Finite Sets Of Natural Numbers Research Articles

Disjoint hypercyclicity, Sidon sets and weakly mixing operators

AbstractWe prove that a finite set of natural numbers J satisfies that $J\cup \{0\}$ is not Sidon if and only if for any operator T, the disjoint hypercyclicity of $\{T^j:j\in J\}$ implies that T is weakly mixing. As an application we show the existence of a non-weakly mixing operator T such that $T\oplus T^2\oplus\cdots \oplus T^n$ is hypercyclic for every n.

There are Infinitely Many Fibonacci Primes

We invent a novel algorithm and solve the Fibonacci prime conjecture by an interaction between proof and algorithm. From the entire set of natural numbers successively deleting the residue class 0 mod a prime, we retain this prime and possibly delete another one prime retained, then we invent a recursive sieve method, a modulo algorithm on finite sets of natural numbers, for indices of Fibonacci primes. The sifting process mechanically yields a sequence of sets of natural numbers, which converges to the index set of all Fibonacci primes. The corresponding cardinal sequence is strictly increasing. The algorithm reveals a structure of particular order topology of the index set of all Fibonacci primes, then we readily prove that the index set of all Fibonacci primes is an infinite set based on the existing theory of the structure. Some mysteries of primes are hidden in second order arithmetics.

The FAN principle and weak König's lemma in herbrandized second-order arithmetic

We introduce a herbrandized functional interpretation of a first-order semi-intuitionistic extension of Heyting Arithmetic and study its main properties. We then extend the interpretation to a certain system of second-order arithmetic which includes a (classically false) formulation of the FAN principle and weak König's lemma. It is shown that any first-order formula provable in this system is classically true. It is perhaps worthy of note that, in our interpretation, second-order variables are interpreted by finite sets of natural numbers.

Balance problems for integer circuits

We investigate the computational complexity of balance problems for {∖,⋅}-circuits computing finite sets of natural numbers. These problems naturally build on problems for integer expressions and integer circuits studied by Stockmeyer and Meyer (1973), McKenzie and Wagner (2007), and Glaßer et al. (2010).Our work shows that the balance problem for {∖,⋅}-circuits is undecidable which is the first natural problem for integer circuits or related constraint satisfaction problems that admits only one arithmetic operation and is proven to be undecidable.Starting from this result we precisely characterize the complexity of balance problems for proper subsets of {∖,⋅}. These problems turn out to be complete for one of the classes L, NL, and NP.

There Are Infinitely Many Mersnne Composite Numbers with Prime Exponents

By extending both arithmetical operations into finite sets of natural numbers, from the entire set of natural numbers successively deleting some residue classes modulo a prime, we invented a recursive sieve method or algorithm on natural numbers and their sets. The algorithm mechanically yields a sequence of sets, which converges to the set of all primes p such that 2p + 1 divides the Mersenne number Mp. The cardinal sequence corresponding to the sequence of sets is strictly increasing. So that we have captured enough usable structures, without any estimation, the existing theories of those structures allow us to prove an exact result: there are infinitely many Mersenne composite numbers with prime exponents Mp.

On integer sequences in product sets

Let B be a finite set of natural numbers or complex numbers. Product set corresponding to B is defined by B.B:={ab:a,b∈B}. In this paper we give an upper bound for longest length of consecutive terms of a polynomial sequence present in a product set accurate up to a positive constant. We give a sharp bound on the maximum number of Fibonacci numbers present in a product set when B is a set of natural numbers and a bound which is accurate up to a positive constant when B is a set of complex numbers.

On the Universality and Non-Universality of Spiking Neural P Systems With Rules on Synapses.

