Conservation Laws

In this paper we model pedestrian flows evacuating a narrow corridor through an exit by a one-dimensional hyperbolic conservation law with a non-local constraint. Existence and stability results for the Cauchy problem with Lipschitz constraint are achieved by a procedure that combines the wave-front tracking algorithm with the operator splitting method. The Riemann problem with piecewise constant constraint is discussed in details, stressing the possible lack of uniqueness, self-similarity and L 1 loc -continuity. One explicit example of application is provided.

Optimal control models of biological movements introduce external task factors to specify the pace of movements. Here, we present the dual to the principle of optimality based on a conserved quantity, called “drive”, that represents the influence of internal motivation level on movement pace. Optimal control and drive conservation provide equivalent descriptions for the regularities observed within individual movements. For regularities across movements, drive conservation predicts a previously unidentified scaling law between the overall size and speed of various self-paced hand movements in the absence of any external tasks, which we confirmed with psychophysical experiments. Drive can be interpreted as a high-level control variable that sets the overall pace of movements and may be represented in the brain as the tonic levels of neuromodulators that control the level of internal motivation, thus providing insights into how internal states affect biological motor control.

We introduce a novel systematic construction for integrable (3+1)-dimensional dispersionless systems using nonisospectral Lax pairs that involve contact vector fields.

In particular, we present new large classes of (3+1)-dimensional integrable dispersionless systems associated to the Lax pairs which are polynomial and rational in the spectral parameter.

Despite broad recognition of the value of social sciences and increasingly vocal calls for better engagement with the human element of conservation, the conservation social sciences remain misunderstood and underutilized in practice. The conservation social sciences can provide unique and important contributions to society's understanding of the relationships between humans and nature and to improving conservation practice and outcomes. There are 4 barriers-ideological, institutional, knowledge, and capacity-to meaningful integration of the social sciences into conservation. We provide practical guidance on overcoming these barriers to mainstream the social sciences in conservation science, practice, and policy. Broadly, we recommend fostering knowledge on the scope and contributions of the social sciences to conservation, including social scientists from the inception of interdisciplinary research projects, incorporating social science research and insights during all stages of conservation planning and implementation, building social science capacity at all scales in conservation organizations and agencies, and promoting engagement with the social sciences in and through global conservation policy-influencing organizations. Conservation social scientists, too, need to be willing to engage with natural science knowledge and to communicate insights and recommendations

Contemporary physics states that an electron neutrino takes part in all events of beta decay. Here we show that electron-positron pairs participate in all beta decay events. For this reason, the hypothesis of electron neutrino presence is not necessary in β¯, β + and electron capture reactions.

Convolving the output of Discontinuous Galerkin computations with symmetric Smoothness-Increasing Accuracy-Conserving (SIAC) filters can improve both smoothness and accuracy. To extend convolution to the boundaries, several one-sided spline filters have recently been developed. This paper interprets these filters as instances of a general class of position-dependent (PSIAC) spline filters that can have non-uniform knot sequences and skip B-splines of the sequence. PSIAC filters with rational knot sequences have rational coefficients. For prototype knot sequences , such as integer sequences that may have repeated entries, PSIAC filters can be expressed in symbolic form. Based on the insight that filters for shifted or scaled knot sequences are easily derived by non-uniform scaling of one prototype filter, a single filter can be re-used in different locations and at different scales. Computing a value of the convolution then simplifies to forming a scalar product of a short vector with the local output data. Restating one-sided filters in this form improves both stability and efficiency compared to their original formulation via numerical integration. PSIAC filtering is demonstrated for several established and one new boundary filter.

" Economics needs a scientific revolution. " This is not the title of an obscure blog or a heterodox article. It is a title published in one of the world's most respected science journals. While there has been a lot of discussions on the failures of mainstream economics, a coherent framework to reform the discipline is yet to emerge. This paper is an attempt to outline a framework for a new paradigm in economics, and how it aligns with universal moral principles.

