Discrete Chain Research Articles

Chaotic Discrete Fractional-Order Food Chain Model and Hybrid Image Encryption Scheme Application

Using the discrete fractional calculus, a novel discrete fractional-order food chain model for the case of strong pressure on preys map is proposed. Dynamical behaviors of the model involving stability analysis of its equilibrium points, bifurcation diagrams and phase portraits are investigated. It is demonstrated that the model can exhibit a variety of dynamical behaviors including stable steady states, periodic and quasiperiodic dynamics. Then, a hybrid encryption scheme based on chaotic behavior of the model along with elliptic curve key exchange scheme is proposed for colored plain images. The hybrid scheme combines the characteristics of noise-like chaotic dynamics of the map, including high sensitivity to values of parameters, with the advantages of reliable elliptic curves-based encryption systems. Security analysis assures the efficiency of the proposed algorithm and validates its robustness and efficiency against possible types of attacks.

Modeling and Validation of the Hospital’s Ambulatory and Inpatients Operations Using a Non-Homogenous Discrete Time Markovian Chains

Overcrowded hospitals in its different process's levels is a very common issue all over the globe. Research through different works have studied this problem from different angles. Most of those focused on a very specific part of the internal flows. This study presents a mathematical model aiming to reflect patient flow and resources usage. This model presents a new perspective on how it is possible to organize a hospital flow for inpatients of different pathologies. It is based on Non-Homogenous Discrete Time Markovian chains. It takes into consideration the patient's pathology, survival function and the current beds distribution and discharges. To model the different variations of a patient's flow, our work uses an actual case study to validate that our mathematical model based on Markov Chains can represent a real scenario of patients flow in a hospital and can predict the resources occupancy. We used as entry to this model the arrival distribution of patients as represented in the NHS dataset. Served patients and overflowing patients relate proportionally to admission rate based on how much time every diagnosis and preliminary treatment will take, then using the previous beds availability ration will permit, with knowledge of all present patients in the system, calculate the beds occupancy in the actual time. Only based on inputs from the NHS database, bed occupancy can be calculated using our model. Then compare this calculus with published data of the bed's occupancy in the NHS. in the period between March 2020 and February 2021. The results of the simulation model were then compared to the dataset using the chi-square goodness of fit test. Non-homogenous discrete Markovian model, survival function, patient's pathology, time-dependent, pathology, primary allocations, secondary allocations.

High-dimensional MCMC with a standard splitting scheme for the underdamped Langevin diffusion.

The efficiency of a Markov sampler based on the underdamped Langevin diffusion is studied for high dimensional targets with convex and smooth potentials. We consider a classical second-order integrator which requires only one gradient computation per iteration. Contrary to previous works on similar samplers, a dimension-free contraction of Wasserstein distances and convergence rate for the total variance distance are proven for the discrete time chain itself. Non-asymptotic Wasserstein and total variation efficiency bounds and concentration inequalities are obtained for both the Metropolis adjusted and unadjusted chains. In particular, for the unadjusted chain, in terms of the dimension d and the desired accuracy ε, the Wasserstein efficiency bounds are of order d∕ε in the general case, d∕ε if the Hessian of the potential is Lipschitz, and d1∕4∕ε in the case of a separable target, in accordance with known results for other kinetic Langevin or HMC schemes.

Microscopic theory for elastic modulus of colloidal polymers: Effect of bond length

Colloidal polymers have attracted increasing attention in condensed physics, statistical mechanics and polymer science and engineering due to their advances in synthesis and visualization. Many useful properties and applications of colloidal polymers make them an ideal model to explore fundamental problems in slow dynamics and rheology of chain-like molecules in supercooled regime. With temperature decreasing or density rapidly increasing, amorphous materials often exhibit nonzero shear moduli. In this article, we are to investigate the nonzero shear modulus and bulk modulus of colloidal polymer in supercooled regime based on recent microscopic theoretical development. At the segmental-scale level, an analytical derivation for elastic modulus of colloidal polymer is constructed based on the standard approximation in naïve mode-coupling theory (NMCT). In the framework of nonlinear Langevin equation theory (NLET), the derivation combines the concept of dynamic free energy, localization and NMCT crossover volume fraction. Taking the chain connectivity into account, an explicit expression for shear modulus including intrachain structure factor, interchain correlation and localized length is formulated. Bulk modulus can be obtained by relating it to long wavelength part of static structure factor. Firstly, our calculation for discrete wormlike chain shows that intrachain structure factor has a power law decay at intermediate wavevector which is similar to flexible linear chain. Secondly, we find that colloidal polymer with long bond length has a lower NMCT crossover volume fraction. Furthermore, inputting the localized length, long wavelength density fluctuation and static intrachain and interchain structures into the theoretical expression, the effect of bond length on shear modulus and bulk modulus are investigated. Interestingly, we find the bond length plays a critical and unique role in localized length and bulk modulus. For instance, when the supercooling degree is used as an independent variable, the local length and bulk elastic modulus of the chain with the same bond length can be collapsed onto a universal curve, which is independent of chain length and local bending energy. However, in the aspect of shear modulus, the bond length is not a unique quantity and the above universal curve cannot be found. The shear modulus depends on other parameters of chain, such as chain length and rigidity. According to the universal behavior of zero-wavevector static structure factor versus bond length, we guess that the nonuniversal curve of shear modulus is due to the bond length effect on long wavevector static structure factor. This work provides a theoretical foundation for controlling various properties of chain-like supercooled materials in the future.

