scholarly journals Remarks about inhomogeneous pair correlations

2021 ◽  
pp. 1-18
Author(s):  
FELIPE A. RAMÍREZ
Keyword(s):  
Pair Correlation ◽  
Infinite Subset ◽  
Pair Correlations ◽  
Carry Over

Abstract Given an infinite subset $\mathcal{A} \subseteq\mathbb{N}$ , let A denote its smallest N elements. There is a rich and growing literature on the question of whether for typical $\alpha\in[0,1]$ , the pair correlations of the set $\alpha A (\textrm{mod}\ 1)\subset [0,1]$ are asymptotically Poissonian as N increases. We define an inhomogeneous generalisation of the concept of pair correlation, and we consider the corresponding doubly metric question. Many of the results from the usual setting carry over to this new setting. Moreover, the double metricity allows us to establish some new results whose singly metric analogues are missing from the literature.

Correlation Funktions of a System with Different Temperatures of the Particle Components

1964 ◽  
Vol 19 (13) ◽  
pp. 1447-1451 ◽  
Author(s):  
G. Ecker ◽  
W. Kröll
Keyword(s):  
Correlation Function ◽  
Electron Correlation ◽  
Pair Correlation ◽  
Bbgky Hierarchy ◽  
Ion Pair ◽  
Pair Correlations ◽  
The Poisson Equation ◽  
Average Potential ◽  
Different Temperatures ◽  
Dressed Particles

We consider a plasma consisting of particle components with different temperatures. The components are uniformly distributed in the configuration space and MAXWELLIAN in the velocity space. Pair correlations are assumed to be small and higher order correlations negligible. It is shown from the BBGKY-hierarchy that the influence of the electrons on the ion kinetics can be taken into account by treating the ions as dressed particles. The hierarchy for these dressed particles provides the ion-ion correlation function. The electron-ion pair correlation is calculated from the POISSON equation using the ion-ion correlation and relating the electron-ion pair distribution to the average potential. By the same procedure we derive the electron-electron correlation making use of the electron-ion correlation. The results are compared with those of other authors.

Genetic and Environmental Causes of Variation in the Difference Between Biological Age Based on DNA Methylation and Chronological Age for Middle-Aged Women

2015 ◽  
Vol 18 (6) ◽  
pp. 720-726 ◽  
Author(s):  
Shuai Li ◽  
Ee Ming Wong ◽  
JiHoon E. Joo ◽  
Chol-Hee Jung ◽  
Jessica Chung ◽  
...  
Keyword(s):  
Dna Methylation ◽  
Genetic Factors ◽  
Pair Correlation ◽  
Weighted Average ◽  
Biological Age ◽  
Sibling Pair ◽  
Chronological Age ◽  
Familial Correlation ◽  
Middle Aged ◽  
Pair Correlations

The disease- and mortality-related difference between biological age based on DNA methylation and chronological age (Δage) has been found to have approximately 40% heritability by assuming that the familial correlation is only explained by additive genetic factors. We calculated two different Δage measures for 132 middle-aged female twin pairs (66 monozygotic and 66 dizygotic twin pairs) and their 215 sisters using DNA methylation data measured by the Infinium HumanMethylation450 BeadChip arrays. For each Δage measure, and their combined measure, we estimated the familial correlation for MZ, DZ and sibling pairs using the multivariate normal model for pedigree analysis. We also pooled our estimates with those from a former study to estimate weighted average correlations. For both Δage measures, there was familial correlation that varied across different types of relatives. No evidence of a difference was found between the MZ and DZ pair correlations, or between the DZ and sibling pair correlations. The only difference was between the MZ and sibling pair correlations (p < .01), and there was marginal evidence that the MZ pair correlation was greater than twice the sibling pair correlation (p < .08). For weighted average correlation, there was evidence that the MZ pair correlation was greater than the DZ pair correlation (p < .03), and marginally greater than twice the sibling pair correlation (p < .08). The varied familial correlation of Δage is not explained by additive genetic factors alone, implying the existence of shared non-genetic factors explaining variation in Δage for middle-aged women.