One major lesson to learn from modern science is the impossibility of perpetual motion. By ignoring this lesson, economic theory is prone to logical inconsistencies, which translate into recurrent market crises and environmental catastrophes.

Contents: Review of conservation laws in GR from both mathematical and physical perspective; maximally symmetric spacetimes; Noether theorem in GR; KBL superpotential and its generalization to modified gravity; Horndeski theory and construction of superpotentials for spacetimes representing black holes and cosmological models;

Is it possible that the most famous critic of quantum mechanics actually
invented most of its fundamentally important concepts?

In his 1905 Brownian motion paper, Einstein quantized matter,
proving the existence of atoms. His light-quantum hypothesis showed
that energy itself comes in particles (photons). He showed energy and
matter are interchangeable, E = mc2. In 1905 Einstein was first to see
nonlocality and instantaneous action-at-a-distance. In 1907 he saw
quantum “jumps” between energy levels in matter, six years before
Bohr postulated them in his atomic model. Einstein saw wave-particle
duality and the “collapse” of the wave in 1909. And in 1916 his transition
probabilities for emission and absorption processes introduced ontological chance when matter and radiation interact, making quantum
mechanics statistical. He discovered the indistinguishability and
odd quantum statistics of elementary particles in 1925 and in 1935
speculated about the nonseparability of interacting identical particles.

It took physicists over twenty years to accept Einstein’s light-quantum.
He explained the relation of particles to waves fifteen years before
Heisenberg matrices and Schrödinger wave functions. He saw
indeterminism ten years before the uncertainty principle. And he saw
nonlocality as early as 1905, presenting it formally in 1927, but was
ignored. In the 1935 Einstein-Podolsky-Rosen paper, he explored nonseparability, which was dubbed “entanglement” by Schrödinger. EPR
has gone from being ignorable to become Einstein’s most cited work:
the basis for today’s “second revolution in quantum mechanics.”

In a radical revision of the history of quantum physics, Bob Doyle
develops Einstein’s idea of objective reality to resolve several of
today’s most puzzling quantum mysteries, including the two-slit
experiment, quantum entanglement, and microscopic irreversibility.

This dissertation details the development of active flux schemes, a new class of methods for solving conservation laws. Active flux methods address three issues plaguing production-level computational fluid dynamics (CFD) codes: reliance on one-dimensional Riemann solvers, second-order accuracy, and computational stencils that do not easily parallelize. The key concept is that edge and vertex values are updated and evolved independently from the conserved cell-average quantities. Interface values are then used to calculate fluxes that conservatively update the cell-averages. Because the edge updates do not need to be conservative, any convenient method can be used and proper attention can be given to multidimensional physics. The scheme uses parabolic reconstructions, with a cubic bubble function to maintain con- servation in two dimensions, making it third-order accurate by construction. All of the reconstructions and updates are local to a single element, giving AF schemes a very compact stencil suitable for parallelization. Additionally, the AF method is fully discrete, advancing from time-level n to n + 1 in a single step.

The method is demonstrated on the linear advection, linear acoustics, and linearized Euler equations in one and two dimensions. The AF method has several advantages over more traditional schemes. For one, the extra degrees of freedom within the cell mean that frequencies up to 2π can be resolved, which is double the frequency range for comparable finite volume (FV) schemes. The AF scheme has superior dissi- pation and dispersion properties, especially as the Courant number approaches one. Its compact stencil makes the AF solution far less sensitive to irregular meshes than a third-order FV scheme. The AF scheme economically achieves third-order accu- racy using two degree(s) of freedom (DOF) per element in one dimension and three DOF in two dimensions. This is comparable to the DOF in a discontinuous Galerkin scheme using linear reconstructions. The AF method achieves third-order accuracy for all of the equation sets using randomized, unstructured meshes. The multidimensional treatment of the acoustics system allows the AF method to preserve excellent symmetry properties on an irregular triangular mesh.

