Algebra Commons^™

Unitary Equivalence To A Complex Symmetric Matrix: Low Dimensions, Stephan Ramon Garcia, Daniel E. Poore '11, James E. Tener '08

Pomona Faculty Publications and Research

A matrix T∈M_n(C) is UECSM if it is unitarily equivalent to a complex symmetric (i.e., self-transpose) matrix. We develop several techniques for studying this property in dimensions three and four. Among other things, we completely characterize 4×4 nilpotent matrices which are UECSM and we settle an open problem which has lingered in the 3×3 case. We conclude with a discussion concerning a crucial difference which makes dimension three so different from dimensions four and above.

On The Matrix Equation Xa + Ax_T = 0, Stephan Ramon Garcia, Amy L. Shoemaker '14

Pomona Faculty Publications and Research

The matrix equation XA+AX^T=0, which has relevance to the study of Lie algebras, was recently studied by De Terán and Dopico (Linear Algebra Appl. 434 (2011), 44–67). They reduced the study of this equation to several special cases and produced explicit solutions in most instances. In this note we obtain an explicit solution in one of the difficult cases, for which only the dimension of the solution space and an algorithm to find a basis of this space were known previously

Review: Classification Of Four And Six Dimensional Drinfel'd Superdoubles, Gizem Karaali

Pomona Faculty Publications and Research

No abstract provided.

Review: Lusztig Symmetries And Automorphisms Of Quantum Superalgebras, Gizem Karaali

Pomona Faculty Publications and Research

No abstract provided.

Review: An Operator Approach To The Rational Solutions Of The Classical Yang-Baxter Equation, Gizem Karaali

Pomona Faculty Publications and Research

No abstract provided.

Two Remarks About Nilpotent Operators Of Order Two, Stephan Ramon Garcia, Bob Lutz '13, D. Timotin

Pomona Faculty Publications and Research

We present two novel results about Hilbert space operators which are nilpotent of order two. First, we prove that such operators are indestructible complex symmetric operators, in the sense that tensoring them with any operator yields a complex symmetric operator. In fact, we prove that this property characterizes nilpotents of order two among all nonzero bounded operators. Second, we establish that every nilpotent of order two is unitarily equivalent to a truncated Toeplitz operator.

Spaces Of Sections Of Banach Algebra Bundles, Emmanuel Dror Farjoun, Claude Schochet

Mathematics Faculty Research Publications

Suppose that B is a G-Banach algebra over 𝔽 = ℝ or ℂ, X is a finite dimensional compact metric space, ζ : P → X is a standard principal G-bundle, and A_ζ = Γ(X,P x_G B) is the associated algebra of sections. We produce a spectral sequence which converges to π_∗(GL_oA_ζ) with

E_{_}²_p,q ≅ Ȟ^p(X ; π_q(GL_oB)).

A related spectral sequence converging to K_∗+1(A_ζ) (the real or complex topological …

Squaring, Cubing, And Cube Rooting, Arthur T. Benjamin

All HMC Faculty Publications and Research

I still recall my thrill and disappointment when I read Mathematical Carnival, by Martin Gardner. I was thrilled because, as my high school teacher had recommended, mathematics was presented in a playful way that I had never seen before. I was disappointed because it contained a formula that I thought I had "invented" a few years earlier. I have always had a passion for mental calculation, and the following formula appears in Gardner's chapter on "Lightning Calculators." It was used by the mathematician A. C. Aitken to mentally square large numbers.

Strongly Complete Logics For Coalgebras, Alexander Kurz, Jiří Rosický

Engineering Faculty Articles and Research

Coalgebras for a functor model different types of transition systems in a uniform way. This paper focuses on a uniform account of finitary logics for set-based coalgebras. In particular, a general construction of a logic from an arbitrary set-functor is given and proven to be strongly complete under additional assumptions. We proceed in three parts.

