Nonlinear hyperbolic equations

The Einstein equations may safely be regarded as one of the highest triumphs of 20 th century physics. Through them , a deep and non-trivial connection is established between the curvature of spacetime and the matter and energy content of the universe: Rµν − 1 2 Rgµν = 8πGTµν These tensorial equations form the cornerstone of the theory of general relativity, much like Newton's F = mα does for Newtonian theory. One of the most fruitful strategies that were adopted to understand those equations was studying them through an initial value problem. This viewpoint culminated, in 1969, in the proof of the existence of a maximal globally hyperbolic development (MGHD) for suitable initial data. It is the purpose of this essay to discuss the meaning of the above sentence and provide the tools and theorems necessary to present its proof. The penultimate chapter of the essay focuses on a new proof that does away with an often frowned-upon characteristic of solutions to mathematical problems, namely Zorn's lemma.

We consider new implicit-explicit (IMEX) Runge-Kutta methods for hyperbolic systems of conservation laws with stiff relaxation terms. The explicit part is treated by a strong-stability-preserving (SSP) scheme, and the implicit part is treated by an L-stable diagonally implicit Runge-Kutta method (DIRK). The schemes proposed are asymptotic preserving (AP) in the zero relaxation limit. High accuracy in space is obtained by Weighted Essentially Non Oscillatory (WENO) reconstruction. After a description of the mathematical properties of the schemes, several applications will be presented.

Among diverse models that are used to describe and interpret the changes in global biodiversity through the
Phanerozoic, the exponential and logistic models (traditionally used in population biology) are the most popular.
As we have recently demonstrated (Markov, Korotayev, 2007), the growth of the Phanerozoic marine biodi-
versity at genus level correlates better with the hyperbolic model (widely used in demography and macrosociology).
Here we show that the hyperbolic model is also applicable to the Phanerozoic continental biota at genus
and family levels, and to the marine biota at species, genus, and family levels. There are many common features
in the evolutionary dynamics of the marine and continental biotas that imply similarity and common nature of
the factors and mechanisms underlying the hyperbolic growth. Both marine and continental biotas are characterized
by continuous growth of the mean longevity of taxa, by decreasing extinction and origination rates, by
similar pattern of replacement of dominant groups, by stepwise accumulation of evolutionary stable, adaptable
and “physiologically buffered” taxa with effective mechanisms of parental care, protection of early developmental
stages, etc.
At the beginning of the development of continental biota, the observed taxonomic diversity was substantially
lower than that predicted by the hyperbolic model. We suggest that this is due, firstly, to the fact that, during
the earliest stages of the continental biota evolution, the groups that are not preserved in the fossil record (such
as soil bacteria, unicellular algae, lichens, etc.) played a fundamental role, and secondly, to the fact that the continental
biota initially formed as a marginal portion of the marine biota, rather than a separate system. The hyperbolic
dynamics is most prominent when both marine and continental biotas are considered together. This
fact can be interpreted as a proof of the integrated nature of the biosphere. In the macrosociological models, the
hyperbolic pattern of the world population growth arises from a non-linear second-order positive feedback between
the demographic growth and technological development (more people – more potential inventors – faster
technological growth – the carrying capacity of the Earth grows faster – faster population growth – more people –
more potential inventors, and so on).
Based on the analogy with macrosociological models and diverse paleontological data, we suggest that the
hyperbolic character of biodiversity growth can be similarly accounted for by a non-linear second-order positive
feedback between the diversity growth and community structure complexity. The feedback can work via
two parallel mechanisms: 1) decreasing extinction rate (more taxa – higher alpha diversity, or mean number of
taxa in a community – communities become more complex and stable – extinction rate decreases – more taxa,
and so on) and 2) increasing origination rate (new taxa facilitate niche construction; newly formed niches can
be occupied by the next “generation” of taxa). The latter possibility makes the mechanisms underlying the hyperbolic
growth of biodiversity and human population even more similar, because the total ecospace of the biota
is analogous to the “carrying capacity of the Earth” in demography. As far as new species can increase
ecospace and facilitate opportunities for additional species entering the community, they are analogous to the
“inventors” of the demographic models whose inventions increase the carrying capacity of the Earth. The hyperbolic
growth of the Phanerozoic biodiverstiy suggests that “cooperative” interactions between taxa can play
an important role in evolution, along with generally accepted competitive interactions. Due to this “cooperation”,
the evolution of biodiversity acquires some features of a self-accelerating process. Macroevolutionary
“cooperation” reveals itself in: 1) increasing stability of communities that arises from alpha diversity growth;
2) ability of species to facilitate opportunities for additional species entering the community

