Mathematics Commons^™

The Poisson Variation Of Montmort's Matching Problem, Don Rawlings

Mathematics

No abstract provided.

On The Baroclinic Instability Of Axisymmetric Rotating Gravity Currents With Bottom Slope, Paul F. Choboter, Gordone E. Swaters

Mathematics

The baroclinic stability characteristics of axisymmetric gravity currents in a rotating system with a sloping bottom are determined. Laboratory studies have shown that a relatively dense fluid released under an ambient fluid in a rotating system will quickly respond to Coriolis effects and settle to a state of geostrophic balance. Here we employ a subinertial two-layer model derived from the shallow-water equations to study the stability characteristics of such a current after the stage at which geostrophy is attained. In the model, the dynamics of the lower layer are geostrophic to leading order, but not quasi-geostrophic, since the height deflections …

Nonstandard Analysis In Topology: Nonstandard And Standard Compactifications, S. Salbany, Todor D. Todorov

Mathematics

Let (X, T) be a topological space, and ^∗X a non–standard extension of X. There is a natural “standard” topology ^ST on ^∗X generated by ^∗G,where G ∈ T . The topological space (^∗X,^ST) will be used to study, in a systematic way, compactifications of (X, T).

Noncommutative Computer Algebra In The Control Of Singularly Perturbed Dynamical Systems, J. W. Helton, F. Dell Kronewitter, Mark Stankus

Mathematics

Most algebraic calculations which one sees in linear systems theory, for example in IEEE TAC, involve block matrices and so are highly noncommutative. Thus conventional commutative computer algebra packages, as in Mathematica and Maple, do not address them. Here we investigate the usefulness of noncommutative computer algebra in a particular area of control theory-singularly perturbed dynamic systems-where working with the noncommutative polynomials involved is especially tedious. Our conclusion is that they have considerable potential for helping practitioners with such computations. For example, the methods introduced here take the most standard textbook singular perturbation calculation, [KK086], one step further than had …

Computer Assistance For "Discovering'' Formulas In System Engineering And Operator Theory, J. W. Helton, Mark Stankus

Mathematics

The objective of this paper is two-fold. First we present a methodology for using a combination of computer assistance and human intervention to discover highly algebraic theorems in operator, matrix, and linear systems engineering theory. Since the methodology allows limited human intervention, it is slightly less rigid than an algorithm. We call it a strategy. The second objective is to illustrate the methodology by deriving four theorems. The presentation of the methodology is carried out in three steps. The first step is introducing an abstraction of the methodology which we call an idealized strategy. This abstraction facilitates a high level …

Time Delay In The Kuramoto Model Of Coupled Oscillators, Stephen M.K. Yeung, S H. Strogatz

Mathematics

We generalize the Kuramoto model of coupled oscillators to allow time-delayed interactions. New phenomena include bistability between synchronized and incoherent states, and unsteady solutions with time-dependent order parameters. We derive exact formulas for the stability boundaries of the incoherent and synchronized states, as a function of the delay, in the special case where the oscillators are identical. The experimental implications of the model are discussed for populations of chirping crickets, where the finite speed of sound causes communication delays, and for physical systems such as coupled phase-locked loops or lasers.

A Sheaf Theoretic Approach To Consciousness, Goro Kato, Daniele C. Struppa

Mathematics

A new fundamental mathematical model of consciousness based on category theory is presented. The model is based on two philosophical-theological assumptions: a) the universe is a sea of consciousness, and b) time is multi-dimensional and non-linear.

A Probabilistic Approach To Some Of Euler's Number Theoretic Identities, Don Rawlings

Mathematics

Probabilistic proofs and interpretations are given for the q-binomial theorem, q-binomial series, two of Euler's fundamental partition identities, and for q-analogs of product expansions for the Riemann zeta and Euler phi functions. The underlying processes involve Bernoulli trials with variable probabilities. Also presented are several variations on the classical derangement problem inherent in the distributions considered.

Classification Of Hereditary Matrices, Jim Agler, J. W. Helton, Mark Stankus

Mathematics

A classical approach used to obtain basic facts in the theory of square matrices involves an analysis of the relationship between polynomials p in one variable and square matrices T such that p(T) = 0. We consider matrices and operators which satisfy a different type of polynomial constraint. Let H be a complex Hilbert space, T be a bounded linear transformation of H, T* be the adjoint of T, and C[x, y] be the algebra of polynomials in x and y with complex coefficients. For a polynomial p E C[x, y] in …

Computer Simplification Of Formulas In Linear Systems Theory, J. W. Helton, Mark Stankus, John J. Wavrik

Mathematics

Currently, the three most popular commercial computer algebra systems are Mathematica, Maple, and MACSYMA. These systems provide a wide variety of symbolic computation facilities for commutative algebra and contain implementations of powerful algorithms in that domain. The Gröbner Basis Algorithm, for example, is an important tool used in computation with commutative algebras and in solving systems of polynomial equations.

