On the existence of noncontinuous irreducible representations of a locally compact group.
Nondiscrete locally compact group having noncontinuous irreducible representations, proving theorem
Bichteler, K.↗
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A Benes-like theorem for the shuffle-exchange graph
One of the first theorems on permutation routing, proved by V. E. Beness (1965), shows that given a set of source-destination pairs in an N-node butterfly network with at most a constant number of sources or destinations in each column of the butterfly, there exists a set of paths of lengths O(log N) connecting each pair such that the total congestion is constant. An analogous theorem yielding constant-congestion paths for off-line routing in the shuffle-exchange graph is proved here. The necklaces of the shuffle-exchange graph play the same structural role as the columns of the butterfly in Beness' theorem.
Schwabe, Eric J.↗
Formal convergence characteristics of elliptically constrained incremental Newton-Raphson algorithms
Various aspects of the convergence, uniqueness, and existence properties associated with solutions generated via the elliptically constrained incremental Newton-Raphson (ECINR) algorithm are analyzed. Several theorems are developed, and the formal behavior of the elliptically constrained scheme developed by Padovan (1981) is discussed in detail. Consideration is given to global and local rates of convergence, to the determination of the occurrence of safety zones wherein the algorithm yields inherently convergent results, to formal limitations on the class of functions which the scheme can be applied to solve, and to single and multidimensional formalisms on existence uniqueness and convergence. Special attention is given to functions whose Jacobian matrix exhibit positive, negative, semi and indefinite properties. Several significant advantages of ECINR over the classical INR are mentioned.
Padovan, J.↗
Generalized Kinetic Description of Steady-State Collisionless Plasmas
We present a general solution to the collisionless Boltzmann (Vlasov) equation for a free-flowing plasma along a magnetic field line using Liouville's theorem, allowing for an arbitrary potential structure including non-monotonicities. The constraints of the existing collisionless kinetic transport models are explored, and the need for a more general approach to the problem of self- consistent potential energy calculations is described. Then a technique that handles an arbitrary potential energy distribution along the field line is presented and discussed. For precipitation of magnetospherically trapped hot plasma, this model yields moment calculations that vary by up to a factor of two for various potential energy structures with the same total potential drop. The differences are much greater for the high-latitude outflow scenario, giving order of magnitude variations depending on the shape of the potential energy distribution.
Khazanov, G. V.↗
Optimum testing of multiple hypotheses in quantum detection theory
The problem of specifying the optimum quantum detector in multiple hypotheses testing is considered for application to optical communications. The quantum digital detection problem is formulated as a linear programming problem on an infinite-dimensional space. A necessary and sufficient condition is derived by the application of a general duality theorem specifying the optimum detector in terms of a set of linear operator equations and inequalities. Existence of the optimum quantum detector is also established. The optimality of commuting detection operators is discussed in some examples. The structure and performance of the optimal receiver are derived for the quantum detection of narrow-band coherent orthogonal and simplex signals. It is shown that modal photon counting is asymptotically optimum in the limit of a large signaling alphabet and that the capacity goes to infinity in the absence of a bandwidth limitation.
Yuen, H. P.↗
The existence of certain linear approximations within a Tchebycheff norm
Existence of certain linear approximations within chebyshev norm
THEOREM PROVING↗
Periodic motion in a class of nth-order autonomous differential equations
Sufficient conditions are obtained for the existence of periodic motion in a class of autonomous nonlinear differential equations of order greater than two. The approach is based on the decomposition of an equation into a linear and a nonlinear part. The analysis relies on some basic ideas from linear analysis and geometry. Sufficient conditions for a periodic solution are derived by means of a general topological principle referred to as the torus principle. The existence of a periodic solution is concluded by an appropriate use of the Brouwer fixed-point theorem.
