Extremal rays of Betti cones

2019 ◽  
Vol 19 (02) ◽  
pp. 2050027
Author(s):  
H. Ananthnarayan ◽  
Rajiv Kumar
Keyword(s):  
Hilbert Series ◽  
Graded Algebra ◽  
One Dimensional ◽  
The Given ◽  
Algebra Over A Field

We study the Betti cone of modules of a standard graded algebra over a field, and find a class of modules whose Betti diagrams span extremal rays of the Betti cone. We also identify a class of one-dimensional Cohen–Macaulay rings with certain Hilbert series, for which the Betti cone is spanned by these extremal rays. These results lead to an algorithm for the decomposition of Betti table of modules, into the extremal rays, over such rings, and also help to obtain bounds for the multiplicity of the given module.

Short Generating Functions for some Semigroup Algebras

10.37236/1729 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Graham Denham
Keyword(s):  
Rational Function ◽  
Generating Functions ◽  
Hilbert Series ◽  
Simple Expression ◽  
Arbitrary Field ◽  
Graded Algebra ◽  
Semigroup Algebras ◽  
Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

scholarly journals BLOCK CIPHERS ON THE BASIS OF REVERSIBLE CELLULAR AUTOMATA

2020 ◽  
Vol 10 (1) ◽  
pp. 8-11
Author(s):  
Yuliya Tanasyuk ◽  
Petro Burdeinyi
Keyword(s):  
Cellular Automata ◽  
Block Cipher ◽  
Key Generation ◽  
One Dimensional ◽  
C Programming Language ◽  
C Programming ◽  
Shared Secret ◽  
Reversible Cellular Automata ◽  
The Given ◽  
First Time

The given paper is devoted to the software development of block cipher based on reversible one-dimensional cellular automata and the study of its statistical properties. The software implementation of the proposed encryption algorithm is performed in C# programming language in Visual Studio 2017. The paper presents specially designed approach for key generation. To ensure desired cryptographic stability, the shared secret parameters can be adjusted to contain information needed for creating substitution tables, defining reversible rules, and hiding final data. For the first time, it is suggested to create substitution tables based on iterations of a cellular automaton that is initialized by the key data.

Dimensionality and Construct Validity of a Video-Based, Objective Personality Test for the Assessment of Willingness to Take Risks in Road Traffic

2005 ◽  
Vol 97 (1) ◽  
pp. 309-320 ◽  
Author(s):  
Martin E. Arendasy ◽  
Andreas Hergovich ◽  
Markus Sommer ◽  
Bettina Bognar
Keyword(s):  
Construct Validity ◽  
Personality Trait ◽  
Road Traffic ◽  
Personality Test ◽  
Traffic Situation ◽  
One Dimensional ◽  
First Results ◽  
The One ◽  
The Given ◽  
Homeostasis Theory

The study at hand reports first results about the dimensionality and construct validity of a newly developed objective, video-based personality test, which assesses the willingness to take risks in traffic situations. On the basis of the theory of risk homeostasis developed by Wilde, different traffic situations with varying amounts of objective danger were filmed. These situations mainly consisted of situations with passing maneuvers and speed choice or traffic situations at intersections. Each of these traffic situations describes an action which should be carried out. The videos of the traffic situations are presented twice. Before the first presentation, a short written explanation of the preceding traffic situation and a situation-contingent reaction is provided. The respondents are allowed to obtain an overview of the given situations during the first presentation of each traffic situation. During the second presentation the respondents are asked to indicate at which point the action that is contingent on the described situation will become too dangerous to carry out. Latencies for items were recorded as a measure for the magnitude of the person's subjectively accepted willingness to take risks in the sense of the risk homeostasis theory by Wilde. In a study with 243 people with different education and sex, the one-dimensionality of the test corresponding to the latency model by Scheiblechner was investigated. Analysis indicated that the new measure assesses a one-dimensional latent personality trait which can be interpreted as subjectively accepted amount of risk (target risk value). First indicators for the construct validity of the test are given by a significant correlation with the construct-related secondary scale, adventurousness of the Eysenck Personality Profiler with, at the same time, nonsignificant correlations to the two secondary scales, extroversion and emotional stability, that are not linked to the construct.

