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- Keyword
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- Kerzman-Stein operator (3)
- Krein matrix (3)
- Cauchy integral (2)
- Distance function (2)
- Dual number (2)
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- Eigenvalue (2)
- Hamiltonian eigenvalue problem (2)
- Higher homotopy operations (2)
- Homotopy-commutative diagram (2)
- Laguerre arc length (2)
- Laguerre geometry (2)
- Nonlinear Schrödinger equation (2)
- Solitons (2)
- Vortices (2)
- *-even matrix polynomial (1)
- Alternative ring (1)
- André-Quillen cohomology (1)
- Automorphism (1)
- Bifurcations (1)
- Binding constant (1)
- Bose-Einstein condensates (1)
- Cantor set (1)
- Cantorval (1)
- Cartan's method of equivalence (1)
- Cauchy operator (1)
- Centrality (1)
- Consensus (1)
- Eigenvector centrality (1)
- Engel manifolds (1)
- Existence (1)
Articles 1 - 30 of 43
Full-Text Articles in Applied Mathematics
Concordance Of 2-Knots, Nathan Sunukjian
Concordance Of 2-Knots, Nathan Sunukjian
University Faculty Publications and Creative Works
In this paper we investigate the 0-concordance classes of 2-knots in S4, an equivalence relation that is related to understanding smooth structures on 4-manifolds. Using Rochlin’s invariant, and invariants arising from Heegaard–Floer homology, we will prove that there are infinitely many 0-concordance classes of 2-knots.
A Reformulated Krein Matrix For Star-Even Polynomial Operators With Applications, Todd Kapitula, Ross Parker, Bjorn Sandstede
A Reformulated Krein Matrix For Star-Even Polynomial Operators With Applications, Todd Kapitula, Ross Parker, Bjorn Sandstede
University Faculty Publications and Creative Works
In its original formulation the Krein matrix was used to locate the spectrum of first-order star-even polynomial operators where both operator coefficients are nonsingular. Such operators naturally arise when considering first-order-in-time Hamiltonian PDEs. Herein the matrix is reformulated to allow for operator coefficients with nontrivial kernel. Moreover, it is extended to allow for the study of the spectral problem associated with quadratic star-even operators, which arise when considering the spectral problem associated with second-order-in-time Hamiltonian PDEs. In conjunction with the Hamiltonian-Krein index (HKI) the Krein matrix is used to study two problems: conditions leading to Hamiltonian-Hopf bifurcations for small spatially …
Null-Homologous Exotic Surfaces In 4–Manifolds, Nathan Sunukjian, Neil R. Hoffman
Null-Homologous Exotic Surfaces In 4–Manifolds, Nathan Sunukjian, Neil R. Hoffman
University Faculty Publications and Creative Works
We exhibit infinite families of embedded tori in 4–manifolds that are topologically isotopic but smoothly distinct. The interesting thing about these tori is that they are topologically trivial in the sense that each bounds a topologically embedded solid handlebody. This implies that there are stably ribbon surfaces in 4–manifolds that are not ribbon.
Properly Handling Negative Values In The Calculation Of Binding Constants By Physicochemical Modeling Of Spectroscopic Titration Data, Nathanael P. Kazmierczak, Douglas A. Vander Griend
Properly Handling Negative Values In The Calculation Of Binding Constants By Physicochemical Modeling Of Spectroscopic Titration Data, Nathanael P. Kazmierczak, Douglas A. Vander Griend
University Faculty Publications and Creative Works
To implement equilibrium hard-modeling of spectroscopic titration data, the analyst must make a variety of crucial data processing choices that address negative absorbance and molar absorptivity values. The efficacy of three such methodological options is evaluated via high-throughput Monte Carlo simulations, root-mean-square error surface mapping, and two mathematical theorems. Accuracy of the calculated binding constant values constitutes the key figure of merit used to compare different data analysis approaches. First, using singular value decomposition to filter the raw absorbance data prior to modeling often reduces the number of negative values involved but has little effect on the calculated binding constant …
A Family Of Cantorvals, John Ferdinands, Timothy Ferdinands
A Family Of Cantorvals, John Ferdinands, Timothy Ferdinands
University Faculty Publications and Creative Works
The set of subsums of the series Σn=1∞ xn is known to be one of three types: a finite union of intervals, homeomorphic to the Cantor set, or of the type known as a Cantorval. Bartoszewicz, Filipczak and Szymonik have described a family of series which contained all known examples of subsum sets which are Cantorvals. We construct another family of series which produces new examples of subsum sets which are Cantorvals.
