scholarly journals Integer part polynomial correlation sequences

2016 ◽  
Vol 38 (4) ◽  
pp. 1525-1542 ◽  
Author(s):  
ANDREAS KOUTSOGIANNIS
Keyword(s):  
Error Term ◽  
Uniform Density ◽  
Integer Part ◽  
Ergodic Averages ◽  
Multiple Correlation ◽  
Transference Principle ◽  
The Mean ◽  
Correlation Sequences ◽  
Measure Preserving Actions ◽  
Invent Math

Following an approach presented by Frantzikinakis [Multiple correlation sequences and nilsequences. Invent. Math. 202(2) (2015), 875–892], we prove that any multiple correlation sequence defined by invertible measure preserving actions of commuting transformations with integer part polynomial iterates is the sum of a nilsequence and an error term, which is small in uniform density. As an intermediate result, we show that multiple ergodic averages with iterates given by the integer part of real-valued polynomials converge in the mean. Also, we show that under certain assumptions the limit is zero. A transference principle, communicated to us by M. Wierdl, plays an important role in our arguments by allowing us to deduce results for $\mathbb{Z}$-actions from results for flows.

scholarly journals Weighted multiple ergodic averages and correlation sequences

2016 ◽  
Vol 38 (1) ◽  
pp. 81-142 ◽  
Author(s):  
NIKOS FRANTZIKINAKIS ◽  
BERNARD HOST
Keyword(s):  
Bounded Sequence ◽  
Regularity Conditions ◽  
Uniform Density ◽  
Ergodic Averages ◽  
Multiple Correlation ◽  
Mean Convergence ◽  
Integer Polynomials ◽  
Several Variables ◽  
Convergence Results ◽  
Correlation Sequences

We study mean convergence results for weighted multiple ergodic averages defined by commuting transformations with iterates given by integer polynomials in several variables. Roughly speaking, we prove that a bounded sequence is a good universal weight for mean convergence of such averages if and only if the average of this sequence times any nilsequence converges. Two decomposition results of independent interest play key roles in the proof. The first states that every bounded sequence in several variables satisfying some regularity conditions is a sum of a nilsequence and a sequence that has small uniformity norm (this generalizes a result of the second author and Kra); and the second states that every multiple correlation sequence in several variables is a sum of a nilsequence and a sequence that is small in uniform density (this generalizes a result of the first author). Furthermore, we use these results in order to establish mean convergence and recurrence results for a variety of sequences of dynamical and arithmetic origin and give some combinatorial implications.

scholarly journals Nilsequences, null-sequences, and multiple correlation sequences

2013 ◽  
Vol 35 (1) ◽  
pp. 176-191 ◽  
Author(s):  
A. LEIBMAN
Keyword(s):  
Finite Measure ◽  
Null Sequence ◽  
Uniform Density ◽  
Multiple Correlation ◽  
Uniform Limit ◽  
Polynomial Sequence ◽  
Correlation Sequences

AbstractA ($d$-parameter) basic nilsequence is a sequence of the form $\psi (n)= f({a}^{n} x)$,$n\in { \mathbb{Z} }^{d} $, where $x$ is a point of a compact nilmanifold $X$, $a$ is a translation on $X$, and$f\in C(X)$; a nilsequence is a uniform limit of basic nilsequences. If $X= G/ \Gamma $ is a compact nilmanifold, $Y$ is a subnilmanifold of $X$, $\mathop{(g(n))}\nolimits_{n\in { \mathbb{Z} }^{d} } $ is a polynomial sequence in $G$, and $f\in C(X)$, we show that the sequence $\phi (n)= \int \nolimits \nolimits_{g(n)Y} f$ is the sum of a basic nilsequence and a sequence that converges to zero in uniform density (a null-sequence). We also show that an integral of a family of nilsequences is a nilsequence plus a null-sequence. We deduce that for any invertible finite measure preserving system $(W, \mathcal{B} , \mu , T)$, polynomials ${p}_{1} , \ldots , {p}_{k} : { \mathbb{Z} }^{d} \longrightarrow \mathbb{Z} $, and sets ${A}_{1} , \ldots , {A}_{k} \in \mathcal{B} $, the sequence $\phi (n)= \mu ({T}^{{p}_{1} (n)} {A}_{1} \cap \cdots \cap {T}^{{p}_{k} (n)} {A}_{k} )$, $n\in { \mathbb{Z} }^{d} $, is the sum of a nilsequence and a null-sequence.

