scholarly journals On the Directed Oberwolfach Problem with Equal Cycle Lengths

10.37236/2982 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Andrea Burgess ◽  
Mateja Šajna
Keyword(s):  
Complete Solution ◽  
Sufficient Conditions ◽  
Partial Solution ◽  
Necessary And Sufficient Conditions ◽  
Cycle Lengths ◽  
Necessary And Sufficient ◽  
Directed Cycles ◽  
Oberwolfach Problem

We examine the necessary and sufficient conditions for a complete symmetric digraph $K_n^\ast$ to admit a resolvable decomposition into directed cycles of length $m$. We give a complete solution for even $m$, and a partial solution for odd $m$.

A NEW KIND OF FUZZY RELATIONAL EQUATIONS

2010 ◽  
Vol 18 (03) ◽  
pp. 333-342 ◽  
Author(s):  
FENG QIN ◽  
PING FANG
Keyword(s):  
Unique Solvability ◽  
Complete Solution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Solution Set ◽  
Fuzzy Relational Equations ◽  
Necessary And Sufficient

In this paper, a new kind of fuzzy relational equations (FREs for short) A ∘R*x = b is first introduced, and then the problem of solving solution to the FREs is discussed, where A is an m × n matrix, x and b are an n and an m dimensional column vectors, respectively. More specifically, their solvability and unique solvability are investigated, the corresponding necessary and sufficient conditions are presented, the complete solution set is obtained. It is worth noting the method to construct the complete solution set.

Summability methods which include the Riesz typical means. I

1971 ◽  
Vol 69 (1) ◽  
pp. 99-106 ◽  
Author(s):  
Dennis C. Russell
Keyword(s):  
Complete Solution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Alternative Form ◽  
Cesàro Summability ◽  
Abel Summability ◽  
Summability Methods ◽  
Cesaro Summability ◽  
Riesz Summability ◽  
Necessary And Sufficient

A number of special results exist for summability methods B which, include Riesz summability (R,λ,k)—for example, when B is generalized Abel summability (A,λ,ρ) [Kuttner(5)], or Riemann summability (,λ,μ) [Russell(14)], or Riemann-Cesàro summability (,λ,p,α) [Rangachari(12)], or generalized Cesàro summability (C,λ,k) [Meir (9); Borwein and Russell (l)]. The question of necessary and sufficient conditions to be satisfied by an arbitrary method B in order that B ⊇ (R,λ,k) has received an answer only for limited values of λ and k—for example, by Lorentz [(6), Theorem 10] for k = 1; the restrictions on λ in this case were removed by Maddox [(8), Theorem 1]. Thus (apart from the well-known case k = 0) the case k = 1 is the only one for which a complete solution exists, though application of a theorem of Russell [(13), Theorem 1A] yields one form of a result for 0 < k ≤ 1. Maddox's results, however, suggest an alternative form capable of generalization to all k ≥ 0, and in this paper we obtain a complete solution for 0 < k ≤ 1 in that form, without restriction on λ. We first recall the following definitions.

On the Existence of Mayer’s Potential

1997 ◽  
Vol 64 (3) ◽  
pp. 606-612 ◽  
Author(s):  
V. M. Cˇovic´ ◽  
M. M. Lukacˇevic´
Keyword(s):  
Dynamical Systems ◽  
Complete Solution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Nonconservative Systems ◽  
Necessary And Sufficient ◽  
The Given ◽  
Special Case

A complete solution of the well-known Mayer’s problem, which is concerned with the possibility of extending Hamilton’s principle expressed in the form valid for conservative dynamical systems to one special case of nonconservative systems (Appell, 1911), is obtained. Namely, the necessary and sufficient conditions which have to be satisfied by the coefficients of the given nonconservative generalized forces so that the Mayer’s potential (and, as a consequence, the descriptive function of the system) can be constructed, are established. This result is illustrated by an example.

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&gt; 0 for eachj&gt; 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.

On the domain of Riesz mean in the space Ls

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 925-940 ◽  
Author(s):  
Medine Yeşilkayagil ◽  
Feyzi Başar
Keyword(s):  
Banach Space ◽  
Sequence Space ◽  
Double Sequence ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Double Sequences ◽  
Riesz Mean ◽  
Double Sequence Space ◽  
Necessary And Sufficient ◽  
Barrelled Space

Let 0 < s < ?. In this study, we introduce the double sequence space Rqt(Ls) as the domain of four dimensional Riesz mean Rqt in the space Ls of absolutely s-summable double sequences. Furthermore, we show that Rqt(Ls) is a Banach space and a barrelled space for 1 ? s < 1 and is not a barrelled space for 0 < s < 1. We determine the ?- and ?(?)-duals of the space Ls for 0 < s ? 1 and ?(bp)-dual of the space Rqt(Ls) for 1 < s < 1, where ? ? {p, bp, r}. Finally, we characterize the classes (Ls:Mu), (Ls:Cbp), (Rqt(Ls) : Mu) and (Rqt(Ls):Cbp) of four dimensional matrices in the cases both 0 < s < 1 and 1 ? s < 1 together with corollaries some of them give the necessary and sufficient conditions on a four dimensional matrix in order to transform a Riesz double sequence space into another Riesz double sequence space.

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