Given a real valued and time-inhomogeneous martingale diffusion X , we investigate the properties of functions defined by the conditional expectation f(t,Xt) = E [g(XT )|Ft]. We show that whenever g is monotonic or Lipschitz continuous then f(t, x) will also be monotonic or Lipschitz continuous in x. If g is convex then f(t, x) will be convex in x and decreasing in t. We also define the margina...

where U ranges over the finite-index open normal subgroups of G. There is a natural continuous homomorphism G → Ĝ through which every other continuous homomorphism from G to a profinite group factors uniquely. If G is profinite already, then G → Ĝ is an isomorphism. In general, G→ Ĝ need not be injective or surjective. Nevertheless, we think of G as being almost isomorphic to Ĝ: The finite-inde...

LetG be a Hausdorff topological locally compact group. LetM(G) denote the Banach algebra of all complex and bounded measures on G. For all integers n ≥ 1 and all μ ∈ M(G), we consider the functional equations ∫ G f(xty)dμ(t) = ∑n i=1gi(x)hi(y), x,y ∈ G, where the functions f , {gi}, {hi}: G → C to be determined are bounded and continuous functions on G. We show how the solutions of these equati...

For a continuous nonvanishing complex-valued function g on the real line, several notions of a mean winding number are introduced. We give necessary conditions for a Toeplitz operator with matrix-valued symbol G to be semi-Fredholm in terms of mean winding numbers of det G. The matrix function G is assumed to be continuous on the real line, and no other apriori assumptions on it are made.

and Applied Analysis 3 A2 there exist g x , h x , and k t 0, such that 0 ≤ f t, x ≤ k t g x h x , ∀x ∈ 0,∞ , a.e. t ∈ 0, 1 , 1.6 here g : 0,∞ → 0,∞ is continuous and nonincreasing, h : 0,∞ → 0,∞ is continuous, and h/g is nondecreasing; A3 There exist two positive constants R > r > 0 such that R > ΦR1 γ∗ ≥ r > 0, ∫1 0 k s g ( rsα−1 ) ds < ∞, R ≥ ( 1 h R g R )∫1 0 1 − s α−β−1 Γ α ( 1 − aξα−β−1 s ...

We investigate possible preduals of the measure algebra M(G) of a locally compact group and the Fourier algebra A(G) of a separable compact group. Both of these algebras are canonically dual spaces and the canonical preduals make the multiplication separately weak∗-continuous so that these algebras are dual Banach algebras. In this paper we find additional conditions under which the preduals C0...

ISSN 2277 – 5048 | © 2012 Bonfring Abstract--In this paper we introduce the concept of − γ generalized α -continuous mappings in topological spaces and study its relationship with other mappings. Further we declare the concepts of αγ -continuous mappings and − γ gα-continuous mappings which coincide when the space is α− γ T1/2. In addition, we define the concept of − γ gα-irresolute mappings in...

We present several examples of feebly compact Hausdorff paratopological groups (i.e., groups with continuous multiplication) which provide answers to a number of questions posed in the literature. It turns out that a 2-pseudocompact, feebly compact Hausdorff paratopological group G can fail to be a topological group. Our group G has the Baire property, is Fréchet–Urysohn, but it is not precompa...