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Search: All articles in the CMB digital archive with keyword stability

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1. CMB 2017 (vol 61 pp. 423)

Zhang, Guo-Bao; Tian, Ge
 Stability of Traveling Wavefronts for a Two-Component Lattice Dynamical System Arising in Competition Models In this paper, we study a two-component Lotka-Volterra competition system on an one-dimensional spatial lattice. By the method of the comparison principle together with the weighted energy, we prove that the traveling wavefronts with large speed are exponentially asymptotically stable, when the initial perturbation around the traveling wavefronts decays exponentially as $j+ct \rightarrow -\infty$, where $j\in\mathbb{Z}$, $t\gt 0$, but the initial perturbation can be arbitrarily large on other locations. This partially answers an open problem by J.-S. Guo and C.-H. Wu. Keywords:lattice dynamical system, competition model, traveling wavefront, stabilityCategories:34A33, 34K20, 92D25

2. CMB 2017 (vol 60 pp. 436)

Weng, Peixuan; Liu, Li
 Globally Asymptotic Stability of a Delayed Integro-Differential Equation with Nonlocal Diffusion We study a population model with nonlocal diffusion, which is a delayed integro-differential equation with double nonlinearity and two integrable kernels. By comparison method and analytical technique, we obtain globally asymptotic stability of the zero solution and the positive equilibrium. The results obtained reveal that the globally asymptotic stability only depends on the property of nonlinearity. As application, an example for a population model with age structure is discussed at the end of the article. Keywords:integro-differential equation, nonlocal diffusion, equilibrium, globally asymptotic stability, population model with age structureCategories:45J05, 35K57, 92D25

3. CMB 2017 (vol 60 pp. 364)

Preda, Ciprian
 On the Roughness of Quasinilpotency Property of Oneâparameter Semigroups Let $\mathbf{S}:=\{S(t)\}_{t\geq0}$ be a C$_0$-semigroup of quasinilpotent operators (i.e. $\sigma(S(t))=\{0\}$ for each $t\gt 0$). In the dynamical systems theory the above quasinilpotency property is equivalent to a very strong concept of stability for the solutions of autonomous systems. This concept is frequently called superstability and weakens the classical finite time extinction property (roughly speaking, disappearing solutions). We show that under some assumptions, the quasinilpotency, or equivalently, the superstability property of a C$_0$-semigroup is preserved under the perturbations of its infinitesimal generator. Keywords:one-parameter semigroups, quasinilpotency, superstability, essential spectrumCategories:34D05, 34D10, 34E10

4. CMB 2016 (vol 60 pp. 95)

Choi, Chang-Kwon; Chung, Jaeyoung; Ju, Yumin; Rassias, John
 Cubic Functional Equations on Restricted Domains of Lebesgue Measure Zero Let $X$ be a real normed space, $Y$ a Bancch space and $f:X \to Y$. We prove the Ulam-Hyers stability theorem for the cubic functional equation \begin{align*} f(2x+y)+f(2x-y)-2f(x+y)-2f(x-y)-12f(x)=0 \end{align*} in restricted domains. As an application we consider a measure zero stability problem of the inequality \begin{align*} \|f(2x+y)+f(2x-y)-2f(x+y)-2f(x-y)-12f(x)\|\le \epsilon \end{align*} for all $(x, y)$ in $\Gamma\subset\mathbb R^2$ of Lebesgue measure 0. Keywords:Baire category theorem, cubic functional equation, first category, Lebesgue measure, Ulam-Hyers stabilityCategory:39B82

5. CMB 2016 (vol 60 pp. 173)

Oubbi, Lahbib
 On Ulam Stability of a Functional Equation in Banach Modules Let $X$ and $Y$ be Banach spaces and $f : X \to Y$ an odd mapping. For any rational number $r \ne 2$, C. Baak, D. H. Boo, and Th. M. Rassias have proved the Hyers-Ulam stability of the following functional equation: \begin{align*} r f \left(\frac{\sum_{j=1}^d x_j}{r} \right) & + \sum_{\substack{i(j) \in \{0,1\} \\ \sum_{j=1}^d i(j)=\ell}} r f \left( \frac{\sum_{j=1}^d (-1)^{i(j)}x_j}{r} \right) = (C^\ell_{d-1} - C^{\ell -1}_{d-1} + 1) \sum_{j=1}^d f(x_j) \end{align*} where $d$ and $\ell$ are positive integers so that $1 \lt \ell \lt \frac{d}{2}$, and $C^p_q := \frac{q!}{(q-p)!p!}$, $p, q \in \mathbb{N}$ with $p \le q$. In this note we solve this equation for arbitrary nonzero scalar $r$ and show that it is actually Hyers-Ulam stable. We thus extend and generalize Baak et al.'s result. Different questions concerning the *-homomorphisms and the multipliers between C*-algebras are also considered. Keywords:linear functional equation, Hyers-Ulam stability, Banach modules, C*-algebra homomorphisms.Categories:39A30, 39B10, 39A06, 46Hxx