Part I argues that sifted colimit preserving functors are those functors that preserve universal algebraic structure. Our main theorem here states that a functor preserves sifted colimits if and only if it has a finitary presentation by operations and equations. Moreover, the presentation of the …

Semigroup As Graphs, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book the authors study the zero divisor graph and unit graph of a semigroup. The zero divisor graphs of semigroups Zn under multiplication is studied and characterized.

Abelian Groups With Partial Decomposition Bases In LΔ∞Ω, Part Ii, Carol Jacoby, Peter Loth

Mathematics Faculty Publications

We consider abelian groups with partial decomposition bases in L^δ_∞ω for ordinals δ. Jacoby, Leistner, Loth and Str¨ungmann developed a numerical invariant deduced from the classical global Warfield invariant and proved that if two groups have identical modified Warfield invariants and Ulm-Kaplansky invariants up to ωδ for some ordinal δ, then they are equivalent in L^δ_∞ω. Here we prove that the modified Warfield invariant is expressible in L^δ_∞ω and hence the converse is true for appropriate δ.

Schur Functions And Their Realizations In The Slice Hyperholomorphic Setting, Daniel Alpay, Fabrizio Colombo, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we start the study of Schur analysis in the quaternionic setting using the theory of slice hyperholomorphic functions. The novelty of our approach is that slice hyperholomorphic functions allows to write realizations in terms of a suitable resolvent, the so called S-resolvent operator and to extend several results that hold in the complex case to the quaternionic case. We discuss reproducing kernels, positive definite functions in this setting and we show how they can be obtained in our setting using the extension operator and the slice regular product. We define Schur multipliers, and find their co-isometric realization …

On The Class Rsi Of J-Contractive Functions Intertwining Solutions Of Linear Differential Equations, Daniel Alpay, Andrey Melnikov, Victor Vinnikov

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we extend and solve in the class of functions RSI mentioned in the title, a number of problems originally set for the class RS of rational functions contractive in the open right-half plane, and unitary on the imaginary line with respect to some preassigned self-adjoint matrix. The problems we consider include the Schur algorithm, the partial realization problem and the Nevanlinna-Pick interpolation problem. The arguments rely on the one-to-one correspondence between elements in a given subclass of RSI and elements in RS. Another important tool in the arguments is a new result pertaining to the classical tangential …

New Topological C-Algebras With Applications In Linear Systems Theory, Daniel Alpay, Guy Salomon

Mathematics, Physics, and Computer Science Faculty Articles and Research

Motivated by the Schwartz space of tempered distributions S′ and the Kondratiev space of stochastic distributions S−1 we define a wide family of nuclear spaces which are increasing unions of (duals of) Hilbert spaces H′p,p∈N, with decreasing norms |⋅|p. The elements of these spaces are functions on a free commutative monoid. We characterize those rings in this family which satisfy an inequality of the form |f∗g|p≤A(p−q)|f|q|g|p for all p≥q+d, where * denotes the convolution in the monoid, A(p−q) is a strictly positive number and d is a fixed natural number (in this case we obtain commutative topological C-algebras). Such an …

Abelian Groups With Partial Decomposition Bases In LΔ∞Ω, Part I, Carol Jacoby, Katrin Leistner, Peter Loth, Lutz Strungmann

Mathematics Faculty Publications

We consider the class of abelian groups possessing partial decomposition bases in L^δ_∞ω for δ an ordinal. This class contains the class of Warfield groups which are direct summands of simply presented groups or, alternatively, are abelian groups possessing a nice decomposition basis with simply presented cokernel. We prove a classification theorem using numerical invariants that are deduced from the classical Ulm-Kaplansky and Warfield invariants. This extends earlier work by Barwise-Eklof, Göbel and the authors.

