Langrangian Support Vector Machine (LSVM) and Least Squares Support Vector Machine (LSSVM) are two quick and effective classification methods. In this paper, we first introduce the mathematical models for LSVM and LSSVM and analyze their properties. In the nonlinear case, Sherman-Morrison-Woodbury identity is not used to compute the inversion of a matrix. According to block computation of a mat...

Recent evidence has shown that face space represents facial identity information using two-pool opponent coding. Here we ask whether the shape of the monotonic neural response functions underlying such coding is linear (i.e. face space codes all equal-sized physical changes with equal sensitivity) or nonlinear (e.g. face space shows greater coding sensitivity around the average face). Using ada...

The concept of “identity” of a complex network system is proposed based on the method of the gauge field theory. The system is assumed to consist of many elements interacting with each other. The interaction weight is treated as a gauge potential. Change in the external environment surrounding the system is assumed to induce a unitary transformation of the state vector of the system, which is r...

We discuss and prove validity of the Hubbard-Stratonovich (HS) identities over hyperbolic domains which are used frequently in the studies on disordered systems and random matrices. We also introduce a counterpart of the HS identity arising in disordered systems with ”chiral” symmetry. Apart from this we outline a way of deriving the nonlinear σ-model from the gauge-invariant Wegner k−orbital m...

We discuss and prove validity of the Hubbard-Stratonovich (HS) identities over hyperbolic domains which are used frequently in the studies on disordered systems and random matrices. We also introduce a counterpart of the HS identity arising in disordered systems with ”chiral” symmetry. Apart from this we outline a way of deriving the nonlinear σ-model from the gauge-invariant Wegner k−orbital m...

The energy-critical, focusing nonlinear Schrödinger equation in the nonradial case reads as follows: i∂tu = −∆u− |u| 4 N−2 u, u(x, 0) = u0 ∈ H(R ), N ≥ 3. Under a suitable assumption on the maximal strong solution, using a compactness argument and a virial identity, we establish the global well-posedness and scattering in the nonradial case, which gives a positive answer to one open problem pro...

We consider maps defined on an open set of Rnþm having a fixed point whose linear part is the identity. We provide sufficient conditions for the existence of a stable manifold in terms of the nonlinear part of the map. These maps arise naturally in some problems of Celestial Mechanics. We apply the results to prove the existence of parabolic orbits of the spatial elliptic three-body problem. r ...

Hamiltonian particle systems may exhibit nonlinear hydrodynamic phenomena as the time evolution of the density fields of energy, momentum, and mass. In this Letter, an exact equation describing the time evolution is derived assuming the local Gibbs distribution at initial time. The key concept in the derivation is an identity similar to the fluctuation theorems. The Navier-Stokes equation is ob...

We study condensate solutions of a nonlinear elliptic equation in R2, which models a W -boson with a cosmic string background. The existence and an energy identity of condensate solutions are discussed, based on which the refined asymptotic behavior of condensate solutions is established by using infinite-dimensional dynamical systems theory. Applying the “shrinking-sphere" method, the symmetry...