##### News Update on Common Fixed Point April-21

**[1] Common Fixed Point Theorems for Contractive Maps**

Two common fixed point theorems have been proved by using minimal type commutativity and contractive conditions. The last theorem extends known results on compatible maps to a wider class of mappings.

**[2] Compatible mappings and common fixed points**

A generalization of the commuting mapping concept is introduced. Properties of this “weakened commutativity” are derived and used to obtain results which generalize a theorem by Park and Bae, a theorem by Hadzic, and others.

**[3] RENARKS ON REWEAKLY COMMUTING MAPPONGS AND COMMON FIXED POINT THEOREMS**

It was the turning point in the “fixed point arena” when the notion of weak commutativity was introduced by Sessa [9] as a sharper tool to obtain common fixed points of mappings. As a result, all the results on fixed point theorems for commuting mappings were easily transformed in the setting of the new notion of weak commutativity of mappings. It gives a new impetus to the studying of common fixed points of mappings satisfying some contractive type conditions and a number of interesting results have been found by various authors. A bulk of results were produced and it was the centre of vigorous research activity in “Fixed Point Theory and its Application in various other Branches of Mathematical Sciences” in last two decades. A major break through was done by Jungck [3] when he proclaimed the new notion what he called “compatibility” of mapping and its usefulness for obtaining common fixed points of mappings was shown by him. There-after a flood of common fixed point theorems was produced by various researchers by using the improved notion of compatibility of mappings. of compatibility of mappings.

**[4] Common Fixed Point Results of Weakly Compatible Maps in G-metric Spaces**

We prove the existence of a unique common fixed point for two weakly compatible maps satisfying Ø- conditions in G-metric spaces. Our result extends and generalizes some results in the literature.

**[5] Common Fixed Points of Kannan and Chatterjea Types of Mappings in a Complete Metric Space**

This paper contains several generalizations of the theorems for common fixed point of R. Kannan, S. K. Chatterjea and P. V. Koparde & B. B. Waghmode types of mappings. These generalizations are done by using a sequentially convergent mappings. Trough several examples, we have shown that the generalized claims are inapplicable, and that the obtained generalized claims prove the existence of a unique common fixed point of considered mappings.

**Reference**

**[1] **Pant, R.P., 1998. Common fixed point theorems for contractive maps. *Journal of Mathematical Analysis and Applications*, *226*(1), pp.251-258.

**[2] **Jungck, G., 1986. Compatible mappings and common fixed points. *International Journal of Mathematics and Mathematical Sciences*, *9*(4), pp.771-779.

**[3] **Pathak, H.K., Cho, Y.J. and Kang, S.M., 1997. RENARKS ON REWEAKLY COMMUTING MAPPONGS AND COMMON FIXED POINT THEOREMS. *Bulletin of the Korean Mathematical Society*, *34*(2), pp.247-257.

**[4] **Eke, K.S., 2015. Common Fixed Point Results of Weakly Compatible Maps in G-metric Spaces. *Journal of Advances in Mathematics and Computer Science*, pp.341-348.

**[5] **Malčeski, R., Malčeski, A., Anevska, K. and Malčeski, S., 2016. Common Fixed Points of Kannan and Chatterjea Types of Mappings in a Complete Metric Space. *Journal of Advances in Mathematics and Computer Science*, pp.1-11.