First-order network flow models are coupled systems of differential equations which describe the build-up and dissipation of congestion along network road segments, known as link models. Models describing flows across network junctions, referred to as node models, play the role of the coupling between the link models and are responsible for capturing the propagation of traffic dynamics through the network. Node models are typically stated as optimization problems, so that the coupling between the link dynamics is not known explicitly. This renders network flow models analytically intractable. This paper examines the properties of node models for urban networks. Solutions to node models that are free of traffic holding, referred to as holding-free solutions , are formally defined and it is shown that flow maximization is only a sufficient condition for holding-free solutions. A simple greedy algorithm is shown to produce holding-free solutions while also respecting the invariance principle. Staging movements through nodes in a manner that prevents conflicting flows from proceeding through the nodes simultaneously is shown to simplify the node models considerably and promote unique solutions. The staging also models intersection capacities in a more realistic way by preventing unrealistically large flows when there is ample supply in the downstream and preventing artificial blocking when some of the downstream supplies are restricted.

This is a growing list of Universal Constants, Variations, and Identities I am compiling. Please continue to check this document, as it will be expanded through time.

Some of these may seem to you to be trivial or obvious, but there's more to them than what we normally expect. They are so simple and yet so profound. We may take them for granted or be oblivious to them! They are embedded so deep in the Universe that they may even seem silly or humorous to the uninitiated and/or the sceptical.

Please give them time to sink in. Then decide for yourself if they are to be ignored, rejected, or reviled.

Allow me time to develop this list and I'm sure you will agree that they comprise everything we know.

Prerequisite thinking leading to more general and complete knowledge representations:
● Awareness is primary and fundamental.
● All Awareness is non-dual unless it is dual.
● There is no inside without an outside nor outside without an inside.
● Duality is bounded, non-duality is boundless.
● Boundaries arise in a spectrum from diffuse to concise.
● Reality is composed of whole parts (Holons).
● All Holons have at least six fundamental capacities.
● Observation is communication between one or more participants.
● The ends are determined by their means.
● Knowledge is the meaning of meaning.
● Knowledge is what awareness does.
● Singular and plural arise together.
● Time is a temporally 'linear' (directed) form of change that is not limited by dimension. Time cannot be reified.
● Creation and discovery compliment each other and are the means in which the Universe fundamentally unfolds and enfolds itself.
● Interiority and Exteriority arise together.
● Dimension is a spectrum or domain of awareness: they essentially build an additional point of view or perspective.

We prove that the unique entropy solution to the macroscopic Lighthill-Witham-Richards model for traffic flow can be rigorously obtained as the large particle limit of the microscopic follow-the-leader model, which is interpreted as the discrete Lagrangian approximation of the former. More precisely, we prove that the empirical measure (respectively the discretised density) obtained from the follow-the-leader system converges in the 1-Wasserstein topology (respectively in L1 loc) to the unique entropy solution of the Lighthill-Witham-Richards equation in the Kruzkov sense. The initial data are taken in L infinity with compact support, hence we are able to handle densities with vacuum. Our result holds for a reasonably general class of velocity maps (including all the relevant examples in the applications) with possible degenerate slope near the vacuum state. The proof of the result is based on discrete BV estimates and on a discrete version of the one-sided Oleinik-type condition. In particular, we prove that the L infinity to BV regularizing effect is intrinsic of the discrete model.

This study applies evolutionary algorithm-based (EA-based) symbolic regression to assess the ability of metacognitive strategy use tested by the metacognitive awareness listening questionnaire (MALQ) and lexico-grammatical knowledge to predict listening comprehension proficiency among English learners. Initially, the psychometric validity of the MALQ subscales, the lexico-grammatical test, and the listening test was examined using the logistic Rasch model and the Rasch-Andrich rating scale model. Next, linear regression found both sets of predictors to have weak or inconclusive effects on listening comprehension; however, the results of EA-based symbolic regression suggested that both lexico-grammatical knowledge and two of the five metacognitive strategies tested predicted strongly and nonlinearly listening proficiency (R2 = .64). Constraining prediction modeling to linear relationships is argued to jeopardize the validity of language assessment studies, potentially leading these studies to inaccurately contradict otherwise well-established language assessment hypotheses and theories.

