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Theoretical and numerical analysis of unsteady fractional viscoelastic flows in simple geometries.In this work we discuss the connection between classical and fractional viscoelastic Maxwell models, presenting the basic theory supporting these constitutive equations, and establishing some background on the admissibility of the fractional Maxwell model. We then develop a numerical method for the solution of two coupled fractional differential equations (one for the velocity and the other for the stress), that appear in the pure tangential annular ow of fractional viscoelastic fluids. The numerical method is based on finite differences, with the approximation of fractional derivatives of the velocity and stress being inspired by the method proposed by Sun and Wu for the fractional diffusionwave equation [ Z.Z. Sun, X. Wu, A fully discrete difference scheme for a diffusionwave system, Applied Numerical Mathematics 56 (2006) 193209]. We prove solvability, study numerical convergence of the method, and also discuss the applicability of this method for simulating the rheological response of complex fluids in a real concentric cylinder rheometer. By imposing a torsional stepstrain, we observe the different rates of stress relaxation obtained with different values of \alpha and \beta (the fractional order exponents that regulate the viscoelastic response of the complex fluids).

Theory and numerics for multiterm periodic delay differential equations, small solutions and their detectionWe summarise a theoretical treatment that analyses whether the equation has small solutions. We consider discrete equations that arise when a numerical method with fixed step size is applied to approximate the solution to (†) and we develop a corresponding theory. Our results show that small solutions can be detected reliably by the numerical scheme. We conclude with some numerical examples.

A time discretization scheme for a nonlocal degenerate problem modelling resistance spot weldingIn the current work we construct a nonlocal mathematical model describing the phase transition occurs during the resistance spot welding process in the industry of metallurgy. We then consider a time discretization scheme for solving the resulting nonlocal moving boundary problem. The scheme consists of solving at each time step a linear elliptic partial differential equation and then making a correction to account for the nonlinearity. The stability and error estimates of the developed scheme are investigated. Finally some numerical results are presented confirming the efficiency of the developed numerical algorithm.

Torsion Units for a Ree group, Tits group and a Steinberg triality groupWe investigate the Zassenhaus conjecture for the Steinberg triality group ${}^3D_4(2^3)$, Tits group ${}^2F_4(2)'$ and the Ree group ${}^2F_4(2)$. Consequently, we prove that the Prime Graph question is true for all three groups.

Torsion Units for Some Almost Simple GroupsWe prove that the Zassenhaus conjecture is true for $Aut(PSL(2,11))$. Additionally we prove that the Prime graph question is true for $Aut(PSL(2,13))$.

Torsion Units for some Projected Special Linear GroupsIn this paper, we investigate the Zassenhaus conjecture for $PSL(4,3)$ and $PSL(5,2)$. Consequently, we prove that the Prime graph question is true for both groups.

Torsion units for some untwisted exceptional groups of lie typeIn this paper, we investigate the Zassenhaus conjecture for exceptional groups of lie type $G_2(q)$ for $q=\{3,4\}$. Consequently, we prove that the Prime graph question is true for these groups.

Torsion units in the integral group ring of PSL(3,4)We investigate the Zassenhaus Conjecture for the integral group ring of the simple group PSL(3,4).

Two highorder time discretization schemes for subdiffusion problems with nonsmooth dataTwo new highorder time discretization schemes for solving subdiffusion problems with nonsmooth data are developed based on the corrections of the existing time discretization schemes in literature. Without the corrections, the schemes have only a first order of accuracy for both smooth and nonsmooth data. After correcting some starting steps and some weights of the schemes, the optimal convergence orders $O(k^{3 \alpha})$ and $O(k^{4 \alpha})$ with $0< \alpha <1$ can be restored for any fixed time $t$ for both smooth and nonsmooth data, respectively. The error estimates for these two new highorder schemes are proved by using Laplace transform method for both homogeneous and inhomogeneous problem. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

Underwhelming the immune response: Effect of slow virus growth on CD8+Tlymphocyte responsesThe speed of virus replication has typically been seen as an advantage for a virus in overcoming the ability of the immune system to control its population growth. Under some circumstances, the converse may also be true: more slowly replicating viruses may evoke weaker cellular immune responses and therefore enhance their likelihood of persistence. Using the model of lymphocytic choriomeningitis virus (LCMV) infection in mice, we provide evidence that slowly replicating strains induce weaker cytotoxicTlymphocyte (CTL) responses than a more rapidly replicating strain. Conceptually, we show a "bellshaped" relationship between the LCMV growth rate and the peak CTL response. Quantitative analysis of human hepatitis C virus infections suggests that a reduction in virus growth rate between patients during the incubation period is associated with a spectrum of disease outcomes, from fulminant hepatitis at the highest rate of viral replication through acute resolving to chronic persistence at the lowest rate. A mathematical model for virusCTL population dynamics (analogous to predator [CTL]prey [virus] interactions) is applied in the clinical datadriven analysis of acute hepatitis B virus infection. The speed of viral replication, through its stimulus of host CTL responses, represents an important factor influencing the pathogenesis and duration of virus persistence within the human host. Viruses with lower growth rates may persist in the host because they "sneak through" immune surveillance.

Units of the group algebra of the group $C_n\times D_6$ over any finite field of characteristic $3$In this paper, we establish the structure of the unit group of the group algebra ${\FF}_{3^t}(C_n\times D_6)$ for $n \geq 1$.

Using approximations to Lyapunov exponents to predict changes in dynamical behaviour in numerical solutions to stochastic delay differential equationsThis book chapter explores the parameter values at which there are changes in qualitative behaviour of the numerical solutions to parameterdependent linear stochastic delay differential equations with multiplicative noise. A possible tool in this analysis is the calculation of the approximate local Lyapunov exponents. We show that estimates for the maximal local Lyapunov exponent have predictable distributions dependent upon the parameter values and the fixed step length of the numerical method, and that changes in the qualitative behaviour of the solutions occur at parameter values that depend on the step length.

Volterra integral equations and fractional calculus: Do neighbouring solutions intersect?This journal article considers the question of whether or not the solutions to two Volterra integral equations which have the same kernel but different forcing terms may intersect at some future time.