DEGREES OF CATEGORICITY ON A CONE VIAη-SYSTEMS

2017 ◽  
Vol 82 (1) ◽  
pp. 325-346 ◽  
Author(s):  
BARBARA F. CSIMA ◽  
MATTHEW HARRISON-TRAINOR
Keyword(s):  
Turing Degrees ◽  
Computable Structure ◽  
System Framework ◽  
Computable Structures ◽  
Degree Of Categoricity ◽  
Image Position ◽  
In The Beginning ◽  
The Given

AbstractWe investigate the complexity of isomorphisms of computable structures on cones in the Turing degrees. We show that, on a cone, every structure has a strong degree of categoricity, and that degree of categoricity is${\rm{\Delta }}_\alpha ^0 $-complete for someα. To prove this, we extend Montalbán’sη-system framework to deal with limit ordinals in a more general way. We also show that, for any fixed computable structure, there is an ordinalαand a cone in the Turing degrees such that the exact complexity of computing an isomorphism between the given structure and another copy${\cal B}$in the cone is a c.e. degree in${\rm{\Delta }}_\alpha ^0\left( {\cal B} \right)$. In each of our theorems the cone in question is clearly described in the beginning of the proof, so it is easy to see how the theorems can be viewed as general theorems with certain effectiveness conditions.

DEGREES OF CATEGORICITY AND SPECTRAL DIMENSION

2018 ◽  
Vol 83 (1) ◽  
pp. 103-116 ◽  
Author(s):  
NIKOLAY A. BAZHENOV ◽  
ISKANDER SH. KALIMULLIN ◽  
MARS M. YAMALEEV
Keyword(s):  
Natural Number ◽  
Open Problem ◽  
Quantitative Characteristic ◽  
Turing Degree ◽  
Spectral Dimension ◽  
Computable Structure ◽  
Rigid Structure ◽  
Degree Of Categoricity ◽  
Image Position

AbstractA Turing degreedis the degree of categoricity of a computable structure${\cal S}$ifdis the least degree capable of computing isomorphisms among arbitrary computable copies of${\cal S}$. A degreedis the strong degree of categoricity of${\cal S}$ifdis the degree of categoricity of${\cal S}$, and there are computable copies${\cal A}$and${\cal B}$of${\cal S}$such that every isomorphism from${\cal A}$onto${\cal B}$computesd. In this paper, we build a c.e. degreedand a computable rigid structure${\cal M}$such thatdis the degree of categoricity of${\cal M}$, butdis not the strong degree of categoricity of${\cal M}$. This solves the open problem of Fokina, Kalimullin, and Miller [13].For a computable structure${\cal S}$, we introduce the notion of the spectral dimension of${\cal S}$, which gives a quantitative characteristic of the degree of categoricity of${\cal S}$. We prove that for a nonzero natural numberN, there is a computable rigid structure${\cal M}$such that$0\prime$is the degree of categoricity of${\cal M}$, and the spectral dimension of${\cal M}$is equal toN.

scholarly journals XXV.—The Generating Function of the Reciprocal of a Determinant

1905 ◽  
Vol 40 (3) ◽  
pp. 615-629
Author(s):  
Thomas Muir
Keyword(s):  
Generating Function ◽  
Specific Character ◽  
Partial Fraction ◽  
Homogeneous Functions ◽  
Series Form ◽  
General Proposition ◽  
In Series ◽  
Image Position ◽  
The Given

(1) This is a subject to which very little study has been directed. The first to enunciate any proposition regarding it was Jacobi; but the solitary result which he reached received no attention from mathematicians,—certainly no fruitful attention,—during seventy years following the publication of it.Jacobi was concerned with a problem regarding the partition of a fraction with composite denominator (u1 − t1) (u2 − t2) … into other fractions whose denominators are factors of the original, where u1, u2, … are linear homogeneous functions of one and the same set of variables. The specific character of the partition was only definable by viewing the given fraction (u1−t1)−1 (u2−t2)−1…as expanded in series form, it being required that each partial fraction should be the aggregate of a certain set of terms in this series. Of course the question of the order of the terms in each factor of the original denominator had to be attended to at the outset, since the expansion for (a1x+b1y+c1z−t)−1 is not the same as for (b1y+c1z+a1x−t)−1. Now one general proposition to which Jacobi was led in the course of this investigation was that the coefficient ofx1−1x2−1x3−1…in the expansion ofy1−1u2−1u3−1…, whereis |a1b2c3…|−1, provided that in energy case the first term of uris that containing xr.

scholarly journals Note on the Application of Compound Poisson Processes to Sickness and Accident Statistics

1960 ◽  
Vol 1 (4) ◽  
pp. 224-237 ◽  
Author(s):  
Carl Philipson
Keyword(s):  
Random Process ◽  
Conditional Probability ◽  
Compound Poisson Process ◽  
Compound Poisson ◽  
Continuous Parameter ◽  
Absolute Probability ◽  
Image Position ◽  
The Given ◽  
Accident Statistics ◽  
Uniformly Bounded

