Zigzag and Breakup Instabilities of Stripes and Rings in the Two-Dimensional Gray–Scott Model

Two different types of instabilities of equilibrium stripe and ring solutions are studied for the singularly perturbed two‐component Gray–Scott (GS) model in a two‐dimensional domain. The analysis is performed in the semi‐strong interaction limit where the ratio O(?^?2) of the two diffusion coefficients is asymptotically large. For ?→ 0 , an equilibrium stripe solution is constructed where the singularly perturbed component concentrates along the mid‐line of a rectangular domain. An equilibrium ring solution occurs when this component concentrates on some circle that lies concentrically within a circular cylindrical domain. For both the stripe and the ring, the spectrum of the linearized problem is studied with respect to transverse (zigzag) and varicose (breakup) instabilities. Zigzag instabilities are associated with eigenvalues that are asymptotically small as ?→ 0 . Breakup instabilities, associated with eigenvalues that are O(1) as ?→ 0 , are shown to lead to the disintegration of a stripe or a ring into spots. For both the stripe and the ring, a combination of asymptotic and numerical methods are used to determine precise instability bands of wavenumbers for both types of instabilities. The instability bands depend on the relative magnitude, with respect to ?, of a nondimensional feed‐rate parameter A of the GS model. Both the high feed‐rate regime A=O(1) , where self‐replication phenomena occurs, and the intermediate regime O(?^1/2) ?A?O(1) are studied. In both regimes, it is shown that the instability bands for zigzag and breakup instabilities overlap, but that a zigzag instability is always accompanied by a breakup instability. The stability results are confirmed by full numerical simulations. Finally, in the weak interaction regime, where both components of the GS model are singularly perturbed, it is shown from a numerical computation of an eigenvalue problem that there is a parameter set where a zigzag instability can occur with no breakup instability. From full‐scale numerical computations of the GS, it is shown that this instability leads to a large‐scale labyrinthine pattern.     

The Existence and Stability of Spike Equilibria in the One-Dimensional Gray–Scott Model: The Low Feed-Rate Regime

In a singularly perturbed limit of small diffusivity ɛ of one of the two chemical species, equilibrium spike solutions to the Gray–Scott (GS) model on a bounded one-dimensional domain are constructed asymptotically using the method of matched asymptotic expansions. The equilibria that are constructed are symmetric k -spike patterns where the spikes have equal heights. Two distinguished limits in terms of a dimensionless parameter in the reaction-diffusion system are considered: the low feed-rate regime and the intermediate regime. In the low feed-rate regime, the solution branches of k -spike equilibria are found to have a saddle-node bifurcation structure. The stability properties of these branches of solutions are analyzed with respect to the large eigenvalues λ in the spectrum of the linearization. These eigenvalues, which have the property that  λ= O (1)  as  ɛ→ 0  , govern the stability of the solution on an O (1) time scale. Precise conditions, in terms of the nondimensional parameters, for the stability of symmetric k -spike equilibrium solutions with respect to this class of eigenvalues are obtained. In the low feed-rate regime, it is shown that a large eigenvalue instability leads either to a competition instability, whereby certain spikes in a sequence are annihilated, or to an oscillatory instability (typically synchronous) of the spike amplitudes as a result of a Hopf bifurcation. In the intermediate regime, it is shown that only oscillatory instabilities are possible, and a scaling-law determining the onset of such instabilities is derived. Detailed numerical simulations are performed to confirm the results of the stability theory. It is also shown that there is an equivalence principle between spectral properties of the GS model in the low feed-rate regime and the Gierer–Meinhardt model of morphogenesis. Finally, our results are compared with previous analytical work on the GS model.     

On Legendre's equation ax2 + by2 + cz2 = 0

Let a, b, c be nonzero integers having no prime factors ≡ 3 (mod 4), not all of the same sign, abc squarefree, and for which Legendre's equation ax² + by² + cz² = 0 is solvable in nonzero integers x, y, z. A property is proved yielding a congruence which must be satisfied by any solution x, y, z.     

Possible values for 2

From GCH and P^{m(κ)-hypermeasurable} (1 <m<gw), we construct a model satisfying 2_ⁿ = _a(n) and 2_^ω = _ω+m for a monotone a:ω→ω satisfying a(n)>n.     

The simultaneous diophantine equations 5Y2 − 20 = X2 and 2Y2 + 1 = Z2

The three numbers 1, 5, 10 have the property that the product of any two numbers decreased by 1 is a perfect square. In this paper it is proved that there is no other positive integer N which shares this property with 1, 5, and 10.     

The quadratic form x2−2py2

Using known properties of continued fractions, we give a very simple and elementary proof of the theorem of Epstein and Rédei on the impossibility in a certain case of representing ?1 by the quadratic form x² ? 2py². Two of our theorems, which concern the representation of a² and ?2a², serve to extend our method to an unknown case in which ?1 is not representable.     

Diophantine方程y~2=px(x~2+2)

设p是大于3的奇素数.本文证明了:当p≡5或7(mod 8)时,方程y~2=px(x~2+2)无正整数解(x,y);当p≡1(mod 8)时,该方程至多有1组解;当p≡3(mod 8)时,该方程至多有2组解.     

A_α~2上复合算子的Hilbert-Schmidt差分

本文设A_α~2为定义在n维复空间单位多圆柱上的加权Bergman空间,L为Bergman空间上有界复合算子的全体并赋予算子范数拓扑,应用复合算子的HilbertSchmidt差分研究L的拓扑连通性.     