6. CMB 2016 (vol 59 pp. 849)

Nah, Kyeongah; Röst, Gergely
 Stability Threshold for Scalar Linear Periodic Delay Differential Equations We prove that for the linear scalar delay differential equation $$\dot{x}(t) = - a(t)x(t) + b(t)x(t-1)$$ with non-negative periodic coefficients of period $P\gt 0$, the stability threshold for the trivial solution is $r:=\int_{0}^{P} \left(b(t)-a(t) \right)\mathrm{d}t=0,$ assuming that $b(t+1)-a(t)$ does not change its sign. By constructing a class of explicit examples, we show the counter-intuitive result that in general, $r=0$ is not a stability threshold. Keywords:delay differential equation, stability, periodic systemCategories:34K20, 34K06

7. CMB 2016 (vol 60 pp. 154)

Liu, Ye
 On Chromatic Functors and Stable Partitions of Graphs The chromatic functor of a simple graph is a functorization of the chromatic polynomial. M. Yoshinaga showed that two finite graphs have isomorphic chromatic functors if and only if they have the same chromatic polynomial. The key ingredient in the proof is the use of stable partitions of graphs. The latter is shown to be closely related to chromatic functors. In this note, we further investigate some interesting properties of chromatic functors associated to simple graphs using stable partitions. Our first result is the determination of the group of natural automorphisms of the chromatic functor, which is in general a larger group than the automorphism group of the graph. The second result is that the composition of the chromatic functor associated to a finite graph restricted to the category $\mathrm{FI}$ of finite sets and injections with the free functor into the category of complex vector spaces yields a consistent sequence of representations of symmetric groups which is representation stable in the sense of Church-Farb. Keywords:chromatic functor, stable partition, representation stabilityCategories:05C15, 20C30

8. CMB 2016 (vol 59 pp. 858)

Osserman, Brian
 Stability of Vector Bundles on Curves and Degenerations We introduce a weaker notion of (semi)stability for vector bundles on reducible curves which does not depend on a choice of polarization, and which suffices for many applications of degeneration techniques. We explore the basic properties of this alternate notion of (semi)stability. In a complementary direction, we record a proof of the existence of semistable extensions of vector bundles in suitable degenerations. Keywords:vector bundle, stability, degenerationCategories:14D06, 14H60

9. CMB 2016 (vol 59 pp. 363)

Li, Dan; Ma, Wanbiao
 Dynamical Analysis of a Stage-Structured Model for Lyme Disease with Two Delays In this paper, a nonlinear stage-structured model for Lyme disease is considered. The model is a system of differential equations with two time delays. The basic reproductive rate, $R_0(\tau_1,\tau_2)$, is derived. If $R_0(\tau_1,\tau_2)\lt 1$, then the boundary equilibrium is globally asymptotically stable. If $R_0(\tau_1,\tau_2)\gt 1$, then there exists a unique positive equilibrium whose local asymptotical stability and the existence of Hopf bifurcations are established by analyzing the distribution of the characteristic values. An explicit algorithm for determining the direction of Hopf bifurcations and the stability of the bifurcating periodic solutions is derived by using the normal form and the center manifold theory. Some numerical simulations are performed to confirm the correctness of theoretical analysis. At last, some conclusions are given. Keywords:Lyme disease, stage-structure, time delay, Lyapunov functional stability Hopf bifurcation.Category:34D20

10. CMB 2014 (vol 58 pp. 30)

Chung, Jaeyoung
 On an Exponential Functional Inequality and its Distributional Version Let $G$ be a group and $\mathbb K=\mathbb C$ or $\mathbb R$. In this article, as a generalization of the result of Albert and Baker, we investigate the behavior of bounded and unbounded functions $f\colon G\to \mathbb K$ satisfying the inequality $\Bigl|f \Bigl(\sum_{k=1}^n x_k \Bigr)-\prod_{k=1}^n f(x_k) \Bigr|\le \phi(x_2, \dots, x_n),\quad \forall\, x_1, \dots, x_n\in G,$ where $\phi\colon G^{n-1}\to [0, \infty)$. Also, as a distributional version of the above inequality we consider the stability of the functional equation \begin{equation*} u\circ S - \overbrace{u\otimes \cdots \otimes u}^{n-\text {times}}=0, \end{equation*} where $u$ is a Schwartz distribution or Gelfand hyperfunction, $\circ$ and $\otimes$ are the pullback and tensor product of distributions, respectively, and $S(x_1, \dots, x_n)=x_1+ \dots +x_n$. Keywords:distribution, exponential functional equation, Gelfand hyperfunction, stabilityCategories:46F99, 39B82