Integration of one-dimensional equations of motions

2020 ◽  
pp. 79-90
Author(s):  
Gleb L. Kotkin ◽  
Valeriy G. Serbo
Keyword(s):  
Potential Energy ◽  
Field Change ◽  
One Dimensional ◽  
System A ◽  
Small Correction ◽  
The Law ◽  
The One ◽  
The Given ◽  
Equations Of Motions

If the potential energy is independent of time, the energy of the system remains constant during the motion of a closed system. A system with one degree of freedom allows for the determination of the law of motion in quadrature. In this chapter, the authors consider motion of the particles in the one-dimensional fields. They discuss also how the law and the period of a particle moving in the potential field change due to adding to the given field a small correction.

Uncertainty analysis of a delineated floodplain

1984 ◽  
Vol 11 (3) ◽  
pp. 387-395 ◽  
Author(s):  
Edward McBean ◽  
Jacques Penel ◽  
Kwok-Lui Siu
Keyword(s):  
Uncertainty Analysis ◽  
Energy Equation ◽  
Data Uncertainty ◽  
Small Computer ◽  
One Dimensional ◽  
The One ◽  
Floodplain Delineation ◽  
Uncertainty Method ◽  
The Given ◽  
Water Profile

The delineation of floodplains involves, in most circumstances, solving the one-dimensional energy equation. However, uncertainties in the identified floodplain arise from both computational and data uncertainties; data uncertainties are concluded to be generally more significant than computational uncertainties.A method is developed to calculate the uncertainty in floodplain delineation arising from data uncertainties. The proposed method requires only HEC-2 computer output and a small computer program. Application of the method to two case studies and comparison with another uncertainty method suggest that the proposed uncertainty theory is applicable to practical situations within the given constraints. Key words: data uncertainty, floodplain, uncertainty analysis, water profile computation.

scholarly journals THE MINIMIZATION OF MOMENTS AND REACTIVE FORCES IN GRILLAGES WITH A GENETIC ALGORITHM

2011 ◽  
Vol 3 (2) ◽  
pp. 56-63
Author(s):  
Rimantas Belevičius ◽  
Darius Mačiūnas ◽  
Dmitrij Šešok
Keyword(s):  
Finite Element ◽  
Bending Moment ◽  
Design Parameters ◽  
Finite Element Program ◽  
One Dimensional ◽  
Absolute Value ◽  
Element Program ◽  
The Absolute ◽  
The One ◽  
The Given

The aim of the article is to report a technology for the optimization of grillage-type foundations seeking for the least possible reactive forces in the piles for a given number of piles and in the absolute value of the bending moments when connecting beams of the grillage. Mathematically, this seems to be the global optimization problem possessing a large number of local minima points. Both goals can be achieved choosing appropriate pile positions under connecting beams; however, these two problems contradict to each other and lead to diff erent schemes for pile placement. Therefore, we suggest using a compromise objective function (to be minimized) that consists of the largest reactive force arising in all piles and that occurring in the absolute value of the bending moment when connecting beams, both with the given weights. Bending moments are calculated at three points of each beam. The design parameters of the problem are positions of the piles. The feasible space of design parameters is determined by two constraints. First, during the optimization process, piles can move only along connecting beams. Therefore, the two-dimensional grillage is “unfolded” to the one-dimensional construct, and supports are allowed to range through this space freely. Second, the minimum allowable distance between two adjacent piles is introduced due to the specific capacities of a pile driver. Also, due to some considerations into the scheme of pile placement, the designer sometimes may introduce immovable supports (usually at the corners of the grillage) that do not participate in the optimization process and always retain their positions. However, such supports hinder to achieve a global solution to a problem and are not treated in this paper. The initial data for the problem are as follows: a geometrical scheme of the grillage, the given number of piles, a cross-section and material data on connecting beams, the minimum possible distance between adjacent supports and loading data given in the form of concentrated loads or trapezoidal distributed loadings. The results of the solution are the required positions of piles. This solution can serve as a pilot project for more detailed design. The entire optimization problem is solved in two steps. First, the grillage is transformed into the one-dimensional construct and the optimizer decides about a routine solution (i.e. the positions of piles in this construct). Second, backward transformation returns pile positions into the two-dimensional grillage and the “black-box” finite element program returns the corresponding objective function value. On the basis of this value, the optimizer predicts new positions of piles etc. The finite element program idealizes connecting beams as beam elements and piles – as mesh nodes of the finite element with a given boundary conditions in the form of vertical and rotational stiff ness. Since the problem may have several tens of design parameters, the only choice for optimization algorithms is using stochastic optimization algorithms. In our case, we use the original elitist real-number genetic algorithm and launch the program sufficient number of times in order to exclude large scattering of results. Three numerical examples are presented for the optimization of 10-pile grillage: when optimizing purely the largest reactive force, purely the largest in the absolute value of the bending moment and both parameters with equal weights.

