scholarly journals Two-Vertex Generators of Jacobians of Graphs

10.37236/7302 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
David Brandfonbrener ◽  
Pat Devlin ◽  
Netanel Friedenberg ◽  
Yuxuan Ke ◽  
Steffen Marcus ◽  
...  
Keyword(s):  
Empirical Evidence ◽  
Random Graphs ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Difference ◽  
Necessary And Sufficient

We give necessary and sufficient conditions under which the Jacobian of a graph is generated by a divisor that is the difference of two vertices. This answers a question posed by Becker and Glass and allows us to prove various other propositions about the order of divisors that are the difference of two vertices. We conclude with some conjectures about these divisors on random graphs and support them with empirical evidence.

A Direct Rotatability Criterion for Spherical Four-Bar Linkages

1995 ◽  
Vol 117 (4) ◽  
pp. 597-600 ◽  
Author(s):  
K. C. Gupta ◽  
R. Ma
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Absolute Value ◽  
The Difference ◽  
Necessary And Sufficient ◽  
Four Bar Linkage

The necessary and sufficient conditions for the full input rotatability in a spherical four-bar linkage are proved. The direct criterion is: for all twist angles α in the range [0, π], the excess (deficit) of the sum of the frame and input twist angles over (from) π should, in absolute value, be greater than that for the coupler and follower twist angles; the difference between the follower and input twist angles, in absolute value, should be greater than that for the coupler and follower twist angles. Application of the direct criterion to full rotatability of other links are discussed and some variations in the form of the criterion are developed.

A Direct Rotatability Criterion for Spherical Four-Bar Linkages

1994 ◽  
Author(s):  
R. Ma ◽  
K. C. Gupta
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Absolute Value ◽  
The Difference ◽  
Necessary And Sufficient ◽  
Four Bar Linkage

Abstract The necessary and sufficient conditions for the full input rotatability in a spherical four bar linkage are proved. The direct criterion is: for all twist angles α in the range [0, π], the excess (deficit) of the sum of the frame and input twist angles over (from) π should, in absolute value, be greater than that for the coupler and follower twist angles; the difference between the follower and input twist angles, in absolute value, should be greater than that for the coupler and follower twist angles. Application of the direct criterion to full rotatability of other links are discussed and some variations in the form of the criterion are developed.

scholarly journals Convex sequences of higher order

Filomat ◽  
2018 ◽  
Vol 32 (13) ◽  
pp. 4655-4663
Author(s):  
Daniel Sofonea ◽  
Ioan Ţincu ◽  
Ana Acu
Keyword(s):  
Real Number ◽  
Sufficient Conditions ◽  
Higher Order ◽  
Necessary And Sufficient Conditions ◽  
Difference Operators ◽  
Real Sequence ◽  
Convex Sequence ◽  
Different Types ◽  
The Difference ◽  
Necessary And Sufficient

In this paper we study the class of convex sequences of higher order defined using the difference operators and investigate their properties. The notion of the convex sequence of order r ? N will be extended for r a real number. Some necessary and sufficient conditions such that a real sequence belongs to the class of convex sequences of higher order r ? R are introduced. Using different types of means we will investigate the convexity of higher order for real sequences.

scholarly journals An Analogue for Strong Summability of Abel's Summability Method

1953 ◽  
Vol 9 (1) ◽  
pp. 28-34
Author(s):  
C. F. Harington ◽  
J. M. Hyslop
Keyword(s):  
Sufficient Conditions ◽  
Binomial Coefficient ◽  
Summability Method ◽  
Strong Summability ◽  
Necessary And Sufficient Conditions ◽  
The Difference ◽  
Image Position ◽  
Necessary And Sufficient

Given a series Σan, we define , by the relationwhere is the binomial coefficient . Let . If , the series Σan is said to be summable (C; k) to the sum s. If k > 0, p ≥ 1 and if, as n → ∞,we say that the series Σan is summable [C; k, p] to the sum s, or that the series is strongly summable (C; k) with index p to the sum s. If denotes the difference , it is known that necessary and sufficient conditions for summability [C; k, p], k > 0, p ≥ 1, to the sum s, are that Σan be summable (C; k) to the sum s and that

A Liapunov functional for a scalar differential difference equation

1978 ◽  
Vol 79 (3-4) ◽  
pp. 307-316 ◽  
Author(s):  
E. F. Infante ◽  
J. A. Walker
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Quadratic Functional ◽  
Liapunov Functional ◽  
Differential Difference Equation ◽  
The Difference ◽  
Rates Of Decay ◽  
Necessary And Sufficient

SynopsisGiven the scalar, retarded differential difference equation x'(t)=ax(t) +bx(t−τ), a quadratic functional in explicit form is obtained that yields necessary and sufficient conditions for the asymptotic stability of this equation. This functional a Liapunov functional, is obtained through the study of the Liapunov functions associated with a difference equation approximation of the difference differential equation. The functional then obtained not only yields necessary and sufficient conditions for asymptotic stability, but provides estimates for rates of decay of the solutions as well as conditions, for asymptotic stability independent of the magnitude of the delay τ.

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.

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