Generalized diagonalization of matrices over quaternion field

Ｉｎｒｅｃｅｎｔｙｅａｒｓ,ａｐｐｌｉｃａｔｉｏｎｓｏｆｑｕａｔｅｒｎｉｏｎｍａｔｒｉｃｅｓａｒｅｂｅｃｏｍｉｎｇｍｏｒｅａｎｄｍｏｒｅｉｍｐｏｒｔａｎｔａｎｄｅｘｔｅｎｓｉｖｅｉｎｒｉｇｉｄｍｅｃｈａｎｉｃｓ,ｑｕａｎｔｕｍｍｅｃｈａｎｉｃｓ,ｃｏｎｔｒｏｌｔｈｅｏｒｙａｎｄｈｅｌｉｃａｌｔｅｃｈｎｏｌｏｇｙ［１～３］．Ｗｉｔｈｔｈｅｒａｐｉｄｄｅｖｅｌｏｐｍｅｎｔｏｆｔｈｅａｂｏｖｅｄｉｓｃｉｐｌｉｎｅｓ,ｉｔｉｓｇｅｔｔｉｎｇｍｏｒｅａｎｄｍｏｒｅｎｅｃｅｓｓａｒｙｆｏｒｕｓｔｏｆｕｒｔｈ…     

Linearized oscillations for autonomous delay differential equations

ＩｎｔｒｏｄｕｃｔｉｏｎＣｏｎｓｉｄｅｒｔｈｅｎｏｎｌｉｎｅａｒａｕｔｏｎｏｍｏｕｓｄｅｌａｙｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｘ′(ｔ) ∑ｍｉ=1ｐｉｆｉ(ｘ(ｔ-τｉ) ) =0 (1 )ａｎｄｔｈｅｌｉｎｅａｒｅｑｕａｔｉｏｎｘ′(ｔ) ∑ｍｉ=1ｐｉｘ(ｔ-τｉ) =0 ,(2 )ｗｈｅｒｅ ,ｐｉ ∈ (0 ,∞ ) ,τｉ ∈ [0 ,∞ )ａｎｄｆｉ ∈Ｃ(Ｒ ,Ｒ)ｆｏｒｉ =1 ,… ,ｍ .ＩｎＲｅｆ.[1 ] ,ｔｈｅｌｉｎｅａｒｉｚｅｄｏｓｃｉｌｌａｔｉｏｎｓｏｆＥｑ .(1 )ｗａｓｓｔｕｄｉｅｄａｎｄｉｔｗａｓｐｒｏｖｅｄｔｈａｔｉｆｔ…     

Solution of A 2-D weak singular integral equation with constraint

In this paper, the solution of a 2-D weak singular integral equation of the first kind

Closedness and normal solvability of an operator generated by a degenerate linear differential equation with variable coefficients

For a linear operator generated by the differential equation

Existence of a Solution “in the Large” for Ocean Dynamics Equations

For the system of equations describing the large-scale ocean dynamics, an existence and uniqueness theorem is proved “in the large”. This system is obtained from the 3D Navier–Stokes equations by changing the equation for the vertical velocity component u ₃ under the assumption of smallness of a domain in z-direction, and a nonlinear equation for the density function ρ is added. More precisely, it is proved that for an arbitrary time interval [0, T], any viscosity coefficients and any initial conditions

An L∞ bound for solutions of the Cahn-Hilliard equation

In this paper we consider the equation

About a condition for blow up of solutions of cauchy problem for a wave equation

ＣｏｎｓｉｄｅｒｔｈｅＣａｕｃｈｙｐｒｏｂｌｅｍｆｏｒｔｈｅｗａｖｅｅｑｕａｔｉｏｎｉｎＲＮ×Ｒ＋（Ｎ≥２）：２ｕ（ｘ,ｔ）ｔ２－ｘｉａｉｊ（ｘ）ｘｊｕ＝｜ｕ｜ｐ－１·ｕ　　（（ｘ,ｔ）∈ＲＮ×（０,Ｔ））,ｕ（ｘ,０）＝ｇ（ｘ）　（ｘ∈ＲＮ）,ｕｔ（ｘ,０）＝ｈ（ｘ）　（ｘ∈ＲＮ）,（１）ｗｈｅｒｅｕ（ｘ,ｔ）ｉｓｎｏｎｔｒｉｖｉａｌｓｏｌｕｔｉｏｎｗｉｔｈｆｉｎｉｔｅｓｐｅｅｄｏｆｐｒｏｐａｇａｔｉｏｎａｎｄｉｓｓｕｐｐｏｒｔｅｄｏｎａｆｏｒｗａｒｄｃｏｎｅ（ｘ,ｔ）·ｔ≥０,｜…     

