Suppose that A and B are nonempty subsets of a complete metric space $$(\mathcal {M},d)$$ $$\phi ,\psi :A\rightarrow B$$ mappings. The aim this work is to investigate some conditions on $$ $$\psi such the two functions, one assigns each $$x\in A$$ exactly $$d(x,\phi x)$$ other $$d(x,\psi , attain global minimum value at same point in A. We have introduced notion proximally F-weakly dominated pa...

The weak approximation principal fails for the forms x + y + z = kw, when k = 2 or 3. The question therefore arises as to what asymptotic density one should predict for the rational zeros of these forms. Evidence, both numerical and theoretical, is presented, which suggests that, for forms of the above type, the product of the local densities still gives the correct global density. Let f(x1, . ...

It is known that ternary complexes of myosin subfragment 1 (S1) with ADP and the Pi analogs beryllium fluoride (BeFx) and aluminum fluoride (AlF4-) are stable analogs of the myosin ATPase intermediates M* x ATP and M** x ADP x Pi, respectively. Using kinetic approaches, we compared the rate of formation of the complexes S1 x ADP x BeFx and S1 x ADP x AlF4- in the absence and in the presence of ...

The notion of a weak ∆-system was introduced and studied by Erdős, Milner and Rado [5] in 1974. A weak ∆-system is a family of sets where all pairs of sets have the same intersection size. Erdős and Szemerédi [7] investigated the behavior of the function F (n,r)—the largest integer so that there exists a family F of subsets of an n-element set which does not contain a ∆-system of r sets. Answer...

The classical Yao principle states that the complexity R (f) of an optimal randomized algorithm for a function f with success probability 1− equals the complexity maxμ D (f) of an optimal deterministic algorithm for f that is correct on a fraction 1 − of the inputs, weighed according to the hardest distribution μ over the inputs. In this paper we investigate to what extent such a principle hold...

Let E , E∗ be Hilbert spaces. Given a Hilbert space H of holomorphic functions in a domain Ω in C, consider the multiplier space MH(E , E∗). It is shown that for “nice enough” H, the following statements are equivalent for f ∈ MH(E , E∗): (1) There exists a g ∈ MH(E∗, E) such that g(z)f(z) = IE for all z ∈ Ω. (2) There exists a Hilbert space Ec and fc ∈ MH(Ec, E∗) such that F (z) := [ f(z) fc(z...

Let G = (V,E) be a graph and let f be a function f : E → {0, 1, 2}. An edge x with f(x) = 0 is said to be undefended with respect to f if it is not incident to an edge with positive weight. The function f is a weak edge Roman dominating function (WERDF) if each edge x with f(x) = 0 is incident to an edge y with f(y) > 0 such that the function f ′ : E → {0, 1, 2}, defined by f ′(x) = 1, f ′(y) =...

Since the discovery of the TCF/LEF family of transcription factors, their functions have been under intensive investigation in the area of cancer biology. The work presented in this paper focused on the changes in TCF-1 and LEF-1 expression levels in a set of astrocytic brain tumors. Protein expression was detected using immunohistochemistry and then evaluated by Ellipse software (ViDiTo, Slova...