Spiking neural P systems with rules on synapses are a new variant of spiking neural P systems. In the systems, the neuron contains only spikes, while the spiking/forgetting rules are moved on the synapses. It was obtained that such system with 30 neurons (using extended spiking rules) or with 39 neurons (using standard spiking rules) is Turing universal. In this work, this number is improved to 6. Specifically, we construct a Turing universal spiking neural P system with rules on synapses having 6 neurons, which can generate any set of Turing computable natural numbers. As well, it is obtained that spiking neural P system with rules on synapses having less than two neurons are not Turing universal: i) such systems having one neuron can characterize the family of finite sets of natural numbers; ii) the family of sets of numbers generated by the systems having two neurons is included in the family of semi-linear sets of natural numbers.

Intersections of adelic groups on a surface

We solve a technical problem related to adeles on an algebraic surface. Given a finite set of natural numbers, one can associate with it an adelic group. We show that this operation commutes with taking intersections if the surface is defined over an uncountable field, and we provide a counterexample otherwise.Bibliography: 12 titles.

Shift graphs on precompact families of finite sets of natural numbers

We study graphs defined on families of finite sets of natural numbers and their chromatic properties. Of particular interest are graphs for which the edge relation is given by the shift. We show that when considering shift graphs with infinite chromatic number, one can center attention on graphs defined on precompact thin families. We define a quasi-order relation on the collection of uniform families defined in terms of homomorphisms between their corresponding shift graphs, and show that there are descending ω1-sequences. Specker graphs are also considered and their relation with shift graphs is established. We characterize the family of Specker graphs which contain a homomorphic image of a shift graph.

On the Sophie Germain prime conjecture

By extending the operations +,× on natural numbers to the operations on finite sets of natural numbers, we founded a new formal system of a second order arithmetic 〈P(N), N,+,×,0,1, ∈〉. We designed...

Theories for TC0 and Other Small Complexity Classes

We present a general method for introducing finitely axiomatizable "minimal" two-sorted theories for various subclasses of P (problems solvable in polynomial time). The two sorts are natural numbers and finite sets of natural numbers. The latter are essentially the finite binary strings, which provide a natural domain for defining the functions and sets in small complexity classes. We concentrate on the complexity class TC^0, whose problems are defined by uniform polynomial-size families of bounded-depth Boolean circuits with majority gates. We present an elegant theory VTC^0 in which the provably-total functions are those associated with TC^0, and then prove that VTC^0 is "isomorphic" to a different-looking single-sorted theory introduced by Johannsen and Pollet. The most technical part of the isomorphism proof is defining binary number multiplication in terms a bit-counting function, and showing how to formalize the proofs of its algebraic properties.

On the System of Differences in Finite Sets of Natural Numbers

In the paper [1] we investigated the following question: If n, a, d are natural numbers and if A is a set of a integers of the interval [0, n], how many (at most resp. at least) pairs (x, y) ∈ A × A exist with y − x = d ? Here we consider the question which arises from the previous one by exchanging y − x = d by y − x ≥ d. We establish an upper bound for the number of these pairs.

A partition theorem for α-large sets

Working with Hardy hierarchy and the notion of largeness determined by it, we define the notion of a partition of a finite set of natural numbers $A=∪_{i

Effective Enumerations of Families of Finite Sets of Natural Numbers

Effective Enumerations of Families of Finite Sets of Natural Numbers

How good can a nonhereditary family be?

A family of vectors of a Hilbert space H is said to be hereditarily complete, if it has biorthogonal\(\mathfrak{X}'\) (minimally) and any element of H can be reconstructed from its Fourier series:. In this paper we describe all pairs of spaces A, B, which contain minimal mutually biorthogonal and complete families\(\mathfrak{X}, \mathfrak{X}' ( V\left( \mathfrak{X} \right) = A, V\left( {\mathfrak{X}'} \right) = B\) and: for this it is necessary and sufficient that the operator PAPBPA not be completely continuous. This assertion allows one to prove that: 1) if dn > 0,, then there exist an orthonormal basis {ϕn}v⩾1 and complete but not hereditarily complete biorthogonal families\(\mathfrak{X}, \mathfrak{X}'\) in H, such that ∥ Xn-ϕn∥⩽dn, ∥x′n-ϕn∥⩽ dr, (n⩾1), 2) if, then there exist families of the type described in the preceding assertion for which, where σ is any finite set of natural numbers and is the spectral projector corresponding to it. One of the auxiliary assertions is the description of all real collections α=(αk)n k=1, representable in the form, where q is a Hilbert seminorm defined in the Euclidean space En, {fk)n k=1 is a suitable orthonormal basis. This set is the convex hull of all permutations of the eigenvalues (λ1, ..., λn) of the seminorm q.