Unitary Equivalence To A Truncated Toeplitz Operator: Analytic Symbols, William T. Ross, Stephan Ramon Garcia, Daniel E. Poore

Department of Math & Statistics Faculty Publications

Unlike Toeplitz operators on H², truncated Toeplitz operators do not have a natural matricial characterization. Consequently, these operators are difficult to study numerically. In this note we provide criteria for a matrix with distinct eigenvalues to be unitarily equivalent to a truncated Toeplitz operator having an analytic symbol. This test is constructive and we illustrate it with several examples. As a byproduct, we also prove that every complex symmetric operator on a Hilbert space of dimension < 3 is unitarily equivalent to a direct sum of truncated Toeplitz operators.

Non Associative Linear Algebras, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors for the first time introduce the notion of non associative vector spaces and non associative linear algebras over a field. We construct non associative space using loops and groupoids over fields. In general in all situations, which we come across to find solutions may not be associative; in such cases we can without any difficulty adopt these non associative vector spaces/linear algebras. Thus this research is a significant one.

This book has six chapters. First chapter is introductory in nature. The new concept of non associative semilinear algebras is introduced in chapter two. This structure is …

Dual Numbers, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

Dual numbers was first introduced by W.K. Clifford in 1873. This nice concept has lots of applications; to screw systems, modeling plane joint, iterative methods for displacement analysis of spatial mechanisms, inertial force analysis of spatial mechanisms etc. In this book the authors study dual numbers in a special way. The main aim of this book is to find rich sources of new elements g such that g2 = 0. The main sources of such new elements are from Zn, n a composite number. We give algebraic structures on them. This book is organized into six chapters. The final chapter …

Exploring The Extension Of Natural Operations On Intervals, Matrices And Complex Numbers, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

This book extends the natural operations defined on intervals, finite complex numbers and matrices. The intervals [a, b] are such that a ≤ b. But the natural class of intervals [a, b] introduced by the authors are such that a ≥ b or a need not be comparable with b. This way of defining natural class of intervals enables the authors to extend all the natural operations defined on reals to these natural class of intervals without any difficulty. Thus with these natural class of intervals working with interval matrices like stiffness matrices finding eigenvalues takes the same time as …

Natural Product Xn On Matrices, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book the authors introduce a new product on matrices called the natural product. ...

Thus by introducing natural product we can find the product of column matrices and product of two rectangular matrices of same order. Further this product is more natural which is just identical with addition replaced by multiplication on these matrices. Another fact about natural product is this enables the product of any two super matrices of same order and with same type of partition. We see on supermatrices products cannot be defined easily which prevents from having any nice algebraic structure on the collection …

Innovative Uses Of Matrices, Florentin Smarandache, W.B. Vasantha Kandasamy, Indra Venkatbabu

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors bring out the innovative applications of matrices defined, described and developed by them. Here they do not include the natural product on matrices newly described and defined by them in the book on ‘natural product ×n on matrices’.

This book is organized into seven chapters. The first one is introductory in nature. In the second chapter authors give the unique and new way of analyzing the data which is time dependent. We construct three types of matrices called Average Time Dependent data matrix (ATD matrix), Refined Time Dependent Data matrix (RTD matrix) and Combined Effective Time …

Supermodular Lattices, Florentin Smarandache, Iqbal Unnisa, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In lattice theory the two well known equational class of lattices are the distributive lattices and the modular lattices. All distributive lattices are modular however a modular lattice in general is not distributive.

In this book, new classes of lattices called supermodular lattices and semi-supermodular lattices are introduced and characterized as follows: A subdirectly irreducible supermodular lattice is isomorphic to the two element chain lattice C2 or the five element modular lattice M3. A lattice L is supermodular if and only if L is a subdirect union of a two element chain C2 and the five element modular lattice M3.

Special Quasi Dual Numbers And Groupoids, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book the authors introduce a new notion called special quasi dual number, x = a + bg.

