On the fiber structure of symmetry invariance sets of solutions to quasilinear equations

Known results about the unique continuation of solutions of anisotropic linear and weakly nonlinear differential equations and the extension of the invariance property of these solutions are applied to prove theorems about the fiberwise quasianalyticity and the fiber structure of sets of symmetry invariance of solutions to quasilinear equations.     

Invariance Principles for Autonomous Infinite Delay Difference Systems

The invariance principles for autonomous difference systems with infinite delay are established.As applications of the obtained invariance principles, criteria for asymptotic stability and asymptotic constancy of solutions are also given.     

Inequality averse multi-utilitarian bargaining solutions

This paper introduces and analyzes the class of inequality averse multi-utilitarian solutions for cooperative bargaining problems. We show that generalized Gini solutions and inequality averse Choquet solutions are particular cases of this new multi-valued solution concept and provide a complete characterization in which an invariance property, consisting of a weakening of both the linear invariance axiom in Blackorby et al. (Econometrica 62:1161–1178, 1994) and the restricted invariance axiom in Ok and Zhou (Games Econ Behav 33:249–264, 2000), plays an important role. Moreover, by relaxing the assumptions involved in the characterization, the class is extended to include inequality loving multi-utilitarian solutions which are also studied in the paper.     

AN INVARIANCE OF CDF EQUATION

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Characterizing properties of approximate solutions for optimization problems

Approximate solutions for optimization problems become of interest if the ‘true’ optimum cannot be found: this may happen for the simple reason that an optimum does not exist or because of the ‘bounded rationality’ (or bounded accuracy) of the optimizer. This paper characterizes several approximate solutions by means of consistency and additional requirements. In particular we consider invariance properties. We prove that, where the domain contains optimization problems without maximum, there is no non-trivial consistent solution satisfying non-emptiness, translation and multiplication invariance. Moreover, we show that the class of ‘satisficing’ solutions is obtained, if the invariance axioms are replaced with Chernoff’s Choice Axiom.     

Nonsmooth analysis and fractional differential equations

In this paper we study Euler solutions, strong and weak invariance of solutions for fractional differential equations.     

Converging strong shock waves in magnetogasdynamics under isothermal condition

The similarity solutions to the problem of cylindrically symmetric strong shock waves in an ideal gas with a constant azimuthal magnetic field are presented. The flow behind the shock wave is assumed to spatially isothermal rather than adiabatic. We use the method of Lie group invariance to determine the possible class of self-similar solutions. Infinitesimal generators of Lie group transformations are determined by using the invariance surface conditions to the system and on the basis of arbitrary constants occurring in the expressions for the generators, four different possible cases of the solutions are reckoned and we observed that only two out of all possibilities hold self-similar solutions, one of which follows the power law and another follows the exponential law. To obtain the similarity exponents numerical calculations have been performed and comparison is made with the existing results in the literature. The flow patterns behind the shock are analyzed graphically.

     

Blow-Up Solutions of Modified Schrödinger Maps

We study blow-up solutions of modified Schrödinger maps. We observe the pseudo-conformal invariance by which explicit blow-up solutions can be constructed.     

Interior Condition on Suitable Weak Solutions to the 3D MHD Equations Via Pressure

We present an interior regularity condition for suitable weak solutions with respect to the magnetic pressure under the class of scaling invariance.     

脉冲时滞Hassell-Varley-Holling功能性反应捕食者—食饵系统周期解存在的充要条件

研究一类具Hassell-Varley-Holling功能性反应的非自治脉冲时滞捕食者一食饵系统的周期解存在性问题.基于重合度理论的延拓定理,发展了一种新的解的估计技巧,并运用拓扑度的同伦不变性,得到了这类系统周期解存在的充要条件.     

EXISTENCE OF PERIODIC SOLUTIONS FOR A DISCRETE-TIME MODEL OF TWO-CELL CNNS

We investigate a class of discrete-time model of two-cell cellular neural networks with symmetric template. By using the Lyapunov direct method, La-Salle's invariance principle, we discuss the existence and the stability of periodic solutions. The model considered has attractive 2-periodic and unstable 2-periodic solutions.     

Weak and Strong Solutions of the 3D Navier–Stokes Equations and Their Relation to a Chessboard of Convergent Inverse Length Scales

Journal of Nonlinear Science - Using the scale invariance of the Navier–Stokes equations to define appropriate space-and-time-averaged inverse length scales associated with weak solutions of...     

Remarks on proofs of conservation laws for nonlinear Schrödinger equations

Conservation laws of the charge and of the energy are proved for nonlinear Schrödinger equations with nonlinearities of gauge invariance in a way independent of approximate solutions.     

Symmetry analysis of a system of modified shallow-water equations

We revise the symmetry analysis of a modified system of one-dimensional shallow-water equations (MSWE) recently considered by Raja Sekhar and Sharma [Commun Nonlinear Sci Numer Simulat 2012;20:630–36]. Only a finite dimensional subalgebra of the maximal Lie invariance algebra of the MSWE, which in fact is infinite dimensional, was found in the aforementioned paper. The MSWE can be linearized using a hodograph transformation. An optimal list of inequivalent one-dimensional subalgebras of the maximal Lie invariance algebra is constructed and used for Lie reductions. Non-Lie solutions are found from solutions of the linearized MSWE.     

Exact Solutions of Linear Partial Differential Equations with Variable Coefficients

Bäcklund transformations relating the solutions of linear PDE with variable coefficients to those of PDE with constant coefficients are found, generalizing the study of Varley and Seymour [2]. Auto-Bäcklund transformations are also determined. To facilitate the generation of new solutions via Bäcklund transformation, explicit solutions of both classes of the PDE just mentioned are found using invariance properties of these equations and other methods. Some of these solutions are new.     

On scaling invariance and type-I singularities for the compressible Navier-Stokes equations

Science China Mathematics - We find a new scaling invariance of the barotropic compressible Navier-Stokes equations. Then it is shown that type-I singularities of solutions with $$\mathop {\lim...     

Variational-iterative Calculations for Symmetric Operators on Real and Complex Spaces

Approximate solutions to problems from quantum scattering theoryand the theory of gauge invariance are obtained. The methodused is a variational-iterative technique applied to operatorson a complex space, not necessarily with a discrete spectrum.     

On the Reduction of Some Dispersionless Integrable Systems

We study the invariance, reduction, exact solutions and conservation laws of the dispersionless Kadomtsev-Petviashivili and the heavenly equation. The existence of nontrivial conservation laws lead to repeated reductions paving the way for determining exact solutions. A variety of such solutions exist dependent on the subalgebra of Lie point symmetries that is chosen in the reduction procedure.     

Oscillation theorems for nth order nonlinear functional differential equations

In the modern geometric approach partial differential equations are cast into equivalent ideals of differential forms. The invariance of forms under transformation groups is used for constructing invariant solutions by geometric methods. In the present paper the concept of partially invariant solutions introduced earlier by Ovsjannikov is studied in order to obtain geometric methods for partially invariant solutions too.     

Linear equations and heights over division algebras

We make a start on the investigation of “small” solutions to systems of homogeneous linear equations over non-commutative division algebras. In this paper we prove some upper and lower bounds for the sizes of solutions to such systems. To measure solutions and coefficient matrices we define heights which satisfy natural invariance and finiteness properties.