The automorphism group of the Toeplitz algebra onH2(T2)

The automorphism group of the C^* -algebra generated by the Toeplitz operatos with symbols being continuous functions on the bicircle is characterized completely. The investigation is based on the analysis of the behaviour of an automorphism of the Toeplitz algebra on itsC* -ideal chains, and the state of the closed ideals in .     

广义函数x+λ和x-μ的乘积

For any complcx λ≠integral, and any natural number n, the product is calcuated and Hadmard''s finite part of obtained. The product is also obtained when λ+μ is a complex number of any other possible kind, and μ is a complex number ≠ integral. The products are also calculated. Any special example of these kinds of products has not yet been calculated.     

On the Diophantine System a^2 ＋ b^2 = c^3 and a^x ＋ b^y = c^z for b is an Odd Prime

Let a, b and c be fixed coprime positive integers. In this paper we prove that if a^2 ＋ b^2 = c^3 and b is an odd prime, then the equation a^x ＋ b^y = c^z has only the positive integer solution （x, y, z） = （2,2,3）.     

Pell方程组x~2-(c~2-1)y~2=y~2-2p_1p_2p_3z~2=1的公解

设p_1,p_2,p_3为不同的奇素数,c1是整数.给出了Pell方程组x~2-(c~2-1)y~2=y~2-2p_1p_2p_3z~2=1的所有非负整数解(x,y,z),从而推广了Keskin (2017)和Cipu(2018)等人的结果.     

Evaluation of a cubic character sum using the √−19 division points of the curve Y2 = X3 − 23 · 19X + 2 · 192

The √?19 division points on the curve y² = f(x) of the title are calculated explicitly and the effect of the Frobenius map on these points is found in order to evaluate the cubic character sum $\text{Σ}{}_{\mathrm{<mtext>x(</mtext><mtext>mod</mtext><mtext>\; p)</mtext>}}\text{(}\text{f(x)}\text{p}\text{)}$.     

Helly-type theorems for pseudoline arrangements in P2

We prove duals of Radon's theorem, Helly's theorem, Carathéodory's theorem, and Kirchberger's theorem for arrangements of pseudolines in the real projective plane, which generalize the original versions of those theorems for plane configurations of points. We also prove a topological generalization of the pseudoline-dual of Helly's theorem.     

关于F2n中的和集

设F₂为两个元素组成的有限域, F₂ⁿ 为F₂上的n维向量空间. 对于集合A, B ⊆ F₂ⁿ , 它们的和集定义为所有两两互异的和a+b所组成的集合, 其中a∈A, b∈B. Green 和Tao 证明了: 设K > 1,如果A, B ? F₂ⁿ 且|A + B|≤K|A|^1/2|B|^1/2, 则存在一个子空间H?F₂ⁿ 满足
|H|>>exp(-O(√KlogK))|A|
以及x,y∈F₂ⁿ, 使得
|A∩(x+H)|^1/2|B∩(y+H)|^1/2≥1/2K|H|.
本文我们将使用Green 和Tao 的方法并作一些修改, 证明如果|H|>>exp(-O(√K))|A|,
则以上的结论仍然成立.     

b~(2)空间及b~(2)空间上的等距映射

等距映射在空间结构的研究中起着很重要的作用,是泛函分析研究的有利工具.本文将介绍一类特殊的F空间,b~(2)空间,然后给出该空间单位球面间满等距映射的表现定理,进而得出b~(2)空间单位球面上满等距映射的线性延拓结论.     

A Study Guide for the l2 Decoupling Theorem

This paper contains a detailed, self contained and more streamlined proof of the l ² decoupling theorem for hypersurfaces from the paper of Bourgain and Demeter in 2015. The authors hope this will serve as a good warm up for the readers interested in understanding the proof of Vinogradov’s mean value theorem from the paper of Bourgain, Demeter and Guth in 2015.     

The Existence and Stability of Asymmetric Spike Patterns for the Schnakenberg Model

Asymmetric spike patterns are constructed for the two-component Schnakenburg reaction–diffusion system in the singularly perturbed limit of a small diffusivity of one of the components. For a pattern with k spikes, the construction yields   k ₁  spikes that have a common small amplitude and   k ₂= k − k ₁  spikes that have a common large amplitude. A k -spike asymmetric equilibrium solution is obtained from an arbitrary ordering of the small and large spikes on the domain. Explicit conditions for the existence and linear stability of these asymmetric spike patterns are determined using a combination of asymptotic techniques and spectral properties associated with a certain nonlocal eigenvalue problem. These asymmetric solutions are found to bifurcate from symmetric spike patterns at certain critical values of the parameters. Two interesting conclusions are that asymmetric patterns can exist for a reaction–diffusion system with spatially homogeneous coefficients under Neumann boundary conditions and that these solutions can be linearly stable on an O (1) time scale.     

The diophantine equation x3 + 3y3 = 2n

It is proved that the equation of the title has a finite number of integral solutions (x, y, n) and necessary conditions are given for (x, y, n) in order that it can be a solution (Theorem 2). It is also proved that for a given odd x₀ there is at most one integral solution (y, n), n ≥ 3, to x₀³ + 3y³ = 2ⁿ and for a given odd y₀ there is at most one integral solution (x, n), n ≥ 3, to x³ + 3y₀³ = 2ⁿ.