Lattice statistics in a magnetic field. II. Order and correlations of a two-dimensional super-exchange antiferromagnet

1960 ◽  
Vol 256 (1287) ◽  
pp. 502-513 ◽  
Keyword(s):  
Magnetic Field ◽  
Long Range ◽  
Pair Correlation ◽  
Configurational Entropy ◽  
Square Lattice ◽  
Lattice Statistics ◽  
Zero Field ◽  
Two Dimensional ◽  
Ising Lattice ◽  
Pair Correlations

The long-range order and pair correlation functions of a two-dimensional super-exchange antiferromagnet in an arbitrary magnetic field are derived rigorously from properties of the standard square Ising lattice in zero field. (The model investigated was described in part I: it is a decorated square lattice with magnetic spins on the bonds coupled antiferromagnetically via non-magnetic spins on the vertices.) The behaviour near the transition temperature in a finite field is similar to that of the normal plane lattice, i. e. the long-range orders or spontaneous magnetizations of the sublattices vanish as ( T t – T ) ⅛ and the pair correlations behave as ω c + W ( T – T t ) ln | T – T t |. The configurational entropy is discussed and the anomalous entropy in the critical field at zero temperature is calculated exactly.

scholarly journals Pair Correlations and Random Walks on the Integers

2016 ◽  
Vol 11 (1) ◽  
pp. 159-164
Author(s):  
Radhakrishnan Nair ◽  
Entesar Nasr
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Real Number ◽  
Pair Correlation ◽  
Real Numbers ◽  
Pair Correlations ◽  
Nonzero Real Number ◽  
Correlation Estimate

AbstractThe paper gives conditions for a sequence of fractional parts of real numbers $\left( {\{ a_n x\} } \right)_{n = 1}^\infty $ to satisfy a pair correlation estimate. Here x is a fixed nonzero real number and $\left( {a_n } \right)_{n = 1}^\infty $ is a random walk on the integers.

Pair correlation function in a fluid with density inhomogeneities: results of the Percus‒Yevick and hypernetted chain approximations for hard spheres near a hard wall

1986 ◽  
Vol 404 (1827) ◽  
pp. 323-337 ◽  
Keyword(s):  
Correlation Function ◽  
Pair Correlation Function ◽  
Hard Spheres ◽  
Pair Correlation ◽  
Hard Wall ◽  
Bulk Fluid ◽  
Pair Correlations ◽  
Hypernetted Chain ◽  
Density Inhomogeneities ◽  
Bulk Case

Solutions for the pair correlation function and density profile of a system of hard spheres near a hard wall are obtained by using the Percus‒Yevick and hypernetted chain approximations, generalized for inhomogeneous fluids. The Percus‒Yevick (PY) results are similar in accuracy to those obtained for the bulk fluid. The PY pair correlation function is generally too small near contact but quite good overall. The hypernetted chain (h. n. c.) results are difficult to obtain numerically and are poorer than in the bulk. Often the h. n. c. pair correlations are too small at contact, in contrast to the bulk case where they are too large, although there are configurations where the contact values of the pair correlation function are too large. Nearly always the error in the h. n. c. results is much worse than is the case for the bulk. Both approximations are qualitatively satisfactory in that they predict the correct asymmetries between the values of the pair correlation functions for pairs of hard spheres whose line of centres is parallel or normal to the surface of the wall.