On the other hand, most of the computation involved in linear control theory is performed on matrices, and these do not commute. A typical issue of IEEE TRANSACTIONS ON AUTOMATIC CONTROL is full of linear systems and computations with their coefficient …

Exceptional Objects, John Stillwell

Mathematics

No abstract provided.

Interactions Of Topological Kinks In Two Coupled Rings Of Nonlinear Oscillators, A E. Duwel, C P. Heij, J C. Weisenfeld, Stephen M.K. Yeung, E Trias, S. J.K. Vardy, H. S.J. Van Der Zant, S H. Strogatz, T P. Orlando

Mathematics

Two discrete rings of nonlinear oscillators with topologically trapped kinks exhibit features due to coupling interactions between the rings. These interaction effects include phase locking between kinks in different rings, precession of the kink/antikink collision region, excitation of kink/antikink pairs, and time-dependent switching. We study these phenomena in simulations of two coupled discrete sine-Gordon equations, and in experiments on two inductively coupled rings of niobium Josephson junctions.

An Embedding Of Schwartz Distributions In The Algebra Of Asymptotic Functions, Michael Oberguggenberger, Todor D. Todorov

Mathematics

We present a solution of the problem of multiplication of Schwartz distributions by embedding the space of distributions into a differential algebra of generalized functions, called in the paper "asymptotic function," similar to but different from J. F Colombeau's algebras of new generalized functions.

Orbifolds With Lower Ricci Curvature Bounds, Joseph E. Borzellino

Mathematics

We show that the first betti number of a compact Riemannian orbifold with Ricci curvature and diameter is bounded above by a constant , depending only on dimension, curvature and diameter. In the case when the orbifold has nonnegative Ricci curvature, we show that the is bounded above by the dimension , and that if, in addition, , then is a flat torus .

A Generalized Mahonian Statistic On Absorption Ring Mappings, Don Rawlings

Mathematics

Based on a coin-tossing scheme, a generalized Mahonian statistic is defined on absorption ring mappings and applied in obtaining combinatorial interpretations of the coefficient ofq^jin the expansion of ∏^k_i=1 (1+q+q²+…+q^m_i). In the permutation case, the statistic coincides with one studied by Han that specializes many known Mahonian statistics.

Magneto-Optic Trapping Of Β-Decaying 38km, 37k From An On-Line Isotope Separator, J. A. Behr, A. Gorelov, T. Swanson, O. Hausser, K. P. Jackson, M. Trinczek, U. Giesen, J. M. D'Auria, R. Hardy, T. Wilson, P. Choboter, F, Leblond, L. Buchmann, M. Dombsky, C. D.P. Levy, G. Roy, B. A. Brown, J. Dilling

Mathematics

A magneto-optic trap (MOT) can provide a well-polarized, backing-free, localized source of radioactive atoms for β-decay experiments. We have trapped approximately 6000 atoms of ³⁸K^m ( t_1/2 = 0.925s) and 2000 atoms of ³⁷K (1.226 s) produced at the TRIUMF on-line separator TISOL in a vapor-cell MOT. We have measured optical isotope shifts and deduced the nuclear charge radii, which show an unusual lack of change at the neutron number N = 20 shell closure. Plans include a search for scalar contributions to the β⁺- ν correlation in the 0⁺→0⁺ decay …

Absorption Processes: Models For Q-Identities, Don Rawlings

Mathematics

Several extensions of Blomqvist's absorption process are presented. Inherent in some of the associated distributions is a method for establishingq-identities ranging from properties of Gaussian polynomials to product expansions of basic hypergeometric series to extensions of results on Mahonian statistics. One process links the comajor index to Russian roulette. Also given are examples involving the Rogers–Ramanujan identities that demonstrate howq-expressions may be modeled with absorption processes.

Computer Assistance In Discovering Formulas And Theorems In System Engineering Ii, J. W. Helton, Mark Stankus, Kurt Schneider

Mathematics

[HSWcdc94] focused on procedures for simplifying complicated expressions automatically. [HScdc95] turned to the adventurous pursuit of developing a highly computer assisted method for “discovering” certain types of formulas and theorems.

