker inequality and the reverse Pinsker inequality. We use these bounds to prove the asymptotic equivalence of the maximum likelihood estimate and minimum Jensen-Shannon divergence estimate as well as the asymptotic consistency of the minimum Jensen-Shannon divergence estimate. These are key properties for likelihood-free simulator-based inference.
Let $Q$ be the lexicographic sum of finite ordered sets $Q_x$ over a finite ordered set $P$. For some $P$ we can give a formula for the jump number of $Q$ in terms of the jump numbers of $Q_x$ and $P$, that is, $s(Q)=s(P)+ \sum _{x\in P} s(Q_x)$, where $s(X)$ denotes the jump number of an ordered set $X$. We first show that $w(P)-1+\sum _{x\in P} s(Q_x)\le s(Q) \le s(P)+ \sum _{x\in P} s(Q_x)$, where $w(X)$ denotes the width of an ordered set $X$. Consequently, if $P$ is a Dilworth ordered set, that is, $s(P) = w(P)-1$, then the formula holds. We also show that it holds again if $P$ is bipartite. Finally, we prove that the lexicographic sum of certain jump-critical ordered sets is also jump-critical.
For any positive integer k ≥ 3, it is easy to prove that the k-polygonal numbers are an(k) = (2n+n(n−1)(k−2))/2. The main purpose of this paper is, using the properties of Gauss sums and Dedekind sums, the mean square value theorem of Dirichlet L-functions and the analytic methods, to study the computational problem of one kind mean value of Dedekind sums S(an(k)ām(k), p) for k-polygonal numbers with 1 ≤ m, n ≤ p − 1, and give an interesting computational formula for it., Jing Guo, Xiaoxue Li., and Obsahuje seznam literatury
The aim of this paper is to construct an L-valued category whose objects are L-E-ordered sets. To reach the goal, first, we construct a category whose objects are L-E-ordered sets and morphisms are order-preserving mappings (in a fuzzy sense). For the morphisms of the category we define the degree to which each morphism is an order-preserving mapping and as a result we obtain an L-valued category. Further we investigate the properties of this category, namely, we observe some special objects, special morphisms and special constructions.
In this paper we consider the energy of a simple graph with respect to its Laplacian eigenvalues, and prove some basic properties of this energy. In particular, we find the minimal value of this energy in the class of all connected graphs on $n$ vertices $(n=1,2,\ldots )$. Besides, we consider the class of all connected graphs whose Laplacian energy is uniformly bounded by a constant $\alpha \ge 4$, and completely describe this class in the case $\alpha =40$.
The Laplacian, signless Laplacian and normalized Laplacian characteristic polynomials of a graph are the characteristic polynomials of its Laplacian matrix, signless Laplacian matrix and normalized Laplacian matrix, respectively. In this paper, we mainly derive six reduction procedures on the Laplacian, signless Laplacian and normalized Laplacian characteristic polynomials of a graph which can be used to construct larger Laplacian, signless Laplacian and normalized Laplacian cospectral graphs, respectively.
Life-history parameters of Barbus peloponnesius and Barbus cyclolepis were studied in two streams in Macedonia, Greece. In B. peloponnesius age ranged from 0+ to 4+ in males and 0+ to 9+ in females, while in B. cyclolepis from 0+ to 5+ in males and 0+ to 9+ in females. In both species, after the first year of life, females exhibited longer mean lengths at age and greater maximum length than the males, while between species B. cyclolepis showed greater mean lengths at age and greater maximum length than B. peloponnesius. Total mortality rates were higher in the males of each species than in females. Significant difference in the sex ratio was found only for B. cyclolepis and this species population was male dominated. Gonad maturation began at the age of 1+ in males and 3+ in females of both species. Both species exhibited a protracted multi-spawning season, which started at the end of March-beginning of April and lasted until mid July. Despite differences in growth and body size, the two species are characterized by similar life-history styles: (1) similar age structure, (2) early maturation and same age at maturity, (3) males have a shorter life span, higher rate of mortality, decreased growth and smaller body size and mature earlier than the females and (4) elongated multi-spawning season, which shows a high investment in reproduction. The life-history style of the two stocks seems to be in concordance with the environmental conditions of their habitats, which are typical of the fluctuating Mediterranean streams.