Positive Definite Distributions Research Articles

A novel two-sample test within the space of symmetric positive definite matrix distributions and its application in finance

A novel two-sample test within the space of symmetric positive definite matrix distributions and its application in finance

Deblurring decay-energy distribution from an invariant-mass measurement

Measured decay-energy spectra from invariant mass spectroscopy can give insights into the shell structure of particle-unbound systems. However, it is challenging to extract the underlying physics from a measured spectrum due to detector resolution and acceptance effects. In this work, we introduce a deblurring method used to restore the decay energy spectrum measured for the three-body decay of 26O. The method utilizes the Richardson-Lucy algorithm, which has proven to be successful in optics, and does not require any prior knowledge about the resonance states in the observed spectrum. It also circumvents the singularity issue by iteratively adjusting a positive definite distribution. The only inputs are the observed energy spectrum and the detector’s response matrix also referred to as the Transfer Matrix (TM). We test the method’s performance on a simulated spectrum using the MoNA-LISA-Sweeper setup and the associated TM. Finally, the approach is applied to the decay energy spectrum of the 26O system measured in an experiment conducted at the National Superconducting Cyclotron Laboratory (NSCL) by the Modular Neutron Array (MoNA) Collaboration. The deblurring approach suggests three peaks in the decay energy spectrum of 26O at 0.15, 1.5 MeV, and a broad peak between 4 and 6 MeV.

Quantum Central Limit Theorems, Emergence of Classicality and Time-Dependent Differential Entropy.

We derive some quantum central limit theorems for the expectation values of macroscopically coarse-grained observables, which are functions of coarse-grained Hermitian operators consisting of non-commuting variables. Thanks to the Hermiticity constraints, we obtain positive-definite distributions for the expectation values of observables. These probability distributions open some pathway for the emergence of classical behaviours in the limit of an infinitely large number of identical and non-interacting quantum constituents. This is in contradistinction to other mechanisms of classicality emergence due to environmental decoherence and consistent histories. The probability distributions thus derived also enable us to evaluate the non-trivial time-dependence of certain differential entropies.

Hybrid discrete-continuous truncated Wigner approximation for driven, dissipative spin systems

We present a systematic approach for the semiclassical treatment of many-body dynamics of interacting, open spin systems. Our approach overcomes some of the shortcomings of the recently developed discrete truncated Wigner approximation (DTWA) based on Monte-Carlo sampling in a discrete phase space that improves the classical treatment by accounting for lowest-order quantum fluctuations. We provide a rigorous derivation of the DTWA by embedding it in a continuous phase space, thereby introducing a hybrid discrete-continuous truncated Wigner approximation (DCTWA). We derive a set of operator-differential mappings that yield an exact equation of motion (EOM) for the continuous SU(2) Wigner function of spins. The standard DTWA is then recovered by a systematic neglection of specific terms in this exact EOM. The hybrid approach permits us to determine the validity conditions and to gain detailed understanding of the quality of the approximation, paving the way for systematic improvements. Furthermore, we show that the continuous embedding allows for a straightforward extension of the method to open spin systems subject to dephasing, losses and incoherent drive, while preserving the key advantages of the discrete approach, such as a positive definite Wigner distribution of typical initial states. We derive exact stochastic differential equations for processes which cannot be described by the standard DTWA due to the presence of non-classical noise. We illustrate our approach by applying it to the dissipative dynamics of Rydberg excitation of one-dimensional arrays of laser-driven atoms and compare it to exact results for small systems.

Aperiodic order and spherical diffraction, II: translation bounded measures on homogeneous spaces

We study the auto-correlation measures of invariant random point processes in the hyperbolic plane which arise from various classes of aperiodic Delone sets. More generally, we study auto-correlation measures for large classes of Delone sets in (and even translation bounded measures on) arbitrary locally compact homogeneous metric spaces. We then specialize to the case of weighted model sets, in which we are able to derive more concrete formulas for the auto-correlation. In the case of Riemannian symmetric spaces we also explain how the auto-correlation of a weighted model set in a Riemannian symmetric space can be identified with a (typically non-tempered) positive-definite distribution on mathbb {R}^n. This paves the way for a diffraction theory for such model sets, which will be discussed in the sequel to the present article.

Modified equilibrium distributions for Cooper-Frye particlization

We introduce a positive definite single-particle distribution that is suitable for describing the transition from a macroscopic hydrodynamic to a microscopic kinetic description during the late stages of heavy-ion collisions in the presence of moderately large viscous corrections. The modified equilibrium distribution function can be constructed with hydrodynamic input from either relativistic viscous fluid dynamics or anisotropic fluid dynamics. We test the modified equilibrium distribution's hydrodynamic output for a stationary hadron resonance gas subject to either shear stress, bulk pressure, or baryon diffusion current at a given freeze-out temperature and baryon chemical potential. While it does not reproduce all components of the net baryon current and energy-momentum tensor exactly, it significantly improves upon the customary linearized approximations for the non-equilibrium correction $\delta f_n$ which typically lead to unphysical negative distribution functions at large particle momenta. A comparison of particle spectra and $p_T$-differential elliptic flow coefficients from the Cooper-Frye formula computed with the modified equilibrium distribution and with linearized $\delta f_n$ corrections is presented, for two different (2+1)-dimensional hypersurfaces corresponding to central and non-central Pb+Pb collisions at the Large Hadron Collider (LHC).

