scholarly journals Universality and Almost Decidability

2017 ◽  
Author(s):  
Cristian S. Calude ◽  
Damien Desfontaines
Keyword(s):  
Open Problem ◽  
Universal Functions ◽  
Open Problems ◽  
Converse Implication ◽  
Density Zero ◽  
Natural Density ◽  
Positive Integers

We present and study new definitions of universal and programmable universal unary functions and consider a new simplicity criterion: almost decidability of the halting set. A set of positive integers S is almost decidable if there exists a decidable and generic (i.e. a set of natural density one) set whose intersection with S is decidable. Every decidable set is almost decidable, but the converse implication is false. We prove the existence of infinitely many universal functions whose halting sets are generic (negligible, i.e. have density zero) and (not) almost decidable. One result—namely, the existence of infinitely many universal functions whose halting sets are generic (negligible) and not almost decidable—solves an open problem in [9]. We conclude with some open problems.

scholarly journals The Smallest One-Realization of a Given Set

10.37236/1171 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Ping Zhao ◽  
Kefeng Diao ◽  
Kaishun Wang
Keyword(s):  
Open Problem ◽  
Feasible Set ◽  
Minimum Number ◽  
Chromatic Spectrum ◽  
Positive Integers ◽  
Mixed Hypergraph

For any set $S$ of positive integers, a mixed hypergraph ${\cal H}$ is a realization of $S$ if its feasible set is $S$, furthermore, ${\cal H}$ is a one-realization of $S$ if it is a realization of $S$ and each entry of its chromatic spectrum is either 0 or 1. Jiang et al. showed that the minimum number of vertices of a realization of $\{s,t\}$ with $2\leq s\leq t-2$ is $2t-s$. Král proved that there exists a one-realization of $S$ with at most $|S|+2\max{S}-\min{S}$ vertices. In this paper, we  determine the number  of vertices of the smallest one-realization of a given set. As a result, we partially solve an open problem proposed by Jiang et al. in 2002 and by Král  in 2004.

Open Problems Column

2021 ◽  
Vol 52 (3) ◽  
pp. 25-25
Author(s):  
William Gasarch
Keyword(s):  
21St Century ◽  
Open Problem ◽  
Open Problems

This issue's Open Problem Column is by Lance Fortnow and its titled Worlds to Die For: Open Oracle Questions for the 21st Century.

scholarly journals On ranges of variants of the divisor functions that are dense

2015 ◽  
Vol 11 (06) ◽  
pp. 1905-1912 ◽  
Author(s):  
Colin Defant
Keyword(s):  
Real Number ◽  
Open Problem ◽  
Dense Subset ◽  
Arithmetic Function ◽  
Divisor Functions ◽  
Positive Integers ◽  
Multiplicative Arithmetic ◽  
Multiplicative Arithmetic Function

For a real number t, let st be the multiplicative arithmetic function defined by [Formula: see text] for all primes p and positive integers α. We show that the range of a function s-r is dense in the interval (0, 1] whenever r ∈ (0, 1]. We then find a constant ηA ≈ 1.9011618 and show that if r > 1, then the range of the function s-r is a dense subset of the interval [Formula: see text] if and only if r ≤ ηA. We end with an open problem.

scholarly journals On the Parity of the Order of Appearance in the Fibonacci Sequence

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1928
Author(s):  
Pavel Trojovský
Keyword(s):  
Fibonacci Sequence ◽  
Natural Density ◽  
Almost All ◽  
Positive Integers

Let (Fn)n≥0 be the Fibonacci sequence. The order of appearance function (in the Fibonacci sequence) z:Z≥1→Z≥1 is defined as z(n):=min{k≥1:Fk≡0(modn)}. In this paper, among other things, we prove that z(n) is an even number for almost all positive integers n (i.e., the set of such n has natural density equal to 1).

scholarly journals The design of statements and the (re)formulation and resolution of open problems that address issues of social relevance with the use of digital technologies in the initial formation of Mathematics teachers

2019 ◽  
Vol 21 (2) ◽  
Author(s):  
Fabiane Fischer Figueiredo ◽  
Claudia Lisete Oliveira Groenwald
Keyword(s):  
Mathematics Teachers ◽  
Open Problem ◽  
Basic Education ◽  
Digital Technologies ◽  
Pedagogical Practice ◽  
Pedagogical Workshop ◽  
Social Relevance ◽  
Open Problems ◽  
Initial Formation ◽  
Design Activities

This paper presents the results obtained with an investigation, in which a pair of Mathematics graduates, participants of an Extension Course, carried out the design of a statement of open problems. This investigation also addressed a theme of social relevance, in which technologies were used so that these problems were (re)formulated and solved, with the use of technological resources, by students of Basic Education. In order for this objective to be achieved, the training teachers carried out the design activities of problems with the use of digital technologies, planning of the pedagogical practice, in which these problems would be proposed, and of execution of this practice, which occurred through a pedagogical workshop. After completing these activities, the graduates had the opportunity to discuss and reflect on the experiences of designer and teacher, so that they contributed to producing knowledge about the design of open problem statements that address issues of social relevance, (re)formulation and resolution using digital technologies, and how to propose such problems.

scholarly journals The Exact Security of PMAC with Two Powering-Up Masks

2019 ◽  
pp. 125-145 ◽  
Author(s):  
Yusuke Naito
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Open Problem ◽  
Random Function ◽  
Block Cipher ◽  
Random Permutation ◽  
Authentication Code ◽  
Open Problems ◽  
Main Research ◽  
Main Research Topic

