On the Inequality

1974 ◽  
Vol 17 (2) ◽  
pp. 193-199 ◽  
Author(s):  
Pal Fischer
Keyword(s):  
Positive Integer ◽  
Renyi’S Entropy ◽  
Image Position ◽  
Fixed Positive Integer ◽  
Special Case

In this article, we are concerned with the following inequality(1)where 0<pi<1, 0<q<1, (i=l, 2,…,n), n is a fixed positive integer, n≥2 and f(p)≠0 for <p<l.This inequality was first considered by A. Renyi, who gave the general differentiate solution of (1) for n≥3, [1]. With the help of this inequality one can characterize Renyi’s entropy [2].We shall state later the Renyi’s result, which will be a special case of the Theorem 3.

scholarly journals On the Inequality

1979 ◽  
Vol 22 (4) ◽  
pp. 483-489 ◽  
Author(s):  
Peter Kardos
Keyword(s):  
Positive Integer ◽  
Functional Inequality ◽  
Renyi’S Entropy ◽  
Image Position ◽  
Fixed Positive Integer ◽  
Special Case

In this paper, we are concerned with the functional inequality1where 0 < Pi < l, 0 < qi < l, fi(p)≠0, for 0 < P < 1, (i = 1, 2,..., n) and n is a fixed positive integer, n ≥ 2.Inequality (1) was studied by Rényi and Fischer, (see [1], [3]) in the special case2and this provided a characterization of Rényi's entropy.

A Combinatorial Theorem

1966 ◽  
Vol 9 (4) ◽  
pp. 515-516
Author(s):  
Paul G. Bassett
Keyword(s):  
Positive Integer ◽  
Combinatorial Theorem ◽  
Image Position ◽  
Fixed Positive Integer ◽  
Positive Integers

Let n be an arbitrary but fixed positive integer. Let Tn be the set of all monotone - increasing n-tuples of positive integers:1Define2In this note we prove that ϕ is a 1–1 mapping from Tn onto {1, 2, 3,…}.

Paradox of the class of all grounded classes

1953 ◽  
Vol 18 (2) ◽  
pp. 114-114 ◽  
Author(s):  
Shen Yuting
Keyword(s):  
Positive Integer ◽  
Natural Number ◽  
Image Position ◽  
Special Case

A class A for which there is an infinite progression of classes A1, A2, … (not necessarily all distinct) such thatis said to be groundless. A class which is not groundless is said to be grounded. Let K be the class of all grounded classes.Let us assume that K is a groundless class. Then there is an infinite progression of classes A1, A2, … such thatSince A1 ϵ K, A1 is a grounded class; sinceA1 is also a groundless class. But this is impossible.Therefore K is a grounded class. Hence K ϵ K, and we haveTherefore K is also a groundless class.This paradox forms a sort of triplet with the paradox of the class of all non-circular classes and the paradox of the class of all classes which are not n-circular (n a given natural number). The last of the three includes as a special case the paradox of the class of all classes which are not members of themselves (n = 1).More exactly, a class A1 is circular if there exists some positive integer n and classes A2, A3, …, An such thatFor any given positive integer n, a class A1 is n-circular if there are classes A2, …, An, such thatQuite obviously, by arguments similar to the above, we get a paradox of the class of all non-circular classes and a paradox of the class of all classes which are not n-circular, for each positive integer n.

scholarly journals GENERALIZED CONTINUED FRACTION EXPANSIONS WITH CONSTANT PARTIAL DENOMINATORS

2018 ◽  
Vol 107 (02) ◽  
pp. 272-288
Author(s):  
TOPI TÖRMÄ
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Continued Fraction ◽  
Positive Real Number ◽  
Rational Numbers ◽  
Positive Real ◽  
Image Position ◽  
Fixed Positive Integer ◽  
Continued Fraction Expansions ◽  
Positive Integers

We study generalized continued fraction expansions of the form $$\begin{eqnarray}\frac{a_{1}}{N}\frac{}{+}\frac{a_{2}}{N}\frac{}{+}\frac{a_{3}}{N}\frac{}{+}\frac{}{\cdots },\end{eqnarray}$$ where $N$ is a fixed positive integer and the partial numerators $a_{i}$ are positive integers for all $i$ . We call these expansions $\operatorname{dn}_{N}$ expansions and show that every positive real number has infinitely many $\operatorname{dn}_{N}$ expansions for each $N$ . In particular, we study the $\operatorname{dn}_{N}$ expansions of rational numbers and quadratic irrationals. Finally, we show that every positive real number has, for each $N$ , a $\operatorname{dn}_{N}$ expansion with bounded partial numerators.

scholarly journals The Number of $f$-Matchings in Almost Every Tree is a Zero Residue

10.37236/517 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Noga Alon ◽  
Simi Haber ◽  
Michael Krivelevich
Keyword(s):  
Positive Integer ◽  
Recent Result ◽  
Independent Sets ◽  
Fixed Positive Integer ◽  
Vertex Disjoint ◽  
Special Case ◽  
Fixed Tree ◽  
Labeled Tree