Consensus And Clustering In Opinion Formation On Networks, Julia Bujalski, Grace Dwyer, Todd Kapitula, Quang Nhat Le
Consensus And Clustering In Opinion Formation On Networks, Julia Bujalski, Grace Dwyer, Todd Kapitula, Quang Nhat Le
University Faculty Publications and Creative Works
Ideas that challenge the status quo either evaporate or dominate. The study of opinion dynamics in the socio-physics literature treats space as uniform and considers individuals in an isolated community, using ordinary differential equation (ODE) models. We extend these ODE models to include multiple communities and their interactions. These extended ODE models can be thought of as being ODEs on directed graphs. We study in detail these models to determine conditions under which there will be consensus and pluralism within the system. Most of the consensus/pluralism analysis is done for the case of one and two cities. However, we numerically …
Eigenvector Centrality: Illustrations Supporting The Utility Of Extracting More Than One Eigenvector To Obtain Additional Insights Into Networks And Interdependent Structures, Dawn Iacobucci, Rebecca Mcbride, Deidre L. Popovich
Eigenvector Centrality: Illustrations Supporting The Utility Of Extracting More Than One Eigenvector To Obtain Additional Insights Into Networks And Interdependent Structures, Dawn Iacobucci, Rebecca Mcbride, Deidre L. Popovich
University Faculty Publications and Creative Works
Among the many centrality indices used to detect structures of actors’ positions in networks is the use of the first eigenvector of an adjacency matrix that captures the connections among the actors. This research considers the seeming pervasive current practice of using only the first eigenvector. It is shows that, as in other statistical applications of eigenvectors, subsequent vectors can also contain illuminating information. Several small examples, and Freeman’s EIES network, are used to illustrate that while the first eigenvector is certainly informative, the second (and subsequent) eigenvector(s) can also be equally tractable and informative.
Roles Of A Teacher And Researcher During In Situ Professional Development Around The Implementation Of Mathematical Modeling Tasks, Hyunyi Jung, Corey Brady
Roles Of A Teacher And Researcher During In Situ Professional Development Around The Implementation Of Mathematical Modeling Tasks, Hyunyi Jung, Corey Brady
University Faculty Publications and Creative Works
Partnership with teachers for professional development has been considered beneficial because of the potential of collaborative work in the teacher’s own classroom to be relevant to practice. From this perspective, both teachers and researchers can draw on their own expertise and work as authentic partners. In this study, we address the need for such collaboration and focus on how a teacher and a researcher performed their roles when collaboratively implementing mathematical modeling tasks within a context of in situ professional development. Using multi-tier design-based research, as a framework, a researcher worked in a teacher’s classroom to implement a series of …
Local Characterization Of A Class Of Ruled Hypersurfaces In C2, Michael Bolt
Local Characterization Of A Class Of Ruled Hypersurfaces In C2, Michael Bolt
University Faculty Publications and Creative Works
Let M3⊂C2 be a three times differentiable real hypersurface. The Levi form of M transforms under biholomorphism, and when restricted to the complex tangent space, the skew-hermitian part of the second fundamental form transforms under fractional linear transformation. The surfaces for which these forms are constant multiples of each other were identified in previous work, but when the constant had unit modulus there was a global requirement. Here we give a local characterization of hypersurfaces for which the constant has unit modulus.
The Kerzman–Stein Operator For Piecewise Continuously Differentiable Regions, Michael Bolt, Andrew Raich
The Kerzman–Stein Operator For Piecewise Continuously Differentiable Regions, Michael Bolt, Andrew Raich
University Faculty Publications and Creative Works
The Kerzman–Stein operator is the skew-hermitian part of the Cauchy operator defined with respect to an unweighted hermitian inner product on a rectifiable curve. If the curve is continuously differentiable, the Kerzman–Stein operator is compact on the Hilbert space of square integrable functions; when there is a corner, the operator is noncompact. Here, we give a complete description of the spectrum for a finite symmetric wedge and we show how this reveals the essential spectrum for curves that are piecewise continuously differentiable. We also give an explicit construction for a smooth curve whose Kerzman–Stein operator has large norm.