ON A NEW CIRCLE PROBLEM

2016 ◽  
Vol 103 (2) ◽  
pp. 231-249
Author(s):  
JUN FURUYA ◽  
MAKOTO MINAMIDE ◽  
YOSHIO TANIGAWA
Keyword(s):  
Asymptotic Formula ◽  
Error Term ◽  
Dirichlet Character ◽  
Arithmetical Function ◽  
Circle Problem ◽  
The Riemann Zeta Function ◽  
Primitive Dirichlet Character ◽  
Riemann Zeta ◽  
The Mean ◽  
Voronoi Formula

We attempt to discuss a new circle problem. Let $\unicode[STIX]{x1D701}(s)$ denote the Riemann zeta-function $\sum _{n=1}^{\infty }n^{-s}$ ($\text{Re}\,s>1$) and $L(s,\unicode[STIX]{x1D712}_{4})$ the Dirichlet $L$-function $\sum _{n=1}^{\infty }\unicode[STIX]{x1D712}_{4}(n)n^{-s}$ ($\text{Re}\,s>1$) with the primitive Dirichlet character mod 4. We shall define an arithmetical function $R_{(1,1)}(n)$ by the coefficient of the Dirichlet series $\unicode[STIX]{x1D701}^{\prime }(s)L^{\prime }(s,\unicode[STIX]{x1D712}_{4})=\sum _{n=1}^{\infty }R_{(1,1)}(n)n^{-s}$$(\text{Re}\,s>1)$. This is an analogue of $r(n)/4=\sum _{d|n}\unicode[STIX]{x1D712}_{4}(d)$. In the circle problem, there are many researches of estimations and related topics on the error term in the asymptotic formula for $\sum _{n\leq x}r(n)$. As a new problem, we deduce a ‘truncated Voronoï formula’ for the error term in the asymptotic formula for $\sum _{n\leq x}R_{(1,1)}(n)$. As a direct application, we show the mean square for the error term in our new problem.

scholarly journals The asymptotic behaviour of the mean-square of fractional part sums

2000 ◽  
Vol 43 (2) ◽  
pp. 309-323 ◽  
Author(s):  
Manfred Kühleitner ◽  
Werner Georg Nowak
Keyword(s):  
Asymptotic Formula ◽  
Asymptotic Behaviour ◽  
Error Term ◽  
Fractional Part ◽  
Mean Square ◽  
The Mean

AbstractIn this article we consider sums S(t) = Σnψ (tf(n/t)), where ψ denotes, essentially, the fractional part minus ½ f is a C4-function with f″ non-vanishing, and summation is extended over an interval of order t. For the mean-square of S(t), an asymptotic formula is established. If f is algebraic this can be sharpened by the indication of an error term.

I. Dynamical considerations - Dynamics of the Moon

1967 ◽  
Vol 296 (1446) ◽  
pp. 245-247 ◽  
Keyword(s):  
Atmospheric Pressure ◽  
Uniform Density ◽  
The Moon ◽  
Homogeneous Body ◽  
Cubic Form ◽  
The Earth ◽  
Change Of State ◽  
Rate Of Increase ◽  
The Mean ◽  
Single Material

We know the mass of the Moon very well from the amount it pulls the Earth about in the course of a month; this is measured by the resulting apparent displacements of an asteroid when it is near us. Combining this with the radius shows that the mean density is close to 3.33 g/cm 3 . The velocities of earthquake waves at depths of 30 km or so are too high for common surface rocks but agree with dunite, a rock composed mainly of olivine (Mg, Fe II ) 2 SiO 4 . This has a density of about 3.27 at ordinary pressures. The veloci­ties increase with depth, the rate of increase being apparently a maximum at depth about 0.055 R in Europe and 0.075 R in Japan. It appeared at one time that there was a discontinuity in the velocities at that depth, corresponding to a transition of olivine from a rhombic to a cubic form under pressure. It now seems that the transition, though rapid, is continuous, presumably owing to impurities, but the main point is that the facts are explained by a change of state, and that the pressure at the relevant depth is reached nowhere in the Moon, on account of its smaller size. There will, however, be some compression, and we can work out how much it would be if the Moon is made of a single material. It turns out that the actual mean density of the Moon would be matched if the density at atmospheric pressure is 3.27—just agreeing with the specimen of dunite originally used for comparison. The density at the centre would be 3.41. Thus for most purposes the Moon can be treated as of uniform density. With a few small corrections the ratio 3 C /2 Ma 2 would be 0.5956 ± 0.0010, as against 0.6 for a homogeneous body. To make it appreciably less would require a much greater thickness of lighter surface rocks than in the Earth.

