Exact Analytical Results Research Articles

Mining top-k approximate closed patterns in an imprecise database

Over the last few years, the growth of data is exponential, leading to colossal amounts of information being produced by computational systems. Meanwhile, the data in real-life applications are usually incomplete and imprecise, which poses big challenges for researchers to obtain exact and valid analytical results with traditional frequent pattern mining methods. Since the potential faults can break the original characteristics of data patterns into multiple small fragments, it is impossible to recover the long true patterns from these fragments. To explore the huge amount of imprecise data by means of frequent pattern mining, we propose a service-oriented model that enables a new way of service provisioning based on users' QoS (quality of service) requirements. The novel model is developed to solve the problem of mining top-k approximate closed patterns in imprecise databases and will be further applied to diagnosis and treatment of potential patients in online medical applications. We test the novel model in an imprecise medical database and the experimental results show that the new model can successfully improve the health services for online customers.

Phase structure of XX0 spin chain and nonintersecting Brownian motion

We study finite size and temperature XX0 Heisenberg spin chain in weak and strong coupling regimes. By using an elegant connection of the model to integrable combinatorics and probability, we explore and interpret a possible phase structure of the model in asymptotic limit: the limit of large inverse temperature and size. First, the partition function and free energy of the model are derived by using techniques and results from random matrix models and nonintersecting Brownian motion. We show that, in the asymptotic limit, partition function of the model, written in terms of matrix integral, is governed by the Tracy–Widom distribution. Second, the exact analytic results for the free energy, which is obtained by the asymptotic analysis of the Tracy–Widom distribution, indicate a completely new and sophisticated phase structure of the model. This phase structure consists of second- and third-order phase transitions. Finally, to shed light on our new results, we provide a possible new interpretation of the phase structure in terms of dynamical behaviour of magnons in the spin chain. We demonstrate distinct features of the phases with schematic spin configurations which have definite features in each region of the phase diagram.

Effect of post-transcriptional feedback on the stochastic effect in gene expression

The post-transcriptional feedback mediated by microRNA is ubiquitous in a wide range of biological organisms, but how it impacts the dynamic of gene expression remains to be fully explored. A qualitative understanding the effect of the post-transcriptional feedback requires analysis of the corresponding stochastic models, the exact analytic results of which have not been obtained so far. Here, we analyzed a model of gene expression with post-transcriptional and feedback regulation and obtain exact results for the corresponding mRNA steady-state distribution. The results obtained provide new insights into the role of post-transcriptional regulation and feedback in controlling noise and stochastic deviation. We demonstrate that the effect of post-transcriptional feedback on noise depends on the time scale of enzyme switching and the reference model. Our results also indicate that deviation in post-transcriptional feedback can be decomposed into two parts: deviation caused by enzyme fluctuation and deviation caused by feedback and intrinsic noise. More importantly, we show that the gain in stochastic deviation is generally obtained at the cost of large noise. Our results provide insight into the role of post-transcriptional regulation in controlling the dynamic of gene expression.

Phenotypic heterogeneity in modeling cancer evolution.

The unwelcome evolution of malignancy during cancer progression emerges through a selection process in a complex heterogeneous population structure. In the present work, we investigate evolutionary dynamics in a phenotypically heterogeneous population of stem cells (SCs) and their associated progenitors. The fate of a malignant mutation is determined not only by overall stem cell and non-stem cell growth rates but also differentiation and dedifferentiation rates. We investigate the effect of such a complex population structure on the evolution of malignant mutations. We derive exactly calculated results for the fixation probability of a mutant arising in each of the subpopulations. The exactly calculated results are in almost perfect agreement with the numerical simulations. Moreover, a condition for evolutionary advantage of a mutant cell versus the wild type population is given in the present study. We also show that microenvironment-induced plasticity in invading mutants leads to more aggressive mutants with higher fixation probability. Our model predicts that decreasing polarity between stem and non-stem cells’ turnover would raise the survivability of non-plastic mutants; while it would suppress the development of malignancy for plastic mutants. The derived results are novel and general with potential applications in nature; we discuss our model in the context of colorectal/intestinal cancer (at the epithelium). However, the model clearly needs to be validated through appropriate experimental data. This novel mathematical framework can be applied more generally to a variety of problems concerning selection in heterogeneous populations, in other contexts such as population genetics, and ecology.

