Main Mathematical Results Research Articles

ANALYSIS SEQUENTIAL FRACTIONAL DIFFERENCES AND RELATED MONOTONICITY RESULTS

This paper employs the monotonicity analysis for non-negativity to derive a class of sequential fractional backward differences of Riemann–Liouville type [Formula: see text] based on a certain subspace in the parameter space [Formula: see text]. Auxiliary and restriction conditions are included in the monotonicity results obtained in this paper and they confirm the monotonicity of the function on [Formula: see text]. A non-monotonicity result is also established based on the main conditions together with further dual conditions, and this confirms that the main theorem is almost sharp. Furthermore, we recast the dual conditions in a sing condition, and then we represent the sharpness result in a new corollary. Finally, numerical results via MATLAB software are used to illustrate the main mathematical results for some special cases.

The role of stabilization in the virtual element method: A survey

The virtual element method was introduced 10 years ago, and has generated a large number of theoretical results and applications ever since. Here, we overview the main mathematical results concerning the stabilization of the method as an introduction for newcomers in the field. In particular, we summarize the proofs of some results for two dimensional “nodal” conforming and nonconforming virtual element spaces to pinpoint the essential tools used in the stability analysis. We discuss their extensions to several other virtual elements. Finally, we show several ways to prove the interpolation estimates, including a recent one that is based on employing the stability bounds.

Nonlinear Einstein paradigm of Brownian motion and localization property of solutions

We employ a generalization of Einstein's random walk paradigm for diffusion to derive a class of multidimensional degenerate nonlinear parabolic equations in nondivergence form. Specifically, in these equations, the diffusion coefficient can depend on both the dependent variable and its gradient, and it vanishes when either one of the latter does. It is known that solutions of such degenerate equations can exhibit finite speed of propagation (so‐called localization property of solutions). We give a proof of this property using a De Giorgi–Ladyzhenskaya iteration procedure for nondivergence form equations. A mapping theorem is then established to a divergence‐form version of the governing equation for the case of one spatial dimension. Numerical results via a finite‐difference scheme are used to illustrate the main mathematical results for this special case. For completeness, we also provide an explicit construction of the one‐dimensional self‐similar solution with finite speed of propagation function, in the sense of Kompaneets–Zel'dovich–Barenblatt. We thus show how the finite speed of propagation quantitatively depends on the model's parameters.

Modeling the interaction of birds and small fish in a coastal lagoon

Abstract Coastal lagoons are high value productive and important systems for different projects. For example, aquaculture, fisheries and tourism are few of them. The quality of coastal waters in the ecosystems of lagoons can be greatly influenced by the growth of unwanted elements, e.g., excessive fisheries, tourism, etc. In this paper, a mathematical model is proposed and analysed to study the general and simplified form of an ecosystem of Chilika Lake, India. Chilika Lake (19°28′N–19°54′N and 85°06′E–85°36′E) is the largest wintering ground for migrating water fowl found anywhere on the Indian sub-continent. These migratory birds utilize the Chilika Lake for feeding, resting and breeding. The interaction of birds and small fish in the Chilika Lake is considered to be Leslie–Gower Holling type II. Since big fish are being sourced as income for local fishermen and the population of big fish is highly variable, and hence birds and small fishes are mainly the two types of biomass considered for this study. It must be noted that, in this study, we have considered the case of Chilika lake theoretically only and no practical data is collected for this study, and the name of Chilika is used only for better ecological understanding. Therefore, this theoretical study maybe linked to any such ecosystem. Their interaction is found mathematically, a two-dimensional continuous-time dynamical system modeling a simple predator–prey food chain. The dynamical system is represented in the form of two nonlinear coupled ordinary differential equation (ODE) systems. The main mathematical results are given in terms of boundedness of solutions, existence of equilibria, local and global stability of the coexisting interior point. An ecosystem in Indian coastal lagoons may suffer immediate environmental perturbations, such as depressions, tropical cyclones, earthquakes, epidemics, etc. To model such situations, the ODE model is further extended to a stochastic model driven by L e ́ $\check{d}{e}$ vy noise. The stochastic analysis includes the existence of the unique global solution, stability in mean, and extinction of the population. The proposed model is numerically simulated with the help of an assumed set of parameters for the possible pictorial behavior of the theoretical model. The proposed model may be used for planning purposes by using the data on meteorological and weather shocks such as heavy rainfall, heat-waves, cold-waves, depressions, tropical cyclones, earthquakes, etc. from India Meteorological Department (IMD).