On cyclic sets

A SYSTEM of subsets of the Cartesian product of two finite sets of natural numbers, called cyclic sets, is studied. Particular attention is devoted to the study of the power characteristics of one of the subsets of the system, called the set of cycles. It is shown that cycles can be used to establish the optimality of the transfer plan in the transport problem.

Combinatorial systems I. Cylindrical problems

The reduction of one combinatorial decision problem PI to another P2 can be achieved by constructing an effective map r : Pa --~ P2 such that each problem p in /'1 is reducible to the corresponding problem r in P2 (el. [4]). Judicious choice of ~b can sometimes ensure that p ~,~ r If, in addition, all the problems in P2, are cylinders [2] then p 41 r so that the reduction of P1 to P2 is best possible. This is because any cylinder has maximum one-one degree within its many-one degree--two cylinders which are many-one equivalent are actually one-one equivalent. The primary aim of this paper is, therefore, to identify some combinatorial problems which are always cylinders. Our investigation is effected by considering those functions on the natural numbers whose values are finite sets of natural numbers--system functions--which possess certain formal properties in common with those arising naturally from G6del numbering of many kinds of combinatorial system. This approach also enables us to construct system functions with noncylindrical problems--a matter reserved for a later paper. Many aspects of combinatorial systems (partial propositional calculi, Markov algorithms semi-Thue systems, etc.) are comprehended in Tarski's definition of deductive systems [5]. Accordingly we first expound a general method of recognizing cylindrical combinatorial problems of deductive systems and then apply it to system functions. For an explanation of the concepts of recursive-function theory as used in this paper the reader is referred to [2]; the notion of a cylinder, which is central to our discussion, and its elementary properties are presented on p. 89. The idea of using system functions in the theory of combinatorial systems is due to W. E. Singletary. The points in the ensuing development at which I am indebted to him are too numerous to mention.

Post’s problem, admissible ordinals, and regularity

1. Introduction. The basic notions of metarecursion theory were introduced in [7]. Metarecursioni theory is an attempt to generalize the ideas and arguments of recursion theory from the natural numbers to the recursive ordinals. Ordinary recursion theory concerns itself with finite sets of natural numbers. Metarecursion theory deals in an analogous fashion with metafinite sets of recursive ordinals. In [7] it was seen that two of the deepest results of ordinary recursion theory, the solution of Post's problem [3] and the maximal set construction [4], generalize to the metarecursive case. Theorem 4 of [7] states that there exist two metarecursively enumerable sets of recursive ordinals such that neither is metarecursive in the other. Kreisel [6] casts doubt on the contention that Theorem 4 of [7] is the correct generalization of Post's problem. He wonders how to correctly formulate the notion of Turing reducibility for recursive ordinals and he discusses four possible choices. It is not our place to decide the correct notion of Turing reducibility for metarecursion theory; however, in ?4 we show that there exist two metarecursively enumerable sets such that neither is reducible to the other by any of the four methods discussed by Kreisel. In [7] two sets of recursive ordinals are said to have the same metadegree if each is metarecursive in the other. It is not known if the metadegrees of metarecursively enumerable sets are order-isomorphic to the Turing degrees of recursively enumerable sets, but all the existing evidence seems to favor an affirmative answer. Driscoll [2] has shown the metadegrees of metarecursively