364 research outputs found
Generalized Satisfiability Problems via Operator Assignments
Schaefer introduced a framework for generalized satisfiability problems on
the Boolean domain and characterized the computational complexity of such
problems. We investigate an algebraization of Schaefer's framework in which the
Fourier transform is used to represent constraints by multilinear polynomials
in a unique way. The polynomial representation of constraints gives rise to a
relaxation of the notion of satisfiability in which the values to variables are
linear operators on some Hilbert space. For the case of constraints given by a
system of linear equations over the two-element field, this relaxation has
received considerable attention in the foundations of quantum mechanics, where
such constructions as the Mermin-Peres magic square show that there are systems
that have no solutions in the Boolean domain, but have solutions via operator
assignments on some finite-dimensional Hilbert space. We obtain a complete
characterization of the classes of Boolean relations for which there is a gap
between satisfiability in the Boolean domain and the relaxation of
satisfiability via operator assignments. To establish our main result, we adapt
the notion of primitive-positive definability (pp-definability) to our setting,
a notion that has been used extensively in the study of constraint satisfaction
problems. Here, we show that pp-definability gives rise to gadget reductions
that preserve satisfiability gaps. We also present several additional
applications of this method. In particular and perhaps surprisingly, we show
that the relaxed notion of pp-definability in which the quantified variables
are allowed to range over operator assignments gives no additional expressive
power in defining Boolean relations
Uncertainty inequalities on groups and homogeneous spaces via isoperimetric inequalities
We prove a family of uncertainty inequalities on fairly general groups
and homogeneous spaces, both in the smooth and in the discrete setting. The
crucial point is the proof of the endpoint, which is derived from a
general weak isoperimetric inequality.Comment: 17 page
Laguerre-Gaussian Modes and the Wigner Transform
Recent developments in laser physics have called renewed attention to
Laguerre-Gaussian (LG) beams of paraxial light. In this paper we consider the
corresponding LG modes for the two-dimensional harmonic oscillator, which
appear in the transversal plane at the laser beam's waist. We see how they
arise as Wigner transforms of Hermite-Gaussian modes, and we proceed to find a
closed form for their own Wigner transforms, providing an alternative to the
methods of Simon and Agarwal. Our main observation is that the Wigner transform
intertwines the creation and annihilation operators for the two classes of
modes.Comment: 12 pages, minor corrections; submitted, Journal of Modern Optic
Haar expectations of ratios of random characteristic polynomials
We compute Haar ensemble averages of ratios of random characteristic
polynomials for the classical Lie groups K = O(N), SO(N), and USp(N). To that
end, we start from the Clifford-Weyl algebera in its canonical realization on
the complex of holomorphic differential forms for a C-vector space V. From it
we construct the Fock representation of an orthosymplectic Lie superalgebra osp
associated to V. Particular attention is paid to defining Howe's oscillator
semigroup and the representation that partially exponentiates the Lie algebra
representation of sp in osp. In the process, by pushing the semigroup
representation to its boundary and arguing by continuity, we provide a
construction of the Shale-Weil-Segal representation of the metaplectic group.
To deal with a product of n ratios of characteristic polynomials, we let V =
C^n \otimes C^N where C^N is equipped with its standard K-representation, and
focus on the subspace of K-equivariant forms. By Howe duality, this is a
highest-weight irreducible representation of the centralizer g of Lie(K) in
osp. We identify the K-Haar expectation of n ratios with the character of this
g-representation, which we show to be uniquely determined by analyticity, Weyl
group invariance, certain weight constraints and a system of differential
equations coming from the Laplace-Casimir invariants of g. We find an explicit
solution to the problem posed by all these conditions. In this way we prove
that the said Haar expectations are expressed by a Weyl-type character formula
for all integers N \ge 1. This completes earlier work by Conrey, Farmer, and
Zirnbauer for the case of U(N).Comment: LaTeX, 70 pages, Complex Analysis and its Synergies (2016) 2:
Harnack inequality for fractional sub-Laplacians in Carnot groups
In this paper we prove an invariant Harnack inequality on
Carnot-Carath\'eodory balls for fractional powers of sub-Laplacians in Carnot
groups. The proof relies on an "abstract" formulation of a technique recently
introduced by Caffarelli and Silvestre. In addition, we write explicitly the
Poisson kernel for a class of degenerate subelliptic equations in product-type
Carnot groups
Quantum-Classical Correspondence of Dynamical Observables, Quantization and the Time of Arrival Correspondence Problem
We raise the problem of constructing quantum observables that have classical
counterparts without quantization. Specifically we seek to define and motivate
a solution to the quantum-classical correspondence problem independent from
quantization and discuss the general insufficiency of prescriptive
quantization, particularly the Weyl quantization. We demonstrate our points by
constructing time of arrival operators without quantization and from these
recover their classical counterparts
Structure Characterization with Thermal Wave Imaging
Thermal imaging is a technique of recent interest for the nondestructive evaluation of materials. This method attempts to characterize the internal structure of a sample (perhaps to locate flaws-cracks, bubbles, corrosion, etc.) by using its surface temperature response to an external heating. Some recent work on this subject is detailed in [2], [3], [4] and [6]
Property (RD) for Hecke pairs
As the first step towards developing noncommutative geometry over Hecke
C*-algebras, we study property (RD) (Rapid Decay) for Hecke pairs. When the
subgroup H in a Hecke pair (G,H) is finite, we show that the Hecke pair (G,H)
has (RD) if and only if G has (RD). This provides us with a family of examples
of Hecke pairs with property (RD). We also adapt Paul Jolissant's works in 1989
to the setting of Hecke C*-algebras and show that when a Hecke pair (G,H) has
property (RD), the algebra of rapidly decreasing functions on the set of double
cosets is closed under holomorphic functional calculus of the associated
(reduced) Hecke C*-algebra. Hence they have the same K_0-groups.Comment: A short note added explaining other methods to prove that the
subalgebra of rapidly decreasing functions is smooth. This is the final
version as published. The published version is available at: springer.co
On the equivalence of Eulerian and Lagrangian variables for the two-component Camassa-Holm system
The Camassa-Holm equation and its two-component Camassa-Holm system
generalization both experience wave breaking in finite time. To analyze this,
and to obtain solutions past wave breaking, it is common to reformulate the
original equation given in Eulerian coordinates, into a system of ordinary
differential equations in Lagrangian coordinates. It is of considerable
interest to study the stability of solutions and how this is manifested in
Eulerian and Lagrangian variables. We identify criteria of convergence, such
that convergence in Eulerian coordinates is equivalent to convergence in
Lagrangian coordinates. In addition, we show how one can approximate global
conservative solutions of the scalar Camassa-Holm equation by smooth solutions
of the two-component Camassa-Holm system that do not experience wave breaking
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