EXPLICIT DISTRIBUTION OF SELECTED TWO-DIMENSIONAL AND THREE-DIMENSIONAL STATISTICS OF THE (0,1)-SEQUENCE

2021 ◽  
Vol 5 ◽  
pp. 72-81
Author(s):  
Vladimir Masol ◽  
◽  
Svetlana Popereshnyak ◽  
Keyword(s):  
Random Number ◽  
Three Dimensional ◽  
Fixed Number ◽  
Two Dimensional ◽  
Random Number Generators ◽  
Joint Distributions ◽  
Basic Assumptions ◽  
Number Of Zeros ◽  
The Given ◽  
Explicit Expressions

The joint distributions of the given number of 2-chains and the given number of 3-chains of a fixed form of a random bit sequence are considered, which allow performing a statistical analysis of local sections of this sequence. All configurations consisting of two consecutive zeros or ones of a bit sequence of a given length act as 2-chains. In turn, 3-chains are all configurations consisting of three consecutive either ones (provided that the 2-chains are zero) or zeros (provided that the 2-chains are one), as well as 3-chains all configurations are considered that consist either of three consecutive digits: one, zero and one (provided that the 2- chains are zero), or of three consecutive digits: zero, one and zero (provided that the 2- chains are one). The paper establishes explicit expressions for two-dimensional and three-dimensional joint distributions of events, reflecting the number of some combinations of the indicated chains in a finite random bit sequence. One of the basic assumptions is that zeros and ones in a bit sequence are independent, equally distributed random variables. The proofs of the formulas for the distributions of these events are based on counting the number of corresponding favorable events, provided that the bit sequence contains a fixed number of zeros and ones. As examples of using explicit expressions of joint distributions, tables are given in which the values of the probabilities of the events listed above for a random bit sequence of length 40 (tables 1–3) and length 24 (table 4) are given for some fixed values of the number of 2-chains and the number 3-chains under the assumption that zeros and ones appear independently and uniformly. For clarity, tables 1‑3 are illustrated with bubble charts. The established formulas may be of interest for the problems of testing local sections formed at the output of pseudo-random number generators, for some problems of protecting information from unauthorized access, as well as in other areas where it becomes necessary to analyze bit sequences.

Design of a Two-Piece Brassiere Cup From its Data Points Toward its Automation

2020 ◽  
Author(s):  
Kotaro Yoshida ◽  
Hidefumi Wakamatsu ◽  
Eiji Morinaga ◽  
Takahiro Kubo
Keyword(s):  
Three Dimensional ◽  
Developable Surface ◽  
Two Dimensional ◽  
Trial And Error ◽  
Design Efficiency ◽  
Developable Surfaces ◽  
Data Points ◽  
Dimensional Shape ◽  
The Given ◽  
Three Dimensional Shape

Abstract A method to design the two-dimensional shapes of patterns of two piece brassiere cup is proposed when its target three-dimensional shape is given as a cloud of its data points. A brassiere cup consists of several patterns and their shapes are designed by repeatedly making a paper cup model and checking its three-dimensional shape. For improvement of design efficiency of brassieres, such trial and error must be reduced. As a cup model for check is made of paper not cloth, it is assumed that the surface of the model is composed of several developable surfaces. When two lines that consist in the developable surface are given, the surface can be determined. Then, the two-piece brassiere cup can be designed by minimizing the error between the surface and given data points. It was mathematically verified that the developable surface calculated by our propose method can reproduce the given data points which is developable surface.

On Three and Two-Dimensional Disturbances of Pipe Flow

1960 ◽  
Vol 27 (3) ◽  
pp. 381-389 ◽  
Author(s):  
Kurt Spielberg ◽  
Hans Timan
Keyword(s):  
Order Differential Equation ◽  
Three Dimensional ◽  
Circular Cross Section ◽  
Main Flow ◽  
Two Dimensional ◽  
Straight Pipe ◽  
Special Cases ◽  
Set Up ◽  
The Stability ◽  
The Given

A system of ordinary, coupled differential equations is set up for three-dimensional disturbances of Poiseuille flow in a straight pipe of circular cross section. The commonly treated equations are shown to be special cases arising from particular assumptions. It is shown that in the nonviscous, and therefore also in the general case, there exists, in contrast to the analogous problem in Cartesian co-ordinates, no transformation reducing the given problem to a two-dimensional one. A fourth-order differential equation is derived for disturbances independent of the direction of the main flow. The solutions, which are obtained, show that those two-dimensional disturbances, termed cross disturbances, decay with time and do therefore not disturb the stability of the main flow. Explicit expressions for the cross disturbances are obtained and a discussion of their nature is given.