##### Antimicrobial Activity of Poly(ester urea) Electrospun Fibers Loaded with Bacteriophages

The capacity of two representative biocompatible polymers to load bacteriophages into electrospun nanofibers was evaluated, paying particular attention to the possibility of retaining their antibacterial function. Specifically, the work involves the following steps: (a) evaluation of the impact of the electric field applied on the phage operation; (b) evaluation of the activity where the process of lyophilization may be prevented by the use of water-soluble polymers (e.g. poly(ethylene glycol); (c) evaluation of the activity when organic solvent is needed for the dissolution of the polymer and theoretical lyophilization is theoretical A poly(ester urea) (PEU) derived from the natural L-leucine amino acid was considered in this case. It has been shown that the adsorption of commercial bacteriophage preparations into calcium carbonate particles is a promising technique to prevent lyophilization and keep the initial operation of the bactericide to a limit. Due to their particular activity against Staphylococci (e.g., S. aureus) and Streptococci (e.g., S. pyogenes) bacteria, the Phagestaph and Fersis bacteriophage commercial preparations were selected for this analysis. The biocompatibility of both unloaded and bacteriophage-loaded PEU scaffolds was demonstrated by adhesion and proliferation assays using epithelial cells, although some minor variations were observed depending on the form of bacteriophage and the selected preparation methodology.

**Author(s) Details**

**Angélica Díaz
**Departament d’Enginyeria Química, EEBE, Universitat Politècnica de Catalunya, Edifici I.2, C/Eduard Maristany, 10-14, 08019 Barcelona, Spain.

**Luis J. del Valle
**Departament d’Enginyeria Química, EEBE, Universitat Politècnica de Catalunya, Edifici I.2, C/Eduard Maristany, 10-14, 08019 Barcelona, Spain.

**Departament d’Enginyeria Química, EEBE, Universitat Politècnica de Catalunya, Edifici I.2, C/Eduard Maristany, 10-14, 08019 Barcelona, Spain.**

Noel Rodrigo

Noel Rodrigo

**Departament d’Enginyeria Química, EEBE, Universitat Politècnica de Catalunya, Edifici I.2, C/Eduard Maristany, 10-14, 08019 Barcelona, Spain.**

María Teresa Casas

María Teresa Casas

**Center for Medical Biotechnology & Bioengineering, Georgian Technical University, 77 Kostava str., Tbilisi 0175, Georgia.**

George Chumburidze

George Chumburidze

**Center for Medical Biotechnology & Bioengineering, Georgian Technical University, 77 Kostava str., Tbilisi 0175, Georgia and Institute of Chemistry and Molecular Engineering, Agricultural University of Georgia, # 240 David Aghmashenebeli Alley, Tblisi 0159, Georgia.**

Ramaz Katsarava

Ramaz Katsarava

**Jordi Puiggalí
**Departament d’Enginyeria Química, EEBE, Universitat Politècnica de Catalunya, Edifici I.2, C/Eduard Maristany, 10-14, 08019 Barcelona, Spain and Research Center for Multiscale Science and Engineering, Universitat Politècnica de Catalunya, C/Eduard Maristany, 10-14, 08019 Barcelona, Spain.

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##### The Impact of Computer-mediated Communication and Interactional Feedback on the Correct Use of Past Tense

This study aims at investigating the effect of types of feedback (explicit vs. implicit) on the acquisition

of simple past tense by Iranian EFL learners and its retention. The parametric independent t-tests

were run to investigate the research questions posed in this study. To this end, 60 pre-intermediate

EFL learners from Sa`adat high school who were chosen based on their performance on KET English

Test took part in the study. The students whose scores were between one standard deviation above

and below the mean were selected for the study. The students were assigned into two groups:

experimental group 1(N=30) that received explicit corrective feedback and experimental group 2

(N=30) that got implicit feedback after sending their mail to the researcher. The data were analyzed

through the parametric statistical analysis of Pearson Correlation and independent t-test. The results

show that the first experimental group (explicit group) outperformed the second group (implicit group)

so, Iranian EFL learners have acquired past tense better using explicit method of teaching grammar.

Also the findings revealed that Iranian EFL learners have retained past tense better using explicit

method of teaching grammar. Theoretical and pedagogical implications of the findings of this study

are reported at the end of the thesis.