This paper obtains exact solutions of a new coupled Kadomtsev-Petviashvili system, which arises in the analysis of various problems in fluid mechanics, theoretical physics and many scientific applications. Lie symmetry method along with the ðG 0 =GÞ-expansion method is employed to find the travelling wave solutions of the underlying system. In addition, we derive the conservation laws of the coupled Kadomtsev-Petviashvili system using the multiplier method.

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.

In several papers of Bouchut, Bourdarias, Perthame and Coquel, Le Floch ((13), (7)...), a general methodology has been developed to construct second order flnite volume schemes for hyperbolic systems of conservation laws satisfying the entire family of entropy inequalities. This approach is mainly based on the construction of an entropy diminishing projection. Unfor- tunately, the explicit computation of this projection

Is there a woman in mathematics? Of course there is. I will assure you of one important discovery of one of them, the brilliant Emmy (Amalie Emmy Noether, 1882-1935), which, because of the theorem, is called a kind of icon of algebra and physics, but whose significance will only grow, I hope for the theory of information I'm just doing.

We present the package SADE (Symmetry Analysis of Differential Equations) for the determination of symmetries and related properties of systems of differential equations. The main methods implemented are: Lie, non classical, Lie-Bäcklund and potential symmetries, invariant solutions, first-integrals, Nöther theorem for both discrete and continuous systems, solution of ordinary differential equations, order and dimension reductions using Lie symmetries, classification of differential equations, Casimir invariants, and the quasi-polynomial formalism for ODE's (previously implemented by the authors in the package QPSI) for the determination of quasipolynomial first-integrals, Lie symmetries and invariant surfaces. Examples of use of the package are given.

Gravitation is mass related. Mass slows down time depending on its distance. Mass itself is 90+ % of kinetic energy of sub atomic particles. It is postulated that the amount of twisting, warping of space and time within matter must be equal to the twisting and warping of space-time outside of an object. Basically saying that there is a conservation law at work for both space and time that causes the effect of gravitation. Which is confirming Noethers theorem as well.

One of the most serious challenges (if not the most serious challenge) for interactive psycho-physical dualism (henceforth interactive dualism or ID) is the so-called 'interaction problem'. It has two facets, one of which this article focuses on, namely the apparent tension between interactions of non-physical minds in the physical world and physical laws of nature. One family of approaches to alleviate or even dissolve this tension is based on a collapse solution ('consciousness collapse/CC) of the measurement problem in quantum mechanics (QM). The idea is that the mind brings about the collapse of a superposed wave function onto one of its eigenstates. Thus, it is claimed, can the mind change the course of things without violating any law figuring in physical theory. I will first show that this hope is premature because energy and momentum are probably not conserved in collapse processes, and that even if this can be dealt with, the violations are either severe or produce further ontological problems. Second, I point out several conceptual difficulties for interactionist CC. I will also present solutions for those problems, but it will become clear that those solutions come at a high cost. Third, I shall briefly list some empirical problems which make life even harder for interactionist CC. I conclude with remarks about why no-collapse interpretations of QM don't help either and what the present study has shown is the real issue for ID: namely to find a plausible integrative view of dualistic mental causation and laws of nature.

Nessyahu and TadmorÕs central scheme [J. Comput. Phys. 87 (1990)] has the benefit of not using Riemann solvers for solving hyperbolic conservation laws. But the staggered averaging causes large dissipation when the time step size is small compared to the mesh size. The recent work of Kurganov and Tadmor [J. Comput. Phys. 160 (2000)] overcomes this problem by using a variable control volume and results in semi-discrete and fully discrete non-staggered schemes. Motivated by this work, we introduce overlapping cell averages of the solution at the same discrete time level, and develop a simple alternative technique to control the O(1/Dt) dependence of the dissipation. The semi-discrete form of the central scheme can also be obtained to which the TVD Runge-Kutta time discretization methods of Shu and Osher [J. Comput. Phys. 77 (1988)] or other stable and sufficiently accurate ODE solvers can be applied. This technique is essentially independent of the reconstruction and the shape of the mesh. The overlapping cell representation of the solution also opens new possibilities for reconstructions. Generally speaking, more compact reconstruction can be achieved. In the following, schemes of up to fifth order in 1D and third order in 2D have been developed. We demonstrate through numerical examples that by combining two classes of the overlapping cells in the reconstruction we can achieve higher resolution.