PMAC is a rate-1, parallelizable, block-cipher-based message authentication code (MAC), proposed by Black and Rogaway (EUROCRYPT 2002). Improving the security bound is a main research topic for PMAC. In particular, showing a tight bound is the primary goal of the research, since Luykx et al.’s paper (EUROCRYPT 2016). Regarding the pseudo-random-function (PRF) security of PMAC, a collision of the hash function, or the difference between a random permutation and a random function offers the lower bound Ω(q2/2n) for q queries and the block cipher size n. Regarding the MAC security (unforgeability), a hash collision for MAC queries, or guessing a tag offers the lower bound Ω(q2m /2n + qv/2n) for qm MAC queries and qv verification queries (forgery attempts). The tight upper bound of the PRF-security O(q2/2n) of PMAC was given by Gaži et el. (ToSC 2017, Issue 1), but their proof requires a 4-wise independent masking scheme that uses 4 n-bit random values. Open problems from their work are: (1) find a masking scheme with three or less random values with which PMAC has the tight upper bound for PRF-security; (2) find a masking scheme with which PMAC has the tight upper bound for MAC-security.In this paper, we consider PMAC with two powering-up masks that uses two random values for the masking scheme. Using the structure of the powering-up masking scheme, we show that the PMAC has the tight upper bound O(q2/2n) for PRF-security, which answers the open problem (1), and the tight upper bound O(q2m /2n + qv/2n) for MAC-security, which answers the open problem (2). Note that these results deal with two-key PMACs, thus showing tight upper bounds of PMACs with single-key and/or with one powering-up mask are open problems.

scholarly journals On(a,1)-Vertex-Antimagic Edge Labeling of Regular Graphs

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Martin Bača ◽  
Andrea Semaničová-Feňovčíková ◽  
Tao-Ming Wang ◽  
Guang-Hui Zhang
Keyword(s):  
Regular Graphs ◽  
Open Problems ◽  
Arithmetic Sequence ◽  
Bijective Mapping ◽  
Vertex Weight ◽  
Positive Integers ◽  
Edge Labeling

An(a,s)-vertex-antimagic edge labeling(or an(a,s)-VAElabeling, for short) ofGis a bijective mapping from the edge setE(G)of a graphGto the set of integers1,2,…,|E(G)|with the property that the vertex-weights form an arithmetic sequence starting fromaand having common differences, whereaandsare two positive integers, and the vertex-weight is the sum of the labels of all edges incident to the vertex. A graph is called(a,s)-antimagic if it admits an(a,s)-VAElabeling. In this paper, we investigate the existence of(a,1)-VAE labeling for disconnected 3-regular graphs. Also, we define and study a new concept(a,s)-vertex-antimagic edge deficiency, as an extension of(a,s)-VAE labeling, for measuring how close a graph is away from being an(a,s)-antimagic graph. Furthermore, the(a,1)-VAE deficiency of Hamiltonian regular graphs of even degree is completely determined. More open problems are mentioned in the concluding remarks.

On a recursive sequence

Kybernetes ◽  
2007 ◽  
Vol 36 (1) ◽  
pp. 98-115
Author(s):  
Mehdi Dehghan ◽  
Reza Mazrooei‐Sebdani
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Open Problem ◽  
Design Methodology ◽  
Initial Conditions ◽  
Open Problems ◽  
Content Type ◽  
Rational Difference Equations ◽  
Global Behaviour ◽  
Recursive Sequence

PurposeThe aim in this paper is to investigate the dynamics of difference equation yn+1=(pyn+yn−k)/(qyn+yn−k), n=0,1,2,… where k∈{1,2,3,…}, the initial conditions y−k, … ,y−1,y0 and the parameters p and q are non‐negative.Design/methodology/approachThe paper studies characteristics such as the character of semicycles, periodicity and the global stability of the above mentioned difference equation.FindingsIn particular, the results solve the open problem introduced by Kulenovic and Ladas in their monograph, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures.Originality/valueThe global behaviour of the solutions of equation yn+1=(pyn+yn−k)/(qyn+yn−k), n=0,1,2,… were investigated providing valuable conclusions on practical data.

PRIMES IN A PRESCRIBED ARITHMETIC PROGRESSION DIVIDING THE SEQUENCE $\{a^k+b^k\}_{k=1}^{\infty}$

2009 ◽  
Vol 05 (04) ◽  
pp. 641-665 ◽  
Author(s):  
P. MOREE ◽  
B. SURY
Keyword(s):  
Arithmetic Progression ◽  
Natural Density ◽  
Positive Integers ◽  
Made In

Given positive integers a,b,c and d such that c and d are coprime, we show that the primes p ≡ c ( mod d) dividing ak+bkfor some k ≥ 1 have a natural density and explicitly compute this density. We demonstrate our results by considering some claims of Fermat that he made in a 1641 letter to Mersenne.

scholarly journals Good approximation and characterization of subgroups of R = Z

2001 ◽  
Vol 38 (1-4) ◽  
pp. 97-113 ◽  
Author(s):  
A. Bíró ◽  
J. M. Deshouillers ◽  
Vera T. Sós
Keyword(s):  
Continued Fraction ◽  
Irrational Number ◽  
Explicit Construction ◽  
Continued Fraction Expansion ◽  
Open Problems ◽  
Positive Integers

Let be a real irrational number and A =(xn) be a sequence of positive integers. We call A a characterizing sequence of or of the group Z mod 1 if lim n 2A n !1 k k =0 if and only if 2 Z mod 1. In the present paper we prove the existence of such characterizing sequences, also for more general subgroups of R = Z . Inthespecialcase Z mod 1 we give explicit construction of a characterizing sequence in terms of the continued fraction expansion of. Further, we also prove some results concerning the growth and gap properties of such sequences. Finally, we formulate some open problems.

Sign in / Sign up

Export Citation Format

Share Document