sequencesHölder inequalityMinkowski inequality.In this paper, a general theorem on absolute Riesz summability factors of infinite series is proved under weaker conditions. Also we have obtained some new and known results.doi:10.1515/tmj-2017-0025Hüseyin Bor...
where α ( ξ ) is bounded and non-decreasing and the integral converges for 0 ≤ x < ∞ . The proof of the Bernstein theorem can be found in [21,22]. We notice that the formulation of the Bernstein theorem can confuse the reader because some of the authors use the Laplace transform...
monotone convergence theorem monotone decreasing function monotone decreasing sequence monotone function monotone increasing function monotone increasing sequence monotone nondecreasing function monotone nondecreasing sequence monotone nonincreasing function monotone nonincreasing sequence ...
Let {Xn} be a sequence of iid random variables. If the common charac-teristic function is absolutely integrable inmth power for some integerm≥ 1, thenZn=n(X1+ … +Xn) has a pdffnfor alln≥m. Here we give a necessary and sufficient condition for supasn. →∞, where φ (x) is...
A negator on L is any decreasing mapping N : L → L satisfying N (0L) = 1L and N (1L) = 0L. Definition 2.9. If N (N (x)) = x for all x ∈ L, then N is called an involutive negator. In this paper, the involutive negator N is fixed. Definition 2.10. A sequence (xn...
Thus, bringing a quasi-deterministic system to a more stochastic regime by decreasing β to the critical value βc (with ΔJ smaller than ΔJc(∞)) increases the diversity of irreversible patterns and hence entropy production. However, further reduction of β makes the system more random (i....
This depends on the observation that each ω has an essentially unique (uniformly bounded multiplicity) sequence of return depths. Thus the estimate can be approached via purely combinatorial arguments very similar to those used in relation to Equation (17). Choosing δ small means the sequences ...
Clearly, r1 is non-negative and non-decreasing. We want to show r1 is continuous. Fix t0 ≥ 0. We first show the left-continuity of r1 at t0. Let {tn}∞n=1 be a sequence in [0, t0) such that tn ↗ t0. Suppose there exists t' ...
if {Zn : n < ω} is a decreasing (with respect to inclusion) family of sets belonging to dom(v), then the set ∩{Zn : n < ω} also belongs to dom(u), and v(∩{Zn:n<w})≥inf{v(Zn):n<w}. Evidently, if v is a finite diffused measure on E, then v satisfies conditio...
The Monotone Convergence Theorem is a fundamental theorem in real analysis that states that if a sequence is monotone (either increasing or decreasing) and bounded, then it converges to a limit. This theorem provides a way to prove the convergence of a sequence by only knowing its...