315,947 research outputs found
n-Linear Algebra of type II
This book is a continuation of the book n-linear algebra of type I and its
applications. Most of the properties that could not be derived or defined for
n-linear algebra of type I is made possible in this new structure: n-linear
algebra of type II which is introduced in this book. In case of n-linear
algebra of type II we are in a position to define linear functionals which is
one of the marked difference between the n-vector spaces of type I and II.
However all the applications mentioned in n-linear algebras of type I can be
appropriately extended to n-linear algebras of type II. Another use of n-linear
algebra (n-vector spaces) of type II is that when this structure is used in
coding theory we can have different types of codes built over different finite
fields whereas this is not possible in the case of n-vector spaces of type I.
Finally in the case of n-vector spaces of type II, we can obtain n-eigen values
from distinct fields; hence, the n-characteristic polynomials formed in them
are in distinct different fields.
An attractive feature of this book is that the authors have suggested 120
problems for the reader to pursue in order to understand this new notion. This
book has three chapters. In the first chapter the notion of n-vector spaces of
type II are introduced. This chapter gives over 50 theorems. Chapter two
introduces the notion of n-inner product vector spaces of type II, n-bilinear
forms and n-linear functionals. The final chapter suggests over a hundred
problems. It is important that the reader is well-versed not only with linear
algebra but also n-linear algebra of type I.Comment: 229 page
On the differential structure of metric measure spaces and applications
The main goals of this paper are: i) To develop an abstract differential
calculus on metric measure spaces by investigating the duality relations
between differentials and gradients of Sobolev functions. This will be achieved
without calling into play any sort of analysis in charts, our assumptions
being: the metric space is complete and separable and the measure is Borel, non
negative and locally finite. ii) To employ these notions of calculus to
provide, via integration by parts, a general definition of distributional
Laplacian, thus giving a meaning to an expression like , where
is a function and is a measure. iii) To show that on spaces with
Ricci curvature bounded from below and dimension bounded from above, the
Laplacian of the distance function is always a measure and that this measure
has the standard sharp comparison properties. This result requires an
additional assumption on the space, which reduces to strict convexity of the
norm in the case of smooth Finsler structures and is always satisfied on spaces
with linear Laplacian, a situation which is analyzed in detail.Comment: Clarified the dependence on the Sobolev exponent of various
objects built in the paper. Updated bibliography. Corrected typo
Principal Floquet subspaces and exponential separations of type II with applications to random delay differential equations
Producción CientíficaThis paper deals with the study of principal Lyapunov exponents, principal Floquet subspaces, and exponential separation for positive random linear dynamical systems in ordered Banach spaces. The main contribution lies in the introduction of a new type of exponential separation, called of type II, important for its application to random differential equations with delay. Under weakened assumptions, the existence of an exponential separation of type II in an abstract general setting is shown, and an illustration of its application to dynamical systems generated by scalar linear random delay differential equations with finite delay is given.2020-01-012020-01-01Ministerio de Economía, Industria y Competitividad - FEDER (Project MTM2015-66330-P
Extension of some theorems of complex functional analysis to linear spaces over the quaternions and Cayley numbers
In this work certain aspects of Functional Analysis are considered in the setting of linear spaces over the division rings of the real Quaternions and the real Cayley algebra. The basic structure of Banach spaces over these division rings and the rings of bounded operators on these spaces is developed. Examples of finite and infinite dimensional spaces over these division rings are given. Questions concerning linear functionals, the Hahn-Banach Theorem and Reflexivity are considered. The Stone-Weierstrass Theorem is proven for functions with values in a real Cayley Dickson algebra of dimension n. The concepts of inner product spaces and Hilbert spaces over the Quaternions and the Cayley algebra are developed. An extensive study of Hilbert spaces over the Quaternions is carried out. In the case of Hilbert spaces over the Quaternions, the Riesz-Representation Theorem and the Jordan-von Neumann Theorem are proven. In addition, spectral theorems for both self-adjoint and normal operators are proven for finite dimensional Hilbert spaces. These results are extended to infinite dimensional spaces for the cases of compact self-adjoint operators and compact normal operators. The spectrum of an arbitrary bounded Hermitian operator on a Hilbert space over the Quaternions is shown to be non-void. A generalization of the Fourier Transform for functions in L[1 over Q](-infinity, infinity) and L[2 over Q]( ](-infinity, infinity) is given. The Plancherel Theorem is proven for functions in L[2 over Q](-infinity, infinity). Finally, the Jordan-von Neumann theorem is proven for a Hilbert space over the Cayley algebra --Abstract, pages ii-iii
The Stochastic Shortest Path Problem : A polyhedral combinatorics perspective
In this paper, we give a new framework for the stochastic shortest path
problem in finite state and action spaces. Our framework generalizes both the
frameworks proposed by Bertsekas and Tsitsikli and by Bertsekas and Yu. We
prove that the problem is well-defined and (weakly) polynomial when (i) there
is a way to reach the target state from any initial state and (ii) there is no
transition cycle of negative costs (a generalization of negative cost cycles).
These assumptions generalize the standard assumptions for the deterministic
shortest path problem and our framework encapsulates the latter problem (in
contrast with prior works). In this new setting, we can show that (a) one can
restrict to deterministic and stationary policies, (b) the problem is still
(weakly) polynomial through linear programming, (c) Value Iteration and Policy
Iteration converge, and (d) we can extend Dijkstra's algorithm
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