Williamson, D.↗
Dynamics of the Venus atmosphere
The superrotation of the Venus atmosphere is a major unanswered problem in planetary science. At cloud-top levels (65-70 km altitude) the atmosphere rotates with a five-day period, corresponding to an equatorial wind speed of 90 m/s. Angular velocity is roughly constant on spherical shells, and decreases linearly with altitude to zero at the surface. The direction of rotation is the same as that of the solid planet, which is retrograde--opposite to the direction of orbital motion, but the 5-day period is short compared to the 243-day spin period of the solid planet or to the mean solar day, which is 117 Earth-days at the surface. The problem with the superrotation is that shearing stresses tend to transfer angular momentum downward, and would slow the atmosphere until it is spinning with the solid planet. Some organized circulation pattern is counteracting the tendency, but the pattern has not been identified. A simple Hadley-type circulation cannot do it because such a circulation is zonally symmetric and Hide's Theorem states that in an axisymmetric circulation an extremum in angular momentum per unit mass M can exist only at the surface. Venus violates the last condition, having a maximum of retrograde M on the equator at 70-80 km altitude. This leaves waves and eddies to maintain the superrotation but the length scales and forcing mechanisms for these motions need to be specified. Possible forcing mechanisms associated with waves, eddies and tides are discussed.
Ingersoll, A. P.↗
Violation of the Vogt-Russell theorem for homogeneous nondegenerate stars
A systematic study is made of the number and types of solutions of the equilibrium equations of stellar structure, in the case of homogeneous stars of Population I over the mass range 2-1000 solar masses, with four different opacity representations. A variant of the usual fitting method permits the simultaneous investigation of convergence and tendency toward multiplicity of the solutions. Quadratic interpolation and extrapolation of Carson's new opacity tables produces a very large opacity at low temperatures that greatly affects the loose outer layers of massive stars, while leaving the cores practically unaffected. As a result, over a small mass range, well above 100 solar masses, triple solutions exist, always near an effective temperature of log Te = 4.73. A simple classification of the known exceptions to the Vogt-Russell theorem on the uniqueness of stellar structure is given.
Stothers, R.↗
Granular Contact Forces: Proof of "Self-Ergodicity" by Generalizing Boltzmann's Stosszahlansatz and H Theorem
Ergodicity is proved for granular contact forces. To obtain this proof from first principles, this paper generalizes Boltzmann's stosszahlansatz (molecular chaos) so that it maintains the necessary correlations and symmetries of granular packing ensembles. Then it formally counts granular contact force states and thereby defines the proper analog of Boltzmann's H functional. This functional is used to prove that (essentially) all static granular packings must exist at maximum entropy with respect to their contact forces. Therefore, the propagation of granular contact forces through a packing is a truly ergodic process in the Boltzmannian sense, or better, it is self-ergodic. Self-ergodicity refers to the non-dynamic, internal relationships that exist between the layer-by-layer and column-by-column subspaces contained within the phase space locus of any particular granular packing microstate. The generalized H Theorem also produces a recursion equation that may be solved numerically to obtain the density of single particle states and hence the distribution of granular contact forces corresponding to the condition of self-ergodicity. The predictions of the theory are overwhelmingly validated by comparison to empirical data from discrete element modeling.
Metzger, Philip T.↗
A torus bifurcation theorem with symmetry
Hopf bifurcation in the presence of symmetry, in situations where the normal form equations decouple into phase/amplitude equations is described. A theorem showing that in general such degeneracies are expected to lead to secondary torus bifurcations is proved. By applying this theorem to the case of degenerate Hopf bifurcation with triangular symmetry it is proved that in codimension two there exist regions of parameter space where two branches of asymptotically stable two-tori coexist but where no stable periodic solutions are present. Although a theory was not derived for degenerate Hopf bifurcations in the presence of symmetry, examples are presented that would have to be accounted for by any such general theory.
