Separation of variables for soliton equations via their binary constrained flows

Binary constrained flows of soliton equations admitting $2\times 2$ Lax matrices have 2N degrees of freedom, which is twice as many as degrees of freedom in the case of mono-constrained flows. For their separation of variables only N pairs of canonical separated variables can be introduced via their Lax matrices by using the normal method. A new method to introduce the other N pairs of canonical separated variables and additional separated equations is proposed. The Jacobi inversion problems for binary constrained flows are established. Finally, the factorization of soliton equations by two commuting binary constrained flows and the separability of binary constrained flows enable us to construct the Jacobi inversion problems for some soliton hierarchies.Comment: 39 pages, Amste

Algebraic Structure of Discrete Zero Curvature Equations and Master Symmetries of Discrete Evolution Equations

An algebraic structure related to discrete zero curvature equations is established. It is used to give an approach for generating master symmetries of first degree for systems of discrete evolution equations and an answer to why there exist such master symmetries. The key of the theory is to generate nonisospectral flows $(\lambda_t=\lambda ^l, l\ge0)$ from the discrete spectral problem associated with a given system of discrete evolution equations. Three examples are given.Comment: 24 pages, LaTex, revise

Monoidal Hom-Hopf algebras

Hom-structures (Lie algebras, algebras, coalgebras, Hopf algebras) have been investigated in the literature recently. We study Hom-structures from the point of view of monoidal categories; in particular, we introduce a symmetric monoidal category such that Hom-algebras coincide with algebras in this monoidal category, and similar properties for coalgebras, Hopf algebras and Lie algebras.Comment: 25 pages; extended version: compared to the version that appeared in Comm. Algebra, the Section Preliminary Results and Remarks 5.1 and 6.1 have been adde

A multiple exp-function method for nonlinear differential equations and its application

A multiple exp-function method to exact multiple wave solutions of nonlinear partial differential equations is proposed. The method is oriented towards ease of use and capability of computer algebra systems, and provides a direct and systematical solution procedure which generalizes Hirota's perturbation scheme. With help of Maple, an application of the approach to the $3+1$ dimensional potential-Yu-Toda-Sasa-Fukuyama equation yields exact explicit 1-wave and 2-wave and 3-wave solutions, which include 1-soliton, 2-soliton and 3-soliton type solutions. Two cases with specific values of the involved parameters are plotted for each of 2-wave and 3-wave solutions.Comment: 12 pages, 16 figure

Exact one-periodic and two-periodic wave solutions to Hirota bilinear equations in 2+1 dimensions

Riemann theta functions are used to construct one-periodic and two-periodic wave solutions to a class of (2+1)-dimensional Hirota bilinear equations. The basis for the involved solution analysis is the Hirota bilinear formulation, and the particular dependence of the equations on independent variables guarantees the existence of one-periodic and two-periodic wave solutions involving an arbitrary purely imaginary Riemann matrix. The resulting theory is applied to two nonlinear equations possessing Hirota bilinear forms: $u_t+u_{xxy}-3uu_y-3u_xv=0$ and $u_t+u_{xxxxy}-(5u_{xx}v+10u_{xy}u-15u^2v)_x=0$ where $v_x=u_y$, thereby yielding their one-periodic and two-periodic wave solutions describing one dimensional propagation of waves

A Census of Eddy Activities in the South China Sea During 1993-2007

Numerous mesoscale eddies occur each year in the South China Sea (SCS), but their statistical characteristics are still not well documented. A Pacific basin-wide three dimensional physical-biogeochemical model has been developed and the result in the SCS subdomain is used to quantify the eddy activities during the period of 1993-2007. The modeled results are compared with a merged and gridded satellite product of sea level anomaly by using the same eddy identification and tracking method. On average, there are about 32.9 +/- 2.4 eddies predicted by the model and 32.8 +/- 3.4 eddies observed by satellite each year, and about 52% of them are cyclonic eddies. The radius of these eddies ranges from about 46.5 to 223.5 km, with a mean value of 87.4 km. More than 70% of the eddies have a radius smaller than 100 km. The mean area covered by these eddies each year is around 160,170 km(2), equivalent to 9.8% of the SCS area with water depths greater than 1000 m. Linear relationships are found between eddy lifetime and eddy magnitude and between eddy vertical extent and eddy magnitude, showing that strong eddies usually last longer and penetrate deeper than weak ones. Interannual variations in eddy numbers and the total eddy-occupied area indicate that eddy activities in the SCS do not directly correspond to the El Nino-Southern Oscillation events. The wind stress curls are thought to be an important but not the only mechanism of eddy genesis in the SCS

Support vector machine classifier via L0/1 soft-margin loss

Support vector machines (SVM) have drawn wide attention for the last two decades due to its extensive applications, so a vast body of work has developed optimization algorithms to solve SVM with various soft-margin losses. To distinguish all, in this paper, we aim at solving an ideal soft-margin loss SVM: L0/1 soft-margin loss SVM (dubbed as L0/1 -SVM). Many of the existing (non)convex soft-margin losses can be viewed as one of the surrogates of the L0/1 soft-margin loss. Despite its discrete nature, we manage to establish the optimality theory for the L0/1 -SVM including the existence of the optimal solutions, the relationship between them and P-stationary points. These not only enable us to deliver a rigorous definition of L0/1 support vectors but also allow us to define a working set. Integrating such a working set, a fast alternating direction method of multipliers is then proposed with its limit point being a locally optimal solution to the L0/1 -SVM. Finally, numerical experiments demonstrate that our proposed method outperforms some leading classification solvers from SVM communities, in terms of faster computational speed and a fewer number of support vectors. The bigger the data size is, the more evident its advantage appears

ECON: Explicit Clothed humans Optimized via Normal integration

The combination of artist-curated scans, and deep implicit functions (IF), is enabling the creation of detailed, clothed, 3D humans from images. However, existing methods are far from perfect. IF-based methods recover free-form geometry but produce disembodied limbs or degenerate shapes for unseen poses or clothes. To increase robustness for these cases, existing work uses an explicit parametric body model to constrain surface reconstruction, but this limits the recovery of free-form surfaces such as loose clothing that deviates from the body. What we want is a method that combines the best properties of implicit and explicit methods. To this end, we make two key observations:(1) current networks are better at inferring detailed 2D maps than full-3D surfaces, and (2) a parametric model can be seen as a “canvas” for stitching together detailed surface patches. ECON infers high-fidelity 3D humans even in loose clothes and challenging poses, while having realistic faces and fingers. This goes beyond previous methods. Quantitative, evaluation of the CAPE and Renderpeople datasets shows that ECON is more accurate than the state of the art. Perceptual studies also show that ECON’s perceived realism is better by a large margin

Neutrino oscillations in de Sitter space-time

We try to understand flavor oscillations and to develop the formulae for describing neutrino oscillations in de Sitter space-time. First, the covariant Dirac equation is investigated under the conformally flat coordinates of de Sitter geometry. Then, we obtain the exact solutions of the Dirac equation and indicate the explicit form of the phase of wave function. Next, the concise formulae for calculating the neutrino oscillation probabilities in de Sitter space-time are given. Finally, The difference between our formulae and the standard result in Minkowski space-time is pointed out.Comment: 13 pages, no figure