‎Let $D$ be an integral domain with quotient field $K$‎, ‎$E$ be a $K$-vector space‎, ‎$R = D propto E$ be the trivial extension of $D$ by $E$‎, ‎and $w$ be the so-called $w$-operation‎. ‎In this paper‎, ‎we show that‎ ‎$R$ is a $w$-FF ring if and only if $D$ is a $w$-FF domain; and‎ ‎in this case‎, ‎each $w$-flat $w$-ideal of $R$ is $w$-invertible.

In this paper, we establish the global $C^{2,\alpha}$ and $W^{2,p}$ regularity for Monge-Ampère equation $\mathrm{det}\ D^2u = f$ subject to boundary condition $Du(\Omega) \Omega^\ast$, where $\Omega$ $\Omega^\ast$ are bounded convex domains in Euclidean space $\mathbb{R}^n$ with $C^{1,1}$ boundaries, $f$ is a Hölder continuous function. This value problem arises naturally optimal transportatio...

u, = (0, for (x, f) E R x lR+, (1.1) W 0) = %(X) for x E IR, (1.2) where m > 1 and u0 is a given nonnegative function. It is well known [l, 11, 151 that if u. is continuous and bounded, problem (1. l), (1.2) has a unique nonnegative solution defined in a generalized sense, u(x, 1). Furthermore, if u,, is nontrivial and has compact support one can define the free boundary c(t) = sup(x E m: u(x, ...

Abstract For an integral functional defined on functions ( u , v stretchy="false">) ∈ W 1 × L {(u,v)\in W^{1,1}\times L^{1}} featuring a prototypical stro...

ion being made, in case the representation g is improper of the first kind, of the values of 0, at most denumerably infinite in number, where gi(8) is discontinuous. (19.9) rests on the fact (whose proof is trivial) that if fiim)(l) tends uniformly to the continuous f((t) when m—>, then if tm—H as m—>«>, we have lim/i(m)(í„) =/i(i) for m—>°o. The assertion is now easily proved that if (19.10...

We consider the solution of (∗) XA + BX = C for bounded operators A, B, C and X on a Hilbert space, A normal. We establish the existence of a polynomial p and a bounded operator W with the property that the unique solution X of (∗) also solves X − p(−B)Xp(A)−1 = W uniquely. A known iterative algorithm can be applied to the latter equation to solve (∗).

Let be a semiiow on a complete metric space W. Assume that is-contracting (in the sense of Kuratowski) and point dissipative. We show here that has a global attractor in W. This is an improvement over earlier results in which a further assumption, wherein the orbit of each bounded set is bounded, is made for. Furthermore, this result includes the Billotti-LaSalle Theorem, where is assumed to be...

‎In this paper we characterize Bergman spaces with‎ ‎respect to double integral of the functions $|f(z)‎ ‎-f(w)|/|z-w|$,‎ ‎$|f(z)‎ -‎f(w)|/rho(z,w)$ and $|f(z)‎ ‎-f(w)|/beta(z,w)$,‎ ‎where $rho$ and $beta$ are the pseudo-hyperbolic and hyperbolic metrics‎. ‎We prove some necessary and sufficient conditions that implies a function to be...

   The objective of the present investigation was to develop and evaluate a contraceptive vagino-adhesive propranolol hydrochloride gel. To achieve this, various mucoadhesive polymers including guar gum (1-4% w/w), sodium alginate (4-7% w/w), xanthan gum (2-5% w/w ), HPMC 4000 (3-5% w/w), Na CMC (4-7% w/w), carbomer 934 and carbomer 940 both in the range of 0.5-2.0% w/w, were dispersed in an aq...