Some Properties of Beatty Sequences II

1960 ◽  
Vol 3 (1) ◽  
pp. 17-22 ◽  
Author(s):  
Ian G. Connell
Keyword(s):  
Complementary Sequences ◽  
Beatty Sequences ◽  
Greatest Integer Function ◽  
Image Position

In a previous paper [l] we discussed a property of the complementary sequences1where square brackets denote the greatest integer function and α is any positive irrational. We called {un} and {vn} Beatty sequences of argument α.

scholarly journals A spectral refinement of the Bergelson–Host–Kra decomposition and new multiple ergodic theorems

2017 ◽  
Vol 39 (4) ◽  
pp. 1042-1070 ◽  
Author(s):  
JOEL MOREIRA ◽  
FLORIAN KARL RICHTER
Keyword(s):  
Spectral Properties ◽  
Correspondence Principle ◽  
Prime Numbers ◽  
Arithmetic Progressions ◽  
Ergodic Theorems ◽  
Ergodic Averages ◽  
Natural Conditions ◽  
Beatty Sequences ◽  
Image Position

We investigate how spectral properties of a measure-preserving system$(X,{\mathcal{B}},\unicode[STIX]{x1D707},T)$are reflected in the multiple ergodic averages arising from that system. For certain sequences$a:\mathbb{N}\rightarrow \mathbb{N}$, we provide natural conditions on the spectrum$\unicode[STIX]{x1D70E}(T)$such that, for all$f_{1},\ldots ,f_{k}\in L^{\infty }$,$$\begin{eqnarray}\lim _{N\rightarrow \infty }\frac{1}{N}\mathop{\sum }_{n=1}^{N}\mathop{\prod }_{j=1}^{k}T^{ja(n)}f_{j}=\lim _{N\rightarrow \infty }\frac{1}{N}\mathop{\sum }_{n=1}^{N}\mathop{\prod }_{j=1}^{k}T^{jn}f_{j}\end{eqnarray}$$in$L^{2}$-norm. In particular, our results apply to infinite arithmetic progressions,$a(n)=qn+r$, Beatty sequences,$a(n)=\lfloor \unicode[STIX]{x1D703}n+\unicode[STIX]{x1D6FE}\rfloor$, the sequence of squarefree numbers,$a(n)=q_{n}$, and the sequence of prime numbers,$a(n)=p_{n}$. We also obtain a new refinement of Szemerédi’s theorem via Furstenberg’s correspondence principle.

scholarly journals ALMOST AUTOMORPHIC SOLUTIONS FOR DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENT

2013 ◽  
Vol 90 (1) ◽  
pp. 99-112 ◽  
Author(s):  
LI-LI ZHANG ◽  
HONG-XU LI
Keyword(s):  
Differential Equations ◽  
Existence And Uniqueness ◽  
Existence And Uniqueness Theorem ◽  
Piecewise Constant ◽  
Exponential Dichotomies ◽  
Almost Automorphic ◽  
Piecewise Constant Argument ◽  
Greatest Integer Function ◽  
Image Position ◽  
Constant Argument

AbstractUsing the method of exponential dichotomies, we establish a new existence and uniqueness theorem for almost automorphic solutions of differential equations with piecewise constant argument of the form $$\begin{eqnarray*}{x}^{\prime } (t)= A(t)x(t)+ B(t)x(\lfloor t\rfloor )+ f(t), \quad t\in \mathbb{R} ,\end{eqnarray*}$$ where $\lfloor \cdot \rfloor $ denotes the greatest integer function, and $A(t), B(t): \mathbb{R} \rightarrow { \mathbb{R} }^{q\times q} $, $f(t): \mathbb{R} \rightarrow { \mathbb{R} }^{q} $ are all almost automorphic.

Some Properties of Beatty Sequences I*

1959 ◽  
Vol 2 (3) ◽  
pp. 190-197 ◽  
Author(s):  
Ian G. Connell
Keyword(s):  
Natural Numbers ◽  
Beatty Sequences ◽  
Image Position ◽  
Positive Integers

Two sequences of natural numbers are said to be complementary if they contain all the positive integers without repetition or omission. S. Beatty [l] observed that the sequences(1)(2)(where square brackets denote the integral part function) are complementary if and only if α > 0 and α is irrational. We call the pair (1),(2) Beatty sequences of argument α.

A Transformation Formula for a Class of Arithmetic Sums

1981 ◽  
Vol 24 (1) ◽  
pp. 73-74
Author(s):  
M. V. Subbarao ◽  
V. C. Harris
Keyword(s):  
Arithmetic Function ◽  
Transformation Formula ◽  
Greatest Integer Function ◽  
Image Position ◽  
Theory Of Numbers

Arithmetic sums of the formwhere f is an arithmetic function and [ ] is the greatest integer function frequently occur in various situations in the theory of numbers and have much of interest in their own right. Two instances appear in the well-known results.

Beyond Beatty sequences: Complementary lattices

2020 ◽  
pp. 1-13
Author(s):  
Sam Vandervelde
Keyword(s):  
Fundamental Property ◽  
Parameter Family ◽  
Two Dimensional ◽  
Geometric Proof ◽  
Complementary Sequences ◽  
Beatty Sequences

Abstract By taking square lattices as a two-dimensional analogue to Beatty sequences, we are motivated to define and explore the notion of complementary lattices. In particular, we present a continuous one-parameter family of complementary lattices. This main result then yields several novel examples of complementary sequences, along with a geometric proof of the fundamental property of Beatty sequences.

