Lie symmetry group transformation for MHD natural convection flow of nanofluid over linearly porous stretching sheet in presence of thermal stratification

The magnetohydrodynamics （MHD） convection flow and heat transfer of an incompressible viscous nanofluid past a semi-infinite vertical stretching sheet in the pres- ence of thermal stratification are examined. The partial differential equations governing the problem under consideration are transformed by a special form of the Lie symmetry group transformations, i.e., a one-parameter group of transformations into a system of ordinary differential equations which are numerically solved using the Runge-Kutta-Gill- based shooting method. It is concluded that the flow field, temperature, and nanoparticle volume fraction profiles are significantly influenced by the thermal stratification and the magnetic field.

Symmetry and general symmetry groups of the coupled Kadomtsev-Petviashvili equation

In this paper, the Lie symmetry algebra of the coupled Kadomtsev-Petviashvili （cKP） equation is obtained by the classical Lie group method and this algebra is shown to have a Kac-Moody-Virasoro loop algebra structure. Then the general symmetry groups of the cKP equation is also obtained by the symmetry group direct method which is proposed by Lou et alo From the general symmetry groups, the Lie symmetry group can be recovered and a group of discrete transformations can be derived simultaneously. Lastly, from a known simple solution of the cKP equation, we can easily obtain two new solutions by the general symmetry groups.

Einstein’s Dark Energy via Similarity Equivalence, ‘tHooft Dimensional Regularization and Lie Symmetry Groups

Realizing the physical reality of ‘tHooft’s self similar and dimensionaly regularized fractal-like spacetime as well as being inspired by a note worthy anecdote involving the great mathematician of Alexandria, Pythagoras and the larger than life man of theoretical physics Einstein, we utilize some deep mathematical connections between equivalence classes of equivalence relations and E-infinity theory quotient space. We started from the basic principles of self similarity which came to prominence in science with the advent of the modern theory of nonlinear dynamical systems, deterministic chaos and fractals. This fundamental logico-mathematical thread related to partially ordered sets is then applied to show how the classical Newton’s kinetic energy E = 1/2mv2 leads to Einstein’s celebrated maximal energy equation E = mc2 and how in turn this can be dissected into the ordinary energy density E(O) = mc2/22 and the dark energy density E(D) = mc2(21/22) of the cosmos where m is the mass;v is the velocity and c is the speed of light. The important role of the exceptional Lie symmetry groups and ‘tHooft-Veltman-Wilson dimensional regularization in fractal spacetime played in the above is also highlighted. The author hopes that the unusual character of the analysis and presentation of the present work may be taken in a positive vein as seriously attempting to propose a different and new way of doing theoretical physics by treating number theory, set theory, group theory, experimental physics as well as conventional theoretical physics on the same footing and letting all these diverse tools lead us to the answer of fundamental questions without fear of being labelled in one way or another.

Symmetry groups and Gauss kernels of Schrdinger equations

The relationship between symmetries and Gauss kernels for the SchrSdinger equation iut = uxx ＋ f（x）u is established. It is shown that if the Lie point symmetries of the equation are nontrivial, a classical integral transformations of the Gauss kernels can be obtained. Then the Gauss kernels of Schroedinger equations are derived by inverting the integral transformations. Furthermore, the relationship between Gauss kernels for two equations related by an equivalence transformation is identified.

New infinite-dimensional symmetry groups for the stationary axisymmetric Einstein-Maxwell equations with multiple Abelian gauge fields

The so-called extended hyperbolic complex （EHC） function method is used to study further the stationary axisymmetric Einstein Maxwell theory with p Abelian gauge fields （EM-p theory, for short）, Two EHC structural Riemann- Hilbert （RH） transformations are constructed and are then shown to give an infinite-dimensional symmetry group of the EM-p theory. This symmetry group is verified to have the structure of semidirect product of Kac-Moody group SU（p ＋ 1, 1） and Virasoro group. Moreover, the infinitesimal forms of these two RH transformations are calculated and found to give exactly the same infinitesimal transformations as in previous author＇s paper by a different scheme, This demonstrates that the results obtained in the present paper provide some exponentiations of all the infinitesimal symmetry transformations obtained before.

