UNIQUENESS AND NONUNIQUENESS RESULTS FOR A CERTAIN CLASS OF ALMOST PERIODIC DISTRIBUTIONS

We consider distributions u is an element of S'(R) of the form u(t) = Sigma(n is an element of N) a(n)e(i lambda nt), where (a(n))(n is an element of N) subset of C and Lambda = (lambda n)(n is an element of N) subset of R have the following properties: (a(n))(n is an element of N) is an element of s', that is, there is a q is an element of N such that (n(-q) a(n))(n is an element of N) is an element of l(1); for the real sequence., there are n(0) is an element of N, C > 0, and alpha > 0 such that n >= n(0) double right arrow vertical bar lambda(n)vertical bar >= Cn(alpha). Let I(epsilon) subset of R be an interval of length epsilon. We prove that for given Lambda, (1) if Lambda = O(n(alpha)) with alpha < 1, then there exists epsilon > 0 such that u vertical bar I(epsilon) = 0 double right arrow u 0; (2) if Lambda = O(n) is uniformly discrete, then there exists epsilon > 0 such that u vertical bar I(epsilon) = 0 double right arrow u 0; (3) if alpha > 1 and. is uniformly discrete, then for all epsilon > 0, u vertical bar I(epsilon) = 0 double right arrow u = 0. Since distributions of the above mentioned form are very common in engineering, as in the case of the modeling of ocean waves, signal processing, and vibrations of beams, plates, and shells, those uniqueness and nonuniqueness results have important consequences for identification problems in the applied sciences. We show an identification method and close this article with a simple example to show that the recovery of geometrical imperfections in a cylindrical shell is possible from a measurement of its dynamics.

Quantum Lie algebras, their existence, uniqueness and q-antisymmetry

Quantum Lie algebras are generalizations of Lie algebras which have the quantum parameter h built into their structure. They have been defined concretely as certain submodules L-h(g) of the quantized enveloping algebras U-h(g). On them the quantum Lie product is given by the quantum adjoint action. Here we define for any finite-dimensional simple complex Lie algebra g an abstract quantum Lie algebra g(h) independent of any concrete realization. Its h-dependent structure constants are given in terms of inverse quantum Clebsch-Gordan coefficients. We then show that all concrete quantum Lie algebras L-h(g) are isomorphic to an abstract quantum Lie algebra g(h). In this way we prove two important properties of quantum Lie algebras: 1) all quantum Lie algebras L-h(g) associated to the same g are isomorphic, 2) the quantum Lie product of any Ch(B) is q-antisymmetric. We also describe a construction of L-h(g) which establishes their existence.

The uniqueness of solutions to discrete, victor, two-point boundary value problems

Difference equations which may arise as discrete approximations to two-point boundary value problems for systems of second-order, ordinary differential equations are investigated and conditions are formulated under which solutions to the discrete problem are unique. Some existence, uniqueness implies existence, and convergence theorems for solutions to the discrete problem are also presented.

On the existence and uniqueness of limit cycles in Liénard differential equations allowing discontinuities

"Vegeu el resum a l´inici del document del fitxer adjunt."

Existence and uniqueness of viscosity solution for Hamilton-Jacobi equation with discontinuous coefficients dependent on time

The main result is a proof of the existence of a unique viscosity solution for Hamilton-Jacobi equation, where the hamiltonian is discontinuous with respect to variable, usually interpreted as the spatial one. Obtained generalized solution is continuous, but not necessarily differentiable.

First price auctions: Monotonicity and uniqueness

I study monotonicity and uniqueness of the equilibrium strategies in a two-person first price auction with affiliated signals. I show thatwhen the game is symmetric there is a unique Nash equilibrium thatsatisfies a regularity condition requiring that the equilibrium strategies be{\sl piecewise monotone}. Moreover, when the signals are discrete-valued, the equilibrium is unique. The central part of the proof consists of showing that at any regular equilibrium the bidders' strategies must be monotone increasing within the support of winning bids. The monotonicity result derived in this paper provides the missing link for the analysis of uniqueness in two-person first price auctions. Importantly, this result extends to asymmetric auctions.

Comprehensive spatiotemporal transcriptomic analyses of the ganglionic eminences demonstrate the uniqueness of its caudal subdivision.

The elucidation of mechanisms underlying telencephalic neural development has been limited by the lack of knowledge regarding the molecular and cellular aspects of the ganglionic eminence (GE), an embryonic structure that supplies the brain with diverse sets of GABAergic neurons. Here, we report a comprehensive transcriptomic analysis of this structure including its medial (MGE), lateral (LGE) and caudal (CGE) subdivisions and its temporal dynamics in 12.5 to 16 day-old rat embryos. Surprisingly, comparison across subdivisions showed that CGE gene expression was the most unique providing unbiased genetic evidence for its differentiation from MGE and LGE. The molecular signature of the CGE comprised a large set of genes, including Rwdd3, Cyp26b1, Nr2f2, Egr3, Cpta1, Slit3, and Hod, of which several encode cell signaling and migration molecules such as WNT5A, DOCK9, VSNL1 and PRG1. Temporal analysis of the MGE revealed differential expression of unique sets of cell specification and migration genes, with early expression of Hes1, Lhx2, Ctgf and Mdk, and late enrichment of Olfm3, SerpinE2 and Wdr44. These GE profiles reveal new candidate regulators of spatiotemporally governed GABAergic neuronogenesis.

Uniqueness in chess studies

Van der Heijden’s ENDGAME STUDY DATABASE IV, HhdbIV, is the definitive collection of 76,132 chess studies. In each one, White is to achieve the stipulated goal, win or draw: study solutions should be essentially unique with minor alternatives at most. In this second note on the mining of the database, we use the definitive Nalimov endgame tables to benchmark White’s moves in sub-7-man chess against this standard of uniqueness. Amongst goal-compatible mainline positions and goal-achieving moves, we identify the occurrence of absolutely unique moves and analyse the frequency and lengths of absolutely-unique-move sequences, AUMSs. We identify the occurrence of equi-optimal moves and suboptimal moves and refer to a defined method for classifying their significance.

On the strong uniqueness of some finite dimensional Dirichlet operators

We prove essential self-adjointness of a class of Dirichlet operators in ℝn using the hyperbolic equation approach. This method allows one to prove essential self-adjointness under minimal conditions on the logarithmic derivative of the density and a condition of Muckenhoupt type on the density itself.