scholarly journals Some Results on the Cohomology of Line Bundles on the Three Dimensional Flag Variety

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 295
Author(s):  
Muhammad Anwar
Keyword(s):  
Borel Subgroup ◽  
Three Dimensional ◽  
Linear Algebraic Group ◽  
Flag Variety ◽  
Line Bundles ◽  
Closed Field ◽  
Two Dimensional ◽  
Prime Characteristic ◽  
Simply Connected ◽  
The Given

Let k be an algebraically closed field of prime characteristic and let G be a semisimple, simply connected, linear algebraic group. It is an open problem to find the cohomology of line bundles on the flag variety G / B , where B is a Borel subgroup of G. In this paper we consider this problem in the case of G = S L 3 ( k ) and compute the cohomology for the case when ⟨ λ , α ∨ ⟩ = − p n a − 1 , ( 1 ≤ a ≤ p , n > 0 ) or ⟨ λ , α ∨ ⟩ = − p n − r , ( r ≥ 2 , n ≥ 0 ) . We also give the corresponding results for the two dimensional modules N α ( λ ) . These results will help us understand the representations of S L 3 ( k ) in the given cases.

An Efficient Technique for Three-Dimensional Image Visualization Through Two-Dimensional Images for Medical Data

2017 ◽  
Vol 29 (1) ◽  
pp. 100-109
Author(s):  
Ganesan Gunasekaran ◽  
Meenakshisundaram Venkatesan
Keyword(s):  
Three Dimensional ◽  
Main Idea ◽  
Two Dimensional ◽  
3D Image ◽  
Image Visualization ◽  
Good Determination ◽  
The Given ◽  
Geometric Primitives ◽  
2D Images

Abstract The main idea behind this work is to present three-dimensional (3D) image visualization through two-dimensional (2D) images that comprise various images. 3D image visualization is one of the essential methods for excerpting data from given pieces. The main goal of this work is to figure out the outlines of the given 3D geometric primitives in each part, and then integrate these outlines or frames to reconstruct 3D geometric primitives. The proposed technique is very useful and can be applied to many kinds of images. The experimental results showed a very good determination of the reconstructing process of 2D images.

scholarly journals Counting Gluings of Octahedra

10.37236/6503 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Valentin Bonzom ◽  
Luca Lionni
Keyword(s):  
Three Dimensional ◽  
Fixed Number ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Plane Trees ◽  
Universality Classes

Three-dimensional colored triangulations are gluings of tetrahedra whose faces carry the colors 0, 1, 2, 3 and in which the attaching maps between tetrahedra are defined using the colors. This framework makes it possible to generalize the notion of two-dimensional $2p$-angulations to three dimensions in a way which is suitable for combinatorics and enumeration. In particular, universality classes of three-dimensional triangulations can be investigated within this framework. Here we study colored triangulations obtained by gluing octahedra. Those which maximize the number of edges at fixed number of octahedra are fully characterized and are shown to have the topology of the 3-sphere. They are further shown to be in bijection with a family of plane trees. The enumeration is performed both directly and using this bijection.

High Resolution Microscopy of Multi-Layered Biological Crystals

1982 ◽  
Vol 40 ◽  
pp. 80-83
Author(s):  
H.A. Cohen ◽  
T.W. Jeng ◽  
W. Chiu
Keyword(s):  
High Resolution ◽  
Low Dose ◽  
Three Dimensional ◽  
Data Interpretation ◽  
Two Dimensional ◽  
Biological Objects ◽  
Density Maps ◽  
Density Map ◽  
Complex Crystal ◽  
High Resolution Microscopy

This tutorial will discuss the methodology of low dose electron diffraction and imaging of crystalline biological objects, the problems of data interpretation for two-dimensional projected density maps of glucose embedded protein crystals, the factors to be considered in combining tilt data from three-dimensional crystals, and finally, the prospects of achieving a high resolution three-dimensional density map of a biological crystal. This methodology will be illustrated using two proteins under investigation in our laboratory, the T4 DNA helix destabilizing protein gp32*I and the crotoxin complex crystal.