Among the reals – 1 behaves in this way, for (– 1)2 = 1 = – (– 1). Likewise –I behaves in such a way (– I)2 = – (– I). These special quasi dual numbers can be generated from matrices with entries from 1 or I using only the natural product ×n. Another rich source of these special quasi dual numbers or quasi special dual numbers is Zn, n a composite number. For instance 8 in Z12 is such that …

Special Dual Like Numbers And Lattices, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book the authors introduce a new type of dual numbers called special dual like numbers. These numbers are constructed using idempotents in the place of nilpotents of order two as new element. That is x = a + bg is a special dual like number where a and b are reals and g is a new element such that g2 =g. The collection of special dual like numbers forms a ring. Further lattices are the rich structures which contributes to special dual like numbers. These special dual like numbers x = a + bg; when a and b …

Non Associative Algebraic Structures Using Finite Complex Numbers, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

Authors in this book for the first time have constructed nonassociative structures like groupoids, quasi loops, non associative semirings and rings using finite complex modulo integers. The Smarandache analogue is also carried out. We see the nonassociative complex modulo integers groupoids satisfy several special identities like Moufang identity, Bol identity, right alternative and left alternative identities. P-complex modulo integer groupoids and idempotent complex modulo integer groupoids are introduced and characterized. This book has six chapters. The first one is introductory in nature. Second chapter introduces complex modulo integer groupoids and complex modulo integer loops.

Neutrosophic Super Matrices And Quasi Super Matrices, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors study neutrosophic super matrices. The concept of neutrosophy or indeterminacy happens to be one the powerful tools used in applications like FCMs and NCMs where the expert seeks for a neutral solution. Thus this concept has lots of applications in fuzzy neutrosophic models like NRE, NAM etc. These concepts will also find applications in image processing where the expert seeks for a neutral solution. Here we introduce neutrosophic super matrices and show that the sum or product of two neutrosophic matrices is not in general a neutrosophic super matrix. Another interesting feature of this book is …

Quasi Set Topological Vector Subspaces, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book the authors introduce four types of topological vector subspaces. All topological vector subspaces are defined depending on a set. We define a quasi set topological vector subspace of a vector space depending on the subset S contained in the field F over which the vector space V is defined. These quasi set topological vector subspaces defined over a subset can be of finite or infinite dimension. An interesting feature about these spaces is that there can be several quasi set topological vector subspaces of a given vector space. This property helps one to construct several spaces with …

Centric Cardinal Sine Function, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

According to any standard dictionary, the word "cardinal" is synonymous with "principal", "essential", "fundamental". In centric mathematics (CM), or ordinary mathematics, cardinal is, on the one hand, a number equal to a number of finite aggregate, called the power of the aggregate, and on the other hand, known as the sine cardinal sinc(x) or cosine cardinal cosc(x), is a special function defined by the centric circular function (CCF). sin(x) and cos(x) are commonly used in undulatory physics (see Figure 1) and whose graph, the graph of cardinal sine, which is called as "Mexican hat" (sombrero) because of its shape.

Enhancements Of Rack Counting Invariants Via Dynamical Cocycles, Alissa S. Crans, Sam Nelson, Aparna Sarkar

Mathematics, Statistics and Data Science Faculty Works

We introduce the notion of N-reduced dynamical cocycles and use these objects to define enhancements of the rack counting invariant for classical and virtual knots and links. We provide examples to show that the new invariants are not determined by the rack counting invariant, the Jones polynomial or the generalized Alexander polynomial.

Odd Or Even: Uncovering Parity Of Rank In A Family Of Rational Elliptic Curves, Anika Lindemann

Honors Theses

Puzzled by equations in multiple variables for centuries, mathematicians have made relatively few strides in solving these seemingly friendly, but unruly beasts. Currently, there is no systematic method for finding all rational values, that satisfy any equation with degree higher than a quadratic. This is bizarre. Solving these has preoccupied great minds since before the formal notion of an equation existed. Before any sort of mathematical formality, these questions were nested in plucky riddles and folded into folk tales. Because they are so simple to state, these equations are accessible to a very general audience. Yet an astounding amount of …