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A CHARACTERIZATION OF UNIFORMLY DISTRIBUTED SEQUENCES OF INTEGERS

Miyoko Uchiyama, Saburo Uchiyama
1962 Journal of Faculty of Science Hokkaido University Series I Mathematics  
The main purpose 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 $2\pi\alpha x.$ )  ... 
doi:10.14492/hokmj/1530691262 fatcat:56i32mahtjaatd2r7c3q6imiba

Ergodic Measure Preserving Transformations of Finite Type

Stanley EIGEN, Arshag HAJIAN, Yuji ITO
1988 Tokyo Journal of Mathematics  
$p$ . transformation $S$ 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  ... 
doi:10.3836/tjm/1270133989 fatcat:ikm55maaqzhghnmtv62gqduldu

Unknotting Operations of Polygonal Type

Haruko AIDA
1992 Tokyo Journal of Mathematics  
An equivalence relation 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.  ... 
doi:10.3836/tjm/1270130254 fatcat:i2fummt7avgedlofevz2bflyfe

Shortest Unique Queries on Strings [chapter]

Xiaocheng Hu, Jian Pei, Yufei Tao
2014 Lecture Notes in Computer Science  
Let D be a long input string 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.  ... 
doi:10.1007/978-3-319-11918-2_16 fatcat:ehsayfmy4nfj7fpzfikxx5xqvm

RATIONAL APPROXIMATIONS TO ALGEBRAIC FUNCTIONS

Saburô Uchiyama
1961 Journal of Faculty of Science Hokkaido University Series I Mathematics  
For any positive 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).  ... 
doi:10.14492/hokmj/1530756192 fatcat:f23vuz7yf5dkhcfrsjlvzgj3xe

On the Topological Orbit Equivalence in a Class of Substitution Minimal Systems

Hisatoshi YUASA
2002 Tokyo Journal of Mathematics  
The characterization is given in terms 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  ... 
doi:10.3836/tjm/1244208850 fatcat:wj2s6anhzzdj7iwrpesdlbc3ia

Axiom-oriented Reasoning to Deal with Inconsistency Between Ontology and Knowledge Base

Anh Tuan Luu, Tho T. Quan, Tru H. Cao, Jin Song Dong
2007 International Semantic Web Conference  
Moreover, to improve the execution speed 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.  ... 
dblp:conf/semweb/LuuQCD07 fatcat:fgrf3iyvlnc3bhum62wucf5w4i

Frequently oscillatory solutions for multi-level partial difference equations

Zhi-Qiang Zhu, Sui Sun Cheng
2006 International Mathematical Forum  
By introducing frequency measures for double sequences 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  ... 
doi:10.12988/imf.2006.06126 fatcat:dioijfbxwrc6pm3rzzgpmrsobe

A theorem concerning locally peripherally separable spaces

F. B. Jones
1935 Bulletin of the American Mathematical Society  
A space is locally peripherally separable provided that, if P is a point 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.  ... 
doi:10.1090/s0002-9904-1935-06111-7 fatcat:jiusfgsu2jgcnnct4ng4vtzhoi

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

J.C. Bradfield
1998 Theoretical Computer Science  
One 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.  ... 
doi:10.1016/s0304-3975(97)00217-x fatcat:5s5cbfkq3ncdngny4qtoyqcl3e

The modal mu-calculus alternation hierarchy is strict [chapter]

J. C. Bradfield
1996 Lecture Notes in Computer Science  
One 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.  ... 
doi:10.1007/3-540-61604-7_58 fatcat:rqvgzyzj3neo7f7rpt465s5iga

Selection networks with 8n log2 n size and O(log n) depth [chapter]

Shuji Jimbo, Akira Maruoka
1992 Lecture Notes in Computer Science  
First we 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.  ... 
doi:10.1007/3-540-56279-6_69 fatcat:ftwtkip4jvcftp76vaa42mcrbq

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 (  ... 

Independent Sets in Hypergraphs and Ramsey Properties of Graphs and the Integers

Robert Hancock, Katherine Staden, Andrew Treglown
2019 SIAM Journal on Discrete Mathematics  
For a 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 ).  ... 
doi:10.1137/18m119642x fatcat:ad77ckfgrncrne2dmb3fmtawny
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