General Mathematical Framework Research Articles

Elastostatics within multi-layer metamaterial structures and an algebraic framework for polariton resonances

Multi-layer structures are ubiquitous in constructing metamaterial devices to realise various frontier applications including super-resolution imaging and invisibility cloaking. In this paper, we develop a general mathematical framework for studying elastostatics within multi-layer material structures in $\mathbb{R}^d$, $d=2,3$. The multi-layer structure is formed by concentric balls and each layer is filled by either a regular elastic material or an elastic metamaterial. The number of layers can be arbitrary and the material parameters in each layer may be different from one another. In practice, the multi-layer structure can serve as the building block for various material devices. Considering the impingement of an incident field on the multi-layer structure, we first derive the exact perturbed field in terms of an elastic momentum matrix, whose dimension is the same as the number of layers. By highly intricate and delicate analysis, we derive a comprehensive study of the spectral properties of the elastic momentum matrix. This enables us to establish a handy algebraic framework for studying polariton resonances associated with multi-layer metamaterial structures, which forms the fundamental basis for many metamaterial applications.

On variable-order fractional linear viscoelasticity

AbstractA generalization of fractional linear viscoelasticity based on Scarpi’s approach to variable-order fractional calculus is presented. After reviewing the general mathematical framework, a variable-order fractional Maxwell model is analysed as a prototypical example for the theory. Some physical considerations are then provided concerning the fractionalisation procedure and the choice of the transition functions. Lastly, the material functions for the considered model are derived and numerically evaluated for exponential-type and Mittag-Leffler-type order functions.

A mathematical framework for evo-devo dynamics

Natural selection acts on phenotypes constructed over development, which raises the question of how development affects evolution. Classic evolutionary theory indicates that development affects evolution by modulating the genetic covariation upon which selection acts, thus affecting genetic constraints. However, whether genetic constraints are relative, thus diverting adaptation from the direction of steepest fitness ascent, or absolute, thus blocking adaptation in certain directions, remains uncertain. This limits understanding of long-term evolution of developmentally constructed phenotypes. Here we formulate a general, tractable mathematical framework that integrates age progression, explicit development (i.e., the construction of the phenotype across life subject to developmental constraints), and evolutionary dynamics, thus describing the evolutionary and developmental (evo-devo) dynamics. The framework yields simple equations that can be arranged in a layered structure that we call the evo-devo process, whereby five core elementary components generate all equations including those mechanistically describing genetic covariation and the evo-devo dynamics. The framework recovers evolutionary dynamic equations in gradient form and describes the evolution of genetic covariation from the evolution of genotype, phenotype, environment, and mutational covariation. This shows that genotypic and phenotypic evolution must be followed simultaneously to yield a dynamically sufficient description of long-term phenotypic evolution in gradient form, such that evolution described as the climbing of a fitness landscape occurs in “geno-phenotype” space. Genetic constraints in geno-phenotype space are necessarily absolute because the phenotype is related to the genotype by development. Thus, the long-term evolutionary dynamics of developed phenotypes is strongly non-standard: (1) evolutionary equilibria are either absent or infinite in number and depend on genetic covariation and hence on development; (2) developmental constraints determine the admissible evolutionary path and hence which evolutionary equilibria are admissible; and (3) evolutionary outcomes occur at admissible evolutionary equilibria, which do not generally occur at fitness landscape peaks in geno-phenotype space, but at peaks in the admissible evolutionary path where “total genotypic selection” vanishes if exogenous plastic response vanishes and mutational variation exists in all directions of genotype space. Hence, selection and development jointly define the evolutionary outcomes if absolute mutational constraints and exogenous plastic response are absent, rather than the outcomes being defined only by selection. Moreover, our framework provides formulas for the sensitivities of a recurrence and an alternative method to dynamic optimization (i.e., dynamic programming or optimal control) to identify evolutionary outcomes in models with developmentally dynamic traits. These results show that development has major evolutionary effects.