The process of coagulation is associated with scalar conservation laws, where the adhesion particle dynamics results from shock waves. Conversely, the fragmentation of a massive particle into a number of smaller ones, or into a continuous (dust) distribution, is associated with rarefaction waves. It is generally agreed that a reversible solution of a conservation law can include neither shock waves nor the spontaneous emergence of rarefaction waves. The present paper is an attempt to demonstrate that both coagulation and fragmentation may coexist for a reversible solution, under a natural generalization of the system of conservation law. This is done by introducing an action principle which includes, in addition to the inertial (kinetic energy) term, also an appropriately defined internal energy. The above generalization of the system of conservation law appears as the Euler-Lagrange equations for this action.

Numerical techniques development for the modeling and simulation of free surface flows has generated great interests over the last decades. In hydraulic engineering, the objectives include the predictions of dam break waves' propagation, fluvial floods and other catastrophic flooding phenomenon, the modeling of estuarine and coastal circulations, and the design and optimization of hydraulic structures. Most of the flooding events involve wetting and drying lands which are critical for the numerical modeling, especially when dealing with complex topographies. Extreme slopes and abrupt changes in irregular geometries, have often led to significant numerical errors and stability difficulties, and these are more critical for propagations over complex dry beds. This paper presents a simple and efficient numerical model for the wetting and drying effects over complex bathymetries. An overview of the key methods that have been suggested since the pioneering studies is first presented. A 2-D cell-centered finite-volume scheme is then proposed for solving the shallow-water equations using both structured and unstructured fixed meshes. Steady state C-property and global mass conservation properties are satisfied using appropriate numerical fluxes and wet/ dry interfaces treatments. The resulting numerical model proved stable and robust and was validated through some benchmarks tests, including comparisons with exact solutions and experimental data, and a real case of wetting-drying simulation in a portion of the river "Rivière des Prairies" in a suburb of Laval, Quebec.

In this paper, we consider the two phases macroscopic traffic model introduced in [P. Goatin, The Aw-Rascle vehicular traffic flow with phase transitions, Mathematical and Computer Modeling 44 (2006) 287-303]. We first apply the wave-front tracking method to prove existence and a priori bounds for weak solutions. Then, in the case the characteristic field corresponding to the free phase is linearly degenerate, we prove that the obtained weak solutions are in fact entropy solutions \`a la Kruzhkov. The case of solutions attaining values at the vacuum is considered. We also present an explicit numerical example to describe some qualitative features of the solutions.

In liquid-liquid contacting equipment such as completely mixed and di erential contactors, droplet population balance based modeling is now being used to describe the complex hydrodynamic behavior of the dispersed phase. For the hydrodynamics of these interacting dispersions this model accounts for droplet breakage, droplet coalescence, axial dispersion, exit and entry events. The resulting population balance equations are integro-partial di erential equations (IPDE) that rarely have an analytical solution, especially when they show spatial dependency, and hence numerical solutions are sought in general. To do this, these IPDEs are projected onto a system of convective dominant partial di erential equations by discretizing the droplet diameter (internal coordinate). This is accomplished by generalizing the ÿxed-pivot (GFP) technique of Kumar and Ramkrishna (Chem. Eng. Sci. 51 (1996a) 1311) handling any two integral properties of the population number density for continuous ow systems by treating the inlet feed distribution as a source term. Moreover, the GFD technique has the advantage of being free of repeated or double integral evaluation resulting from the weighted residual approaches such as the Galerkin's method. This allows the time-dependent breakage and coalescence functions to be easily handled without appreciable increase in the computational time. The resulting system of PDEs is spatially discretized in conservative form using a simpliÿed ÿrst order upwind scheme as well as ÿrst-and second-order non-oscillatory central di erencing schemes. This spatial discretization avoids the characteristic decomposition of the convective ux based on the approximate Riemann solvers and the operator splitting technique required by classical upwind schemes. The time variable is discretized using an implicit strongly stable approach that is formulated by careful lagging of the non-linear parts of the convective and source terms. The algorithm is tested against analytical solutions of the simpliÿed population balance equation for a di erential liquid-liquid extraction column through four case studies. In all these case studies the discrete models converge successfully to the available analytical solutions and to solutions on relatively ÿne grids when the analytical solution is not available. Realization of the algorithm is accomplished by comparing its predictions to experimental steady-state hydrodynamic data of a laboratory scale rotating disc contactor of 0:15 m diameter. Practically, the combined algorithm is found fast enough for the computation of the transient and steady-state hydrodynamic behavior of the continuously and spatially distributed interacting liquidliquid dispersions.