Discreteness and integrality in Conformal Field Theory

Various observables in compact CFTs are required to obey positivity, discreteness, and integrality. Positivity forms the crux of the conformal bootstrap, but understanding of the abstract implications of discreteness and integrality for the space of CFTs is lacking. We systematically study these constraints in two-dimensional, non-holomorphic CFTs, making use of two main mathematical results. First, we prove a theorem constraining the behavior near the cusp of integral, vector-valued modular functions. Second, we explicitly construct non-factorizable, non-holomorphic cuspidal functions satisfying discreteness and integrality, and prove the non-existence of such functions once positivity is added. Application of these results yields several bootstrap-type bounds on OPE data of both rational and irrational CFTs, including some powerful bounds for theories with conformal manifolds, as well as insights into questions of spectral determinacy. We prove that in rational CFT, the spectrum of operator twists tge frac{c}{12} is uniquely determined by its complement. Likewise, we argue that in generic CFTs, the spectrum of operator dimensions Delta >frac{c-1}{12} is uniquely determined by its complement, absent fine-tuning in a sense we articulate. Finally, we discuss implications for black hole physics and the (non-)uniqueness of a possible ensemble interpretation of AdS3 gravity.

Michell Structures within L-Shaped Domains

By recalling the main mathematical results concerning the theory of Michell structures, the present paper puts forward an interpretation of the selected numerical methods for constructing their approximants, that is, trusses with a large number of nodes. The efficiency of one of these methods: the ground structure method in its adaptive version is shown in the context of the L-shaped design domain problem. A large family of highly accurate truss approximants corresponding to the point loads acting at selected vertices is constructed and discussed.

Regularized ab initio molecular force fields for key biological molecules: melatonin and pyridoxal-5′-phosphate methylamine Shiff base (Vitamin B6)

The main mathematical results on the data processing in vibrational spectroscopy are presented. The approaches and algorithms proposed for molecular force field calculations have been constructed on a base of regularizing methods for solving nonlinear ill-posed problems and have been implemented in the software package SPECTRUM. These algorithms were used for constructing the regularized ab initio force fields of important biological molecules including the melatonin and Vitamin B6 derivatives.

The Exponentiated Truncated Inverse Weibull-Generated Family of Distributions with Applications

In this paper, we propose a generalization of the so-called truncated inverse Weibull-generated family of distributions by the use of the power transform, adding a new shape parameter. We motivate this generalization by presenting theoretical and practical gains, both consequences of new flexible symmetric/asymmetric properties in a wide sense. Our main mathematical results are about stochastic ordering, uni/multimodality analysis, series expansions of crucial probability functions, probability weighted moments, raw and central moments, order statistics, and the maximum likelihood method. The special member of the family defined with the inverse Weibull distribution as baseline is highlighted. It constitutes a new four-parameter lifetime distribution which brightensby the multitude of different shapes of the corresponding probability density and hazard rate functions. Then, we use it for modelling purposes. In particular, a complete numerical study is performed, showing the efficiency of the corresponding maximum likelihood estimates by simulation work, and fitting three practical data sets, with fair comparison to six notable models of the literature.

