Compact orthoalgebras

We initiate a study of topological orthoalgebras (TOAs), concentrating on the compact case. Examples of TOAs include topological orthomodular lattices, and also the projection lattice of a Hilbert space. As the latter example illustrates, a lattice-ordered TOA need not be a topological lattice. However, we show that a compact Boolean TOA is a topological Boolean algebra. Using this, we prove that any compact regular TOA is atomistic , and has a compact center. We prove also that any compact TOA with isolated is of finite height. We then focus on stably ordered TOAs: those in which the upper set generated by an open set is open. These include both topological orthomodular lattices and interval orthoalgebras - in particular, projection lattices. We show that the topology of a compact stably-ordered TOA with isolated is determined by that of its space of atoms.

     

Quantum generalized Toda system

We construct a ??spectral curve?? for the generalized Toda system, which allows efficiently finding its quantization. In turn, the quantization is realized using the technique of the quantum characteristic polynomial for the Gaudin system and an appropriate Alder-Kostant-Symes reduction. We also discuss some relations of this result to the recent consideration of the Drinfeld Zastava space, the monopole space, and corresponding symmetries of the Borel Yangian.     

Tensor products of orthoalgebras

We define a tensor product via a universal mapping property on the class oforthoalgebras, which are both partial algebras and orthocomplemented posets. We show how to construct such a tensor product forunital orthoalgebras, and use the Fano plane to show that tensor products do not always exist.     

Quantum -modules and generalized mirror transformations

In the previous paper [Hiroshi Iritani, Quantum D-modules and equivariant Floer theory for free loop spaces, Math. Z. 252 (3) (2006) 577–622], the author defined equivariant Floer cohomology for a complete intersection in a toric variety and showed that it is isomorphic to the small quantum D-module after a mirror transformation when the first Chern class c₁(M) of the tangent bundle is nef. In this paper, even when c₁(M) is not nef, we show that the equivariant Floer cohomology reconstructs the big quantum D-module under certain conditions on the ambient toric variety. The proof is based on a mirror theorem of Coates and Givental [T. Coates, A.B. Givental, Quantum Riemann — Roch, Lefschetz and Serre, Ann. of Math. (2) 165 (1) (2007) 15–53]. The reconstruction procedure here gives a generalized mirror transformation first observed by Jinzenji in low degrees [Masao Jinzenji, On the quantum cohomology rings of general type projective hypersurfaces and generalized mirror transformation, Internat. J. Modern Phys. A 15 (11) (2000) 1557–1595; Masao Jinzenji, Co-ordinate change of Gauss–Manin system and generalized mirror transformation, Internat. J. Modern Phys. A 20 (10) (2005) 2131–2156].     

Quantum cohomology of the infinite-dimensional generalized flag manifolds

Consider the infinite-dimensional flag manifold LK/T corresponding to the simple Lie group K of rank l and with maximal torus T. We show that, for K of type A, B or C, if we endow the space (where q₁,…,q_l+1 are multiplicative variables) with an -bilinear product satisfying some simple properties analogous to the quantum product on QH∗(K/T), then the isomorphism type of the resulting ring is determined by the integrals of motion of a certain periodic Toda lattice system, in exactly the same way as the isomorphism type of QH∗(K/T) is determined by the integrals of motion of the non-periodic Toda lattice (see (Ann. Math. 149 (1999) 129)). This is an infinite-dimensional extension of the main result of Mare (Relations in the quantum cohomology ring of G/B, preprint math. DG/0210026) and at the same time a generalization of M.A. Guest and T. Otofuji (Comm. Math. Phys. 217 (2001) 475).     

Quantum generalized cluster algebras and quantum dilogarithms of higher degrees

We extend the notion of quantizing the coefficients of ordinary cluster algebras to the generalized cluster algebras of Chekhov and Shapiro. In parallel to the ordinary case, it is tightly integrated with certain generalizations of the ordinary quantum dilogarithm, which we call the quantum dilogarithms of higher degrees. As an application, we derive the identities of these generalized quantum dilogarithms associated with any period of quantum Y -seeds.     

Quantum mechanics and the hydrogen atom in a generalized Wigner-Seitz cell

We investigate the energy spectrum of a nonrelativistic quantum particle and a hydrogen-like atom placed in a vacuum cavity with general boundary conditions ensuring confinement. When these conditions, as in the Wigner-Seitz model, admit a large amplitude of the wave function on the boundary of the cavity, a nonperturbative rearrangement of lower energy levels of the spectrum occurs, which is essentially different from the case of the confinement by a potential barrier. A nontrivial role in this spectrum rearrangement is played by the von Neumann-Wigner effect of repulsion of nearby levels. For such a confined state of a hydrogen-like atom in a spherical cavity of radius R with the boundary formed by a potential layer of depth d, we show that the lowest energy level of the atom has a pronounced minimum at physically meaningful layer parameters and that the binding energy can be much greater than E _1s , the energy of the 1s level of a free-standing atom, and that the regime where the atom binding is much greater than E _1s becomes possible for a cavity with R ～ 10–100 nm.     

