On the number of fixed points of a sofic shift-flip system

2013 ◽  
Vol 35 (2) ◽  
pp. 482-498 ◽  
Author(s):  
YOUNG-ONE KIM ◽  
SIEYE RYU
Keyword(s):  
Finite Number ◽  
Fixed Points ◽  
Sofic Shift ◽  
Joint State

AbstractIf $X$ is a sofic shift and $\varphi : X\rightarrow X$ is a homeomorphism such that ${\varphi }^{2} = {\text{id} }_{X} $ and $\varphi {\sigma }_{X} = { \sigma }_{X}^{- 1} \varphi $, the number of points in $X$ that are fixed by ${ \sigma }_{X}^{m} $ and ${ \sigma }_{X}^{n} \varphi , m= 1, 2, \ldots , n\in \mathbb{Z} $, is expressed in terms of a finite number of square matrices: the matrices are obtained from Krieger’s joint state chain of a sofic shift which is conjugate to $X$.

A Note on Involutions with a Finite Number of Fixed Points

1968 ◽  
Vol 20 ◽  
pp. 1522-1530
Author(s):  
John D. Miller
Keyword(s):  
Finite Number ◽  
Fixed Points ◽  
Connected Manifold ◽  
Simply Connected

LetMbe a smooth, closed, simply connected manifold of dimension greater than 5. LetTbe an involution onMwith a positive, finite number of fixed points. Our aim in this paper is to prove the following theorem (which is somewhat like that of Wasserman (7)).

Topological Transitivity on the Torus

1994 ◽  
Vol 37 (4) ◽  
pp. 549-551 ◽  
Author(s):  
Sol Schwartzman
Keyword(s):  
Vector Field ◽  
Finite Number ◽  
Fixed Points ◽  
Analytic Vector ◽  
Topological Transitivity ◽  
Topologically Transitive ◽  
Transitive Flow ◽  
Real Analytic ◽  
Analytic Vector Field

AbstractT. Ding has shown that a topologically transitive flow on the torus given by a real analytic vector field is orbitally equivalent to a Kronecker flow on the torus, modified so as to have a finite number of fixed points, provided the original flow had only a finite number of fixed points. In this paper it is shown that the assumption that there are only finitely many fixed points is unnecessary.

scholarly journals Morse-Bott energy function for surface Ω-stable flows

2020 ◽  
Vol 22 (4) ◽  
pp. 434-441
Author(s):  
Anna E. Kolobyanina ◽  
Vladislav E. Kruglov
Keyword(s):  
Lyapunov Function ◽  
Finite Number ◽  
Fixed Points ◽  
Structural Stability ◽  
Limit Cycles ◽  
Energy Function ◽  
Hyperbolic Fixed Points ◽  
Smale Flows ◽  
Flows On Surfaces

In this paper, we consider the class of Ω-stable flows on surfaces, i.e. flows on surfaces with the non-wandering set consisting of a finite number of hyperbolic fixed points and a finite number of hyperbolic limit cycles. The class of Ω -stable flows is a generalization of the class of Morse-Smale flows, admitting the presence of saddle connections that do not form cycles. The authors have constructed the Morse-Bott energy function for any such flow. The results obtained are an ideological continuation of the classical works of S. Smale, who proved the existence of the Morse energy function for gradient-like flows, and K. Meyer, who established the existence of the Morse-Bott energy function for Morse-Smale flows. The specificity of Ω-stable flows takes them beyond the framework of structural stability, but the decrease along the trajectories of such flows is still tracked by the regular Lyapunov function.

Lambda theories allowing terms with a finite number of fixed points

2015 ◽  
Vol 27 (3) ◽  
pp. 405-427
Author(s):  
BENEDETTO INTRIGILA ◽  
RICHARD STATMAN
Keyword(s):  
Finite Number ◽  
Fixed Points ◽  
Related Question ◽  
Natural Question ◽  
Open Problems ◽  
Complete Answer ◽  
Recursively Enumerable

A natural question in the λ-calculus asks what is the possible number of fixed points of a combinator (closed term). A complete answer to this question is still missing (Problem 25 of TLCA Open Problems List) and we investigate the related question about the number of fixed points of a combinator in λ-theories. We show the existence of a recursively enumerable lambda theory where the number is always one or infinite. We also show that there are λ-theories such that some terms have only two fixed points. In a first example, this is obtained by means of a non-constructive (more precisely non-r.e.) λ-theory where the range property is violated. A second, more complex example of a non-r.e. λ-theory (with a higher unsolvability degree) shows that some terms can have only two fixed points while the range property holds for every term.