In this paper, analytically investigated is a higher-order dispersive nonlinear Schrödinger equation. Based on the linear eigenvalue problem associated with this equation, the integrability is identified by admitting an infinite number of conservation laws. By using the Darboux transformation method, the explicit multi-soliton solutions are generated in a recursive manner. The propagation characteristic of solitons and their interactions under the periodic plane wave background are discussed. Finally, the modulational instability of solutions is analyzed in the presence of small perturbation.

Marta RUSNAK Polskie prawo o obiektach dziedzictwa technicznego adaptowanych dla potrzeb muzealnych i wystawienniczych Proces przekształcania obiektów dziedzictwa technicznego dla potrzeb muzealnych jest regulowany trzema grupami dokumentów. Pierwsza grupa dotyczy zagadnień budowlanych, głównie Ustawy o prawie budowlanym. Druga to akty dotyczące aspektu zachowania walorów zabytkowych np. Ustawa o ochronie zabytków i opiece nad nimi. Trzecią pulę stanowią wytyczne sporządzane przez Ministerstwo Kultury i Dziedzictwa Narodowego. Głównym dokumentem należącym do tej ostatniej grupy jest Ustawa o muzeach. W przypadku niektórych transformacji główny zestaw będą uzupełniają inne akty prawne, często dotyczące dziedzin których skojarzenie z przystosowaniem obiektu techniki dla potrzeb ekspozycji nie jest tak oczywiste. Ze względu na historyczny charakter opisywanych obiektów podejmowane w nich współczesne interwencje reguluje szereg dokumentów konserwatorskich zaczynając od pierwszej Karty Ateńskiej, a kończąc na obowiązującej Ustawie o ochronie zabytków i opiece nad zabytkami 1 . W punktach 1e) i 2d) artykułu 6 ustawa nakazuje opiekę nad obiektami nieruchomymi i ruchomymi wytworami techniki, bez względu na ich stan zachowania. Sposób opisania obiektu techniki, definiowanego przez ustawę wprost jako elektrownia, kopalnia lub huta, ukazuje pośrednio ogólną tendencję według której, społeczeństwo nie ma świadomości kulturowej wartości tych reliktów. Omawiany fragment ustawy zwraca tym większą uwagę gdyż poprzedzające je podpunkty są ujęte w sposób bardziej ogólnikowy np. dzieła architektury i budownictwa; bez egzemplifikacji reprezentantów tych grup np. pałaców. Tworzący ustawę odczuwali potrzebę doprecyzowania tego zagadnienia, gdyż zabytki techniki były długo traktowane jako obiekty gorszej klasy 2 . 1 z dnia 23 lipca 2003 r. 2 W dawnym rozumieniu wyznacznikami wartości zabytkowej były wartości estetyczne lub historyczne. za Orłowski Bolesław, Dlaczego przyszła moda na starocie techniczno-przemysłowe?, w: Formy ochrony zabytków techniki. Doświadczenia zebrane w trakcie projektu Consist zawartego w 6. Programie ramowym", Wrocław 2008, s.20

The collisionless Boltzmann equation is generalized herein using the Green function theory proposed recently by A.K. Rajagopal et al. [Phys. Rev. Lett. 80, 3911 (1998)] to describe nonextensive systems based on Tsallis formalism. Its invariance with the nonextensive index, q, and some conservative laws useful in the description of sound propagation are verified. In this context, the Wigner distribution has also been introduced. Furthermore, the formalism developed in the research is applied for fermionic system, to obtain the dynamic dielectric response function.

