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2^n Bordered Constructions of SelfDual codes from Group RingsSelfdual codes, which are codes that are equal to their orthogonal, are a widely studied family of codes. Various techniques involving circulant matrices and matrices from group rings have been used to construct such codes. Moreover, families of rings have been used, together with a Gray map, to construct binary selfdual codes. In this paper, we introduce a new bordered construction over group rings for selfdual codes by combining many of the previously used techniques. The purpose of this is to construct selfdual codes that were missed using classical construction techniques by constructing selfdual codes with diﬀerent automorphism groups. We apply the technique to codes over ﬁnite commutative Frobenius rings of characteristic 2 and several group rings and use these to construct interesting binary selfdual codes. In particular, we construct some extremal selfdual codes length 64 and 68, constructing 30 new extremal selfdual codes of length 68.

A Comprehensive Review of the Composition, Nutritional Value, and Functional Properties of Camel Milk FatRecently, camel milk (CM) has been considered as a healthpromoting icon due to its medicinal and nutritional benefits. CM fat globule membrane has numerous healthpromoting properties, such as antiadhesion and antibacterial properties, which are suitable for people who are allergic to cow’s milk. CM contains milk fat globules with a small size, which accounts for their rapid digestion. Moreover, it also comprises lower amounts of cholesterol and saturated fatty acids concurrent with higher levels of essential fatty acids than cow milk, with an improved lipid profile manifested by reducing cholesterol levels in the blood. In addition, it is rich in phospholipids, especially plasmalogens and sphingomyelin, suggesting that CM fat may meet the daily nutritional requirements of adults and infants. Thus, CM and its dairy products have become more attractive for consumers. In view of this, we performed a comprehensive review of CM fat’s composition and nutritional properties. The overall goal is to increase knowledge related to CM fat characteristics and modify its unfavorable perception. Future studies are expected to be directed toward a better understanding of CM fat, which appears to be promising in the design and formulation of new products with significant healthpromoting benefits.

A Novel Averaging Principle Provides Insights in the Impact of Intratumoral Heterogeneity on Tumor ProgressionTypically stochastic differential equations (SDEs) involve an additive or multiplicative noise term. Here, we are interested in stochastic differential equations for which the white noise is nonlinearly integrated into the corresponding evolution term, typically termed as random ordinary differential equations (RODEs). The classical averaging methods fail to treat such RODEs. Therefore, we introduce a novel averaging method appropriate to be applied to a specific class of RODEs. To exemplify the importance of our method, we apply it to an important biomedical problem, in particular, we implement the method to the assessment of intratumoral heterogeneity impact on tumor dynamics. Precisely, we model gliomas according to a wellknown Go or Grow (GoG) model, and tumor heterogeneity is modeled as a stochastic process. It has been shown that the corresponding deterministic GoG model exhibits an emerging Allee effect (bistability). In contrast, we analytically and computationally show that the introduction of white noise, as a model of intratumoral heterogeneity, leads to monostable tumor growth. This monostability behavior is also derived even when spatial cell diffusion is taken into account.

Addendum to the article: On the Dirichlet to Neumann Problem for the 1dimensional Cubic NLS Equation on the HalfLineWe present a short note on the extension of the results of [1] to the case of nonzero initial data. More specifically, the defocusing cubic NLS equation is considered on the halfline with decaying (in time) Dirichlet data and sufficiently smooth and decaying (in space) initial data. We prove that for this case also, and for a large class of decaying Dirichlet data, the Neumann data are sufficiently decaying so that the Fokas unified method for the solution of defocusing NLS is applicable.

An algorithm for the numerical solution of twosided spacefractional partial differential equations.We introduce an algorithm for solving twosided spacefractional partial differential equations. The spacefractional derivatives we consider here are lefthanded and righthanded Riemann–Liouville fractional derivatives which are expressed by using Hadamard finitepart integrals. We approximate the Hadamard finitepart integrals by using piecewise quadratic interpolation polynomials and obtain a numerical approximation of the spacefractional derivative with convergence order

An algorithm to detect small solutions in linear delay differential equationsThis preprint discusses an algorithm that provides a simple reliable mechanism for the detection of small solutions in linear delay differential equations.

Algorithms for the fractional calculus: A selection of numerical methodsThis article discusses how numerical algorithms can help engineers work with fractional models in an efficient way.

An Altered Four Circulant Construction for SelfDual Codes from Group Rings and New Extremal Binary Selfdual Codes IWe introduce an altered version of the four circulant construction over group rings for selfdual codes. We consider this construction over the binary field, the rings F2 + uF2 and F4 + uF4; using groups of order 4 and 8. Through these constructions and their extensions, we find binary selfdual codes of lengths 16, 32, 48, 64 and 68, many of which are extremal. In particular, we find forty new extremal binary selfdual codes of length 68, including twelve new codes with \gamma=5 in W68,2, which is the first instance of such a value in the literature.

An analysis of the L1 scheme for stochastic subdiffusion problem driven by integrated spacetime white noiseWe consider the strong convergence of the numerical methods for solving stochastic subdiffusion problem driven by an integrated spacetime white noise. The time fractional derivative is approximated by using the L1 scheme and the time fractional integral is approximated with the Lubich's first order convolution quadrature formula. We use the Euler method to approximate the noise in time and use the truncated series to approximate the noise in space. The spatial variable is discretized by using the linear finite element method. Applying the idea in Gunzburger \et (Math. Comp. 88(2019), pp. 17151741), we express the approximate solutions of the fully discrete scheme by the convolution of the piecewise constant function and the inverse Laplace transform of the resolvent related function. Based on such convolution expressions of the approximate solutions, we obtain the optimal convergence orders of the fully discrete scheme in spatial multidimensional cases by using the Laplace transform method and the corresponding resolvent estimates.