Pair correlations of the leVeque sequence on the polydisc

2008 ◽  
Vol 145 (1) ◽  
pp. 197-203
Author(s):  
R. NAIR
Keyword(s):  
Correlation Function ◽  
Pair Correlation Function ◽  
Resonance Condition ◽  
Pair Correlation ◽  
Natural Numbers ◽  
Pair Correlations ◽  
The Real ◽  
Image Position

AbstractWe consider a system of “forms” defined for ẕ = (zij) on a subset of $\Bbb C^d$ by where d = d1 + ⋅ ⋅ ⋅ + dl and for each pair of integers (i,j) with 1 ≤ i ≤ l, 1 ≤ j ≤ di we denote by $(v_{ij}(k))_{k=1}^{\infty}$ a strictly increasing sequence of natural numbers. Let ${\Bbb C}_1$ = {z ∈ ${\Bbb C}$ : |z| < 1} and let ${\underline X} \ = \ \times _{i=1}^l \times _{j=1}^{d_i}X_{ij}$ where for each pair (i, j) we have Xij = ${\Bbb C}\backslash {\Bbb C}_1$. We study the distribution of the sequence on the l-polydisc $({\Bbb C}_1)^l$ defined by the coordinatewise polar fractional parts of the sequence Xk(ẕ) = (L1(ẕ)(k),. . ., Ll(ẕ)(k)) for typical ẕ in ${\underline X}$ More precisely for arcs I1, . . ., I2l in $\Bbb T$, let B = I1 × ⋅ ⋅ ⋅ × I2l be a box in $\Bbb T^{2l}$ and for each N ≥ 1 define a pair correlation function by and a discrepancy by ΔN = $\sup_{B \subset \Bbb T^{2l}}${VN(B) − N(N−1)leb(B)}, where the supremum is over all boxes in $\Bbb T^{2l}$. We show, subject to a non-resonance condition on $(v_{ij}(k))_{k=1}^{\infty}$, that given ε > 0 we have ΔN = o(N$(log N)^{l + {1\over 2}}$(log log N)1+ε) for almost every $\underline x(\underline z)\in \Bbb T^{2l}$. Similar results on extremal discrepancy are also proved. Our results complement those of I. Berkes, W. Philipp, M. Pollicott, Z. Rudnick, P. Sarnak, R Tichy and the author in the real setting.

Collision-induced and multiple light scattering by simple fluids

1979 ◽  
Vol 293 (1402) ◽  
pp. 359-375 ◽  
Keyword(s):  
Light Scattering ◽  
Incident Light ◽  
Pair Correlation ◽  
Point Dipole ◽  
Scattered Intensity ◽  
Pair Correlations ◽  
Atomic Fluids ◽  
The Many ◽  
Correlation Properties ◽  
Multiple Light Scattering

Most discussions of light scattering by simple (e.g. classical, atomic) fluids have treated only ‘first order’ processes, i.e. those where the incident light is scattered only once and the atomic polarizabilities are undistorted by interaction. Correspondingly the scattered intensity is related by Fourier transform to the time- and space- pair correlations. In this paper we describe instead the ‘second order’ processes of collision-induced scattering (c.i.s.), in which the incident light is scattered only once but the relevant polarizability is that of an interacting cluster of atoms, and multiple light scattering (m.l.s.), in which only undistorted polarizabilities are involved but the incident light is scattered more than once. In both cases the scattered intensity is determined by correlations involving more than two particles. In addition, the c.i.s. experiments provide information about the many-atom polarization while the m.l.s. studies offer new probes of large fluctuations in critical and nucleating fluids. We discuss in particular theoretical and experimental c.i.s. investigations of the two-body polarizability anisotropy induced by collision; it is concluded that the nature and origin of non-point-dipole behaviour has yet to be satisfactorily explained. Similarly, we consider how various depolarization m.l.s. studies suggest improved analyses of pair correlation properties in classical systems.

scholarly journals Alteration in Gene Pair Correlations in Tryptophan Metabolism as a Hallmark in Cancer Diagnosis

2020 ◽  
Vol 13 ◽  
pp. 117864692097701
Author(s):  
Meena Kishore Sakharkar ◽  
Sarinder Kaur Dhillon ◽  
Karthic Rajamanickam ◽  
Benjamin Heng ◽  
Nady Braidy ◽  
...  
Keyword(s):  
Metabolic Pathway ◽  
Gene Pair ◽  
Pair Correlation ◽  
Correlation Coefficients ◽  
Aldehyde Oxidase ◽  
Tryptophan Metabolism ◽  
Cancer Development ◽  
Strongly Correlated ◽  
Pair Correlations ◽  
Pairwise Correlation

