Uniform Accuracy of the Quasicontinuum Method [PowerPoint Submission]

2006 ◽  
Vol 978 ◽  
Author(s):  
Jerry Z. Yang
Keyword(s):  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Order Accuracy ◽  
Quasicontinuum Method ◽  
First Order ◽  
Local Atomic Environment ◽  
Necessary And Sufficient ◽  
Nonlocal Approach ◽  
Special Case ◽  
Uniform Accuracy

AbstractThe accuracy of the quasicontinuum method is studied by reformulating the summation rules in terms of reconstruction schemes for the local atomic environment of the representative atoms. The necessary and sufficient condition for uniform first order accuracy and consequently the elimination of the “ghost force” is formulated in terms of the reconstruction schemes.The quasi-nonlocal approach is discussed as a special case of this condition.Examples of reconstruction schemes that satisfy this condition are presented.Transition between atom-based and element-based summation rules are studied.

A Classification and Closure Properties of Languages for Describing Concurrent System Behaviours

1981 ◽  
Vol 4 (3) ◽  
pp. 531-549 ◽  
Author(s):  
Miklós Szijártó
Keyword(s):  
Formal Languages ◽  
Parallel Program ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Concurrent System ◽  
Closure Properties ◽  
Sequential Program ◽  
Necessary And Sufficient ◽  
Special Case ◽  
Independence Relations

The correspondence between sequential program schemes and formal languages is well known (Blikle and Mazurkiewicz (1972), Engelfriet (1974)). The situation is more complicated in the case of parallel program schemes, and trace languages (Mazurkiewicz (1977)) have been introduced to describe them. We introduce the concept of the closure of a language on a so called independence relation on the alphabet of the language, and formulate several theorems about them and the trace languages. We investigate the closedness properties of Chomsky classes under closure on independence relations, and as a special case we derive a new necessary and sufficient condition for the regularity of the commutative closure of a language.

scholarly journals Existence of nonoscillatory solutions of first order nonlinear neutral equations

1990 ◽  
Vol 32 (2) ◽  
pp. 180-192 ◽  
Author(s):  
Lu Wudu
Keyword(s):  
Nonoscillatory Solution ◽  
Sufficient Condition ◽  
Neutral Equation ◽  
Necessary And Sufficient Condition ◽  
Nonoscillatory Solutions ◽  
Neutral Equations ◽  
First Order ◽  
Image Position ◽  
Necessary And Sufficient

AbstractConsider the nonlinear neutral equationwhere pi(t), hi(t), gj(t), Q(t) Є C[t0, ∞), limt→∞hi(t) = ∞, limt→∞gj(t) = ∞ i Є Im = {1, 2, …, m}, j Є In = {1, 2, …, n}. We obtain a necessary and sufficient condition (2) for this equation to have a nonoscillatory solution x(t) with limt→∞ inf|x(t)| > 0 (Theorems 5 and 6) or to have a bounded nonoscillatory solution x(t) with limt→∞ inf|x(t)| > 0 (Theorem 7).

scholarly journals On B- Covariant Derivative of First Order for Some Tensors in different Spaces

2021 ◽  
Vol 2 (2) ◽  
pp. 30-37
Author(s):  
Alaa A. Abdallah ◽  
A. A. Navlekar ◽  
Kirtiwant P. Ghadle
Keyword(s):  
Curvature Tensor ◽  
Covariant Derivative ◽  
Torsion Tensor ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
First Order ◽  
Recurrence Property ◽  
Necessary And Sufficient ◽  
The Relationship

In this paper, we study the relationship between Cartan's second curvature tensor $P_{jkh}^{i}$ and $(h) hv-$torsion tensor $C_{jk}^{i}$ in sense of Berwald. Moreover, we discuss the necessary and sufficient condition for some tensors which satisfy a recurrence property in $BC$-$RF_{n}$, $P2$-Like-$BC$-$RF_{n}$, $P^{\ast }$-$BC$-$RF_{n}$ and $P$-reducible-$BC-RF_{n}$.

scholarly journals On composition of formal power series

2002 ◽  
Vol 30 (12) ◽  
pp. 761-770 ◽  
Author(s):  
Xiao-Xiong Gan ◽  
Nathaniel Knox
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Open Problem ◽  
Constant Term ◽  
Radius Of Convergence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Formal Power ◽  
Special Case

Given a formal power seriesg(x)=b0+b1x+b2x2+⋯and a nonunitf(x)=a1x+a2x2+⋯, it is well known that the composition ofgwithf,g(f(x)), is a formal power series. If the formal power seriesfabove is not a nonunit, that is, the constant term offis not zero, the existence of the compositiong(f(x))has been an open problem for many years. The recent development investigated the radius of convergence of a composed formal power series likefabove and obtained some very good results. This note gives a necessary and sufficient condition for the existence of the composition of some formal power series. By means of the theorems established in this note, the existence of the composition of a nonunit formal power series is a special case.