Numerical study of a nonlinear integrodifferential equation

I.IntroductionItiswell-knobal.nthatKorteweg-deVriesequationisacanonicalmodeltodescribethebalanceofthenonlineareffectandthedispersiveeffectofaphysicalsystem.Thisequationpossessestheso-called'soliton"solution,whichhasbeenfoundnumericallybyZabuskyandKruskall'].Ho-c'Jlever,sometimesthebalanceofnonlinearityanddispersionofasystemmayleadtoa,integroditTerentialequationinsteadofadifferentialequation.Forinstance,inthestudyofvortexbreakdownofanunboundedrotatingfluidLeibovich12]derivedfollowingnonline…     

On nonlinear hyperbolic equation in unbounded domain

The following nonlinear hyperbolic equation is discussed in this paper: where The model comes from the transverse deflection equation of an extensible beam. We prove that there exists a unique local solution of the above equation as M depends on x.     

The extended Jordan's lemma and the relation between laplace transform and fourier transform

I.IntroductionItiswell4n0wnthatthecontourintergrationofcomp1exvariableftinctioncanmakealotintegrationveryconvenient.Jordan'slemmahasaveryimportantstatusintheonec0mplexvariableintegration,anditisveryusefulforavarityofintegration.Withthetheoryoffunctionsofo…     

A Helmholtz/Weyl Decomposition in Energy Space

For a bounded region in a Helmholtz/Weyl decomposition of the Sobolev space is given,with orthogonality with respect to the strain-energy inner product of elasticity (anisotropic or isotropic).     

On the existence of periodic solutions to higher dimensional periodic system with delay

Ｃｏｎｓｉｄｅｒｉｎｇｔｈｅｈｉｇｈｅｒｄｉｍｅｎｓｉｏｎａｌｐｅｒｉｏｄｉｃｓｙｓｔｅｍｓｗｉｔｈｄｅｌａｙｏｆｔｈｅｆｏｒｍｘ′（ｔ）＝Ａ（ｔ,ｘ（ｔ））ｘ（ｔ）＋ｆ（ｔ,ｘ（ｔ－τ））,（１）ｘ′（ｔ）＝ｇｒａｄＧ（ｘ（ｔ））＋ｆ（ｔ,ｘ（ｔ－...     

Stability of a class of neural network models with delay

ＩｎｔｒｏｄｕｃｔｉｏｎＲｅｃｅｎｔｌｙ,ｔｈｅｏｒｅｔｉｃａｌａｎｄａｐｌｉｅｄｓｔｕｄｉｅｓｏｆｎｅｕｒａｌｎｅｔｗｏｒｋｍｏｄｅｌｈａｖｅｂｅｎｔｈｅｎｅｗｆｏｃｕｓｏｆｓｔｕｄｉｅｓｉｎｔｈｅｗｏｒｌｄ．Ｉｔｉｓｗｅｌ＿ｋｎｏｗｎｔｈａｔｑｕ...     

Initial value problems for a class of nonlinear integro-partial differential equations

１ＰｒｏｂｌｅｍｓａｎｄＭａｉｎＲｅｓｕｌｔｓＩｎｔｈｉｓｐａｐｅｒ,ｗｅｓｔｕｄｙｔｈｅｎｏｎｌｉｎｅａｒｖｉｂｒａｔｉｏｎｓｏｆｉｎｆｉｎｉｔｅｒｏｄｓｗｉｔｈｖｉｓｃｏｅｌａｓｔｉｃｉｔｙ．Ｔｈｅｃｏｎｓｔｉｔｕｔｉｏｎｌａｗｏｆｔｈｅｒｏｄｓ...     

