Distribution Of Degeneration Research Articles

A note on degenerate derangement polynomials and numbers

In this paper, we study the degenerate derangement polynomials and numbers, investigate some properties of those polynomials and numbers and explore their connections with the degenerate gamma distributions. In more detail, we derive their explicit expressions, recurrence relations and some identities involving the degenerate derangement polynomials and numbers and other special polynomials and numbers, which include the fully degenerate Bell polynomials, the degenerate Fubini polynomials and the degenerate Stirling numbers of both kinds. We also show that those polynomials and numbers are connected with the moments of some variants of the degenerate gamma distributions.

Commentary: Clinical profile and demographic distribution of Terrien's marginal degeneration in a multitier ophthalmology network in India.

Commentary: Clinical profile and demographic distribution of Terrien's marginal degeneration in a multitier ophthalmology network in India.

Subset simulation for problems with strongly non-Gaussian, highly anisotropic, and degenerate distributions

Results from subset simulation often have significant variability that can be attributed to sample fluctuation and correlation among the conditional samples. In extreme cases, such as when conditional distributions are highly anisotropic or degenerate, sample correlation can cause conditional sampling to break down, resulting in failed subset simulations. To address the extreme cases where subset simulation breaks down, we propose to use an affine invariant ensemble MCMC sampler for conditional sampling. Unlike traditional MCMC algorithms used in subset simulation that use a single proposal density per subset or adapts the proposal in a heuristic manner, the proposed scheme automatically varies the step size with each move. The algorithm is particularly effective for estimating failure probabilities when the conditional probability density is strongly non-Gaussian and degenerates to possess a lower effective dimension. Two added benefits are that it allows subset simulation to be performed directly with non-Gaussian, highly dependent, or implicitly defined random variables and the method has only a single parameter. Therefore it is not sensitive to the many parameters that must be calibrated for the proposal density in conventional algorithms. Several examples are considered, each illustrating the benefit of the proposed methodology for different classes of problems.

Neuropsychological, clinico‐pathologic, neuroimaging, and biomarker profiles of the MGH FTD Unit posterior cortical atrophy (PCA) cohort

AbstractBackgroundObjective: To present comprehensive data on the MGH FTD Unit PCA cohort. PCA is a neurodegenerative clinical syndrome usually arising from Alzheimer’s Disease (AD) neuropathologic changes (ADNC). While visuoperceptual and spatial problems are typical early presenting symptoms, patients also report problems with spelling, calculation, and word retrieval. Due to significant heterogeneity in the syndrome, further classification of PCA by subtypes based on predominant cognitive impairments (dorsal, ventral, caudal, dominant parietal) has also been proposed. We launched our multidisciplinary MGH PCA program in 2008 to address the special care needs of this population, and to advance our scientific understanding of PCA and its treatment.Methods39 participants were recruited for research participation based on clinical diagnostic criteria consistent with PCA, including a dominant hemisphere variant (i.e., progressive Gerstmann‐like syndrome minimal visuospatial disturbance). Participants underwent structured review of symptoms and history; systematic neurological exam; comprehensive neuropsychological and speech‐language assessments; multimodal imaging (MRI and/or functional MRI and FDG, amyloid, and tau PET); and CSF analysis. Some patients have come to autopsy. We used the Neuropsychological Assessment Rating (NAR) scale to quantify performance in cognitive domains pertinent to PCA and for subtype classification.ResultsOf seven individuals for whom we have autopsy‐confirmed neuropathology, five had ADNC; one had lobar atrophy and tau pathology in a cortico‐basal degeneration (CBD) distribution; and one had Frontotemporal Lobar Degeneration TDP43 Type A (GRN mutation). 22 patients had biomarkers consistent with ADNC (17 with elevated PiB PET imaging signal, 5 with AD‐like CSF ). Mean NAR domain scores (impairment level ratings: 0=Normal, .5=questionable/very mild, 1=mild, 2= moderate, 3=severe) revealed relatively normal Attention/Processing Speed (0.2); mildly impaired Executive Function (0.8) and Memory (1.1); questionably/very mildly impaired Language (0.5); and moderately impaired Visuospatial Function (2).ConclusionThe MGH PCA program has a robust cohort of patients with a comprehensive data set that includes standardized characterization of the PCA neurocognitive and neurologic profiles, high‐resolution MRI (structural, functional), multimodal PET imaging, CSF biomarkers, and in some cases neuropathology. This data set will allow for detailed examination and improved understanding of this atypical neurodegenerative syndrome. Additional analyses are ongoing, including tau imaging profiles.

