We present the NumericalImplicitization.m2 package for Macaulay2, which allows for user-friendly computation of the invariants of the image of a polynomial map, such as dimension, degree, and Hilbert function values. ...
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A recent numerical study observed that neural network classifiers enjoy a large degree of symmetry in the penultimate layer. Namely, if h(x) = Af(x) + b where A is a linear map and f is the output of the penultimate l...
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The advantages in formulation and use of semi-discretized approaches to the numerical solution of initial/boundary value problems are well known. The aim of this paper is to demonstrate that it is feasible to obtain a...
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The advantages in formulation and use of semi-discretized approaches to the numerical solution of initial/boundary value problems are well known. The aim of this paper is to demonstrate that it is feasible to obtain accurate results even with a coarse spatial mesh. A method is developed which produces in a simple manner matrix representations for high-order central difference operators. Dirichlet, Neumann and mixed boundary conditions are considered, both homogeneous and non-homogeneous. It is shown in all cases that, for linear problems at least, there is no need to use a finer mesh than that dictated by the essential frequency content of the initial function data.
The global attractor problem of a dissipative generalized KdV equation with periodic boundary condition was studied. The existence of global attractors of this problem was proved by means of a uniform a priori estimat...
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The global attractor problem of a dissipative generalized KdV equation with periodic boundary condition was studied. The existence of global attractors of this problem was proved by means of a uniform a priori estimate for time.
An approach to deal with movings and collisions of arbitrary many discontinuities in the conservative front tracking method is developed. Using this approach one may develop an ＂all-purposed and robust＂ front-tracki...
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An approach to deal with movings and collisions of arbitrary many discontinuities in the conservative front tracking method is developed. Using this approach one may develop an ＂all-purposed and robust＂ front-tracking algorithm. The algorithm with this approach may have some inconsistency and thus will have 0（1） magnitude errors in some grid cells at sometimes. Nevertheless, these errors will be eliminated by the conservation-preserving property of the front-tracking method in the following computation. Numerical examples are presented to illustrate the efficiency of the approach.
SiO^+ and SiO, which play vital roles in astrophysics and astrochemistry, have long attracted considerable ***, accurate information about excited states of SiO^+ is still limited. In this work, the structures of 14 ...
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SiO^+ and SiO, which play vital roles in astrophysics and astrochemistry, have long attracted considerable ***, accurate information about excited states of SiO^+ is still limited. In this work, the structures of 14 Λ–S states and 30? states of SiO^+ are computed with explicitly correlated configuration interaction method. On the basis of the calculated potential energy curves of those Λ–S states and ? states, the spectroscopic constants of bound states are evaluated, which are in good agreement with the latest experimental results. The predissociation mechanism of B^2Σ^+ state is illuminated with the aid of spin–orbit coupling matrix elements. On the basis of the calculated potential energy curves and transition dipole moments, the radiative lifetime for each of low-lying vibrational states B^2Σ^+and A^2Π is estimated. The laser cooling scheme of SiO^+ is proposed by employing B^2Σ^+–X^2Σ^+ transition. Finally, the vertical ionization energy values from SiO(X^1Σ^+) to ionic states: SiO^+ , X^2Σ^+, B^2Σ^+, and A^2Π are calculated, which agree well with experimental measurements.
Let G be a simple, connected, triangle-free graph with minimum degree δ, order n and leaf number L(G). If G has a cut-vertex, we prove that L(G)≥4δ-4 and n≥4δ-1. Both lower bounds are sharp. The lower bound on th...
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In this paper,we study the initial-boundary value problem of one class of nonlinear Schr(o)dinger equations described in molecular ***,the existence of the global solution is obtained by means of interpolation inequal...
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In this paper,we study the initial-boundary value problem of one class of nonlinear Schr(o)dinger equations described in molecular ***,the existence of the global solution is obtained by means of interpolation inequality and a priori estimation.
It is known that ion channel can effectively limit the radial expansion of an artificial electron beam during its longrange propagation in the space plasma *** prior studies discussed the focusing characteristics of t...
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It is known that ion channel can effectively limit the radial expansion of an artificial electron beam during its longrange propagation in the space plasma *** prior studies discussed the focusing characteristics of the beam in the ion channel,but the establishment process and transient properties of the ion channel itself,which also plays a crucial role during the propagation of the relativistic electron beam in the plasma environment,were commonly *** this study,a series of two-dimensional(2D)particle-in-cell simulations is performed and an analytical model of ion channel oscillation is constructed according to the single-particle *** results showed that when the beam density is higher than the density of plasma environment,ion channel can be established and always continues to oscillate periodically over the entire *** factors,including the beam electron density,initial beam radius,and the plasma density can affect the oscillation properties of ion *** velocity of the beam oscillates synchronously with the ion channel and this phenomenon will finally develop into a two-stream instability which can seriously affect the effective transport for relativistic electron *** appropriate beam parameters based on various plasma environments may contribute to the improvement of the stability of ion ***,radial expansion of the beam can be limited by ion channel and a stable long-range propagation in terrestrial atmosphere may be achieved.
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