Unrelated Quantities Research Articles

A novel way to determine the scale of inflation

We show that in the Feebly Interacting Massive Particle (FIMP) model of Dark Matter (DM), one may express the inflationary energy scale H* as a function of three otherwise unrelated quantities, the DM isocurvature perturbation amplitude, its mass and its self-coupling constant, independently of the tensor-to-scalar ratio. The FIMP model assumes that there exists a real scalar particle that alone constitutes the DM content of the Universe and couples to the Standard Model via a Higgs portal. We consider carefully the various astrophysical, cosmological and model constraints, accounting also for variations in inflationary dynamics and the reheating history, to derive a robust estimate for H* that is confined to a relatively narrow range. We point out that, within the context of the FIMP DM model, one may thus determine H* reliably even in the absence of observable tensor perturbations.

Comparing Apples and Oranges: Query Trade-off in Submodular Maximization

Fast algorithms for submodular maximization problems have a vast potential use in applicative settings, such as machine learning, social networks, and economics. Though fast algorithms were known for some special cases, only recently such algorithms were considered in the general case of maximizing a monotone submodular function subject to a matroid independence constraint. The known fast algorithm matches the best possible approximation guarantee, while trying to reduce the number of value oracle queries the algorithm performs. Our main result is a new algorithm for this general case that establishes a surprising trade-off between two seemingly unrelated quantities: the number of value oracle queries and the number of matroid independence queries performed by the algorithm. Specifically, one can decrease the former by increasing the latter, and vice versa, while maintaining the best possible approximation guarantee. Such a trade-off is very useful since various applications might incur significantly different costs in querying the value and matroid independence oracles. Furthermore, in case the rank of the matroid is O(nc), where n is the size of the ground set and c is an absolute constant smaller than 1, the total number of oracle queries our algorithm uses can be made to have a smaller magnitude compared to that needed by the current best known algorithm. We also provide even faster algorithms for the well-studied special cases of a cardinality constraint and a partition matroid independence constraint, both of which capture many real-world applications and have been widely studied both theoretically and in practice.

On a relation between roughening and coarsening

We argue that a strict relation exists between two in principle unrelated quantities: The size of the growing domains in a coarsening system, and the kinetic roughening of an interface. This relation is confirmed by extensive simulations of the Ising model with different forms of quenched disorder, such as random bonds, random fields andstochastic dilution.

Condensed phase of Bose-Fermi mixtures with a pairing interaction

We study the condensed phase of a Bose-Fermi mixture with a tunable pairing interaction between bosons and fermions with many-body diagrammatic methods and fixed-node diffusion Quantum Monte Carlo simulations. A universal behavior of the condensate fraction and bosonic momentum distribution with respect to the boson concentration is found to hold in an extended range of boson-fermion couplings and concentrations. For vanishing boson density, we prove that the bosonic condensate fraction reduces to the quasiparticle weight Z of the Fermi polaron studied in the context of polarized Fermi gases, unifying in this way two apparently unrelated quantities.

Risk-free rate effects on conditional variances and conditional correlations of stock returns

This paper investigates whether the risk-free rate may explain the movements observed in the conditional second moments of asset returns. Original results are derived, within the C-CAPM framework, that attest the existence of a channel connecting these seemingly unrelated quantities. The empirical results, involving 165 time series of stock returns quoted at the NYSE, show that the risk-free rate does contain information that is relevant in predicting the 165 conditional variances and 13,530 conditional correlations. These findings are particularly pronounced at lower frequencies where the persistence of the conditional second moments is significantly weaker.

Efficient Protocols for Generating Bipartite Classical Distributions and Quantum States

We investigate the fundamental problem of generating bipartite classical distributions or quantum states. By designing efficient communication protocols and proving their optimality, we establish a number of intriguing connections to fundamental measures in optimization, convex geometry, and information theory. 1) To generate a classical distribution P(x,y), we tightly characterize the minimum amount of quantum communication needed by the psd-rank of P (as a matrix), a measure recently proposed by Fiorini et al. (Proc. 44th ACM Symp. Theory Comput., pp. 95-106, 2012) in studies of the minimum size of extended formulations of optimization problems such as TSP. This echos the previous characterization for the optimal classical communication cost by the nonnegative rank of P. The result is obtained via investigating the more general case of bipartite quantum state generation and designing an optimal protocol for it. 2) When an approximation ϵ is allowed to generate a distribution (X,Y)~P, we present a classical protocol of the communication cost O((C(X,Y)+1)/ϵ, where C(X,Y) is common information, a well-studied measure in information theory introduced by Wyner (IEEE Trans. Inf. Theory, 21 (2):163-179, 1975). This also links nonnegative rank and common information, two seemingly unrelated quantities in different fields. 3) For approximately generating a quantum pure state |ψ〉, we completely characterize the minimum cost by a corresponding approximate rank, closing a possibly exponential gap left in Ambainis etal. (SIAM J. Comput., 32 (6):1570-1585, 2003).

