In this article, the contraction mapping principle and Liapunov’s method are used to study qualitative properties of nonlinear Volterra equations of the form x(t) = a(t)− ∫ t 0 C(t, s)g(s, x(s)) ds, t ≥ 0. In particular, the existence of bounded solutions and solutions with various L properties are studied under suitable conditions on the functions involved with this equation.

K e y w o r d s M a x w e l l ' s single-particle probability density, Rayleigh's dissipation function, Heisenberg's uncertainty principle, Distribution function for nonzero-volume disparate particles with the spin mixtures, It6's stochastic differential equation. 0895-7177/05/$ see front mat ter @ 2005 Elsevier Ltd. All rights reserved. doi: 10.1016/j.mcm.2003.09.044 Typeset by AA4S-TEX

In this second recitation, we will address the following topics: • Review of the projection theorem. • Derivation of the projection operator on the range of a matrix, using completion of squares. • Solutions of under and over constrained linear equations. Notice that this recitation note is an alternative discussion of the least-squares problem which is more difficult than the one you can find ...

Can far-from-equilibrium material response under arbitrary loading be inferred from equilibrium data and vice versa? the effect of element transmutation on mechanical behavior predicted? Remarkably, such extrapolations are possible in principle for systems governed by stochastic differential equations, thanks to a set exact relations between probability densities trajectories derived path integ...

where K, h ∈ L(R), a(x) is a positive bounded function, and f(s) is asymptotically linear with respect to s at infinity, that is f(s)/s goes to a constant as s → +∞. Under suitable assumptions on K, a and f , we prove that the problem (P) has at least two positive solutions for |K|2 and |h|2 small by using Ekeland’s variational principle and Mountain pass theorem.

In the present study, comparative micro-morphology of fruits of above five species and estimation of active principle solasodine with HPTLC-densitometric method has been carried out. An artificial key based on macroscopic and microscopic characters were developed for identification and authentication of fruits. The trend of solasodine content was as: S. virginianum > S. nigrum > S. torvum > S. ...

The critical points of the generalized complementary energy variational principles are clariied. An open problem left by Hellinger and Reissner is solved completely. A pure complementary energy (involving the Kirchhoo type stress only) is constructed. We prove that the well-known generalized Hellinger-Reissner's energy L(u; s) is a saddle point functional if and only is the Gao-Strang gap funct...

applying two definitions of the union of if-events, p. grzegorzewski gave two generalizations of the inclusion-exclusion principle for if-events.in this paper we prove an inclusion-exclusion principle for if-states based on a method which can also be used to prove grzegorzewski's inclusion-exclusion principle for probabilities on if-events.finally, we give some applications of this principle by...