Hartree-Fock treatment of Fermi polarons using the Lee-Low-Pine transformation

We consider the Fermi polaron problem at zero temperature, where a single impurity interacts with non-interacting host fermions. We approach the problem starting with a Frohlich-like Hamiltonian where the impurity is described with canonical position and momentum operators. We apply the Lee-Low-Pine (LLP) transformation to change the fermionic Frohlich Hamiltonian into the fermionic LLP Hamiltonian which describes a many-body system containing host fermions only. We adapt the self-consistent Hartree-Fock (HF) approach, first proposed by Edwards, to the fermionic LLP Hamiltonian in which a pair of host fermions with momenta $\mathbf{k}$ and $\mathbf{k}'$ interact with a potential proportional to $\mathbf{k}\cdot\mathbf{k}'$. We apply the HF theory, which has the advantage of not restricting the number of particle-hole pairs, to repulsive Fermi polarons in one dimension. When the impurity and host fermion masses are equal our variational ansatz, where HF orbitals are expanded in terms of free-particle states, produces results in excellent agreement with McGuire's exact analytical results based on the Bethe ansatz. This work raises the prospect of using the HF ansatz and its time-dependent generalization as building blocks for developing all-coupling theories for both equilibrium and nonequilibrium Fermi polarons in higher dimensions

Spontaneous collective synchronization in the Kuramoto model with additional non-local interactions

In the context of the celebrated Kuramoto model of globally-coupled phase oscillators of distributed natural frequencies, which serves as a paradigm to investigate spontaneous collective synchronization in many-body interacting systems, we report on a very rich phase diagram in presence of thermal noise and an additional non-local interaction on a one-dimensional periodic lattice. Remarkably, the phase diagram involves both equilibrium and non-equilibrium phase transitions. In two contrasting limits of the dynamics, we obtain exact analytical results for the phase transitions. These two limits correspond to (i) the absence of thermal noise, when the dynamics reduces to that of a non-linear dynamical system, and (ii) the oscillators having the same natural frequency, when the dynamics becomes that of a statistical system in contact with a heat bath and relaxing to a statistical equilibrium state. In the former case, our exact analysis is based on the use of the so-called Ott-Antonsen ansatz to derive a reduced set of nonlinear partial differential equations for the macroscopic evolution of the system. Our results for the case of statistical equilibrium are on the other hand obtained by extending the well-known transfer matrix approach for nearest-neighbor Ising model to consider non-local interactions. The work offers a case study of exact analysis in many-body interacting systems. The results obtained underline the crucial role of additional non-local interactions in either destroying or enhancing the possibility of observing synchrony in mean-field systems exhibiting spontaneous synchronization.

Diffusion in Deterministic Interacting Lattice Systems.

We study reversible deterministic dynamics of classical charged particles on a lattice with hard-core interaction. It is rigorously shown that the system exhibits three types of transport phenomena, ranging from ballistic, through diffusive to insulating. By obtaining an exact expressions for the current time-autocorrelation function we are able to calculate the linear response transport coefficients, such as the diffusion constant and the Drude weight. Additionally, we calculate the long-time charge profile after an inhomogeneous quench and obtain diffusive profilewith the Green-Kubo diffusion constant. Exact analytical results are corroborated by MonteCarlo simulations.

Noise Figure Degradation in Balanced Amplifiers

Balanced amplifiers suffer from the noise figure performance in comparison with a stand-alone amplifier even with an ideal input divider. The noise parameters degrade further with an imperfect divider. We present exact and approximate analytical results for the noise figure and noise parameters of the balanced amplifier in divider topology. Measurement results are also presented as a verification.

