Asymptotic Behavior of Resolvents of a Convergent Sequence of Convex Functions on Complete Geodesic Spaces

The asymptotic behavior of resolvents of a proper convex lower semicontinuous function is studied in the various settings of spaces. In this paper, we consider the asymptotic behavior of the resolvents of a sequence of functions defined in a complete geodesic space. To obtain the result, we assume the Mosco convergence of the sets of minimizers of these functions.

Sets of Fractional Operators and Numerical Estimation of the Order of Convergence of a Family of Fractional Fixed-Point Methods

Considering the large number of fractional operators that exist, and since it does not seem that their number will stop increasing soon at the time of writing this paper, it is presented for the first time, as far as the authors know, a simple and compact method to work the fractional calculus through the classification of fractional operators using sets. This new method of working with fractional operators, which may be called fractional calculus of sets, allows generalizing objects of conventional calculus, such as tensor operators, the Taylor series of a vector-valued function, and the fixed-point method, in several variables, which allows generating the method known as the fractional fixed-point method. Furthermore, it is also shown that each fractional fixed-point method that generates a convergent sequence has the ability to generate an uncountable family of fractional fixed-point methods that generate convergent sequences. So, it is presented a method to estimate numerically in a region Ω the mean order of convergence of any fractional fixed-point method, and it is shown how to construct a hybrid fractional iterative method to determine the critical points of a scalar function. Finally, considering that the proposed method to classify fractional operators through sets allows generalizing the existing results of the fractional calculus, some examples are shown of how to define families of fractional operators that satisfy some property to ensure the validity of the results to be generalized.

Gene Body Methylation Confers Transcription Robustness in Mangroves During Long-Term Stress Adaptation

Whether induced epigenetic changes contribute to long-term adaptation remains controversial. Recent studies indicate that environmentally cued changes in gene body methylation (gbM) can facilitate acclimatization. However, such changes are often associated with genetic variation and their contribution to long-term stress adaptation remains unclear. Using whole-genome bisulfite sequencing, we examined evolutionary gains and losses of gbM in mangroves that adapted to extreme intertidal environments. We treated mangrove seedlings with salt stress, and investigated expression changes in relation with stress-induced or evolutionarily-acquired gbM changes. Evolution and function of gbM was compared with that of genetic variation. Mangroves gained much more gbM than their terrestrial relatives, mainly through convergent evolution. Genes that convergently gained gbM during evolution are more likely to become methylated in response to salt stress in species where they are normally not marked. Stress-induced and evolutionarily convergent gains of gbM both correlate with reduction in expression variation, conferring genome-wide expression robustness under salt stress. Moreover, convergent gbM evolution is uncoupled with convergent sequence evolution. Our findings suggest that transgenerational inheritance of acquired gbM helps environmental canalization of gene expression, facilitating long-term stress adaptation of mangroves in the face of a severe reduction in genetic diversity.

ANALYSIS OF TIME-FRACTIONAL BURGERS AND DIFFUSION EQUATIONS BY USING MODIFIED q-HATM

In this paper, the q-homotopy analysis transform technique is implemented to analyze the solution of fractional-order Burgers and diffusion equations with the help of Caputo operator. The results of the proposed method are shown and analyzed with the help of figures. This approach is used to determine the solution in a convergent sequence and illustrate the q-homotopy analysis transform technique solutions convergence to the exact result. Several examples showed the reliability and simplicity of the technique and highlighted the significance of this work. Therefore, the proposed method is successful in investigating other fractional-order linear and nonlinear partial differential equations.

Hyperspace of finite unions of convergent sequences

The symbol S(X) denotes the hyperspace of finite unions of convergent sequences in a Hausdor˛ space X. This hyper-space is endowed with the Vietoris topology. First of all, we give a characterization of convergent sequence in S(X). Then we consider some cardinal invariants on S(X), and compare the character, the pseudocharacter, the sn-character, the so-character, the network weight and cs-network weight of S(X) with the corresponding cardinal function of X. Moreover, we consider rank k-diagonal on S(X), and give a space X with a rank 2-diagonal such that S(X) does not Gδ -diagonal. Further, we study the relations of some generalized metric properties of X and its hyperspace S(X). Finally, we pose some questions about the hyperspace S(X).

On paranormed Ideal convergent sequence spaces defined by Jordan totient function

The study of sequence spaces and summability theory has been an important aspect in defining new notions of convergence for the sequences that do not converge in the usual sense. Paving the way into the applications of law of large numbers and theory of functions, it has proved to be an essential tool. In this paper we generalise the classical Maddox sequence spaces $c_{0}(p)$ c 0 ( p ) , $c(p)$ c ( p ) , $\ell (p)$ ℓ ( p ) and $\ell _{\infty }(p)$ ℓ ∞ ( p ) and define new ideal paranormed sequence spaces $c^{I}_{0}(\Upsilon ^{r}, p)$ c 0 I ( ϒ r , p ) , $c^{I}(\Upsilon ^{r}, p)$ c I ( ϒ r , p ) , $\ell ^{I}_{ \infty }(\Upsilon ^{r}, p)$ ℓ ∞ I ( ϒ r , p ) and $\ell _{\infty }(\Upsilon ^{r}, p)$ ℓ ∞ ( ϒ r , p ) defined with the aid of Jordan’s totient function and a bounded sequence of positive real numbers. We develop isomorphism between certain maps and also find their α-, β- and γ-duals. We examine algebraic and topological properties of these corresponding spaces. Further we study some standard inclusion relations and prove the decomposition theorem.

Ideal convergent sequence spaces of fuzzy star–shaped numbers

In 1990, Diamond [16] primarily established the base of fuzzy star–shaped sets, an extension of fuzzy sets and numerous of its properties. In this paper, we aim to generalize the convergence induced by an ideal defined on natural numbers ℕ , introduce new sequence spaces of fuzzy star–shaped numbers in ℝ n and examine various algebraic and topological properties of the new corresponding spaces as well. In support of our results, we provide several examples of these new resulting sequences.

sp− Closedness and sp − Convergent Sequences in Intuitionistic Fuzzy Metric Space

The aim of present article is to introduce the concepts of sp– convergent sequence in intuitionistic fuzzy metric spaces and analyze relations of convergence, sp– convergence, s∞– convergence and st– convergence in intuitionistic fuzzy metric spaces. Further, we examine sp– convergence, s∞– convergence and st– convergence using the subsequence of convergent sequence in intuitionistic fuzzy metric spaces. Stationary intuitionistic fuzzy metric spaces are defined and investigated. We Finally define sp– closed sets, s∞– closed sets and st– closed sets in intuitionistic fuzzy metric spaces and investigate relations of them.