For every Hausdorff space X the space XΘ is introduced. If X is H-closed, then XΘ is a quasy-compact T1-space. If f : X → Y is a mapping, then there exists the mapping fΘ : XΘ → YΘ. We say that a mapping f : X → Y is Θ-closed if fΘ is a closed mapping. If X and Y are H-closed and if f : X → Y is a HJ-mapping, then fΘ is Θ-closed. Let X = {Xa, pab, A} be an inverse system of H-closed spaces Xa a...

DP-reduction F ❀ DPv(F ), applied to a clause-set F and a variable v, replaces all clauses containing v by their resolvents (on v). A basic case, where the number of clauses is decreased (i.e., c(DPv(F )) < c(F )), is singular DP-reduction (sDP-reduction), where v must occur in one polarity only once. For minimally unsatisfiable F ∈ MU , sDPreduction produces another F ′ := DPv(F ) ∈ MU with th...

Let k be a field of characteristic two. We prove that monic polynomial f∈k[X] degree n≥1 is the minimal/characteristic symmetric matrix with entries in if and only it not product pairwise distinct inseparable irreducible polynomials. In this case, we f minimal size n. also any element α∈kalg eigenvalue n or n+1, first case happening α separable.

We prove that for any collection F of n ≥ 2 pairwise disjoint compact convex sets in the plane there is a pair A and B such line separates from either or subcollection with at least /18 sets.

Given a family F of pairwise almost disjoint (ad) sets on a countable set S, we study families F̃ of maximal almost disjoint (mad) sets extending F . We define a+(F) to be the minimal possible cardinality of F̃ \F for such F̃ and a+(κ) = max{a+(F) : | F | ≤ κ}. We show that all infinite cardinal less than or equal to the continuum c can be represented as a+(F) for some ad F (Theorem 21) and that t...

(4.1) Let E,F be nonempty subsets of R, E is compact and F is closed. Then there exist (e, f) ∈ E ×F such that d(E,F ) = d(e, f). proof Let α = d(E,F ) = inf(x,y)∈E×F d(x, y) be the distance between the two sets. Let > 0, and let x0 be a given element of E. Since E is compact, it is in particular bounded, and there exists r > 0 such that E ⊂ B(r, x0). Now consider the closed ball B(r + α+ , x0)...

Let ρ be an ultra-naturally Sylvester, hyper-everywhere complex functional. In [2], the main result was the characterization of trivial planes. We show that z′′ is globally anti-Gaussian and Artinian. Therefore recent interest in surjective isometries has centered on characterizing geometric random variables. The work in [2, 2, 13] did not consider the pairwise free, closed, sub-tangential case.

Let X be a metric space with metric p, let f(X)QX be a continuous mapping, and let h(X) ^X be a homeomorphism. For x&X, the s e t 23^-°o/(^) i called the semi-orbit of # under ƒ and the set n£XZh{x) is called the orbit of # under h. For # £ X , the closure of the semi-orbit of x under ƒ is called the semi-orbit-closure of x under ƒ and the closure of the orbit of x under h is called the orbit-c...

On a Lebesgue measure space with measure element do, and total measure finite or infinite, we consider the complex-valued measurable functions f, y, so, ., each determined only a.e., and each belonging to all L,-classes simultaneously, 1<r< a. We donote (i) by F = {f} a ring of functions with complex constants and ordinary multiplication and closed under the involution: if f -= f + if2 e F then...

We present a closed form upper bound for the average pairwise error probabilities (PEP) of space-time codes for a keyhole channel. It is derived from the exact conditional PEP for given fading channel coefficients, using a moment generating function-based approach. We include simulation results for varying numbers of antennas which affirm that the proposed PEP serves as a tight bound for codes ...