Exact Analytical Formula Research Articles

Non-Gaussian signatures and collective effects in charge noise affecting a dynamically decoupled qubit

The effects of a collection of classical two-level charge fluctuators on the coherence of a dynamically-decoupled qubit are studied. Distinct dynamics are found at different qubit working positions. Exact analytical formulae are derived at pure dephasing and approximate solutions are found at the general working position, for weakly- and strongly-coupled fluctuators. Analysis of these solutions, combined with numerical simulations of the multiple random telegraph processes, reveal the scaling of the noise with the number of fluctuators and the number of control pulses, as well as dependence on other parameters of the qubit-fluctuators system. These results can be used to determine potential microscopic models for the charge environment by performing noise spectroscopy.

Nonequispaced grid sampling in photoacoustics with a nonuniform fast Fourier transform.

To obtain the initial pressure from the collected data on a planar sensor arrangement in photoacoustic tomography, there exists an exact analytic frequency domain reconstruction formula. An efficient realization of this formula needs to cope with the evaluation of the data's Fourier transform on a non-equispaced mesh. In this paper, we use the non-uniform fast Fourier transform to handle this issue and show its feasibility in 3D experiments with real and synthetic data. This is done in comparison to the standard approach that uses linear, polynomial or nearest neighbor interpolation. Moreover, we investigate the effect and the utility of flexible sensor location to make optimal use of a limited number of sensor points. The computational realization is accomplished by the use of a multi-dimensional non-uniform fast Fourier algorithm, where non-uniform data sampling is performed both in frequency and spatial domain. Examples with synthetic and real data show that both approaches improve image quality.

Many-qubit quantum state transfer via spin chains

The transfer of an unknown quantum state, from a sender to a receiver, is one of the main requirements to perform quantum information processing (QIP) tasks. In this respect, the state transfer of a single qubit by means of spin chains has been widely discussed, and many protocols aiming at performing this task have been proposed. Nevertheless, the state transfer of more than one qubit has not been properly addressed so far. In this paper, we present a modified version of a recently proposed quantum state transfer (QST) protocol (Lorenzo S, Apollaro T J G, Sindona A and Plastina F 2013 Phys. Rev. A 87 042313) to obtain a quantum channel for the transfer of two qubits. This goal is achieved by exploiting Rabi-like oscillations due to excitations induced by means of strong and localized magnetic fields. We derive exact analytical formulae for the fidelity of the QST and obtain a high-quality transfer for general quantum states as well as for specific classes of states relevant for QIP.

The anomalous scaling exponents of turbulence in general dimension from random geometry

We propose an exact analytical formula for the anomalous scaling exponents of inertial range structure functions in incompressible fluid turbulence. The formula is a gravitational Knizhnik-Polyakov-Zamolodchikov (KPZ)-type relation, and is valid in any number of space dimensions. It incorporates intermittency by gravitationally dressing the Kolmogorov linear scaling via a coupling to a random geometry. The formula has one real parameter $\gamma$ that depends on the number of space dimensions. The scaling exponents satisfy the convexity inequality, and the supersonic bound constraint. They agree with the experimental and numerical data in two and three space dimensions, and with numerical data in four space dimensions. Intermittency increases with $\gamma$, and in the infinite $\gamma$ limit the scaling exponents approach the value one, as in Burgers turbulence. At large $n$ the $n$th order exponent scales as $\sqrt{n}$. We discuss the relation between fluid flows and black hole geometry that inspired our proposal.