Discrete magnetic breathers in monoaxial chiral helimagnet

We analyze spatially localized breather excitations for the model of a discrete Heisenberg spin chain, which includes the antisymmetric exchange interaction and single-ion anisotropy of the easy-plane type. In a finite size chain the breather modes may be indexed by number of embedded kink-antikink pairs forming the regular breather lattice. The influence of the antisymmetric exchange on properties of the discrete breather modes is examined.

A Convergent Structure-Preserving Finite-Volume Scheme for the Shigesada--Kawasaki--Teramoto Population System

An implicit Euler finite-volume scheme for an n-species population cross-diffusion system of Shigesada-Kawasaki-Teramoto-type in a bounded domain with no-flux boundary conditions is proposed and analyzed. The scheme preserves the formal gradient-flow or entropy structure and preserves the nonnegativity of the population densities. The key idea is to consider a suitable mean of the mobilities in such a way that a discrete chain rule is fulfilled and a discrete analog of the entropy inequality holds. The existence of finite-volume solutions, the convergence of the scheme, and the large-time asymptotics to the constant steady state are proven. Furthermore, numerical experiments in one and two space dimensiona for two and three species are presented. The results are valid for a more general class of cross-diffusion systems satisfying some structural conditions.

Discrete Giant Polymeric Chains Based on Nanosized Monomers.

As size-amplified analogues of canonical macromolecules, polymeric chains built up by “giant” monomers represent an experimental realization of the “beads-on-a-string” model at larger length scales, which could provide insights into fundamental principles of polymer science. In this work, we modularly constructed discrete giant polymeric chains using nanosized building blocks (polyhedral oligomeric silsesquioxane, POSS) as basic repeat units through an efficient and robust iterative exponential growth approach, with precise control on molecular parameters, including size, composition, regioconfiguration, and surface functionalities. Their chemical structures were fully characterized by nuclear magnetic resonance spectroscopy, size-exclusion chromatography, and matrix-assisted laser desorption ionization time-of-flight mass spectrometry. With elaborately designed amphiphilic block POSS chains and their analogues made of conventional monomers, the size effects were delicately studied and highlighted. Interesting assembly behaviors emerge as a result of distinct interactions and molecular dynamics. This category of molecules shares general self-assembly characteristics as the conventional counterparts in terms of phase transition and evolution. Meanwhile, it turns out that the monomer size has profound impacts on phase stability, as a trade-off between entropic and enthalpic contributions. It may open up a door for modular and programmable design of interesting materials with complex structures and diverse functions.

Exact solutions for the vibration of finite granular beam using discrete and gradient elasticity cosserat models

The present study theoretically investigates the free vibration problem of a discrete granular system. This problem can be considered as a simple model to rigorously study the effects of the microstructure on the dynamic behavior of the equivalent continuum structural model. The model consists of uniform grains confined by discrete elastic interactions, to take into account the lateral granular contributions. This repetitive discrete system can be referred to discrete Cosserat chain or a lattice elastic model with shear interaction. First for the simply supported granular beam resting on Winkler foundations, due to the critical frequencies which concern the nature of the dynamic results, the natural frequencies are exactly calculated, starting from the resolution of the linear difference eigenvalue problem. The natural frequencies of such a granular model are analytically calculated for whatever modes. It is shown that the difference equations governed to the discrete system converge to the differential equations of the Bresse-Timoshenko beam resting on Winkler foundation (also classified as a continuous Cosserat beam model) for an infinite number of grains. A gradient Bresse-Timoshenko model is constructed from continualization of the difference equations. This continuous gradient elasticity Cosserat model is obtained from a polynomial or a rational expansion of the pseudo-differential operators, stemming from the continualization process. Scale effects of the granular chain are captured by the continuous gradient elasticity model. The natural frequencies of the continuous gradient Cosserat models are compared with those of the discrete Cosserat model associated with the granular chain. The results clarify the dependency of the beam dynamic responses to the beam length ratio.

