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- Yang-Baxter equation (10)
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Articles 1 - 30 of 54
Full-Text Articles in Physical Sciences and Mathematics
Review: A C*-Algebra Approach To Complex Symmetric Operators, Stephan Ramon Garcia
Review: A C*-Algebra Approach To Complex Symmetric Operators, Stephan Ramon Garcia
Pomona Faculty Publications and Research
No abstract provided.
Review: Transitivity And Bundle Shifts, Stephan Ramon Garcia
Review: Transitivity And Bundle Shifts, Stephan Ramon Garcia
Pomona Faculty Publications and Research
No abstract provided.
Lattices From Hermitian Function Fields, Albrecht Böttcher, Lenny Fukshansky, Stephan Ramon Garcia, Hiren Maharaj
Lattices From Hermitian Function Fields, Albrecht Böttcher, Lenny Fukshansky, Stephan Ramon Garcia, Hiren Maharaj
Pomona Faculty Publications and Research
We consider the well-known Rosenbloom-Tsfasman function field lattices in the special case of Hermitian function fields. We show that in this case the resulting lattices are generated by their minimal vectors, provide an estimate on the total number of minimal vectors, and derive properties of the automorphism groups of these lattices. Our study continues previous investigations of lattices coming from elliptic curves and finite Abelian groups. The lattices we are faced with here are more subtle than those considered previously, and the proofs of the main results require the replacement of the existing linear algebra approaches by deep results of …
Review: The Classical Hom-Yang-Baxter Equation And Hom-Lie Bialgebras, Gizem Karaali
Review: The Classical Hom-Yang-Baxter Equation And Hom-Lie Bialgebras, Gizem Karaali
Pomona Faculty Publications and Research
No abstract provided.
Review: Crystal Bases Of Q-Deformed Kac Modules Over The Quantum Superalgebras Uq(Gl(Mln)), Gizem Karaali
Review: Crystal Bases Of Q-Deformed Kac Modules Over The Quantum Superalgebras Uq(Gl(Mln)), Gizem Karaali
Pomona Faculty Publications and Research
No abstract provided.
Review: The Relationships Among Multiplicities Of A J-Self-Adjoint Differential Operator's Eigenvalue, Stephan Ramon Garcia
Review: The Relationships Among Multiplicities Of A J-Self-Adjoint Differential Operator's Eigenvalue, Stephan Ramon Garcia
Pomona Faculty Publications and Research
No abstract provided.
Splitting Fields And Periods Of Fibonacci Sequences Modulo Primes, Sanjai Gupta, Parousia Rockstroh '08, Francis E. Su
Splitting Fields And Periods Of Fibonacci Sequences Modulo Primes, Sanjai Gupta, Parousia Rockstroh '08, Francis E. Su
All HMC Faculty Publications and Research
We consider the period of a Fibonacci sequence modulo a prime and provide an accessible, motivated treatment of this classical topic using only ideas from linear and abstract algebra. Our methods extend to general recurrences with prime moduli and provide some new insights. And our treatment highlights a nice application of the use of splitting fields that might be suitable to present in an undergraduate course in abstract algebra or Galois theory.
Squaring, Cubing, And Cube Rooting, Arthur T. Benjamin
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.
On The Quantization Of Zero-Weight Super Dynamical R-Matrices, Gizem Karaali
On The Quantization Of Zero-Weight Super Dynamical R-Matrices, Gizem Karaali
Pomona Faculty Publications and Research
Solutions of the classical dynamical Yang-Baxter equation on a Lie superalgebra are called super dynamical r-matrices. A super dynamical r-matrix r satisfies the zero weight condition if
[h ⊗ 1 + 1 ⊗ h, r(λ)] = 0 for all h ∈ ɧ, λ ∈ ɧ ∗ .
In this paper we explicitly quantize zero-weight super dynamical r-matrices with zero coupling constant for the Lie superalgebra gl(m, n) . We also answer some questions about super dynamical R-matrices. In particular, we prove a classification theorem and offer some support for one particular …
Two Remarks About Nilpotent Operators Of Order Two, Stephan Ramon Garcia, Bob Lutz '13, D. Timotin
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.
Unitary Equivalence To A Truncated Toeplitz Operator: Analytic Symbols, Stephan Ramon Garcia, Daniel E. Poore '11, William T. Ross
Unitary Equivalence To A Truncated Toeplitz Operator: Analytic Symbols, Stephan Ramon Garcia, Daniel E. Poore '11, William T. Ross
Pomona Faculty Publications and Research
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 paper 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.
Unitary Equivalence To A Complex Symmetric Matrix: Low Dimensions, Stephan Ramon Garcia, Daniel E. Poore '11, James E. Tener '08
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∈Mn(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.
Review: Classification Of Four And Six Dimensional Drinfel'd Superdoubles, Gizem Karaali
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
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
Review: An Operator Approach To The Rational Solutions Of The Classical Yang-Baxter Equation, Gizem Karaali
Pomona Faculty Publications and Research
No abstract provided.
On The Matrix Equation Xa + Ax_T = 0, Stephan Ramon Garcia, Amy L. Shoemaker '14
On The Matrix Equation Xa + Ax_T = 0, Stephan Ramon Garcia, Amy L. Shoemaker '14
Pomona Faculty Publications and Research
The matrix equation XA+AXT=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
Classical Kloosterman Sums: Representation Theory, Magic Squares, And Ramanujan Multigraphs, Patrick S. Fleming, Stephan Ramon Garcia, Gizem Karaali
Classical Kloosterman Sums: Representation Theory, Magic Squares, And Ramanujan Multigraphs, Patrick S. Fleming, Stephan Ramon Garcia, Gizem Karaali
Pomona Faculty Publications and Research
We consider a certain finite group for which Kloosterman sums appear as character values. This leads us to consider a concrete family of commuting hermitian matrices which have Kloosterman sums as eigenvalues. These matrices satisfy a number of “magical” combinatorial properties and they encode various arithmetic properties of Kloosterman sums. These matrices can also be regarded as adjacency matrices for multigraphs which display Ramanujan-like behavior.
