Differential equations and Lie group representations

We discuss the role of differential equations in Lie group representation theory. We use Kashiwara’s pentagon as a reference frame for the real representation theory and then report on some work arising from its p-adic analogue by Emerton, Kisin, Patel, Huyghe, Schmidt, Strauch using Berthelot’s theory of arithmetic D-modules and Schneider–Stuhler theory of sheaves on buildings.     

Lipschitzian semigroups and abstract functional differential equations

The paper is devoted to studying an abstract functional differential equation by a nonlinear semigroup approach. We first prove in details the equivalence of the well posedness of an abstract functional differential equation and an associated abstract Cauchy problem in the sense of strong solutions. Secondly, a sufficient condition is derived for well posedness of the abstract functional differential equation. Thirdly, we present principles of linearized stability for the abstract functional differential equation. Finally, the results obtained are applied to a reaction-diffusion equation with delays.     

Some well-posed functional equations which generate semigroups

Consider the abstract linear functional equation (FE) (Dx)(t) = f(t) (t ? 0), x(t) = ?(t) (t ? 0) in a Banach space B. A theorem is proven which contains the following result as a special case. Let Y(R; B; η) be a L^p-space or C₀-space on R = (?t8, ∞), with a suitable weight function η, and with values in B. Let D be a closed (unbounded) causal linear operator in Y(R; B; η), which commutes with translations. Suppose that D + λI has a continuous causal inverse for some complex λ, and that D restricted to those functions in Y(R;B;η) which vanish on R^? = (?∞, 0] has a continuous causal inverse. Then (FE) generates a strongly continuous semigroup of translation type on a Banach space, which is essentially the cross product of the restriction of the domain of D to R^? and Y(R⁺; B; η). Examples with B = Cⁿ on how the theory applies to a neutral functional differential equation, a difference equation, a Volterra integrodifferential equation (with nonintegrable kernel but integrable resolvent), and a fractional order functional differential equation are given. Also, an abstract neutral functional differential equation in a Hilbert space is studied and applications to an abstract Volterra integrodifferential equation in a Banach space are indicated.     

Levi-Civita functional equations and the status of spectral synthesis on semigroups

We show, contrary to some published statements, that spectral synthesis does not generally hold for commutative semigroups that are not groups. On the positive side we prove that it holds if the semigroup is a monoid with no prime ideal. For semigroups with a prime ideal, the picture is not so clear. On the negative side we provide a variety of examples illustrating the failure of spectral synthesis for many semigroups with prime ideals, but we also give examples of semigroups with prime ideals on which spectral synthesis holds.

     

Some recent applications of functional equations to geometry

The results of applications of solutions of functional equations or of methods used in the theory of functional equations to the following subjects are discussed in this paper. Determination of all Cremona transformations which reduce linear transformations with triangular matrices to translations. One-parameter subsemigroups of affine transformations and their homomorphisms. Extensions of homomorphisms from sub-semigroups to groups generated by them. Determination of all collineations on subsets of general projective planes and their extensions to the entire plane.     

On functional equations related to a dihedral group

We propose a method for regularization of two three-element functional equations generated by a special case of dihedral group of rotations. These results are applied to the moment problem for entire functions of exponential type and for evaluation of biorthogonally dual systems of analytic functions.     

A perturbation theorem for positive contraction semigroups on L 1-spaces with applications to transport equations and Kolmogorov's differential equations

In this paper, a criterion is given for assuring that a linear positive contraction C ₀-semigroup defined on an L ¹-space is generated by the closure of A+B, A and B being suitable unbounded linear operators. Furthermore, this criterion is applied to the transport equation, Kolmogorov's differential equations, and a transport equation modelling cell growth.     

Functional equations on semigroups

Summary. We determine the general solution g:S? F g:S\to F of the d'Alembert equation¶¶g(x+y)+g(x+sy)=2g(x)g(y)       (x,y ? S) g(x+y)+g(x+\sigma y)=2g(x)g(y)\qquad (x,y\in S) ,¶the general solution g:S? G g:S\to G of the Jensen equation¶¶g(x+y)+g(x+sy)=2g(x)       (x,y ? S) g(x+y)+g(x+\sigma y)=2g(x)\qquad (x,y\in S) ,¶and the general solution g:S? H g:S\to H of the quadratic equation¶¶g(x+y)+g(x+sy)=2g(x)+2g(y)       (x,y ? S) g(x+y)+g(x+\sigma y)=2g(x)+2g(y)\qquad (x,y\in S) ,¶ where S is a commutative semigroup, F is a quadratically closed commutative field of characteristic different from 2, G is a 2-cancellative abelian group, H is an abelian group uniquely divisible by 2, and s \sigma is an endomorphism of S with s(s(x)) = x \sigma(\sigma(x)) = x .     

Irreducible representations of general group algebras and their applications

In this paper we obtain some properties of irreducible representations of general group algebras, then as applications, we extend Gluck-Wolf Theorem for ordinary characters to the case of Brauer characters; we investigate the relations between the degree of an indecomposable module and that of its source.     

A linearity theorem for group actions on spheres with applications to homotopy representations

LetG be a finite group andX an equivariantZ/|G|-homology sphere. By Smith-theory the fixed point setX ^H for ap-subgroupH is aZ/p-homology sphere of dimensiond(H).     

Automorphisms on a Lie group contracting modulo a compact subgroup and applications to semistable convolution semigroups

Let ( _t ) _t0 be a -semistable convolution semigroup of probability measures on a Lie groupG whose idempotent ₀ is the Haar measure on some compact subgroupK. Then all the measures ₁ are supported by theK-contraction groupC _K() of the topological automorphism ofG. We prove here the structure theoremC _K()=C()K, whereC() is the contraction group of . Then it turns out that it is sufficient to study semistable convolution semigroups on simply connected nilpotent Lie groups that have Lie algebras with a positive graduation.     

Projectors on the generalized eigenspaces for functional differential equations using integrated semigroups

The aim of this article is to derive explicit formulas for the projectors on the generalized eigenspaces associated to some eigenvalues for linear functional differential equations (FDE) by using integrated semigroup theory. The idea is to formulate the FDE as a non-densely defined Cauchy problem and obtain an explicit formula for the integrated solutions of the non-densely defined Cauchy problem, from which we then derive explicit formulas for the projectors on the generalized eigenspaces associated to some eigenvalues. The results are useful in studying bifurcations in some semi-linear problems.     

R-bounded approximating sequences and applications to semigroups

It is shown that on certain Banach spaces, including C[0,1] and L₁[0,1], there is no strongly continuous semigroup (T_t)_0<t<1 consisting of weakly compact operators such that (T_t)_0<t<1 is an R-bounded family. More general results concerning approximating sequences are included and some variants of R-boundedness are also discussed.