scholarly journals Extensions of strict partial orders in N-Groups

1978 ◽  
Vol 25 (2) ◽  
pp. 241-249 ◽  
Author(s):  
K. B. Prabhakara Rao
Keyword(s):  
Partial Order ◽  
Sufficient Conditions ◽  
Partial Orders ◽  
Necessary And Sufficient Conditions ◽  
Full Extension ◽  
Partially Ordered ◽  
Strict Partial Order ◽  
Positive Elements ◽  
Necessary And Sufficient

AbstractAn attempt is made to extend the theory of extensions of partial orders in groups to strict partially ordered N-groups. Necessary and sufficient conditions, for a strict partial order of an N-group to have a strict full extension, and for a strict partial order of an N-group to be an intersection of strict full orders, are obtained when the partially ordered near-ring N and the N-group G satisfy the condition (− x) n = − xn for all elements x in G and positive elements n in N.

scholarly journals A metrizable semitopological semilattice with non-closed partial order

2020 ◽  
Vol 8 (1) ◽  
pp. 67-75
Author(s):  
Taras Banakh ◽  
Serhii Bardyla ◽  
Alex Ravsky
Keyword(s):  
Partial Order ◽  
Dense Subset ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Countable Family ◽  
Hausdorff Topology ◽  
Necessary And Sufficient ◽  
Convergent Sequences

AbstractWe construct a metrizable semitopological semilattice X whose partial order P = {(x, y) ∈ X × X : xy = x} is a non-closed dense subset of X × X. As a by-product we find necessary and sufficient conditions for the existence of a (metrizable) Hausdorff topology on a set, act, semigroup or semilattice, having a prescribed countable family of convergent sequences.

Simple algebraic necessary and sufficient conditions for Lyapunov stability of a Chen system and their applications

2017 ◽  
Vol 40 (7) ◽  
pp. 2200-2210 ◽  
Author(s):  
Guopeng Zhou ◽  
Xiaoxin Liao ◽  
Bingji Xu ◽  
Pei Yu ◽  
Guanrong Chen
Keyword(s):  
Asymptotic Stability ◽  
Exponential Stability ◽  
Lyapunov Stability ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Feedback ◽  
Chen System ◽  
Local Asymptotic Stability ◽  
Positive Elements ◽  
Necessary And Sufficient

In this paper, we study the Lyapunov stability problem of a Chen chaotic system. Because of the positive elements of the main diagonal of a linearized Chen system, compared to the coefficient of a linearized Lorenz system which are all negative, it is more difficult to deal with the stability analysis. Since it has the properties of invariance and symmetry, different Lyapunov functions in different regions are constructed to solve stability problems with geometric and algebraic methods. Then, simple algebraic necessary and sufficient conditions of global exponential stability, global asymptotic stability and global instability of equilibrium [Formula: see text] are proposed. We obtain the relevant expression of corresponding parameters for local exponential stability, local asymptotic stability and local instability of equilibria [Formula: see text]. Furthermore, the smallest conservative linear feedback controllers are used to globally exponentially stabilize equilibria.

scholarly journals UPPER BOUNDS IN THE OHTSUKI–RILEY–SAKUMA PARTIAL ORDER ON 2-BRIDGE KNOTS

2012 ◽  
Vol 21 (09) ◽  
pp. 1250084 ◽  
Author(s):  
SCOTT M. GARRABRANT ◽  
JIM HOSTE ◽  
PATRICK D. SHANAHAN
Keyword(s):  
Partial Order ◽  
Upper Bound ◽  
Continued Fractions ◽  
Sufficient Conditions ◽  
Upper Bounds ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

In this paper we use continued fractions to study a partial order on the set of 2-bridge knots derived from the work of Ohtsuki, Riley, and Sakuma. We establish necessary and sufficient conditions for any set of 2-bridge knots to have an upper bound with respect to the partial order. Moreover, given any 2-bridge knot K1 we characterize all other 2-bridge knots K2 such that {K1, K2} has an upper bound. As an application we answer a question of Suzuki, showing that there is no upper bound for the set consisting of the trefoil and figure-eight knots.

scholarly journals On a Separation Theorem for Delta-Convex Functions

2020 ◽  
Vol 34 (1) ◽  
pp. 133-141
Author(s):  
Andrzej Olbryś
Keyword(s):  
Convex Function ◽  
Partial Order ◽  
Convex Functions ◽  
Stability Problem ◽  
Sufficient Conditions ◽  
Separation Theorem ◽  
Necessary And Sufficient Conditions ◽  
Lorentz Cone ◽  
Necessary And Sufficient

AbstractIn the present paper we establish necessary and sufficient conditions under which two functions can be separated by a delta-convex function. This separation will be understood as a separation with respect to the partial order generated by the Lorentz cone. An application to a stability problem for delta-convexity is also given.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

Nonlinear boundary-value problems for degenerate differential-algebraic systems

2020 ◽  
Vol 17 (3) ◽  
pp. 313-324
Author(s):  
Sergii Chuiko ◽  
Ol'ga Nesmelova
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Algebraic System ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weakly Nonlinear ◽  
Differential Algebraic System ◽  
Linear Differential ◽  
Necessary And Sufficient

The study of the differential-algebraic boundary value problems, traditional for the Kiev school of nonlinear oscillations, founded by academicians M.M. Krylov, M.M. Bogolyubov, Yu.A. Mitropolsky and A.M. Samoilenko. It was founded in the 19th century in the works of G. Kirchhoff and K. Weierstrass and developed in the 20th century by M.M. Luzin, F.R. Gantmacher, A.M. Tikhonov, A. Rutkas, Yu.D. Shlapac, S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, O.A. Boichuk, V.P. Yacovets, C.W. Gear and others. In the works of S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and V.P. Yakovets were obtained sufficient conditions for the reducibility of the linear differential-algebraic system to the central canonical form and the structure of the general solution of the degenerate linear system was obtained. Assuming that the conditions for the reducibility of the linear differential-algebraic system to the central canonical form were satisfied, O.A.~Boichuk obtained the necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and constructed a generalized Green operator of this problem. Based on this, later O.A. Boichuk and O.O. Pokutnyi obtained the necessary and sufficient conditions for the solvability of the weakly nonlinear differential algebraic boundary value problem, the linear part of which is a Noetherian differential algebraic boundary value problem. Thus, out of the scope of the research, the cases of dependence of the desired solution on an arbitrary continuous function were left, which are typical for the linear differential-algebraic system. Our article is devoted to the study of just such a case. The article uses the original necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and the construction of the generalized Green operator of this problem, constructed by S.M. Chuiko. Based on this, necessary and sufficient conditions for the solvability of the weakly nonlinear differential-algebraic boundary value problem were obtained. A typical feature of the obtained necessary and sufficient conditions for the solvability of the linear and weakly nonlinear differential-algebraic boundary-value problem is its dependence on the means of fixing of the arbitrary continuous function. An improved classification and a convergent iterative scheme for finding approximations to the solutions of weakly nonlinear differential algebraic boundary value problems was constructed in the article.

Sign in / Sign up

Export Citation Format

Share Document