Abstract:
A method of quantization of classical soliton cellular automata (QSCA) is put forward that provides a description of their time evolution operator by means of quantum circuits that involve quantum gates from which the associated Hamiltonian describing a quantum chain model is constructed. The intrinsic parallelism of QSCA, a phenomenon first known from quantum computers, is also emphasized.

Abstract:
We investigate the dynamics of the critical isoscalar condensate, formed during heavy-ion collisions. Our analysis is based on a simplified model where the sigma and the pions are the only degrees of freedom. In field description, both in physical and momentum space, we find that the freeze-out profile presents a structure which reveals clear traces of the critical fluctuations in the sigma-component. In particle representation, using Monte-Carlo simulations and factorial moment analysis, we show that signatures of the initial criticality survive at the detected pions. We propose the distribution of suitably defined intermittency indices, incorporating dynamical effects due to sigma-pion interaction, as the basic observable for the exploration of critical fluctuations in heavy-ion collision experiments.

Abstract:
We investigate the evolution of the density-density correlations in the isoscalar critical condensate formed at the QCD critical point. The initial equilibrium state of the system is characterized by a fractal measure determining the distribution of isoscalar particles (sigmas) in configuration space. Non-equilibrium dynamics is induced through a sudden symmetry breaking leading gradually to the deformation of the initial fractal geometry. After constructing an ensemble of configurations describing the initial state of the isoscalar field we solve the equations of motion and show that remnants of the critical state and the associated fractal geometry survive for time scales larger than the time needed for the mass of the isoscalar particles to reach the two-pion threshold. This result is more transparent in an event-by-event analysis of the phenomenon. Thus, we conclude that the initial fractal properties can eventually be transferred to the observable pion-sector through the decay of the sigmas even in the case of a quench.

Abstract:
In this paper we are studying the Cartesian space robot manipulator control problem by using Neural Networks (NN). Although NN compensation for model uncertainties has been traditionally carried out by modifying the joint torque/force of the robot, it is also possible to achieve the same objective by using the NN to modify other quantities of the controller. We present and evaluate four different NN controller designs to achieve disturbance rejection for an uncertain system. The design perspectives are dependent on the compensated position by NN. There are four quantities that can be compensated: torque , force F, control input U and the input trajectory Xd. By defining a unified training signal all NN control schemes have the same goal of minimizing the same objective functions. We compare the four schemes in respect to their control performance and the efficiency of the NN designs, which is demonstrated via simulations.

Abstract:
For the group G = PGL(2) we prove nonstandard matching and the fundamental lemma between two relative trace formulas: on one hand, the relative trace formula of Jacquet for the quotient T\G/T, where T is a nontrivial torus; on the other, the Kuznetsov trace formula with nonstandard test functions. The matching is nonstandard in the sense that orbital integrals are related to each other not one-by-one, but via an explicit integral transform. These results will be used in the sequel to compare the corresponding global trace formulas and reprove the celebrated result of Waldspurger on toric periods.

Abstract:
The description of irreducible representations of a group G can be seen as a question in harmonic analysis; namely, decomposing a suitable space of functions on G into irreducibles for the action of G x G by left and right multiplication. For a split p-adic reductive group G over a local non-archimedean field, unramified irreducible smooth representations are in bijection with semisimple conjugacy classes in the ``Langlands dual'' group. We generalize this description to an arbitrary spherical variety X of G as follows: Irreducible unramified quotients of the space $C_c^\infty(X)$ are in natural ``almost bijection'' with a number of copies of $A_X^*/W_X$, the quotient of a complex torus by the ``little Weyl group'' of X. This leads to a description of the Hecke module of unramified vectors (a weak analog of geometric results of Gaitsgory and Nadler), and an understanding of the phenomenon that representations ``distinguished'' by certain subgroups are functorial lifts. In the course of the proof, rationality properties of spherical varieties are examined and a new interpretation is given for the action, defined by F. Knop, of the Weyl group on the set of Borel orbits.

Abstract:
The Casselman-Shalika method is a way to compute explicit formulas for periods of irreducible unramified representations of p-adic groups that are associated to unique models (i.e. multiplicity-free induced representations). We apply this method to the case of the Shalika model of GL_n, which is known to distinguish lifts from odd orthogonal groups. In the course of our proof, we further develop a variant of the method, that was introduced by Y.Hironaka, and in effect reduce many such problems to straightforward calculations on the group.

Abstract:
We present a conceptual and uniform interpretation of the methods of integral representations of L-functions (period integrals, Rankin-Selberg integrals). This leads to: (i) a way to classify of such integrals, based on the classification of certain embeddings of spherical varieties (whenever the latter is available), (ii) a conjecture which would imply a vast generalization of the method, and (iii) an explanation of the phenomenon of "weight factors" in a relative trace formula. We also prove results of independent interest, such as the generalized Cartan decomposition for spherical varieties of split groups over p-adic fields (following an argument of Gaitsgory and Nadler).

Abstract:
Let X=H\G be a homogeneous spherical variety for a split reductive group G over the integers o of a p-adic field k, and K=G(o) a hyperspecial maximal compact subgroup of G=G(k). We compute eigenfunctions ("spherical functions") on X=X(k) under the action of the unramified (or spherical) Hecke algebra of G, generalizing many classical results of "Casselman-Shalika" type. Under some additional assumptions on X we also prove a variant of the formula which involves a certain quotient of L-values, and we present several applications such as: (1) a statement on "good test vectors" in the multiplicity-free case (namely, that an H-invariant functional on an irreducible unramified representation \pi is non-zero on \pi^K), (2) the unramified Plancherel formula for X, including a formula for the "Tamagawa measure" of X(o), and (3) a computation of the most continuous part of H-period integrals of principal Eisenstein series.

Abstract:
Schwartz functions, or measures, are defined on any smooth semi-algebraic ("Nash") manifold, and are known to form a cosheaf for the semi-algebraic restricted topology. We extend this definition to smooth semi-algebraic stacks, which are defined as geometric stacks in the category of Nash manifolds. Moreover, when those are obtained from algebraic quotient stacks of the form $X/G$, with $X$ a smooth affine variety and $G$ a reductive group defined over a global field $k$, we define, whenever possible, an "evaluation map" at each semisimple $k$-point of the stack, without using truncation methods. This corresponds to a regularization of orbital integrals. These evaluation maps produce, in principle, a distribution which generalizes the Arthur-Selberg trace formula and Jacquet's relative trace formula, although the former, and many instances of the latter, cannot actually be defined by the purely geometric methods of this paper. In any case, the stack-theoretic point of view provides an explanation for the pure inner forms that appear in many versions of the Langlands, and relative Langlands, conjectures.