Long time dynamics for the Landau-Fermi-Dirac equation with hard potentials

In this document we discuss the long time behaviour for the homogeneous Landau-Fermi-Dirac equation in the hard potential case. Uniform in time estimates for statistical moments and Sobolev regularity are presented and used to prove exponential relaxation of non degenerate distributions to the Fermi-Dirac statistics. All these results are valid for rather general initial datum. An important feature of the estimates is the independence with respect to the quantum parameter. Consequently, in the classical limit the same estimates are recovered for the Landau equation.

Geographic distributions of age-related macular degeneration incidence: a systematic review and meta-analysis

PurposeWe performed a systematic review and meta-analysis to summarise the geographic distribution of age-related macular degeneration (AMD) incidence.MethodsDatabases including PubMed, Embase and Web of Science were searched for publications of...

Analysis of Estimation Error of Skewness and Kurtosis of Bunimovich-Rice Processes with Exponentially Power Waveform of Pulses

Mathematical expectations and variances of estimates of skewness and kurtosis coefficients of the noise signal model have been derived in this study. The specified noise signals represent the Bunimovich–Rice processes with exponentially power waveform of pulses expressed in terms of cumulant coefficients of the specified processes. It is shown that the distribution of instantaneous values of Bunimovich–Rice processes is quite different from the Gaussian distribution. The root-mean-square and relative estimation errors of skewness and kurtosis coefficients depending on of the time constant and waveform parameter of elementary pulses, as well as distributions of pulse amplitudes (degenerate and gamma distributions) and their intensities are analyzed. Expressions for finding the minimal sample volumes are obtained that ensure the specified values of relative errors in estimating the skewness and kurtosis coefficients of Bunimovich–Rice processes. The minimal sample volumes depending on parameters of these processes have been determined that ensure the relative estimation errors, which do not exceed 1%.

Regularized Transport Between Singular Covariance Matrices

We consider the problem of steering a linear stochastic system between two endpoint <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">degenerate</i> Gaussian distributions in finite time. This accounts for those situations in which some but not all of the state entries are uncertain at the initial, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$t=0$</tex-math></inline-formula> , and final time, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$t=T$</tex-math></inline-formula> . This problem entails nontrivial technical challenges, as the singularity of terminal state covariance causes the control to grow unbounded at the final time <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$T$</tex-math></inline-formula> . Consequently, the entropic interpolation ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Schrödinger bridge</i> ) is provided by a diffusion process, which is not <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">finite energy</i> , thereby placing this case outside of most of the current theory. In this article, we show that a feasible interpolation can be derived as a limiting case of earlier results for nondegenerate cases, and that it can be expressed in closed form. Moreover, we show that such interpolation belongs to the same <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">reciprocal class</i> of the uncontrolled evolution. By doing so, we also highlight a time symmetry of the problem, contrasting dual formulations in the forward and reverse time directions, where in each, the control grows unbounded as time approaches the endpoint (in the forward and reverse time direction, respectively).

A NOTE ON THE ASYMPTOTIC BEHAVIOR OF THE HEIGHT FOR A BIRTH-AND-DEATH PROCESS

Recently, the asymptotic mean value of the height for a birth-and-death process is given in Videla [Videla, L.A. (2020)]. We consider the asymptotic variance of the height in the case when the number of states tends to infinity. Further, we prove that the heights exhibit a cutoff phenomenon and that the normalized height converges to a degenerate distribution.

Analysis of the convergence of the degree distribution of contracting random networks towards a Poisson distribution using the relative entropy.

We present analytical results for the structural evolution of random networks undergoing contraction processes via generic node deletion scenarios, namely, random deletion, preferential deletion, and propagating deletion. Focusing on configuration model networks, which exhibit a given degree distribution P_{0}(k) and no correlations, we show using a rigorous argument that upon contraction the degree distributions of these networks converge towards a Poisson distribution. To this end, we use the relative entropy S_{t}=S[P_{t}(k)||π(k|〈K〉_{t})] of the degree distribution P_{t}(k) of the contracting network at time t with respect to the corresponding Poisson distribution π(k|〈K〉_{t}) with the same mean degree 〈K〉_{t} as a distance measure between P_{t}(k) and Poisson. The relative entropy is suitable as a distance measure since it satisfies S_{t}≥0 for any degree distribution P_{t}(k), while equality is obtained only for P_{t}(k)=π(k|〈K〉_{t}). We derive an equation for the time derivative dS_{t}/dt during network contraction and show that the relative entropy decreases monotonically to zero during the contraction process. We thus conclude that the degree distributions of contracting configuration model networks converge towards a Poisson distribution. Since the contracting networks remain uncorrelated, this means that their structures converge towards an Erdős-Rényi (ER) graph structure, substantiating earlier results obtained using direct integration of the master equation and computer simulations [Tishby etal., Phys. Rev. E 100, 032314 (2019)2470-004510.1103/PhysRevE.100.032314]. We demonstrate the convergence for configuration model networks with degenerate degree distributions (random regular graphs), exponential degree distributions, and power-law degree distributions (scale-free networks).