This paper presents the generalized differential quadrature (GDQ) simulation for analysis of a nanofluid
over a nonlinearly stretching sheet. The obtained governing equations of flow and heat transfer are
discretized by GDQ method and then are solved by Newton-Raphson method. The effects of stretching
parameter, Brownian motion number (Nb), Thermophoresis number (Nt) and Lewis number (Le), on the
concentration distribution and temperature distribution are evaluated. The obtained results exhibit that the
heat transfer rate can be controlled by choosing different nanoparticles and stretching parameter.

Keywords: noble gases shale gas two-phase flow migration–fractionation geochemical tracers geological carbon storage Environmental monitoring of shale gas production and geological carbon dioxide (CO 2) storage requires identification of subsurface gas sources. Noble gases provide a powerful tool to distinguish different sources if the modifications of the gas composition during transport can be accounted for. Despite the recognition of compositional changes due to gas migration in the subsurface, the interpretation of geochemical data relies largely on zero-dimensional mixing and fractionation models. Here we present two-phase flow column experiments that demonstrate these changes. Water containing a dissolved noble gas is displaced by gas comprised of CO 2 and argon. We observe a characteristic pattern of initial co-enrichment of noble gases from both phases in banks at the gas front, followed by a depletion of the dissolved noble gas. The enrichment of the co-injected noble gas is due to the dissolution of the more soluble major gas component, while the enrichment of the dissolved noble gas is due to stripping from the groundwater. These processes amount to chromatographic separations that occur during two-phase flow and can be predicted by the theory of gas injection. This theory provides a mechanistic basis for noble gas fractionation during gas migration and improves our ability to identify subsurface gas sources after post-genetic modification. Finally, we show that compositional changes due to two-phase flow can qualitatively explain the spatial compositional trends observed within the Bravo Dome natural CO 2 reservoir and some regional compositional trends observed in drinking water wells overlying the Marcellus and Barnett shale regions. In both cases, only the migration of a gas with constant source composition is required, rather than multi-stage mixing and fractionation models previously proposed.

In this paper, we present an analytical study, in the one space dimensional case, of the fluid dynamics system proposed in [4] to model the formation of biofilms. After showing the hyperbolicity of the system, we show that, in a open neighborhood of the physical parameters, the system is totally dissipative near its unique non vanishing equilibrium point. Using this property, we are able to prove existence and uniqueness of global smooth solutions to the Cauchy problem on the whole line for small perturbations of this equilibrium point and the solutions are shown to converge exponentially in time at the equilibrium state.

We develop a polygonal mesh simplification algorithm based on a novel analysis of the mesh geometry. Particularly, we propose first a characterization of vertices as hyperbolic or non-hyperbolic depend-ing upon their discrete local geometry. Subsequently, the simplification process computes a volume cost for each non-hyperbolic vertex, in anal-ogy with spherical volume, to capture the loss of fidelity if that vertex is decimated. Vertices of least volume cost are then successively deleted and the resulting holes re-triangulated using a method based on a novel heuristic. Preliminary experiments indicate a performance comparable to that of the best known mesh simplification algorithms.

In a paper [Appl. Math. Comput., 188 (2) (2007) 1587--1591], authors have suggested and analyzed a method for solving nonlinear equations. In the present work, we modified this method by using the finite difference scheme, which has a quintic convergence. We have compared this modified Halley method with some other iterative of fifth-orders convergence methods, which shows that this new method having convergence of fourth order, is efficient.

In this article, we study in details the fluid dynamics system proposed in Clarelli et al (2013) to model the formation of cyanobacteria biofilms. After analyzing the linear stability of the unique non trivial equilibrium of the system, we introduce in the model the influence of light and temperature, which are two important factors for the development of cyanobacteria biofilm. Since the values of the coefficients we use for our simulations are estimated through information found in the literature, some sensitivity and robustness analyses on these parameters are performed. All these elements enable us to control and to validate the model we have already derived and to present some numerical simulations in the 2D and the 3D cases.

In this paper, the performance of single-tone Radio over Fiber (RoF) system has been analyzed by employing different duobinary modulation formats. This single-tone RoF system has been modeled and analyzed using OptiSystem (14.0) software. To evaluate the transmission performance of RoF system, various performance metrics such as Q-factor, BER, and Eye Height are considered. Simulation results indicate that duobinary Hyperbolic-Secant pulse generator format with Single Drive Mach-Zehnder modulator provides better Q-factor and minimum BER as compared to existing modulation format in RoF system.

In this paper we study a semilinear hyperbolic-parabolic system modeling biological phenomena evolving on a network composed by oriented arcs. We prove the existence of global (in time) smooth solutions to this problem. The result is obtained by using energy estimates with suitable transmission conditions at nodes.