This paper obtains solutions to the Zakharov-Kuznetsov modified equal width equation with power law nonlinearity. The Lie symmetry approach and the simplest equation method are used to obtain these solutions. Moreover, conservation laws are derived for the underlying equation by employing two approaches: the new conservation theorem and the multiplier method.

In this article we present a high resolution hybrid central finite difference—WENO scheme for the solution of conservation laws, in particular, those related to shock–turbulence interaction problems. A sixth order central finite difference scheme is conjugated with a fifth order weighted essentially non-oscillatory WENO scheme in a grid-based adaptive way. High order multi-resolution analysis is used to detect the high gradients regions of the numerical solution in order to capture the shocks with the WENO scheme while the smooth regions are computed with the more efficient and accurate central finite difference scheme. The application of high order filtering to mitigate the dispersion error of central finite difference schemes is also discussed. Numerical experiments with the 1D compressible Euler equations are shown.

A fundamental result of classical electromagnetism is that Maxwell's equations imply that electric charge is locally conserved. Here we show the converse: Local charge conservation implies the local existence of fields satisfying Maxwell's equations. This holds true for any conserved quantity satisfying a continuity equation. It is obtained by means of a strong form of the Poincaré lemma presented here that states: Divergence-free multivector fields locally possess curl-free antiderivatives on flat manifolds. The above converse is an application of this lemma in the case of divergence-free vector fields in spacetime. We also provide conditions under which the result generalizes to curved manifolds.

This paper aims to construct conservation laws for a Benjamin-Bona-Mahony equation with variable coefficients, which is a third-order partial differential equation. This equation does not have a Lagrangian and so we transform it to a fourth-order partial differential equation, which has a Lagrangian. The Noether approach is then employed to construct the conservation laws. It so happens that the derived conserved quantities fail to satisfy the divergence criterion and so one needs to make adjustments to the derived conserved quantities in order to satisfy the divergence condition. The conservation laws are then expressed in the original variable. Finally, a conservation law is used to obtain exact solution of a special case of the Benjamin-Bona-Mahony equation.

Dynamic process simulation differs from purely steady-state simulation in that the former requires the mechanical construction of process items be taken into account; the amount of mechanical detail being dependant upon the particular application. The reason for this is that dynamic mass, energy and momentum balances have to be continuously updated. These calculations are fundamental to dynamic process simulation and they require knowledge of volumes, metal mass, etc., to predict the proper dynamic behaviour of a particular plant. ...

We build on active flux (AF) schemes and extend the method to the two-dimensional linear advection, linear acoustics, and linearized Euler equations. The active flux method independently updates edge and centroid values. Because the interface fluxes are calculated from independently updated quantities, we say that they are actively updated, as opposed to a more traditional finite-volume scheme where the fluxes are updated passively from the conserved values. The one-dimensional active flux method is reformulated using Lagrange basis functions and the basic features of the method are reviewed. The two-dimensional formulation also uses standard basis functions, but includes a bubble function to maintain conservation. A novel approach for solving the linear wave system is presented that uses spherical means to compute the edge updates used to calculate fluxes. We demonstrate that the AF method is third-order accurate for advection, acoustics, and the linearized Euler equations.

We obtain a generalization of E. Noether's theorem for the optimal control problems. The generalization involves a one-parameter family of smooth maps which may depend also on the control and a Lagrangian which is invariant up to an addition of an exact differential.

