Spectral Theory For Gaussian Processes: Reproducing Kernels, Random Functions, Boundaries, And L2-Wavelet Generators With Fractional Scales,
2015
Chapman University
Spectral Theory For Gaussian Processes: Reproducing Kernels, Random Functions, Boundaries, And L2-Wavelet Generators With Fractional Scales, Daniel Alpay
Mathematics, Physics, and Computer Science Faculty Articles and Research
A recurrent theme in functional analysis is the interplay between the theory of positive definite functions, and their reproducing kernels, on the one hand, and Gaussian stochastic processes, on the other. This central theme is motivated by a host of applications, e.g., in mathematical physics, and in stochastic differential equations, and their use in financial models. In this paper, we show that, for three classes of cases in the correspondence, it is possible to obtain explicit formulas which are amenable to computations of the respective Gaussian stochastic processes. For achieving this, we first develop two functional analytic tools. They are: …
The Unimodality Of Pure O-Sequences Of Type Two In Four Variables,
2015
Sacred Heart University
The Unimodality Of Pure O-Sequences Of Type Two In Four Variables, Bernadette Boyle
Mathematics Faculty Publications
Since the 1970's, great interest has been taken in the study of pure O-sequences, which, due to Macaulay's theory of inverse systems, have a bijective correspondence to the Hilbert functions of Artinian level monomial algebras. Much progress has been made in classifying these according to their shape. Macaulay's theorem immediately gives us that all Artinian algebras in two variables have unimodal Hilbert functions. Furthermore, it has been shown that all Artinian level monomial algebras of type two in three variables have unimodal Hilbert functions. This paper will classify all Artinian level monomial algebras of type two in four variables into …
Pure Injective And *-Pure Injective Lca Groups,
2015
Sacred Heart University
Pure Injective And *-Pure Injective Lca Groups, Peter Loth
Mathematics Faculty Publications
No abstract provided.
Multidimensional Mod Planes,
2015
University of New Mexico
Multidimensional Mod Planes, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral
Branch Mathematics and Statistics Faculty and Staff Publications
In this book authors name the interval [0, m); 2 ≤ m ≤ ∞ as mod interval. We have studied several properties about them but only here on wards in this book and forthcoming books the interval [0, m) will be termed as the mod real interval, [0, m)I as mod neutrosophic interval, [0,m)g; g2 = 0 as mod dual number interval, [0, m)h; h2 = h as mod special dual like number interval and [0, m)k, k2 = (m − 1) k as mod special quasi dual number interval. However there is only one real interval (∞, ∞) but …
Neutrosophic Graphs: A New Dimension To Graph Theory,
2015
University of New Mexico
Neutrosophic Graphs: A New Dimension To Graph Theory, Florentin Smarandache, Wb. Vasantha Kandasamy, K. Ilanthenral
Branch Mathematics and Statistics Faculty and Staff Publications
In this book authors for the first time have made a through study of neutrosophic graphs. This study reveals that these neutrosophic graphs give a new dimension to graph theory. The important feature of this book is it contains over 200 neutrosophic graphs to provide better understanding of this concepts. Further these graphs happen to behave in a unique way inmost cases, for even the edge colouring problem is different from the classical one. Several directions and dimensions in graph theory are obtained from this study. Finally certainly these new notions of neutrosophic graphs in general and in particular the …
Inner Product Spaces And Krein Spaces In The Quaternionic Setting,
2015
Chapman University
Inner Product Spaces And Krein Spaces In The Quaternionic Setting, Daniel Alpay, Fabrizio Colombo, Irene Sabadini
Mathematics, Physics, and Computer Science Faculty Articles and Research
In this paper we provide a study of quaternionic inner product spaces. This includes ortho-complemented subspaces, fundamental decompositions as well as a number of results of topological nature. Our main purpose is to show that a closed uniformly positive subspace in a quaternionic Krein space is ortho-complemented, and this leads to our choice of the results presented in the paper.