Decentralized intersection control techniques have received recent attention in the literature as means to overcome scalability issues associated with network-wide intersection control. Chief among these techniques are backpressure (BP) control algorithms, which were originally developed of for large wireless networks. In addition to being lightweight computationally, they come with guarantees of performance at the network level, specifically in terms of network-wide stability. The dynamics in backpressure control are represented using networks of point queues and this also applies to all of the applications to traffic control. As such, BP in traffic fail to capture the spatial distribution of vehicles along the intersection links and, consequently, spill-back dynamics. This paper derives a position weighted backpressure (PWBP) control policy for network traffic applying continuum modeling principles of traffic dynamics and thus capture the spatial distribution of vehicles along network roads and spill-back dynamics. PWBP inherits the computational advantages of traditional BP. To prove stability of PWBP, (i) a Lyapunov functional that captures the spatial distribution of vehicles is developed; (ii) the capacity region of the network is formally defined in the context of macroscopic network traffic; and (iii) it is proved, when exogenous arrival rates are within the capacity region, that PWBP control is network stabilizing. We conduct comparisons against a real-world adap-tive control implementation for an isolated intersection. Comparisons are also performed against other BP approaches in addition to optimized fixed timing control at the network level. These experiments demonstrate the superiority of PWBP over the other control policies in terms of capacity region, network-wide delay, congestion propagation speed, recov-erability from heavy congestion (outside of the capacity region), and response to incidents.

Motivated by geological carbon dioxide (CO2) storage, we present a vertical-equilibrium sharp-interface model for the migration of immiscible gravity currents with constant residual trapping in a two-dimensional confined aquifer. The residual acts as a loss term that reduces the current volume continuously. In the limit of a horizontal aquifer, the interface shape is self-similar at early and at late times. The spreading of the current and the decay of its volume are governed by power-laws. At early times the exponent of the scaling law is independent of the residual, but at late times it decreases with increasing loss. Owing to the self-similar nature of the current the volume does not become zero, and the current continues to spread. In the hyperbolic limit, the leading edge of the current is given by a rarefaction and the trailing edge by a shock. In the presence of residual trapping, the current volume is reduced to zero in finite time. Expressions for the up-dip migration distance and the final migration time are obtained. Comparison with numerical results shows that the hyperbolic limit is a good approximation for currents with large mobility ratios even far from the hyperbolic limit. In gently sloping aquifers, the current evolution is divided into an initial near-parabolic stage, with power-law decrease of volume, and a later near-hyperbolic stage, characterized by a rapid decay of the plume volume. Our results suggest that the efficient residual trapping in dipping aquifers may allow CO2 storage in aquifers lacking structural closure, if CO2 is injected far enough from the outcrop of the aquifer.

In this paper we model pedestrian flows evacuating a narrow corridor through an exit by a one-dimensional hyperbolic conservation law with a point constraint in the spirit of [Colombo and Goatin, J. Differential Equations, 2007]. We introduce a nonlocal constraint to restrict the flux at the exit to a maximum value p(ξ), where ξ is the weighted averaged instantaneous density of the crowd in an upstream vicinity of the exit. Choosing a non-increasing constraint function p(⋅), we are able to model the capacity drop phenomenon at the exit.

Existence and stability results for the Cauchy problem with Lipschitz constraint function p(⋅) are achieved by a procedure that combines the wave-front tracking algorithm with the operator splitting method. In view of the construction of explicit examples (one is provided), we discuss the Riemann problem with discretized piecewise constant constraint p(⋅). We illustrate the fact that nonlocality induces loss of self-similarity for the Riemann solver; moreover, discretization of p(⋅) may induce non-uniqueness and instability of solutions.
Keywords: Crowd dynamics; capacity drop; nonlocal constrained hyperbolic PDEs
AMSC: 35L65, 90B20

We consider new implicit–explicit (IMEX) Runge–Kutta methods for hyperbolic systems of conservation laws with stiff relaxation terms. The explicit part is treated by a strong-stability-preserving (SSP) scheme, and the implicit part is treated by an L-stable diagonally implicit Runge–Kutta method (DIRK). The schemes proposed are asymptotic preserving (AP) in the zero relaxation limit. High accuracy in space is obtained by Weighted Essentially Non Oscillatory (WENO) reconstruction. After a description of the mathematical properties of the schemes, several applications will be presented

Many studies have shown that Physarum polycephalum slime mold is able to find the shortest path in a maze. In this paper we study this behavior in a network, using a hyperbolic model of chemotaxis. Suitable transmission and boundary conditions at each node are considered to mimic the behavior of such an organism in the feeding process. Several numerical tests are presented for special network geometries to show the qualitative agreement between our model and the observed behavior of the mold.