Revisiting the Θ Point

Using the first-order perturbation theory, we compute the osmotic second and third virial coefficients, the mean-square end-to-end distance ⟨Re2⟩, and the mean-square radius of gyration ⟨Rg2⟩ of a polymer near the Θ point. Our model is based on the discrete Gaussian chain model and includes a square-gradient term accounting for the finite-range interaction (characterized by κ), in addition to the usual monomer second and third virial coefficients (characterized by v and w, respectively). The use of the discrete model avoids the divergence problems encountered in previous studies using the continuous model. Our study identifies four special temperatures in the Θ regime: the temperature ΘN where the osmotic second virial coefficient vanishes, the critical temperature ΘNcr for phase separation, and two compensation temperatures ΘN(e) and ΘN(g) at which ⟨Re2⟩ and ⟨Rg2⟩ reach their respective ideal values. In the infinite chain-length limit N → ∞, all of these four temperatures approach Θ∞, the Θ temperature for the infinitely long chain. These temperatures differ from each other by terms of order N–1/2. In general, these temperatures follow the order ΘN > ΘNcr and ΘN > ΘN(e) > ΘN(g). Furthermore, ΘN > Θ∞, in agreement with the result obtained by Khokhlov some time ago. On the other hand, depending on the ratio w/κb, Θ∞ can be higher than ΘN(e) (for w/κb < 9.45), lower than ΘN(g) (for w/κb > 11.63), or in between ΘN(e) and ΘN(g) (for 9.45 < w/κb < 11.63). ΘNcr can be either higher or lower than Θ∞ depending on whether w/b6 is larger or smaller than 0.574. From the order of these temperatures, we conclude that the chain is mostly expanded relative to the ideal chain at its ΘN. However, at Θ∞, the chain can be either expanded or contracted, depending on the relative position of Θ∞ with respect to ΘN(e) and ΘN(g) and depending on whether the chain dimension is measured by ⟨Re2⟩ or ⟨Rg2⟩.

Physics-Based Computational Protein Design: An Update.

We describe methods for physics-based protein design and some recent applications from our work. We present the physical interpretation of a MC simulation in sequence space and show that sequences and conformations form a well-defined statistical ensemble, explored with Monte Carlo and Boltzmann sampling. The folded state energy combines molecular mechanics for solutes with continuum electrostatics for solvent. We usually assume one or a few fixed protein backbone structures and discrete side chain rotamers. Methods based on molecular dynamics, which introduce additional backbone and side chain flexibility, are under development. The redesign of a PDZ domain and an aminoacyl-tRNA synthetase enzyme were successful. We describe a versatile, adaptive, Wang-Landau MC method that can be used to design for substrate affinity, catalytic rate, catalytic efficiency, or the specificity of these properties. The methods are transferable to all biomolecules, can be systematically improved, and give physical insights.

Buckling of granular systems with discrete and gradient elasticity Cosserat continua

The stability of a granular column composed of a finite number of grains is investigated through an exact and some approximated continuum models. Shear and rotational interactions are taken into account at the rigid grain interfaces. This system can be considered as a discrete Cosserat chain with two independent degrees-of-freedom, namely the deflection and the rotation of each grain. The buckling of this discrete granular system on elastic foundation with translational and rotational stiffness (to account for some possible transversal grain interactions) is calculated whatever the number of grains. The formulation of the discrete boundary value problem is based on the exact resolution of a fourth-order linear difference equation. This solution is compared to the one of a continuous Cosserat chain asymptotically obtained for an infinite number of grains. In this last case, the asymptotic solution converges towards the one of a Bresse–Timoshenko beam under Winkler–Pasternak foundation. A more refined Cosserat continuum is built by continualization of the difference equations valid for the discrete Cosserat medium. It is shown that this more refine continuous model can be classified as a gradient elasticity Cosserat continuum, which is able to reproduce the scale effects observed for the buckling of the discrete granular system. These scale effects are related to the grain size, as compared to the structural length of the granular system. The key role played by the shear interaction in the instabilities of granular structural system is revealed, especially when the bending interaction can be neglected.