An accurate analysis of the nonlinear power amplifier effects on SC-FDMA signals

Single carrier-frequency division multiple access (SC-FDMA) is a multiple access technique in broadband wireless networks which has been adapted by 3GPP for uplink transmission in 4G mobile communications. In this paper, the nonlinear effect of practical power amplifier (PA) is studied on the power allocated SC-FDMA signals. The interference power on the estimated symbols of all users are derived by two approaches based on the polynomial model of nonlinear PA and allocated power of subcarriers. In the first approach, an accurate analysis is followed.An approximation of the accurate result is presented in the second approach to provide a closed form formula. A simulation study is conducted to verify the analytical outcomes. The simulation and the exact analytical results are significantly matched. Conversely, the approximate relations are extremely suitable for allocating power in system design due to their closed form nature where they provide acceptable accuracy for practical applications.

Equilibrium charge distribution on a finite straight one-dimensional wire

The electrostatic properties of uniformly charged regular bodies are prominently discussed on college-level electromagnetism courses. However, one of the most basic problems of electrostatics that deals with how a continuous charge distribution reaches equilibrium is rarely mentioned at this level. In this work we revisit the problem of equilibrium charge distribution on a straight one-dimensional (1D) wire with finite length. The majority of existing treatments in the literature deal with the 1D wire as a limiting case of a higher-dimensional structure that can be treated analytically for a Coulomb interaction potential between point charges. Surprisingly, different models (for instance, an ellipsoid or a cylinder model) may lead to different results, thus there is even some ambiguity on whether the problem is well-posed. In this work we adopt a different approach where we do not start with any higher-dimensional body that reduces to a 1D wire in the appropriate limit. Instead, our starting point is the obvious one, a finite straight 1D wire that contains charge. However, the new tweak in the model is the assumption that point charges interact with each other via a non-Coulomb power-law interaction potential. This potential is well-behaved, allows exact analytical results and approaches the standard Coulomb interaction potential as a limit. The results originating from this approach suggest that the equilibrium charge distribution for a finite straight 1D wire is a uniform charge density when the power-law interaction potential approaches the Coulomb interaction potential as a suitable limit. We contrast such a finding to results obtained using a different regularised logarithmic interaction potential which allows exact treatment in 1D. The present self-contained material may be of interest to instructors teaching electromagnetism as well as students who will discover that simple-looking problems may sometimes pose important scientific challenges.

Diffusing diffusivity: a new derivation and comparison with simulations

Many experiments are now available where it has been shown that the probability distribution function (pdf) for the position of a Brownian particle diffusing in a heterogeneous medium is not Gaussian. However, in spite of this non-Gaussianity, the mean square displacement (MSD) still remains Fickian, i.e., $$\left\langle x^{2}\right\rangle \propto T$$ . One possible explanation of this non-Gaussian yet Brownian behavior is that the diffusivity of the particle itself is “diffusing”. Chubynsky and Slater (Phys. Rev. Lett. 113 098302 2014) proposed a model of “diffusing diffusivity” which they were able to solve analytically at small time scales, but simulations were performed for intermediate to large time scales. We present here a class of diffusing diffusivity models and show that the problem of calculating pdf for the position of diffusing particle is equivalent to calculating the survival probability of a particle undergoing Brownian motion in the presence of a sink. We give exact analytical results for all time scales and show that the pdf is non-Gaussian at short times which crosses over to a Gaussian at long times. The MSD is also shown to vary linearly with time at all times. We find that our results reproduce the numerical results of Chubynsky and Slater quite well. SYNOPSIS A model of diffusing diffusivity is analyzed and exact analytical results are obtained. The MSD is shown to vary linearly with time at all times yet the pdf is not Gaussian at all times. A comparison of these results is also made with the numerical results of Chubynsky and Slater.

Exact analytical results for integrate-and-fire neurons driven by excitatory shot noise.

