Based on the analytic property of the symmetric q-exponent e q(x), a new symmetric q-deformed Kadomtsev-Petviashvili (q-KP for short) hierarchy associated with the symmetric q-derivative operator ∂ q is constructed. Furthermore, the symmetric q-CKP hierarchy and symmetric q-BKP hierarchy are defined. The authors also investigate the additional symmetries of the symmetric q-KP hierarchy.
The groups as mentioned in the title are classified up to isomorphism. This is an answer to a question proposed by Berkovich and Janko.
Cheng-type inequality, Cheeger-type inequality and Faber-Krahn-type inequality are generalized to Finsler manifolds. For a compact Finsler manifold with the weighted Ricci curvature bounded from below by a negative constant, Li-Yau’s estimation of the first eigenvalue is also given.
For a family of smooth functions, the author shows that, under certain generic conditions, all extremal (minimal and maximal) points are non-degenerate.
Let M be a connected orientable compact irreducible 3-manifold. Suppose that ∂M consists of two homeomorphic surfaces F 1 and F 2, and both F 1 and F 2 are compressible in M. Suppose furthermore that g(M,F 1) = g(M) + g(F 1), where g(M,F 1) is the Heegaard genus of M relative to F 1. Let M f be the closed orientable 3-manifold obtained by identifying F 1 and F 2 using a homeomorphism f: F 1 → F 2. The authors show that if f is sufficiently complicated, then g(M f) = g(M,∂M) + 1.
The authors obtain some gradient estimates for positive solutions to the following nonlinear parabolic equation: \frac{{\partial u}}{{\partial t}} = \Delta u - b(x,t)u^\sigma on complete noncompact manifolds with Ricci curvature bounded from below, where 0 < σ < 1 is a real constant, and b(x, t) is a function which is C 2 in the x-variable and C 1 in the t-variable.
In this paper, the authors prove a Schwarz-Pick lemma for bounded complex-valued harmonic functions in the unit ball of ℝ n.
Let n = p 1 p 2 ⋯ p k, where p i (1 ≤ i ≤ k) are primes in the descending order and are not all equal. Let Ωk(n) = P(p 1+p 2)P(p 2+p 3) ⋯ P(p k−1+p k)P(p k +p 1), where P(n) is the largest prime factor of n. Define w 0(n) = n and w i(n) = w(w i−1(n)) for all integers i ≥ 1. The smallest integer s for which there exists a positive integer t such that Ω k s(n) = Ω k s+t(n) is called the index of periodicity of n. The authors investigate the index of periodicity of n.
A high-codimension homoclinic bifurcation is considered with one orbit flip and two inclination flips accompanied by resonant principal eigenvalues. A local active coordinate system in a small neighborhood of homoclinic orbit is introduced. By analysis of the bifurcation equation, the authors obtain the conditions when the original flip homoclinic orbit is kept or broken. The existence and the existence regions of several double periodic orbits and one triple periodic orbit bifurcations are proved. Moreover, the complicated homoclinic-doubling bifurcations are found and expressed approximately.
The authors integrate two well-known systems, the Rössler and Lorentz systems, to introduce a new chaotic system, called the Lorentz-Rössler system. Then, taking into account the effect of environmental noise, the authors incorporate white noise in both Rössler and Lorentz systems to have a corresponding stochastic system. By deriving the uniform a priori estimates for an approximate system and then taking them to the limit, the authors prove the global existence, uniqueness and the pathwise property of solutions to the Lorentz-Rössler system. Moreover, the authors carried out a number of numerical experiments, and the numerical results demonstrate their theoretic analysis and show some new qualitative properties of solutions which reveal that the Lorentz-Rössler system could be used to design more complex and more secure nonlinear hop-frequence time series.
A horizontal Hodge Laplacian operator ▭ H is defined for Hermitian holomorphic vector bundles over PTM on Kähler Finsler manifold, and the expression of ▭ H is obtained explicitly in terms of horizontal covariant derivatives of the Chern-Finsler connection. The vanishing theorem is obtained by using the \partial _\mathcal{H} \bar \partial _\mathcal{H} -method on Kähler Finsler manifolds.
This paper deals with an alternative proof of Beurling-Lax theorem by adopting a constructive approach instead of the isomorphism technique which was used in the original proof.
In this paper, solutions of the following non-Lipschitz stochastic differential equations driven by G-Brownian motion: X_t = x + \int_0^t {b(s,\omega ,X_s )ds} + \int_0^t {h(s,\omega ,X_s )d\left\langle B \right\rangle _s } + \int_0^t {\sigma (s,\omega ,X_s )dB_s } are constructed. It is shown that they have the cocycle property. Moreover, under some special non-Lipschitz conditions, they are bi-continuous with respect to t, x.