11. CMB 2011 (vol 56 pp. 44)

Biswas, Indranil; Dey, Arijit
 Polystable Parabolic Principal $G$-Bundles and Hermitian-Einstein Connections We show that there is a bijective correspondence between the polystable parabolic principal $G$-bundles and solutions of the Hermitian-Einstein equation. Keywords:ramified principal bundle, parabolic principal bundle, Hitchin-Kobayashi correspondence, polystabilityCategories:32L04, 53C07

12. CMB 2011 (vol 54 pp. 593)

Boersema, Jeffrey L.; Ruiz, Efren
 Stability of Real $C^*$-Algebras We will give a characterization of stable real $C^*$-algebras analogous to the one given for complex $C^*$-algebras by Hjelmborg and RÃ¸rdam. Using this result, we will prove that any real $C^*$-algebra satisfying the corona factorization property is stable if and only if its complexification is stable. Real $C^*$-algebras satisfying the corona factorization property include AF-algebras and purely infinite $C^*$-algebras. We will also provide an example of a simple unstable $C^*$-algebra, the complexification of which is stable. Keywords:stability, real C*-algebrasCategory:46L05

13. CMB 2010 (vol 54 pp. 364)

Preda, Ciprian; Preda, Petre
 Lyapunov Theorems for the Asymptotic Behavior of Evolution Families on the Half-Line Two theorems regarding the asymptotic behavior of evolution families are established in terms of the solutions of a certain Lyapunov operator equation. Keywords:evolution families, exponential instability, Lyapunov equationCategories:34D05, 47D06

14. CMB 2009 (vol 53 pp. 218)

Biswas, Indranil
 Restriction of the Tangent Bundle of $G/P$ to a Hypersurface Let P be a maximal proper parabolic subgroup of a connected simple linear algebraic group G, defined over $\mathbb C$, such that $n := \dim_{\mathbb C} G/P \geq 4$. Let $\iota \colon Z \hookrightarrow G/P$ be a reduced smooth hypersurface of degree at least $(n-1)\cdot \operatorname{degree}(T(G/P))/n$. We prove that the restriction of the tangent bundle $\iota^*TG/P$ is semistable. Keywords:tangent bundle, homogeneous space, semistability, hypersurfaceCategories:14F05, 14J60, 14M15

15. CMB 2006 (vol 49 pp. 358)

Khalil, Abdelouahed El; Manouni, Said El; Ouanan, Mohammed
 On the Principal Eigencurve of the $p$-Laplacian: Stability Phenomena We show that each point of the principal eigencurve of the nonlinear problem $$-\Delta_{p}u-\lambda m(x)|u|^{p-2}u=\mu|u|^{p-2}u \quad \text{in } \Omega,$$ is stable (continuous) with respect to the exponent $p$ varying in $(1,\infty)$; we also prove some convergence results of the principal eigenfunctions corresponding. Keywords:$p$-Laplacian with indefinite weight, principal eigencurve, principal eigenvalue, principal eigenfunction, stabilityCategories:35P30, 35P60, 35J70

16. CMB 2000 (vol 43 pp. 418)

Gong, Guihua; Jiang, Xinhui; Su, Hongbing
 Obstructions to $\mathcal{Z}$-Stability for Unital Simple $C^*$-Algebras Let $\cZ$ be the unital simple nuclear infinite dimensional $C^*$-algebra which has the same Elliott invariant as $\bbC$, introduced in \cite{JS}. A $C^*$-algebra is called $\cZ$-stable if $A \cong A \otimes \cZ$. In this note we give some necessary conditions for a unital simple $C^*$-algebra to be $\cZ$-stable. Keywords:simple $C^*$-algebra, $\mathcal{Z}$-stability, weak (un)perforation in $K_0$ group, property $\Gamma$, finitenessCategory:46L05

17. CMB 1998 (vol 41 pp. 49)

Harrison, K. J.; Ward, J. A.; Eaton, L-J.
 Stability of weighted darma filters We study the stability of linear filters associated with certain types of linear difference equations with variable coefficients. We show that stability is determined by the locations of the poles of a rational transfer function relative to the spectrum of an associated weighted shift operator. The known theory for filters associated with constant-coefficient difference equations is a special case. Keywords:Difference equations, adaptive $\DARMA$ filters, weighted shifts,, stability and boundedness, automatic continuityCategories:47A62, 47B37, 93D25, 42A85, 47N70
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