A framework for fair decision-making over time with time-invariant utilities

Fairness is a major concern in contemporary decision problems. In these situations, the objective is to maximize fairness while preserving the efficacy of the underlying decision-making problem. This paper examines repeated decisions on problems involving multiple stakeholders and a central decision maker. Repetition of the decision-making provides additional opportunities to promote fairness while increasing the complexity from symmetry to finding solutions. This paper presents a general mathematical programming framework for the proposed fairness-over-time (FOT) decision-making problem. The framework includes a natural abstraction of how a stakeholder’s acquired utilities can be aggregated over time. In contrast with a natural, descriptive formulation, we demonstrate that if the aggregation function possesses certain basic properties, a strong reformulation can be written to remove symmetry from the problem, making it amenable to branch-and-cut solvers. Finally, we propose a particular relaxation of this reformulation that can assist in the construction of high-quality approximate solutions to the original problem and can be solved using simultaneous row and column generation techniques.

A Fast Online Planning Under Partial Observability Using Information Entropy Rewards

Motion planning in an unknown environment is a common challenge because of the existing uncertainties. Representatively, the partially observable Markov decision process (POMDP) is a general mathematical framework for planning in uncertain environments. Recent POMDP solvers generally adopt the sparse reward scheme to solve the planning under uncertainty problem. Subsequently, the robot's exploration may be hindered without immediate rewards, resulting in excessively long planning time. In this article, a POMDP method information entropy determinized sparse partially observation tree (IE-DESPOT) is proposed to explore a high-quality solution and efficient planning in unknown environments. First, a novel sample method integrating state distribution and Gaussian distribution is proposed to optimize the quality of the sampled states. Then, an information entropy based on sampled states is established for real-time reward calculation, resulting in the improvement of robot exploration efficiency. Moreover, the near-optimality and convergence of the proposed algorithm are analyzed. As a result, compared with general-purpose POMDP solvers, the proposed algorithm exhibits fast convergence to a near-optimal policy in many examples of interest. Furthermore, the IE-DESPOT's performance is verified in real mobile robot experiments.

Structured population models on Polish spaces: A unified approach including graphs, Riemannian manifolds and measure spaces to describe dynamics of heterogeneous populations

This paper presents a mathematical framework for modeling the dynamics of heterogeneous populations. Models describing local and non-local growth and transport processes appear in a variety of applications, such as crowd dynamics, tissue regeneration, cancer development and coagulation-fragmentation processes. The diverse applications pose a common challenge to mathematicians due to the multiscale nature of the structures that underlie the system’s self-organization and control. Similar abstract mathematical problems arise when formulating problems in the language of measure evolution on a multi-faceted state space. Motivated by these observations, we propose a general mathematical framework for nonlinear structured population models on abstract metric spaces, which are only assumed to be separable and complete. We exploit the structure of the space of non-negative Radon measures with the dual bounded Lipschitz distance (flat metric), which is a generalization of the Wasserstein distance, capable of addressing non-conservative problems. The formulation of models on general metric spaces allows considering infinite-dimensional state spaces or graphs and coupling discrete and continuous state transitions. This opens up exciting possibilities for modeling single-cell data, crowd dynamics or coagulation-fragmentation processes.

Reduced basis stochastic Galerkin methods for partial differential equations with random inputs

We present a reduced basis stochastic Galerkin method for partial differential equations with random inputs. In this method, the reduced basis methodology is integrated into the stochastic Galerkin method, resulting in a significant reduction in the cost of solving the Galerkin system. To reduce the main cost of matrix-vector manipulation involved in our reduced basis stochastic Galerkin approach, the secant method is applied to identify the number of reduced basis functions. We present a general mathematical framework of the methodology, validate its accuracy and demonstrate its efficiency with numerical experiments.

Reduced Lagrange multiplier approach for non-matching coupling of mixed-dimensional domains

Many physical problems involving heterogeneous spatial scales, such as the flow through fractured porous media, the study of fiber-reinforced materials, or the modeling of the small circulation in living tissues — just to mention a few examples — can be described as coupled partial differential equations defined in domains of heterogeneous dimensions that are embedded into each other. This formulation is a consequence of geometric model reduction techniques that transform the original problems defined in complex three-dimensional domains into more tractable ones. The definition and the approximation of coupling operators suitable for this class of problems is still a challenge. We develop a general mathematical framework for the analysis and the approximation of partial differential equations coupled by non-matching constraints across different dimensions, focusing on their enforcement using Lagrange multipliers. In this context, we address in abstract and general terms the well-posedness, stability, and robustness of the problem with respect to the smallest characteristic length of the embedded domain. We also address the numerical approximation of the problem and we discuss the infsup stability of the proposed numerical scheme for some representative configuration of the embedded domain. The main message of this work is twofold: from the standpoint of the theory of mixed-dimensional problems, we provide general and abstract mathematical tools to formulate coupled problems across dimensions. From the practical standpoint of the numerical approximation, we show the interplay between the mesh characteristic size, the dimension of the Lagrange multiplier space, and the size of the inclusion in representative configurations interesting for applications. The latter analysis is complemented with illustrative numerical examples.