Quantum coin flipping,qubit measurement,and generalized Fibonacci numbers

Theoretical and Mathematical Physics - The problem of Hadamard quantum coin measurement in $$n$$ trials, with an arbitrary number of repeated consecutive last states, is formulated in terms of...     

Maximal integral over observable measures

Let f be a continuous transformation on a compact,finite-dimensional manifold M,andψa continuous function on M.This paper establishes the following formula:ess sup lim sup n→∞ 1/n ψ_n(x)=sup{∫ψdμ︱μ∈O_f}≤lim sup n→∞ 1/n ess sup ψ_n(x),where ess sup denotes the essential supremum taken against the Lebesgue measure,ψ_n(x)=n-1 ∑ i=0 ψ(f~ix) and O_f is the set of observable measures.Examples are provided to illustrate that the inequality could be an equality or strict.Moreover,if μ is the unique maximizing observable measure for ψ,it is weakly statistical stable.     

Partially observable multistage stochastic programming

We propose a class of partially observable multistage stochastic programs and describe an algorithm for solving this class of problems. We provide a Bayesian update of a belief-state vector, extend the stochastic programming formulation to incorporate the belief state, and characterize saddle-function properties of the corresponding cost-to-go function. Our algorithm is a derivative of the stochastic dual dynamic programming method.     

Entanglement measures based on observable correlations

By regarding quantum states as communication channels and using observable correlations quantitatively expressed by mutual information, we introduce a hierarchy of entanglement measures that includes the entanglement of formation as a particular instance. We compare the maximal and minimal measures and indicate the conceptual advantages of the minimal measure over the entanglement of formation. We reveal a curious feature of the entanglement of formation by showing that it can exceed the quantum mutual information, which is usually regarded as a theoretical measure of total correlations. This places the entanglement of formation in a broader scenario, highlights its peculiarity in relation to pure-state ensembles, and introduces a competing definition with intrinsic informational significance. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 3, pp. 453–462, June, 2008.     

Martingale measures and partially observable diffusions

In this paper, we solve the problem of existence of an optimal control based on partial observations in the general case where the observation process depends on the control. The method of solution is based on the use of relaxed controls and martingales measures: we associate a martingale problem with the filter and we prove that this problem is equivalent to the initial one     

Miniversal deformations of observable marked matrices

Given the set of vertical pairs of matrices keeping the subspace invariant, we compute miniversal deformations of a given pair when it is observable, and the subspace is marked. Moreover, we obtain the dimension of the orbit, characterize the structurally stable vertical pairs, and study the effect of each deformation parameter. Copyright © 2013 John Wiley & Sons, Ltd.     

On finitely accessible and finitely observable rings

Two classes of rings which occur in linear system theory are introduced and compared. Characterizations of one of them are given in terms of integral extensions and a Cayley-Hamilton type matrix condition.     

Adjoint translation,adjoint observable and uncertainty principles

The motivation to this paper stems from signal/image processing where it is desired to measure various attributes or physical quantities such as position, scale, direction and frequency of a signal or an image. These physical quantities are measured via a signal transform, for example, the short time Fourier transform measures the content of a signal at different times and frequencies. There are well known obstructions for completely accurate measurements formulated as “uncertainty principles”. It has been shown recently that “conventional” localization notions, based on variances associated with Lie-group generators and their corresponding uncertainty inequality might be misleading, if they are applied to transformation groups which differ from the Heisenberg group, the latter being prevailing in signal analysis and quantum mechanics. In this paper we describe a generic signal transform as a procedure of measuring the content of a signal at different values of a set of given physical quantities. This viewpoint sheds a light on the relationship between signal transforms and uncertainty principles. In particular we introduce the concepts of “adjoint translations” and “adjoint observables”, respectively. We show that the fundamental issue of interest is the measurement of physical quantities via the appropriate localization operators termed “adjoint observables”. It is shown how one can define, for each localization operator, a family of related “adjoint translation” operators that translate the spectrum of that localization operator. The adjoint translations in the examples of this paper correspond to well-known transformations in signal processing such as the short time Fourier transform (STFT), the continuous wavelet transform (CWT) and the shearlet transform. We show how the means and variances of states transform appropriately under the translation action and compute associated minimizers and equalizers for the uncertainty criterion. Finally, the concept of adjoint observables is used to estimate concentration properties of ambiguity functions, the latter being an alternative localization concept frequently used in signal analysis.     

A two-population game in observable double-ended queuing systems

In this study, the existing game theoretical framework is extended to strategic queuing in search of solutions for a two-population game in observable double-ended queuing systems with zero matching times. We show that multiple Nash equilibria and one unique subgame perfect Nash equilibrium exist in this game.