scholarly journals Deleting digits

2017 ◽  
Vol 101 (550) ◽  
pp. 60-68 ◽  
Author(s):  
Ioulia N. Baoulina ◽  
Martin Kreh ◽  
Jörn Steuding
Keyword(s):  
Negative Integer ◽  
Digit Sequence ◽  
Positive Integers

We consider here the positive integers with respect to their unique decimal expansions, where each n ∈ ℕ is given by for some non-negative integer k and digit sequence αkαk-1 … α0. With slight abuse of notation, we also use n to denote αkαk-1 … α0. For such sequences of digits (as well as for the numbers represented by the corresponding expansions) we write x ⊲ y if x is a subsequence of y, which means that either x = y or x can be obtained from y by deleting some digits of y. For example, 514 ⊲ 352148. The main problem is as follows: Given a set S ⊂ ℕ, find the smallest possible set M ⊂ S such that for all s ∈ S there exists m ∈ M with m ⊲ s.

A physical model for teaching multiplication of integers

1968 ◽  
Vol 15 (6) ◽  
pp. 525-528
Author(s):  
Warren H. Hill
Keyword(s):  
Positive Integer ◽  
Physical Model ◽  
Number Line ◽  
Negative Integer ◽  
Physical Models ◽  
Addition And Subtraction ◽  
Positive Integers ◽  
Partial Success

One of the pedagogical pitfalls involved in teaching the multiplication of integers is the scarcity of physical models that can be used to illustrate their product. This problem becomes even more acute if a phyiscal model that makes use of a number line is desired. If the students have previously encountered the operations of addition and subtraction of integers using a number line, then the desirability of developing the operation of multiplication in a similar setting is obvious. Many of the more common physical models that can be used to represent the product of positive and negative integers are unsuitable for interpretation on a number line. Partial success is possible in the cases of the product of two positive integers and the product of a positive and a negative integer. But, how does one illustrate that the product of two negative integers is equal to a positive integer?

The “unknotting number” associated with other local moves than the crossing-change

2018 ◽  
Vol 27 (10) ◽  
pp. 1850051
Author(s):  
Eiji Ogasa
Keyword(s):  
Positive Integer ◽  
Negative Integer ◽  
Positive Integers

The ordinary unknotting number of 1-dimensional knots, which is defined by using the crossing-change, is a very basic and important invariant. Let [Formula: see text] be a positive integer. It is very natural to consider the “unknotting number” associated with other local moves on [Formula: see text]-dimensional knots. In this paper, we prove the following. For the ribbon-move on 2-knots, which is a local move on knots, we have the following: There is a 2-knot which is changed into the unknot by two times of the ribbon-move not by one time. The “unknotting number” associated with the ribbon-move is unbounded. For the pass-move on 1-knots, which is a local move on knots, we have the following: There is a 1-knot such that it is changed into the unknot by two times of the pass-move not by one time and such that the ordinary unknotting number is [Formula: see text]. For any positive integer [Formula: see text], there is a 1-knot whose “unknotting number” associated with the pass-move is [Formula: see text] and whose ordinary unknotting number is [Formula: see text]. Let [Formula: see text] and [Formula: see text] be positive integers. For the [Formula: see text]-move on [Formula: see text]-knots, which is a local move on knots, we have the following: Let [Formula: see text] be a non-negative integer. There is a [Formula: see text]-knot which is changed into the unknot by two times of the [Formula: see text]-move not by one time. The “unknotting number” associated with the [Formula: see text]-move is unbounded. There is a [Formula: see text]-knot which is changed into the unknot by two times of the [Formula: see text]-move not by one. The “unknotting number” associated with the [Formula: see text]-move is unbounded. We prove the following: For any positive integer [Formula: see text] and any positive integer [Formula: see text], there is a [Formula: see text]-knot which is changed into the unknot by [Formula: see text] times of the twist-move not by [Formula: see text] times.

scholarly journals Two elementary commutativity theorems for generalized boolean rings

1997 ◽  
Vol 20 (2) ◽  
pp. 409-411
Author(s):  
Vishnu Gupta
Keyword(s):  
Positive Integer ◽  
Identity Element ◽  
Negative Integer ◽  
Free Ring ◽  
Boolean Rings ◽  
Commutativity Theorems ◽  
Fixed Positive Integer ◽  
Positive Integers ◽  
Torsion Free

In this paper we prove that ifRis a ring with1as an identity element in whichxm−xn∈Z(R)for allx∈Rand fixed relatively prime positive integersmandn, one of which is even, thenRis commutative. Also we prove that ifRis a2-torsion free ring with1in which(x2k)n+1−(x2k)n∈Z(R)for allx∈Rand fixed positive integernand non-negative integerk, thenRis commutative.