**Author (s) Details**

**Fatemeh Alipanahi
**Zanjan University, Zanjan, Iran.

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##### The Effect of Anisotropy on the Structure Optimization Using BEM-GSS and BEM-NGGP Algorithms

*Aims:** A shape optimization technique is developed, using the boundary element method, for two-dimensional anisotropic structures to study the effects of anisotropy on the displacements and stresses, then minimize weight while satisfying certain constraints upon stresses and geometry.*

*Study Design:** Original Research Paper.*

*Place and Duration of Study:** Jamoum University College, Mathematics Department, between June 2016 and July 2017.*

*Methodology:** The shape design sensitivity analysis of a two-dimensional anisotropic structure using a singular formulation of the boundary element method is investigated to study the effects of anisotropy on the displacements and stresses. An Implicit differentiation technique of the discretized boundary integral equations is performed to produce terms that contain derivatives of the fundamental solutions employed in the analysis. This technique allows the coupling between optimization technique and numerical boundary element method (BEM) to form an optimum shape design algorithm that yields shape design sensitivities of the displacement and stress fields for anisotropic materials with very high accuracy. The fundamental solutions of displacements and tractions in terms of complex variables employed in the analysis. The boundary element method was developed and implemented for use with the golden-section search (GSS) algorithm and neutrosophic goal geometric programming (NGGP) algorithm as a numerical optimization technique for minimizing weight while satisfying all of the constraints.*

*Results:** The proposed method has been verified by using the two-dimensional plate with an elliptical hole as the numerical example. The numerical results show that the proposed method is suitable and effective tool for the computer implementation of the solution.*

*Conclusion:** From the research that has been performed, it is possible to conclude that the optimal shape of the two-dimensional plate with an elliptical hole is crucial when elastic field is sensitive to boundary shape. Also from this knowledge of the effects of anisotropy on the displacements and stresses, we can design various anisotropic structures to meet specific engineering requirements and utilize within which to place new information can be more effective.*

**Read full article:** http://bp.bookpi.org/index.php/bpi/catalog/view/46/228/389-2

##### A New Definition of Limit of Periodic Function and Periodic g-Contractive Mapping at Infinity

*Limit is a basic concept of calculus. However, according to the updated definition, the limit of periodic function at infinity is not in existence. This conclusion of description does not suit with the periodic phenomenon. For example, the temperature on earth is changed periodically every year since the birth of the earth (viewed as t=0). Today (viewed as t →∞) the temperature on earth is continuing. Continuation means that the limit exists. In this paper, a new definition of limit of periodic function and periodic g-contractive mapping at infinity is defined by the value of its initial point based on transformation of variables. Similar definition is made for g- contractive ratio of periodic g-contractive mapping with k-related fixed points. These definitions can be used to describe the k-polar problems and calculation the limit of combinations of periodic functions at infinity. Furthermore, the new definition on contractive ratio of periodic iterative g-contractive mapping at infinity can help us to find the constant G and improves the application of the periodic iterative g-contractive mapping theorem.*

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##### Recursive Computation of Binomial and Multinomial Coefficients and Probabilities

*This chapter studies a prominent class of recursively-defined combinatorial functions, namely, the binomial and multinomial coefficients and probabilities. The chapter reviews the basic notions and mathematical definitions of these four functions. Subsequently, it characterizes each of these functions via a recursive relation that is valid over a certain two-dimensional or multi-dimensional region and is supplemented with certain boundary conditions. Visual interpretations of these characterizations are given in terms of regular acyclic signal flow graphs. The graph for the binomial coefficients resembles a Pascal Triangle, while that for trinomial or multinomial coefficients looks like a Pascal Pyramid, Tetrahedron, or Hyper-Pyramid. Each of the four functions is computed using both its conventional and recursive definitions. Moreover, the recursive structures of the binomial coefficient and the corresponding probability are utilized in an iterative scheme, which is substantially more efficient than the conventional or recursive evaluation. Analogous iterative evaluations of the multinomial coefficient and probability can be constructed similarly. Applications to the reliability evaluation for two-valued and multi-valued k-out-of-n systems are also pointed out. *

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##### On the Proof Complexities of Strongly Equal Non-classical Tautologies

