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...
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 ...
Given a non-decreasing sequence of real numbers \(\{r_0,r_1,\ldots , r_m\}\) (called knots), the B-spline basis functions of degree p, denoted by \(\{N_{i,p} \}_{i=0}^{m-p}\), are defined recursively as $$\begin{aligned} \begin{aligned}&N_{i,0}(r) = \left\...
In this case, by property (P3) of the previous lemma, we obtain that the non-decreasing sequence \(\{\alpha_{n}\}\) is contained in \([\frac{1}{2},1)\). Also, since T is non-expansive and with at least one fixed point, it follows by standard arguments that \(\{x_{n}\...
The set of constraints for this problem is not just a single linear knapsack constraint of the form (2) but a set of n nested linear constraints with the right-hand sides forming a non-decreasing sequence d1 ≤ d2 ≤ ⋅⋅⋅ ≤ dn. As above, here for each j, 1...
Instead of the abrupt drop at this value, M(T) continues decreasing almost linearly with the temperature increase. This behavior is explained by the segregation of the Cr atoms, see lower inset in Fig. 2.4 for the experimental images before and after annealing. Eventually the TC of NiCr ...
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....
Theorem 1.1 Let M be a 3–manifold with empty or toroidal boundary. Assume that ı n . / ¤ 1 for some 2 H 1 .MI /, then ı n W H 1 .MI / ! 0 is a semi-norm. In particular, this allows us to show that the sequence fı n g is eventually constant. That is, the...
Instead, the picture change drastically if we switch to semi-linear elliptic equations with non-Lipschitz nonlinearities: given, let us consider for example the class of equations (1) whereis the unit ball inandandare respectively the positive and negative part ofu. Notice that the main feature...
is a decreasing sequence of non-empty, closed and bounded subsets of \(\mathrm{pc}(j,e)\) . set \(b=\bigcap_{n=1}^{ \infty}b_{n}\) . now, by arguing as in steps 1, 3 and 6 in the proof of theorem 3.1 in [ 41 ], we see that the values of r are closed, the set...