Universal Generalized Computable Numberings and Hyperimmunity

Let x : C → P n C x:{\mathbf {C}} \to {P_n}{\mathbf {C}} be a holomorphic curve of finite lower order μ \mu , and let A = { α } A = \{ \alpha \} be an arbitrary finite family of holomorphic curves α : C → ( P n C ) ∗ \alpha :{\mathbf {C}} \to {({P_n}{\mathbf {C}})^\ast } satisfying T ( r , α ) = o ( T ( r , x ) ) ( r → ∞ ) T(r,\alpha ) = o(T(r,x))\;(r \to \infty ) . Suppose x x is nondegenerate with respect to A A , and A A is in general position. We show the following general defect relations: (1) x x has at most n n deficient curves in A A if μ = 0 \mu = 0 . (2) ∑ α ∈ A δ ( α ) ≤ n if 0 > μ ≤ 1 / 2 \sum \nolimits _{\alpha \in A} {\delta (\alpha ) \leq n\;{\text {if}}\;0 > \mu \leq 1/2} . (3) ∑ α ∈ A δ ( α ) ≤ [ 2 n μ ] + 1 if 1 / 2 > μ > + ∞ \sum \nolimits _{\alpha \in A} {\delta (\alpha ) \leq [2n\mu ] + 1\;{\text {if}}\;1/2 > \mu > + \infty } .

We derive the joint asymptotic distribution of the outlier eigenvalues of an additively deformed Wigner matrix $H$. Our only assumptions on the deformation are that its rank be fixed and its norm bounded. Our results extend those of [The isotropic semicircle law and deformation of Wigner matrices. Preprint] by admitting overlapping outliers and by computing the joint distribution of all outliers. In particular, we give a complete description of the failure of universality first observed in [Ann. Probab. 37 (2009) 1–47; Ann. Inst. Henri Poincaré Probab. Stat. 48 (1013) 107–133; Free convolution with a semi-circular distribution and eigenvalues of spiked deformations of Wigner matrices. Preprint]. We also show that, under suitable conditions, outliers may be strongly correlated even if they are far from each other. Our proof relies on the isotropic local semicircle law established in [The isotropic semicircle law and deformation of Wigner matrices. Preprint]. The main technical achievement of the current paper is the joint asymptotics of an arbitrary finite family of random variables of the form $\langle\mathbf{v},(H-z)^{-1}\mathbf{w}\rangle$.

In his book Transversal Theory (4) Mirsky introduces the problem of finding a necessary and sufficient condition in order that a finite family of sets have a unique independent transversal. The purpose of this note is to give one necessary and sufficient condition. More generally, we give a necessary and sufficient condition in order that a common basis of two rank-finite matroids be unique. We then specialize this result to give an answer to the above question.

Viscosity approximation methods for a finite family of nonexpansive mappings in Banach spaces

Sur la compatibilité des extensions ponctuelles d'un matroïde

In this paper, we present a sufficient condition for the global asymptotic stability of a switched nonlinear system composed of a finite family of subsystems. We show that the global asymptotic stability of each subsystem and the pairwise commutation of the vector fields that define the subsystems (i.e., the Lie bracket of any pair of them is zero) are sufficient for the global asymptotic stability of the switched system. We also show that these conditions are sufficient for the existence of a common Lyapunov function.

In this paper, we will establish several strong convergence theorems for the approximation of common fixed points of r−strictly asymptotically pseudocontractive mappings in uniformly convex Banach spaces using the modiied implicit iteration sequence with errors, and prove the necessary and sufficient conditions for the convergence of the sequence. Our results generalize, extend and improve the recent work, in this topic[9,10].

Strong Convergence for a Finite Family of Generalized Asymptotically Nonexpansive Mappings

This paper develops some new results on semistability and asymptotic stability of switched systems using multiple Lyapunov functions. The main results of the paper involve sufficient conditions for guaranteeing semistability and asymptotic stability of switched systems using multiple Lyapunov functions and a class of switching conditions. Moreover, converse Lyapunov theorems for a finite family of pairwise commuting semistable systems are also developed. Using these stability results and an extended notion of homogeneity for switched systems, we derive some robustness results for asymptotic stability and semistability of switched systems.

Let E be a real Banach space, {Ti} N=1 be a finite family of continuous pseudocon- tractive self mappings of E and G : E → E be a mapping which is both δ -strongly accretive and λ -strictly pseudocontractive of Browder-Petryshyn type such that δ +λ 1. We propose a new implicit iteration scheme with perturbed mapping G for the approximation of common fixed points of {Ti} N=1 . For an arbitrary initial point x0 ∈ E , the sequence {xn} ∞=1 is defined by xn = αn(xn−1 − λnG(xn−1)) + (1 − αn)Tnxn where Tn = Tn mod N , {αn} ∞=1 ⊂ (a,b) ⊂)0,1( and {λn} ∞=1 ⊂ (0,1(. We establish some weak convergence theorems for this implicit iteration scheme. Also, necessary and sufficient conditions for strong convergence of this implicit iteration scheme are obtained.

In this paper, we discuss several stronger forms of sensitivities for iterated function systems (IFSs), such as strong sensitivity and syndetical sensitivity, and obtain some sufficient conditions for an IFS to have such sensitivities. We pay special attention to IFSs acting on the circle S1. We prove that each sensitive IFS acting on the circle S1 generating by a finite family of circle homeomorphisms is strongly sensitive. However, to obtain syndetical sensitivity, we impose some extra conditions on the IFS. Finally, we study sensitivity properties for syndetical transitive IFSs. We show that each syndetically transitive IFS is topologically ergodic.

On Guaspari's problem about partially conservative sentences

In this article, A demiclosed principle is proved for generalized asymptotically nonexpansive mapping in a Banach space. Additionally, it proves strong convergence of a composite implicit iteration scheme to a common fixed point for two finite families of generalized asymptotically quasi-nonexpansive mappings in a nonempty closed convex subset of a uniformly convex Banach space. We derive a necessary and sufficient condition for the strong convergence of this iteration process to a common fixed point of these mappings. The results of this article improve and extend the corresponding results of Xu and Ori [23], Zhou and Chang [26], Chang et al. [3], Yang and Yu [25], Shahzad and Zegeye [20].

In this paper, we propose a new explicit iteration scheme with perturbed mapping for approximation of common fixed points of a finite family of I-asymptotically nonexpansive mappings. We establish some convergence theorems for this explicit iteration scheme. In particular, necessary and sufficient conditions for strong convergence of this explicit iteration scheme were obtained.

We use a graph-theory-based argument to propose a novel Lyapunov construction for continuous-time switching systems. Starting with a finite family of continuously differentiable functions, the inequalities involving these functions and the vector fields of the switching system are encoded in a direct and labeled graph. Relaying on the (path-)completeness of this graph, we introduce a signal-dependent Lyapunov function, providing sufficient conditions for stability under fixed-time or dwell-time switching hypothesis. For the case of linear systems, our conditions turn into linear matrix inequalities (LMI), and thus they are compared with previous results, via numerical examples.