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A CHARACTERIZATION OF UNIFORMLY DISTRIBUTED SEQUENCES OF INTEGERS

1962
*
Journal of Faculty of Science Hokkaido University Series I Mathematics
*

The main purpose

doi:10.14492/hokmj/1530691262
fatcat:56i32mahtjaatd2r7c3q6imiba
*of*this note is to present another characterization*of*uniformly distributed sequences*of**integers*, making use*of*a kind*of*integrals*defined*over the space*of**integers*. ... This, as well as the criterion quoted above, will have some analogy with the well-known characterization*of*uniform distribution $(mod 1)$*of*sequences*of*real numbers (cf. [3; Chap. IV]). ... .$ for every measurable*set*$E$*of*positive measure. But this is in substance contained in the proof*of*Proposition 4. (*Of*course, a similar result holds for sin $2\pi\alpha x.$ ) ...##
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Ergodic Measure Preserving Transformations of Finite Type

1988
*
Tokyo Journal of Mathematics
*

$p$ . transformation $S$

doi:10.3836/tjm/1270133989
fatcat:ikm55maaqzhghnmtv62gqduldu
*defined*on the measure space $(Y, c\mathscr{F}\*mu*)$ and the exh. w.w.*set*$V$ for $S$ under $B$ , with $\*mu*(V)=1$ . ... Moreover, it is possible to discuss direct sum decompositions*of*$Z$ , the*set**of*all*integers*. ... YUJI ITO DEPARTMENT*OF*MATHEMATICS, FACULTY*OF*SCIENCE AND TECHNOLOGY, KEIO UNrVERSITY HIYOSHI, KOHOKU-KU, YOKOHAMA 223, JAPAN ...##
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Unknotting Operations of Polygonal Type

1992
*
Tokyo Journal of Mathematics
*

An equivalence relation

doi:10.3836/tjm/1270130254
fatcat:i2fummt7avgedlofevz2bflyfe
*of*links. Let $L^{\*mu*}$ be the*set**of*all $\*mu*$ -component links. ... $\square $ $L^{\*mu*}/\sim n$ denotes the*set**of*equivalence classes*of*$L^{\*mu*}$ by $\sim n$ COROLLARY 3.2. ...##
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Shortest Unique Queries on Strings
[chapter]

2014
*
Lecture Notes in Computer Science
*

Let D be a long input string

doi:10.1007/978-3-319-11918-2_16
fatcat:ehsayfmy4nfj7fpzfikxx5xqvm
*of*n characters (from an alphabet*of*size up to 2 w , where w is the number*of*bits in a machine word). ... Given a substring q*of*D, a shortest unique query returns a shortest unique substring*of*D that contains q. ... As before, let M be the*set**of*MUS's*of*D. For each*MUS*D[i : j] in M, create an interval [i, j]. Denote by I the*set**of*all the intervals created this way. ...##
###
RATIONAL APPROXIMATIONS TO ALGEBRAIC FUNCTIONS

1961
*
Journal of Faculty of Science Hokkaido University Series I Mathematics
*

For any positive

doi:10.14492/hokmj/1530756192
fatcat:f23vuz7yf5dkhcfrsjlvzgj3xe
*integers*$\gamma_{1}\cdots,$ $\gamma_{m}$ and a real number $\lambda>0$ the number*of**sets**of**integers*$i_{1},$ $\cdots,$ $i_{m}$ such that $\sum_{\*mu*=1}^{m}\frac{i_{\*mu*}}{r_{\*mu*}}\leqq ... Denote by $E$ the*set**of*all such solutions $(p, q)$*of*(5). ...##
###
On the Topological Orbit Equivalence in a Class of Substitution Minimal Systems

2002
*
Tokyo Journal of Mathematics
*

The characterization is given in terms

doi:10.3836/tjm/1244208850
fatcat:wj2s6anhzzdj7iwrpesdlbc3ia
*of*the Perron-Frobenius eigenvalue*of*a matrix associated with a substitution. ... In this note, a partial answer to the problem to characterize the topological orbit equivalence class*of*substitution minimal systems. ...*Define*$p_{\sigma}$ : $K^{0}(T)\sigmaarrow \mathbb{R}$ by $p_{\sigma}(\psi_{n}(a))=\*mu*^{-()}n-1\beta(a))a\in \mathbb{Z}^{r}$ , where $\*mu*$ is the Perron-Frobenius eigenvalue*of*$M(\tau)$ and $\beta$ is ...##
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Axiom-oriented Reasoning to Deal with Inconsistency Between Ontology and Knowledge Base

2007
*
International Semantic Web Conference
*

Moreover, to improve the execution speed

dblp:conf/semweb/LuuQCD07
fatcat:fgrf3iyvlnc3bhum62wucf5w4i
*of*algorithms employed in the framework, we also discuss an axiom-oriented strategy to reason on a reduced space*of*formula to be inferred in Ontology and KB. ... When deployed in practical applications, Ontologies and KBs often suffer various kinds*of*inconsistency, which limit the applications performances significantly. ... Unsatisfied concept c with*set**of*formulae ∑. Output:*set*MUPS corresponding to c. ...##
###
Frequently oscillatory solutions for multi-level partial difference equations