CROSS-CORRELATIONS OF THE Lyα FOREST WITH WEAK-LENSING CONVERGENCE. ANALYTICAL ESTIMATES OF SIGNAL-TO-NOISE RATIO AND IMPLICATIONS FOR NEUTRINO MASS AND DARK ENERGY

We expect a detectable correlation between two seemingly unrelated quantities: the four point function of the cosmic microwave background (CMB) and the amplitude of flux decrements in quasar (QSO) spectra. The amplitude of CMB convergence in a given direction measures the projected surface density of matter. Measurements of QSO flux decrements trace the small-scale distribution of gas along a given line-of-sight. While the cross-correlation between these two measurements is small for a single line-of-sight, upcoming large surveys should enable its detection. This paper presents analytical estimates for the signal to noise (S/N) for measurements of the cross-correlation between the flux decrement and the convergence and for measurements of the cross-correlation between the variance in flux decrement and the convergence. For the ongoing BOSS (SDSS III) and Planck surveys, we estimate an S/N of 30 and 9.6 for these two correlations. For the proposed BigBOSS and ACTPOL surveys, we estimate an S/N of 130 and 50 respectively. Since the cross-correlation between the variance in flux decrement and the convergence is proportional to the fourth power of $\sigma_8$, the amplitude of these cross-correlations can potentially be used to measure the amplitude of $\sigma_8$ at z~2 to 2.5% with BOSS and Planck and even better with future data sets. These measurements have the potential to test alternative theories for dark energy and to constrain the mass of the neutrino. The large potential signal estimated in our analytical calculations motivate tests with non-linear hydrodynamical simulations and analyses of upcoming data sets.

Strong pairbreaking in anisotropic superconductors

Abrikosov-Gor'kov work on pairbreaking in isotropic materials is generalized to anisotropic Fermi surfaces and order parameters. New scaling relations for states with a strong pairbreaking are found for the specific-heat jump $\ensuremath{\Delta}C\ensuremath{\propto}{T}_{c}^{3}$; for the penetration depth that deviates from the zero-$T$ value as $\ensuremath{\Delta}\ensuremath{\lambda}=\ensuremath{\beta}{T}^{2}$ at low temperatures with $\ensuremath{\beta}\ensuremath{\propto}{T}_{c}^{\ensuremath{-}3}$, and for the slopes of the upper critical fields ${H}_{c2}^{\ensuremath{'}}({T}_{c})\ensuremath{\propto}{T}_{c}$. A remarkably simple relation between these at first sight unrelated quantities is found: $\ensuremath{\Delta}C{\ensuremath{\beta}}^{2}{T}_{c}^{4}/|{H}_{c2}^{\ensuremath{'}}|={\ensuremath{\phi}}_{0}/16{\ensuremath{\pi}}^{2}$ is a universal constant. The predictions are checked on ${\text{CeCoIn}}_{5}$ and the possibility to apply them to iron-based materials is discussed.

English

A novel formulation to study the large amplitude free vibration behavior of uniform moderately thick square plates with axially immovable edges is presented. The formulation is very simple as the large amplitude free vibration behavior is accurately evaluated knowing the totally unrelated quantities like the tensions developed in the plate due to large deflections and the corresponding buckling load. The basis for this formulation is the recent work of the authors on predicting the free vibration behavior of initially loaded plates. This novel formulation involves three basic steps, through which the large amplitude free vibration behavior of square plates can be very easily and accurately evaluated by following the assumed space mode (accurate single term admissible functions depending on the boundary conditions) approach used by many researches. Numerical results, with emphasis on the fundamental frequency, for the uniform moderately thick square plates, are presented in the digital form in this paper along with the standard solutions of other researchers, to conclusively demonstrate the efficacy of the startling simplicity of the present novel formulation.

Quantum noise and entanglement generated by a local quantum quench

We examine the growth of entanglement under a quantum quench at point contacts of simple fractional quantum Hall fluids and its relation with the measurement of local observables. Recently Klich and Levitov proposed that the noise generated from a local quantum quench provides a measure of the entanglement entropy. Their methods were specific to noninteracting electrons and the generalization to interacting systems was left as an open question. In this work, we generalize their result to the Laughlin states. We investigate the noise generated in the current along the edge of a fractional quantum Hall state at filling factors $\ensuremath{\nu}=1/m$, when a quantum point contact, initially closed, is fully opened at some initial time ${t}_{0}=0$. We find that local quenching in these systems gives time dependent correlation functions that have universal behavior on sufficiently long time and length scales. We calculate the noise and full counting statistics for $\ensuremath{\nu}=1/m$ and find that in general, the entanglement entropy and noise generated are unrelated quantities. We also discuss a generalization of this problem to the critical quantum Ising spin chain.

