Global solution of a diffusive predator–prey model with prey-taxis

We consider the prey-taxis system:

$\left\{\begin{array}{cc}{u}_t=d_1\Delta u?\chi ??(u?v)+u(a?\mu u)+buf\left(v\right),\hfill & x\in \Omega ,t>0,\hfill \\ v_t=d_2\Delta v+v(c?\beta v)?uf\left(v\right),\hfill & x\in \Omega ,t>0\hfill \end{array}\right.$

in a smoothly bounded domain $\Omega ?{\mathbb{R}}^n$, with zero-flux boundary condition, where $a,d_1,d_2,\chi ,\mu ,b,c$ are positive constants and $\beta$ is a non-negative constant. We first investigate the global existence and local boundedness of solution for the case $\beta =0$. Moreover, when $\beta >0$, we show that the solution exists globally and is uniformly bounded provided $\mu$ is large enough.     

On the absence of global solutions for quantum versions of Schrödinger equations and systems

We, first, consider the quantum version of the nonlinear Schrödinger equation

$i^q{D}_{q|t}u(t,x)+\Delta u(qt,x)=\lambda |u(qt,x){|}^p,t>0,x\in {\mathbb{R}}^N,$

where $0<q<1$, $i^q$ is the principal value of $i^q$, $D_{q|t}$ is the $q$-derivative with respect to $t$, $\Delta$ is the Laplacian operator in ${\mathbb{R}}^N$, $\lambda \in ??\left\{0\right\}$, $p>1$, and $u(t,x)$ is a complex-valued function. Sufficient conditions for the nonexistence of global weak solution to the considered equation are obtained under suitable initial data. Next, we study the system of nonlinear coupled equations

$i^q{D}_{q|t}u(t,x)+\Delta u(qt,x)=\lambda |v(qt,x){|}^p,t>0,x\in {\mathbb{R}}^N,$

$i^q{D}_{q|t}v(t,x)+\Delta v(qt,x)=\lambda |u(qt,x){|}^m,t>0,x\in {\mathbb{R}}^N,$     

Note on nonparametric spectral analysis of wideband spectrum with missing data via sample-and-hold interpolation and deconvolution

In [1] a procedure for bias-free estimation of the autocorrelation function is introduced for equidistantly sampled data with randomly occurring samples being invalid. The method incorporates sample-and-hold interpolation of the missing data points. The occurring dynamic error of the primary estimate of the correlation function is treated by a deconvolution procedure with two parameters $c_0$ and $c_1$ with $c_0+2c_1=1$, which are the on-diagonal and the aside-diagonal parameters of a specific correction matrix (at all lag times except zero). The parameters $c_0$ and $c_1$ were obtained as a function of the probability α of a sample to be valid by numerical simulation. However, explicit expressions for the parameters $c_0(\alpha )=1-\frac{2}{\alpha }+\frac{2}{{\alpha }^2}$ and $c_1(\alpha )=\frac{1}{\alpha }-\frac{1}{{\alpha }^2}$ can be derived, which might improve the usability of the deconvolution procedure in [1].     

On the system of word equations x i 1 x i 2…x i m=y i 1 y i 2…y i n (i=1, 2, …) in a free monoid

 It is proved that the system of word equations x ⁱ ₁=y ⁱ ₁ y ⁱ ₂…y ⁱ _n , i=1, 2,…, ⌈n/2⌉ +1, has only cyclic solutions. Some sharpenings concerning the cases n=5, 7 and n≥9 are derived as well as results concerning the general system of equations x ⁱ ₁ x ⁱ ₂…x ⁱ _m =y ⁱ ₁ y ⁱ ₂…y ⁱ _n , i=1, 2,… . Applications to test sets of certain bounded languages are considered. Received: 18 May 1995/2 January 1996     

Mason: morphological simplification

The traditional rounding and filleting morphological filters are biased. Hence, as r grows, the rounding R_r (S) of S shrinks and the filleting F_r (S) grows. A shape S is r-regular when R_r (S) = F_r (S) = S. The combinations F_r (R_r (S)) and R_r (F_r (S)) produce nearly r-regular shapes, but retain a bias: F_r (R_r (S)) is usually smaller than S and R_r (F_r (S)) is larger. To overcome this bias, we propose a new filter, called Mason. The r-mortar M_r (S) of S is F_r (S)–R_r (S), and the stability of a point P with respect to S is the smallest value of r for which P belongs to M_r (S). Stability provides important information about the shape’s imbedding that cannot be obtained through traditional topological or differential analysis tools. F_r (R_r (S)) and R_r (F_r (S)) only affect space in M_r (S). For each maximally connected component of M_r (S), Mason performs either F_r (R_r (S)) or R_r (F_r (S)), choosing the combination that alters the smallest portion of that component. Hence, Mason acts symmetrically on the shape and on its complement. Its output is guaranteed to have a smaller symmetric difference with the original shape than that of either combination F_r (R_r (S)) or R_r (F_r (S)). Many previously proposed shape simplification algorithms were focused on reducing the combinatorial storage or processing costs of a shape at the expense of the smoothness and regularity or altered the shape in regular portions that did not exhibit any high frequency complexity. Mason is the first shape simplification operator that is independent of the particular representation and offers the advantage of preserving portions of the boundary of S that are regular at the desired scale.     

Rational Solutions of Riccati-like Partial Differential Equations

When factoring linear partial differential systems with a finite-dimensional solution space or analysing symmetries of nonlinear ODEs, we need to look for rational solutions of certain nonlinear PDEs. The nonlinear PDEs are called Riccati-like because they arise in a similar way as Riccati ODEs. In this paper we describe the structure of rational solutions of a Riccati-like system, and an algorithm for computing them. The algorithm is also applicable to finding all rational solutions of Lie’s system { ∂_xu + u² + a₁u + a₂v + a₃, ∂_yu + uv + b₁u + b₂v + b₃, ∂_xv + uv + c₁u + c₂v + c₃, ∂_yv + v² + d₁u + d₂v + d₃},where a₁, . . . , d₃are rational functions of x and y.     

Curves of Genus 2 with (N,N) Decomposable Jacobians

Let C be a curve of genus 2 and ψ₁: C  −  → E ₁  a map of degree n, from C to an elliptic curveE₁ , both curves defined over C. This map induces a degree n map φ₁:P¹  −  → P ¹  which we call a Frey–Kani covering. We determine all possible ramifications for φ₁. If ψ₁:C  −  → E ₁  is maximal then there exists a maximal map ψ₂: C  −  → E ₂  , of degree n, to some elliptic curveE₂ such that there is an isogeny of degree n²from the JacobianJ_C to E₁ × E₂. We say thatJ_C is (n, n)-decomposable. If the degree n is odd the pair (ψ₂, E₂) is canonically determined. For n =  3, 5, and 7, we give arithmetic examples of curves whose Jacobians are (n, n)-decomposable.     

Superconvergence analysis of nonconforming finite element method for time-fractional nonlinear parabolic equations on anisotropic meshes

In this paper, we prove a novel result of the consistency error estimate with order $O\left(h^2\right)$ for $E{Q}_1^{rot}$ element (see Lemma 2) on anisotropic meshes. Then, a linearized fully discrete Galerkin finite element method (FEM) is studied for the time-fractional nonlinear parabolic problems, and the superclose and superconvergent estimates of order $O(\tau +h^2)$ in broken $H^1$-norm on anisotropic meshes are derived by using the proved character of $E{Q}_1^{rot}$ element, which improve the results in the existing literature. Numerical results are provided to confirm the theoretical analysis.