An analysis of the modified L1 scheme for timefractional partial differential equations with nonsmooth dataWe introduce a modified L1 scheme for solving time fractional partial differential equations and obtain error estimates for smooth and nonsmooth initial data in both homogeneous and inhomogeneous cases. Jin \et (2016, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA J. of Numer. Anal., 36, 197221) established an $O(k)$ convergence rate for the L1 scheme for smooth and nonsmooth initial data for the homogeneous problem, where $k$ denotes the time step size. We show that the modified L1 scheme has convergence rate $O(k^{2\alpha}), 0< \alpha <1$ for smooth and nonsmooth initial data in both homogeneous and inhomogeneous cases. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

Analysis of transient RivlinEricksen fluid and irreversibility of exothermic reactive hydromagnetic variable viscosityThe study analysed unsteady RivlinEricksen fluid and irreversibility of exponentially temperature dependent variable viscosity of hydromagnetic twostep exothermic chemical reactive flow along the channel axis with walls convective cooling. The nonNewtonian HeleShaw flow of RivlinErickson fluid is driven by bimolecular chemical kinetic and unvarying pressure gradient. The reactive fluid is induced by periodic changes in magnetic field and time. The Newtons law of cooling is satisfied by the constant heat coolant convection exchange at the wall surfaces with the neighboring regime. The dimensionless nonNewtonian reactive fluid equations are numerically solved using a convergent and consistence semiimplicit finite difference technique which are confirmed stable. The response of the reactive fluid flow to variational increase in the values of some entrenched fluid parameters in the momentum and energy balance equations are obtained. A satisfying equations for the ratio of irreversibility, entropy generation and Bejan number are solved with the results presented graphically and discussed quantitatively. From the study, it was obtained that the thermal criticality conditions with the right combination of thermofluid parameters, the thermal runaway can be prevented. Also, the entropy generation can minimize by at low dissipation rate and viscosity.

An analytic approach to the normalized Ricci flowlike equation: RevisitedIn this paper we revisit Hamilton’s normalized Ricci flow, which was thoroughly studied via a PDE approach in Kavallaris and Suzuki (2010). Here we provide an improved convergence result compared to the one presented Kavallaris and Suzuki (2010) for the critical case λ=8πλ=8π. We actually prove that the convergence towards the stationary normalized Ricci flow is realized through any time sequence.

Analytical and numerical investigation of mixedtype functional differential equationsThis journal article is concerned with the approximate solution of a linear nonautonomous functional differential equation, with both advanced and delayed arguments.

Analytical and numerical treatment of oscillatory mixed differential equations with differentiable delays and advancesThis article discusses the oscillatory behaviour of the differential equation of mixed type.

An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth dataIn this paper, we shall review an approach by which we can seek higher order time discretisation schemes for solving time fractional partial differential equations with nonsmooth data. The low regularity of the solutions of time fractional partial differential equations implies standard time discretisation schemes only yield first order accuracy. To obtain higher order time discretisation schemes when the solutions of time fractional partial differential equations have low regularities, one may correct the starting steps of the standard time discretisation schemes to capture the singularities of the solutions. We will consider these corrections of some higher order time discretisation schemes obtained by using Lubich's fractional multistep methods, L1 scheme and its modification, discontinuous Galerkin methods, etc. Numerical examples are given to show that the theoretical results are consistent with the numerical results.

Bifurcations in numerical methods for volterra integrodifferential equationsThis article discusses changes in bifurcations in the solutions. It extends the work of Brunner and Lambert and Matthys to consider other bifurcations.

Binary selfdual and LCD codes from generator matrices constructed from two group ring elements by a heuristic search scheme<p style='textindent:20px;'>We present a generator matrix of the form <inlineformula><texmath id="M1">\begin{document}$ [ \sigma(v_1) \  \ \sigma(v_2)] $\end{document}</texmath></inlineformula>, where <inlineformula><texmath id="M2">\begin{document}$ v_1 \in RG $\end{document}</texmath></inlineformula> and <inlineformula><texmath id="M3">\begin{document}$ v_2\in RH $\end{document}</texmath></inlineformula>, for finite groups <inlineformula><texmath id="M4">\begin{document}$ G $\end{document}</texmath></inlineformula> and <inlineformula><texmath id="M5">\begin{document}$ H $\end{document}</texmath></inlineformula> of order <inlineformula><texmath id="M6">\begin{document}$ n $\end{document}</texmath></inlineformula> for constructing selfdual codes and linear complementary dual codes over the finite Frobenius ring <inlineformula><texmath id="M7">\begin{document}$ R $\end{document}</texmath></inlineformula>. In general, many of the constructions to produce selfdual codes forces the code to be an ideal in a group ring which implies that the code has a rich automorphism group. Unlike the traditional cases, codes constructed from the generator matrix presented here are not ideals in a group ring, which enables us to find selfdual and linear complementary dual codes that are not found using more traditional techniques. In addition to that, by using this construction, we improve <inlineformula><texmath id="M8">\begin{document}$ 10 $\end{document}</texmath></inlineformula> of the previously known lower bounds on the largest minimum weights of binary linear complementary dual codes for some lengths and dimensions. We also obtain <inlineformula><texmath id="M9">\begin{document}$ 82 $\end{document}</texmath></inlineformula> new binary linear complementary dual codes, <inlineformula><texmath id="M10">\begin{document}$ 50 $\end{document}</texmath></inlineformula> of which are either optimal or near optimal of lengths <inlineformula><texmath id="M11">\begin{document}$ 41 \leq n \leq 61 $\end{document}</texmath></inlineformula> which are new to the literature.</p>