For graphs $F$ and $G$ an $F$-matching in $G$ is a subgraph of $G$ consisting of pairwise vertex disjoint copies of $F$. The number of $F$-matchings in $G$ is denoted by $s(F,G)$. We show that for every fixed positive integer $m$ and every fixed tree $F$, the probability that $s(F,\mathcal{T}_n) \equiv 0 \pmod{m}$, where $\mathcal{T}_n$ is a random labeled tree with $n$ vertices, tends to one exponentially fast as $n$ grows to infinity. A similar result is proven for induced $F$-matchings. As a very special special case this implies that the number of independent sets in a random labeled tree is almost surely a zero residue. A recent result of Wagner shows that this is the case for random unlabeled trees as well.

scholarly journals Pair correlation for fractional parts of αn2

2010 ◽  
Vol 148 (3) ◽  
pp. 385-407 ◽  
Author(s):  
D. R. HEATH–BROWN
Keyword(s):  
Positive Integer ◽  
Poisson Distribution ◽  
Algebraic Number ◽  
Rational Number ◽  
Pair Correlation ◽  
Partial Quotients ◽  
Integer Exponent ◽  
Image Position ◽  
Fixed Positive Integer ◽  
Diophantine Numbers

It was proved by Weyl [8] in 1916 that the sequence of values of αn2 is uniformly distributed modulo 1, for any fixed real irrational α. Indeed this result covered sequences αnd for any fixed positive integer exponent d. However Weyl's work leaves open a number of questions concerning the finer distribution of these sequences. It has been conjectured by Rudnick, Sarnak and Zaharescu [6] that the fractional parts of αn2 will have a Poisson distribution provided firstly that α is “Diophantine”, and secondly that if a/q is any convergent to α then the square-free part of q is q1+o(1). Here one says that α is Diophantine if one has (1.1) for every rational number a/q and any fixed ϵ > 0. In particular every real irrational algebraic number is Diophantine. One would predict that there are Diophantine numbers α for which the sequence of convergents pn/qn contains infinitely many squares amongst the qn. If true, this would show that the second condition is independent of the first. Indeed one would expect to find such α with bounded partial quotients.

scholarly journals The essential norms of composition operators

1992 ◽  
Vol 34 (2) ◽  
pp. 143-155 ◽  
Author(s):  
Boo Rim Choe
Keyword(s):  
Positive Integer ◽  
Hardy Space ◽  
Probability Measure ◽  
Composition Operators ◽  
Invariant Probability Measure ◽  
Open Unit ◽  
Rotation Invariant ◽  
Open Unit Ball ◽  
Image Position ◽  
Fixed Positive Integer

Throughout the paper n denotes a fixed positive integer unless otherwise specified. Let B = Bn denote the open unit ball of ℂn and let S = Sn denote its boundary, the unit sphere. The unique rotation-invariant probability measure on 5 will be denoted by σ = σn. For n = l, we use more customary notations D = B1, T = S1 and dσ1= dθ/2π. The Hardy space on B, denoted by H2(B), is then the space of functions f holomorphic on B for which

A Class of Polynomials in Self-Adjoint Operators in Spaces with an Indefinite Metric

1968 ◽  
Vol 20 ◽  
pp. 673-678 ◽  
Author(s):  
C.-Y. Lo
Keyword(s):  
Hilbert Space ◽  
Positive Integer ◽  
Orthogonal Decomposition ◽  
Inner Product ◽  
Indefinite Metric ◽  
Indefinite Inner Product ◽  
Image Position ◽  
Fixed Positive Integer ◽  
Adjoint Operators ◽  
Usual Product

Let H be a Hilbert space with the usual product [x, y] and with an indefinite inner product (x, y) which, for some orthogonal decompositionin H, is defined bywhereand dim H1 = κ, a fixed positive integer.

On Certain Sequences of Plus and Minus Ones

1978 ◽  
Vol 30 (01) ◽  
pp. 170-179 ◽  
Author(s):  
D. Borwein ◽  
W. Gawronski
Keyword(s):  
Positive Integer ◽  
Image Position ◽  
Fixed Positive Integer

Suppose throughout that c is a fixed positive integer, that and that

Relations between arithmetical functions

1967 ◽  
Vol 63 (4) ◽  
pp. 1027-1029
Author(s):  
C. J. A. Evelyn
Keyword(s):  
Positive Integer ◽  
Möbius Function ◽  
Natural Numbers ◽  
Arithmetical Functions ◽  
Mobius Function ◽  
Image Position ◽  
Fixed Positive Integer

In a recent note(1) I proved that if μ(n) denotes the usual Möbius function, N denotes a fixed positive integer and ifthenwhere T runs through all natural numbers ≤ x which are not divisible by an Nth power. In the present paper I shall establish some further relations of this character and, in particular, I shall prove that ifwherethenThus, in some respects, L(x) appears more regular than M(x), the sum over L(x/T) being multiplicative, whereas M(x1/N) is not.

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