In order to fix our ideas an illustration of the theory for (a) a general elementary random process, (b) a compound Poisson process and (c) a Polya process shall be given here below following Ove Lundberg (On Random Processes and Their Application to Accident and Sickness Statistics, Inaug. Diss., Uppsala 1940).Let the continuous parameter t* be measured on an absolute scale from a given point of zero and consider the random function N* (t*) which takes only non-negative and integer values with N* (o) = o. This function constitutes a general elementary random process for which the conditional probability that N* (t*) = n relative to the hypothesis that shall be denoted , while the absolute probability that N* (t) = n i.e. shall be written If quantities of lower order than dt* are neglected, we may write for the conditional probability that N* (t* + dt*) = n + 1 relative to thehyp othesis that N* (t*) = n, i.e. is the intensity function of the process which is assumed to be a continuous function of t* (the condition of existence for the integral over the given interval of t* for every n > m may be substituted for the condition of continuity). The expectations for an arbitrary but fix value of t* of N* (t*) and p* (t*) will be denoted by the corresponding symbol with a bar so thatIf is uniformly bounded for all n in the interval o ≤ t* < T*, where T* is an arbitrary but fix value of t*, we have i.a. that

scholarly journals Some Problems Connected with the Calculation of Stop Loss Premiums for Large Portfolios

1967 ◽  
Vol 4 (2) ◽  
pp. 170-174 ◽  
Author(s):  
Fredrik Esscher
Keyword(s):  
Distribution Function ◽  
Expected Number ◽  
Limit Values ◽  
Empirical Determination ◽  
The Mean ◽  
Premium Rates ◽  
Image Position ◽  
Stop Loss ◽  
The Given

When experience is insufficient to permit a direct empirical determination of the premium rates of a Stop Loss Cover, we have to fall back upon mathematical models from the theory of probability—especially the collective theory of risk—and upon such assumptions as may be considered reasonable.The paper deals with some problems connected with such calculations of Stop Loss premiums for a portfolio consisting of non-life insurances. The portfolio was so large that the values of the premium rates and other quantities required could be approximated by their limit values, obtained according to theory when the expected number of claims tends to infinity.The calculations were based on the following assumptions.Let F(x, t) denote the probability that the total amount of claims paid during a given period of time is ≤ x when the expected number of claims during the same period increases from o to t. The net premium II (x, t) for a Stop Loss reinsurance covering the amount by which the total amount of claims paid during this period may exceed x, is defined by the formula and the variance of the amount (z—x) to be paid on account of the Stop Loss Cover, by the formula As to the distribution function F(x, t) it is assumed that wherePn(t) is the probability that n claims have occurred during the given period, when the expected number of claims increases from o to t,V(x) is the distribution function of the claims, giving the conditioned probability that the amount of a claim is ≤ x when it is known that a claim has occurred, andVn*(x) is the nth convolution of the function V(x) with itself.V(x) is supposed to be normalized so that the mean = I.

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 THE COMPUTATIONAL CONTENT OF INTRINSIC DENSITY

2018 ◽  
Vol 83 (2) ◽  
pp. 817-828 ◽  
Author(s):  
ERIC P. ASTOR
Keyword(s):  
Reverse Mathematics ◽  
Previous Article ◽  
Asymptotic Density ◽  
Turing Degrees ◽  
Image Position ◽  
Immune Set

AbstractIn a previous article, the author introduced the idea of intrinsic density—a restriction of asymptotic density to sets whose density is invariant under computable permutation. We prove that sets with well-defined intrinsic density (and particularly intrinsic density 0) exist only in Turing degrees that are either high (${\bf{a}}\prime { \ge _{\rm{T}}}\emptyset \prime \prime$) or compute a diagonally noncomputable function. By contrast, a classic construction of an immune set in every noncomputable degree actually yields a set with intrinsic lower density 0 in every noncomputable degree.We also show that the former result holds in the sense of reverse mathematics, in that (over RCA0) the existence of a dominating or diagonally noncomputable function is equivalent to the existence of a set with intrinsic density 0.

The Meromorphisms of an Elliptic Function-Field

1936 ◽  
Vol 32 (2) ◽  
pp. 212-215 ◽  
Author(s):  
H. Davenport
Keyword(s):  
Finite Field ◽  
Elliptic Function ◽  
Function Field ◽  
Unit Element ◽  
Number Of Solutions ◽  
Complicated Case ◽  
Zero Divisors ◽  
The Subject ◽  
Image Position ◽  
The Given

1. Hasse's second proof of the truth of the analogue of Riemann's hypothesis for the congruence zeta-function of an elliptic function-field over a finite field is based on the consideration of the normalized meromorphisms of such a field. The meromorphisms form a ring of characteristic 0 with a unit element and no zero divisors, and have as a subring the natural multiplications n (n = 0, ± 1, …). Two questions concerning the nature of meromorphisms were left open, first whether they are commutative, and secondly whether every meromorphism μ satisfies an algebraic equation with rational integers n0, … not all zero. I have proved that except in the case (which is equivalent to |N−q|=2 √q, where N is the number of solutions of the Weierstrassian equation in the given finite field of q elements), both these results are true. This proof, of which I give an account in this paper, suggested to Hasse a simpler treatment of the subject, which throws still more light on the nature of meromorphisms. Consequently I only give my proof in full in the case in which the given finite field is the mod p field, and indicate briefly in § 4 how it generalizes to the more complicated case.