Tryptophan metabolism plays essential roles in both immunomodulation and cancer development. Indoleamine 2,3-dioxygenase, a rate-limiting enzyme in the metabolic pathway, is overexpressed in different types of cancer. To get a better understanding of the involvement of tryptophan metabolism in cancer development, we evaluated the expression and pairwise correlation of 62 genes in the metabolic pathway across 12 types of cancer. Only gene AOX1, encoding aldehyde oxidase 1, was ubiquitously downregulated, Furthermore, we observed that the 62 genes were widely and strongly correlated in normal controls, however, the gene pair correlations were significantly lost in tumor patients for all 12 types of cancer. This implicated that gene pair correlation coefficients of the tryptophan metabolic pathway could be applied as a prognostic and/or diagnostic biomarker for cancer.

scholarly journals A pair correlation problem, and counting lattice points with the zeta function

2021 ◽  
Author(s):  
Christoph Aistleitner ◽  
Daniel El-Baz ◽  
Marc Munsch
Keyword(s):  
Real Number ◽  
Pair Correlation ◽  
Unit Interval ◽  
Lattice Points ◽  
Pair Correlations ◽  
Integer Sequence ◽  
Correlation Problem ◽  
Random Behavior ◽  
Additive Energy ◽  
Almost All

AbstractThe pair correlation is a localized statistic for sequences in the unit interval. Pseudo-random behavior with respect to this statistic is called Poissonian behavior. The metric theory of pair correlations of sequences of the form $$(a_n \alpha )_{n \ge 1}$$ ( a n α ) n ≥ 1 has been pioneered by Rudnick, Sarnak and Zaharescu. Here $$\alpha $$ α is a real parameter, and $$(a_n)_{n \ge 1}$$ ( a n ) n ≥ 1 is an integer sequence, often of arithmetic origin. Recently, a general framework was developed which gives criteria for Poissonian pair correlation of such sequences for almost every real number $$\alpha $$ α , in terms of the additive energy of the integer sequence $$(a_n)_{n \ge 1}$$ ( a n ) n ≥ 1 . In the present paper we develop a similar framework for the case when $$(a_n)_{n \ge 1}$$ ( a n ) n ≥ 1 is a sequence of reals rather than integers, thereby pursuing a line of research which was recently initiated by Rudnick and Technau. As an application of our method, we prove that for every real number $$\theta >1$$ θ > 1 , the sequence $$(n^\theta \alpha )_{n \ge 1}$$ ( n θ α ) n ≥ 1 has Poissonian pair correlation for almost all $$\alpha \in {\mathbb {R}}$$ α ∈ R .

Angular pair correlations of observed and simulated galaxy distributions

1990 ◽  
Vol 68 (9) ◽  
pp. 827-830
Author(s):  
G. Wiedenmann ◽  
H. Atmanspacher ◽  
H. Scheingraber
Keyword(s):  
Correlation Function ◽  
Large Scale ◽  
Pair Correlation ◽  
Quantitative Information ◽  
Small Scale ◽  
Main Body ◽  
Pair Correlations ◽  
Large Scale Structures ◽  
Galaxy Distribution ◽  
The Galaxy

The main body of quantitative information about galaxy statistics is obtained from correlation studies. It has recently turned out that a modified correlation formalism can provide details about large-scale structure in the galaxy distribution, which are obscured by artefacts of the conventional correlation function. The modified pair correlation function, as applied to the Zwicky catalogue of galaxies, shows two distinct power-law regimes at small scales (< 1°) and large scales (around 10°). Based on the comparison of simulated bubblelike large-scale structures with the Zwicky sample, these regimes are interpreted to correspond to the distribution of galaxies within the shells of the bubbles (small scale), and the distribution of the bubbles themselves (large scale).

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