On the Newton polygons of twisted L-functions of binomials

Let χ be an order c multiplicative character of a finite field and $f(x)=x^d+\lambda x^e$ a binomial with $(d,e)=1$. We study the twisted classical and T-adic Newton polygons of f. When $p>(d-e)(2d-1)$, we give a lower bound of Newton polygons and show that they coincide if p does not divide a certain integral constant depending on $p\mathrm{mod}cd$.We conjecture that this condition holds if p is large enough with respect to $c,d$ by combining all known results and the conjecture given by Zhang-Niu. As an example, we show that it holds for $e=d-1$.     

Foundational ways of thinking for understanding the idea of logarithm

This study explored two undergraduate precalculus students’ understandings of the idea of logarithm as they completed conceptually oriented exploratory lessons on exponential and logarithmic functions. The students participated in consecutive, individual teaching experiments that focused on Sparky – an animated mystical saguaro that doubled in height every week. The exponential lesson focused on developing students’ conceptions of growth factors and tupling (e.g., doubling) periods as a foundational understanding for conceptualizing logarithms and logarithmic properties meaningfully. This paper characterizes conceptions that are productive for students to acquire in introductory lessons on exponential and logarithmic growth, and discusses two understandings that were revealed to be foundational for students’ development of productive meanings for exponents, logarithms and logarithmic properties.     

Asymptotic Properties for a Semilinear Edge-Degenerate Parabolic Equation

The author deals with a semi-linear edge-degenerate parabolic equation, and proves that the solution increases exponentially under the initial energy J(u_0) ≤ d, where d is the mountain-pass level. Moreover, the author estimates the blow-up time and the blow-up rate for the solution under J(u_0) 0.     

Exponential sums and rigid cohomology

In this article, we prove a comparison theorem between the Dwork cohomology introduced by Adolphson and Sperber and the rigid cohomology. As a corollary, we can calculate the rigid cohomology of Dwork isocrystal on torus.     

Numerical treatment of Casson nanofluid Bioconvectional flow with heat transfer due to stretching cylinder/plate: Variable physical properties

PurposeThe purpose of the current framework is to scrutinize the two-dimensional flow and heat transfer of Casson nanofluid over cylinder/plate along with impacts of thermophoresis and Brownian motion effects. Also, the effects of exponential thermal sink/source, bioconvection, and motile microorganisms are taken.Methodology/ApproachThe resulting non-linear equations (PDEs) are reformed into nonlinear ODEs by using appropriate similarity variables. The resultant non-linear (ODEs) were numerically evaluated by the use of the Bvp4c package in the mathematical solver MATLAB.FindingsThe numerical and graphical illustration regarding outcomes represents the performance of flow-involved physical parameters on velocity, temperature, concentration, and microorganism profiles. Additionally, the skin friction coefficient, local Nusselt number, local Sherwood number, and local microorganism density number are computed numerically for the current presented system. We noted that the velocity profile diminishes for the rising estimations of magnetic and mixed convection parameters. The Prandtl number corresponds with the declining performance of the temperature profile observed. The enhancement in the values of the Solutal Biot number and Brownian motion parameter increased in the concentration profile.OriginalityIn specific, this framework focuses on the rising heat transfer of Casson nanofluid with bioconvection by using a shooting mathematical model. The novel approach of the presented study is the use of motile microorganisms with exponential thermal sink/source in a Casson nano-fluid through a cylinder/plate. A presented study performed first time in the author’s opinion. Understanding the flow characteristics and behaviors of these nanofluids is crucial for the scientific community in the developing subject of nanofluids.     

Bound state solutions of D‐dimensional schrödinger equation with exponential‐type potentials

In this article, using an exactly‐solvable multiparameter exponential‐type potential we propose a unified treatment of the analytical bound—state solutions of the Schrödinger equation for exponential‐type potentials in D‐dimensions. Our proposal accepts different approximations to the centrifugal term; however, its usefulness is exemplified in the frame of the Green and Aldrich approach. This fact enables us to compare our results with specific potentials found in the literature and that are obtained here as particular cases of our proposal. That is, instead of solving a specific exponential‐type potential, by resorting each time to a specialized method, the energy spectra and wavefunctions are derived straightforward from the proposed approach. Furthermore, our proposal can be used as an alternative way in the search of solutions to new exponential‐type potentials besides that one can study different approximations to the term . © 2014 Wiley Periodicals, Inc.