Abstract. A landmark theorem in the metric theory of continued fractions begins this way: Select a non-negative real function f defined on the positive integers and a real number x, and form the partial sums sn of f evaluated at the partial quotients a1, . . . , an in the continued fraction expansion for x. Does the sequence {sn/n} have a limit as n → ∞? In 1935 A. Y. Khinchin proved that the a...

Using Theorem 1 (or, rather, the analogous result for higher-dimensional Euclidean spaces) we deduce4' 6 THEOREM 2. Let D be any partial differential operator with constant coefficients, and let T be any distribution. Then we can find a distribution S such that DS = T. This result is also valid if D is a partial differential-difference operator with constant coefficients. Remark: We can also gi...

The variational principle states that if a differentiable functional F attains its minimum at some point zi, then F’(C) = 0; it has proved a valuable tool for studying partial differential equations. This paper shows that if a differentiable function F has a finite lower bound (although it need not attain it), then, for every E > 0, there exists some point u( where 11 F’(uJj* < l , i.e., its de...

For simplicity, we follow the rules: X denotes a non empty set, Y denotes a set, S denotes a σ-field of subsets of X, F denotes a function from N into S, f , g denote partial functions from X to R, A, B denote elements of S, r, s denote real numbers, a denotes a real number, and n denotes a natural number. Let X be a non empty set, let f be a partial function from X to R, and let a be a real nu...

For simplicity, we adopt the following rules: X is a non empty set, S is a σ-field of subsets of X, M is a σ-measure on S, E is an element of S, F , G are sequences of partial functions from X into R, I is a sequence of extended reals, f , g are partial functions from X to R, s1, s2, s3 are sequences of extended reals, p is an extended real number, n, m are natural numbers, x is an element of X...

Abstract We deal with multiplicity of solutions to the following Schrödinger-Poisson-type system in this article: Δ H u −</m:mo...

We introduce a new Galois connection for partial operations on a finite set, which induces a natural quasi-order on the collection of all partial algebras on this set. The quasi-order is compatible with the basic concepts of natural duality theory, and we use it to turn the set of all alter egos of a given finite algebra into a doubly algebraic lattice. The Galois connection provides a framewor...

Partial metric spaces were introduced by Matthews in 1994 as a part of the study of denotational semantics of data flow networks. In 2014 Asadi and {it et al.} [New Extension of $p$-Metric Spaces with Some fixed point Results on $M$-metric paces, J. Ineq. Appl. 2014 (2014): 18] extend the Partial metric spaces to $M$-metric spaces. In this work, we introduce the class of $F(psi,varphi)$-contrac...

Let R(·) stand for the bounded-error randomized query complexity. We show that for any relation f ⊆ {0, 1}×S and partial Boolean function g ⊆ {0, 1}×{0, 1}, R1/3(f ◦g ) = Ω(R4/9(f)·

The terminology and notation used here have been introduced in the following articles: [3], [7], [17], [2], [4], [12], [13], [14], [16], [1], [5], [9], [15], [18], [10], [8], [20], [21], [19], [11], [22], and [6]. For simplicity, we use the following convention: n, i denote elements of N, X, X1 denote sets, r, p, s, x0, x1, x2 denote real numbers, f , f1, f2 denote partial functions from R to R...