PERIODIC RESULTS FOR THE COLOURED (GENERALISED) JONES POLYNOMIAL

1995 ◽  
Vol 04 (04) ◽  
pp. 633-672
Author(s):  
BOHDAN I. KURPITA ◽  
KUNIO MURASUGI
Keyword(s):  
Jones Polynomial ◽  
Negative Integer ◽  
Vertex Model ◽  
Model Interpretation ◽  
Positive Integers

Using the vertex model interpretation of the coloured (generalised) Jones polynomial of a link L, we show that if the colour of the ith component is Ni+mir, then modulo tr−1 this coloured Jones polynomial is congruent, up to a product of calculable factors, to the coloured Jones polynomial with the colour of the ith component Ni, where Ni and r are positive integers, and mi is a non-negative integer. The proof depends on the fact that, up to a known factor, the coloured Jones polynomial of a link may be calculated from a (1, 1)-tangle, the closure of which represents the link.

scholarly journals Variations on a Conjecture of C. C. Yang Concerning Periodicity

2021 ◽  
Author(s):  
Xinling Liu ◽  
Risto Korhonen ◽  
Kai Liu
Keyword(s):  
Entire Function ◽  
Periodic Function ◽  
Meromorphic Functions ◽  
Negative Integer ◽  
Positive Integers

AbstractThe generalized Yang’s Conjecture states that if, given an entire function f(z) and positive integers n and k, $$f(z)^nf^{(k)}(z)$$ f ( z ) n f ( k ) ( z ) is a periodic function, then f(z) is also a periodic function. In this paper, it is shown that the generalized Yang’s conjecture is true for meromorphic functions in the case $$k=1$$ k = 1 . When $$k\ge 2$$ k ≥ 2 the conjecture is shown to be true under certain conditions even if n is allowed to have negative integer values.

scholarly journals The Radial Radio Number and the Clique Number of a Graph

2019 ◽  
Vol 9 (1S4) ◽  
pp. 1002-1006
Keyword(s):  
Connected Graph ◽  
Clique Number ◽  
Negative Integer ◽  
Radio Number ◽  
The Difference ◽  
The Given ◽  
Positive Integers ◽  
Radio Labeling

Let G(V(G),E(G)) be a graph. A radial radio labeling, f, of a connected graph G is an assignment of positive integers to the vertices satisfying the following condition: d(u, v) | f (u)  f (v) | 1  r(G) , for any two distinct vertices u, v V(G) , where d(u,v) and r(G) denote the distance between the vertices u and v and the radius of the graph G, respectively. The span of a radial radio labeling f is the largest integer in the range of f and is denoted by span(f). The radial radio number of G, r(G) , is the minimum span taken over all radial radio labelingsof G. In this paper, we construct a graph a graph for which the difference between the radial radio number and the clique number is the given non negative integer.

scholarly journals Enumerating Contingency Tables via Random Permanents

2008 ◽  
Vol 17 (1) ◽  
pp. 1-19 ◽  
Author(s):  
ALEXANDER BARVINOK
Keyword(s):  
Random Matrix ◽  
Contingency Tables ◽  
Randomized Algorithm ◽  
Concave Function ◽  
Accurate Estimate ◽  
Negative Integer ◽  
Total Weight ◽  
Positive Integers