Spectrum Of The Kerzman-Stein Operator For A Family Of Smooth Regions In The Plane, Michael Bolt
Spectrum Of The Kerzman-Stein Operator For A Family Of Smooth Regions In The Plane, Michael Bolt
University Faculty Publications and Creative Works
The Kerzman-Stein operator is the skew-hermitian part of the Cauchy operator defined with respect to an unweighted hermitian inner product on the boundary. For bounded regions with smooth boundary, the Kerzman-Stein operator is compact on the Hilbert space of square integrable functions. Here we give an explicit computation of its Hilbert-Schmidt norm for a family of simply connected regions. We also give an explicit computation of the Cauchy operator acting on an orthonormal basis, and we give estimates for the norms of the Kerzman-Stein and Cauchy operators on these regions. The regions are the first regions that display no apparent …
Szego Kernel Transformation Law For Proper Holomorphic Mappings, Michael Bolt
Szego Kernel Transformation Law For Proper Holomorphic Mappings, Michael Bolt
University Faculty Publications and Creative Works
Let ω1;ω2 be smoothly bounded doubly connected regions in the complex plane. We establish a transformation law for the Szego kernel under proper holomorphic mappings. This extends known results concerning biholomorphic mappings between multiply connected regions as well as proper holomorphic mappings from multiply connected regions to simply connected regions.
Szego Kernel Transformation Law For Proper Holomorphic Mappings, Michael Bolt
Szego Kernel Transformation Law For Proper Holomorphic Mappings, Michael Bolt
University Faculty Publications and Creative Works
Let ω1;ω2 be smoothly bounded doubly connected regions in the complex plane. We establish a transformation law for the Szego kernel under proper holomorphic mappings. This extends known results concerning biholomorphic mappings between multiply connected regions as well as proper holomorphic mappings from multiply connected regions to simply connected regions.
Instability Indices For Matrix Polynomials, Todd Kapitula, Elizabeth Hibma, Hwa Pyeong Kim, Jonathan Timkovich
Instability Indices For Matrix Polynomials, Todd Kapitula, Elizabeth Hibma, Hwa Pyeong Kim, Jonathan Timkovich
University Faculty Publications and Creative Works
There is a well-established instability index theory for linear and quadratic matrix polynomials for which the coefficient matrices are Hermitian and skew-Hermitian. This theory relates the number of negative directions for the matrix coefficients which are Hermitian to the total number of unstable eigenvalues for the polynomial. Herein we extend the theory to *-even matrix polynomials of any finite degree. In particular, unlike previously known cases we show that the instability index depends upon the size of the matrices when the degree of the polynomial is greater than two. We also consider Hermitian matrix polynomials, and derive an index which …
The Krein Matrix: General Theory And Concrete Applications In Atomic Bose-Einstein Condensates, Todd Kapitula, Panayotis G. Kevrekidis, Dong Yan
The Krein Matrix: General Theory And Concrete Applications In Atomic Bose-Einstein Condensates, Todd Kapitula, Panayotis G. Kevrekidis, Dong Yan
University Faculty Publications and Creative Works
When finding the nonzero eigenvalues for Hamiltonian eigenvalue problems it is especially important to locate not only the unstable eigenvalues (i.e., those with positive real part) but also those which are purely imaginary but have negative Krein signature. These latter eigenvalues have the property that they can become unstable upon collision with other purely imaginary eigenvalues; i.e., they are a necessary building block in the mechanism leading to the so-called Hamiltonian-Hopf bifurcation. In this paper we review a general theory for constructing a meromorphic matrix-valued function, the so-called Krein matrix, which has the property of not only locating the unstable …
The Krein Matrix: General Theory And Concrete Applications In Atomic Bose-Einstein Condensates, Todd Kapitula, Panayotis G. Kevrekidis, Dong Yan
The Krein Matrix: General Theory And Concrete Applications In Atomic Bose-Einstein Condensates, Todd Kapitula, Panayotis G. Kevrekidis, Dong Yan
University Faculty Publications and Creative Works