The stability and Hopf bifurcation for an HIV model with saturated infection rate and double delays

An HIV infection model with saturated infection rate and double delays is investigated. First, the existence of the infection-free equilibrium [Formula: see text], the immune-exhausted equilibrium [Formula: see text] and the infected equilibrium [Formula: see text] with immunity in different conditions is shown. By analyzing the characteristic equation, we study the locally asymptotical stability of the trivial equilibrium, and the existence of Hopf bifurcations when two delays are used as the bifurcation parameter. Furthermore, we apply the Nyquist criterion to estimate the length of delay for which stability continues to hold. Then with suitable Lyapunov function and LaSalle’s invariance principle, the global stability of the three equilibriums is obtained. Finally, numerical simulations are presented to illustrate the main mathematical results.

Physico-mathematical modeling methods for the pressure distribution determination in the gas-dynamic bearing gap of the ball gyroscope

The considering problem in the article connects with solution methods of the specific questions in gas-dynamic lubrication theory. The comparative analysis of analytical and numerical methods for the calculating gas-dynamic bearing characteristics is provided. The main applying aspects of them for the solving of specific gas lubrication theory points are presented. This research is carried out for the investigated gap bearing geometry for the designed ball gyroscope construction. The main mathematical equation and results of developed numerical simulation for the pressure distribution determination are shown.

A History, the Main Mathematical Results and Applications for the Mathematics of Harmony

We give a survey on the history, the main mathematical results and applications of the Mathematics of Harmony as a new interdisciplinary direction of modern science. In its origins, this direction goes back to Euclid’s “Elements”. According to “Proclus hypothesis”, the main goal of Euclid was to create a full geometric theory of Platonic solids, associated with the ancient conception of the “Universe Harmony”. We consider the main periods in the development of the “Mathematics of Harmony” and its main mathematical results: algorithmic measurement theory, number systems with irrational bases and their applications in computer science, the hyperbolic Fibonacci functions, following from Binet’s formulas, and the hyperbolic Fibonacci l-functions (l = 1, 2, 3, …), following from Gazale’s formulas, and their applications for hyperbolic geometry, in particular, for the solution of Hilbert’s Fourth Problem.

An introduction to migration-selection PDE models

This expositoryarticle concerns a system of semilinear parabolic partial differential equations that describesthe evolution of the gene frequencies at a single locus under the joint actionof migration and selection.We shall review mathematical techniques suited for the models under investigation; discuss some of the main mathematical results, including most recent developments; and also propose some open problems.

A mathematical review of the analysis of the betaplane model and equatorial waves

This review paper is devoted to the presentation of recent progress in the mathematical analysis of equatorial waves. After a short presentation of the physical background, we present some of the main mathematical results related to the problem. <br>&nbsp; &nbsp More precisely we are interested in the study of the shallow water equations set in the vicinity of the equator: in that situation the Coriolis force vanishes and its linearization near zero leads to the so-called betaplane model. Our aim is to study the asymptotics of this model in the limit of small Rossby and Froude numbers. We show in a first part the existence and uniqueness of bounded (strong) solutions on a uniform time, and we study their weak limit. In a second part we give a more precise account of the asymptotics by characterizing the possible defects of compactness to that limit, in the framework of weak solutions only. <br>&nbsp; &nbsp These results are based on the studies [6]-[8] on the one hand, and [11] on the other.

Monotone and near-monotone biochemical networks.

Monotone subsystems have appealing properties as components of larger networks, since they exhibit robust dynamical stability and predictability of responses to perturbations. This suggests that natural biological systems may have evolved to be, if not monotone, at least close to monotone in the sense of being decomposable into a “small” number of monotone components, In addition, recent research has shown that much insight can be attained from decomposing networks into monotone subsystems and the analysis of the resulting interconnections using tools from control theory. This paper provides an expository introduction to monotone systems and their interconnections, describing the basic concepts and some of the main mathematical results in a largely informal fashion.

