Acta Mathematica Vietnamica
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Issue 2, Volume 22, December 1997
1997 · Open access
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ON THE LANGLANDS TYPE DISCRETE GROUPS III. THE CONTINUOUS COHOMOLOGY DO NGOC DIEP PDF379-394
▸ Abstract
For a fixed percuspidal subgroup $P=MAU$ and a fixed finite spectrum $\Gamma$-module $V$, the associated spectral sequence for the fibration $$U/\Gamma\cap U \rightarrowtail {}^{\circ}P\to M/\Gamma_M$$ converges and the cohomology group $H^*(K_M\backslash {}^{\circ}P/\Gamma\cap P; V)$ is isomorphic to the direct sum of $E_2$-terms. Every cohomology class of this type can be represented by an $V$-valued automorphic form. The restriction map sends the cohomology classes at infinity of $H^*(\Gamma; V)$, represented by singular values of the associated Eisenstein series to the cohomology classes of the boundary $\partial(\overline{X}_{cusp}/\Gamma)$, compatible with its weight decomposition. All together these give us a decomposition of the cohomology of Langlands type discrete groups into the cuspidal and Eisenstein parts.
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395-405
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BOUNDARY VALUE PROBLEMS IN $\mathbf{C}$ AND $\mathbf{C}^n$ HEINRICH BEGEHR PDF407-425
▸ Abstract
Boundary value problems which are unconditionally solvable for one complex variable are in general not solvable for several complex variables. This phenomenon will be explained in the case of the Schwarz problem for polydiscs. Besides analytic functions, inhomogeneous Cauchy-Riemann systems are investigated. These systems in several complex variables are
overdetermined. Another overdetermined system in two complex variables is considered by introducing a proper hypercomplex variable and solved under Riemann-Hilbert boundary conditions on some submanifold of the boundary under consideration.
The theory of bianalytic functions is used to reduce the stress boundary value problem in orthotropic elasticity to boundary value problems for analytic functions in plane domains. This paper is the improved version of [17] in which formula (16) was incorrect. As for the Poisson equation $(n = 1)$ the solution should contain fundamental solutions (Green functions) in the kernels of the integrals. -
SOME RESULTS ON REDUCTION PRINCIPLES, BIFURCATION AND HOPF BIFURCATION OF EQUATIONS CONCERNING LIPSCHITZ CONTINUOUS MAPPINGS KLAUSS SCHNEIDER, NGUYEN XUAN TAN PDF427-465
▸ Abstract
Some reduction principles of equations independing or depending on a parameter and concerning Lipschitz continuous mappings
are introduced and then are applied to consider the existence of nontrivial solutions and nontrivial periodic solutions with a small norm and the existence of bifurcation and Hopf bifurcation points of equations concerning Lipschitz continuous mappings in Banach spaces, investigating the definition and nonvanishing of the topological degree of mappings or the existence of regular nonzero solutions of algebraic equations in a finite-dimensional space. Some well-known results of other authors are generalized. -
GLOBAL SOLUTION OF A TWO PHASE FREE BOUNDARY PROBLEM DANG DINH ANG, NGUYEN DINH TRI, DANG DUC TRONG PDF467-491
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493-500
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INVARIANCE OF THE GLOBAL MONODROMIES IN FAMILIES OF POLYNOMIALS OF TWO COMPLEX VARIABLES HA HUY VUI, PHAM TIEN SON PDF515-526
▸ Abstract
We consider global monodromy fibrations defined by a family of polynomials of two complex variables. The main result gives certain
sufficient conditions for the conjugacy of global monodromies. -
ON AN EXPANSION OF THE SPECIAL LAGRANGIAN FORM DAO TRONG THI, NGUYEN DUY BINH PDF527-540
▸ Abstract
In this paper we present a class of $k$-forms on $C^n$ induced by $k$-forms on $R^n$ and investigate their comass and set of maximal directions. In particular, the set of maximal directions of the real part of powers of the complex symplectic form on $H^n$ is described completely.
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DUALITY FOR RADIANT AND SHADY PROGRAMS J.-P. PENOT PDF541-566
▸ Abstract
We consider the two classes of closed convex sets which are stable under dilatations and shrinkings respectively (homotheties of rates greater than one and less than one respectively). We define conjugacies for the classes of functions whose sublevel sets belong to these classes. For such functions, the conjugate can be defined on the dual space and an extra parameter is not needed. We apply these notions to the maximization of a convex function on a convex set and to the minimization of a convex function on the set of points outside a convex subset. We introduce several dual problems related to each of these problems and we give conditions
ensuring there is no duality gap. -
ON THE EQUIVALENCE, VIA RELAXATION{PENALIZATION, BETWEEN VECTOR GENERALIZED SYSTEMS C. ANTONI, F. GIANNESSI PDF567-588
▸ Abstract
A vector generalized system is considered, and a condition is given under which a relaxation of the domain and a penalization of the vector function do not change the set of solutions of the system. Applications are made to Vector Optimization and to Vector Variational Inequalities in a discrete space.
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APPROXIMATE CONTROLLABILITY WITH POSITIVE CONTROLS NGUYEN KHOA SON PDF589-620
▸ Abstract
In this paper, controllability of the linear discrete-time systems $(A,B,\Omega):x_{k+1}=Ax_k+Bu_k,\ x_k\in X, u_k\in\Omega$, is studied, where $X$ is a Banach space and the control set is assumed to be a cone in a Banach space $U$. Some criteria for approximate controllability are given. The case where the operator $A$ is compact is examined in detail by using the spectral decomposition of the state space $X$. As a result, a criterion for approximate controllability of $(A,B,\Omega)$ is obtained without imposing a restrictive condition that the system with no control constraints $(A,B,U)$ is exactly controllable. The obtained results are then applied to consider the problem of controllability for linear functional differential equations with positive controls. Some necessary and sufficient conditions of approximate controllability to the state space $\mathbf{R}^n\times L_p$ are presented and some illustrating examples are given.