When finding the nonzero eigenvalues for Hamiltonian eigenvalue problems it is especially important to locate not only the unstable eigenvalues (i.e., those with positive real part) but also those which are purely imaginary but have negative Krein signature. These latter eigenvalues have the property that they can become unstable upon collision with other purely imaginary eigenvalues; i.e., they are a necessary building block in the mechanism leading to the so-called Hamiltonian-Hopf bifurcation. In this paper we review a general theory for constructing a meromorphic matrix-valued function, the so-called Krein matrix, which has the property of not only locating the unstable …
Higher Homotopy Operations And André-Quillen Cohomology, David Blanc, Mark W. Johnson, James M. Turner
Higher Homotopy Operations And André-Quillen Cohomology, David Blanc, Mark W. Johnson, James M. Turner
University Faculty Publications and Creative Works
There are two main approaches to the problem of realizing a Π-algebra (a graded group Λ equipped with an action of the primary homotopy operations) as the homotopy groups of a space X. Both involve trying to realize an algebraic free simplicial resolution G . of Λ by a simplicial space W ., and proceed by induction on the simplicial dimension. The first provides a sequence of André-Quillen cohomology classes in H n+2(Λ;Ω nΛ) (n≥1) as obstructions to the existence of successive Postnikov sections for W . (cf. Dwyer et al. (1995) [27]). The second gives a sequence of geometrically …
Stability Indices For Constrained Self-Adjoint Operators, Todd Kapitula, Keith Promislow
Stability Indices For Constrained Self-Adjoint Operators, Todd Kapitula, Keith Promislow
University Faculty Publications and Creative Works
A wide class of problems in the study of the spectral and orbital stability of dispersive waves in Hamiltonian systems can be reduced to understanding the so-called "energy spectrum", that is, the spectrum of the second variation of the Hamiltonian evaluated at the wave shape, which is constrained to act on a closed subspace of the underlying Hilbert space. We present a substantially simplified proof of the negative eigenvalue count for such constrained, self-adjoint operators, and extend the result to include an analysis of the location of the point spectra of the constrained operator relative to that of the unconstrained …
Stability Indices For Constrained Self-Adjoint Operators, Todd Kapitula, Keith Promislow
Stability Indices For Constrained Self-Adjoint Operators, Todd Kapitula, Keith Promislow
University Faculty Publications and Creative Works
A wide class of problems in the study of the spectral and orbital stability of dispersive waves in Hamiltonian systems can be reduced to understanding the so-called "energy spectrum", that is, the spectrum of the second variation of the Hamiltonian evaluated at the wave shape, which is constrained to act on a closed subspace of the underlying Hilbert space. We present a substantially simplified proof of the negative eigenvalue count for such constrained, self-adjoint operators, and extend the result to include an analysis of the location of the point spectra of the constrained operator relative to that of the unconstrained …
The Möbius Geometry Of Hypersurfaces, Ii, Michael Bolt
The Möbius Geometry Of Hypersurfaces, Ii, Michael Bolt
University Faculty Publications and Creative Works
No abstract provided.
The Möbius Geometry Of Hypersurfaces, Ii, Michael Bolt
The Möbius Geometry Of Hypersurfaces, Ii, Michael Bolt
University Faculty Publications and Creative Works
No abstract provided.
A Generalised Kummer's Conjecture, M. J.R. Myers
A Generalised Kummer's Conjecture, M. J.R. Myers
University Faculty Publications and Creative Works
Kummer's conjecture predicts the rate of growth of the relative class numbers of cyclotomic fields of prime conductor. We extend Kummer's conjecture to cyclotomic fields of conductor n, where n is any natural number. We show that the Elliott-Halberstam conjecture implies that this generalised Kummer's conjecture is true for almost all n but is false for infinitely many n. Copyright © 2010 Glasgow Mathematical Journal Trust.
A Generalised Kummer's Conjecture, M. J.R. Myers
A Generalised Kummer's Conjecture, M. J.R. Myers
University Faculty Publications and Creative Works
Kummer's conjecture predicts the rate of growth of the relative class numbers of cyclotomic fields of prime conductor. We extend Kummer's conjecture to cyclotomic fields of conductor n, where n is any natural number. We show that the Elliott-Halberstam conjecture implies that this generalised Kummer's conjecture is true for almost all n but is false for infinitely many n.