Vangils, S. A.↗
A Machine-Checked Proof of A State-Space Construction Algorithm
This paper presents the correctness proof of Saturation, an algorithm for generating state spaces of concurrent systems, implemented in the SMART tool. Unlike the Breadth First Search exploration algorithm, which is easy to understand and formalise, Saturation is a complex algorithm, employing a mutually-recursive pair of procedures that compute a series of non-trivial, nested local fixed points, corresponding to a chaotic fixed point strategy. A pencil-and-paper proof of Saturation exists, but a machine checked proof had never been attempted. The key element of the proof is the characterisation theorem of saturated nodes in decision diagrams, stating that a saturated node represents a set of states encoding a local fixed-point with respect to firing all events affecting only the node s level and levels below. For our purpose, we have employed the Prototype Verification System (PVS) for formalising the Saturation algorithm, its data structures, and for conducting the proofs.
Catano, Nestor↗
Long-time predictions in nonlinear dynamics
It is known that nonintegrable dynamical systems do not allow precise predictions concerning their behavior for arbitrary long times. The available series solutions are not uniformly convergent according to Poincare's theorem and numerical integrations lose their meaningfulness after the elapse of arbitrary long times. Two approaches are the use of existing global integrals and statistical methods. This paper presents a generalized method along the first approach. As examples long-time predictions in the classical gravitational satellite and planetary problems are treated.
Szebehely, V.↗
Possibilities for improving the efficiency of linear error-correcting codes
Results are presented in the form of 14 theorems specifying sufficient conditions under which it is possible to construct new more efficient single- and multi error correcting codes from existing one when the method used to construct existing codes is known.
Varshamov, R. R.↗
Oscillations in neutral functional differential equations
Fredholm alternative for the existence of periodic solutions of a nonhomogeneous linear neutral functional differential equation was announced. Boundary value problems were used to solve the theorems.
Hale, J. K.↗
Comparison between Liouville's theorem and observed latitudinal distributions of trapped ions in the plasmapause region
The presence of anisotropic plasma distributions, trapped at the Earth's magnetic equator, has consequences for the electric field structure which must exist in equilibrium along the magnetic field line. Data from SCATHA and Dynamics Explorer 1 indicate that the core ion distributions at the magnetic equator can be well described as bi-Maxwellian distributions, with a perpendicular temperature an order of magnitude larger than the parallel temperature. A collisionless model is developed for the variation in plasma parameters, following the forms developed by Whipple (1977). If the core electron anisotropy is low, the resulting electric field of approximately 0.1 microV/m is pointed away from the equator. Under these conditions the self-consistent electric field will not overcome the effects of magnetic trapping. The resulting potential distribution results in a local maximum in total plasma density at the equator. Only when the electron distribution is primarily field-aligned can there be a density minimum at the equator. Comparisons are made between this model and the observed variations in DE 1 plasma parameters with latitude.
Olsen, R. C.↗
Simple proof of the concavity of the entropy power with respect to Gaussian noise
A very simple proof of M. H. Costa's result that the entropy power of Xt = X + N (O, tI) is concave in t, is derived as an immediate consequence of an inequality concerning Fisher information. This relationship between Fisher information and entropy is found to be useful for proving the central limit theorem. Thus, one who seeks new entropy inequalities should try first to find new inequalities about Fisher information, or at least to exploit the existing ones in new ways.
Dembo, Amir↗
Feedback stabilization and control of linear neutral systems
The first problem treated here is the realization and stabilization of linear neutral systems with discrete delays. It is shown that any autonomous linear neutral system with discrete delays is zero-state equivalent to an abstract linear system over a local ring of operators. Using the abstract model, the basic existence question for neutral realization is then settled. For general infinite dimensional linear systems, there is no precise analog of the finite dimensional state space isomorphism theorem. Because of this, the notion of spectral minimality must be introduced. For the case of single input-single output systems, realizations are obtained that are both minimal and spectrally minimal. Using the Cruz-Hale theory of stable D-operators, conditions are given that ensure that any poles introduced into the realization are strictly contained in the left half plane and indeed are characterized as characteristic values of the D-operator. The problem of the feedback stabilization of neutral systems is then considered using the abstract model. It is shown that, for neutral systems with commensurable delays and a stable D-operator in the sense of Cruz and Hale, Morses theorem (1976) on pole assignment over a PID implies stabilizability in the reachable case.
Tarn, T. J.↗