Ab Initio Surface-Hopping Simulation of Femtosecond Transient-Absorption Pump-Probe Signals of Nonadiabatic Excited-State Dynamics Using the Doorway-Window Representation.

An ab initio theoretical framework for the simulation of femtosecond time-resolved transient absorption (TA) pump-probe (PP) spectra with quasi-classical trajectories is presented. The simulations are based on the classical approximation to the doorway-window (DW) representation of third-order four-wave-mixing signals. The DW formula accounts for the finite duration and spectral shape of the pump and probe pulses. In the classical DW formalism, classical trajectories are stochastically sampled from a positive definite doorway distribution, and the signals are evaluated by averaging over a positive definite window distribution. Nonadiabatic excited-state dynamics is described by a stochastic surface-hopping algorithm. The method has been implemented for the pyrazine molecule with the second-order algebraic-diagrammatic construction (ADC(2)) ab initio electronic-structure method. The methodology is illustrated by ab initio simulations of the ground-state bleach, stimulated emission, and excited-state absorption contributions to the TA PP spectrum of gas-phase pyrazine.

Dilated Convolutional Neural Networks for Sequential Manifold-valued Data.

Efforts are underway to study ways via which the power of deep neural networks can be extended to non-standard data types such as structured data (e.g., graphs) or manifold-valued data (e.g., unit vectors or special matrices). Often, sizable empirical improvements are possible when the geometry of such data spaces are incorporated into the design of the model, architecture, and the algorithms. Motivated by neuroimaging applications, we study formulations where the data are sequential manifold-valued measurements. This case is common in brain imaging, where the samples correspond to symmetric positive definite matrices or orientation distribution functions. Instead of a recurrent model which poses computational/technical issues, and inspired by recent results showing the viability of dilated convolutional models for sequence prediction, we develop a dilated convolutional neural network architecture for this task. On the technical side, we show how the modules needed in our network can be derived while explicitly taking the Riemannian manifold structure into account. We show how the operations needed can leverage known results for calculating the weighted Fréchet Mean (wFM). Finally, we present scientific results for group difference analysis in Alzheimer's disease (AD) where the groups are derived using AD pathology load: here the model finds several brain fiber bundles that are related to AD even when the subjects are all still cognitively healthy.

Distributional boundary values of analytic functions and positive definite distributions

We propose necessary and sufficient conditions for a distribution (generalized function) fof several variables to be positive definite. For this purpose, certain analytic extensions of f to tubular domains in complex space Cn are studied. The main result is given in terms of the Cauchy transform and completely monotonic functions.

On positive definite distributions with compact support

We propose necessary and sufficient conditions for a distribution (generalized function) $f$ of several variables to be positive definite. For this purpose, certain analytic extensions of $f$ to tubular domains in complex space $\mathbb{C}^n$ are studied. The main result is given in terms of completely monotonic functions on convex cones in $\mathbb{R}^n$.

The λ-intersection bodies and an analytic generalized Busemann-Petty problem

In this paper, we obtain an extension of connections between an analytic generalization of the Busemann-Petty problem and positive definite distributions. We show that the class of the λ -intersection bodies is closely related to the analytic generalized Busemann-Petty problem. Mathematics subject classification (2010): 52A40, 52A20.

Canonical Correlation Analysis on Riemannian Manifolds and Its Applications.

Canonical correlation analysis (CCA) is a widely used statistical technique to capture correlations between two sets of multi-variate random variables and has found a multitude of applications in computer vision, medical imaging and machine learning. The classical formulation assumes that the data live in a pair of vector spaces which makes its use in certain important scientific domains problematic. For instance, the set of symmetric positive definite matrices (SPD), rotations and probability distributions, all belong to certain curved Riemannian manifolds where vector-space operations are in general not applicable. Analyzing the space of such data via the classical versions of inference models is rather sub-optimal. But perhaps more importantly, since the algorithms do not respect the underlying geometry of the data space, it is hard to provide statistical guarantees (if any) on the results. Using the space of SPD matrices as a concrete example, this paper gives a principled generalization of the well known CCA to the Riemannian setting. Our CCA algorithm operates on the product Riemannian manifold representing SPD matrix-valued fields to identify meaningful statistical relationships on the product Riemannian manifold. As a proof of principle, we present results on an Alzheimer's disease (AD) study where the analysis task involves identifying correlations across diffusion tensor images (DTI) and Cauchy deformation tensor fields derived from T1-weighted magnetic resonance (MR) images.

