A Pappus Type Theorem in the Affine Group

1968 ◽  
Vol 11 (4) ◽  
pp. 547-554
Author(s):  
R. Paré
Keyword(s):  
Projective Plane ◽  
Final Section ◽  
Type Theorem ◽  
Affine Group ◽  
Real Numbers ◽  
One Dimensional ◽  
Master's Thesis ◽  
Geometrical Concepts ◽  
The One ◽  
The Relationship

In [3] H. Schwerdtfeger embedded the one-dimensional affine group over the real numbers in the projective plane. The relationship between group-theoretical properties and geometrical concepts was studied.In this paper the methods of [3] are used to prove Pappus' theorem. In the final section we give a similar theorem for (4n+2)-gons.This paper is a generalization of part of my master's thesis, written under the direction of Professor H. Schwerdtfeger.

Projective Geometry in the One-Dimensional Affine Group

1964 ◽  
Vol 16 ◽  
pp. 683-700 ◽  
Author(s):  
Hans Schwerdtfeger
Keyword(s):  
Projective Space ◽  
Projective Plane ◽  
Simple Group ◽  
Final Section ◽  
Theoretical Interpretation ◽  
Special Group ◽  
One Dimensional ◽  
The One ◽  
Straight Lines ◽  
Geometrical Investigation

The idea of considering the set of the elements of a group as a space, provided with a topology, measure, or metric, connected somehow with the group operation, has been used often in the work of E. Cartan and others. In the present paper we shall study a very special group whose space can be embedded naturally into a projective plane and where the straight lines have a very simple group-theoretical interpretation. It remains to be seen whether this geometrical embedding in a projective space can be extended to other classes of groups and whether the method could become an instrument of geometrical investigation, like co-ordinates or reflections. In the final section it is shown how a geometrical theorem may lead to relations within the group.

scholarly journals A general renormalization procedure on the one-dimensional lattice and decay of correlations

2019 ◽  
Vol 19 (01) ◽  
pp. 1950003
Author(s):  
Artur O. Lopes
Keyword(s):  
Fixed Point ◽  
Final Section ◽  
Polynomial Decay ◽  
Renormalization Procedure ◽  
Real Numbers ◽  
One Dimensional ◽  
Point Potential ◽  
Renormalization Operator ◽  
The One ◽  
Natural Way

We present a general form of renormalization operator [Formula: see text] acting on potentials [Formula: see text]. We exhibit the analytical expression of the fixed point potential [Formula: see text] for such operator [Formula: see text]. This potential can be expressed in a natural way in terms of a certain integral over the Hausdorff probability on a Cantor type set on the interval [0,1]. This result generalizes a previous one by Baraviera, Leplaideur and Lopes where the fixed point potential [Formula: see text] was of Hofbauer type. For the potentials of Hofbauer type (a well-known case of phase transition) the decay is like [Formula: see text], [Formula: see text]. Among other things we present the estimation of the decay of correlation of the equilibrium probability associated to the fixed potential [Formula: see text] of our general renormalization procedure. In some cases we get polynomial decay like [Formula: see text], [Formula: see text], and in others a decay faster than [Formula: see text], when [Formula: see text]. The potentials [Formula: see text] we consider here are elements of the so-called family of Walters’ potentials on [Formula: see text] which generalizes a family of potentials considered initially by Hofbauer. For these potentials some explicit expressions for the eigenfunctions are known. In a final section we also show that given any choice [Formula: see text] of real numbers varying with [Formula: see text] there exists a potential [Formula: see text] on the class defined by Walters which has a invariant probability with such numbers as the coefficients of correlation (for a certain explicit observable function).

scholarly journals A Novel Approach to the Analysis of the Soil Consolidation Problem by Using Non-Classical Rheological Schemes

2021 ◽  
Vol 11 (5) ◽  
pp. 1980
Author(s):  
Kazimierz Józefiak ◽  
Artur Zbiciak ◽  
Karol Brzeziński ◽  
Maciej Maślakowski
Keyword(s):  
Constitutive Models ◽  
Organic Soil ◽  
Organic Soils ◽  
Soil Consolidation ◽  
Flexible Tool ◽  
One Dimensional ◽  
Novel Approach ◽  
Soil Skeleton ◽  
The One ◽  
The Relationship

