Extended Kung–Traub Methods for Solving Equations with Applications

Kung and Traub (1974) proposed an iterative method for solving equations defined on the real line. The convergence order four was shown using Taylor expansions, requiring the existence of the fifth derivative not in this method. However, these hypotheses limit the utilization of it to functions that are at least five times differentiable, although the methods may converge. As far as we know, no semi-local convergence has been given in this setting. Our goal is to extend the applicability of this method in both the local and semi-local convergence case and in the more general setting of Banach space valued operators. Moreover, we use our idea of recurrent functions and conditions only on the first derivative and divided difference, which appear in the method. This idea can be used to extend other high convergence multipoint and multistep methods. Numerical experiments testing the convergence criteria complement this study.

Unified Convergence Analysis of Two-Step Iterative Methods for Solving Equations

In this paper we consider unified convergence analysis of two-step iterative methods for solving equations in the Banach space setting. The convergence order four was shown using Taylor expansions requiring the existence of the fifth derivative not on this method. But these hypotheses limit the utilization of it to functions which are at least five times differentiable although the method may converge. As far as we know no semi-local convergence has been given in this setting. Our goal is to extend the applicability of this method in both the local and semi-local convergence case and in the more general setting of Banach space valued operators. Moreover, we use our idea of recurrent functions and conditions only on the first derivative and divided differences which appear on the method. This idea can be used to extend other high convergence multipoint and multistep methods. Numerical experiments testing the convergence criteria complement this study.

Generalized Almost Periodicity in Lebesgue Spaces with Variable Exponents

In this paper, we introduce and analyze Stepanov uniformly recurrent functions, Doss uniformly recurrent functions and Doss almost-periodic functions in Lebesgue spaces with variable exponents. We investigate the invariance of these types of generalized almost-periodicity in Lebesgue spaces with variable exponents under the actions of convolution products, providing also some illustrative applications to the abstract semilinear integro-differential inclusions in Banach spaces.

Logarithmic Spirals and Right-angled Triangles

This article presents some recurrent functions which we can use to make Right-angledTriangles and through their special arrangement we can obtain logarithmic spirals,and their discrete form. The recurrent functions are obtained through recurrencerelations which can be expressed as a linear combination of fibonacci numbers.

Extended Convergence Analysis of the Newton–Hermitian and Skew–Hermitian Splitting Method

Many problems in diverse disciplines such as applied mathematics, mathematical biology, chemistry, economics, and engineering, to mention a few, reduce to solving a nonlinear equation or a system of nonlinear equations. Then various iterative methods are considered to generate a sequence of approximations converging to a solution of such problems. The goal of this article is two-fold: On the one hand, we present a correct convergence criterion for Newton–Hermitian splitting (NHSS) method under the Kantorovich theory, since the criterion given in Numer. Linear Algebra Appl., 2011, 18, 299–315 is not correct. Indeed, the radius of convergence cannot be defined under the given criterion, since the discriminant of the quadratic polynomial from which this radius is derived is negative (See Remark 1 and the conclusions of the present article for more details). On the other hand, we have extended the corrected convergence criterion using our idea of recurrent functions. Numerical examples involving convection–diffusion equations further validate the theoretical results.

On Third-Order Linear Recurrent Functions

A function ψ:R→R is said to be a Tribonacci function with period p if ψ(x+3p)=ψ(x+2p)+ψ(x+p)+ψ(x), for all x∈R. In this paper, we present some properties on the Tribonacci functions with period p. We show that if ψ is a Tribonacci function with period p, then limx→∞ψ(x+p)/ψ(x)=β, where β is the root of the equation x3-x2-x-1=0 such that 1<β<2.

Flip chip reliability and intermetallic compounds for SIP module

In the aeronautical field, the electronic integration roadmaps show that the weight and the volume dedicated to on-board electronics must be reduced by a factor of 4 to 10 compared to the existing ones for the most recurrent functions in the next years. This work is an opening to new technological solutions to increase our ability to save space while improving the overall reliability of the system. The first part of this work is dedicated to the study of “system in package” (SiP) solutions based on different substrates, namely organic or silicon. Generally speaking a SIP is composed by several active and passive components stacked on an interposer. Benchmarks done by our laboratory have demonstrated that in terms of substrate, embedded die technology leads to several advantages compared to 3D TSV or TGV based packaging approaches. The benefits provided by this substrate is the possibility to embed some surface mount technologies (SMT), some bare chips or some integrated passives devices (IPD) banks directly above or below the stacked active components. This way, top and bottom surface of the substrate can be used to integrate several heterogeneous dies side by side while using low profile flip-chip assemblies on the C4 side. Finally, in this kind of 3D architecture, this embedded technology enable a gain of integration, without using costly TSV connections. Substrates of high quality allow a reduction of I/Os interconnection pitches leading to very aggressive integration down to 50μm. Secondly, a 3D stack with 3 levels of components, as described above, leads to 2 or 3 REACH compliant sequential assembly processes, depending of the needs. In order to consider all the solutions for an optimized overall integration with high reliability, this work focuse on the study one simple SIP which includes the top die assembled by flip-chip. For the flip chip hybridization on organic interposers copper pillars technologies will be studied. The objective is to understand in depth the processes and to obtain information on the reliability aspect after thermal cycling stress of the flip chip assembly. Thirdly, we built many silicon test chips with different characteristics with a dedicated daisy chain test vehicle. The different parameters are: chip's thicknesses (50 to 200 μm), chip's sizes (2 to 8 mm), bump structures (diameter), the pitches of the interconnection (from 50 to 250 μm) and the number of interconnection rows. Designs were chosen in order to fit real operational configurations. Moreover, these configurations are interesting to build a comprehensive model in order to understand the failure mechanisms. These chips are then stacked by flip chip on the silicon and on the organic substrate. We are also designing the both configurations of substrates. Only the production of the organics part is outsourced. Fourth, for all assemblies thermos-cycling test results will be evaluated with thermo mechanical simulations done by finite elements. 3D models will take into account the different geometries in order to understand and quantify the various key parameters. The analysis will mainly focus on 3D interconnections. Design rules based on the results will be derivated. The aim is to obtain dimensional criteria based on stress versus deformation responses. Lastly intermetallic formation will be evaluated using EBSD analysis to obtain better understanding of copper pillar failures for this specific bumps size. Issued information's will be exploited for designing the future functional SIP. The ultimate goal of this work is finally to define mechanical design rules that can then be used in functional SiP modules.

Expanding the applicability of secant-like methods for solving nonlinear equations

We use the method of recurrent functions to provide a new semilocal convergence analysis for secant-like methods in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Our sufficient convergence criteria are weaker than in earlier studies such as [18, 19, 20, 21, 25, 26]. Therefore, the new approach has a larger convergence domain and uses the same constants. A numerical example involving a nonlinear integral equation of mixed Hammerstein type is given to illustrate the advantages of the new approach. Another example of nonlinear integral equations is presented to show that the old convergence criteria are not satisfied but the new convergence are satisfied.