Newton’s method for generalized equations: a sequential implicit function theorem

In an extension of Newton’s method to generalized equations, we carry further the implicit function theorem paradigm and place it in the framework of a mapping acting from the parameter and the starting point to the set of all associated sequences of Newton’s iterates as elements of a sequence space. An inverse function version of this result shows that the strong regularity of the mapping associated with the Newton sequences is equivalent to the strong regularity of the generalized equation mapping.     

On generalized Rubel’s equation

We solve generalized the generalized Rubel equation on the space of analytic functions in domains.     

The period function of generalized Loudʼs centers

In this paper a three parameter family of planar differential systems with homogeneous nonlinearities of arbitrary odd degree is studied. This family is an extension to higher degree of Loud?s systems. The origin is a nondegenerate center for all values of the parameter and we are interested in the qualitative properties of its period function. We study the bifurcation diagram of this function focusing our attention on the bifurcations occurring at the polycycle that bounds the period annulus of the center. Moreover we determine some regions in the parameter space for which the corresponding period function is monotonous or it has at least one critical period, giving also its character (maximum or minimum). Finally we propose a complete conjectural bifurcation diagram of the period function of these generalized Loud?s centers.     

Newton’s method on generalized Banach spaces

We present a weaker convergence analysis of Newton’s method than in Kantorovich and Akilov (1964), Meyer (1987), Potra and Ptak (1984), Rheinboldt (1978), Traub (1964) on a generalized Banach space setting to approximate a locally unique zero of an operator. This way we extend the applicability of Newton’s method. Moreover, we obtain under the same conditions in the semilocal case weaker sufficient convergence criteria; tighter error bounds on the distances involved and an at least as precise information on the location of the solution. In the local case we obtain a larger radius of convergence and higher error estimates on the distances involved. Numerical examples illustrate the theoretical results.     

Regev’s and Amitsur’s conjectures for codimensions of generalized polynomial identities

Let A be a finite-dimensional associative algebra over a field of characteristic 0. Then there exist C ∈ ℚ₊ and t ∈ ℤ₊ such that gc _n (A) ∼ Cn ^t d ⁿ as n → ∞, where d = PIexp(A). In particular, Amitsur’s and Regev’s conjectures hold for the codimensions gc _n (A) of generalized polynomial identities.     

Solving multi-period interdiction via generalized Bender’s decomposition

This paper considers a novel formulation of the multi-period network interdiction problem. In this model, delivery of the maximum flow as well as the act of interdiction happens over several periods, while the budget of resource for interdiction is limit. It is assumed that when an edge is interdicted in a period, the evader considers a rate of risk of detection at consequent periods. Application of the generalized Benders decomposition algorithm considers solving the resulting mixed-integer nonlinear programming problem. Computational experiences denote reasonable consistency with expectations.     

Jacobson’s lemma for the generalized Drazin inverse

0truemm0truemm We study properties of elements in a ring which admit the generalized Drazin inverse. It is shown that the element 1-ab is generalized Drazin invertible if and only if so is 1-ba and a formula for the generalized Drazin inverse of 1-ba in terms of the generalized Drazin inverse and the spectral idempotent of 1-ab is provided. Further, recent results relating to the Drazin index can be recovered from our theorems.     

The analyticity of a generalized Ruelle’s operator

In this work we propose a generalization of the concept of Ruelle’s operator for one dimensional lattices used in thermodynamic formalism and ergodic optimization, which we call generalized Ruelle’s operator. Our operator generalizes both the Ruelle operator proposed in [2] and the Perron Frobenius operator defined in [7]. We suppose the alphabet is given by a compact metric space, and consider a general a-priori measure to define the operator. We also consider the case where the set of symbols that can follow a given symbol of the alphabet depends on such symbol, which is an extension of the original concept of transition matrices from the theory of subshifts of finite type. We prove the analyticity of the Ruelle’s operator and present some examples.     

Ramanujan’s cubic transformation and generalized modular equation

We study the quotient of hypergeometric functionsμ_a~*(r)=π/2sin(πa) F(a,1-a;1;1-r~3)/F(a,1-a;1;r~3),r∈(0,1)in the theory of Ramanujan's generalized modular equation for a ∈(0,1/2],and find an infinite product formula for μ_(1/3)~*(r) by use of the properties of μ_a~*(r) and Ramanujan's cubic transformation.Besides,a new cubic transformation formula of hypergeometric function is given,which complements the Ramanujan's cubic transformation.     

The ring of Colombeau’s full generalized quaternions

In this paper, we introduce the ring of Colombeau full generalized quaternion, and we study its algebraic and topological properties. We prove that the ring of Colombeau full generalized quaternion is Gelfand, normal, duo and some related properties. Moreover, we study the essential ideals of this ring.     

Combinatorial interpretations of Ramanujan’s tau function

We use a $q$-series identity by Ramanujan to give a combinatorial interpretation of Ramanujan’s tau function which involves $t$-cores and a new class of partitions which we call $(m,k)$-capsids. The same method can be applied in conjunction with other related identities yielding alternative combinatorial interpretations of the tau function.