Physical Sciences and Mathematics Commons^™

Holography, A Covariant C Function, And The Geometry Of The Renormalization Group, Vatche Sahakian

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We propose a covariant geometrical expression for the c function for theories which admit dual gravitational descriptions. We state a c theorem with respect to this quantity and prove it. We apply the expression to a class of geometries, from domain walls in gauged supergravities, to extremal and near extremal Dp-branes, and the AdS Schwarzschild black hole. In all cases, we find agreement with expectations.

The Phases Of 2d Ncos, Vatche Sahakian

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We study the phases of the (1+1)-dimensional Non-Commutative Open String theory on a circle. We find that the length scale of non-commutativity increases at strong coupling, the coupling in turn being dressed by a power of D-string charge. The system is stringy at around this length scale, with dynamics involving an interplay between the open and wrapped closed strings sectors. Above this energy scale and at strong coupling, and below it at weak coupling, the system acquires a less stringy character. The near horizon geometry of the configuration exhibits several intriguing features, such as a flip in the dilaton field …

Existence Of Many Positive Nonradial Solutions For A Superlinear Dirichlet Problem On Thin Annuli, Alfonso Castro, Marcel B. Finan

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We study the existence of many positive nonradial solutions of a superlinear Dirichlet problem in an annulus in R^N. Our strategy consists of finding the minimizer of the energy functional restricted to the Nehrai manifold of a subspace of functions with symmetries. The minimizer is a global critical point and therefore is a desired solution. Then we show that the minimal energy solutions in different symmetric classes have mutually different energies. The same approach has been used to prove the existence of many sign-changing nonradial solutions (see [5]).

Optimal Token Allocations In Solitaire Knock'm Down, Arthur T. Benjamin, Mark L. Huber, Matthew T. Fluet '99

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In the game Knock 'm Down, tokens are placed in N bins. At each step of the game, a bin is chosen at random according to a fixed probability distribution. If a token remains in that bin, it is removed. When all the tokens have been removed, the player is done. In the solitaire version of this game, the goal is to minimize the expected number of moves needed to remove all the tokens. Here we present necessary conditions on the number of tokens needed for each bin in an optimal solution, leading to an asymptotic solution.

Random Approaches To Fibonacci Identities, Arthur T. Benjamin, Gregory M. Levin, Karl Mahlburg '01, Jennifer J. Quinn

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No abstract provided in this article.

Comments On D Branes And The Renormalization Group, Vatche Sahakian

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We review the de Boer-Verlinde-Verlinde formalism for the renormalization group in the context of Dp-brane vacua for p < 5. We comment on various aspects of the dictionary between bulk and boundary and relate the discussion to the Randall-Sundrum scenario. We find that the gravitational coupling for the Randall-Sundrum gravity on the Dp-brane worldvolume is dressed by the c-function of the Yang-Mills theory. We compute the beta function and find the expected uneventful flow prescribed by the classical dimension of the Yang-Mills operator.

Phased Tilings And Generalized Fibonacci Identities, Arthur T. Benjamin, Jennifer J. Quinn, Francis E. Su

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Fibonacci numbers arise in the solution of many combinatorial problems. They count the number of binary sequences with no consecutive zeros, the number of sequences of 1's and 2's which sum to a given number, and the number of independent sets of a path graph. Similar interpretations exist for Lucas numbers. Using these interpretations, it is possible to provide combinatorial proofs that shed light on many interesting Fibonacci and Lucas identities (see [1], [3]). In this paper we extend the combinatorial approach to understand relationships among generalized Fibonacci numbers.

Given G₀ and G₁ a generalized Fibonacci sequence G …

What's Best?, Arthur T. Benjamin, Matthew T. Fluet '99

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No abstract provided in this article.

Evolution Of Positive Solution Curves In Semipositone Problems With Concave Nonlinearities, Alfonso Castro, Sudhasree Gadam, Ratnasingham Shivaji

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We study the existence, multiplicity, and stability of positive solutions to -u''(x) = λf(u(x)) for x є (-1,1), u(-1) = 0 = u(1), where λ > 0 and f:[0,∞)→R is monotonically increasing and concave with f(0) < 0 (semipositone). We establish that f should be appropriately concave (by establishing conditions on f) to allow multiple positive solutions. For any λ > 0, we obtain the exact number of positive solutions as a function of f(t)/t. We follow several families of nonlinearities f for which f(∞) := lim t→∞ f(t) > 0 and study how the positive solution curves to the above problem evolve. Also, we give examples where our results apply. This work extends the work of A. Castro and R. Shivaji (1988, Proc. Roy. Soc. Edinburgh …

Counting On Continued Fractions, Arthur T. Benjamin, Francis E. Su, Jennifer J. Quinn

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No abstract provided in this article.