scholarly journals Ekman Layer Rectification

2006 ◽  
Vol 36 (8) ◽  
pp. 1646-1659 ◽  
Author(s):  
James C. McWilliams ◽  
Edward Huckle
Keyword(s):  
Wind Speed ◽  
Surface Wind ◽  
Ekman Layer ◽  
Surface Velocity ◽  
Mixing Efficiency ◽  
Uniform Density ◽  
Surface Boundary ◽  
Very High Frequency ◽  
Rectification Effect ◽  
The Mean

Abstract The phenomenon of oceanic Ekman layer rectification refers to how the time-mean, Ekman layer velocity profile with depth differs as a consequence of variability in the surface wind in addition to the time-mean wind. This study investigates rectification using the K-Profile Parameterization (KPP) model for the turbulent surface boundary layer under simple conditions of uniform density and no surface buoyancy flux or surface wave influences. The rectification magnitude is found to be significant under typical conditions. Its primary effects are to extend the depth profile deeper into the interior, reduce the mean shear, increase the effective eddy viscosity due to turbulent momentum mixing, and rotate slightly the surface velocity farther away from the mean wind direction. These effects are partly due to the increase in mean stress because of its quadratic dependence on wind speed but also are due to the nonlinearity of the turbulent mixing efficiency. The strongest influence on the rectification magnitude is the ratio of transient wind amplitude to mean wind speed. It is found that an accurate estimate of the mean current usually can be obtained by using a quasi-stationary approximation that is a weighted integral of the steady Ekman layer response over the probability density function for the wind, independent of the detailed wind history. Rectification occurs even for very high frequency wind fluctuations, though the accuracy of the quasi-steady approximation degrades in this limit (as does the validity of the KPP model). This theory is extended to include the effects of the horizontal component of the Coriolis frequency, f y. Based on published computational turbulence solutions, a simple parameterization is proposed that amplifies the turbulent eddy diffusivity in KPP by a factor that decreases with latitude and depends on the wind orientation. The effect of f y ≠ 0 is to increase both the shear and the surface speed in the time-mean Ekman current for winds directed to the northeast and decrease both quantities for winds to the southwest, with weaker influences on these properties for the orthogonal directions of southeast and northwest. Furthermore, with transient winds there is significant coupling between f y ≠ 0 and the rectification effect; for example, the mean surface current direction, relative to the mean wind, is significantly changed for these orthogonal directions.

MMPI Characteristics of Clergymen in Counseling Training and Their Relationship to Supervisor's and Peers' Ratings of Counseling Effectiveness

1973 ◽  
Vol 33 (3) ◽  
pp. 695-698 ◽  
Author(s):  
David G. Jansen ◽  
Edward C. Bonk ◽  
Frank J. Garvey
Keyword(s):  
State Hospital ◽  
Multiple Correlation ◽  
Counseling Training ◽  
Clinical Scales ◽  
Validity Scale ◽  
The Mean

Normative MMPI data for 85 clergymen entering counseling training at a state hospital were computed. Basically, the mean MMPI scores of Ss were similar to those reported previously for male marriage counselors. The correlation between supervisor's and peers' ratings of counseling effectiveness was .64. Three MMPI clinical scales showed negative correlations of more than .30 with supervisor's ratings, whereas six clinical scales and one validity scale correlated –.30 or more with peers' ratings of effectiveness. Two two-scale combinations showed a negative multiple correlation of .50 or more with supervisor's ratings of effectiveness, while six such combinations correlated –.50 or more with peers' ratings of effectiveness.

scholarly journals دراسة مقارنة لمعايير المعلومات لتحديد رتبة نماذج الانحدار الذاتي

2012 ◽  
Vol 18 (65) ◽  
pp. 323
Author(s):  
جنان عباس ناصر
Keyword(s):  
Autoregressive Model ◽  
Error Term ◽  
Information Criterion ◽  
Information Criteria ◽  
Autoregressive Models ◽  
Sample Sizes ◽  
Data Generating Process ◽  
The Mean ◽  
Normally Distributed

In this study, we compare between the traditional Information Criteria (AIC, SIC, HQ, FPE) with The Modified Divergence Information Criterion (MDIC) which used to determine the order of Autoregressive model (AR) for the data generating process, by using the simulation by generating data from several of Autoregressive models, when the error term is normally distributed with different values for its parameters (the mean and the variance),and for different sample  sizes.

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