Physical Sciences and Mathematics Commons^™

Stability Of Steady Cross-Waves: Theory And Experiment, Seth Lichter, Andrew J. Bernoff

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A bifurcation analysis is performed in the neighborhood of neutral stability for cross waves as a function of forcing, detuning, and viscous damping. A transition is seen from a subcritical to a supercritical bifurcation at a critical value of the detuning. The predicted hysteretic behavior is observed experimentally. A similarity scaling in the inviscid limit is also predicted. The experimentally observed bifurcation curves agree with this scaling.

A Semilinear Wave Equation With Nonmonotone Nonlinearity, Alfonso Castro, Sumalee Unsurangsie

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We prove that a semilinear wave equation in which the range of the derivative of the nonlinearity includes an eigenvalue of infinite multiplicity has a solution. The solution is obtained through an iteration scheme which provides a priori estimates.

Fault Tolerance In Networks Of Bounded Degree, Cynthia Dwork, David Peleg, Nicholas Pippenger, Eli Upfal

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Achieving processor cooperation in the presence of faults is a major problem in distributed systems. Popular paradigms such as Byzantine agreement have been studied principally in the context of a complete network. Indeed, Dolev [J. Algorithms, 3 (1982), pp. 14–30] and Hadzilacos [Issues of Fault Tolerance in Concurrent Computations, Ph.D. thesis, Harvard University, Cambridge, MA, 1984] have shown that Ω(t) connectivity is necessary if the requirement is that all nonfaulty processors decide unanimously, where t is the number of faults to be tolerated. We believe that in forseeable technologies the number of faults will grow with the size of the …

Nonnegative Solutions For A Class Of Nonpositone Problems, Alfonso Castro, Ratnasingham Shivaji

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In the recent past many results have been established on non-negative solutions to boundary value problems of the form

-u''(x) = λf(u(x)); 0 < x < 1,

u(0) = 0 = u(1)

where λ>0, f(0)>0 (positone problems). In this paper we consider the impact on the non-negative solutions when f(0)<0. We find that we need f(u) to be convex to guarantee uniqueness of positive solutions, and f(u) to be appropriately concave for multiple positive solutions. This is in contrast to the case of positone problems, where the roles of convexity and concavity were interchanged to obtain similar results. We further establish the existence of non-negative solutions with interior zeros, which did not exist in positone problems.

An Ab Initio Investigation Of The Stabilization Of Selected Β-Substituted Ethyl Cations And Α-Substituted Methyl Cations, Julia C. White, Robert J. Cave, Ernest R. Davidson

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In this study, we calculate the stabilization of β-substituted ethyl cations (R = H, Li, BeH, BH_2, CH_3, NH_2, OH, F, Na, MgH, AlH_2, SiH_3, PH_2, SH, Cl) and α-substituted methyl cations (R = H, Li, CH_3, NH_2, OH, F, Na, SiH_3, PH_2, SH, Cl) in order to obtain a relationship between the nature of the substituent and the degree of stabilization of the cation. Results show that the stabilization energy is related to the electronegativity of the β substituents, but not the α substituents. The rotational barrier of the β-substituted ethyl cation is linearly related to the Mulliken population …

Ab Initio Investigation Of Several Low-Lying States Of All-Trans Octatetraene, Robert J. Cave, Ernest R. Davidson

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The results of ab initio calculations that examine vertical and nonvertical transitions to several low-lying states of all-trans-octatetraene are presented. It is found that the lowest vertical excitation is to a valence π → π* ¹B_u state, and the nominally doubly excited 2¹A_g state occurs approximately 0.4 eV higher at the geometry of the ground state. Using estimated excited-state equilibrium geometries, we find that the 2¹A_g state is indeed the lowest singlet excited state, having a 0-0 tarnsition energy of 4.15 eV. The 0-0 transition energy for the 1¹B_u state is calculated to be 4.56 eV. The present results …

A Theoretical Investigation Of Several Low-Lying States Of Trans, Trans-1,3,5-Hexatriene, Robert J. Cave, Ernest R. Davidson

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Results from ab initio calculations concerning several low-lying electronic states of trans,trans-1,3,5-hexatriene are presented and compared with experimental and previous theoretical results. The lowest excited singlet state is predicted to be the ¹B_u state, having essentially valencelike π → π* character. The nominally doubly excited 2¹A_g state is found to lie approximately 0.6-0.9 eV above the 1¹B_u state. Results are also presented for several Rydberg states. The implications of the present results for current parametrizations of semiempirical π molecular orbital schemes are discussed.

Wide-Sense Nonblocking Networks, Paul Feldman, Joel Friedman, Nicholas Pippenger

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A new method for constructing wide-sense nonblocking networks is presented. Application of this method yields (among other things) wide-sense nonblocking generalized connectors with n inputs and outputs and size O( n log n ), and with depth k and size O( n^{1 + 1/k} ( log n )^{1 - 1/k} ).

Reliable Computation By Formulas In The Presence Of Noise, Nicholas Pippenger

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It is shown that if formulas are used to compute Boolean functions in the presence of randomly occurring failures then: (1) there is a limit strictly less than 1/2 to the failure probability per gate that can be tolerated, and (2) formulas that tolerate failures must be deeper (and, therefore, compute more slowly) than those that do not. The heart of the proof is an information-theoretic argument that deals with computation and errors in very general terms. The strength of this argument is that it applies with equal ease no matter what types of gate are available. Its weaknesses is …