Invariants of 3-manifolds derived from covering presentations

2010 ◽  
Vol 149 (2) ◽  
pp. 263-295 ◽  
Author(s):  
ERI HATAKENAKA
Keyword(s):  
Finite Sequence ◽  
Covering Space ◽  
Topological Invariants ◽  
Witten Invariant ◽  
Monodromy Representation ◽  
Topological Invariance ◽  
The Given

AbstractBy a covering presentation of a 3-manifold, we mean a labelled link (i.e., a link with a monodromy representation), which presents the 3-manifold as the simple 4-fold covering space of the 3-sphere branched along the link with the given monodromy. It is known that two labelled links present a homeomorphic 3-manifold if and only if they are related by a finite sequence of some local moves. This paper presents a method for constructing topological invariants of 3-manifolds based on their covering presentations. The proof of the topological invariance is shown by verifying the invariance under the local moves. As an example of such invariants, we present the Dijkgraaf–Witten invariant of 3-manifolds.

scholarly journals Modifications and Cobounding Manifolds

1960 ◽  
Vol 12 ◽  
pp. 503-528 ◽  
Author(s):  
Andrew H. Wallace
Keyword(s):  
Algebraic Variety ◽  
Finite Sequence ◽  
Differentiable Manifold ◽  
The Other ◽  
Algebraic Varieties ◽  
The Given ◽  
Hyperplane Sections

The object of this paper is to establish a simple connection between Thorn's theory of cobounding manifolds and the theory of modifications. The former theory is given in detail in (8) and sketched in (3), while the latter is worked out in (1). In particular in (1) it is shown that the only modifications which can transform one differentiable manifold into another are what I call below spherical modifications, which consist in taking out a sphere from the given manifold and replacing it by another. The main result is that manifolds cobound if and only if each is obtainable from the other by a finite sequence of spherical modifications.The technique consists in approximating the manifolds by pieces of algebraic varieties. Thus if M1 and M2 form the boundary of M, the last is taken to be part of an algebraic variety such that M1 and M2 are two members of a pencil of hyperplane sections.

Differences, Derivatives, and Decreasing Rearrangements

1967 ◽  
Vol 19 ◽  
pp. 1153-1178 ◽  
Author(s):  
G. F. D. Duff
Keyword(s):  
Finite Sequence ◽  
Real Numbers ◽  
Decreasing Rearrangement ◽  
Image Position ◽  
Decreasing Rearrangements ◽  
The Given

The decreasing rearrangement of a finite sequence a1, a2, … , an of real numbers is a second sequence aπ(1), aπ(2), … , aπ(n), where π(l), π(2), … , π(n) is a permutation of 1, 2, … , n and(1, p. 260). The kth term of the rearranged sequence will be denoted by . Thus the terms of the rearranged sequence correspond to and are equal to those of the given sequence ak, but are arranged in descending (non-increasing) order.

scholarly journals Definition and some Properties of Information Entropy

2007 ◽  
Vol 15 (3) ◽  
pp. 111-119
Author(s):  
Bo Zhang ◽  
Yatsuka Nakamura
Keyword(s):  
Information Entropy ◽  
Finite Sequence ◽  
Basic Properties ◽  
Maximum Property ◽  
Finite Sequences ◽  
Transformation Functions ◽  
Probability Matrices ◽  
Concept Of Information ◽  
Properties Of Information ◽  
The Given

Definition and some Properties of Information EntropyIn this article we mainly define the information entropy [3], [11] and prove some its basic properties. First, we discuss some properties on four kinds of transformation functions between vector and matrix. The transformation functions are LineVec2Mx, ColVec2Mx, Vec2DiagMx and Mx2FinS. Mx2FinS is a horizontal concatenation operator for a given matrix, treating rows of the given matrix as finite sequences, yielding a new finite sequence by horizontally joining each row of the given matrix in order to index. Then we define each concept of information entropy for a probability sequence and two kinds of probability matrices, joint and conditional, that are defined in article [25]. Further, we discuss some properties of information entropy including Shannon's lemma, maximum property, additivity and super-additivity properties.

scholarly journals Necklaces, Self-Reciprocal Polynomials, and q-Cycles

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Umarin Pintoptang ◽  
Vichian Laohakosol ◽  
Suton Tadee
Keyword(s):  
Positive Integer ◽  
Equivalence Class ◽  
Prime Power ◽  
Finite Sequence ◽  
Color Matching ◽  
The Given