Spatial Isomorphisms Of Algebras Of Truncated Toeplitz Operators, Stephan Ramon Garcia, William T. Ross, Warren R. Wogen
Spatial Isomorphisms Of Algebras Of Truncated Toeplitz Operators, Stephan Ramon Garcia, William T. Ross, Warren R. Wogen
Pomona Faculty Publications and Research
We examine when two maximal abelian algebras in the truncated Toeplitz operators are spatially isomorphic. This builds upon recent work of N. Sedlock, who obtained a complete description of the maximal algebras of truncated Toeplitz operators.
Review: Massey Products On Cycles Of Projective Lines And Trigonometric Solutions Of The Yang-Baxter Equations, Gizem Karaali
Review: Massey Products On Cycles Of Projective Lines And Trigonometric Solutions Of The Yang-Baxter Equations, Gizem Karaali
Pomona Faculty Publications and Research
No abstract provided.
Review: The Semi-Dynamical Reflection Equation: Solutions And Structure Matrices, Gizem Karaali
Review: The Semi-Dynamical Reflection Equation: Solutions And Structure Matrices, Gizem Karaali
Pomona Faculty Publications and Research
No abstract provided.
Recognizing Graph Theoretic Properties With Polynomial Ideals, Jesus A. De Loera, Christopher J. Hillar, Peter N. Malkin, Mohamed Omar
Recognizing Graph Theoretic Properties With Polynomial Ideals, Jesus A. De Loera, Christopher J. Hillar, Peter N. Malkin, Mohamed Omar
All HMC Faculty Publications and Research
Many hard combinatorial problems can be modeled by a system of polynomial equations. N. Alon coined the term polynomial method to describe the use of nonlinear polynomials when solving combinatorial problems. We continue the exploration of the polynomial method and show how the algorithmic theory of polynomial ideals can be used to detect k-colorability, unique Hamiltonicity, and automorphism rigidity of graphs. Our techniques are diverse and involve Nullstellensatz certificates, linear algebra over finite fields, Gröbner bases, toric algebra, convex programming, and real algebraic geometry.
Some New Classes Of Complex Symmetric Operators, Stephan Ramon Garcia, Warren R. Wogen
Some New Classes Of Complex Symmetric Operators, Stephan Ramon Garcia, Warren R. Wogen
Pomona Faculty Publications and Research
We say that an operator $T \in B(H)$ is complex symmetric if there exists a conjugate-linear, isometric involution $C:H\to H$ so that $T = CT^*C$. We prove that binormal operators, operators that are algebraic of degree two (including all idempotents), and large classes of rank-one perturbations of normal operators are complex symmetric. From an abstract viewpoint, these results explain why the compressed shift and Volterra integration operator are complex symmetric. Finally, we attempt to describe all complex symmetric partial isometries, obtaining the sharpest possible statement given only the data $(\dim \ker T, \dim \ker T^*)$.
Review: Classification Of Quasi-Trigonometric Solutions Of The Classical Yang-Baxter Equation, Gizem Karaali
Review: Classification Of Quasi-Trigonometric Solutions Of The Classical Yang-Baxter Equation, Gizem Karaali
Pomona Faculty Publications and Research
No abstract provided.
Review: Intertwining Symmetry Algebras Of Quantum Superintegrable Systems, Gizem Karaali
Review: Intertwining Symmetry Algebras Of Quantum Superintegrable Systems, Gizem Karaali
Pomona Faculty Publications and Research
No abstract provided.
Review: Quantization Of Hamiltonian-Type Lie Algebras, Gizem Karaali
Review: Quantization Of Hamiltonian-Type Lie Algebras, Gizem Karaali
Pomona Faculty Publications and Research
No abstract provided.
Review: A Class Of Solutions To The Quantum Colored Yang-Baxter Equation, Gizem Karaali
Review: A Class Of Solutions To The Quantum Colored Yang-Baxter Equation, Gizem Karaali
Pomona Faculty Publications and Research
No abstract provided.
Search Bounds For Zeros Of Polynomials Over The Algebraic Closure Of Q, Lenny Fukshansky
Search Bounds For Zeros Of Polynomials Over The Algebraic Closure Of Q, Lenny Fukshansky
CMC Faculty Publications and Research
We discuss existence of explicit search bounds for zeros of polynomials with coefficients in a number field. Our main result is a theorem about the existence of polynomial zeros of small height over the field of algebraic numbers outside of unions of subspaces. All bounds on the height are explicit.
Review: On Quantum Yang-Baxter Coherent Algebra Sheaves, Gizem Karaali
Review: On Quantum Yang-Baxter Coherent Algebra Sheaves, Gizem Karaali
Pomona Faculty Publications and Research
No abstract provided.
Review: Graded Structure And Hopf Structures In Parabosonic Algebra. An Alternative Approach To Bosonisation, Gizem Karaali
Review: Graded Structure And Hopf Structures In Parabosonic Algebra. An Alternative Approach To Bosonisation, Gizem Karaali
Pomona Faculty Publications and Research
No abstract provided.
Review: Dynamical Yang-Baxter Maps With An Invariance Condition, Gizem Karaali
Review: Dynamical Yang-Baxter Maps With An Invariance Condition, Gizem Karaali
Pomona Faculty Publications and Research
No abstract provided.