Distribution modeling quantifies collective TH cell decision circuits in chronic inflammation.

Immune responses are tightly regulated by a diverse set of interacting immune cell populations. Alongside decision-making processes such as differentiation into specific effector cell types, immune cells initiate proliferation at the beginning of an inflammation, forming two layers of complexity. Here, we developed a general mathematical framework for the data-driven analysis of collective immune cell dynamics. We identified qualitative and quantitative properties of generic network motifs, and we specified differentiation dynamics by analysis of kinetic transcriptome data. Furthermore, we derived a specific, data-driven mathematical model for T helper 1 versus T follicular helper cell-fate decision dynamics in acute and chronic lymphocytic choriomeningitis virus infections in mice. The model recapitulates important dynamical properties without model fitting and solely by using measured response-time distributions. Model simulations predict different windows of opportunity for perturbation in acute and chronic infection scenarios, with potential implications for optimization of targeted immunotherapy.

Continuous-Spectrum-Polarization Recombinant Optical Encryption with a Dielectric Metasurface.

The Jones matrix, with eight degrees of freedom (DoFs), provides a general mathematical framework for the multifunctional design of metasurfaces. Theoretically, the maximum eight DoFs can be further extended in the spectrum dimension to endow unique encryption capabilities. However, the topology and intrinsic spectral responses of meta-atoms constrains the continuous engineering of polarization evolution over wavelength dimension. In this work, a forward evolution strategy to quickly establish the mapping relationships between the solutions of the dispersion Jones matrix and the spectral responses of meta-atoms is reported. Based on the eigenvector transformation method, arbitrary conjugate polarization channels over the continuous-spectrum dimension are successfully reconstructed. As a proof-of-concept, a silicon metadevice is demonstrated for optical information encryption transmission. Remarkably, the arbitrary combination forms of polarization and wavelength dimension increase the information capacity (210 ), and the measured polarization contrasts of the conjugate polarization conversion are >94% in the entire wavelength range (3-4µm). It is believed that the proposed approach will benefit secure optical and quantum information technologies.

Emergent information dynamics in many-body interconnected systems.

The information implicitly represented in the state of physical systems allows for their analysis using analytical techniques from statistical mechanics and information theory. This approach has been successfully applied to complex networks, including biophysical systems such as virus-host protein-protein interactions and whole-brain models in health and disease, drawing inspiration from quantum statistical physics. Here we propose a general mathematical framework for modeling information dynamics on complex networks, where the internal node states are vector valued, allowing each node to carry multiple types of information. This setup is relevant for various biophysical and sociotechnological models of complex systems, ranging from viral dynamics on networks to models of opinion dynamics and social contagion. Instead of focusing on node-node interactions, we shift our attention to the flow of information between network configurations. We uncover fundamental differences between widely used spin models on networks, such as voter and kinetic dynamics, which cannot be detected through classical node-based analysis. We illustrate the mathematical framework further through an exemplary application to epidemic spreading on a low-dimensional network. Our model provides an opportunity to adapt powerful analytical methods from quantum many-body systems to study the interplay between structure and dynamics in interconnected systems.

A Unified Switched Nonlinear Dynamic Model of an Electric Vehicle for Performance Evaluation

The advanced modeling and estimation of overall system dynamics play a vital role in electric vehicles (EVs), as the selection of components in the powertrain and prediction of performance are the key market qualifiers. The state-space averaged model and small-signal transfer function model are useful for assessing long-term behavior in system-level analysis and for designing the controller parameters, respectively. Both models take less computation time but ignore the high-frequency switching dynamics. Therefore, these two models could be impractical for the development and testing of EV prototypes. On the other hand, the multi-domain model in available simulation tools gives in-depth information about the short-term behavior and loss analysis of power electronic devices in each subsystem, considering the switching dynamics in a long computation time. In this paper, a general mathematical framework for the dynamical analysis of complete EVs is presented using a unified, switched nonlinear model. This equation-based model runs faster than the available module-based simulation tools. Two other models, namely the time domain state-space averaged model and frequency domain small-signal transfer function model, are also developed from the switched nonlinear model for the analysis with less computation time. The design and performance of an EV with two different motors and its controllers are evaluated using the general mathematical framework.