Penalized multiple distribution selection method for imbalanced data classification

In reality, the amount of data from different categories varies significantly, which results in learning bias towards prominent classes, hindering the overall classification performance. In this paper, by proving that traditional classification methods that use single softmax distribution are limited for modeling complex and imbalanced data, we propose a general Multiple Distribution Selection (MDS) method for imbalanced data classification. MDS employs a mixture distribution that is composed of a single softmax distribution and a set of degenerate distributions to model imbalanced data. Furthermore, a dynamic distribution selection method, based on L1 regularization, is also proposed to automatically determine the weights of distributions. In addition, the corresponding two-stage optimization algorithm is designed to estimate the parameters of models. Extensive experiments conducted on three widely used benchmark datasets (IMDB, ACE2005, 20NewsGroups) show that our proposed mixture method outperforms previous methods. Moreover, under highly imbalanced setting, our method achieves up to a 4.1 absolute F1 gain over high-performing baselines.

Higher order weighted Sobolev spaces on the real line for strongly degenerate weights. Application to variational problems in elasticity of beams

For one-dimensional interval and integrable weight function w we define via completion a weighted Sobolev space Hμwm,p of arbitrary integer order m. The weights in consideration may suffer strong degeneration so that, in general, functions u from Hμwm,p do not have weak derivatives. This contribution is focussed on studying the continuity properties of functions u at a chosen internal point x0 to which we attribute a notion of criticality of order k and with respect to the weight w. For non-critical points x0 we formulate a local embedding result that guarantees continuity of functions u or their derivatives. Conversely, we employ duality theory to show that criticality of x0 furnishes a smooth approximation of functions in Hμwm,p admitting jump-type discontinuities at x0. The work concludes with demonstration of established results in the context of variational problem in elasticity theory of beams with degenerate width distribution.

Fecundity regulation in a spatial birth-and-death process

We study a Markov birth-and-death process on a space of locally finite configurations, which describes an ecological model with a density-dependent fecundity regulation mechanism. We establish existence and uniqueness of this process and analyze its properties. In particular, we show global time-space boundedness of the population density and, using a constructed Foster–Lyapunov-type function, we study return times to certain level sets of tempered configurations. We also find sufficient conditions that the degenerate invariant distribution is unique for the considered process.

Statistical inference for heavy tailed series with extremal independence

We consider stationary time series $\{X_j, j \in Z\} whose finite dimensional distributions are regularly varying with extremal independence. We assume that for each $h \geq 1$, conditionally on $X_0$ to exceed a threshold tending to infinity, the conditional distribution of $X_h$ suitably normalized converges weakly to a non degenerate distribution. We consider in this paper the estimation of the normalization and of the limiting distribution.

Statistical tests for latent class in censored data due to detection limit.

Measures of substance concentration in urine, serum or other biological matrices often have an assay limit of detection. When concentration levels fall below the limit, the exact measures cannot be obtained. Instead, the measures are censored as only partial information that the levels are under the limit is known. Assuming the concentration levels are from a single population with a normal distribution or follow a normal distribution after some transformation, Tobit regression models, or censored normal regression models, are the standard approach for analyzing such data. However, in practice, it is often the case that the data can exhibit more censored observations than what would be expected under the Tobit regression models. One common cause is the heterogeneity of the study population, caused by the existence of a latent group of subjects who lack the substance measured. For such subjects, the measurements will always be under the limit. If a censored normal regression model is appropriate for modeling the subjects with the substance, the whole population follows a mixture of a censored normal regression model and a degenerate distribution of the latent class. While there are some studies on such mixture models, a fundamental question about testing whether such mixture modeling is necessary, i.e. whether such a latent class exists, has not been studied yet. In this paper, three tests including Wald test, likelihood ratio test and score test are developed for testing the existence of such latent class. Simulation studies are conducted to evaluate the performance of the tests, and two real data examples are employed to illustrate the tests.

An heterogeneous-agent New-Monetarist model with an application to unemployment

A New-Monetarist model is constructed with expenditure and unemployment risks that generates equilibria with non-degenerate distribution of money holdings. Distributional effects can overturn key insights of the model with degenerate distributions, e.g., the value of money depends on the income distribution; a one-time money injection raises aggregate real balances in the short run – price adjustments look sluggish; anticipated inflation can raise output and welfare; there can be a long-run trade-off between inflation and unemployment. Distributional effects also generate a quantitatively significant aggregate demand channel through which transfers financed with money creation can raise employment, and productivity shocks are amplified.

Electrostatic Topographical Viewpoint of π-Conjugation and Aromaticity of Hydrocarbons.