scholarly journals Classification from a Computable Viewpoint

2006 ◽  
Vol 12 (2) ◽  
pp. 191-218 ◽  
Author(s):  
Wesley Calvert ◽  
Julia F. Knight
Keyword(s):  
Set Theory ◽  
Model Theory ◽  
Large Body ◽  
Computable Structure ◽  
Descriptive Set Theory ◽  
Computable Structures ◽  
Structure Theorems ◽  
Sample Structure ◽  
Descriptive Set ◽  
Scott Rank

Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism or other important equivalence, in terms of relatively simple invariants. Where this is impossible, it is useful to have concrete results saying so. In model theory and descriptive set theory, there is a large body of work showing that certain classes of mathematical structures admit classification while others do not. In the present paper, we describe some recent work on classification in computable structure theory.Section 1 gives some background from model theory and descriptive set theory. From model theory, we give sample structure and non-structure theorems for classes that include structures of arbitrary cardinality. We also describe the notion of Scott rank, which is useful in the more restricted setting of countable structures. From descriptive set theory, we describe the basic Polish space of structures for a fixed countable language with fixed countable universe. We give sample structure and non-structure theorems based on the complexity of the isomorphism relation, and on Borel embeddings.Section 2 gives some background on computable structures. We describe three approaches to classification for these structures. The approaches are all equivalent. However, one approach, which involves calculating the complexity of the isomorphism relation, has turned out to be more productive than the others. Section 3 describes results on the isomorphism relation for a number of mathematically interesting classes—various kinds of groups and fields. In Section 4, we consider a setting similar to that in descriptive set theory. We describe an effective analogue of Borel embedding which allows us to make distinctions even among classes of finite structures. Section 5 gives results on computable structures of high Scott rank. Some of these results make use of computable embeddings. Finally, in Section 6, we mention some open problems and possible directions for future work.

Degree spectra of relations on computable structures in the presence of Δ20 isomorphisms

2002 ◽  
Vol 67 (2) ◽  
pp. 697-720 ◽  
Author(s):  
Denis R. Hirschfeldt
Keyword(s):  
Point Of View ◽  
The Other ◽  
Computable Structure ◽  
Sufficient Condition ◽  
Computable Structures ◽  
Linear Orderings ◽  
Degree Spectra ◽  
Other Hand ◽  
Infinite Degree ◽  
Degree Spectrum

AbstractWe give some new examples of possible degree spectra of invariant relations on Δ20-categorical computable structures, which demonstrate that such spectra can be fairly complicated. On the other hand, we show that there are nontrivial restrictions on the sets of degrees that can be realized as degree spectra of such relations. In particular, we give a sufficient condition for a relation to have infinite degree spectrum that implies that every invariant computable relation on a Δ20-categorical computable structure is either intrinsically computable or has infinite degree spectrum. This condition also allows us to use the proof of a result of Moses [23] to establish the same result for computable relations on computable linear orderings.We also place our results in the context of the study of what types of degree-theoretic constructions can be carried out within the degree spectrum of a relation on a computable structure, given some restrictions on the relation or the structure. From this point of view we consider the cases of Δ20-categorical structures, linear orderings, and 1-decidable structures, in the last case using the proof of a result of Ash and Nerode [3] to extend results of Harizanov [14].

Solution of a problem of Tarski

1956 ◽  
Vol 21 (1) ◽  
pp. 49-51 ◽  
Author(s):  
John Myhill
Keyword(s):  
Unary Operation ◽  
Negative Answer ◽  
Operation Symbol ◽  
Counter Example ◽  
Recursively Enumerable ◽  
Consistent Extension ◽  
Axiomatizable Theory ◽  
Definition Of ◽  
Image Position ◽  
The Given

We presuppose the terminology of [1], and we give a negative answer to the following problem ([1], p. 19): Does every essentially undecidable axiomatizable theory have an essentially undecidable finitely axiomatizable subtheory?We use the following theorem of Kleene ([2], p. 311). There exist two recursively enumerable sets α and β such that (1) α and β are disjoint (2) there is no recursive set η for which α ⊂ η, β ⊂ η′. By the definition of recursive enumerability, there are recursive predicates Φ and Ψ for whichWe now specify a theory T which will afford a counter-example to the given problem of Tarski. The only non-logical constants of T are two binary predicates P and Q, one unary operation symbol S, and one individual constant 0. As in ([1], p. 52) we defineThe only non-logical axioms of T are the formulae P(Δm, Δn) for all pairs of integers m, n satisfying Δ(m, n); the formulae Q(Δm, Δn) for all pairs of integers m, n satisfying Ψ(m, n); and the formulaT is consistent, since it has a model. It remains to show that (1) every consistent extension of T is undecidable (2) if T1 is a finitely axiomatizable subtheory of T, there exists a consistent and decidable extension of T1 which has the same constants as T1.

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