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1.
SHEN SHIYI 《数学年刊A辑(中文版)》1980,1(2):223-234
In this paper, we give a coding theorem for general source sequence. A source
sequence
\[{\mathcal{T}^{(n)}} = \{ [{X^{(n)}},{p^{(n)}}({X^{(n)}})],[{X^{(n)}} \otimes {Y^{(n)}},{\rho ^{(n)}}({X^{(n)}},{Y^{(n)}})]\} \]
is said to be \[({R^{(n)}},{d^{(n)}})\]-compress, if (i)\[{R^{(n)}}\], \[{d^{(n)}}\] are two sequences of real numbers
\[{R^{(n)}} \to \infty \]; (ii) there exist \[{\varepsilon ^{(n)}} > 0({\varepsilon ^{(n)}} \to 0)\] and set \[{B^{(n)}} \subset {Y^{(n)}}\] so that \[|{B^{(n)}}| \leqslant {2^{{R^{(n)}}(1 + {\varepsilon ^{(n)}})}}\]
and
\[{p^{(n)}}{\text{\{ }}({X^{(n)}}){\rho ^{(n)}}({X^{(n)}},{B^{(n)}}) \leqslant {d^{(n)}}\} \geqslant 1 - {\varepsilon ^{(n)}}\]
where \[{\rho ^{(n)}}({X^{(n)}},{B^{(n)}}) = \mathop {\min }\limits_{{y^{(n)}} \in {B^{(n}}} {\rho ^{(n)}}({X^{(n)}},{Y^{(n)}})\].
A \[{\mathcal{T}^{(n)}}\] is said to be \[({\mathcal{F}^{(n)}}{D^{(n)}})\]-information bounded, if (i)\[{\mathcal{F}^{(n)}} \to \infty \];(ii) there exist \[{\varepsilon ^{(n)}} > 0\] and conditional probability \[{Q^{(n)}}({Y^{(n)}}/{X^{(n)}})\] so that probability distribution
\[{p^{(n)}}({X^{(n)}}{Y^{(n)}}) = {p^{(n)}}({X^{(n)}}){Q^{(n)}}({Y^{(n)}}/{X^{(n)}})\] is satisfied by
\[\begin{gathered}
{p^{(n)}}\{ ({X^{(n)}},{Y^{(n)}}):i({X^{(n)}},{Y^{(n)}}) \leqslant {\mathcal{F}^{(n)}}(1 + {\varepsilon ^{(n)}}), \hfill \ {\rho ^{(n)}}({X^{(n)}},{Y^{(n)}}) \leqslant {\rho ^{(n)}}\} \geqslant 1 - {\varepsilon ^{(n)}} \hfill \\
\end{gathered} \]
where
\[i({X^{(n)}},{Y^{(n)}}) = \log \frac{{{p^{(n)}}({X^{(n)}},{Y^{(n)}})}}{{{p^{(n)}}({X^{(n)}}){p^{(n)}}({Y^{(n)}})}}\]
Theorem The necessary and sufficient conditions for a source sequence \[{\mathcal{F}^{(n)}}\] to be
\[({R^{(n)}},{a^{(n)}})\]-compress is that \[{\mathcal{F}^{(n)}}\] must be \[({R^{(n)}},{a^{(n)}})\]-information bounded.
From the theorem we obtain immediately the coding theorem and its converse for stationary and unstationary sources with memory. 相似文献
2.
<正>贵刊2013年6月下第20页上洪振铎老师的《为神奇的分数m7增添魅力》一文非常有趣,也很有启发.本文目的是将该文后半部分数字等式,推广成一般等式.在此,与朋友们交流,并请指正.图1性质1已知abcdef是一个六位数,将图1各顶点处的数字按顺时针或逆时针排成数列,其和及平方和都会相等,即(Ⅰ)二位数:ab n+bc n+cd n+de n+ef n+fa n=af n+fe n+ed n+dc n+cb n+ba n.(Ⅱ)三位数:abc n+bcd n+cde n+def n+efa n+fab n=afe n+fed n+edc n+dcb n+cba n+baf n. 相似文献
3.
本文研究了麦比乌斯梯子C(2n,n)的强边染色问题.利用组合分析的方法,得到了如下结果:当n=3时,χ'_s(C(2n,n))=9;当n=4时,χ'_s(C(2n,n))=10;当n=5,8时,χ'_s(C(2n,n))=8;当n 3且n≡2(mod 4)时,χ'_s(C(2n,n))=6;当n 7且n≡0,1或3(mod 4)时,χ'_s(C(2n,n))=7. 相似文献
4.