On the existence and stability of periodic solutions for hopfield neural network equations with delay

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｅｒｅｃｅｎｔｙｅａｒｓ,ｗｉｔｈａｌｏｔｏｆａｐｐｌｉｃａｔｉｏｎｓｏｆｎｅｕｒａｌｎｅｔｗｏｒｋｍｏｄｅｌｓ,ｍａｎｙａｕｔｈｏｒｓ［１～３］ａｒｅｉｎｔｅｒｅｓｔｅｄｉｎｔｈｅｒｅｓｅａｒｃｈｏｆｔｈｅｓｔｒｕｃｔｕｒｅａｎｄｐｅｒｆｏｒｍａｎｃｅｆｏｒｔｈｅｓｅｎｅｔｗｏｒｋｓ．ＢｅｃａｕｓｅＨｏｐｆｉｅｌｄｎｅｕｒａｌｎｅｔｗｏｒｋｗｅｌｌｓｉｍｕｌａｔｅｔｈｅｅｃｏｌｏｇｉｃａｌｓｙｓｔｅｍ,ｍａｎｙｓｔｕｄｉｅｓａｒｅｃｏｎｃｅｎｔｒａｔｅｄｏｎｔｈ…     

Asymptotic and oscillation of the solutions for a class of the first order neutral differential equations

ＩｎｔｒｏｄｕｃｔｉｏｎＯｗｉｎｇｔｏｔｈｅｅｘｔｅｎｓｉｖｅａｐｐｌｉｃａｔｉｏｎｏｆｎｅｕｔｒａｌｅｑｕａｔｉｏｎｓ,ｍｏｒｅａｎｄｍｏｒｅｓｔｕｄｉｅｓｈａｖｅｂｅｎｍａｄｅｏｎｔｈｅｂｅｈａｖｉｏｒｏｆｔｈｅｓｏｌｕｔｉｏｎｓ［１,２］．Ｆｏ...     

Existence of Weak Solutions to the Equations of Non-Stationary Motion of Non-Newtonian Fluids with Shear Rate Dependent Viscosity

In the present paper we prove the existence of weak solutions to the equations of non-stationary motion of an incompressible fluid with shear rate dependent viscosity in a cylinder Q = Ω × (0,T), where denotes an open set. For the power-low model with we are able to construct a weak solution with ∇ · u = 0.     

On the Local Smoothness of Solutions of the Navier–Stokes Equations

We consider the Cauchy problem for incompressible Navier–Stokes equations with initial data in , and study in some detail the smoothing effect of the equation. We prove that for T < ∞ and for any positive integers n and m we have , as long as stays finite.     

On the normal forms of Hamiltonian systems

We propose a novel method to analyze the dynamics of Hamiltonian systems with a periodically modulated Hamiltonian. The method is based on a special parametric form of the canonical transformation , using Poincaré generating function Ψ (t,x,y). As a result, stability problem of a periodic solution is reduced to finding a minimum of the Poincaré function. The proposed method can be used to find normal forms of Hamiltonians. It should be emphasized that we apply the modified concept of Zhuravlev [Introduction to Theoretical Mechanics. Nauka Fizmatlit, Moscow (1997); Prikladnaya Matematika i Mekhanika 66(3), (2002) in Russian] to define an invariant normal form, which does not require any partition to either autonomous – non-autonomous, or resonance – non-resonance cases, but it is treated in the frame of one approach. In order to find the corresponding normal form asymptotics, a system of equations is derived analogous to Zhuravlev's chain of equations. Instead of the generator method and guiding Hamiltonian, a parametrized guiding function is used. It enables a direct (without the transformation to an autonomous system as in Zhuravlev's method) computation of the chain of equations for non-autonomous Hamiltonians. For autonomous systems, the methods of computation of normal forms coincide in the first and second approximations. Using this method we will present solutions of the following problems: nonlinear Duffing oscillator; oscillation of a swinging spring; dynamics of solid particles in the acoustic wave of viscous liquid, and other problems.     

An Example of Finite-time Singularities in the 3d Euler Equations

Let be the exterior of the closed unit ball. Consider the self-similar Euler system