Lateral inhibitory effect is a well-known feature of information processing in neural systems. This paper presents a neural array model with simple lateral inhibitory connections. After detailed examining into the dynamics of this kind of neural array, the author gives the sufficient conditions under which the outputs of the network will tend to a special stable pattern called spatial sparse pattern in which if the output of a neuron is 1, then the outputs of the neurons in its neighborhood are 0. This ability called spatial sparse coding plays an important role in self-coding, self-organization and associative memory for patterns and pattern sequences. The main conclusions about the dynamics of this kind of neural array which is related to spatial sparse coding are introduced.
The author establishes the long-time existence and convergence results of the mean curvature flow of entire Lagrangian graphs in the pseudo-Euclidean space, which is related to the logarithmic Monge-Ampère flow.
A new class of Gorenstein algebras T m,n(A) is introduced, their module categories are described, and all the Gorenstein-projective T m,n(A)-modules are explicitly determined.
The authors express the essential norms of composition operators between Hardy spaces of the unit disc in terms of the natural Nevanlinna counting function.
The Cauchy problem to the Oldroyd-B model is studied. In particular, it is shown that if the smooth solution (u, τ) to this system blows up at a finite time T*, then ∫0 T* ‖▿u(t)‖ L ∞dt = ∞. Furthermore, the global existence of smooth solution to this system is given with small initial data.
This paper mainly deals with the type II singularities of the mean curvature flow from a symplectic surface or from an almost calibrated Lagrangian surface in a Kähler surface. The relation between the maximum of the Kähler angle and the maximum of |H|2 on the limit flow is studied. The authors also show the nonexistence of type II blow-up flow of a symplectic mean curvature flow which is normal flat or of an almost calibrated Lagrangian mean curvature flow which is flat.
In this paper, the sharp estimates of all homogeneous expansions for f are established, where f(z) = (f 1(z), f 2(z), …, f n(z))′ is a k-fold symmetric quasi-convex mapping defined on the unit polydisk in ℂ n and $\begin{gathered} \frac{{D^{tk + 1} + f_p \left( 0 \right)\left( {z^{tk + 1} } \right)}}{{\left( {tk + 1} \right)!}} = \sum\limits_{l_1 ,l_2 ,...,l_{tk + 1} = 1}^n {\left| {apl_1 l_2 ...l_{tk + 1} } \right|e^{i\tfrac{{\theta pl_1 + \theta pl_2 + ... + \theta pl_{tk + 1} }}{{tk + 1}}} zl_1 zl_2 ...zl_{tk + 1} ,} \hfill \\ p = 1,2,...,n. \hfill \\ \end{gathered} $ Here i = $\sqrt { - 1} $, θ plq ∈ (−θ,θ] (q = 1, 2, …, tk + 1), l 1, l 2, …, l tk+1 = 1, 2, …, n, t = 1, 2, …. Moreover, as corollaries, the sharp upper bounds of growth theorem and distortion theorem for a k-fold symmetric quasi-convex mapping are established as well. These results show that in the case of quasi-convex mappings, Bieberbach conjecture in several complex variables is partly proved, and many known results are generalized.
In this paper, a super version of the Hopf quiver theory is developed. The notion of Hopf superquivers is introduced. It is shown that only the path supercoalgebras of Hopf superquivers admit graded Hopf superalgebra structures. A complete classification of such graded Hopf superalgebras is given. A superquiver setting for general pointed Hopf superalgebras is also built up. In particular, a super version of the Gabriel type theorem and the Cartier-Gabriel decomposition theorem is given.
The authors investigate the global existence and semiclassical limit of weak solutions to a sixth-order parabolic system, which is a quantum-corrected macroscopic model derived recently to simulate the quantum effects in miniaturized semiconductor devices.
The authors discuss one type of general forward-backward stochastic differential equations (FBSDEs) with Itô’s stochastic delayed equations as the forward equations and anticipated backward stochastic differential equations as the backward equations. The existence and uniqueness results of the general FBSDEs are obtained. In the framework of the general FBSDEs in this paper, the explicit form of the optimal control for linearquadratic stochastic optimal control problem with delay and the Nash equilibrium point for nonzero sum differential games problem with delay are obtained.
The author establishes operator-valued Fourier multiplier theorems on multi-dimensional Hardy spaces H p($\mathbb{T}$ d;X), where 1 ≤ p < ∞, d ∈ ℕ, and X is an AUMD Banach space having the property (α). The sufficient condition on the multiplier is a Marcinkiewicz type condition of order 2 using Rademacher boundedness of sets of bounded linear operators. It is also shown that the assumption that X has the property (α) is necessary when d ≥ 2 even for scalar-valued multipliers. When the underlying Banach space does not have the property (α), a sufficient condition on the multiplier of Marcinkiewicz type of order 2 using a notion of d-Rademacher boundedness is also given.
In 2004, Tong found bounds for the approximation quality of a regular continued fraction convergent to a rational number, expressed in bounds for both the previous and next approximation. The authors sharpen his results with a geometric method and give both sharp upper and lower bounds. The asymptotic frequencies that these bounds occur are also calculated.