构造函数解决与自然数有关的问题 总被引:1,自引:0,他引:1
解决与自然数有关的命题通常用数学归纳法、二项式定理的展开式 .而数列作为定义在自然数集上的函数 ,若用数学归纳法解题有一定的难度 ,如果将问题转化为函数来处理 ,则往往使问题变得简洁、容易 ,此时常常将 n视为自变量 .下面举几例说明 .例 1 已知 n∈ N ,证明不等式1 12 13 … 1n <2 n .证明 构造函数 f ( n) =1 12 13 … 1n - 2 n ,∵ f ( n 1 ) - f ( n) =1n 1 2 n - 2 n 1 = n - n 1( n 1 n ) n 1 <0 ,∴ f ( n 1 ) 相似文献
5.
证明不等式的几种特殊方法 总被引:2,自引:0,他引:2
文[1]给出了六种证明不等式的特殊方法.这里再给出四种,以解决一些不等式的证明问题.1 利用二项式定理证明对于有些不等式,可根据其结构特点,联想或构造二项式模型,利用二项式定理来证.例1 (第2 1届全苏数学竞赛)求证:对于任意的正整数n ,不等式(2n + 1) n ≥(2n) n + (2n - 1) n成立.证 由二项式定理,有 (2n + 1) n- (2n - 1) n=2 [C1n(2n) n -1+C3n(2n) n -3 +…]≥2C1n(2n) n -1=(2n) n,即(2n + 1) n≥(2n) n+ (2n - 1) n.例2 (1988年全国高中数学联赛)已知a ,b为正实数,且1a+ 1b =1.试证对于每一个n∈N都有(a +b) n-an-bn≥2 2n-… 相似文献
6.
利用正整数n的一类特殊的3分拆n=n1+n2+n3,n1>n2>n3≥1,且n2+n3>n1的Ferrers图将不定方程4x1+3x2+2x3=n(n≥9)的正整数解与这种分拆联系起来,从而得到了该不定方程的正整数解数公式;同时也给出了正整数n的一类4分拆的计数公式.此外,还给出了周长为n的整边三角形的计数公式的一个简单证明. 相似文献
7.
对于任意正整数n,S(n),SL(n),φ(n)分别为Smarandache函数,Smarandache LCM函数和Euler函数.本文利用S(n),SL(n),φ(n)的基本性质结合初等方法推广了方程S(n)=φ(n)和SL(n)=φ(n),研究了方程S(SL(n))=φ(n)的可解性,给出并证明了该方程仅有正整数解n=1,8,9,12,18. 相似文献
8.
9.
文 [1 ]中给出了一个涉及n的不等式 :设正整数n >1 ,则2n + 23·n - 2 (n - 1 ) + 23·n - 1≤n <4n + 36 ·n - 4(n - 1 ) + 36 ·n - 1 ( 1 )由不等式 ( 1 ) ,可推出2n + 23·n - 2 - 13≤∑nk=1k≤4n + 36 ·n - 16 ( 2 )当且仅当n =1 ,2时 ,式中等号成立 .本文给出类似于不等式 ( 1 )的关于 kn的两个不等式 ,并提出一个猜想 .定理 1 设正整数n >1 ,则1 2n + 71 6 ·3n - 1 2 (n - 1 ) + 71 6 ·3n - 1 <3n <3n + 24 ·3n - 3(n - 1 ) + 24 ·3n - 1 ( 3)证 要证上限不等式 3n <3n + 24 ·3n- 3(n - 1 ) + 24 ·3n - 1 ,只要证( 3n - 2… 相似文献
10.
自然数方幂和问题是指Sk(n)=nΣi=1ik(n,k∈N)的计算与表示.早在公元前二百多年,希腊著名科学家阿基米德就已经得出了k=2和k=3时的结果:S2(n)=12+22+…+n2=n(n+1)(2n+1)/6,S3(n)=13+23+…+n3=n2(n+1)2/4,尽管他的证明比较复杂,但S4(n)的结果却始终无法找到,直到一千多年之后的11世纪,阿拉伯数学家才得道:S4(n)=1/30n(n+1)(2n+1)(3n2+3n-1).…… 相似文献
11.
在解析几何中有二次曲线与直线位置关系的讨论、二次曲面与直线位置关系的讨论,而二次曲面与平面相关位置关系的探讨较少.本文给出二次曲面a11x2+a22y2+a33z2+2a12xy+2a13xz+2a23yz+2a14x+2a24y+2a34z+a44=0(1)和平面Ax+By+Cz+D=0(2)的相对位置的判别式Δ=a11a12a13a14Aa21a22a23a24Ba31a32a33a34Ca41a42a43a44DA B C D0(aij=aji).(3)并证明了:若Δ>0,则二次曲面(1)与平面(2)相交;若Δ=0,则(1)和(2)相切;若Δ<0,则(1)和(2)相离. 相似文献
12.