It is often the case that some variables in the formulation of a problem are not the natural “coordinates” for solution of the problem. Gröbner Basis Algorithms, which lie at the core of our method, are very good at eliminating unknowns, but have no way of finding good changes of variables. This paper gives a way of incorporating changes of variables into our method.

As an example, we “discover” the …

M-Isometric Transformations Of Hilbert Space, Iii, Jim Alger, Mark Stankus

Mathematics

No abstract provided.

An Existence Result For Linear Partial Differential Equations With C∞ Coefficients In An Algebra Of Generalized Functions, Todor D. Todorov

Mathematics

We prove the existence of solutions for essentially all linear partial differential equations with C^∞-coefficients in an algebra of generalized functions, defined in the paper. In particular, we show that H. Lewy’s equation has solutions whenever its right-hand side is a classical C^∞-function.

The Closed Geodesic Problem For Compact Riemannian 2-Orbifolds, Joseph E. Borzellino, Benjamin G. Lorica

Mathematics

In this paper it is shown that any compact Riemannian 2-orbifold whose underlying space is a (compact) manifold without boundary has at least one closed geodesic.

Computer Assistance In Discovering Formulas And Theorems In System Engineering, J. W. Helton, Mark Stankus

Mathematics

If one reads a typical article on A,B,C,D systems in the control transactions, one finds that most of the algebra involved is non commutative rather than commutative. Thus, for symbolic computing to have much impact on linear systems research, one needs a program which will do non-commuting operations. Mathematica, Macsyma and Maple do not. We have a package, NCAlgebra, which runs under Mathematica which does the basic operations, block matrix manipulations and other things. The package might be seen as a competitor to a yellow pad. Like Mathematica the emphasis is on interaction with the program and flexibility.

The issue …

M-Isometric Transformations Of Hilbert Space, I, Jim Alger, Mark Stankus

Mathematics

No abstract provided.

Adjacencies In Words, Jean-Marc Fedou, Don Rawlings

Mathematics

Based on two inversion formulas for enumerating words in the free monoid by adjacencies, we present a new approach to a class of permutation problems having Eulerian-type generating functions. We also show that a specialization of one of the inversion formulas gives Diekert's lifting to the free monoid of an inversion theorem due to Cartier and Foata.

M-Isometric Transformations Of Hilbert Space, Ii, Jim Alger, Mark Stankus

Mathematics

No abstract provided.

The Genus 22 Crossing Number Of K9, Adrian Riskin

Mathematics

Our main result is that a 1971 conjecture due to Paul Kainen is false. Kainen's conjecture implies that the genus 2 crossing number of K 9 is 3. We disprove the conjecture by showing that the actual value is 4. The method used is a new one in the study of crossing numbers, involving proof of the impossibility of certain genus 2 embeddings of Ks.

Computer Simplification Of Engineering Systems Formulas, J. W. Helton, Mark Stankus, John Wavrik

Mathematics

Currently, the three most popular commercial computer algebra systems are Mathematica, Maple, and MACSYMA (the 3 M’s). These systems provide a wide variety of symbolic computation facilities for commutative algebra and contain implementations of powerful algorithms in that domain. The Gröbner Basis Algorithm, for example, is an important tool used in computation with commutative algebras and in solving systems of polynomial equations.

On the other hand, most of the computation involved in linear control theory is performed on matrices and these do not commute. A typical issue of IEEE TAC is full of A B C D type linear systems …

Pinching Theorems For Teardrops And Footballs Of Revolution, Joseph E. Borzellino

Mathematics

We give explicit optimal curvature pinching constants for the Riemannian (p, q)- football orbifolds under the assumption that they are realised as surfaces of revolution in ³. We show that sufficiently pinched sectional curvature assumptions imply that a (p, q)-football must be good.

Monads And Realcompactness, Sergio Salbany, Todor D. Todorov

Mathematics

We give a quantifier free characterization of realcompactness and ordered realcompactness in terms of monads. We also present simple proofs of some topological facts concerning realcompact spaces.

Projective Plane Embeddings Of Polyhedral Pinched Maps, Adrian Riskin

Mathematics

We give various conditions on pinched-torus polyhedral maps which are necessary for their graphs to be embeddable in the projective plane. Our other main result is that even if the graph of a polyhedral map in the pinched torus is embeddable in a projective plane, the map induced by the embedding cannot be polyhedral, but must have all faces bounded by cycles. Finally, we give a class of examples of graphs which have polyhedral embeddings on the pinched torus and also on orientable surfaces of arbitrary high genus.