New Type Of Equation Research Articles

Control of multi-agent systems: Results, open problems, and applications

Abstract The purpose of this review article is to present some recent results on the modeling and control of large systems of agents. We focus on particular applications where the agents are capable of independent actions instead of simply reacting to external forces. In the literature, such agents were referred to as autonomous, intelligent, self-propelled, greedy, and others. The main applications we have in mind are social systems (as opinion dynamics), pedestrians’ movements (also called crowd dynamics), animal groups, and vehicular traffic. We note that the last three examples include physical constraints; however, the agents are able to inject energy into the system, thus preventing the typical conservation of momentum and energy. In addition, the control problems posed by such systems are new and require innovative methods. We illustrate some ideas developed recently, including the use of sparse controls, limiting the total variation of controls, and defining new control problems for measures. After reviewing various approaches, we discuss some future research directions of potential interest. The latter encompasses both new types of equations and new types of limiting procedures to connect several scales at which a system can be represented. We conclude by illustrating a recent real-life experiment using autonomous vehicles on an open highway to smooth traffic waves. This opens the door to a new era of interventions to control real-time multi-agent systems and to increase the societal impact of such interventions guided by control research.

Applications of physics informed neural operators

We present a critical analysis of physics-informed neural operators (PINOs) to solve partial differential equations (PDEs) that are ubiquitous in the study and modeling of physics phenomena using carefully curated datasets. Further, we provide a benchmarking suite which can be used to evaluate PINOs in solving such problems. We first demonstrate that our methods reproduce the accuracy and performance of other neural operators published elsewhere in the literature to learn the 1D wave equation and the 1D Burgers equation. Thereafter, we apply our PINOs to learn new types of equations, including the 2D Burgers equation in the scalar, inviscid and vector types. Finally, we show that our approach is also applicable to learn the physics of the 2D linear and nonlinear shallow water equations, which involve three coupled PDEs. We release our artificial intelligence surrogates and scientific software to produce initial data and boundary conditions to study a broad range of physically motivated scenarios. We provide the source code, an interactive website to visualize the predictions of our PINOs, and a tutorial for their use at the Data and Learning Hub for Science.

(2010-6217) Solvability of fuzzy fractional stochastic Pantograph differential system

In this paper, a new type of equation namely fuzzy fractional stochastic Pantograph delay differential system (FSPDDS) is proposed. In our previous work, a first extension of fuzzy stochastic differential system into fuzzy fractional stochasticdifferential system by using Granular differentiability has been established. Here we study the existence and uniqueness results for the fuzzy FSPDDS which are obtained by using generalized Granular differentiability and contraction principlewith weaker conditions. This kind of equation is used in many real world problems. Finally, we provide two numerical examples for the effectiveness of the theoretical results.

A new type of equation of motion and numerical method for a harmonic oscillator with left and right fractional derivatives

The aim of this research is to propose a new fractional Euler-Lagrange equation for a harmonic oscillator. The theoretical analysis is given in order to derive the equation of motion in a fractional framework. The new equation has a complicated structure involving the left and right fractional derivatives of Caputo-Fabrizio type, so a new numerical method is developed in order to solve the above-mentioned equation effectively. As a result, we can see different asymptotic behaviors according to the flexibility provided by the use of the fractional calculus approach, a fact which may be invisible when we use the classical Lagrangian technique. This capability helps us to better understand the complex dynamics associated with natural phenomena.

Relationship between enzyme concentration and Michaelis constant in enzyme assays

The upper bound of enzyme concentration for accurately estimating the parameters in Michaelis-Menten (MM) equation is not completely determined and still under discussion, even though many researchers have investigated the equation’s validity for a long time. In the paper, we broadly investigated the correlation between the system of ordinary differential equations for monosubstrate irreversible enzyme reaction (HMM-system) and its derivative MM equation focusing on the relationship between initial enzyme concentration [E]0 and Michaelis constant Km by numerical simulation. According to the results, the initial reaction velocity v0 is still a function of initial substrate concentration [S]0 at [E]0<Km. The function is identical to the MM equation at [E]0≦0.01Km, while it is described as a new type of equation at 0.01Km≦[E]0<Km. This function is of great significance in enzyme assays as a comprehensive approximation for the HMM-system.

