scholarly journals Normal surfaces in non-compact 3-manifolds

2005 ◽  
Vol 78 (3) ◽  
pp. 305-321 ◽  
Author(s):  
Ensil Kang
Keyword(s):  
Finite Number ◽  
Infinite Number ◽  
Boundary Component ◽  
Unique Feature ◽  
Admissible Solution ◽  
Compact Surface ◽  
Normal Surface ◽  
Ideal Triangulation ◽  
Normal Surfaces ◽  
Ideal Triangulations

AbstractWe extend the normal surface Q-theory to non-compact 3-manifolds with respect to ideal triangulations. An ideal triangulation of a 3-manifold often has a small number of tetrahedra resulting in a system of Q-matching equations with a small number of variables. A unique feature of our approach is that a compact surface F with boundary properly embedded in a non-compact 3-manifold M with an ideal triangulation with torus cusps can be represented by a normal surface in M as follows. A half-open annulus made up of an infinite number of triangular disks is attached to each boundary component of F. The resulting surface , when normalized, will contain only a finite number of Q-disks and thus correspond to an admissible solution to the system of Q-matching equations. The correspondence is bijective.

SEIFERT SURFACES IN KNOT COMPLEMENTS

2007 ◽  
Vol 16 (08) ◽  
pp. 1053-1066 ◽  
Author(s):  
ENSIL KANG
Keyword(s):  
Compact Surface ◽  
Seifert Surface ◽  
Normal Surface ◽  
Incompressible Surface ◽  
Ideal Triangulation ◽  
Seifert Surfaces ◽  
Knot Complements

In the ordinary normal surface for a compact 3-manifold, any incompressible, ∂-incompressible, compact surface can be moved by an isotopy to a normal surface [9]. But in a non-compact 3-manifold with an ideal triangulation, the existence of a normal surface representing an incompressible surface cannot be guaranteed. The figure-8 knot complement is presented in a counterexample in [12]. In this paper, we show the existence of normal Seifert surface under some restriction for a given ideal triangulation of the knot complement.

An Infinite Number of Laminagrams from a Finite Number of Radiographs

Radiology ◽  
1971 ◽  
Vol 98 (2) ◽  
pp. 249-255 ◽  
Author(s):  
Earl R. Miller ◽  
Edward M. MoCurry ◽  
Bernard Hruska
Keyword(s):  
Finite Number ◽  
Infinite Number

Some comments on positron–atom scattering at intermediate energy

1982 ◽  
Vol 60 (4) ◽  
pp. 558-564 ◽  
Author(s):  
F. W. Byron Jr.
Keyword(s):  
Finite Number ◽  
Cross Sections ◽  
Infinite Number ◽  
Intermediate Energy ◽  
Total Cross Sections ◽  
Type Approximation ◽  
Atom Scattering ◽  
Target States

A brief survey of available theoretical techniques is given for positron–atom scattering. The distinction between methods involving a finite number of target states and those with an infinite number of target states is emphasized. The situation regarding total cross sections is summarized, and a new, non-perturbative, eikonal-type approximation, based on the work of Wallace, is introduced.

scholarly journals Research of Register Pressure Aware Loop Unrolling Optimizations for Compiler

2018 ◽  
Vol 228 ◽  
pp. 03008
Author(s):  
Xuehua Liu ◽  
Liping Ding ◽  
Yanfeng Li ◽  
Guangxuan Chen ◽  
Jin Du
Keyword(s):  
Finite Number ◽  
Infinite Number ◽  
Performance Degradation ◽  
Transformation Process ◽  
Fine Grained ◽  
Loop Unrolling ◽  
Average Improvement ◽  
Register Pressure ◽  
Linpack Benchmark ◽  
Loop Optimizations

Register pressure problem has been a known problem for compiler because of the mismatch between the infinite number of pseudo registers and the finite number of hard registers. Too heavy register pressure may results in register spilling and then leads to performance degradation. There are a lot of optimizations, especially loop optimizations suffer from register spilling in compiler. In order to fight register pressure and therefore improve the effectiveness of compiler, this research takes the register pressure into account to improve loop unrolling optimization during the transformation process. In addition, a register pressure aware transformation is able to reduce the performance overhead of some fine-grained randomization transformations which can be used to defend against ROP attacks. Experiments showed a peak improvement of about 3.6% and an average improvement of about 1% for SPEC CPU 2006 benchmarks and a peak improvement of about 3% and an average improvement of about 1% for the LINPACK benchmark.

