A Tychonoff space $X$ is called $\kappa$-pseudocompact if for every continuous mapping $f$ of into $\mathbb{R}^\kappa$ the image $f(X)$ compact. This notion generalizes pseudocompactness and gives a stratification spaces lying between pseudocompact compact spaces. It well known that determined by uniform structure function $C_p(X)$ real-valued functions on endowed with pointwise topology. In re...

For Tychonoff X and α an infinite cardinal, let αdef X := the minimum number of α cozero-sets of the Čech-Stone compactification which intersect to X (generalizing R-defect), and let rtX := minα max(α, α def X). Give C(X) the compact-open topology. It is shown that τC(X) ≤ nχC(X) ≤ rtX = max(L(X), L(X) def X), where: τ is tightness; nχ is the network character; L(X) is the Lindelöf number. For ...

The Schauder-Tychonoff theorem states that a continuous function from a compact convex subset of a locally convex topological vector space into itself must have a fixed point ([1, Chapter V, 10.5], or [2]). Using this theorem, we obtain here a stronger result, stating that a map from such a set into the surrounding vector space has a fixed point if the directions in which the points are moved s...

We consider a numerical space discretization of Maxwell’s system by the socalled Yee’s scheme. For such a scheme, we first show that the boundary observability estimate is not uniform with respect to the mesh size. Using a discrete multiplier method, we prove an observability estimate that separates the low and high frequency contributions. We then describe one of its consequence to control pro...

In the appendix to the book by F. F. Bonsal, Lectures on Some Fixed Point Theorems of Functional Analysis (Tata Institute, Bombay, 1962) a proof by Singbal of the SchauderTychonoff fixed point theorem, based on a locally convex variant of Schauder mapping method, is included. The aim of this note is to show that this method can be adapted to yield a proof of Kakutani fixed point theorem in the ...

If ft is an open subset of Rn+ , the approximation problem is to decide whether every solution of the heat equation on II can be approximated by solutions defined on all of R . The necessary and sufficient condition on il which insures this type of approximation is that every section of Q taken by hyperplanes orthogonal to the Z-axis be an open set without "holes," i.e., whose complement has no...