Metaheuristics: A Review of Algorithms

In science and engineering, many optimization tasks are difficult to solve, and the core concern these days is to apply metaheuristic (MH) algorithms to solve them. Metaheuristics have gained significant attention in recent years, with nature serving as the fundamental inspiration where self-organization property led to collective intelligence emerging from the behavior of a swarm of birds or colony of insects or more and more natural behavior. These swarms or colonies, even with extremely low individual competence, have the ability to accomplish many complicated activities that can be considered necessary for their existence. Accordingly, many MH algorithms have been developed based on natural phenomena. In this article, an analysis review of more than one hundred metaheuristics have been made. Further, the main contributions of this article are to give some vital insights about metaheuristics, presenting and proposing the general mathematical framework of MH algorithms and dividing it into a number of tasks with possible progress for each task. While there are still many open issues in this field, it is worth noting that there have been significant advancements in recent years. As a result, new algorithms are continuously being proposed to address these challenges.

NOMA Assisted Semi-Grant-Free Scheme for Scheduling Multiple Grant-Free Users

Nonorthogonal multiple access (NOMA) assisted semi-grant-free (SGF) transmission is a powerful solution to significantly reduce the end-to-end latency of data transmission and improve spectrum utilization efficiency facing the future Internet of Everything (IoE). Moreover, the recently proposed hybrid successive interference cancellation (HSIC) scheme can avoid the user's outage probability error floor, which is unavoidable in traditional uplink NOMA. However, the existing NOMA assisted SGF transmission schemes using this HSIC scheme admit only one grant-free (GF) user into the grant-based (GB) user's channel at a time. In order to further improve the throughput and reduce the waiting delay of GF users, in this paper, we propose a NOMA assisted SGF scheme with a new HSIC scheme allowing a certain number of GF users to share GB user's channel. First, we propose a new HSIC scheme to decode the signals of multiple GF users and one GB user. Then based on the signal's decoding order at the BS and stochastic geometry, we provide a general mathematical framework to analyze the association success probability of GF users. Besides, in order to intuitively show the throughput improvement brought by multiple admitted GF users, we derive the analytical expressions of admitted GF users' association success probability, outage probability, and throughput when two GF users are allowed to share simultaneously GB user's channel. Finally, the simulation results validate the throughput improvement brought by two GF users being served simultaneously and the accuracy of the theoretical analyses.

Conclusive Discrimination by $$N$$ Sequential Receivers between $$r\geq2$$ Arbitrary Quantum States

In the present paper, we develop a general mathematical framework for discrimination between $$r\geq2$$ quantum states by $$N\geq1$$ sequential receivers for the case in which every receiver obtains a conclusive result. This type of discrimination constitutes an $$N$$ -sequential extension of the minimum-error discrimination by one receiver. The developed general framework, which is valid for a conclusive discrimination between any number $$r\geq2$$ of quantum states, pure or mixed, of an arbitrary dimension and any number $$N\geq1$$ of sequential receivers, is based on the notion of a quantum state instrument, and this allows us to derive new important general results. In particular, we find a general condition on $$r\geq2$$ quantum states under which, within the strategy in which all types of receivers’ quantum measurements are allowed, the optimal success probability of the $$N$$ -sequential conclusive discrimination between these $$r\geq2$$ states is equal to that of the first receiver for any number $$N\geq2$$ of further sequential receivers and specify the corresponding optimal protocol. Furthermore, we extend our general framework to include an $$N$$ -sequential conclusive discrimination between $$r\geq2$$ arbitrary quantum states under a noisy communication. As an example, we analyze analytically and numerically a two-sequential conclusive discrimination between two qubit states via depolarizing quantum channels. The derived new general results are important both from the theoretical point of view and for the development of a successful multipartite quantum communication via noisy quantum channels.