A neuron receives input from other neurons via electrical pulses, so-called spikes. The pulse-like nature of the input is frequently neglected in analytical studies; instead, the input is usually approximated to be Gaussian. Recent experimental studies have shown, however, that an assumption underlying this approximation is often not met: Individual presynaptic spikes can have a significant effect on a neuron's dynamics. It is thus desirable to explicitly account for the pulse-like nature of neural input, i.e. consider neurons driven by a shot noise - a long-standing problem that is mathematically challenging. In this work, we exploit the fact that excitatory shot noise with exponentially distributed weights can be obtained as a limit case of dichotomous noise, a Markovian two-state process. This allows us to obtain novel exact expressions for the stationary voltage density and the moments of the interspike-interval density of general integrate-and-fire neurons driven by such an input. For the special case of leaky integrate-and-fire neurons, we also give expressions for the power spectrum and the linear response to a signal. We verify and illustrate our expressions by comparison to simulations of leaky-, quadratic- and exponential integrate-and-fire neurons.

Stochastic gene expression conditioned on large deviations

The intrinsic stochasticity of gene expression can give rise to large fluctuations and rare events that drive phenotypic variation in a population of genetically identical cells. Characterizing the fluctuations that give rise to such rare events motivates the analysis of large deviations in stochastic models of gene expression. Recent developments in non-equilibrium statistical mechanics have led to a framework for analyzing Markovian processes conditioned on rare events and for representing such processes by conditioning-free driven Markovian processes. We use this framework, in combination with approaches based on queueing theory, to analyze a general class of stochastic models of gene expression. Modeling gene expression as a Batch Markovian Arrival Process (BMAP), we derive exact analytical results quantifying large deviations of time-integrated random variables such as promoter activity fluctuations. We find that the conditioning-free driven process can also be represented by a BMAP that has the same form as the original process, but with renormalized parameters. The results obtained can be used to quantify the likelihood of large deviations, to characterize system fluctuations conditional on rare events and to identify combinations of model parameters that can give rise to dynamical phase transitions in system dynamics.

Heterogeneous diffusion in comb and fractal grid structures

We give exact analytical results for diffusion with a power-law position dependent diffusion coefficient along the main channel (backbone) on a comb and grid comb structures. For the mean square displacement along the backbone of the comb we obtain behavior 〈x2(t)〉∼t1/(2−α), where α is the power-law exponent of the position dependent diffusion coefficient D(x) ∼ |x|α. Depending on the value of α we observe different regimes, from anomalous subdiffusion, superdiffusion, and hyperdiffusion. For the case of the fractal grid we observe the mean square displacement, which depends on the fractal dimension of the structure of the backbones, i.e., 〈x2(t)〉∼t(1+ν)/(2−α), where 0 < ν < 1 is the fractal dimension of the backbones structure. The reduced probability distribution functions for both cases are obtained by help of the Fox H-functions.

Giant acoustic atom: A single quantum system with a deterministic time delay

We investigate the quantum dynamics of a single transmon qubit coupled to surface acoustic waves (SAWs) via two distant connection points. Since the acoustic speed is five orders of magnitude slower than the speed of light, the travelling time between the two connection points needs to be taken into account. Therefore, we treat the transmon qubit as a giant atom with a deterministic time delay. We find that the spontaneous emission of the system, formed by the giant atom and the SAWs between its connection points, initially decays polynomially in the form of pulses instead of a continuous exponential decay behaviour, as would be the case for a small atom. We obtain exact analytical results for the scattering properties of the giant atom up to two-phonon processes by using a diagrammatic approach. We find that two peaks appear in the inelastic (incoherent) power spectrum of the giant atom, a phenomenon which does not exist for a small atom. The time delay also gives rise to novel features in the reflectance, transmittance, and second-order correlation functions of the system. Furthermore, we find the short-time dynamics of the giant atom for arbitrary drive strength by a numerically exact method for open quantum systems with a finite-time-delay feedback loop.

Anomalous diffusion on a fractal mesh.