The Effects of Interest Rate Movements on Assets’ Conditional Second Moments

This paper investigates whether the short term interest rate may explain the movements observed in the conditional second moments of asset returns. The theoretical connections between these seemingly unrelated quantities are studied within the C-CAPM framework. Under the assumption that the product of the relative risk aversion coefficient and the marginal utility is monotonic in consumption, original results are derived that attest the existence of a relation between the risk-free rate and the conditional second moments. The empirical findings, involving 165 stock returns quoted at the NYSE, confirm that, at low frequencies, the interest rate is a determinant of the 165 conditional variances and 13530 conditional correlations.

Density, overcompleteness, and localization of frames

This work presents a quantitative framework for describing the overcompleteness of a large class of frames. It introduces notions of localization and approximation between two frames F = { f i } i ∈ I \mathcal {F} = \{f_i\}_{i \in I} and E = { e j } j ∈ G \mathcal {E} = \{e_j\}_{j \in G} ( G G a discrete abelian group), relating the decay of the expansion of the elements of F \mathcal {F} in terms of the elements of E \mathcal {E} via a map a : I → G a \colon I \to G . A fundamental set of equalities are shown between three seemingly unrelated quantities: the relative measure of F \mathcal {F} , the relative measure of E \mathcal {E} —both of which are determined by certain averages of inner products of frame elements with their corresponding dual frame elements—and the density of the set a ( I ) a(I) in G G . Fundamental new results are obtained on the excess and overcompleteness of frames, on the relationship between frame bounds and density, and on the structure of the dual frame of a localized frame. These abstract results yield an array of new implications for irregular Gabor frames. Various Nyquist density results for Gabor frames are recovered as special cases, but in the process both their meaning and implications are clarified. New results are obtained on the excess and overcompleteness of Gabor frames, on the relationship between frame bounds and density, and on the structure of the dual frame of an irregular Gabor frame. More generally, these results apply both to Gabor frames and to systems of Gabor molecules, whose elements share only a common envelope of concentration in the time-frequency plane.

Density, Overcompleteness, and Localization of Frames. I. Theory

Frames have applications in numerous fields of mathematics and engineering. The fundamental property of frames which makes them so useful is their overcompleteness. In most applications, it is this overcompleteness that is exploited to yield a decomposition that is more stable, more robust, or more compact than is possible using nonredundant systems. This work presents a quantitative framework for describing the overcompleteness of frames. It introduces notions of localization and approximation between two frames \({\frak F} = \{f_i\}_{i\in I}\) and \({\frak E} = \{e_j\}_{j\in G}\) (\(G\) a discrete abelian group), relating the decay of the expansion of the elements of \({\frak F}\) in terms of the elements of \({\frak E}\) via a map \(a : I \rightarrow G\). A fundamental set of equalities are shown between three seemingly unrelated quantities: The relative measure of \({\frak F}\), the relative measure of \({\frak E}\) — both of which are determined by certain averages of inner products of frame elements with their corresponding dual frame elements — and the density of the set \(a(I)\) in \(G\). Fundamental new results are obtained on the excess and overcompleteness of frames, on the relationship between frame bounds and density, and on the structure of the dual frame of a localized frame. In a subsequent article, these results are applied to the case of Gabor frames, producing an array of new results as well as clarifying the meaning of existing results. The notion of localization and related approximation properties introduced in this article are a spectrum of ideas that quantify the degree to which elements of one frame can be approximated by elements of another frame. A comprehensive examination of the interrelations among these localization and approximation concepts is presented.

Calculating the limits of statistical regulation of manufacturing processes, taking into account deviations in the shape of the machined workpiece

Existing methods of statistical regulation of manufacturing processes have been developed without taking into account deviations in the shape of the machined workpiece and the correlation between its current dimensions. The problem can be reduced to the special case of the regulation of a process formed by random mutually unrelated quantities with Gaussian or Maxwellian distribution laws. The analytical solution of this problem in general form, taking the above conditions into consideration, involves a need to operate with systems of random quantities, which is extremely difficult and unsuited to use by personnel, even in cases where the distribution law of the quantities concerned is expressed analytically.

The incorporation of spirit-levelling into a three-dimensional cartesian reference frame

Abstract In modern first-order geodetic work it is customary to measure horizontal and vertical angles, spatial distances and astronomic azimuths, latitudes and longitudes. These are projected onto an ellipsoid of reference and are then adjusted to yield positional co-ordinates on that ellipsoid. The results of spirit-levelling, which yield heights above the geoid, are commonly adjusted as a separate entity. Thus the classical method produces two distinct sets of unrelated quantities. A method for rigorously incorporating the results of spirit-levelling into a three-dimensional Cartesian frame of reference is presented. The use of the geoid and of an ellipsoid is avoided.