Instrumentation For Quantitative Microscopy

1977 ◽  
Vol 35 ◽  
pp. 206-209
Author(s):  
B. Ralph ◽  
A.R. Jones
Keyword(s):  
Volume Fraction ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Automatic Data ◽  
Phase Volume Fraction ◽  
Statistical Confidence ◽  
Resultant Data ◽  
Automatic Data Collection ◽  
Third Dimension ◽  
Evolution Of Microstructure

In all fields of microscopy there is an increasing interest in the quantification of microstructure. This interest may stem from a desire to establish quality control parameters or may have a more fundamental requirement involving the derivation of parameters which partially or completely define the three dimensional nature of the microstructure. This latter categorey of study may arise from an interest in the evolution of microstructure or from a desire to generate detailed property/microstructure relationships. In the more fundamental studies some convolution of two-dimensional data into the third dimension (stereological analysis) will be necessary.In some cases the two-dimensional data may be acquired relatively easily without recourse to automatic data collection and further, it may prove possible to perform the data reduction and analysis relatively easily. In such cases the only recourse to machines may well be in establishing the statistical confidence of the resultant data. Such relatively straightforward studies tend to result from acquiring data on the whole assemblage of features making up the microstructure. In this field data mode, when parameters such as phase volume fraction, mean size etc. are sought, the main case for resorting to automation is in order to perform repetitive analyses since each analysis is relatively easily performed.

A linearity test for thick specimens imaged by HVEM over a 180° angular range

1991 ◽  
Vol 49 ◽  
pp. 552-553
Author(s):  
Yu Liu
Keyword(s):  
Phase Contrast ◽  
Three Dimensional ◽  
Angular Range ◽  
Two Dimensional ◽  
Back Projection ◽  
Line Integral ◽  
Exponential Law ◽  
Transmission Electron ◽  
Absorption Law ◽  
Linearity Test

The image obtained in a transmission electron microscope is the two-dimensional projection of a three-dimensional (3D) object. The 3D reconstruction of the object can be calculated from a series of projections by back-projection, but this algorithm assumes that the image is linearly related to a line integral of the object function. However, there are two kinds of contrast in electron microscopy, scattering and phase contrast, of which only the latter is linear with the optical density (OD) in the micrograph. Therefore the OD can be used as a measure of the projection only for thin specimens where phase contrast dominates the image. For thick specimens, where scattering contrast predominates, an exponential absorption law holds, and a logarithm of OD must be used. However, for large thicknesses, the simple exponential law might break down due to multiple and inelastic scattering.

An Image Reconstruction System Applied to High Voltage Microscopy

1975 ◽  
Vol 33 ◽  
pp. 292-293
Author(s):  
D. E. Johnson
Keyword(s):  
High Voltage ◽  
Prior Information ◽  
Block Diagram ◽  
Three Dimensional ◽  
Real Space ◽  
Two Dimensional ◽  
Efficient Manner ◽  
Fourier Methods ◽  
Tilt Series ◽  
Methods Of Reconstruction

Increased specimen penetration; the principle advantage of high voltage microscopy, is accompanied by an increased need to utilize information on three dimensional specimen structure available in the form of two dimensional projections (i.e. micrographs). We are engaged in a program to develop methods which allow the maximum use of information contained in a through tilt series of micrographs to determine three dimensional speciman structure.In general, we are dealing with structures lacking in symmetry and with projections available from only a limited span of angles (±60°). For these reasons, we must make maximum use of any prior information available about the specimen. To do this in the most efficient manner, we have concentrated on iterative, real space methods rather than Fourier methods of reconstruction. The particular iterative algorithm we have developed is given in detail in ref. 3. A block diagram of the complete reconstruction system is shown in fig. 1.

Computer Assisted Image Reconstruction of Oligomer Structure

1976 ◽  
Vol 34 ◽  
pp. 472-473
Author(s):  
A.M. Jones ◽  
A. Max Fiskin
Keyword(s):  
Three Dimensional ◽  
Depth Of Field ◽  
Computer Assisted ◽  
Projection Data ◽  
Two Dimensional ◽  
Single Particles ◽  
Dimensional Reconstruction ◽  
Computer Assisted Image ◽  
The Fourier Transform ◽  
Space Approach

If the tilt of a specimen can be varied either by the strategy of observing identical particles orientated randomly or by use of a eucentric goniometer stage, three dimensional reconstruction procedures are available (l). If the specimens, such as small protein aggregates, lack periodicity, direct space methods compete favorably in ease of implementation with reconstruction by the Fourier (transform) space approach (2). Regardless of method, reconstruction is possible because useful specimen thicknesses are always much less than the depth of field in an electron microscope. Thus electron images record the amount of stain in columns of the object normal to the recording plates. For single particles, practical considerations dictate that the specimen be tilted precisely about a single axis. In so doing a reconstructed image is achieved serially from two-dimensional sections which in turn are generated by a series of back-to-front lines of projection data.

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