Simultaneous determination of the source terms in a linear hyperbolic problem from the final overdetermination: weak solution approach

The problem of determining the pair w:={F(x, t);f(t)} of sourceterms in the hyperbolic equation u_tt = (k(x)u_x)_x + F(x, t) andin the Neumann boundary condition k(0)u_x(0, t) = f(t) from themeasured data µ(x):=u(x, T) and/or (x):=u_t(x, t) at thefinal time t = T is formulated. It is proved that both componentsof the Fréchet gradient of the cost functionals J₁(w)= ||u(x, t;w) – µ(x)||₀² and J₂(w) = ||u_t(x, T;w)– (x)||₀² can be found via the solutions of correspondingadjoint hyperbolic problems. Lipschitz continuity of the gradientis derived. Unicity of the solution and ill-conditionednessof the inverse problem are analysed. The obtained results permitone to construct a monotone iteration process, as well as toprove the existence of a quasi-solution.     

Evolution Generated by continuous Dissipative Plus Compact Operators

Let B and C be continuous nonlinear operators in a Banach space.Assume that B satisfies a dissipativeness condition, and thatthe range of C is relatively compact. Some sufficient conditionsare given for the existence of solutions of the equation u'(t)= B(t, u(t)) + C(t, u(t)).     

Algebras with Collapsing Monomials

A semigroup S is called collapsing if there exists a positiveinteger n such that for every subset of n elements in S, atleast two distinct words of length n on these letters are equalin S. In particular, S is collapsing whenever it satisfies alaw. Let U(A) denote the group of units of a unitary associativealgebra A over a field k of characteristic zero. If A is generatedby its nilpotent elements, then the following conditions areequivalent: U(A) is collapsing; U(A) satisfies some semigrouplaw; U(A) satisfies the Engel condition; U(A) is nilpotent;A is nilpotent when considered as a Lie algebra.     

On a Topological Property of certain Calkin Algebras

Let X = 1^p, 1 p < , or X = c₀, B(X) be the algebra of allbounded linear operators on X, H(X) be the ideal of compactoperators in B(X), and C(X) = B(X)/H(X) be the Calkin algebraon X. For TB(X), let ||T||_c = dist(T, H(X)) be the essentialnorm of T that is the norm of T+H(X) in C(X). It is shown thatfor any operator TB(X) and any number 0 < t < 1, thereexists a closed infinite dimensional subspace Z Z X such that ||Tx|| t||T||_c, for all x Z. As a consequence, it is shown that every (not necessarily complete)submultiplicative norm on the Calkin algebra C(X) is equivalentto the quotient norm || ||_c on C(X).     

Dualizing Complex and the Canonical Element Conjecture II

In this paper we continue our study of the Canonical ElementConjecture (henceforth C.E.C.) via the dualizing complex. Throughoutthe work (A, m, k) will denote a noetherian complete local ringA of dimension n, m its maximal ideal and k=A/m. Since A iscomplete, we can find a complete local Gorenstein ring (R, m_R,k) (complete intersection) such that dim R=dim A and A=R/I.Let denote the canonical module of A, that is, =Hom_R (A, R),which may be identified with the annihilator of I in R, an idealof R. When A is a domain, we change notation and denote I byP; in this case P is a height 0 prime ideal of R.     

Norms on Free Topological Groups

A norm on a group G is a function N mapping G into the set ofnon-negative real numbers such that for each x and y in G, N(xy^–1) N(x)+N(y) and N(e) = 0, where e is the identity element ofG. It is shown here that if F(X) is the free topological groupon any completely regular Hausdorff space X and H is a subgroupof F(X) generated by a finite subset of X, then any norm onH can be extended to a continuous norm on F(X).     

Locally Finite Groups All of Whose Subgroups are Boundedly Finite over Their Cores

For n a positive integer, a group G is called core-n if H/H_Ghas order at most n for every subgroup H of G (where H_G is thenormal core of H, the largest normal subgroup of G containedin H). It is proved that a locally finite core-n group G hasan abelian subgroup whose index in G is bounded in terms ofn. 1991 Mathematics Subject Classification 20D15, 20D60, 20F30.     

The Uniform Closure of Rational Modules

For a smooth function g, the module R(X)+gR(X) is dense in C(X)if and only if R(Z) is dense in C(Z), where X is compact andnowhere dense and Z is the subset of X on which g vanishes.The "localness" of the module R(X)+gR(X) is also investigated.     

Idempotent Modules in the Stable Category

Let G be a finite group and k be an algebraically closed fieldof prime characteristic. Corresponding to each closed homogeneoussubvariety W of the maximal ideal spectrum of H*(G, k) we construct(usually infinite-dimensional) kG-modules E(W) and F(W) whichare idempotent in the sense that E(W) and F(W) are isomorphic(up to projective summands) to E(W) E(W) and F(W) F(W) respectively.We study the properties of these modules, and as an applicationwe use them to describe natural direct sum decompositions ofmodules in quotient categories.     

Crossed Product Orders Over Valuation Rings II: Tamely Ramified Crossed Product Algebras

Let V be a commutative valuation domain of arbitrary Krull-dimension,with quotient field F, let K be a finite Galois extension ofF with group G, and let S be the integral closure of V in K.Suppose that one has a 2-cocycle on G that takes values in thegroup of units of S. Then one can form the crossed product ofG over S, S*G, which is a V-order in the central simple F-algebraK*G. If S*G is assumed to be a Dubrovin valuation ring of K*G,then the main result of this paper is that, given a suitabledefinition of tameness for central simple algebras, K*G is tamelyramified and defectless over F if and only if K is tamely ramifiedand defectless over F. The residue structure of S*G is alsoconsidered in the paper, as well as its behaviour upon passageto Henselization. 2000 Mathematics Subject Classification 16H05,16S35.