A Note on Smooth Matrices of Constant Rank

We show that, given a $C^h$ time–varying matrix A(t) of constant rank, there exists a $C^h$ matrix H(t) such that the rows of H(t)A(t) are an orthonormal basis of the space spanned by the rows of A(t). We present some consequences of this result and, in particular, we prove a version for $m \times n$ matrices of Doležal's Theorem. These results are not new, and references are given. All the proofs of the results stated in these references, with the exception of those based on the use of differential equations — which holds only for $h \geq 1$ —, find suitable $C^h$ matrices defined on overlapping subsets of the domain and then patch them together without losing regularity and the other required properties. In our approach the patching needs to be done only for matrices consisting of one row and all the remaining results are obtained by usual algebraic tools

A Study on the Early Stages of Degradation of Multi-component Alloy Surfaces in Extreme Environments Using the Multi-cell Monte Carlo Method

A computational toolset is presented and used in two examples that examined the interactions between structural materials and their extreme environments. A multi-cell Monte Carlo algorithm was developed to generate thermodynamically realistic solid-state alloy systems. These structures served as the foundation upon which surface slab models were generated. The tedious procedure of generating surface slab models from bulk structures was automated. The tools were used to study the high temperature surface corrosion resistance of a high-entropy alloy, Al10Nb15Ta5Ti30Zr40, and a nickel-based alloy, Ni70Nb10W20, under an oxygen and chlorine atmosphere, respectively

Geometric analysis of nonlinear differential-algebraic equations via nonlinear control theory

For nonlinear differential-algebraic equations (DAEs), we define two kinds of equivalences, namely, the external and internal equivalence. Roughly speaking, the word "external" means that we consider a DAE (locally) everywhere and "internal" means that we consider the DAE on its (locally) maximal invariant submanifold (i.e., where its solutions exist) only. First, we revise the geometric reduction method in DAEs solution theory and formulate an implementable algorithm to realize that method. Then a procedure named explicitation with driving variables is proposed to connect nonlinear DAEs with nonlinear control systems and we show that the driving variables of an explicitation system can be reduced under some involutivity conditions. Finally, due to the explicitation, we will use some notions from nonlinear control theory to derive two nonlinear generalizations of the Weierstrass form.Comment: 38 page

The finite horizon singular time-varying H∞H_\infty control problem with dynamic measurement feedback

This paper is concerned with the finite horizon version of the $H_\infty$ problem with measurement feedback. Given a finite-dimensional linear , time-varying system, together with a positive real number $\gamma$, we obtain necessary and sufficient conditions for the existence of a possibly time-varying dynamic compensator such that the $L_2([0,t_1])$-induced norm of the closed loop operator is smaller than $\gamma$. These conditions are expressed in terms of a pair of quadratic differential inequalities, generalizing the well-known Riccati differential equations introduced recently in the context of finite horizon $H_\infty$ control

Convolutions of maximal monotone mappings

Bibliography: p. 46-48.Supported by the ITP Foundation, Madrid, Spain and the National Science Foundation under grant ECS 8217668Javier Luque

An extensive English language bibliography on graph theory and its applications

Bibliography on graph theory and its application