Thanks to (1.1), the bilinear form on X×X at the left hand side is bounded, symmetric, and elliptic, and the right-hand side defines a bounded functional on X . From the Lax-Milgram lemma, we conclude that (1.2), and so the least-squares problem, has a unique solution u∈X that depends continuously on f ∈Y ′. Whenever the equation Gu= f has a solution, i.e., f ∈IG (consistency), it is the unique...

The purpose of this paper is to introduce the concept of L-fuzzybilinear operators. We obtain a decomposition theorem for L-fuzzy bilinearoperators and then prove that a L-fuzzy bilinear operator is the same as apowerset operator for the variable-basis introduced by S.E.Rodabaugh (1991).Finally we discuss the continuity of L-fuzzy bilinear operators.

in an earlier work we showed that for ordered fields f not isomorphic to the reals r, there are continuous 1-1 unctions on [0, 1]f which map some interior point to a boundary point of the image (and so are not open). here we show that over closed bounded intervals in the rationals q as well as in all non-archimedean ordered fields of countable cofinality, there are uniformly continuous 1-1 func...

In this paper we construct and consider a new group-based digital signature scheme with evolving secret key, which is built using bilinear map. This map an asymmetric pairing of Type 3, although, for the reason paper, it treated in completely abstract fashion ought to be viewed as being actually defined over $E(\FF_{q^{n}})[p]\times E(\FF_{q^{nk}})[p]\to \FF_{q^{nk}}[p]$. The crucial element ke...

For large classes of infinite-dimensional time-varying control systems, the equivalence between integral input-to-state stability (iISS) and combination global uniform asymptotic under zero input (0-GUAS) property uniformly bounded-energy input/bounded state is established a reasonable assumption continuity trajectories with respect to at input. By particularizing specific instances such as tim...

We prove that a quasi-isometric map, and more generally coarse embedding, between pinched Hadamard manifolds is within bounded distance from unique harmonic map.

A quadratic function q on an Abelian groupG is a map, with values in an Abelian group, such that the map b : (x, y) 7→ q(x + y) − q(x) − q(y) is Z-bilinear. Such a map q satisfies q(0) = 0. If, in addition, q satisfies the relation q(nx) = nq(x) for all n ∈ Z and x ∈ G, then q is homogeneous. In general, a quadratic function cannot be recovered from the associated bilinear pairing b. Homogeneou...

Let F be a field of characteristic p > 0. Let Ω(F ) be the F vector space of n-differentials of F over F . Let K = F (g) be the function field of an irreducible polynomial g in m > 1 variables over F . We derive an explicit description of the kernel of the restriction map Ω(F ) → Ω(K). As an application in the case p = 2, we determine the kernel of the restriction map when passing from the Witt...