On the general bilinear time series model

1988 ◽  
Vol 25 (3) ◽  
pp. 553-564 ◽  
Author(s):  
Jian Liu ◽  
Peter J. Brockwell
Keyword(s):  
Time Series ◽  
Stationary Solution ◽  
Time Series Model ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Special Cases ◽  
Non Linear ◽  
Necessary And Sufficient ◽  
Special Case

A sufficient condition is derived for the existence of a strictly stationary solution of the general bilinear time series equations. The condition is shown to reduce to the conditions of Pham and Tran (1981) and Bhaskara Rao et al. (1983) in the special cases which they consider. Under the condition specified, a solution is constructed which is shown to be causal, stationary and ergodic. It is moreover the unique causal solution and the unique stationary solution of the defining equations. In the special case when the defining equations contain no non-linear terms, our condition reduces to the well-known necessary and sufficient condition for existence of a causal stationary solution.

scholarly journals Oscillation for nonlinear delay difference equations

2001 ◽  
Vol 32 (4) ◽  
pp. 275-280 ◽  
Author(s):  
X. H. Tang
Keyword(s):  
Oscillatory Behavior ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Delay Difference Equation ◽  
First Order ◽  
Condition Of Oscillation ◽  
Necessary And Sufficient ◽  
Delay Difference Equations ◽  
Delay Difference ◽  
Nonlinear Delay

The oscillatory behavior of the first order nonlinear delay difference equation of the form $$ x_{n+1} - x_n + p_n x_{n-k}^{\alpha} = 0, ~~~ n = 0, 1, 2, \ldots ~~~~~~~ \eqno{(*)} $$ is investigated. A necessary and sufficient condition of oscillation for sublinear equation (*) ($ 0 < \alpha < 1 $) and an almost sharp sufficient condition of oscillation for superlinear equation (*) ($ \alpha > 1 $) are obtained.

scholarly journals Existence of minimizers for one-dimensional vectorial non-semicontinuous functionals with second order lagrangian

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Sandro Zagatti
Keyword(s):  
Existence Result ◽  
Variational Problems ◽  
Convex Envelope ◽  
Minimum Problem ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
One Dimensional ◽  
First Order ◽  
Existence Of Minimizers ◽  
Necessary And Sufficient

<p style='text-indent:20px;'>We study the minimum problem for functionals of the form</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} \mathcal{F}(u) = \int_{I} f(x, u(x), u^ \prime(x), u^ {\prime\prime}(x))\,dx, \end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where the integrand <inline-formula><tex-math id="M1">\begin{document}$ f:I\times \mathbb{R}^m\times \mathbb{R}^m\times \mathbb{R}^m \to \mathbb{R} $\end{document}</tex-math></inline-formula> is not convex in the last variable. We provide an existence result assuming that the lower convex envelope <inline-formula><tex-math id="M2">\begin{document}$ \overline{f} = \overline{f}(x,p,q,\xi) $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M3">\begin{document}$ f $\end{document}</tex-math></inline-formula> with respect to <inline-formula><tex-math id="M4">\begin{document}$ \xi $\end{document}</tex-math></inline-formula> is regular and enjoys a special dependence with respect to the i-th single components <inline-formula><tex-math id="M5">\begin{document}$ p_i, q_i, \xi_i $\end{document}</tex-math></inline-formula> of the vector variables <inline-formula><tex-math id="M6">\begin{document}$ p,q,\xi $\end{document}</tex-math></inline-formula>. More precisely, we assume that it is monotone in <inline-formula><tex-math id="M7">\begin{document}$ p_i $\end{document}</tex-math></inline-formula> and that it satisfies suitable affinity properties with respect to <inline-formula><tex-math id="M8">\begin{document}$ \xi_i $\end{document}</tex-math></inline-formula> on the set <inline-formula><tex-math id="M9">\begin{document}$ \{f&gt; \overline{f}\} $\end{document}</tex-math></inline-formula> and with respect to <inline-formula><tex-math id="M10">\begin{document}$ q_i $\end{document}</tex-math></inline-formula> on the whole domain. We adopt refined versions of the integro-extremality method, extending analogous results already obtained for functionals with first order lagrangians. In addition we show that our hypotheses are nearly optimal, providing in such a way an almost necessary and sufficient condition for the solvability of this class of variational problems.</p>

scholarly journals A necessary and sufficient condition for global existence for a quasilinear reaction-diffusion system

2005 ◽  
Vol 2005 (11) ◽  
pp. 1809-1818 ◽  
Author(s):  
Alan V. Lair
Keyword(s):  
Global Solution ◽  
Smooth Boundary ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Special Case

We show that the reaction-diffusion systemut=Δφ(u)+f(v),vt=Δψ(v)+g(u), with homogeneous Neumann boundary conditions, has a positive global solution onΩ×[0,∞)if and only if∫∞ds/f(F−1(G(s)))=∞(or, equivalently,∫∞ds/g(G−1(F(s)))=∞), whereF(s)=∫0sf(r)drandG(s)=∫0sg(r)dr. The domainΩ⊆ℝN(N≥1)is bounded with smooth boundary. The functionsφ,ψ,f, andgare nondecreasing, nonnegativeC([0,∞))functions satisfyingφ(s)ψ(s)f(s)g(s)>0fors>0andφ(0)=ψ(0)=0. Applied to the special casef(s)=spandg(s)=sq,p>0,q>0, our result proves that the system has a global solution if and only ifpq≤1.

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