An Optical Coherence Microscope For 3-Dimensional Imaging In Developmental Biology, Barbara M. Hoeling, Andrew D. Fernandez, Richard C. Haskell, Eric Huang, Whittier R. Myers, Daniel C. Petersen, Sharon E. Ungersma, Ruye Wang, Mary E. Williams, Scott E. Fraser

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An optical coherence microscope (OCM) has been designed and constructed to acquire 3-dimensional images of highly scattering biological tissue. Volume-rendering software is used to enhance 3-D visualization of the data sets. Lateral resolution of the OCM is 5 mm (FWHM), and the depth resolution is 10 mm (FWHM) in tissue. The design trade-offs for a 3-D OCM are discussed, and the fundamental photon noise limitation is measured and compared with theory. A rotating 3-D image of a frog embryo is presented to illustrate the capabilities of the instrument.

A Rational Solution To Cootie, Arthur T. Benjamin, Matthew T. Fluet '99

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No abstract provided in this article.

Efficient Multicast In Heterogeneous Networks Of Workstations, Ran Libeskind-Hadas, Jeff R.K. Hartline '01

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This paper studies the problem of efficient multicast in heterogeneous networks of workstations (HNOWs) using a parameterized communication model [3]. This model associates a sending overhead and a receiving overhead with each node as well as a network latency parameter. The problem of finding optimal multicasts in this model is known to be NP-complete in the strong sense. Nevertheless, we show that for two different properties that arise in typical HNOWs, provably near-optimal and optimal solutions, respectively, can be found in polynomial time. Specifically, we show the following two results: When the ratios of receiving overhead to sending overhead among …

Procedural Support For Cooperative Negotiations: Theoretical Design And Practical Implementation, Matthias G. Raith, Francis Su

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We discuss the theoretical design of algorithms for solving distributional conflicts within groups. We consider an algorithm to be procedural if the implementation of the outcome requires the participation of the players, or if it can even be conducted by the players themselves without computational assistance. We compare two procedures for multilateral problems of fair division; both establish envy-freeness, given the possibility of monetary compensations between players.

Review: Cake-Cutting Algorithms: Be Fair If You Can, Francis E. Su

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No abstract provided in this article.

A Leveque-Type Lower Bound For Discrepancy, Francis E. Su

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A sharp lower bound for discrepancy on R / Z is derived that resembles the upper bound due to LeVeque. An analogous bound is proved for discrepancy on R^k / Z^k. These are discussed in the more general context of the discrepancy of probablity measures. As applications, the bounds are applied to Kronecker sequences and to a random walk on the torus.

A Look At Biseparating Maps From An Algebraic Point Of View, Melvin Henriksen, Frank A. Smith

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In [ABN], Araujo, Beckenstein, and Narici add the capstone to a series of papers by several groups of authors by showing that if ρ is a biseparating map between two algebras of all real or complex-valued functions on realcompact spaces, then it is a continuous multiple of an isomorphism between these rings. Their proof uses relatively powerful analytic and topological techniques. In what follows, the extent to which such a result can be generalized to a wider class of algebras using algebraic techniques is investigated. We are unable, however to obtain the main result of [ABN] using these techniques.

Solvent As Electron Donor:  Donor/Acceptor Electronic Coupling Is A Dynamical Variable, Edward W. Castner Jr., Darcy Kennedy '00, Robert J. Cave

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We combine analysis of measurements by femtosecond optical spectroscopy, computer simulations, and the generalized Mulliken−Hush (GMH) theory in the study of electron-transfer reactions and electron donor−acceptor interactions. Our focus is on ultrafast photoinduced electron-transfer reactions from aromatic amine solvent donors to excited-state acceptors. The experimental results from femtosecond dynamical measurements fall into three categories:  six coumarin acceptors reductively quenched by N,N-dimethylaniline (DMA), eight electron-donating amine solvents reductively quenching coumarin 152 (7-(dimethylamino)-4-(trifluoromethyl)coumarin), and reductive quenching dynamics of two coumarins by DMA as a function of dilution in the nonreactive solvents toluene and chlorobenzene. Applying a combination of molecular …

On The Vertical And Adiabatic Excitation Energies Of The 21a(G), State Of Trans-1,3-Butadiene, Jason Lappe '00, Robert J. Cave

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The excitation energy to the 2¹A_g state of trans-1,3-butadiene is examined using a variety of ab initio electronic structure techniques. While analogous states have been shown to be the lowest singlet excited states for all longer polyenes, for butadiene the position of the 2¹A_g state relative to the HOMO → LUMO excitation (1¹B_u) has been difficult to establish theoretically. We employ a variety of methods (CASSCF, CASPT2, MRSDCI, QDVPT) to examine both the vertical and adiabatic excitation energies for this state. At the ground-state geometry, the vertical excitation energies …

When Is |C(X X Y)| = |C(X)||C(Y)|?, O. T. Alas, W. W. Comfort, S. Garcia-Ferreira, Melvin Henriksen, R. G. Wilson, R. G. Woods

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Sufficient conditions on the Tychonoff spaces X and Y are found that imply that the equation in the title holds. Sufficient conditions on the Tychonoff space X are found that ensure that the equation holds for every Tychonoff space Y . A series of examples (some using rather sophisticated cardinal arithmetic) are given that witness that these results cannot be generalized much.