Sho Suda 《Journal of Combinatorial Theory, Series A》2012,119(1):117-134
We define new parameters, a zero interval and a dual zero interval, of subsets in P- or Q-polynomial association schemes. A zero interval of a subset in a P-polynomial association scheme is a successive interval index for which the inner distribution vanishes, and a dual zero interval of a subset in a Q-polynomial association scheme is a successive interval index for which the dual inner distribution vanishes. We derive bounds of the lengths of a zero interval and a dual zero interval using the degree and dual degree respectively, and show that a subset in a P-polynomial association scheme (resp. a Q-polynomial association scheme) having a large length of a zero interval (resp. a dual zero interval) induces a completely regular code (resp. a Q-polynomial association scheme). Moreover, we consider the spherical analogue of a dual zero interval. 相似文献
13.
Scalarization of Henig Proper Efficient Points in a Normed Space 总被引:1,自引:0,他引:1
In a general normed space equipped with the order induced by a closed convex cone with a base, using a family of continuous monotone Minkowski functionals and a family of continuous norms, we obtain scalar characterizations of Henig proper efficient points of a general set and a bounded set, respectively. Moreover, we give a scalar characterization of a superefficient point of a set in a normed space equipped with the order induced by a closed convex cone with a bounded base. 相似文献
14.
《Journal of Pure and Applied Algebra》2022,226(11):107116
It was proved in [4] that every group ring of a torsion abelian group over a commutative local ring is a semi-clean ring. It was asked in [4] whether every group ring of a torsion abelian group over a commutative clean ring is a semi-clean ring and whether every group ring of a torsion abelian group over a commutative semi-clean ring is a semi-clean ring. In this paper, we give a positive answer to question 1 and a negative answer to question 2. 相似文献
15.
B. Lemaire 《Set-Valued Analysis》2001,9(1-2):111-121
This paper deals with a variant of a dynamical selection scheme introduced by Attouch and Cominetti for ill-posed convex minimization which combines approximation with the steepest descent method by mean of a suitable parameterization of the approximation parameter as a function of the time. This variant applies to a general inclusion with a maximal monotone operator by mean of a staircase parameterization. A discrete analogue is also considered. Applications to selecting a particular zero of a maximal monotone operator or a particular fixed point of a nonexpansive mapping via regularization techniques are presented. Finally, the alternative use of well-posedness by perturbations is discussed. 相似文献
16.
Takesi Kawasaki 《Transactions of the American Mathematical Society》2002,354(1):123-149
The Rees algebra is the homogeneous coordinate ring of a blowing-up. The present paper gives a necessary and sufficient condition for a Noetherian local ring to have a Cohen-Macaulay Rees algebra: A Noetherian local ring has a Cohen-Macaulay Rees algebra if and only if it is unmixed and all the formal fibers of it are Cohen-Macaulay. As a consequence of it, we characterize a homomorphic image of a Cohen-Macaulay local ring. For non-local rings, this paper gives only a sufficient condition. By using it, however, we obtain the affirmative answer to Sharp's conjecture. That is, a Noetherian ring having a dualizing complex is a homomorphic image of a finite-dimensional Gorenstein ring.
17.
We describe the spectrum of the Laplacian for a homogeneous graph acted on by a discrete group. This follows from a more general result which describes the spectrum of a convolution operator on a homogeneous space of a locally compact group. We also prove a version of Harnack inequality for a Schrödinger operator on an invariant homogeneous graph. 相似文献
18.
Michael T. Jury David W. Kribs 《Proceedings of the American Mathematical Society》2005,133(1):213-222
Given a row contraction of operators on a Hilbert space and a family of projections on the space that stabilizes the operators, we show there is a unique minimal joint dilation to a row contraction of partial isometries that satisfy natural relations. For a fixed row contraction the set of all dilations forms a partially ordered set with a largest and smallest element. A key technical device in our analysis is a connection with directed graphs. We use a Wold decomposition for partial isometries to describe the models for these dilations, and we discuss how the basic properties of a dilation depend on the row contraction.
19.
Chung-Yee Lee 《Operations Research Letters》2004,32(6):581-590
Motivated by a practical industrial problem where a manufacturer stipulates a minimum order from each buyer but where a local dealer promises the buyer a just-in-time delivery with a slightly higher unit cost, this paper uses a dynamic lot-sizing model with a stepwise cargo cost function and a minimum order amount constraint to help the buyer select the supplier with minimum total cost. 相似文献
20.
Hong Oh Kim Rae Young Kim Jae Kun Lim Zuowei Shen 《Journal of Approximation Theory》2007,147(2):196-204
We start with a characterization of a pair of frames to be orthogonal in a shift-invariant space and then give a simple construction of a pair of orthogonal shift-invariant frames. This is applied to obtain a construction of a pair of Gabor orthogonal frames as an example. This is also developed further to obtain constructions of a pair of orthogonal wavelet frames. 相似文献