A mean-value theorem and its applications

For a function f defined in an interval I, satisfying the conditions ensuring the existence and uniqueness of the Lagrange mean L[f], we prove that there exists a unique two variable mean M[f] such that

     

To the mean-value theorem

New versions of the mean-value theorem for real and complex-valued functions are presented.     

On the mean-value theorem for analytic functions

A version of the mean-value theorem (formulas of finite increments) for analytic functions is proved. Volyn University, Lutsk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 8, pp. 1143–1147, August, 1997.     

The Bolzano mean-value theorem and partial differential equations

We study the existence of solutions to abstract equations of the form $0=Au+F(u)$, $u\in K?E$, where A is an abstract differential operator acting in a Banach space E, K is a closed convex set of constraints being invariant with respect to resolvents of A and a perturbation F satisfies a certain tangency condition. Such problems are closely related to the so-called Poincaré–Miranda theorem, being the multi-dimensional counterpart of the celebrated Bolzano intermediate value theorem. In fact our main results should be regarded as infinite-dimensional variants of Bolzano and Miranda–Poincaré theorems. Along with single-valued problems we deal with set-valued ones, yielding the existence of the so-called constrained equilibria of set-valued maps. The abstract results are applied to show existence of (strong) steady state solutions to some weakly coupled systems of drift reaction–diffusion equations or differential inclusions of this type. In particular we get the existence of strong solutions to the Dirichlet, Neumann and periodic boundary problems for elliptic partial differential inclusions under the presence of state constraints of different type. Certain aspects of the Bernstein theory for bvp for second order ODE are studied, too. No assumptions concerning structural coupling (monotonicity, cooperativity) are undertaken.     

A generalized Stone-Weierstrass theorem

Archiv der Mathematik -     

A generalized Kakutani-Kodaira theorem

In this note, we present a generalized Kakutani-Kodaira theorem for all locally compact groups.

     

A generalized four-vertex theorem

The classical four-vertex thereom of differential geometry is interpreted in maybe its more natural conformai setting. Under weaker assumptions a generalized stronger result is obtained.     

A generalized Milne-Thomson theorem

Using analytic continuation theory, a new simple proof of a standard generalized circle theorem is given. Additionally, new cases involving complex coefficients in the boundary condition and allowing for an arbitrary singularity of a given complex potential at the interface are considered.     

A generalized sandwich theorem

It is proved for an arbitrary commutative ring A with identity and any integer n3 that if H is a subgroup of GL_n(A) normalized by E _n(A,q), then there is an ideal of A such that E _n(A,) H GL _n (A, (:q⁴⁰).Furthermore, is uniquely determined up to a certain equivalence relation on the set of ideals of A. The result extends a theorem of Bak, by removing a stability condition he uses on A.     

An elementary proof of a converse mean-value theorem

We present a new converse mean value theorem, with a rather elementary proof.     

A generalized principal ideal theorem

The main theorem of this article is an extension of the generalized principal ideal theorem for ideals in Noetherian rings. Instead of requiring the rings to be Noetherian, some natural requirements are imposed on the chains of prime ideals under consideration. The standard (Noetherian) version of the generalized principal ideal theorem is deduced as a corollary and two other applications are presented.     

The mean-value theorem for elliptic operators on stratified sets

In this paper, an analog of the mean-value theorem for harmonic functions is proved for an elliptic operator on the stratified set of “stratified” spheres whose radius is sufficiently small. In contrast to the classical case, the statement of the theorem has the form of a special differential relationship between the mean values over different parts of the sphere. The result is used to prove the strong maximum principle.     

A generalized Macaulay theorem and generalized face rings

We prove that the f-vector of members in a certain class of meet semi-lattices satisfies Macaulay inequalities 0?k^∂(fk)?fk−1 for all k?0. We construct a large family of meet semi-lattices belonging to this class, which includes all posets of multicomplexes, as well as meet semi-lattices with the “diamond property,” discussed by Wegner [G. Wegner, Kruskal-Katona's theorem in generalized complexes, in: Finite and Infinite Sets, vol. 2, in: Colloq. Math. Soc. János Bolyai, vol. 37, North-Holland, Amsterdam, 1984, pp. 821-828], as special cases. Specializing the proof to the later family, one obtains the Kruskal-Katona inequalities and their proof as in [G. Wegner, Kruskal-Katona's theorem in generalized complexes, in: Finite and Infinite Sets, vol. 2, in: Colloq. Math. Soc. János Bolyai, vol. 37, North-Holland, Amsterdam, 1984, pp. 821-828].For geometric meet semi-lattices we construct an analogue of the exterior face ring, generalizing the classic construction for simplicial complexes. For a more general class, which also includes multicomplexes, we construct an analogue of the Stanley-Reisner ring. These two constructions provide algebraic counterparts (and thus also algebraic proofs) of Kruskal-Katona's and Macaulay's inequalities for these classes, respectively.     

Analog of the mean-value theorem for polynomials of special form

We prove a mean-value theorem for polynomials of a special form. We investigate the case of a sum over the vertices of a regular polygon and obtain a criterion for an equation of a special form to be satisfied.