We present exact analytical results for properties of anomalous diffusion on a fractal mesh. The fractal mesh structure is a direct product of two fractal sets, one belonging to a main branch of backbones, the other to the side branches of fingers. Both fractal sets are constructed on the entire (infinite) y and x axes. We suggest a special algorithm in order to construct such sets out of standard Cantor sets embedded in the unit interval. The transport properties of the fractal mesh are studied, in particular, subdiffusion along the backbones is obtained with the dispersion relation 〈x^{2}(t)〉≃t^{β}, where the transport exponent β<1 is determined by the fractal dimensions of both backbone and fingers. Superdiffusion with β>1 has been observed as well when the environment is controlled by means of a memory kernel.

First gap statistics of long random walks with bounded jumps

We obtain exact analytical results for the joint statistics of the gap and time interval between the first two maxima of long, one-dimensional, random walks (RWs) with bounded jumps. Both discrete and continuous time settings are considered. For discrete time RWs, we find that the joint distribution exhibits a concentration effect in the sense that a gap close to its maximum possible value is much more likely to be achieved by a single jump (i.e. by realizations with adjacent first two maxima) rather than by a long walk between the first two maxima. We show that a similar, albeit slightly different, concentration phenomenon also occurs for continuous time random walks (CTRWs). Our numerical simulations confirm this concentration effect.

Numerical treatment of the spatio-temporal electromagnetic beam-wave packet

Abstract Propagation of a two-dimensional spatio-temporal electromagnetic beam wave is analysed. In parabolic (paraxial) approximation the exact analytical results for a spatio-temporal Gaussian impulse can be obtained. For solution of the full wave equation the numerical simulation has to be used. The various facets of this simulation are discussed here.

Interaction-induced two-photon edge states in an extended Hubbard model realized in a cavity array

We study theoretically two-photon states in a periodic array of coupled cavities with both on-site and nonlocal Kerr-type nonlinearities. In the absence of nonlinearity the structure is topologically trivial and possesses no edge states. The interplay of two nonlinear interaction mechanisms described by the extended Hubbard model facilitates formation of edge states of bound photon pairs. Numerical and exact analytical results for the two-photon wave functions are presented. Our findings thus shed light onto the edge states of composite particles and their localization properties.

A Paradigm-Shifting Methodology for Sensitivity Analysis of Critical Multiplying Nuclear Systems

Dispensing with the traditional approach to solving the equations modeling multiplying critical nuclear systems as an eigenvalue system, this work proposes a new and comprehensive mathematical framework (C-Framework) that eliminates the need for solving eigenvalue problems when computing the forward and adjoint neutron flux distributions in critical reactors. Consequently, the C-Framework enables the mathematical and computational analysis of critical and noncritical multiplying systems, with or without external sources, in a unified manner. By eliminating the need for solving eigenvalue problems, the C-Framework also enables the use of more efficient numerical methods (than currently used) for computing the forward and adjoint neutron flux distributions in critical reactors. Furthermore, the C-Framework also enables the application of the Comprehensive Adjoint Sensitivity Analysis Methodology (C-ASAM) as a replacement for the so-called generalized perturbation theory (GPT). The C-ASAM is much simpler to apply than the GPT, while not only yielding all of the results that the GPT can deliver, but also delivering results for all of the many—and not “GPT-allowable”—nonlinear responses of interest in reactor analysis that do not satisfy the very restrictive orthogonality relations required by the GPT’s underlying generalized adjoint equation. By dispensing with the need for solving eigenvalue problems involving the inversion of singular operators, the C-ASAM is vastly more general and more efficient than the GPT. These conclusions are underscored by exact analytical results presented for paradigm illustrative problems, which include problems that are solvable using the GPT (e.g., the system’s multiplication factor, ratios of reaction rates responses), and problems that are not solvable using the GPT (e.g., absolute reaction rates, equilibrium xenon concentration responses); all of these problems are shown to be solvable exactly and most efficiently within the C-ASAM framework.