A fundamental result of classical electromagnetism is that Maxwell's equations imply that electric charge is locally conserved. Here we show the converse: Local charge conservation implies the local existence of fields satisfying Maxwell's equations. This holds true for any conserved quantity satisfying a continuity equation. It is obtained by means of a strong form of the Poincaré lemma presented here that states: Divergence-free multivector fields locally possess curl-free antiderivatives on flat manifolds. The above converse is an application of this lemma in the case of divergence-free vector fields in spacetime. We also provide conditions under which the result generalizes to curved manifolds.

The purpose of this present paper is to present a cognitive framework, coined as “Conservation Law,” which may shed new light on insightful understanding of the structures of sentences involving relative pronouns. Given that there are roughly four types of relative pronouns; namely, general, compound, relative adverbial, and quasi-relative, a conservation law in both structures and semantics exists. We start by combining two simple sentences into a resulting sentence with relative pronouns, and, through careful observation and calculation, we found the equivalence in word counts between original and resulting sentences. The conservation law mainly refers to the structure equivalence on either words or meanings among these four types of relative pronouns, or relative clauses. With the help of conservation law, the understanding of relative clauses can be much easier, because they are, among others, the most difficult and complex structures in English syntax, especially for EFL learners. Hopefully, this cognitive framework may be of great contribution to EFL English instruction.

The flux-corrected-transport paradigm is generalized to finite element schemes based on arbitrary time-stepping. A conservative flux decomposition procedure is proposed for both convective and diffusive terms. Mathematical properties of positivity-preserving schemes are reviewed. A nonoscillatory low-order method is constructed by elimination of negative off-diagonal entries of the discrete transport operator. The linearization of source terms and extension to hyperbolic systems are discussed. Zalesak's multidimensional limiter is employed to switch between linear discretizations of high and low order. A rigorous proof of positivity is provided. The treatment of nonlinearities and iterative solution of linear systems are addressed. The performance of the new algorithm is illustrated by numerical examples for the shock tube problem in 1D and scalar transport equations in 2D.

Wigner found unreasonable the "effectiveness of mathematics in the natural sciences". But if the mathematics we use to describe nature is simply a coded expression of our experience then its effectiveness is quite reasonable. Its effectiveness is built into its design. We consider group theory, the logic of symmetry. We examine the premise that symmetry is identity; that group theory encodes our experience of identification. I propose that, as used to describe the physical world, symmetry is so elemental that it coincides with the concept of identity itself. The theory of symmetry is the mathematical expression of the notion of identification and that is why it is so effective as the basis of science.

The search for new integrable (3+1)-dimensional partial differential systems is among the most important challenges in the modern integrability theory. It turns out that such a system can be associated to any pair of rational functions of one variable in general position, as established below using contact Lax pairs introduced in arXiv:1401.2122v5.

The conservation laws for the variant Boussinesq system are derived by an interesting method of increasing the order of partial differential equations. The variant Boussinesq system is a third-order system of two partial differential equations. The transformations u → U x , v → V x are used to convert the variant Boussinesq system to a fourth order system in U, V variables. It is interesting that a standard Lagrangian exists for the fourthorder system. Noether's approach is then used to derive the conservation laws. Finally, the conservation laws are expressed in the variables u, v and they constitute the conservation laws for the third-order variant Boussinesq system. Infinitely many nonlocal conserved quantities are found for the variant Boussinesq system.

The conservation laws for second order scalar partial differential equations and systems of partial differential equations which occur in fluid mechanics are constructed using different approaches. The direct method, Noether's theorem, the characteristic method, the variational approach (multiplier approach) for arbitrary functions as well as on the solution space, symmetry conditions on the conserved quantities, the direct construction formula approach, the partial Noether approach and the Noether approach for the equation and its adjoint are discussed and explained with the help of an illustrative example on a non-linear field equation describing the relaxation to a Maxwellian distribution. The conservation laws for the non-linear diffusion equation for the spreading of an axisymmetric thin liquid drop, the system of two partial differential equations governing flow in a laminar two-dimensional jet and the system of two partial differential equations governing flow in a laminar radial jet are discussed via these approaches.