Melt extraction from the Earth's mantle through high-porosity channels is required to explain the composition of the oceanic crust. Feedbacks from reactive melt transport are thought to localize melt into a network of high-porosity channels. Recent studies invoke lithological heterogeneities in the Earth's mantle to seed the localization of partial melts. Therefore, it is necessary to understand the reaction fronts that form as melt flows across the lithological interface between the heterogeneity and the ambient mantle. Here we present a chromatographic analysis of reactive melt transport across lithological boundaries, using the theory of hyperbolic conservation laws. This is an extension of linear trace element chromatography to the coupling of major elements and energy transport. Our analysis allows the prediction of the nonlinear feedbacks that arise in reactive melt transport due to changes in porosity. This study considers the special case of a partially molten porous medium with binary solid solution. As melt traverses a lithological contact, binary solid solution leads to the formation of a reacted zone between an advancing reaction front and the initial contact. The analysis also shows that the behavior of a fertile heterogeneity depends on its absolute concentration, in addition to compositional differences between itself and the refractory background. We present a regime diagram that predicts if melt emanating from a fertile heterogeneity localizes into high-porosity channels or develops a zero porosity shell. The theoretical framework presented here provides a useful tool for understanding nonlinear feedbacks in reactive melt transport, because it can be extended to more complex and realistic phase behaviors.

We propose a model of a density-dependent compressible-incompressible fluid, which is intended as a simplified version of models based on mixture theory, as for instance those arising in the study of biofilms, tumor growth, and vasculogenesis. Though our compressible-incompressible model seems to be very close to the density-dependent incompressible Euler equations, it presents some differences from an analytical point of view. In this paper, we establish a result of local existence and uniqueness of solutions in Sobolev spaces to this model, using two different singular perturbation approximations.

We consider new implicit-explicit (IMEX) Runge-Kutta methods for hyperbolic systems of conservation laws with stiff relaxation terms. The explicit part is treated by a strong-stability-preserving (SSP) scheme, and the implicit part is treated by an L-stable diagonally implicit Runge-Kutta method (DIRK). The schemes proposed are asymptotic preserving (AP) in the zero relaxation limit. High accuracy in space is obtained by Weighted Essentially Non Oscillatory (WENO) reconstruction. After a description of the mathematical properties of the schemes, several applications will be presented.

Gathering together some existing results, we show that the solutions to the one-dimensional Burgers equation converge for long times towards the stationary solutions to the steady Burgers equation, whose Fourier spectrum is not integrable. This is one of the main features of wave turbulence.

For the large sparse systems of weakly nonlinear equations arising in the discretizations of many classical differential and integral equations, this paper presents a class of asynchronous parallel mult isplitting two-stage TOR(Two-parameter Over Relaxat ion) methods for getting their solutions by the high-speed multiprocessor systems. Under suitable assumptions, we study the global convergence properties of these asynchronous mult isplitting two-stage TOR methods.

We consider new implicit-explicit (IMEX) Runge-Kutta methods for hyperbolic systems of conservation laws with stiff relaxation terms. The explicit part is treated by a strong-stability-preserving (SSP) scheme, and the implicit part is treated by an L-stable diagonally implicit Runge-Kutta method (DIRK). The schemes proposed are asymptotic preserving (AP) in the zero relaxation limit. High accuracy in space is obtained by Weighted Essentially Non Oscillatory (WENO) reconstruction. After a description of the mathematical properties of the schemes, several applications will be presented.

Nella prima parte si fornisce lo stato dell'arte delle conoscenze sulla fisica delle colate detritiche e dei metodi per risolvere le equazioni del moto. Nella seconda parte del presente lavoro vengono presentati e discussi i risultati delle simulazioni effettuate per due bacini italiani, dove le colate di detriti si verificano abbastanza regolarmente e rappresentano una minaccia per la sicurezza della popolazione locale.

In this paper, we consider a class of models for multiphase fluids, in the framework of mixture theory. The considered system, in its more general form, contains both the gradient of a hydrostatic pressure, generated by an incompressibility constraint, and the gradient of a compressible pressure depending on the volume fractions of some of the different phases. To approach these systems, we define an approximation based on the \emph{Leray} projection, which involves the use of the \emph{Lax} symbolic symmetrizer for hyperbolic systems and paradifferential techniques. In two space dimensions, we prove its well-posedness and convergence to the unique classical solution to the original system. In the last part, we shortly discuss the difficulties in the three dimensional case.