CONSTRUCTING CHAOTIC POLYNOMIAL MAPS

2009 ◽  
Vol 19 (02) ◽  
pp. 531-543 ◽  
Author(s):  
XU ZHANG ◽  
YUMING SHI ◽  
GUANRONG CHEN
Keyword(s):  
Logistic Map ◽  
Special Kind ◽  
Quadratic Polynomials ◽  
One Dimensional ◽  
Polynomial Maps ◽  
Polynomial Map ◽  
Expansion Theory ◽  
Chaotic Parameter ◽  
The Given ◽  
Nonzero Polynomial

This paper studies the construction of one-dimensional real chaotic polynomial maps. Given an arbitrary nonzero polynomial of degree m (≥ 0), two methods are derived for constructing chaotic polynomial maps of degree m + 2 by simply multiplying the given polynomial with suitably designed quadratic polynomials. Moreover, for m + 2 arbitrarily given different positive constants, a method is given to construct a chaotic polynomial map of degree 2m based on the coupled-expansion theory. Furthermore, by multiplying a real parameter to a special kind of polynomial, which has at least two different non-negative or nonpositive zeros, the chaotic parameter region of the polynomial is analyzed based on the snap-back repeller theory. As a consequence, for any given integer n ≥ 2, at least one polynomial of degree n can be constructed so that it is chaotic in the sense of both Li–Yorke and Devaney. In addition, two natural ways of generalizing the logistic map to higher-degree chaotic logistic-like maps are given. Finally, an illustrative example is provided with computer simulations for illustration.

scholarly journals An initial-boundary value problem for Boltzmann’s non-stationary nonlinear one-dimensional four-moment system of equations

2020 ◽  
pp. 91-95
Author(s):  
G. Suleimenov
Keyword(s):  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Moment Equations ◽  
One Dimensional ◽  
The Moment ◽  
Initial Boundary Value ◽  
The Given ◽  
Dynamic Variables ◽  
Microscopic Distribution

In this article, the set of boundary conditions is defined for first and boundary value problems for the second approximation of Boltzmann’s system of one-dimensional nonlinear moment equations and their logic. For the second approximation of Boltzmann’s one-dimensional non-stationary nonlinear moment equations, which satisfies the Maxwell-Auzhan boundary condition, the theorem for the first boundary problem is considered and by proving this theorem, it is proved that there are only solutions to the given problems. It is known that in many problems of gas dynamics there is no need to describe the complete state of the gas by the function of microscopic distribution of molecules. Therefore, it is better to look for an easier way to describe the gas using macroscopic gas – dynamic variables (density, hydrodynamic average velocity, temperature) are determined in this rotations by the moments of the microscopic distribution function of the molecules, the author faced with the problem of analyzing the different moments of the Boltzmann equation. By studying the moment equations, the author obtained some information about the function of the microscopic distribution of molecules and the convergence of the moment method.

scholarly journals DIRICHLET PROBLEM FOR PULKIN’S EQUATION IN A RECTANGULAR DOMAIN

2017 ◽  
Vol 20 (10) ◽  
pp. 91-101
Author(s):  
R.M. Safina
Keyword(s):  
Mixed Type ◽  
Type Equation ◽  
Rectangular Domain ◽  
Regular Solutions ◽  
Singular Coefficient ◽  
Small Denominator ◽  
Small Denominators ◽  
One Dimensional ◽  
Criterion Of Uniqueness ◽  
The Given

In the given article for the mixed-type equation with a singular coefficient the first boundary value problem is studied. On the basis of property of completeness of the system of own functions of one-dimensional spectral problem the criterion of uniqueness is established. The solution the problem is constructed as the sum of series of Fourier - Bessel. At justification of convergence of a row there is a problem of small denominators. In connection with that the assessment about apartness of small denominator from zero with the corresponding asymptotic which allows to prove the convergence of the series constructed in a class of regular solutions under some restrictions is given.

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