*The strong equality of classical tautologies and their proof complexities comparative analysis in certain proof systems were given by first author in previous studies. Here we introduce the analogous notions of strong equality for non-classical (intuitionistic and minimal) tautologies and investigate the relations between the proof complexity measures of strongly equal non-classical tautologies in some proof systems. We prove that 1) the strongly equal tautologies have the same proof complexities in some proof systems and 2) there are such proof systems, in which some measures of proof complexities for strongly equal tautologies are the same, while the other measures differ from each other only as a function of the sizes of tautologies.*

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##### DRBEM Sensitivity Analysis and Shape Optimization of Rotating Magneto-Thermo-Viscoelastic FGA Structures Using DRBEM-GSS and DRBEM-NGGP Algorithms

*Aims:** A practicalshape optimization technique is developed, using the dual reciprocity boundary element method (DRBEM) with the golden-section search algorithm based on uniform bicubic B-splines, for rotating magneto-thermo-viscoelastic functionally graded anisotropic (FGA) structures subjected to a moving heat source in the context of the Green and Naghdi theory of type III.*

*Study Design:** Original Research Paper.*

*Place and Duration of Study:** Jamoum University College, Mathematics Department, between July 2016 and August 2017.*

*Methodology:** An implicit-implicit staggered algorithm was proposed for use with the DRBEM to obtain the final DRBEM coupled linear system of equations for displacements and temperature that describe the magneto-thermo-viscoelastic structural analysis problem. An implicit differentiation of the discretized dual reciprocity boundary integral equation with respect to design variables is used to calculate shape displacement sensitivities of anisotropic materials with very high accuracy. This method allows the coupling between optimization technique and a dual reciprocity boundary element method. The feasible direction method was developed and implemented for use with the one-dimensional golden-section search technique based on uniform bicubic B-splines, as a numerical optimization method for minimizing weight while satisfying all of the constraints. The DRBEM was developed and implemented for use with the golden-section search (DRBEM-GSS) algorithm and also implemented with the neutrosophic goal geometric programming (DRBEM-NGGP) algorithm as a numerical optimization techniques for minimizing weight while satisfying all of the constraints.*

*Results:** The optimum shape design of fillet in tension bars used as the numerical example in order to verify the formulation and the implementation of the proposed technique. The numerical results show our technique is efficient and precise.*

*Conclusion:** From the research that has been performed, it is possible to conclude that the optimal shape of the top half of the fillet under stress constraint based on magneto-thermo-viscoelasticity is crucial when magneto-thermoviscoelastic field is sensitive to boundary shape. Also from this knowledge of the variation of the displacements and temperature sensitivities with time for magneto-thermo-viscoelastic FGA structures, we can design various magneto-thermoviscoelastic structures to meet specific engineering requirements and utilize within which to place new information can be more effective.*

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##### Examples of Simply and Multiply Connected Fatou Sets for a Class of Meromorphic Functions

*Aims: **We give some families of functions which are meromorphic outside a compact countable set B of essential singularities. Our aim is to give some examples of the stable set (called the Fatou set) and the unstable set (called the Julia set) since there are not many examples of parametric family of this class of functions (called in the introduction functions of class K) in complex dynamics.*

*Study design: **We study components of the Fatou set and some theorems related with the iteration of functions in class K and design a computational program to give examples of the Julia and Fatou sets.*

*Place and Duration of Study: **Facultad de Ciencias F__sico Matem_aticas, Benem_erita Universidad Aut_onoma de Puebla, M_exico between June 2011 and July 2012.*

*Methodology: **We use some theorems of complex dynamics in order to study components of the Fatou set. We program some algorithms in C and get some _gures of the Fatou set.*

*Results: **Given a family of functions in class K we get some mathematical results of the Fatou and Julia sets and its _gures for some parameters given.*

*Conclusion: **Taking some families in class K ∩ S _{k} we give examples of the Fatou set which can be either simply-connected or multiply-connected in the last case the Julia set is a totally disconnected set.*

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