Let n≥2 be a positive integer and q a prime power. Consider necklaces consisting of n beads, each of which has one of the given q colors. A primitive Cn-orbit is an equivalence class of n necklaces closed under rotation. A Cn-orbit is self-complementary when it is closed under an assigned color matching. In the work of Miller (1978), it is shown that there is a 1-1 correspondence between the set of primitive, self-complementary Cn-orbits and that of self-reciprocal irreducible monic (srim) polynomials of degree n. Let N be a positive integer relatively prime to q. A q-cycle mod N is a finite sequence of nonnegative integers closed under multiplication by q. In the work of Wan (2003), it is shown that q-cycles mod N are closely related to monic irreducible divisors of xN-1∈Fq[x]. Here, we show that: (1) q-cycles can be used to obtain information about srim polynomials; (2) there are correspondences among certain q-cycles and Cn-orbits; (3) there are alternative proofs of Miller's results in the work of Miller (1978) based on the use of q-cycles.

scholarly journals Arithmetical properties of the digits of the multiples of an irrational number

1973 ◽  
Vol 8 (2) ◽  
pp. 191-203 ◽  
Author(s):  
Kurt Mahler
Keyword(s):  
Irrational Number ◽  
Finite Sequence ◽  
The Other ◽  
Integral Multiple ◽  
Other Hand ◽  
The Given

Little seems to be known about the digits or sequences of digits in the decimal representation of a given irrational number like √2 or π. There is no difficulty in constructing an irrational number such that in its decimal representation certain digits or sequences of digits do not occur. On the other hand, well known theorems by Tchebychef, Kronecker, and Weyl imply that some integral multiple of the given irrational number always has any given finite sequence of digits occuring at least once in its decimal representation: for the fractional parts of the multiplies of the number lie dense in the interval (0, 1).

MASS PROBLEMS AND INITIAL SEGMENT COMPLEXITY

2014 ◽  
Vol 79 (01) ◽  
pp. 20-44 ◽  
Author(s):  
W. M. PHILLIP HUDELSON
Keyword(s):  
Asymptotic Behavior ◽  
Lower Bound ◽  
Lower Bounds ◽  
Kolmogorov Complexity ◽  
Initial Segment ◽  
Infinite Sequence ◽  
Finite Sequence ◽  
Randomness Extraction ◽  
The Given ◽  
Infinite Sequences

Abstract By the complexity of a finite sequence of 0’s and 1’s we mean the Kolmogorov complexity, that is the length of the shortest input to a universal recursive function which returns the given sequence as output. By initial segment complexity of an infinite sequence of 0’s and 1’s we mean the asymptotic behavior of the complexity of its finite initial segments. In this paper, we construct infinite sequences of 0’s and 1’s with given recursive lower bounds on initial segment complexity which do not compute any infinite sequences of 0’s and 1’s with a significantly larger recursive lower bound on initial segment complexity. This improves several known results about randomness extraction and separates many natural degrees in the lattice of Muchnik degrees.

Personal Problem Solving of Partners in Divorce Proceedings

2015 ◽  
Vol 36 (1) ◽  
pp. 82-103 ◽  
Author(s):  
Tereza Soukupova ◽  
Petr Goldmann
Keyword(s):  
Gender Differences ◽  
Problem Solving ◽  
Thematic Apperception Test ◽  
Scoring Systems ◽  
Personal Control ◽  
Personality Characteristics ◽  
Personal Problem ◽  
Thematic Apperception ◽  
And Gender ◽  
The Given

Abstract. The Thematic Apperception Test is one of the most frequently administered apperceptive techniques. Formal scoring systems are helpful in evaluating story responses. TAT stories, made by 20 males and 20 females in the situation of legal divorce proceedings, were coded for detection and comparison of their personal problem solving ability. The evaluating instrument utilized was the Personal Problem Solving System-Revised (PPSS-R) as developed by G. F. Ronan. The results indicate that in relation to card 1, men more often than women saw the cause of the problem as removable. With card 6GF, women were more motivated to resolve the given problem than were men, women had a higher personal control and their stories contained more optimism compared to men’s stories. In relation to card 6BM women, more often than men, used emotions generated from the problem to orient themselves within the problem. With card 13MF, the men’s level of stress was less compared to that of the women, and men were more able to plan within the context of problem-solving. Significant differences in the examined groups were found in those cards which depicted significant gender and parental potentials. The TAT can be used to help identify personality characteristics and gender differences.