Reflection positivity and conformal symmetry

A reflection positive Hilbert space is a triple (E,E+,θ), where E is a Hilbert space, θ a unitary involution and E+ a closed subspace on which the hermitian form 〈v,w〉θ:=〈θv,w〉 is positive semidefinite. For a triple (G,τ,S), where G is a Lie group, τ an involutive automorphism of G and S a subsemigroup invariant under the involution s↦s♯=τ(s)−1, a unitary representation π of G on (E,E+,θ) is called reflection positive if θπ(g)θ=π(τ(g)) and π(S)E+⊆E+. This is the first in a series of papers in which we develop a new and systematic approach to reflection positive representations based on reflection positive distributions and reflection positive distribution vectors. This approach is most natural to obtain classification results, in particular in the abelian case. Among the tools we develop is a generalization of the Bochner–Schwartz Theorem to positive definite distributions on open convex cones. We further illustrate our techniques with a non-abelian example by constructing reflection positive distribution vectors for complementary series representations of the conformal group O1,n+1+(R) of the sphere Sn.

Parameterization of the Extrapolations in the Kreĭn–Schwartz Theorem and the Entropy Maximizer: the Scalar Case

A Kreĭn-de Branges-Kotani space \(\mathbb{H }\) is associated to a given positive-definite distribution \(Q\) on the finite interval \((-a,a)\) of the real line. The Kreĭn-de Branges Theorem is applied to get a spectral measure of \(\mathbb{H }\). In this way a simple proof of the solubility of the Kreĭn’s extension problem related to \(Q\) (Kreĭn–Schwartz Theorem) is obtained. A condition that guarantees the uniqueness of the extrapolation as well as a parameterization of the extrapolations by means of Schur functions are given. The choice of the Schur function \(\eta \equiv 0\) in the parameterization is shown to produce an absolutely continuous measure which maximizes Burg’s entropy. The results are based on the coupling of the Arov–Grossman functional model with a Hilbert space operator built up from the multiplication operator on \(\mathbb{H }\).

A note on positive definite distributions

We provide necessary and sufficient conditions for a tempered distribution F∈S′(R) to be positive definite. A generalized Cauchy transform F˜ of F is used as a numerical continuation of F to the open upper and lower half-planes in C. In fact, our necessary and sufficient conditions for F are determined completely by the properties of the restriction of F˜ to the imaginary axis in C. The main result is given in terms of completely monotonic and absolutely monotonic functions.

Positive definite distributions and normed spaces

We answer a question of Alex Koldobsky. We show that for each − ∞ < p < 2 and each n ⩾ 3 − p there is a normed space X of dimension n which embeds in L s if and only if − n < s ⩽ p .

Time operators in stroboscopic wave-packet basis and the time scales in tunneling

We demonstrate that the time operator that measures the time of arrival of a quantum particle into a chosen state can be defined as a self-adjoint quantum-mechanical operator using periodic boundary conditions and applied to wave functions in energy representation. The time becomes quantized into discrete eigenvalues; and the eigenstates of the time operator, i.e., the stroboscopic wave packets introduced recently [Phys. Rev. Lett. 101, 046402 (2008)], form an orthogonal system of states. The formalism provides simple physical interpretation of the time-measurement process and direct construction of normalized, positive definite probability distribution for the quantized values of the arrival time. The average value of the time is equal to the phase time but in general depends on the choice of zero time eigenstate, whereas the uncertainty of the average is related to the traversal time and is independent of this choice. The general formalism is applied to a particle tunneling through a resonant tunneling barrier in one dimension.

Spin Kinetic Theory—Quantum Kinetic Theory in Extended Phase Space

The concept of phase space distribution functions and their evolution is used in the case of en enlarged phase space. In particular, we include the intrinsic spin of particles and present a quantum kinetic evolution equation for a scalar quasi-distribution function. In contrast to the proper Wigner transformation technique, for which we expect the corresponding quasi-distribution function to be a complex matrix, we introduce a spin projection operator for the density matrix in order to obtain the aforementioned scalar quasi-distribution function. There is a close correspondence between this projection operator and the Husimi (or Q) function used extensively in quantum optics. Such a function is based on a Gaussian smearing of a Wigner function, giving a positive definite distribution function. Thus, our approach gives a Wigner-Husimi quasi-distribution function in extended phase space, for which the reduced distribution function on the Bloch sphere is strictly positive. We also discuss the gauge issue and the fluid moment hierarchy based on such a quantum kinetic theory.

On an analytic generalization of the Busemann–Petty problem

In this paper, we establish an extension of the connections between an analytic generalization of the Busemann–Petty problem and the positive definite distributions. Our results show that the structure of the positive definite distributions in R n is closely related to the analytic generalization of the Busemann–Petty problem which was posed by Koldobsky.

Standard quantum mechanics featuring probabilities instead of wave functions

A new formulation of quantum mechanics (probability representation) is discussed. In this representation, a quantum state is described by a standard positive definite probability distribution (tomogram) rather than by a wave function. An unambiguous relation (analog of Radon transformation) between the density operator and a tomogram is constructed both for continuous coordinates and for spin variables. A novel feature of a state, tomographic entropy, is considered, and its connection with von Neumann entropy is discussed. A one-to-one map of quantum observables (Hermitian operators) on positive probability distributions is found.