Simple Identification of Electron Diffraction Ring Patterns

1969 ◽  
Vol 27 ◽  
pp. 160-161
Author(s):  
Carolyn Nohr ◽  
Ann Ayres
Keyword(s):  
Electron Microscope ◽  
Electron Diffraction ◽  
Diffraction Ring ◽  
Image Position

Texts on electron diffraction recommend that the camera constant of the electron microscope be determine d by calibration with a standard crystalline specimen, using the equation

Atomic orientation effects in proton stopping at low velocity

1983 ◽  
Vol 41 ◽  
pp. 708-709
Author(s):  
Kin Lam
Keyword(s):  
Polycrystalline Material ◽  
Charged Particles ◽  
Target Material ◽  
Stopping Power ◽  
Low Velocity ◽  
Electronic Density ◽  
Interatomic Distances ◽  
Frame Work ◽  
Image Position ◽  
Atomic Orientation

The energy of moving ions in solid is dependent on the electronic density as well as the atomic structural properties of the target material. These factors contribute to the observable effects in polycrystalline material using the scanning ion microscope. Here we outline a method to investigate the dependence of low velocity proton stopping on interatomic distances and orientations.The interaction of charged particles with atoms in the frame work of the Fermi gas model was proposed by Lindhard. For a system of atoms, the electronic Lindhard stopping power can be generalized to the formwhere the stopping power function is defined as

A Magnetic Spectrometer for a Scanning Transmission Electron Microscope

1974 ◽  
Vol 32 ◽  
pp. 436-437
Author(s):  
A. Kosiara ◽  
J. W. Wiggins ◽  
M. Beer
Keyword(s):  
Scanning Transmission Electron Microscope ◽  
Magnetic Spectrometer ◽  
Fringe Field ◽  
Curvature Radius ◽  
Magnetic Pole ◽  
Quadrupole Field ◽  
Transmission Electron ◽  
Scanning Transmission ◽  
Under Construction ◽  
Image Position

A magnetic spectrometer to be attached to the Johns Hopkins S. T. E. M. is under construction. Its main purpose will be to investigate electron interactions with biological molecules in the energy range of 40 KeV to 100 KeV. The spectrometer is of the type described by Kerwin and by Crewe Its magnetic pole boundary is given by the equationwhere R is the electron curvature radius. In our case, R = 15 cm. The electron beam will be deflected by an angle of 90°. The distance between the electron source and the pole boundary will be 30 cm. A linear fringe field will be generated by a quadrupole field arrangement. This is accomplished by a grounded mirror plate and a 45° taper of the magnetic pole.

Optimizing conditions for x-ray microchemical analysis in analytical electron microscopy

1978 ◽  
Vol 36 (1) ◽  
pp. 548-549 ◽  
Author(s):  
N. J. Zaluzec
Keyword(s):  
Solid Angle ◽  
Analytical Electron Microscopy ◽  
Incident Beam ◽  
Thin Foil ◽  
Experimental Conditions ◽  
X Ray ◽  
Microchemical Analysis ◽  
Analytical Electron ◽  
Image Position ◽  
Incident Beam Energy

The ultimate sensitivity of microchemical analysis using x-ray emission rests in selecting those experimental conditions which will maximize the measured peak-to-background (P/B) ratio. This paper presents the results of calculations aimed at determining the influence of incident beam energy, detector/specimen geometry and specimen composition on the P/B ratio for ideally thin samples (i.e., the effects of scattering and absorption are considered negligible). As such it is assumed that the complications resulting from system peaks, bremsstrahlung fluorescence, electron tails and specimen contamination have been eliminated and that one needs only to consider the physics of the generation/emission process.The number of characteristic x-ray photons (Ip) emitted from a thin foil of thickness dt into the solid angle dΩ is given by the well-known equation

X-ray microanalysis of thin specimens in the transmission electron microscope at voltages up to 1000 Kv.

1978 ◽  
Vol 36 (1) ◽  
pp. 540-541
Author(s):  
G. Cliff ◽  
M.J. Nasir ◽  
G.W. Lorimer ◽  
N. Ridley
Keyword(s):  
Electron Microscope ◽  
Transmission Electron Microscope ◽  
Weight Fraction ◽  
Present Series ◽  
Constant Independent ◽  
X Ray ◽  
Transmission Electron ◽  
Series Of Experiments ◽  
Image Position ◽  
X Ray Absorption

In a specimen which is transmission thin to 100 kV electrons - a sample in which X-ray absorption is so insignificant that it can be neglected and where fluorescence effects can generally be ignored (1,2) - a ratio of characteristic X-ray intensities, I1/I2 can be converted into a weight fraction ratio, C1/C2, using the equationwhere k12 is, at a given voltage, a constant independent of composition or thickness, k12 values can be determined experimentally from thin standards (3) or calculated (4,6). Both experimental and calculated k12 values have been obtained for K(11<Z>19),kα(Z>19) and some Lα radiation (3,6) at 100 kV. The object of the present series of experiments was to experimentally determine k12 values at voltages between 200 and 1000 kV and to compare these with calculated values.The experiments were carried out on an AEI-EM7 HVEM fitted with an energy dispersive X-ray detector.

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