An optimal double-magic flip angle for performing the distance measurement REDOR experiment on a spin S=1

Distance measurements between a half-spin and a quadrupolar S=1 spin having a small quadrupolar coupling constant can be performed using the rotational echo double resonance (REDOR) experiment. We derived an analytical expression for the probability of transitions between energy levels resulting from the application of an arbitrary pulse flip angle to the quadrupolar spin and consequently minimized the probability that populations of individual levels do not undergo a spin transition during the pulse. As a result we discovered that if the flip angle of the quadrupolar spin pulse is 109.47°, the maximal recoupling values are the largest possible and the signal reaches a maximum value of 8/9, larger than in the use of either a 90° pulse or a 180° pulse. In addition, the slope of the initial decay is higher than that of the 90° pulse. The recoupling signal can be modeled by an exact analytical formula in the ideal case and simulations show that the advantage of the 109.47° pulse is preserved when the quadrupolar coupling constant CQ has a finite value typical of 2H and 6Li spins (up to CQ~200kHz). Experimental results on two spin pairs, 2H–13C and 6Li–13C, demonstrate the validity and accuracy of this method.

On steady non-breaking downstream waves and the wave resistance

In this work we have obtained exact analytical formulae expressing the wave resistance of a two-dimensional body by the parameters of the downstream non-breaking waves. The body moves horizontally at a constant speed $c$in a channel of finite depth $h$. We have analysed the relationships between the parameters of the upstream flow and the downstream waves. Making use of some results by Keady & Norbury (J. Fluid Mech., vol. 70, 1975, pp. 663–671) and Benjamin (J. Fluid Mech., vol. 295, 1995, pp. 337–356), we have rigorously proved that realistic steady free-surface flows with a positive wave resistance exist only if the upstream flow is subcritical, i.e. the Froude number$\mathit{Fr}=c/\sqrt{gh}<1$. For all solutions with downstream waves obtained by a perturbation of a supercritical upstream uniform flow the wave resistance is negative. Applying a numerical technique, we have calculated accurate values of the wave resistance depending on the wavelength, amplitude and mean depth.

Concise presentation of the Coulomb electrostatic potential of a uniformly charged cube

We use a novel method to calculate in closed form the Coulomb electrostatic potential created by a uniformly charged cube at an arbitrary point in space. We apply a suitable transformation of variables that allows us to obtain a simple presentation of the electrostatic potential in one-dimensional integral form. The final concise closed form expression of the Coulomb electrostatic potential of the uniformly charged cube is obtained after completing the calculation of the resulting one-dimensional integrals. Such integrals consist of combinations of products of error functions and power functions that can be solved exactly despite their intimidating appearance. The exact analytic formula for the Coulomb electrostatic potential that we derive reflects the symmetry of the cube and is easy to implement. We illustrate its use by calculating the exact values of the electrostatic potential at some points of symmetry such as the center of cube, center of face of cube, center of edge of cube and corner of cube.

Monochromaticity of orientation maps in v1 implies minimum variance for hypercolumn size.

In the primary visual cortex of many mammals, the processing of sensory information involves recognizing stimuli orientations. The repartition of preferred orientations of neurons in some areas is remarkable: a repetitive, non-periodic, layout. This repetitive pattern is understood to be fundamental for basic non-local aspects of vision, like the perception of contours, but important questions remain about its development and function. We focus here on Gaussian Random Fields, which provide a good description of the initial stage of orientation map development and, in spite of shortcomings we will recall, a computable framework for discussing general principles underlying the geometry of mature maps. We discuss the relationship between the notion of column spacing and the structure of correlation spectra; we prove formulas for the mean value and variance of column spacing, and we use numerical analysis of exact analytic formulae to study the variance. Referring to studies by Wolf, Geisel, Kaschube, Schnabel, and coworkers, we also show that spectral thinness is not an essential ingredient to obtain a pinwheel density of π, whereas it appears as a signature of Euclidean symmetry. The minimum variance property associated to thin spectra could be useful for information processing, provide optimal modularity for V1 hypercolumns, and be a first step toward a mathematical definition of hypercolumns. A measurement of this property in real maps is in principle possible, and comparison with the results in our paper could help establish the role of our minimum variance hypothesis in the development process.