A molecular electrostatic potential (MESP) topographical study has been conducted for a variety of conjugated hydrocarbons at B3LYP/6-311+G(d,p) level of theory to understand their π-conjugation features and aromaticity. The value of MESP minimum (Vm) is related to the localized and delocalized distribution of π-electron density. The Vm values are located interior to the rings in polycyclic benzenoid hydrocarbons (PBHs), whereas they lie outside the boundary of the rings in antiaromatic and in fused systems consisting of aromatic and antiaromatic moieties. The Vm points lie on top and bottom of the π-regions in linear polyenes and annulenes, while a degenerate distribution of CPs around the midpoint region of triple bonds is observed in alkynes. The eigenvalues λ1, λ2, and λ3 of the Hessian matrix at Vm(MESP minima) are used to define the aromatic character of the cyclic structures. The eigenvalues follow the trend λ1 ≫ λ2 > λ3 ≅ 0 in PBHs, λ1 > λ2 > λ3 ≅ 0 in linear polyenes, and λ1 > λ2 > λ3 ≠ 0 in antiaromatic systems. The difference in the aromatic character of PBHs from that of benzene is related to the deviations Δλ1, Δλ2, and Δλ3. The total deviation ∑i=13Δλi is found to be ≤ 0.011 au for all aromatic systems and lies between 0.011 and 0.035 au for all nonaromatic systems. For antiaromatic systems, its value is found to be above 0.035 au. Further, ∑i=13Δλi gives a direct interpretation of Clar's aromatic sextet structures for PBHs. In summary, MESP topography mapping is a powerful technique to quantify the localized and delocalized π-electron distribution in a variety of unsaturated hydrocarbon systems.

Chameleons and mimics: Progressive supranuclear palsy and corticobasal degeneration.

Progressive supranuclear palsy (PSP) and corticobasal degeneration (CBD) are neurodegenerative disorders that show parkinsonism as their main symptom. Both PSP and CBD are sporadic tauopathies associated with hyperphosphorylated four-repeat tau aggregation in neurons and glial cells. The characteristic pathologies of PSP are midbrain atrophy and the appearance of tufted astrocytes and globose-type neurofibrillary tangles. PSP shows severe degeneration in the globus pallidus, substantia nigra, subthalamic nucleus, and cerebellar dentate nuclei. Conversely, the characteristic pathologies of CBD are cortical atrophy and the appearance of astrocytic plaques and argyrophilic threads. CBD is associated with severe degeneration in the cerebral white matter, substantia nigra, and globus pallidus. Clinical symptoms depend on the topographical distribution and severity of degeneration rather than on the type of aggregated protein or inclusions. PSP and CBD present clinically differential diagnostic difficulties because of their overlapping pathological distributions.

Local and global degree profiles of randomly grown self-similar hooking networks under uniform and preferential attachment

We investigate node degrees in a network grown from a seed by hooking self-similar components under two models of randomness: a uniform attachment model and a model based on preferential attachment. We study two degree profiles: a local profile tracking the evolution of the degree of a particular node over time, and a global profile concerned about counts of the number of nodes of a particular degree.For the local profile, under uniform attachment growth, we have the exact mean, variance and probability distribution in terms of standard combinatorial numbers like generalized harmonic numbers and Stirling numbers of the first kind. Asymptotically, we observe phases: The early nodes have an asymptotically normal distribution, intermediate nodes have a Poisson distribution and the late nodes have a degenerate distribution. In contrast, under preferential attachment, the moments of the degree of a node contain Stirling numbers of the second kind and (under appropriate scaling) the profile has a gamma-type limit law.As for the global profile, we use Pólya urns to derive strong laws. Four regimes arise according to the structure of the seed. Within these regimes, we identify a few degenerate cases. Barring these degenerate cases, we uncover an asymptotically normal joint multivariate distribution for nodes of very small degrees.

Minimum risk sequential point estimation of the stress-strength reliability parameter for exponential distribution

In this article, using purely and two-stage sequential procedures, the problem of minimum risk point estimation of the reliability parameter (R) under the stress–strength model, in case the loss function is squared error plus sampling cost, is considered when the random stress (X) and the random strength (Y) are independent and both have exponential distributions with different scale parameters. The exact distribution of the total sample size and explicit formulas for the expected value and mean squared error of the maximum likelihood estimator of the reliability parameter under the stress–strength model are provided under the two-stage sequential procedure. Using the law of large numbers and Monte Carlo integration, the exact distribution of the stopping rule under the purely sequential procedure is approximated. Moreover, it is shown that both proposed sequential procedures are finite and for special cases the exact distribution of stopping times has a degenerate distribution at the initial sample size. The performances of the proposed methodologies are investigated with the help of simulations. Finally, using a real data set, the procedures are clearly illustrated.