Global Lie–Tresse theorem

We prove a global algebraic version of the Lie–Tresse theorem which states that the algebra of differential invariants of an algebraic pseudogroup action on a differential equation is generated by a finite number of rational-polynomial differential invariants and invariant derivations.     

Graded versions of Goldie’s theorem

We prove graded variants of Goldie’s theorem of existence, structure, and coincidence of right classical quotient ring and right maximal quotient ring of a semiprime (prime) right Goldie’s ring (Theorems 10, 11, and 13). The main difficulty consisting in the problem of existence of a homogeneous regular element in each gr-essential right ideal is solved by posing some additional requirements onto the group grading the ring or onto the homogeneous components of the ring.     

Distributional versions of littlewood’s Tauberian theorem

We provide several general versions of Littlewood’s Tauberian theorem. These versions are applicable to Laplace transforms of Schwartz distributions. We employ two types of Tauberian hypotheses; the first kind involves distributional boundedness, while the second type imposes a one-sided assumption on the Cesàro behavior of the distribution. We apply these Tauberian results to deduce a number of Tauberian theorems for power series and Stieltjes integrals where Cesàro summability follows from Abel summability. We also use our general results to give a new simple proof of the classical Littlewood one-sided Tauberian theorem for power series.     

New perspective on the Kuhn–Tucker theorem

A new proof of the Kuhn–Tucker theorem on necessary conditions for a minimum of a differentiable function of several variables in the case of inequality constraints is given. The proof relies on a simple inequality (common in textbooks) for the projection of a vector onto a convex set.     

A note on the Fuglede–Putnam theorem

We prove the following generalization of the Fuglede–Puntam theorem. Let N be an unbounded normal operator in the Hilbert space, and let A be an unbounded self-adjoint operator such that $D(N)\subseteq D(A)$ . Then, $ AN\subseteq N^*A \Rightarrow AN^*\subseteq NA.$      

A constructive version of the Boyle–Handelman theorem on the spectra of nonnegative matrices

A constructive version of the celebrated Boyle–Handelman theorem on the non-zero spectra of nonnegative matrices is presented.     

Graded versions of Goldie’s theorem, II

We continue the study of graded Goldie rings and their quotient rings. The main results are inverse Goldie’s theorem for graded rings and graded analogs of third Goldie’s theorem.     

Aspects of the Borsuk–Ulam theorem

This is a largely expository account of various aspects of the Borsuk–Ulam theorem, including extensions of the classical theorem to families of maps parametrized by a base space and to multivalued maps. The main technical tool is the Euler class with compact supports.     

Topological versions of passy– prisman’S minimax theorem

A topological version of Passy–Prisman’s minimax Theorem is proved. We introduce to pological cones and we prove our results under connectedness assumptions. We give examples of cones in spaces without any linear structure. Even when interpreted in a linear framework our results are new and improve Passy–Prisman’s minimax Theorem, and consequently Sion’s minimax Theorem     

Some remarks on the parametrized Borsuk–Ulam theorem

Given a locally trivial fibre bundle \(E\rightarrow B\) (with fibres and base finite complexes), an orthogonal real line bundle \(\lambda \) over E and a real vector bundle \(\xi \) over B, we consider a fibrewise map \(f: S(\lambda ) \rightarrow \xi \) over B defined on the unit sphere bundle of \(\lambda \). Following the fundamental work of Jaworowski and Dold on the parametrized Borsuk–Ulam theorem, we investigate lower bounds on the cohomological dimension of the set \(\{ v\in S(\lambda ) \vert f(v)=f(-v)\}\).     

An improvement of the Arzela–Ascoli theorem

For the classical Arzela–Ascoli theorem and its typical modern formulation, we have improved the sufficiency part by weakening the compactness of the domain space, and the necessity part is improved by strengthening the necessity part of the classical version.     

Vanishing theorem for sheaves of microfunctions at the boundary on cr–manifolds

Let X be a complex analytic manifold. Consider S?M?Xreal analytic submonifolds with codium ^R _MS=1,and let ω be a connected component of MS. Let p∈S X_MT_M ^*X where T^* _Xdenotes the conormal bundle to M in X, and denote by ν(p) the complex radial Euler field at p. Denote by μ*(Ox) (for * = M, ω) the microlocalization of the sheaf of holomorphic functions along *.

Under the assumption dim^R(T_pT_M ^*X? ν(p)) = 1, a theorem of vanishing for the cohomology groups H^jμM(Ox)p is proved in [K-S 1, Prop. 11.3.1], j being related to the number of positive and negative eigenvalue for the Levi form of M.

Under the hypothesis dim^R(T_pT_S ^*X∩ν(p))=1, a similar result is proved here for the cohomology groups of the complex of microfunctions at the boundary μω(Ox).Stating this result in terms of regularity at the boundary for CR–hyperfunctions a local Bochner–type theorem is then obtained.     

A proof of the Lyusternik–Graves theorem

In this paper, we give a new proof of the Lyusternik–Graves theorem, based on an intermediate result regarding linear openness inspired by works of Frankowska and Ursescu.     

Sharp version of the Goldberg–Sachs theorem

We reexamine from first principles the classical Goldberg–Sachs theorem from General Relativity. We cast it into the form valid for complex metrics, as well as real metrics of any signature. We obtain the sharpest conditions on the derivatives of the curvature that are sufficient for the implication (integrability of a field of alpha planes)T{Rightarrow} (algebraic degeneracy of the Weyl tensor). With every integrable field of alpha planes, we associate a natural connection, in terms of which these conditions have a very simple form.     

Stability of some versions of the Prékopa–Leindler inequality

We study the behavior of positive solutions of the following Dirichlet problem

$$\left \{ \begin{array}{ll} -\Delta_{p}u=\lambda u^{s-1}+u^{q-1} &\quad {\rm in}\enspace \Omega \\ u_{\mid\partial \Omega}=0 \end{array}\right. $$

when s → p ^?. Here \({p >1 , s\,{\in}\,]1,p]}\) and q > p with \({q\leq\frac{Np}{N-p}}\) if N > p.