Propagation Properties of Finite Olver-Gaussian Beams Passing through a Paraxial ABCD Optical System

In this paper, an exact analytical propagation formula of Finite Olver-Gaussian Beams (FOGBs) passing through a paraxial ABCD optical system is developed and some numerical examples are performed. The propagation properties of the FOGBs through general optical systems characterized by given ABCD matrix are studied on the basis of the generalized Huygens-Fresnel diffraction integral, which permits to show the behavior of this laser beams family and its properties de-pending of the laser parameters. This research is of interest to prove some investigations done in the past by other researchers.

Regularized solution of a nonlinear problem in electromagnetic sounding*

Non destructive investigation of soil properties is crucial when trying to identify inhomogeneities in the ground or the presence of conductive substances. This kind of survey can be addressed with the aid of electromagnetic induction measurements taken with a ground conductivity meter. In this paper, starting from electromagnetic data collected by this device, we reconstruct the electrical conductivity of the soil with respect to depth, with the aid of a regularized damped Gauss–Newton method. We propose an inversion method based on the low-rank approximation of the Jacobian of the function to be inverted, for which we develop exact analytical formulae. The algorithm chooses a relaxation parameter in order to ensure the positivity of the solution and implements various methods for the automatic estimation of the regularization parameter. This leads to a fast and reliable algorithm, which is tested on numerical experiments both on synthetic data sets and on field data. The results show that the algorithm produces reasonable solutions in the case of synthetic data sets, even in the presence of a noise level consistent with real applications, and yields results that are compatible with those obtained by electrical resistivity tomography in the case of field data.

A new general analytical approach for modeling patterns of genetic differentiation and effective size of subdivided populations over time

The main purpose of this paper is to develop a theoretical framework for assessing effective population size and genetic divergence in situations with structured populations that consist of various numbers of more or less interconnected subpopulations. We introduce a general infinite allele model for a diploid, monoecious and subdivided population, with subpopulation sizes varying over time, including local subpopulation extinction and recolonization, bottlenecks, cyclic census size changes or exponential growth. Exact matrix analytic formulas are derived for recursions of predicted (expected) gene identities and gene diversities, identity by descent and coalescence probabilities, and standardized variances of allele frequency change. This enables us to compute and put into a general framework a number of different types of genetically effective population sizes (Ne) including variance, inbreeding, nucleotide diversity, and eigenvalue effective size. General expressions for predictions (gST) of the coefficient of gene differentiation GST are also derived. We suggest that in order to adequately describe important properties of a subdivided population with respect to allele frequency change and maintenance of genetic variation over time, single values of gST and Ne are not enough. Rather, the temporal dynamic patterns of these properties are important to consider. We introduce several schemes for weighting subpopulations that enable effective size and expected genetic divergence to be calculated and described as functions of time, globally for the whole population and locally for any group of subpopulations. The traditional concept of effective size is generalized to situations where genetic drift is confounded by external sources, such as immigration and mutation. Finally, we introduce a general methodology for state space reduction, which greatly decreases the computational complexity of the matrix analytic formulas.

The Analytical Description of Regular LDPC Codes Correcting Ability

Abstract The analytical description of regular LDPC (Low-Density Parity Check) codes correcting ability has been investigated. The statistical dependencies for the maximum number of corrected bits per the code word as a function of LDPC code word length and code rate are given based on multiple experimental analyses of LDPC check matrices. The analytical expressions are proposed for the cases of linear, exponential and polynomial approximations of given results. The most exact analytical formula is proved by criterion of the minimum divergence between the experimental and theoretical results.