Normal functions and constructive ordinal notations

1976 ◽  
Vol 41 (2) ◽  
pp. 439-459 ◽  
Author(s):  
Larry W. Miller
Keyword(s):  
Continuous Function ◽  
Finite Number ◽  
Fixed Points ◽  
Common Fixed Points ◽  
Normal Function ◽  
Normal Functions ◽  
Ordinal Notations ◽  
The Common ◽  
Regular Ordinal ◽  
Number Class

An r-normal function is a strictly increasing continuous function from r to r where r is a regular ordinal > ω (identify an ordinal with the set of smaller ordinals). Given an r-normal function f one can form a sequence {f(x, −)}x<r of r-normal functions—the Veblen hierarchy [33] on f—as follows: f(0, −) = f and, for x > 0, f(x, −) enumerates in order {z ∣ f(y, z) = z for all y < x}, the common fixed points of the f(y, −)'s for y < x. In this paper we give as readable an exposition as we can of Veblen hierarchies and of Bachmann's and Isles's techniques in [3] and [15] of using higher finite number classes for forming sequences {f(x, −)}x<y where y > r of r-normal functions which extend the Veblen hierarchy on f. We will show how these sequences—Bachmann hierarchies—yield extremely natural constructive notations for ordinals in various initial segments of the second number class. We will also consider various other techniques for obtaining constructive ordinal notations and relate them to the notations obtained by Bachmann's and Isles's techniques. In particular, we will use these notations to characterize as directly and as usefully as we can various of Takeuti's systems of constructive ordinal notations, which he calls ordinal diagrams ([31], [32]).

Characteristics of complexes of conics in space of four dimensions

1925 ◽  
Vol 22 (5) ◽  
pp. 621-629
Author(s):  
C. G. F. James
Keyword(s):  
Finite Number ◽  
Fixed Points ◽  
Singular System ◽  
Arbitrary Point ◽  
Singular Surface ◽  
Singular Curve ◽  
Ordinary Space ◽  
Four Dimensions ◽  
Linear Complexes ◽  
Pass Through

In this note we obtain a system of characteristics on which depend the main enumerative properties of complexes, or systems ∞3, of conics in space of four dimensions. The method used is applicable to a large number of problems of this nature, and we select this illustration as being analogous to congruences in ordinary space. In particular a finite number of conics pass through an arbitrary point, thereby defining the order, n, of the system. Linear complexes are those for which this order is unity. The conics of a complex satisfy eight simple conditions, in general conditions of contact with a fixed form or forms, but they may include conditions of incidence with a surface, each such counting once, with a curve, each such counting twice, or with fixed points each such counting thrice. In particular only incidence conditions can occur when the system is linear, for otherwise more than one conic would pass through certain ∞3 points of space. Points through which more than a finite number of conics pass are termed singular, as well as their loci. Directrix constructs are necessarily singular, but they do not necessarily exhaust the singular system, for the complex may possess ∞2 curves lying on a surface. In the linear case through a point of a singular curve pass ∞2 conics, and through a point of a singular surface ∞1 conics. The possibilities in the nonlinear cases are too numerous to be detailed. Similarly the system of planes of the conies may possess a singular curve, through which ∞2 of the planes pass.

On the weak Kleene scheme in Kripke's theory of truth

1991 ◽  
Vol 56 (4) ◽  
pp. 1452-1468 ◽  
Author(s):  
James Cain ◽  
Zlatan Damnjanovic
Keyword(s):  
Fixed Point ◽  
Finite Number ◽  
Fixed Points ◽  
Function Symbol ◽  
Kripke’S Theory Of Truth ◽  
Limit Ordinal ◽  
Definable Sets ◽  
Primitive Recursive ◽  
Minimal Fixed Point ◽  
The Way

AbstractIt is well known that the following features hold of AR + T under the strong Kleene scheme, regardless of the way the language is Gödel numbered:1. There exist sentences that are neither paradoxical nor grounded.2. There are fixed points.3. In the minimal fixed point the weakly definable sets (i.e., sets definable as {n ∣ A(n) is true in the minimal fixed point}, where A(x) is a formula of AR + T) are precisely the sets.4. In the minimal fixed point the totally defined sets (sets weakly defined by formulae all of whose instances are true or false) are precisely the sets.5. The closure ordinal for Kripke's construction of the minimal fixed point is .In contrast, we show that under the weak Kleene scheme, depending on the way the Gödel numbering is chosen:1. There may or may not exist nonparadoxical, ungrounded sentences.2. The number of fixed points may be any positive finite number, ℵ0, or .3. In the minimal fixed point, the sets that are weakly definable may range from a subclass of the sets 1-1 reducible to the truth set of AR to the sets, including intermediate cases.4. Similarly, the totally definable sets in the minimal fixed point range from precisely the arithmetical sets up to precisely the sets.5. The closure ordinal for the construction of the minimal fixed point may be ω, , or any successor limit ordinal in between.In addition we suggest how one may supplement AR + T with a function symbol interpreted by a certain primitive recursive function so that, irrespective of the choice of the Gödel numbering, the resulting language based on the weak Kleene scheme has the five features noted above for the strong Kleene language.

Criteria and Methods for Reliable Three-Dimensional Image Reconstruction

1974 ◽  
Vol 32 ◽  
pp. 330-331
Author(s):  
R. A. Crowther
Keyword(s):  
Image Reconstruction ◽  
Finite Number ◽  
Density Distribution ◽  
Electron Micrographs ◽  
Mathematical Problem ◽  
Three Dimensional ◽  
Ideal Case ◽  
Dimensional Image ◽  
General Object ◽  
The Ideal

The reconstruction of a three-dimensional image of a specimen from a set of electron micrographs reduces, under certain assumptions about the imaging process in the microscope, to the mathematical problem of reconstructing a density distribution from a set of its plane projections.In the absence of noise we can formulate a purely geometrical criterion, which, for a general object, fixes the resolution attainable from a given finite number of views in terms of the size of the object. For simplicity we take the ideal case of projections collected by a series of m equally spaced tilts about a single axis.

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