Bistable attachments for wideband nonlinear vibration attenuation in a metamaterial beam

This work investigates the amplitude-dependent dynamics of a locally resonant metamaterial beam with bistable attachments. The concept that was previously demonstrated for a discrete chain is extended to a continuous system, and the enhancement in vibration attenuation bandwidth is investigated through a cantilever beam under base excitation. The analysis approach combines the harmonic balance method and time-domain numerical integration to capture periodic and aperiodic responses for up-sweep and down-sweep harmonic excitation. The bistable attachments are shown to exhibit linear intrawell, nonlinear intrawell and nonlinear interwell oscillations for low, moderate, and high base acceleration levels. As a result, the metastructure leverages linear locally resonant bandgap under low excitation intensity, and nonlinear wideband attenuation due to chaotic vibrations of the bistable attachments under high excitation intensity. This is first demonstrated through frequency sweep numerical simulations for a broad range of excitation amplitudes. Experimental validations are then presented for a base-excited cantilever beam hosting seven magnetoelastic beam attachments. For moderate-to-high amplitude excitation levels, the interwell oscillations of the attachments produce an attenuation frequency range that is 350% wider than the corresponding linear locally resonant bandgap (observed for low-amplitude excitation levels), yielding the suppression of modes outside the bandgap with increased excitation intensity.

Вероятностные цепочки с полиномиальным ростом как модель распределения ресурсов

The article is devoted to the method of discrete probability chains for constructing the forecast of changes in socio-economic data. Initial data about the distribution of the resource among several participants are presented in the form of the probabilistic vector, and its changing over time is described by a discrete dynamical system which is specified by a certain function. Chains with linear and logarithmic-linear growth have been well studied. In this paper, we consider the probabilistic chains in which the right-hand side is given by polynomials of a certain type. The results of the construction are applied to the research of the dynamics of the distribution of the national income of Canada, Great Britain, and the United States. The accuracy of the results obtained is estimated by using the correlation coefficient, and the dynamics of the process modeled is estimated by using Shannon’s entropy.

Preservation effort effects on retailers and manufacturers in integrated multi-deteriorating item discrete supply chain model

This paper studies the multi-deteriorating item discrete supply chain to realize considerable savings by aggregating the replenishment. The present integrated replenishment policy has already been widely applied in a variety of industries. This study deals with an integrated multi-deteriorating item replenishment problem with preservation effort and discrete demand rate and discrete order quantity. Most existing studies about preservation effort focused on a single-item replenishment policy. However, integrated replenishment has been extensively applied in many industries to take advantage of economies of scale in preservation. Since it is difficult to solve this problem directly, the necessary and sufficient conditions with these properties are derived; a solution procedure and an algorithm using heuristic approach are developed to obtain the optimal solutions. Numerical examples, comparative analysis and sensitivity analyses are also provided and tested to elucidate the multi-deteriorating item discrete supply chain model with preservation efforts. The results reveal that the extensions of the model provide a wider and reasonable situation in practice, so that the annual channel profit can be maximized.

Combinatorial Selection Among Geometrical Isomers of Discrete Long-Carbon-Chain Naphthalenediimides Induces Local Order at the Liquid/Solid Interface.

We report two families of naphthalenediimides (NDIs) symmetrically functionalized with discrete carbon chains comprising up to 55 carbon atoms (Cn-NDI-Cn, n = 39, 44, 50, and 55) and their self-assembly at the 1-phenyloctane/highly oriented pyrolytic graphite interface (1-PO/HOPG interface). The compounds differ by the presence or absence of two or three internal double bonds in the carbon chains (unsaturated and saturated Cn-NDI-Cn, respectively). Combinatorial distributions of geometrical isomers displaying either the E- or Z-configuration at each double bond are obtained for the unsaturated compounds. Analysis of the self-assembled monolayers of equally long unsaturated and saturated Cn-NDI-Cn by scanning tunneling microscopy (STM) reveal that all Cn-NDI-Cn tend to form lamellar systems featuring alternating areas of aromatic cores and carbon chains. Extended chain lengths are found to significantly increase disorder in the self-assembled monolayers due to misalignments and enhanced strength of interchain interactions. This phenomenon is antagonized by the local order-inducing effect of the internal double bonds: unsaturated Cn-NDI-Cn give qualitatively more ordered self-assembled monolayers compared to their saturated counterparts. The use of combinatorial distributions of unsaturated Cn-NDI-Cn geometrical isomers does not represent a limitation to achieve local order in the self-assembled monolayers. The self-assembly process operates a combinatorial search and selects the geometrical isomer(s) affording the most thermodynamically stable pattern, highlighting the adaptive character of the system. Finally, the antagonistic interplay between the extended carbon chain lengths and the presence of internal double bonds brings to the discovery of the lamellar “phase C” morphology for unsaturated Cn-NDI-Cn with n ≥ 50.

