Related papers: Average number of distinct part sizes in a random …
In this paper we study the component structure of random graphs with independence between the edges. Under mild assumptions, we determine whether there is a giant component, and find its asymptotic size when it exists. We assume that the…
Assume that $X$ and $Y$ are independent random variables, each having a Cauchy distribution with a known median. Taking a random independent sample of size $n$ of each $X$ and $Y$, one can then compute their centralized empirical…
For a sufficiently nice 2 dimensional shape, we define its approximating matrix (or patterned matrix) as a random matrix with iid entries arranged according to a given pattern. For large approximating matrices, we observe that the…
The Carlitz $\mathbb{F}_q$-algebra $C=C_\nu$, $\nu \in \mathbb{N}$, is generated by an algebraically closed field $\CK $ (which contains a non-discrete locally compact field of positive characteristic $p>0$, i.e. $K\simeq \mathbb{F}_q[[…
If $a_1, a_2, ..., a_k$ and $n$ are positive integers such that $n = a_1 + a_2 + ... + a_k$, then the sum $a_1 + a_2 + ... + a_k$ is said to be a \emph{partition of $n$} of \emph{length $k$}, and $a_1, a_2, ..., a_k$ are said to be the…
We show that the class of conditional distributions satisfying the coarsening at random (CAR) property for discrete data has a simple and robust algorithmic description based on randomized uniform multicovers: combinatorial objects…
We show that the averaged characteristic polynomial and the averaged inverse characteristic polynomial, associated with Hermitian matrices whose elements perform a random walk in the space of complex numbers, satisfy certain partial…
A sequence of geometric random variables of length $n$ is a sequence of $n$ independent and identically distributed geometric random variables ($\Gamma_1, \Gamma_2, \dots, \Gamma_n$) where $\mathbb{P}(\Gamma_j=i)=pq^{i-1}$ for…
Fix a positive integer $N$. Select an additive composition $\xi$ of $N$ uniformly out of $2^{N-1}$ possibilities. The interplay between the number of parts in $\xi$ and the maximum part in $\xi$ is our focus. It is not surprising that…
We relate the Fermi-Dirac statistics of an ideal Fermi gas in a harmonic trap to partitions of given integers into distinct parts, studied in number theory. Using methods of quantum statistical physics we derive analytic expressions for…
Continued fractions with prescribed structures on sequences of their partial quotients have been intensively studied in the literature. As far as an integer sequence, especially a randomly generated one is concerned, an attractive question…
Estimating the condition numbers of random structured matrices is a well known challenge, linked to the design of efficient randomized matrix algorithms. We deduce such estimates for Gaussian random Toeplitz and circulant matrices. The…
In a sample variance decomposition, with components functions of the sample's spacings, the largest component $\tilde{I}_n$ is used in cluster detection. It is shown for normal samples that the asymptotic distribution of $\tilde{I}_n$ is…
In this work we give precise asymptotic expressions on the probability of the existence of fixed-size components at the threshold of connectivity for random geometric graphs.
The paper is concerned with the asymptotic analysis of a family of Boltzmann (multiplicative) distributions over the set $\check{\varLambda}^{q}$ of strict integer partitions (i.e., with unequal parts) into perfect $q$-th powers. A…
This report presents an expression for the number of a multiset's sub-multisets of a given cardinality as a function of the multiplicity of its elements. This is also the number of distinct samples of a given size that may be produced by…
Cantor sets of integers have a rich set of arithmetic combinatorial properties. We consider classical Cantor sets, with a base and a fixed set of allowed digits. For such sets, we (a) give examples of such sets that satisfy the intersective…
Carlitz rank and index are two important measures for the complexity of a permutation polynomial $f(x)$ over the finite field $\F_q$. In particular, for cryptographic applications we need both, a high Carlitz rank and a high index. In this…
A result of Legendre asserts that the difference between the numbers of (length) even and odd partitions of $n$ into distinct parts is $0$, $1$, or $-1$; this also follows from Euler's pentagonal number theorem. We establish an analogous…
To any finite ordered subset and any finite partition of a group a set of tuples of positive integers, named as configurations, is associated that describes the group's behavior. The present paper provides an exposition of this notion and…