2006
*
International Mathematical Forum
*

By introducing frequency measures for double sequences

doi:10.12988/imf.2006.06126
fatcat:dioijfbxwrc6pm3rzzgpmrsobe
*defined*on half lattice planes, oscillatory solutions can be*defined*for these equations and frequent oscillation criteria found. ... Let us denote the*set**of*all*integers*by Z, the*set**of*nonnegative*integers*by N, the*set*{a, a + 1, ..., b}*of**integers*by Z [a, b] , and the*set*{a, a + 1, a + 2, ...}*of**integers*by Z[a, ∞). ... Introduction In [1] , the two-level partial difference equation u (n+1) m = au (n) m−1 + bu (n) m + cu (n) m+1 , m ∈ Z, n ∈ N (1) where a, b, c ∈ R, Z is the*set**of**integers*and N the*set**of*nonnegative ...##
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A theorem concerning locally peripherally separable spaces

1935
*
Bulletin of the American Mathematical Society
*

A space is locally peripherally separable provided that, if P is a point

doi:10.1090/s0002-9904-1935-06111-7
fatcat:jiusfgsu2jgcnnct4ng4vtzhoi
*of*a region R, there exists in R a domain D containing P such that the boundary*of*D is separable. THEOREM. ... Alexandroff has shown that a connected metric space is completely separable provided it is locally completely separable, f It is the purpose*of*this paper to establish a similar theorem for connected, ... Let D\ denote some region*of*G. For each*integer*i let*Mu*denote the*set**of*all points X*of*the boundary j8i*of*D\ such that n x -i. ...##
###
Page 480 of American Journal of Mathematics Vol. 79, Issue 3
[page]

1957
*
American Journal of Mathematics
*

If M is a maximal 0-proper open semigroup in G, then the

*set*~(*MU*—M.) is identical with the closed subgroup b(M). Proof.*Set*B=~(*MU*—M). Since s(M)N—M=CO, then s(J) C BUM. ... The results*of*this section are valid for non-commutative groups, pro- vided s(A) is*defined*to be {v€ G: (a+ A)U(A+2)C A}. 3. Maximal 0-proper open semigroups. ...##
###
The modal mu-calculus alternation hierarchy is strict

1998
*
Theoretical Computer Science
*

One

doi:10.1016/s0304-3975(97)00217-x
fatcat:5s5cbfkq3ncdngny4qtoyqcl3e
*of*the open questions about the modal*mu*-calculus is whether the alternation hierarchy collapses; that is, whether all modal fixpoint properties can be expressed with only a few alternations*of*least ... It turns out that (~0 is the least ordinal not the order-type*of*a di well-ordering*of*N; this implies that the p-*definable**sets**of**integers*are actually a strict subset*of*the di*sets**of**integers*, which ... The hierarchy*of*the*sets**of**integers*defmable by Zip formulae*of*the arithmetic*mu*-calculus is a strict hierarchy. ...##
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The modal mu-calculus alternation hierarchy is strict
[chapter]

1996
*
Lecture Notes in Computer Science
*

One

doi:10.1007/3-540-61604-7_58
fatcat:rqvgzyzj3neo7f7rpt465s5iga
*of*the open questions about the modal*mu*-calculus is whether the alternation hierarchy collapses; that is, whether all modal fixpoint properties can be expressed with only a few alternations*of*least ... It turns out that (~0 is the least ordinal not the order-type*of*a di well-ordering*of*N; this implies that the p-*definable**sets**of**integers*are actually a strict subset*of*the di*sets**of**integers*, which ... The hierarchy*of*the*sets**of**integers*defmable by Zip formulae*of*the arithmetic*mu*-calculus is a strict hierarchy. ...##
###
Selection networks with 8n log2 n size and O(log n) depth
[chapter]

1992
*
Lecture Notes in Computer Science
*

First we

doi:10.1007/3-540-56279-6_69
fatcat:ftwtkip4jvcftp76vaa42mcrbq
*define*a comparator network called a layer. Notation 5 Let $n$ and $j$ be positive*integers*. ... Notation 1 (The number*of*the elements*of*a finite*set*) For a finite*set*$X,$ $|X|$ denotes the number*of*the elementS*of*X. ...##
###
Page 727 of American Mathematical Society. Proceedings of the American Mathematical Society Vol. 5, Issue 5
[page]

1954
*
American Mathematical Society. Proceedings of the American Mathematical Society
*

Then there exists an increasing infinite sequence

*of*non-negative*integers*mo,*mu*, m2, - - - and an infinite sequence*of*real numbers yo, 91, ye, - - - such that ... The cardinal number*of*the*set**of*all increasing infinite sequences*of*non-negative*integers*is 28°, and we may therefore well-order this*set*to form a transfinite sequence (1) So, Si, S2, i a 5g,*oF*( ...##
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Independent Sets in Hypergraphs and Ramsey Properties of Graphs and the Integers

2019
*
SIAM Journal on Discrete Mathematics
*

For a

doi:10.1137/18m119642x
fatcat:ad77ckfgrncrne2dmb3fmtawny
*set*A and a positive*integer*x, we*define*\bigl( A x \bigr) to be the*set**of*all subsets*of*A*of*size x, and we*define*\bigl( A \leq x \scrI r (\scrH ) := \Biggl\{ (I1, . . . , Ir) \in \scrP (V ... Indeed, consider the r-coloring \sigma*of*G*defined*by*setting*\sigma (e) = i when i is the least*integer*such that e \in f (S i ). ...
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