Stabilized methods and post-processing techniques for miscible displacements

Stabilized finite element formulations and post-processing techniques are presented for Darcy flow and miscible displacement problems. In these techniques all primary unknowns are approximated in space by standard equal-order interpolations. We presented a review of the main mathematical results, including convergence estimates for global post-processing to solve the pressure–velocity problem and for the stabilized solution of the complete set of coupled partial differential equations in both space–time and semi-discrete frameworks. Demonstration problems are solved to show that we obtained numerically high rates of convergence than those predicted, achieving good accuracy in the solution of monophasic flow problems in adverse conditions.

Asymptotics of heavy atoms in high magnetic fields: II. Semiclassical regions

The ground state energy of an atom of nuclear chargeZe in a magnetic fieldB is exactly evaluated to leading order asZ→∞ in the following three regions:B≪Z4/3,B≈Z4/3 andZ4/3≪B≪Z3. In each case this is accomplished by a modified Thomas-Fermi (TF) type theory. We also analyze these TF theories in detail, one of their consequences being the nonintuitive fact that atoms are spherical (to leading order) despite the leading order change in energy due to theB field. This paper complements and completes our earlier analysis [1], which was primarily devoted to the regionsB≈Z3 andB≫Z3 in which a semiclassical TF analysis is numerically and conceptually wrong. There are two main mathematical results in this paper, needed for the proof of the exactitude of the TF theories. One is a generalization of the Lieb-Thirring inequality for sums of eigenvalues to include magnetic fields. The second is a semiclassical asymptotic formula for sums of eigenvalues that isuniform in the fieldB.

A model of concurrency with fair merge and full recursion

We introduce in this paper a model for interaction and asynchronous, concurrent communication, where each agent's perception of the system is represented by a game of interaction. The model combines (strict) fair merge with full recursion, and the main mathematical results provide evidence for the robustness and naturalness of a novel interpretation of recursive definitions of non-deterministic processes. Our conceptual approach is closest to Park's, whose ideas are the starting points for this work.

Analysis of the activity of single neurons in stochastic settings.

This paper presents a new way of modeling the activity of single neurons in stochastic settings. It incorporates in a natural way many physiological mechanisms not usually found in stochastic models, such as spatial integration, non-linear membrane characteristics and non-linear interactions between excitation and inhibition. The model is based on the fact that most of the neuronal inputs have a finite lifetime. Thus, the stochastic input can be modeled as a simple finite markov chain, and the membrane potential becomes a function of the state of this chain. Firing occurs at states whose membrane potential is above threshold. The main mathematical results of the model are: (i) the input-output firing rate curve is convex at low firing rates and is saturated at high firing rates, and (ii) at low firing rates, firing usually occurs when there is synchronous convergence of many excitatory events.

Mathematical and numerical aspects of one-dimensional laminar flame simulation

The aim of this paper is to solve several mathematical and numerical questions related to the simulation of stationary and nonstationary premixed flat flames. Most of the results are obtained in the general context of complex chemical and diffusion mechanisms. The main mathematical results concern: (i) thea priori positivity of the mass fractions, and (ii) the sensitivity of the flame speed to the computational domain. The numerical method proposed for solving the stationary problem is a new combination of the pseudo-nonstationary approach, the Newton iterations, and the adaptive gridding. The computation of H2-O2-N2 flames with various initial concentrations (including the chemical extinction zone) shows the efficiency of this method.

Bifurcation in Nonlinear Hydrodynamic Stability

The appearance of secondary motions in a viscous fluid field can be understood to some extent as a bifurcation phenomenon with exchange of stability between the basic and the secondary flow. This article summarizes the main mathematical results of bifurcation and stability in hydrodynamic stability theory so far obtained. A unified functional-analytic approach is presented which tries to accentuate the ideas and to avoid technicalities. Besides the general results on the existence, the number of solutions and their qualitative behavior, the constructive analytical methods are emphasized. The Taylor and the Benard models are studied in detail. In the latter case, all possible solutions of regular cell pattern are classified. Stability and instability and their exchange at the point of bifurcation are studied.