Finding Symmetry Groups of Some Quadratic Programming Problems

Solution and analysis of mathematical programming problems may be simplified when these problems are symmetric under appropriate linear transformations.In particular,a knowledge of the symmetries may help decrease the problem dimension,reduce the size of the search space by means of linear cuts.While the previous studies of symmetries in the mathematical programming usually dealt with permutations of coordinates of the solutions space,the present paper considers a larger group of invertible linear transformations.We study a special case of the quadratic programming problem,where the objective function and constraints are given by quadratic forms.We formulate conditions,which allow us to transform the original problem to a new system of coordinates,such that the symmetries may be sought only among orthogonal transformations.In particular,these conditions are satisfied if the sum of all matrices of quadratic forms,involved in the constraints,is a positive definite matrix.We describe the structure and some useful properties of the group of symmetries of the problem.Besides that,the methods of detection of such symmetries are outlined for different special cases as well as for the general case.

The symmetry group of Feynman diagrams and consistency of the BPHZ renormalization scheme

We study the relation between the symmetry group of a Feynman diagram and its reduced diagrams.We then prove that the counterterms in the BPHZ renormalization scheme are consistent with adding counterterms to the interaction Hamiltonian in all cases,including that of Feynman diagrams with symmetry factors.

Lie point symmetry algebras and finite transformation groups of the general Broer-Kaup system

Using a new symmetry group theory, the transformation groups and symmetries of the general Broer-Kaup system are obtained. The results are much simpler than those obtained via the standard approaches.

Finite symmetry transformation group of the Konopelchenko Dubrovsky equation from its Lax pair

Starting from a weak Lax pair, the general Lie point symmetry group of the Konopelchenko-Dubrovsky equation is obtained by using the general direct method. And the corresponding Lie algebra structure is proved to be a Kac-Mood-Virasoro type. Furthermore, a new multi-soliton solution for the Konopelchenko-Dubrovsky equation is also given from this symmetry group and a known solution.

Revisions of P-1 Space Groups to Higher Symmetry Space Groups

The space groups of 244 crystal structures originally reported as P-1 are revised to space groups of higher symmetry. The largest number involves revisions to P2/c and C2/c.

Symmetry analysis and explicit solutions of the (3+1)-dimensional baroclinic potential vorticity equation

This paper investigates an important high-dimensional model in the atmospheric and oceanic dynamics-（3＋1）- dimensional nonlinear baroclinic potential vorticity equation by the classical Lie group method. Its symmetry algebra, symmetry group and group-invariant solutions are analysed. Otherwise, some exact explicit solutions are obtained from the corresponding （2＋1）-dimensional equation, the inviscid barotropic nondivergent vorticy equation. To show the properties and characters of these solutions, some plots as well as their possible physical meanings of the atmospheric circulation are given out.

On group structures of strategic-form games

There are two recognized classes of strategic-form symmetric games,both of which can be conveniently defined through the corresponding player symmetry groups.We investigate the basic properties of these groups and several related concepts.We generalize the notion of coveringness and adapt their results to characterize these player symmetry groups.We study the relationships between the coveringnesses of various symmetry groups.Our results demonstrate that these symmetry groups have rich mathematical structures that are of game theoretical and economic interests.

Testing the Results of Measurements of Neutrino Parameters Using the Dirac CPV Phase Formula

Essentially the main intention of this paper was to test the formula for the Dirac CPV phase and see if it can reflect the results of experimental measurements of neutrino parameters. By knowing the mathematical formula for the Dirac CPV phase, a connection was established with some of the residual symmetry groups, which made it possible to develop a procedure for directly determining the range in which the numerical value for the Dirac CPV phase could be found. In this sense, two different sources of information containing measured data for neutrinos were used for the corresponding calculations, and then a comparative overview of the calculated results was presented. It is particularly emphasized that the formula for the Dirac CPV phase does not depend on the mixing angles that are incorporated into the PMNS matrix, but only on the ratio between the corresponding squares of the neutrino mass difference. All the numerous results obtained from the corresponding calculations for the Dirac CPV phase point to the justified introduction of the theory that is related to three neutrinos, and thus the agreement of our results with the STEREO experiment is justified, so that the hypothesis of the possible existence of a sterile neutrino in nature should be excluded.