Presenting Distributive Laws,
2015
Leiden University
Presenting Distributive Laws, Marcello M. Bonsangue, Helle H. Hansen, Alexander Kurz, Jurriaan Rot
Engineering Faculty Articles and Research
Distributive laws of a monad T over a functor F are categorical tools for specifying algebra-coalgebra interaction. They proved to be important for solving systems of corecursive equations, for the specification of well-behaved structural operational semantics and, more recently, also for enhancements of the bisimulation proof method. If T is a free monad, then such distributive laws correspond to simple natural transformations. However, when T is not free it can be rather difficult to prove the defining axioms of a distributive law. In this paper we describe how to obtain a distributive law for a monad with an equational presentation …
Some Problems In The Representation Theory Of Simple Modular Lie Algebras,
2015
University of South Alabama
Some Problems In The Representation Theory Of Simple Modular Lie Algebras, Georgia Benkart, Jorg Feldvoss
University Faculty and Staff Publications
No abstract provided.
Enhancement On Counting Invariant On Symmetric Virtual Biracks,
2015
Scripps College
Enhancement On Counting Invariant On Symmetric Virtual Biracks, Melinda Ho
Scripps Senior Theses
This thesis introduces a new enhancement for virtual birack counting invariants. We first introduce knots and other general types of knots (oriented knots, framed knots, racks, and biracks). Then we’ll discuss the methods, knot invariants, mathematicians use to identify whether two knots are different. Next we’ll look at knots with virtual crossings and knots with a good involution. Finally, we introduce a new symmetric enhancement for virtual birack counting invariants and provide an example.
Positive Fragments Of Coalgebraic Logics,
2015
University Politehnica of Bucharest
Positive Fragments Of Coalgebraic Logics, Adriana Balan, Alexander Kurz, Jirí Velebil
Engineering Faculty Articles and Research
Positive modal logic was introduced in an influential 1995 paper of Dunn as the positive fragment of standard modal logic. His completeness result consists of an axiomatization that derives all modal formulas that are valid on all Kripke frames and are built only from atomic propositions, conjunction, disjunction, box and diamond. In this paper, we provide a coalgebraic analysis of this theorem, which not only gives a conceptual proof based on duality theory, but also generalizes Dunn's result from Kripke frames to coalgebras for weak-pullback preserving functors. To facilitate this analysis we prove a number of category theoretic results on …
Extensions Of Functors From Set To V-Cat,
2015
University Politehnica of Bucharest
Extensions Of Functors From Set To V-Cat, Adriana Balan, Alexander Kurz, Jirí Velebil
Engineering Faculty Articles and Research
We show that for a commutative quantale V every functor Set --> V-cat has an enriched left- Kan extension. As a consequence, coalgebras over Set are subsumed by coalgebras over V-cat. Moreover, one can build functors on V-cat by equipping Set-functors with a metric.
Approximation Of Nested Fixpoints,
2015
Chapman University
Approximation Of Nested Fixpoints, Alexander Kurz
Engineering Faculty Articles and Research
The question addressed in this paper is how to correctly approximate infinite data given by systems of simultaneous corecursive definitions. We devise a categorical framework for reasoning about regular datatypes, that is, datatypes closed under products, coproducts and fixpoints. We argue that the right methodology is on one hand coalgebraic (to deal with possible nontermination and infinite data) and on the other hand 2-categorical (to deal with parameters in a disciplined manner). We prove a coalgebraic version of Bekic lemma that allows us to reduce simultaneous fixpoints to a single fix point. Thus a possibly infinite object of interest is …
Coalgebraic Semantics Of Reflexive Economics (Dagstuhl Seminar 15042),
2015
University of Oxford
Coalgebraic Semantics Of Reflexive Economics (Dagstuhl Seminar 15042), Samson Abramsky, Alexander Kurz, Pierre Lescanne, Viktor Winschel
Engineering Faculty Articles and Research
This report documents the program and the outcomes of Dagstuhl Seminar 15042 “Coalgebraic Semantics of Reflexive Economics”.
Quaternionic Hardy Spaces In The Open Unit Ball And Half Space And Blaschke Products,
2015
Chapman University
Quaternionic Hardy Spaces In The Open Unit Ball And Half Space And Blaschke Products, Daniel Alpay, Fabrizio Colombo, Irene Sabadini
Mathematics, Physics, and Computer Science Faculty Articles and Research
The Hardy spaces H2(B) and H2(H+), where B and H+ denote, respectively, the open unit ball of the quaternions and the half space of quaternions with positive real part, as well as Blaschke products, have been intensively studied in a series of papers where they are used as a tool to prove other results in Schur analysis. This paper gives an overview on the topic, collecting the various results available.