A Global Characterization Of Tubed Surfaces In ℂ2, Michael Bolt
A Global Characterization Of Tubed Surfaces In ℂ2, Michael Bolt
University Faculty Publications and Creative Works
Let M3 S C2 be a three times differentiable real hypersurface. The Levi form of M transforms under biholomorphism, and when restricted to the complex tangent space, the skew-Hermitian part of the second fundamental form transforms under Möbius transformations. The surfaces for which these forms are constant multiples of each other were identified in previous work, provided the constant is not unimodular. Here it is proved that if the surface is assumed to be complete and if the constant is unimodular, then the surface is tubed over a strongly convex curve. The converse statement is true, too, and is easily …
A Global Characterization Of Tubed Surfaces In ℂ2, Michael Bolt
A Global Characterization Of Tubed Surfaces In ℂ2, Michael Bolt
University Faculty Publications and Creative Works
Let M3 S C2 be a three times differentiable real hypersurface. The Levi form of M transforms under biholomorphism, and when restricted to the complex tangent space, the skew-Hermitian part of the second fundamental form transforms under Möbius transformations. The surfaces for which these forms are constant multiples of each other were identified in previous work, provided the constant is not unimodular. Here it is proved that if the surface is assumed to be complete and if the constant is unimodular, then the surface is tubed over a strongly convex curve. The converse statement is true, too, and is easily …
Higher Homotopy Operations And Cohomology, David Blanc, Mark W. Johnson, James M. Turner
Higher Homotopy Operations And Cohomology, David Blanc, Mark W. Johnson, James M. Turner
University Faculty Publications and Creative Works
We explain how higher homotopy operations, defined topologically, may be identified under mild assumptions with (the last of) the Dwyer-Kan-Smith cohomological obstructions to rectifying homotopy-commutative diagrams. © 2010 ISOPP.
Interaction Of Excited States In Two-Species Bose-Einstein Condensates: A Case Study, Todd Kapitula, Kody J. H. Law, Panayotis G. Kevrekidis
Interaction Of Excited States In Two-Species Bose-Einstein Condensates: A Case Study, Todd Kapitula, Kody J. H. Law, Panayotis G. Kevrekidis
University Faculty Publications and Creative Works
In this paper we consider the existence and spectral stability of excited states in two-species Bose-Einstein condensates in the case of a pancake magnetic trap. Each new excited state found in this paper is to leading order a linear combination of two one-species dipoles, each of which is a spectrally stable excited state for one-species condensates. The analysis is done via a Lyapunov-Schmidt reduction and is valid in the limit of weak nonlinear interactions. Some conclusions, however, can be made at this limit which remain true even when the interactions are large.
Laguerre Arc Length From Distance Functions, David E. Barrett, Michael Bolt
Laguerre Arc Length From Distance Functions, David E. Barrett, Michael Bolt
University Faculty Publications and Creative Works
For the Laguerre geometry in the dual plane, invariant arc length is shown to arise naturally through the use of a pair of distance functions. These distances are useful for identifying equivalence classes of curves, within which the extremal curves are proved to be strict maximizers of Laguerre arc length among three-times differentiable curves of constant signature in a prescribed isotopy class. For smoother curves, it is shown that Laguerre curvature determines the distortion of the distance functions. These results extend existing work for the Möbius geometry in the complex plane. © 2010 International Press.
Laguerre Arc Length From Distance Functions, David E. Barrett, Michael Bolt
Laguerre Arc Length From Distance Functions, David E. Barrett, Michael Bolt
University Faculty Publications and Creative Works
For the Laguerre geometry in the dual plane, invariant arc length is shown to arise naturally through the use of a pair of distance functions. These distances are useful for identifying equivalence classes of curves, within which the extremal curves are proved to be strict maximizers of Laguerre arc length among three-times differentiable curves of constant signature in a prescribed isotopy class. For smoother curves, it is shown that Laguerre curvature determines the distortion of the distance functions. These results extend existing work for the Möbius geometry in the complex plane
On The Rigidity Of The Cotangent Complex At The Prime 2, James M. Turner
On The Rigidity Of The Cotangent Complex At The Prime 2, James M. Turner
University Faculty Publications and Creative Works
In [D. Quillen, On the (co)homology of commutative rings, Proc. Symp. Pure Math. 17 (1970) 65-87; L. Avramov, Locally complete intersection homomorphisms and a conjecture of Quillen on the vanishing of cotangent homology, Annals of Math. 2 (150) (1999) 455-487] a conjecture was posed to the effect that if R → A is a homomorphism of Noetherian commutative rings then the flat dimension, as defined in the derived category of A-modules, of the associated cotangent complex LA / R satisfies: fdA LA / R < ∞ {long rightwards double arrow} fdA LA / R ≤ 2. The aim of this paper is to initiate an approach for solving this conjecture when R has characteristic 2 using simplicial algebra techniques. To that end, we obtain two results. First, we prove that the conjecture can be reframed in terms of certain nilpotence properties for the divided square γ2 and the André operation θ{symbol} as it acts on TorR (A, ℓ), ℓ any residue field of A. Second, we prove the conjecture is valid in two cases: when fdR A < ∞ and when R is a Cohen-Macaulay ring.