ترغب بنشر مسار تعليمي؟ اضغط هنا

Characterization of finite metric space by their isometric sequences

226   0   0.0 ( 0 )
 نشر من قبل Masashi Shinohara
 تاريخ النشر 2018
  مجال البحث
والبحث باللغة English




اسأل ChatGPT حول البحث

Let $(X,d)$ be a finite metric space with $|X|=n$. For a positive integer $k$ we define $A_k(X)$ to be the quotient set of all $k$-subsets of $X$ by isometry, and we denote $|A_k(X)|$ by $a_k$. The sequence $(a_1,a_2,ldots,a_{n})$ is called the isometric sequence of $(X,d)$. In this article we aim to characterize finite metric spaces by their isometric sequences under one of the following assumptions: (i) $a_k=1$ for some $k$ with $2leq kleq n-2$; (ii) $a_k=2$ for some $k$ with $4leq kleq frac{1+sqrt{1+4n}}{2}$; (iii) $a_3=2$; (iv) $a_2=a_3=3$. Furthermore, we give some criterion on how to embed such finite metric spaces to Euclidean spaces. We give some maximum cardinalities of subsets in the $d$-dimensional Euclidean space with small $a_3$, which are analogue problems on a sets with few distinct triangles discussed by Epstein, Lott, Miller and Palsson.



قيم البحث

اقرأ أيضاً

We give a new characterization of biinfinite Sturmian sequences in terms of indistinguishable asymptotic pairs. Two asymptotic sequences on a full $mathbb{Z}$-shift are indistinguishable if the sets of occurrences of every pattern in each sequence co incide up to a finitely supported permutation. This characterization can be seen as an extension to biinfinite sequences of Pirillos theorem which characterizes Christoffel words. Furthermore, we provide a full characterization of indistinguishable asymptotic pairs on arbitrary alphabets using substitutions and biinfinite characteristic Sturmian sequences. The proof is based on the well-known notion of derived sequences.
A colored space is the pair $(X,r)$ of a set $X$ and a function $r$ whose domain is $binom{X}{2}$. Let $(X,r)$ be a finite colored space and $Y,Zsubseteq X$. We shall write $Ysimeq_r Z$ if there exists a bijection $f:Yto Z$ such that $r(U)=r(f(U))$ f or each $Uinbinom{Y}{2}$. We denote the numbers of equivalence classes with respect to $simeq_r$ contained in $binom{X}{2}$ and $binom{X}{3}$ by $a_2(r)$ and $a_3(r)$, respectively. In this paper we prove that $a_2(r)leq a_3(r)$ when $5leq |X|$, and show what happens when the equality holds.
180 - Jiangtao Peng , Yuanlin Li 2013
Let $G$ be a finite cyclic group. Every sequence $S$ of length $l$ over $G$ can be written in the form $S=(n_1g)cdotldotscdot(n_lg)$ where $gin G$ and $n_1, ldots, n_lin[1, ord(g)]$, and the index $ind(S)$ of $S$ is defined to be the minimum of $(n_1 +cdots+n_l)/ord(g)$ over all possible $gin G$ such that $langle g rangle =G$. In this paper, we determine the index of any minimal zero-sum sequence $S$ of length 5 when $G=langle grangle$ is a cyclic group of a prime order and $S$ has the form $S=g^2(n_2g)(n_3g)(n_4g)$. It is shown that if $G=langle grangle$ is a cyclic group of prime order $p geq 31$, then every minimal zero-sum sequence $S$ of the above mentioned form has index 1 except in the case that $S=g^2(frac{p-1}{2}g)(frac{p+3}{2}g)((p-3)g)$.
152 - Guoqing Wang 2020
Let $mathcal{S}$ be a finite cyclic semigroup written additively. An element $e$ of $mathcal{S}$ is said to be idempotent if $e+e=e$. A sequence $T$ over $mathcal{S}$ is called {sl idempotent-sum free} provided that no idempotent of $mathcal{S}$ can be represented as a sum of one or more terms from $T$. We prove that an idempotent-sum free sequence over $mathcal{S}$ of length over approximately a half of the size of $mathcal{S}$ is well-structured. This result generalizes the Savchev-Chen Structure Theorem for zero-sum free sequences over finite cyclic groups.
139 - Yuanlin Li , Jiangtao Peng 2013
Let $G$ be a finite cyclic group. Every sequence $S$ over $G$ can be written in the form $S=(n_1g)cdotldotscdot(n_lg)$ where $gin G$ and $n_1, ldots, n_lin[1, ord(g)]$, and the index $ind(S)$ of $S$ is defined to be the minimum of $(n_1+cdots+n_l)/or d(g)$ over all possible $gin G$ such that $langle g rangle =G$. An open problem on the index of length four sequences asks whether or not every minimal zero-sum sequence of length 4 over a finite cyclic group $G$ with $gcd(|G|, 6)=1$ has index 1. In this paper, we show that if $G=langle grangle$ is a cyclic group with order of a product of two prime powers and $gcd(|G|, 6)=1$, then every minimal zero-sum sequence $S$ of the form $S=(g)(n_2g)(n_3g)(n_4g)$ has index 1. In particular, our result confirms that the above problem has an affirmative answer when the order of $G$ is a product of two different prime numbers or a prime power, extending a recent result by the first author, Plyley, Yuan and Zeng.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا