Symmetry Breaking for Black-Scholes Equations

Black-Scholes equation is used to model stock option pricing. In this paper, optimal systems with one to four parameters of Lie point symmetries for Black-Scholes equation and its extension are obtained. Their symmetry breaking interaction associated with the optimal systems is also studied. As a. result, symmetry reductions and corresponding solutions for the resulting equations are obtained.

Exact Geometric Relationships, Symmetry Breaking and Structural Stability for Single-Walled Carbon Nanotubes

We pioneered a study about how the geometric relationship of single-walled carbon nanotubes(SWCNT) is influenced by curvature factor and non-planar geometry factor in cylindrical coordinate system based on the assumption of complete symmetry. The bond length and angle of every carbon-carbon bonds are determined by using the principle of the minimum energy. The results of the paper include(1) From the calculation result, the symmetry breaking appears for chiral carbon nanotubes, while the part symmetry appears for achiral carbon nanotubes with increasing curvature.(2) The synergistic effect of bond lengths and bond angles is first found.(3) We conclude that the influence of non-planar geometry factor can be completely ignored on bond lengths and bond angles when the curvature parameter has been included in the model.(4)The two fractal dimensions are given from the nanoscale to the macroscale for zigzag topology and armchair topology respectively. Fractal dimensions of SWCNT show special characteristics, varying with the length of SWCNT until the lengths approach infinity. The close and inevitable correlations among curvature, symmetry breaking and stability of SWCNTs can be summed up as: the increase of curvature causes symmetry breaking,and such symmetry breaking will further reduce the structural stability.

Existence of spontaneous symmetry breaking in two-lane totally asymmetric simple exclusion processes with an intersection

We study two-lane totally asymmetric simple exclusion processes(TASEPs)with an intersection.Monte Carlo simulations show that only symmetric phases exist in the system.To verify the existence of asymmetric phases,we carry out a cluster mean-field analysis.Analytical results show that the densities of the two upstream segments of the intersection site are always equal,which indicates that the system is not in asymmetric phases.It demonstrates that the spontaneous symmetry breaking does not exist in the system.The density profiles and the boundaries of the symmetric phases are also investigated.We find that the cluster mean-field analysis shows better agreement with simulations than the simple mean-field analysis where the correlation of sites is ignored.

Symmetry Breaking Ground States of Bose-Einstein Condensates in 1D Double Square Well and Optical Lattice Well

We investigate the phenomena of symmetry breaking and phase transition in theground state of Bose-Einstein condensates (BECs) trapped in a double square well and in an opticallattice well, respectively. By using standing-wave expansion method, we present symmetric andasymmetric ground state solutions of nonlinear Schroedinger equation (NLSE) with a symmetric doublesquare well potential for attractive nonlinearity. In particular, we study the ground state wavefunction's properties by changing the depth of potential and atomic interactions (here we restrictourselves to the attractive regime). By using the Fourier grid Hamiltonian method, we also reveal aphase transition of BECs trapped in one-dimensional optical lattice potential.

Dynamic symmetry breaking and structure-preserving analysis on thelongitudinal wave in an elastic rod with a variable cross-section

The longitudinal wave propagating in an elastic rod with a variable cross-section owns wide engineering background,in which the longitudinal wave dissipation determines some important performances of the slender structure.To reproduce the longitudinal wave dissipation effects on an elastic rod with a variable cross-section,a structure-preserving approach is developed based on the dynamic symmetry breaking theory.For the dynamic model controlling the longitudinal wave propagating in the elastic rod with the variable cross-section,the approximate multi-symplectic form is deduced based on the multi-symplectic method,and the expression of the local energy dissipation for the longitudinal wave propagating in the rod is presented,referring to the dynamic symmetry breaking theory.A structure-preserving method focusing on the residual of the multi-symplectic structure and the local energy dissipation of the dynamic model is constructed by using the midpoint difference discrete method.The longitudinal wave propagating in an elastic rod fixed at one end is simulated,and the local/total energy dissipations of the longitudinal wave are investigated by the constructed structure-preserving scheme in two typical cases in detail.

Symmetry breaking in the opinion dynamics of a multi-group project organization

A bounded confidence model of opinion dynamics in multi-group projects is presented in which each group＇s opinion evolution is driven by two types of forces：（i） the group＇s cohesive force which tends to restore the opinion back towards the initial status because of its company culture;and（ii） nonlinear coupling forces with other groups which attempt to bring opinions closer due to collaboration willingness.Bifurcation analysis for the case of a two-group project shows a cusp catastrophe phenomenon and three distinctive evolutionary regimes,i.e.,a deadlock regime,a convergence regime,and a bifurcation regime in opinion dynamics.The critical value of initial discord between the two groups is derived to discriminate which regime the opinion evolution belongs to.In the case of a three-group project with a symmetric social network,both bifurcation analysis and simulation results demonstrate that if each pair has a high initial discord,instead of symmetrically converging to consensus with the increase of coupling scale as expected by Gabbay＇s result（Physica A 378（2007） p.125 Fig.5）,project organization（PO） may be split into two distinct clusters because of the symmetry breaking phenomenon caused by pitchfork bifurcations,which urges that apart from divergence in participants＇ interests,nonlinear interaction can also make conflict inevitable in the PO.The effects of two asymmetric level parameters are tested in order to explore the ways of inducing dominant opinion in the whole PO.It is found that the strong influence imposed by a leader group with firm faith on the flexible and open minded follower groups can promote the formation of a positive dominant opinion in the PO.

Origin of Symmetry Breaking and Confinement in Conducting Polymers with Ring Structures

A model to describe the main features of conjugated polymers with ring structures, such as polythiophene and polypyrrole, is constructed. It is shown that the origin of the symmetry breaking and confinement of a soliton and anti-soliton pair is branch hopping in the polymer rings.

Spontaneous U（1） symmetry breaking and Bose-Einstein condensation

Based on Bogoliubov＇s truncated Hamiltonian HB for a weakly interacting Bose system, and adding a U（1） symmetry breaking term √V（λα0＋λα0^＋） to HB, we show by using the coherent state theory and the mean-field approximation rather than the c-number approximations, that the Bose-Einstein condensation（BEC） occurs if and only if the U（1） symmetry of the system is spontaneously broken. The real ground state energy and the justification of the Bogoliubov c-number substitution are given by solving the Schroedinger eigenvalue equation and using the self-consistent condition.

Spontaneous U(1) Symmetry Breaking in Coupled Bose System

Adding a U（1） symmetry breaking term √V（λ1a0 ＋ λ1＊a0） ＋ √V（λ2b0 ＋ λ2＊b0） to Bogoliubov＇s truncated Hamiltonian HB for a weakly interacting coupled Bose system, by using the mean-field approximation rather than the c-number approximation, we And that, via a Feshbach resonance at zero temperature, the states of the coupled Bose system are generalized SU（1,1） SU（1,1） coherent states. The Bose-Einstein condensation occurs in response to the spontaneous U（1） symmetry breaking.

Mode structure symmetry breaking of reversed shear Alfvén eigenmodes and its impact on the generation of parallel velocity asymmetries in energetic particle distribution

In this work,the gyrokinetic eigenvalue code LIGKA,the drift-kinetic/MHD hybrid code HMGC and the gyrokinetic full-f code TRIMEG-GKX are employed to study the mode structure details of reversed shear Alfvén eigenmodes(RSAEs).Using the parameters from an ASDEXUpgrade plasma,a benchmark with the three different physical models for RSAE without and with energetic particles(EPs)is carried out.Reasonable agreement has been found for the mode frequency and the growth rate.Mode structure symmetry breaking(MSSB)is observed when EPs are included,due to the EPs’non-perturbative effects.It is found that the MSSB properties are featured by a finite radial wave phase velocity,and the linear mode structure can be well described by an analytical complex Gaussian expressionФ(s)=e^(-σ(s-s_(0))^(2))with complex parametersσand s_(0),where s is the normalized radial coordinate.The mode structure is distorted in opposite manners when the EP drive shifted from one side of qminto the other side,and specifically,a non-zero average radial wave number with opposite signs is generated.The initial EP density profiles and the corresponding mode structures have been used as the input of HAGIS code to study the EP transport.The parallel velocity of EPs is generated in opposite directions,due to different values of the average radial wave number,corresponding to different initial EP density profiles with EP drive shifted away from the qmin.

Dynamical Symmetry Breaking of Maximally Generalized Yang-Mills Model and Its Restoration at Finite Temperatures

In terms of the Nambu-Jona-Lasinio mechanism, dynamical breaking of gauge symmetry for the maximally generalized Vang-Mills model is investigated. The gauge symmetry behavior at finite temperature is also investigated and it is shown that the gauge symmetry broken dynamically at zero temperature can be restored at finite temperatures.

Relativistic Hydrodynamics of Color-Flavor Locking Phase with SpontaneousSymmetry Breaking

We study the hydrodynamics of color-flavor locking phase of three flavors of light quarks in high density QCD with spontaneous symmetry breaking. The basic hydrodynamic equations are presented based on the Poisson bracket method and the Goldstone phonon and the thermo phonon are compared. The dissipative equations are constructed in the frame of the first-order theory and all the transport coefficients are also defined, which could be looked on as the general case including the Landau's theory and the Eckart's theory.

Spontaneous symmetry breaking of a Bose-Fermi mixture in a two-dimensional double-well potential

We study the spontaneous symmetry breaking of a superfluid Bose-Fermi mixture in a two-dimensional double- well potential. The mixture is described by a set of coupled Gross-Pitaevskii equations. The symmetry breaking phenomenon is demonstrated in the two-dimensional double-well potential in the mixture. The results are summarized in the phase diagrams of the mixture particle numbers, which are divided into symmetric and asymmetric regions by the asymmetry ratios. The dynamical pictures of the spontaneous symmetry breaking induced by a gradual transformation of the single-well potential into a double-well one are also illustrated. The properties of the quantum degenerate mixture are explored using the realistic parameters for a ^40K-^87Rb system.

Spontaneous symmetry breaking vacuum energy in cosmology

The gravitational effect of spontaneous symmetry breaking vacuum energy density is investigated by subtracting the fiat space-time contribution from the energy in the curved space-time. We find that the remaining effective energy- momentum tensor is too small to cause the acceleration of the universe, although it satisfies the characteristics of dark energy. However, it could provide a promising explanation to the puzzle of why the gravitational effect produced by the huge symmetry breaking vacuum energy in the electroweak theory has not been observed, as it has a sufficiently small value （smaller than the observed cosmic energy density by a factor of 1032）.

Critical Behavior of Dynamical Chiral Symmetry Breaking with Gauge Boson Mass in QED3

Since the massless quantum electrodynamics in 2＋1 dimensions （QEDa） with nonzero gauge boson mass ζ can be used to explain some important traits of high-Tc superconductivity in planar cuprates, it is worthwhile to apply this model to analyze the nature of chiral phase transition at the critical value ζ. Based on the feature of chiral susceptibility, we show that the system at ζ exhibits a second-order phase transition which accords with the nature of appearance of the high-To superconductivity, and the estimated critical exponents around ζ are illustrated.

Time-Dependent Ferrofluid Dynamics in Symmetry Breaking Transverse

We investigate the Taylor-Couette flow of a rotating ferrofluid under the influence of symmetry breaking transverse magnetic field in counter-rotating small-aspect-ratio setup. We find only changing the magnetic field strength can drive the dynamics from time-periodic limit-cycle solution to time-independent steady fixed-point solution and vice versa. Thereby both solutions exist in symmetry related offering mode-two symmetry with left-or right-winding characteristics due to finite transverse magnetic field. Furthermore the time-periodic limit-cycle solutions offer alternately stroboscoping both helical left-and right-winding contributions of mode-two symmetry. The Navier-Stokes equations are solved with a second order time splitting method combined with spatial discretization of hybrid finite difference and Galerkin method.

Symmetry breaking in the opinion dynamics of a multi-group project organization

A bounded confidence model of opinion dynamics in multi-group projects is presented in which each group’s opinion evolution is driven by two types of forces:(i) the group’s cohesive force which tends to restore the opinion back towards the initial status because of its company culture;and(ii) nonlinear coupling forces with other groups which attempt to bring opinions closer due to collaboration willingness.Bifurcation analysis for the case of a two-group project shows a cusp catastrophe phenomenon and three distinctive evolutionary regimes,i.e.,a deadlock regime,a convergence regime,and a bifurcation regime in opinion dynamics.The critical value of initial discord between the two groups is derived to discriminate which regime the opinion evolution belongs to.In the case of a three-group project with a symmetric social network,both bifurcation analysis and simulation results demonstrate that if each pair has a high initial discord,instead of symmetrically converging to consensus with the increase of coupling scale as expected by Gabbay’s result(Physica A 378(2007) p.125 Fig.5),project organization(PO) may be split into two distinct clusters because of the symmetry breaking phenomenon caused by pitchfork bifurcations,which urges that apart from divergence in participants’ interests,nonlinear interaction can also make conflict inevitable in the PO.The effects of two asymmetric level parameters are tested in order to explore the ways of inducing dominant opinion in the whole PO.It is found that the strong influence imposed by a leader group with firm faith on the flexible and open minded follower groups can promote the formation of a positive dominant opinion in the PO.

Crystal engineering regulation achieving inverse temperature symmetry breaking ferroelasticity in a cationic displacement type hybrid perovskite system

Ferroelastic hybrid perovskite materials have been revealed the significance in the applications of switches,sensors,actuators,etc.However,it remains a challenge to design high-temperature ferroelastic to meet the requirements for the practical applications.Herein,we reported an one-dimensional organicinorganic hybrid perovskites(OIHP)(3-methylpyrazolium)CdCl_(3)(3-MBCC),which possesses a mmmF2/m ferroelastic phase transition at 263 K.Moreover,utilizing crystal engineering,we replace-CH_(3) with-NH_(2) and-H,which increases the intermolecular force between organic cations and inorganic frameworks.The phase transition temperature of(3-aminopyrazolium)CdCl_(3)(3-ABCC),and(pyrazolium)CdCl_(3)(BCC)increased by 73 K and 10 K,respectively.Particularly,BCC undergoes an unconventional inverse temperature symmetry breaking(ISTB)ferroelastic phase transition around 273 K.Differently,it transforms from a high symmetry low-temperature paraelastic phase(point group 2/m)to a low symmetry high-temperature ferroelastic phase(point group ī)originating from the rare mechanism of displacement of organic cations phase transition.It means that crystal BCC retains in ferroelastic phase above 273 K until melting point(446 K).Furthermore,characteristic ferroelastic domain patterns on crystal BCC are confirmed with polarized optical microscopy.Our study enriches the molecular mechanism of ferroelastics in the family of organic-inorganic hybrids and opens up a new avenue for exploring high-temperature ferroic materials.

Experimental observation of spontaneous symmetry breaking in a quantum phase transition

Spontaneous symmetry breaking(SSB)plays a central role in understanding a large variety of phenomena associated with phase transitions,such as superfluid and superconductivity.So far,the transition from a symmetric vacuum to a macroscopically ordered phase has been substantially explored.The process bridging these two distinct phases is critical to understanding how a classical world emerges from a quantum phase transition,but so far remains unexplored in experiment.We here report an experimental demonstration of such a process with a quantum Rabi model engineered with a superconducting circuit.We move the system from the normal phase to the superradiant phase featuring two symmetry-breaking field components,one of which is observed to emerge as the classical reality.The results demonstrate that the environment-induced decoherence plays a critical role in the SSB.

Symmetry breaking of highly symmetrical nanoclusters for triggering highly optical activity

Developing new approaches to fulfill the enantioseparation of nanocluster racemates and construct cluster-based nanomaterials with optical activity remains highly desired in cluster science,because it is an essential prerequisite for fundamental research and extensive applications of these nanomaterials.We herein propose a strategy termed“active-site exposing and partly re-protecting”to trigger the symmetry breaking of highly symmetrical nanoclusters and to render cluster crystals optically active.The vertex PPh_(3)of the symmetrical Ag_(29)(SSR)_(12)(PPh_(3))_(4)(SSR=1,3-benzenedithiol)nanocluster was firstly dissociated in the presence of counterions with large steric hindrance,and then the exposed Ag active sites of the obtained Ag_(29)(SSR)_(12)nanocluster were partly re-protected by Ag^(+),yielding an Ag_(29)(SSR)_(12)-Ag_(2)nanocluster with a symmetry-breaking construction.Ag_(29)(SSR)_(12)-Ag_(2)followed a chiral crystallization mode,and its crystal displayed strong optical activity,derived from CD and CPL characterizations.Overall,this work presents a new approach(i.e.,active-site exposing and partly re-protecting)for the symmetry breaking of highly symmetrical nanoclusters,the enantioseparation of nanocluster racemates,and the achievement of highly optical activity.