Monochromatic sums and products in $\mathbb{N}$

An old question in Ramsey theory asks whether any finite coloring of the natural numbers admits a monochromatic pair $\{x+y,xy\}$. We answer this question affirmatively in a strong sense by exhibiting a large new class of non-linear patterns which can be found in a single cell of any finite partition of $\mathbb{N}$. Our proof involves a correspondence principle which transfers the problem into the language of topological dynamics. As a corollary of our main theorem we obtain partition regularity for new types of equations, such as $x^2-y^2=z$ and $x^2+2y^2-3z^2=w$.

Longitudinal and Transverse Phonon Transport in Dielectric Crystals

Longitudinal and transverse modes of thermal disturbances in isotopically pure alkali-halide crystals are derived from phonon hydrodynamics. Guyer-Krumhansl (GK) model of phonon scattering is first recovered by relating the first and second viscosity to the relaxation times. Helmholtz potentials are then introduced to split the two modes from the heat flux vector. It has been found that the scalar potential coincides with the heat equation in the dual-phase-lag model, while the vector potential describes dispersive transverse phonons and introduces a new type of equation in microscale heat transfer.

Functional equations for Feynman integrals

New types of equations for Feynman integrals are found. It is shown that the latter satisfy functional equations that relate integrals with different kinematics. A regular method for obtaining such relations is proposed. A derivation of the functional equations for one-loop two-, three-, and four-point functions with arbitrary masses and external momenta is given. It is demonstrated that the functional equations can be used to analytically continue Feynman integrals to various kinematical domains.

Backward stochastic differential equations with time delayed generators—results and counterexamples

We deal with backward stochastic differential equations with time delayed generators. In this new type of equation, a generator at time t can depend on the values of a solution in the past, weighted with a time delay function, for instance, of the moving average type. We prove existence and uniqueness of a solution for a sufficiently small time horizon or for a sufficiently small Lipschitz constant of a generator. We give examples of BSDE with time delayed generators that have multiple solutions or that have no solutions. We show for some special class of generators that existence and uniqueness may still hold for an arbitrary time horizon and for arbitrary Lipschitz constant. This class includes linear time delayed generators which we study in more detail. We are concerned with different properties of a solution of a BSDE with time delayed generator, including the inheritance of boundedness from the terminal condition, the comparison principle, the existence of a measure solution and the BMO martingale property. We give examples in which they may fail.

On Malliavin’s differentiability of BSDEs with time delayed generators driven by Brownian motions and Poisson random measures

We investigate solutions of backward stochastic differential equations (BSDEs) with time delayed generators driven by Brownian motions and Poisson random measures, that constitute the two components of a Lévy process. In these new types of equations, the generator can depend on the past values of a solution, by feeding them back into the dynamics with a time lag. For such time delayed BSDEs, we prove the existence and uniqueness of solutions provided we restrict on a sufficiently small time horizon or the generator possesses a sufficiently small Lipschitz constant. We study differentiability in the variational or Malliavin sense and derive equations that are satisfied by the Malliavin gradient processes. On the chosen stochastic basis this addresses smoothness both with respect to the continuous part of our Lévy process in terms of the classical Malliavin derivative for Hilbert space valued random variables, as well as with respect to the pure jump component for which it takes the form of an increment quotient operator related to the Picard difference operator.

10.1007/s11497-008-2001-y

Wave equations make it possible to ascribe the quantum numbers to leptons theoretically. Minimal extension of the groups describing stable leptons led to three new types of groups. Each of them has an appropriate structure invariant ±1.0 and its own set of substructures having a physical interpretation in terms of stable leptons. The properties of the above-mentioned groups allow one to generate new types of equations and to relate them with two doublets (particle-antiparticle) of massive, charged unstable leptons and a doublet of massive unstable neutrinos.