Stochastic holin expression can account for lysis time variation in the bacteriophageλ

The inherent stochastic nature of biochemical processes can drive differences in gene expression between otherwise identical cells. While cell-to-cell variability in gene expression has received much attention, randomness in timing of events has been less studied. We investigate event timing at the single-cell level in a simple system, the lytic pathway of the bacterial virus phage λ. In individual cells, lysis occurs on average at 65 min, with an s.d. of 3.5 min. Interestingly, mutations in the lysis protein, holin, alter both the lysis time (LT) mean and variance. In our analysis, LT is formulated as the first-passage time (FPT) for cellular holin levels to cross a critical threshold. Exact analytical formulae for the FPT moments are derived for stochastic gene expression models. These formulae reveal how holin transcription and translation efficiencies independently modulate the LT mean and variation. Analytical expressions for the LT moments are used to evaluate previously published single-cell LT data for λ phages with mutations in the holin sequence or its promoter. Our results show that stochastic holin expression is sufficient to account for the intercellular LT differences in both wild-type phages, and phage variants where holin transcription and the threshold for lysis have been experimentally altered. Finally, our analysis reveals regulatory motifs that enhance the robustness of lysis timing to cellular noise.

Quantum discord for generalized bloch sphere states

In this study, for particular states of bipartite quantum system in 2 n × 2 m dimensional Hilbert space state, similar to m or n-qubit density matrices represented in Bloch sphere which we call them generalized Bloch sphere states (GBSS), we give an efficient optimization procedure so that analytic evaluation of quantum discord can be performed. Using this optimization procedure, we find an exact analytical formula for the optimum positive operator valued measure (POVM) that maximizes the measure of the classical correlation for these states. The presented optimization procedure is also used to show that for any concave entropy function the same POVMs are sufficient for quantum discord of mentioned states. Furthermore, we show that such optimization procedure can be used to calculate the geometric measure of quantum discord (GMQD) and then an explicit formula for GMQD is given. Finally, a complete geometric view is presented for quantum discord of GBSS.

Effective axial forces in offshore lined and clad pipes

To protect pipelines from corrosion attacks, some offshore pipelines have corrosion resistant liners or cladding made from stainless steel. Stainless steels have higher temperature expansion coefficients and lower Poisson’s ratios than ordinary high strength carbon manganese steels. Due to the differences in material properties, liners and cladding have a non-trivial influence on the critical design parameter called the effective axial force in offshore pipelines. Current offshore design codes do not contain guidance on how to include the effects of liners and cladding on the effective axial force. In this paper, an exact analytical formula for the effective axial force in lined and clad pipes is deduced. A simplified approximate formula, which is more suitable for use in engineering contexts, is developed and its validity verified by comparisons to results of the complex exact one.

On the probability of hitting a constant or a time-dependent boundary for a geometric Brownian motion with time-dependent coefficients

This paper presents exact analytical formulae for the crossing of a constant onesided or two-sided boundary by a geometric Brownian motion with timedependent, non-random, drift and diffusion coefficients, under the assumption that the drift coefficient is a constant multiple of the diffusion coefficient, as well as approximate analytical formulae for general time-dependent, non-random, drift and diffusion coefficients and general time-dependent, non-random, boundaries. The numerical implementation of these formulae is very simple. Mathematics Subject Classification: 60J65, 60J60; 58J35

Arbitrarily accurate passband composite pulses for dynamical suppression of amplitude noise

We introduce flexible high-fidelity passband (PB) composite pulse sequences constructed by concatenation of recently derived arbitrarily large and arbitrarily accurate broadband $\mathcal{B}$ and narrowband $\mathcal{N}$ composite sequences. Our PB sequences allow to produce flexible and tunable nearly rectangular two-state inversion profiles as a function of the individual pulse area because the width and the rectangularity of these profiles can be adjusted at will. Moreover, these PB sequences suppress excitation around pulse area $0$ and $2\pi$, and suppress deviations from complete population inversion around pulse area $\pi$ to arbitrarily high orders. These features makes them a valuable tool for high-fidelity qubit operations in the presence of relatively strong amplitude noise. We construct two types of PB pulses: $\mathcal{N}(\mathcal{B})$ in which a broadband pulse is nested into a narrowband pulse, and $\mathcal{B}(\mathcal{N})$ in which a narrowband pulse is nested into a broadband pulse; the latter sequences deliver narrower profiles. We derive exact analytic formulas for the composite phases of the PB pulses and exact analytic formulas for the inversion profiles. These formulas allow an easy estimation of the experimental resources needed for any desired qubit inversion profile.