Pattern Recognition in Histo-Pathology: Basic Considerations

1982 ◽  
Vol 21 (01) ◽  
pp. 15-22 ◽  
Author(s):  
W. Schlegel ◽  
K. Kayser
Keyword(s):  
Pattern Recognition ◽  
Graph Theory ◽  
Normal Tissue ◽  
Diagnostic Procedures ◽  
Minimal Distance ◽  
Malignant Growth ◽  
Colon Tissue ◽  
First Results ◽  
Tissue Structures ◽  
The Given

A basic concept for the automatic diagnosis of histo-pathological specimen is presented. The algorithm is based on tissue structures of the original organ. Low power magnification was used to inspect the specimens. The form of the given tissue structures, e. g. diameter, distance, shape factor and number of neighbours, is measured. Graph theory is applied by using the center of structures as vertices and the shortest connection of neighbours as edges. The algorithm leads to two independent sets of parameters which can be used for diagnostic procedures. First results with colon tissue show significant differences between normal tissue, benign and malignant growth. Polyps form glands that are twice as wide as normal and carcinomatous tissue. Carcinomas can be separated by the minimal distance of the glands formed. First results of pattern recognition using graph theory are discussed.

Coordinating Medical Literature with Patient Care

1977 ◽  
Vol 16 (04) ◽  
pp. 234-240 ◽  
Author(s):  
Joann Gustafson ◽  
J. Nelson ◽  
Ann Buller
Keyword(s):  
Health Care ◽  
Medical Staff ◽  
Medical Information ◽  
Medical Literature ◽  
Medical Information System ◽  
Staff Members ◽  
Information Displays ◽  
Medical Displays ◽  
Computerized Medical Information ◽  
The Given

The contribution of a special library project to a computerized problem-oriented medical information system (PROMIS) is discussed. Medical information displays developed by the PROMIS medical staff are accessible to the health care provider via touch screen cathode terminals. Under PROMIS, members of the library project developed two information services, one concerned with the initial building of the medical displays and the other with the updating of this information. Information from 88 medical journals is disseminated to physicians involved in the building of the medical displays. Articles meeting predetermined selection criteria are abstracted and the abstracts are made available by direct selective dissemination or via a problem-oriented abstract file. The updating service involves comparing the information contained in the selected articles with the computerized medical displays on the given topic. Discrepancies are brought to the attention of PROMIS medical staff members who evaluate the information and make appropriate changes in the displays. Thus a feedback loop is maintained which assures the completeness, accuracy, and currency of the computerized medical information. The development of this library project and its interface with the computerized health care system thus attempts to deal with the problems in the generation, validation, dissemination, and application of medical literature.

Identification and Transformation of Terminal Morphemes in Medical English

1969 ◽  
Vol 8 (02) ◽  
pp. 84-90 ◽  
Author(s):  
A. W. Pratt ◽  
M. Pacak
Keyword(s):  
Word Form ◽  
Subsequent Transformation ◽  
Part Of Speech ◽  
Morphological Segmentation ◽  
Medical English ◽  
Nominal Form ◽  
Admissible Path ◽  
Near Future ◽  
The Given ◽  
Admissible Sequences

The system for the identification and subsequent transformation of terminal morphemes in medical English is a part of the information system for processing pathology data which was developed at the National Institutes of Health.The recognition and transformation of terminal morphemes is restricted to classes of adjectivals including the -ING and -ED forms, nominals and homographic adjective/noun forms.The adjective-to-noun and noun-to-noun transforms consist basically of a set of substitutions of adjectival and certain nominal suffixes by a set of suffixes which indicate the corresponding nominal form(s).The adjectival/nominal suffix has a polymorphosyntactic transformational function if it has the property of being transformed into more than one nominalizing suffix (e.g., the adjectival suffix -IC can be substituted by a set of nominalizing suffixes -Ø, -A, -E, -Y, -IS, -IA, -ICS): the adjectival suffix has a monomorphosyntactic transformational property if there is only one admissible transform (e.g., -CIC → -X).The morphological segmentation and the subsequent transformations are based on the following principles:a. The word form is segmented according to the principle of »double consonant cut,« i.e., terminal characters following the last set of double consonants are analyzed and treated as a potential suffix. For practical purposes only such terminal suffixes of a maximum length of four have been analyzed.b. The principle that the largest segment of a word form common to both adjective and noun or to both noun stems is retained as a word base for transformational operations, and the non-identical segment is considered to be a »suffix.«The backward right-to-left character search is initiated by the identification of the terminal grapheme of the given word form and is extended to certain admissible sequences of immediately preceding graphemes.The nodes which represent fixed sequences of graphemes are labeled according to their recognition and/or transformation properties.The tree nodes are divided into two groups:a. productive or activatedb. non-productive or non-activatedThe productive (activated) nodes are sequences of sets of graphemes which possess certain properties, such as the indication about part-of-speech class membership, the transformation properties, or both. The non-productive (non-activated) nodes have the function of connectors, i.e., they specify the admissible path to the productive nodes.The computer program for the identification and transformation of the terminal morphemes is open-ended and is already operational. It will be extended to other sub-fields of medicine in the near future.

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