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.
We define the linear capacity of an algebraic cone, give basic properties of the notion and new formulations of certain known results of the Matrix Theory. We derive in an explicit way the formula for the linear capacity of an irreducible component of the zero cone of a quadratic form over an algebraically closed field. We also give a formula for the linear capacity of the cone over the conjugacy class of a ''generic'' non-nilpotent matrix.
We study a linear system of equations arising from fluid motion around a moving rigid body, where rotation is included. Originally, the coordinate system is attached to the fluid, which means that the domain is changing with respect to time. To get a problem in the fixed domain, the problem is rewritten in the coordinate system attached to the body. The aim of the present paper is the proof of the existence of a strong solution in a weighted Lebesgue space. In particular, we prove the existence of a global pressure gradient in L2.
In the present work, using a formula describing all scalar spectral functions of a Carleman operator A of defect indices (1, 1) in the Hilbert space L 2 (X, µ) that we obtained in a previous paper, we derive certain results concerning the localization of the spectrum of quasi-selfadjoint extensions of the operator A.
Suppose k + 1 runners having nonzero distinct constant speeds run laps on a unit-length circular track. The Lonely Runner Conjecture states that there is a time at which a given runner is at distance at least 1/(k + 1) from all the others. The conjecture has been already settled up to seven (k ≤ 6) runners while it is open for eight or more runners. In this paper the conjecture has been verified for four or more runners having some particular speeds using elementary tools.
We show that the global-in-time solutions to the compressible Navier-Stokes equations driven by highly oscillating external forces stabilize to globally defined (on the whole real line) solutions of the same system with the driving force given by the integral mean of oscillations. Several stability results will be obtained.