Givenmpositive integersR= (ri),npositive integersC= (cj) such that Σri= Σcj=N, andmnnon-negative weightsW=(wij), we consider the total weightT=T(R, C;W) of non-negative integer matricesD=(dij) with the row sumsri, column sumscj, and the weight ofDequal to$\prod w_{ij}^{d_{ij}}$. For different choices ofR,C, andW, the quantityT(R,C;W) specializes to the permanent of a matrix, the number of contingency tables with prescribed margins, and the number of integer feasible flows in a network. We present a randomized algorithm whose complexity is polynomial inNand which computes a numberT′=T′(R,C;W) such thatT′ ≤T≤ α(R,C)T′ where$\alpha(R,C) = \min \bigl\{\prod r_i! r_i^{-r_i}, \ \prod c_j! c_j^{-c_j} \bigr\} N^N/N!$. In many cases, lnT′ provides an asymptotically accurate estimate of lnT. The idea of the algorithm is to expressTas the expectation of the permanent of anN×Nrandom matrix with exponentially distributed entries and approximate the expectation by the integralT′ of an efficiently computable log-concave function on ℝmn.

scholarly journals A result of commutativity of rings

1994 ◽  
Vol 17 (1) ◽  
pp. 65-72
Author(s):  
Vishnu Gupta
Keyword(s):  
Negative Integer ◽  
Positive Integers

In this paper we prove the following:THEOREM. Letn>1andmbe fixed relatively prime positive integers andkis any non-negative integer. IfRis a ring with unity1satisfyingxk[xn,y]=[x,ym]for allx,y∈RthenRis commutative.

On t-relaxed chromatic number of r-power paths

2018 ◽  
Vol 10 (02) ◽  
pp. 1850017
Author(s):  
Jun Lan ◽  
Wensong Lin
Keyword(s):  
Chromatic Number ◽  
Negative Integer ◽  
Minimum Integer ◽  
Vertex Set ◽  
Positive Integers

Let [Formula: see text] be a graph and [Formula: see text] a non-negative integer. Suppose [Formula: see text] is a mapping from the vertex set of [Formula: see text] to [Formula: see text]. If, for any vertex [Formula: see text] of [Formula: see text], the number of neighbors [Formula: see text] of [Formula: see text] with [Formula: see text] is less than or equal to [Formula: see text], then [Formula: see text] is called a [Formula: see text]-relaxed [Formula: see text]-coloring of [Formula: see text]. And [Formula: see text] is said to be [Formula: see text]-colorable. The [Formula: see text]-relaxed chromatic number of [Formula: see text], denote by [Formula: see text], is defined as the minimum integer [Formula: see text] such that [Formula: see text] is [Formula: see text]-colorable. Let [Formula: see text] and [Formula: see text] be two positive integers with [Formula: see text]. Denote by [Formula: see text] the path on [Formula: see text] vertices and by [Formula: see text] the [Formula: see text]th power of [Formula: see text]. This paper determines the [Formula: see text]-relaxed chromatic number of [Formula: see text] the [Formula: see text]th power of [Formula: see text].

Language, procedures, and the non-perceptual origin of natural number concepts

2016 ◽  
Author(s):  
David Barner
Keyword(s):  
General Framework ◽  
Ad Hoc ◽  
Building Blocks ◽  
Number Words ◽  
Linguistic Markers ◽  
Logical Foundations ◽  
Large Numbers ◽  
Perceptual Representations ◽  
Positive Integers ◽  
Perceptual Systems

Perceptual representations – e.g., of objects or approximate magnitudes –are often invoked as building blocks that children combine with linguisticsymbols when they acquire the positive integers. Systems of numericalperception are either assumed to contain the logical foundations ofarithmetic innately, or to supply the basis for their induction. Here Ipropose an alternative to this general framework, and argue that theintegers are not learned from perceptual systems, but instead arise toexplain perception as part of language acquisition. Drawing oncross-linguistic data and developmental data, I show that small numbers(1-4) and large numbers (~5+) arise both historically and in individualchildren via entirely distinct mechanisms, constituting independentlearning problems, neither of which begins with perceptual building blocks.Specifically, I propose that children begin by learning small numbers(i.e., *one, two, three*) using the same logical resources that supportother linguistic markers of number (e.g., singular, plural). Several yearslater, children discover the logic of counting by inferring the logicalrelations between larger number words from their roles in blind countingprocedures, and only incidentally associate number words with perception ofapproximate magnitudes, in an *ad hoc* and highly malleable fashion.Counting provides a form of explanation for perception but is not causallyderived from perceptual systems.

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