Structure quantum numbers and unstable leptons

Wave equations make it possible to ascribe the quantum numbers to leptons theoretically. Minimal extension of the groups describing stable leptons led to three new types of groups. Each of them has an appropriate structure invariant ±1.0 and its own set of substructures having a physical interpretation in terms of stable leptons. The properties of the above-mentioned groups allow one to generate new types of equations and to relate them with two doublets (particle-antiparticle) of massive, charged unstable leptons and a doublet of massive unstable neutrinos.

The submodels of flow of gas with the linear field of speeds determined by matrixes of a rank 1

The equations of gas dynamics with the arbitrariest equation of state which solutions were searched in the form of the linear field of speeds are considered. Submodels of flow of gas with the linear field of speeds are as a result received and new types of equations of state are found.

Mathematical Simulation of Membrane Processes and Metabolic Fluxes of the Pancreatic β-cell

A new type of equation to describe enzyme-catalyzed reactions was developed, which allows the description of processes both at or near equilibrium and far from equilibrium, as they are both known to occur in the living cell. These equations combine kinetic as well as energetic characteristics within one single equation, and they describe the steady state as well as oscillations, as is shown for the glucose metabolism of the pancreatic beta-cell. A simulation of oxidative glucose metabolism could be elaborated, which allows to analyse in detail, how membrane and metabolic oscillations of the pancreatic beta-cell are generated, and how they are kinetically coupled. Glucose metabolism shows steady-state behaviour at a resting glucose concentration ([Glu]) of 4 mM. The steady state is switched to the oscillatory state by a first increase of the conductance of the glucokinase-catalyzed reaction at an elevated [Glu] of 10 mM. This is in fact sufficient to decrease the cytosolic adenosine diphosphate concentration ([ADP](c)) at constant intracellular [Ca(2+)]. The associated changes of the ATP and ADP species can reduce the conductance of ATP-sensitive K(+) channels (K(ATP)), thereby initiating bursts of the cell membrane potential (Delta(c)phi) with a concomitant influx of Ca(2+) ions from the extracellular space into the cell. The production of oscillations of [ADP](c), [Ca(2+)](c), and all other variables, including those of mitochondria, are brought about on the one hand by a [Ca(2+)](m) dependent activation of mitochondrial ATP production, on the other hand by a [Ca(2+)](c)-dependent activation of ATP utilisation in the cytosol. Both processes must be coordinated in such a way that ATP production slightly precedes its utilisation. Oscillatory frequencies (fast/slow) are determined by the conductance (high/low, respectively) of flux through pyruvate dehydrogenase and/or citric acid cycle. The simulation shows that the so-called pyruvate paradox possibly results from a relatively low membrane conductance of beta-cells for pyruvate.

A Grassmann integral equation

The present study introduces and investigates a new type of equation which is called Grassmann integral equation in analogy to integral equations studied in real analysis. A Grassmann integral equation is an equation which involves Grassmann (Berezin) integrations and which is to be obeyed by an unknown function over a (finite-dimensional) Grassmann algebra Gm (i.e., a sought after element of the Grassmann algebra Gm). A particular type of Grassmann integral equations is explicitly studied for certain low-dimensional Grassmann algebras. The choice of the equation under investigation is motivated by the effective action formalism of (lattice) quantum field theory. In a very general setting, for the Grassmann algebras G2n, n=2,3,4, the finite-dimensional analogues of the generating functionals of the Green functions are worked out explicitly by solving a coupled system of nonlinear matrix equations. Finally, by imposing the condition G[{Ψ̄},{Ψ}]=G0[{λΨ̄},{λΨ}]+const, 0&amp;lt;λ∈R (Ψ̄k, Ψk, k=1,…,n, are the generators of the Grassmann algebra G2n), between the finite-dimensional analogues G0 and G of the (“classical”) action and effective action functionals, respectively, a special Grassmann integral equation is being established and solved which also is equivalent to a coupled system of nonlinear matrix equations. If λ≠1, solutions to this Grassmann integral equation exist for n=2 (and consequently, also for any even value of n, specifically, for n=4) but not for n=3. If λ=1, the considered Grassmann integral equation (of course) has always a solution which corresponds to a Gaussian integral, but remarkably in the case n=4 a further solution is found which corresponds to a non-Gaussian integral. The investigation sheds light on the structures to be met for Grassmann algebras G2n with arbitrarily chosen n.