Mesoscopic behavior from microscopic Markov dynamics and its application to calcium release channels

A major challenge in biology is to understand how molecular processes determine phenotypic features. We address this fundamental problem in a class of model systems by developing a general mathematical framework that allows the calculation of mesoscopic properties from the knowledge of microscopic Markovian transition probabilities. We show how exact analytic formulae for the first and second moments of resident time distributions in mesostates can be derived from microscopic resident times and transition probabilities even for systems with a large number of microstates. We apply our formalism to models of the inositol trisphosphate receptor, which plays a key role in generating calcium signals triggering a wide variety of cellular responses. We demonstrate how experimentally accessible quantities, such as opening and closing times and the coefficient of variation of inter-spike intervals, and other, more elaborated, quantities can be analytically calculated from the underlying microscopic Markovian dynamics. A virtue of our approach is that we do not need to follow the detailed time evolution of the whole system, as we derive the relevant properties of its steady state without having to take into account the often extremely complicated transient features. We emphasize that our formulae fully agree with results obtained by stochastic simulations and approaches based on a full determination of the microscopic system's time evolution. We also illustrate how experiments can be devised to discriminate between alternative molecular models of the inositol trisphosphate receptor. The developed approach is applicable to any system described by a Markov process and, owing to the analytic nature of the resulting formulae, provides an easy way to characterize also rare events that are of particular importance to understand the intermittency properties of complex dynamic systems.

The Distribution of Word Matches Between Markovian Sequences with Periodic Boundary Conditions

Word match counts have traditionally been proposed as an alignment-free measure of similarity for biological sequences. The D(2) statistic, which simply counts the number of exact word matches between two sequences, is a useful test bed for developing rigorous mathematical results, which can then be extended to more biologically useful measures. The distributional properties of the D(2) statistic under the null hypothesis of identically and independently distributed letters have been studied extensively, but no comprehensive study of the D(2) distribution for biologically more realistic higher-order Markovian sequences exists. Here we derive exact formulas for the mean and variance of the D(2) statistic for Markovian sequences of any order, and demonstrate through Monte Carlo simulations that the entire distribution is accurately characterized by a Pólya-Aeppli distribution for sequence lengths of biological interest. The approach is novel in that Markovian dependency is defined for sequences with periodic boundary conditions, and this enables exact analytic formulas for the mean and variance to be derived. We also carry out a preliminary comparison between the approximate D(2) distribution computed with the theoretical mean and variance under a Markovian hypothesis and an empirical D(2) distribution from the human genome.

ASYMPTOTIC EVALUATION OF BOSONIC PROBABILITY AMPLITUDES IN LINEAR UNITARY NETWORKS IN THE CASE OF LARGE NUMBER OF BOSONS

An asymptotic analytical approach is proposed for bosonic probability amplitudes in unitary linear networks, such as the optical multiport devices for photons. The asymptotic approach applies for large number of bosons N ≫ M in the M-mode network, where M is finite. The probability amplitudes of N bosons unitarily transformed from the input modes to the output modes of a unitary network are approximated by a multidimensional integral with the integrand containing a large parameter (N) in the exponent. The integral representation allows an asymptotic estimate of bosonic probability amplitudes up to a multiplicative error of order 1/N by the saddle point method. The estimate depends on solution of the scaling problem for the M × M-dimensional unitary network matrix: to find the left and right diagonal matrices which scale the unitary matrix to a matrix which has specified row and column sums (equal, respectively, to the distributions of bosons in the input and output modes). The scaled matrices give the saddle points of the integral. For simple saddle points, an explicit formula giving the asymptotic estimate of bosonic probability amplitudes is derived. Performance of the approximation and the scaling of the relative error with N are studied for two-mode network (the beam splitter), where the saddle-points are roots of a quadratic and an exact analytical formula for the probability amplitudes is available, and for three-mode network (the tritter).