This communication in uences on magnetohydrodynamic ow of viscoelastic uid with Magnetic  eld induced by oscillating plate. General solutions have been found out for velocity and shear stress pro les using mathematical transformations (integral transforms). The governing partial di erential equations have been solved analytically under boundary conditions u(0; t) = A0H(t)sin( t) and u(0; t) = A0H(t)cos( t) with t  0. For the sake of simplicity of boundary conditions are veri ed on the analytical general solutions and similar solutions have been particularized under three limited cases namely (i) Maxwell uid without Magnetic  eld if 6= 0; M = 0 (ii) Newtonian uid with Magnetic  eld if = 0; M 6= 0 and (iii) Newtonian uid with out Magnetic  eld if = 0; M = 0. Finally various physical parameters with variations of uid behaviors are analyzed and depicted graphical illustrations.

In this article we introduce the concepts of minimal prime z-, lter, essential z-, lter and r-, lter. We investigate and study the be-havior of minimal prime z-, lters and compare them with minimal prime ideals and coz-ultra, lters. We show that X is a P-space if and only if every , xed prime z-, lter is minimal prime. It is observed that if X is a @-space then X is a P-space if and only if Z[Mf ] is an r-, lter, for every f 2 C(X). The collection of all minimal prime z-, lters will be topologized and it is proved that the space of minimal prime z-, lters is homeomorphic with the space of coz-ultra, lters. Finally, it is obtained several properties and relations between the space of minimal prime z-, lters and the space of minimal prime ideals in C(X).

This study presents the characteristics of Cattaneo-Christov heat ux model for the boundary layer ow of Burgers' uid model. Instead of simple Fourier's law of heat conduction, we presented the Cattaneo-Christov model to analyze the thermal relaxation properties when the heat source/sink was present in the system. Mathematical modeling of the laws of momentum and energy were presented under the order analysis approach. It was revealed that the term \ B2 0u= " was for the hydro-Magnetic rheology of the Newtonian model whereas the generalized Magnetic  eld term was for the Burgers' model, which was incorporated in the current analysis. Suitable transformations were utilized for the conversion of partial di erential system into coupled nonlinear set of ordinary di erential equations, which was tackled analytically through homotopy analysis technique. The plots of various physical quantities are presented, showing the dynamics of the considered analysis. Streamlines for Burgers' and Newtonian models are presented, which show their di erence in rheology. Numerical values for skin friction and surface heat transfer rate are presented in tables.

N-(2-pyridylmethyl)-L-histidine functionalized Fe3O4 Magnetic nanoparticles (PMHis@Fe3O4 MNPs) e ciently catalyzed the three-component Mannich-type reaction of ketones, aromatic aldehydes, and anilines to synthesize -amino ketones in good to high yields. Mannich adducts were obtained in moderate to high diastereoselectivity, favoring anti isomers. The imidazole moiety of PMHis residue on a catalyst plays an important role in the diastereoselectivity. PMHis@Fe3O4 MNPs were prepared by the simple coprecipitation from an aqueous solution of Fe2+ and Fe3+ ions using NH4OH in the presence of L-histidine, followed by reductive amination with 2-pyridine carbaldehyde in the presence of NaBH4. Obtained PMHis@Fe3O4 MNPs were characterized by FT-IR, XRD, VSM, BET, TGA, SEM, EDX, and TEM analyses.

The main goal of the present paper is to construct some families of the Carlitz's q-Bernoulli polyno-mials and numbers. We  rstly introduce the modi ed Carlitz's q-Bernoulli polynomials and numbers with weight (p. We then de ne the modi ed degenerate Carlitz's q-Bernoulli polynomials and num-bers with weight ( ;  ) and obtain some recurrence relations and other identities. Moreover, we derive some correlations with the modi ed Carlitz's q-Bernoulli polynomials with weight ( ;  ), the modi ed degenerate Carlitz's q-Bernoulli polynomials with weight ( ;  ), the Stirling numbers of the  rst kind and second kind.

The present paper aims at investigating the manner of two dissipative massive scalar  elds. Two massive scalar  elds that interact with a reservoir were con-sidered and a reservoir was modeled by continuum Klein-Gordon  elds. The Lagrangian of the total system was canonically quantized and the dynamics of the system was de-termined using the Euler-Lagrange equation. Then, the explicit form of the quantum massive scalar  elds in long-time limit were observed. The propagator of the system and correlation functions were calculated at  nite temperature in the thermo eld dynamics formalizem.

In the present paper, we introduce the notions of ,-type and generalized ,--type multi-valued contractive mappings in a Menger PbM-space and obtain some new , xed point theorems for both ,-admissible and (, , )-admissible mappings. Our results extensively gen-eralize previous results in this framework. Furthermore, we support our results by some non-trivial examples and an application.