Chick growth in albatrosses: curve fitting with a twist

We present a new type of equation to describe the growth patterns of procellariiform seabirds and other species whose chicks characteristically lose mass towards the end of the rearing period. Our equation is based on the Gompertz curve; our principles are also applicable to logistic and von Bertalanffy curves. From our model, five coefficients can be derived to characterise the patterns of growth. These are: growth rate, peak mass and age at which it is attained, loss rate and an index describing the overall shape of the curve. We illustrate the use of this new equation with data collected, using automated weighing platforms, on six years of chick growth of Black-browed Diomedea melanophris and Grey-headed D. chrysostoma albatrosses at Bird Island, South Georgia. In comparison with Grey-headed Albatross, Black-browed Albatross chicks grow at a faster rate and to a higher peak mass; they also reach their peak mass at an earlier age, and lose mass at a faster rate in the mass recession period. However, in both species, chicks reached peak mass when 72% of the rearing period had elapsed; within species, only this did not vary between years. This new equation not only enables the period of mass recession to be incorporated into growth analysis, but, because it does not require assumptions about asymptotic mass, greatly facilitates inter-species comparisons.

Rooted maps on orientable surfaces, Riccati's equation and continued fractions

We present a new approach in the study of rooted maps without regard to genus. We prove the existence of a new type of equation for the generating series of these maps enumerated with respect to edges and vertices. This is Riccati's equation. It seems to be the first time that such a differential equation appears in the enumeration of rooted maps. Solving this equation leads to different closed forms of the studied generating series. The most interesting consequence is a development of this generating function in a very nice continued fraction leading to a new equation generalizing the well-known Dyck equation for rooted planar trees. In a second part, we also obtain a differential equation for the generating series of rooted trees regardless of the genus, with respect to edges. This also leads to a continued fraction for the generating series of rooted genus independent trees and to an unexpected relation between both previous generating series of trees and rooted maps.

A formal representation of process model equations

The lack of adequate support for the development of mathematical process models confines the application of model-based techniques in the design and operation of complex chemical processes. To overcome this problem considerable effort has to be put into the development of knowledge-based software tools. The development of such tools requires a proper structuring of process models followed by a formalization of their representation. In this contribution, elementary modeling objects for the representation of the behavioral aspects of chemical processes are defined. The introduced methodology is based on ontological principles and general systems theory. Its formalization in terms of the object-oriented data model VeDa allows the formal representation of the mixed types of equations and formalisms arising in mathematical process models. It provides generality and extensibility since new types of equations or formalisms can be incorporated easily.

A formal representation of process model equations

The lack of adequate support for the development of mathematical process models confines the application of model—based techniques in the design and operation of complex chemical processes. To overcome this problem considerable effort has to be put in the development of knowledge—based software tools. The development of such tools requires a proper structuring of process models followed by a formalization of their representation. In this contribution, elementary modeling objects for the representation of the behavioral aspects of chemical processes are defined. The introduced methodology and its formalization in terms of the object—oriented data model VeDa allows the representation of the mixed types of equations and formalisms arising in mathematical process models. It provides generality and extensibility since new types of equations or formalisms can be incorporated easily.

Heat transfer by nucleate pool boiling—general correlation based on thermodynamic similarity

A ‘general’ equation for nucleate boiling heat transfer valid for various fluids is assumed to exist according to the hypothesis of thermodynamic similarity, modified by taking into account fluid specific parameters. The physical quantities are nondimensionalized in the new type of equation by fluid-specific scaling units being critical data or power products of critical data of the fluid. The equation and the fluid-specific reference values α 0 of the heat transfer coefficient presented by Gorenflo for nearly 50 fluids are the empirical basis, to which the new equation is fitted. A resulting ‘general’ heat transfer correlation for nucleate boiling represents the reference values α 0 of Gorenflo with a mean deviation of about 14%, depending on the number of fluid specific parameters taken into account for. The correlation is supposed to allow for estimating nucleate boiling heat transfer coefficients in poorly known fluids. Fluid properties at the actual boiling condition or specific reference values of the heat transfer coefficient are not required to apply the new type of correlation.