INTERACTIONS BETWEEN CETYLTRIMETHYLAMMONIUM OR TETRAMETYLAMMMONIUM BROMIDE AND MCl2 OR MCl5 (M = Hg, Cu, Zn, Co, Ni, Cd and Nb) IN MONONUCLEAR COMPLEXES: SYNTHESIS AND INFRARED STUDY

On allowing cetyltrimethyl ammonium bromide (C19H42NBr) and HgCl2, CuCl2.2H2O, CoCl2.6H2O, ZnCl2, NiCl2.6H2O or NbCl5 in specific ratio in methanol, new mononuclear have been obtained and studied by infrared. The suggested structures are discrete, of infinite chain with octahedral or tetrahedral environments around the metallic centres. The MX2 behaves as discrete chains. For most of the structures, when intermolecular hydrogen bonds are considered, supramolecular architectures may be obtained.

Phase transformations in substrate-free dissipative multistable metamaterials

We show that propagating transitions fronts observed recently in multistable structural networks are analogous to solid–solid phase transformations in crystals and can therefore be described quantitatively as propagating shock fronts. We demonstrate that the well-established sharp-interface theory from shock physics agrees well with the exact and approximate wave solutions obtained from treating the multistable metamaterial as a discrete chain and as a homogenized continuum, respectively. We further discuss the energy transport that governs the underlying dynamic transition phenomenon. Through numerical examples we showcase the diverse nature of the achievable transition effects depending on the interplay between inertia and dissipation in the multistable network, which enables wave tailoring and guidance. We further confirm applicability of the theory by comparison to experimental data. Though focusing on one-dimensional transition front propagation as the most fundamental problem, our results and conclusions admit extension to higher dimensions.

On the Taylor Expansion of Value Functions

We introduce a framework for approximate dynamic programming that we apply to discrete time chains on $\mathbb{Z}_+^d$ with countable action sets. Our approach is grounded in the approximation of the (controlled) chain's generator by that of another Markov process. In simple terms, our approach stipulates applying a second-order Taylor expansion to the value function to replace the Bellman equation with one in continuous space and time where the transition matrix is reduced to its first and second moments. In some cases, the resulting equation (which we label {\bf TCP}) can be interpreted as corresponding to a Brownian control problem. When tractable, the TCP serves as a useful modeling tool. More generally, the TCP is a starting point for approximation algorithms. We develop bounds on the optimality gap---the sub-optimality introduced by using the control produced by the Taylored equation. These bounds can be viewed as a conceptual underpinning, analytical rather than relying on weak convergence arguments, for the good performance of controls derived from Brownian control problems. We prove that, under suitable conditions and for suitably large initial states, (i) the optimality gap is smaller than a $1-\alpha$ fraction of the optimal value, where $\alpha\in (0,1)$ is the discount factor, and (ii) the gap can be further expressed as the infinite horizon discounted value with a lower-order per period reward. Computationally, our framework leads to an aggregation approach with performance guarantees. While the guarantees are grounded in PDE theory, the practical use of this approach requires no knowledge of that theory.

Dynamics of the Buckling Transition in Double-Stranded DNA and RNA

DNA under torsional strain undergoes a buckling transition that is the fundamental step in plectoneme nucleation and supercoil dynamics, which are critical for the processing of genomic information. Despite its importance, quantitative models of the buckling transition, in particular to also explain the surprising two-orders-of-magnitude difference between the buckling times for RNA and DNA revealed by single-molecule tweezers experiments, are currently lacking. Additionally, little is known about the configurations of the DNA during the buckling transition because they are not directly observable experimentally. Here, we use a discrete worm-like chain model and Brownian dynamics to simulate the DNA/RNA buckling transition. Our simulations are in good agreement with experimentally determined parameters of the buckling transition. The simulations show that the buckling time strongly and exponentially depends on the bending stiffness, which accounts for more than half the measured difference between DNA and RNA. Analyzing the microscopic conformations of the chain revealed by our simulations, we find clear evidence for a solenoid-shaped transition state and a curl intermediate. The curl intermediate features a single loop and becomes increasingly populated at low forces. Taken together, the simulations suggest that the worm-like chain model can account semiquantitatively for the buckling dynamics of both DNA and RNA.

A new solution of the discrete catenary problem

We address the very old and famous catenary problem. However, instead of competing with a flexible and continuous chain, we consider a discrete chain made of N identical links. By using the elementary tools of Newtonian mechanics, we show that there is a simple relation between the angles of the links’ slope that enable us to find the N discrete chain solution. We also investigate the limit of the continuous chain and show that the discrete solution converges into the well-known classical solution of the catenary problem. The advantage of this new method is that it better enables the attainment of a physical insight into the problem in a quite simple way, without using advanced mathematics.