The paper presents classical and non-classical rheological schemes used to formulate constitutive models of the one-dimensional consolidation problem. The authors paid special attention to the secondary consolidation effects in organic soils as well as the soil over-consolidation phenomenon. The systems of partial differential equations were formulated for every model and solved numerically to obtain settlement curves. Selected numerical results were compared with standard oedometer laboratory test data carried out by the authors on organic soil samples. Additionally, plasticity phenomenon and non-classical rheological elements were included in order to take into account soil over-consolidation behaviour in the one-dimensional settlement model. A new way of formulating constitutive equations for the soil skeleton and predicting the relationship between the effective stress and strain or void ratio was presented. Rheological structures provide a flexible tool for creating complex constitutive relationships of soil.

scholarly journals Analysis of Sealing and Leakage Performance of the Subsea Collet Connector with Lens-Type Sealing Structure

2020 ◽  
Vol 8 (6) ◽  
pp. 444
Author(s):  
Feihong Yun ◽  
Gang Wang ◽  
Zheping Yan ◽  
Peng Jia ◽  
Xiujun Xu ◽  
...  
Keyword(s):  
Structural Characteristics ◽  
Gain Coefficient ◽  
Leakage Rate ◽  
One Dimensional ◽  
Mechanics Model ◽  
Sealing Performance ◽  
Width Measurement ◽  
Pressure Experiment ◽  
The One ◽  
The Relationship

The contact mechanics model of the metal lens-type sealing gasket is established on the basis of Hertz theory on the macroscopical scale in this paper. The relationship among sealing width, contact pressure, and preload is solved. Based on the structural characteristics of the subsea collet connector, the self-locking characteristics are analyzed to determine the gain coefficient of the sealing structure for the loading thrust. On the microscopic scale, the contact characteristics of the turning lens-type sealing gasket and the hub structure are analyzed by the equivalent replacement of the peak cut coefficient of the one-dimensional sinusoidal wave. The influence of different leakage forms on sealing performance is discussed from both radial and circumferential leakage, and the leakage rate of the lens-type sealing structure is calculated. The hydrostatic pressure experiment of the subsea collet connector with lens-type sealing gasket is carried out, and the correctness of the theoretical analysis is verified from the results of the pressure maintaining, sealing width measurement, and preload conversion.

scholarly journals Marcinkiewicz-type strong law of large numbers for double arrays of pairwise independent random variables

1999 ◽  
Vol 22 (1) ◽  
pp. 171-177 ◽  
Author(s):  
Dug Hun Hong ◽  
Seok Yoon Hwang
Keyword(s):  
Law Of Large Numbers ◽  
Double Sequence ◽  
Random Variables ◽  
Independent Random Variables ◽  
Real Numbers ◽  
Strong Law ◽  
One Dimensional ◽  
Large Numbers ◽  
Pairwise Independent Random Variables ◽  
The One

Let {Xij}be a double sequence of pairwise independent random variables. If P{|Xmn|≥t}≤P{|X|≥t}for all nonnegative real numbers tandE|X|p(log+|X|)3<∞, for1<p<2, then we prove that∑i=1m∑j=1n(Xij−EXij)(mn)1/p→0    a.s.   as  m∨n→∞.                                     (0.1)Under the weak condition ofE|X|plog+|X|<∞, it converges to 0inL1. And the results can be generalized to anr-dimensional array of random variables under the conditionsE|X|p(log+|X|)r+1<∞,E|X|p(log+|X|)r−1<∞, respectively, thus, extending Choi and Sung's result [1] of the one-dimensional case.

RELATION BETWEEN A CLASS OF QUASIPERIODIC LATTICES GENERATED BY THE SUBSTITUTION METHOD AND THE GENERALIZED CONTINUED FRACTION EXPANSION

1996 ◽  
Vol 10 (17) ◽  
pp. 2081-2101
Author(s):  
TOSHIO YOSHIKAWA ◽  
KAZUMOTO IGUCHI
Keyword(s):  
Continued Fraction ◽  
Positive Real Number ◽  
Real Numbers ◽  
Continued Fraction Expansion ◽  
Positive Real ◽  
Periodic Sequences ◽  
One Dimensional ◽  
Finite Period ◽  
The One ◽  
Positive Integers

The continued fraction expansion for a positive real number is generalized to that for a set of positive real numbers. For arbitrary integer n≥2, this generalized continued fraction expansion generates (n−1) sequences of positive integers {ak}, {bk}, … , {yk} from a given set of (n−1) positive real numbers α, β, …ψ. The sequences {ak}, {bk}, … ,{yk} determine a sequence of substitutions Sk: A → Aak Bbk…Yyk Z, B → A, C → B,…,Z → Y, which constructs a one-dimensional quasiperiodic lattice with n elements A, B, … , Z. If {ak}, {bk}, … , {yk} are infinite periodic sequences with an identical period, then the ratio between the numbers of n elements A, B, … , Z in the lattice becomes a : β : … : ψ : 1. Thereby the correspondence is established between all the sets of (n−1) positive real numbers represented by a periodic generalized continued fraction expansion and all the one-dimensional quasiperiodic lattices with n elements generated by a sequence of substitutions with a finite period.