The classification of travelling wave solutions and superposition of multi-solutions to Camassa-Holm equation with dispersion

Under the travelling wave transformation, the Camassa-Holm equation with dispersion is reduced to an integrable ordinary differential equation （ODE）, whose general solution can be obtained using the trick of one-parameter group. Furthermore, by using a complete discrimination system for polynomial, the classification of all single travelling wave solutions to the Camassa-Holm equation with dispersion is obtained. In particular, an affine subspace structure in the set of the solutions of the reduced ODE is obtained. More generally, an implicit linear structure in the Camassa-Holm equation with dispersion is found. According to the linear structure, we obtain the superposition of multi-solutions to Camassa-Holm equation with dispersion.

Quantum Dark Energy from the Hyperbolic Transfinite Cantorian Geometry of the Cosmos

The quintessence of hyperbolic geometry is transferred to a transfinite Cantorian-fractal setting in the present work. Starting from the building block of E-infinity Cantorian spacetime theory, namely a quantum pre-particle zero set as a core and a quantum pre-wave empty set as cobordism or surface of the core, we connect the interaction of two such self similar units to a compact four dimensional manifold and a corresponding holographic boundary akin to the compactified Klein modular curve with SL(2,7) symmetry. Based on this model in conjunction with a 4D compact hy- perbolic manifold M(4) and the associated general theory, the so obtained ordinary and dark en- ergy density of the cosmos is found to be in complete agreement with previous analysis as well as cosmic measurements and observations such as WMAP and Type 1a supernova.

From Witten’s 462 Supercharges of 5-D Branes in Eleven Dimensions to the 95.5 Percent Cosmic Dark Energy Density behind the Accelerated Expansion of the Universe

The measured 95.5% dark energy density of the cosmos presumed to be behind the observed accelerated cosmic expansion is determined theoretically based upon Witten’s five branes in eleven dimensions theory. We show that the said dark energy density is easily found from the ratio of the 462 states of the five dimensional Branes to the total number of states, namely 528 minus the 44 degrees of freedom of the vacuum, i.e. , almost exactly as found in WMAP and Type 1a supernova measurements.

Three types of generalized Kadomtsev-Petviashvili equations arising from baroclinic potential vorticity equation

By means of the reductive perturbation method, three types of generalized （2＋l）-dimensional Kadomtsev- Petviashvili （KP） equations are derived from the baroclinic potential vorticity （BPV） equation, including the modified KP （mKP） equation, standard KP equation and cylindrical KP （cKP） equation. Then some solutions of generalized cKP and KP equations with certain conditions are given directly and a relationship between the generalized mKP equation and the mKP equation is established by the symmetry group direct method proposed by Lou et al. From the relationship and the solutions of the mKP equation, some solutions of the generalized mKP equation can be obtained. Furthermore, some approximate solutions of the baroclinic potential vorticity equation are derived from three types of generalized KP equations.

Lie Point Symmetry Analysis of the Harry-Dym Type Equation with Riemann-Liouville Fractional Derivative

In this paper, Lie point symmetry group of the Harry-Dym type equation with Riemann-Liouville fractional derivative is constructed. Then complete subgroup classification is obtained by means of the optimal system method. Finally, corresponding group-invariant solutions with reduced fractional ordinary differential equations are presented via similarity reductions.

PERIODIC SOLUTIONS OF DELAY DIFFERENTIAL EQUATIONS WITH TWO DELAY VIA BI-HAMILTONIAN SYSTEMS

Using the theory of existence of periodic solutions of a bi-Hamiltonian system and the method of symmetry groups, the existence of periodic solutions for some two-delay differential equation is obtained. Some new sufficient conditions are given.

A Scheme for Generating Nonisospectral Integrable Hierarchies and Its Related Applications

Under the framework of the complex column-vector loop algebra C^(p),we propose a scheme for generating nonisospectral integrable hierarchies of evolution equations which generalizes the applicable scope of the Tu scheme.As applications of the scheme,we work out a nonisospectral integrable Schrodinger hierarchy and its expanding integrable model.The latter can be reduced to some nonisospectral generalized integrable Schrodinger systems,including the derivative nonlinear Schrodinger equation once obtained by Kaup and Newell.Specially,we obtain the famous Fokker-Plank equation and its generalized form,which has extensive applications in the stochastic dynamic systems.Finally,we investigate the Lie group symmetries,fundamental solutions and group-invariant solutions as well as the representation of the tensor of the Heisenberg group H_(3)and the matrix linear group SL(2,R)for the generalized Fokker-Plank equation(GFPE).