A definition of existence in terms of abstraction and disjunction

1957 ◽  
Vol 22 (4) ◽  
pp. 343-344
Author(s):  
Frederic B. Fitch
Keyword(s):  
Finite Number ◽  
Infinite Number ◽  
Recursive Relation ◽  
Basic Logic ◽  
Class A ◽  
Definition Of ◽  
Image Position

Greater economy can be effected in the primitive rules for the system K of basic logic by defining the existence operator ‘E’ in terms of two-place abstraction and the disjunction operator ‘V’. This amounts to defining ‘E’ in terms of ‘ε’, ‘έ’, ‘o, ‘ό’, ‘W’ and ‘V’, since the first five of these six operators are used for defining two-place abstraction.We assume that the class Y of atomic U-expressions has only a single member ‘σ’. Similar methods can be used if Y had some other finite number of members, or even an infinite number of members provided that they are ordered into a sequence by a recursive relation represented in K. In order to define ‘E’ we begin by defining an operator ‘D’ such thatHere ‘a’ may be thought of as an existence operator that provides existence quantification over some finite class of entities denoted by a class A of U-expressions. In other words, suppose that ‘a’ is such that ‘ab’ is in K if and only if, for some ‘e’ in A, ‘be’ is in K. Then ‘Dab’ is in K if and only if, for some ‘e and ‘f’ in A, ‘be’ or ‘b(ef)’ is in K; and ‘a’, ‘Da’, ‘D(Da)’, and so on, can be regarded as existence operators that provide for existence quantification over successively wider and wider finite classes. In particular, if ‘a’ is ‘εσ’, then A would be the class Y having ‘σ’ as its only member, and we can define the unrestricted existence operator ‘E’ in such a way that ‘Eb’ is in K if and only if some one of ‘εσb’, ‘D(εσ)b’, ‘D(D(εσ))b’, and so on, is in K.

scholarly journals On the Reduction of Ferrers' Associated Legendre Function

1929 ◽  
Vol 1 (4) ◽  
pp. 241-243
Author(s):  
Hrishikesh Sircar
Keyword(s):  
Finite Number ◽  
Infinite Series ◽  
Infinite Number ◽  
Legendre Function ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
Reduced Order ◽  
Zonal Harmonics ◽  
Integral Degree ◽  
Associated Legendre Function

Introduction. In the present paper a formula will be obtained to express a Ferrers' Associated Legendre Function of any integral degree and order as a sum of a finite number of Associated Legendre Functions of an order reduced by an even number. When the order is reduced by unity, an infinite series of the functions of reduced order is required. Thus a Ferrers' function can be expressed as the sum of a finite or infinite number of zonal harmonics according as the order of the function is even or odd.

scholarly journals Solution of the Word Problem for Certain Types of Groups I

1956 ◽  
Vol 3 (1) ◽  
pp. 45-54 ◽  
Author(s):  
J. L. Britton
Keyword(s):  
Finite Number ◽  
Word Problem ◽  
Infinite Number ◽  
Finite System ◽  
Defining Relations ◽  
Number Of Generators ◽  
Infinite Word ◽  
Formed Part ◽  
Free Product Of Groups ◽  
Product Of Groups

The main result of this series of papers is a theorem on the free product of groups (Theorem 1) which formed part of a doctoral thesis. This theorem has an immediate application to the word problem (Theorem 2). Usually the word problem refers to a finite system of generators and a finite number of defining relations, but in this context it is more natural to allow an infinite number of generators and defining relations. This (infinite) word problem is not solvable in general (Example 2).

Periodic forest whose largest clearings are of size 3

1970 ◽  
Vol 266 (1172) ◽  
pp. 113-121 ◽  
Keyword(s):  
Finite Number ◽  
Infinite Number ◽  
Basic Theory

Miller has observed that there are a finite number of periodic forests whose largest clearings are of size 1 or 2, and an infinite number whose largest clearings are of size 4. In this note the basic theory of periodic forests is outlined, and the number of periodic forests whose largest clearings are of size 3 is examined. There are 12 such forests; their corresponding tessellations are sketched.

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