Phase transition in one-dimensional random walk with partially reflecting boundaries

1985 ◽  
Vol 17 (03) ◽  
pp. 594-606 ◽  
Author(s):  
Ora E. Percus
Keyword(s):  
Random Walk ◽  
Transition Probability ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
Boundary Case ◽  
Reflecting Boundaries ◽  
The Mean ◽  
The One ◽  
The Relationship ◽  
Asymptotic Forms

We consider an asymmetric random walk, with one or two boundaries, on a one-dimensional lattice. At the boundaries, the walker is either absorbed (with probability 1–ρ) or reflects back to the system (with probability p). The probability distribution (Pn (m)) of being at position m after n steps is obtained, as well as the mean number of steps before absorption. In the one-boundary case, several qualitatively different asymptotic forms of P n(m) result, depending on the relationship between transition probability and the reflection probability.

Phase transition in one-dimensional random walk with partially reflecting boundaries

1985 ◽  
Vol 17 (3) ◽  
pp. 594-606 ◽  
Author(s):  
Ora E. Percus
Keyword(s):  
Random Walk ◽  
Transition Probability ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
Boundary Case ◽  
Reflecting Boundaries ◽  
The Mean ◽  
The One ◽  
The Relationship ◽  
Asymptotic Forms

We consider an asymmetric random walk, with one or two boundaries, on a one-dimensional lattice. At the boundaries, the walker is either absorbed (with probability 1–ρ) or reflects back to the system (with probability p).The probability distribution (Pn(m)) of being at position m after n steps is obtained, as well as the mean number of steps before absorption. In the one-boundary case, several qualitatively different asymptotic forms of Pn(m) result, depending on the relationship between transition probability and the reflection probability.

scholarly journals STUDY OF THE ONE-DIMENSIONAL FIBROUS PRODUCT’S COMPONENT RATIO OF UNEVENNESS AND ITS DEPENDENCE OF THE LINEAR DENSITY UNEVENNESS

2020 ◽  
pp. 15-21
Author(s):  
P. A. Sevost’yanov
Keyword(s):  
Correlation Functions ◽  
Coefficient Of Variation ◽  
Linear Density ◽  
Spectral Composition ◽  
Component Ratio ◽  
One Dimensional ◽  
Spectral Dispersion ◽  
The One ◽  
The Relationship

The article presents the results of a study of the relationship between irregularity in linear density and the proportion of components of one-dimensional fibrous products (tape, tourniquet, roving, yarn). A formula is obtained for estimating the coefficient of variation for the component fraction depending on the unevenness for the linear density and the average component fraction. The estimate allows one to predict the expected unevenness of the component share in the product based on known information about the unevenness of the components. Regularities are established and examples are given of the relationship between correlation functions and spectral dispersion densities for unevenness by linear density and component share in the product depending on the share of components, their unevenness, and the spectral composition of the unevenness.

scholarly journals GEODYNAMICS

GEODYNAMICS ◽  
2011 ◽  
Vol 2(11)2011 (2(11)) ◽  
pp. 155-157
Author(s):  
B. Ladanivskyy ◽  
◽  
Keyword(s):  
Transfer Function ◽  
Field Source ◽  
One Dimensional ◽  
Conductivity Model ◽  
The Earth ◽  
Mantle Conductivity ◽  
The One ◽  
Relationship Of ◽  
The Relationship ◽  
Source Structure

The regional magneto-variational sounding method (aka Z/H method) was used for estimation of the Earth's mantle conductivity model at the Panagurishte (PAG) observatory region. A magneto-variational transfer function is calculated on the base of the relationship of vertical to horizontal geomagnetic field spectra components recorded on the Earth surface and priori assumptions about a field source structure. Inversions of the transfer function allow to obtain the one dimensional conductivity model.

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