How asymmetric mating patterns affect the rate of neutral genetic substitution

IntroductionA population under neutral drift is expected to accumulate genetic substitutions at a fixed “molecular clock” rate over time. If the population is well-mixed, a classic result equates the rate of substitution per generation to the probability of mutation per birth. However, this substitution rate can be altered if individual birth and death rates vary by class or by spatial location.MethodsHere we investigate how mating patterns affect the rate of neutral genetic substitution in a diploid, sexually reproducing population. We employ a general mathematical modeling framework that allows for arbitrary mating pattern and spatial structure.ResultsWe demonstrate that if survival rates and mating opportunities vary systematically across individuals, the rate of neutral substitution can be either accelerated or slowed. In particular, this can occur in populations with uneven sex ratio at birth, or with reproductive skew.DiscussionOur results suggest that estimates of the rate of neutral substitution, in species with uneven sex ratio and/or reproductive skew, may need to take asymmetries in mating opportunity and survival into account.

Steady-State Indeterminacy in Lossless Switched-Mode Power Converters

In this article, it is shown for the first time that lossless switched-mode power converters may not possess a unique steady-state solution. Rather, they can exhibit an inherent <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">indeterminacy</i> in their steady-state behavior, and a unique solution is only found when losses are included in the analysis. Even more interestingly, it is shown that lossless converters of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">odd order</i> never possess a unique steady-state solution. Indeterminate converters can be intrinsically sensitive to parasitic resistances and other nonideal effects, a phenomenon that practically manifests itself in the form of undesirable voltages or circulating currents possessing, in general, both dc and ac components. This article first sets a general mathematical framework for a systematic assessment of steady-state indeterminacy. This is subsequently linked to the geometry of the state vector periodic motion in the converter's <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -dimensional state space. On the basis of the developed theory, odd-order converters are shown to always be affected by the indeterminacy issue. Besides inherent theoretical importance in the power electronics field, the results of this article provide a deeper justification for the observed behavior of several converter topologies of significant practical relevance, the most notable being multiphase and multilevel dc–dc converters, examples of which are discussed as case studies.

The fitness value of ecological information in a variable world.

Information processing is increasingly recognized as a fundamental component of life in variable environments, including the evolved use of environmental cues, biomolecular networks, and social learning. Despite this, ecology lacks a quantitative framework for understanding how population, community, and ecosystem dynamics depend on information processing. Here, we review the rationale and evidence for 'fitness value of information' (FVOI), and synthesize theoretical work in ecology, information theory, and probability behind this general mathematical framework. The FVOI quantifies how species' per capita population growth rates can depend on the use of information in their environment. FVOI is a breakthrough approach to linking information processing and ecological and evolutionary outcomes in a changing environment, addressing longstanding questions about how information mediates the effects of environmental change and species interactions.

A General Mathematical Framework for Constrained Mixed-variable Blackbox Optimization Problems with Meta and Categorical Variables

A mathematical framework for modelling constrained mixed-variable optimization problems is presented in a blackbox optimization context. The framework introduces a new notation and allows solution strategies. The notation framework allows meta and categorical variables to be explicitly and efficiently modelled, which facilitates the solution of such problems. The new term meta variables is used to describe variables that influence which variables are included or excluded: meta variables may affect the number of variables and constraints. The flexibility of the solution strategies supports the main blackbox mixed-variable optimization approaches: direct search methods and surrogate-based methods (Bayesian optimization). The notation system and solution strategies are illustrated through an example of a hyperparameter optimization problem from the machine learning community.

A scheduling framework for performing resource slicing with guarantees in 6G RIS-enabled smart radio environments

Smart Radio Environments (SRE) transform the wireless propagation phenomenon in a programmable process. Leveraging multiple Reconfigurable Intelligent Surfaces (RIS), the wireless waves emitted by a device can be almost freely routed and manipulated, reaching their end destination via improbable paths, with minimized fading and path losses. This work begins with the observation that each such wireless communication customization occupies a certain number of RIS units, e.g., to form a wireless path with consecutive customized reflections. Therefore, SREs can be modeled as a resource of constrained capacity, which needs to be sliced among interested clients. This work provides a foundational model of SRE-as-a-resource, defining Service Level Agreements (SLAs) and Service Level Objectives (SLOs) for the SRE client requests. Employing this model, we study a class of negative drift dynamic weighted round robin policies, that is able to guarantee specific SRE resource shares to competing user requests. We provide a general mathematical framework where the class of policies map ping user requests to resources does not require statistical knowledge regarding the arrival distribution or the duration of each user communication. We study the meaning of work conserving and non-work conserving modes of SRE operation, and also study the convergence properties of our scheduling framework for both cases. Finally, we perform the feasibility space analysis for our framework and we validate our analysis through extensive simulations.