Self-Mappings Of The Quaternionic Unit Ball: Multiplier Properties, Schwarz-Pick Inequality, And Nevanlinna-Pick Interpolation Problem,
2015
Chapman University
Self-Mappings Of The Quaternionic Unit Ball: Multiplier Properties, Schwarz-Pick Inequality, And Nevanlinna-Pick Interpolation Problem, Daniel Alpay, Vladimir Bolotnikov, Fabrizio Colombo, Irene Sabadini, Fabrizio Colombo
Mathematics, Physics, and Computer Science Faculty Articles and Research
We study several aspects concerning slice regular functions mapping the quaternionic open unit ball B into itself. We characterize these functions in terms of their Taylor coefficients at the origin and identify them as contractive multipliers of the Hardy space H2(B). In addition, we formulate and solve the Nevanlinna-Pick interpolation problem in the class of such functions presenting necessary and sufficient conditions for the existence and for the uniqueness of a solution. Finally, we describe all solutions to the problem in the indeterminate case.
An Extension Of Herglotz's Theorem To The Quaternions,
2015
Chapman University
An Extension Of Herglotz's Theorem To The Quaternions, Daniel Alpay, Fabrizio Colombo, David P. Kimsey, Irene Sabadini, David P. Kimsey
Mathematics, Physics, and Computer Science Faculty Articles and Research
A classical theorem of Herglotz states that a function n↦r(n) from Z into Cs×s is positive definite if and only there exists a Cs×s-valued positive measure dμ on [0,2π] such that r(n)=∫2π0eintdμ(t)for n∈Z. We prove a quaternionic analogue of this result when the function is allowed to have a number of negative squares. A key tool in the argument is the theory of slice hyperholomorphic functions, and the representation of such functions which have a positive real part in the unit ball of the quaternions. We study in great detail the case of positive definite functions.
A New Resolvent Equation For The S-Functional Calculus,
2015
Chapman University
A New Resolvent Equation For The S-Functional Calculus, Daniel Alpay, Fabrizio Colombo, Jonathan Gantner, Irene Sabadini
Mathematics, Physics, and Computer Science Faculty Articles and Research
The S-functional calculus is a functional calculus for (n + 1)-tuples of non necessarily commuting operators that can be considered a higher dimensional version of the classical Riesz-Dunford functional calculus for a single operator. In this last calculus, the resolvent equation plays an important role in the proof of several results. Associated with the S-functional calculus there are two resolvent operators: the left S−1 L (s, T ) and the right one S−1 R (s, T ), where s = (s0, s1, . . . , sn) ∈ Rn+1 and T = (T0, T1, . . . , Tn) is …
Infinite Product Representations For Kernels And Iteration Of Functions,
2015
Chapman University
Infinite Product Representations For Kernels And Iteration Of Functions, Daniel Alpay, Palle Jorgensen, Izchak Lewkowicz, Itzik Marziano
Mathematics, Physics, and Computer Science Faculty Articles and Research
We study infinite products of reproducing kernels with view to their use in dynamics (of iterated function systems), in harmonic analysis, and in stochastic processes. On the way, we construct a new family of representations of the Cuntz relations. Then, using these representations we associate a fixed filled Julia set with a Hilbert space. This is based on analysis and conformal geometry of a fixed rational mapping R in one complex variable, and its iterations.
On Algebras Which Are Inductive Limits Of Banach Spaces,
2015
Chapman University
On Algebras Which Are Inductive Limits Of Banach Spaces, Daniel Alpay, Guy Salomon
Mathematics, Physics, and Computer Science Faculty Articles and Research
We introduce algebras which are inductive limits of Banach spaces and carry inequalities which are counterparts of the inequality for the norm in a Banach algebra. We then define an associated Wiener algebra, and prove the corresponding version of the well-known Wiener theorem. Finally, we consider factorization theory in these algebra, and in particular, in the associated Wiener algebra.
Wiener-Chaos Approach To Optimal Prediction,
2015
Chapman University
Wiener-Chaos Approach To Optimal Prediction, Daniel Alpay, Alon Kipnis
Mathematics, Physics, and Computer Science Faculty Articles and Research
In this work we combine Wiener chaos expansion approach to study the dynamics of a stochastic system with the classical problem of the prediction of a Gaussian process based on part of its sample path. This is done by considering special bases for the Gaussian space G generated by the process, which allows us to obtain an orthogonal basis for the Fock space of G such that each basis element is either measurable or independent with respect to the given samples. This allows us to easily derive the chaos expansion of a random variable conditioned on part of the sample …
