arXiv subjects

Romeo Meštrović

Publications and source records attributed to Romeo Meštrović.

At least 19 recordsLinked to original sources

arXiv2026

A short survey the game Bulgarian solitaire and related games

Let $N$ be an arbitrary positive integer and let $λ=(λ_1, λ_2, \ldots, λ_l)$ be a partition of $N$ of length $l$, i.e., $\sum_{i=1}^lλ_i= N$ with parts $λ_1\ge λ_2\ldots λ_l\ge 1$. Define $T(λ)$ as the partition of $N$ with parts $l,λ_1-1λ_2-1,\ldots λ_l-1$,ignoring any zeros that might occur. Starting with a partition $λ$ of $N$, we describe Bulgarian solitaire by repeatedly applying the shift operation $T$ to obtain the sequence of partitions $$ λ, T(λ), T^2(λ),\ldots . $$ We say a partition $μ$ of $N$ is $T$-cyclic if $T(μ) = μ$ for some $i\ge 1$. In 1982 Brandt [9] characterized all $T$-cyclic partitions for Bulgarian solitaire. Bulgarian solitaire is a dynamical system on integer partition of a positive integer $N$ which converges to a unique fixed point if $N=1+2+\cdots +k$ is a triangular number. In this paper we present a short survey of the game Bulgarian solitaire and several variations of this game.

math.HO↗

arXiv2023

Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 B.C.--2022) and another new proof

{\bf In the fourth extended version of this article, we provide a comprehensive historical survey of 200 different proofs of famous Euclid's theorem on the infinitude of prime numbers (300 {\small B.C.}--2022)}. The author is trying to collect almost all the known proofs on infinitude of primes, including some proofs that can be easily obtained as consequences of some known problems or divisibility properties. Furthermore, here are listed numerous elementary proofs of the infinitude of primes in different arithmetic progressions. All the references concerning the proofs of Euclid's theorem that use similar methods and ideas are exposed subsequently. Namely, presented proofs are divided into the first five subsections of Section 2 in dependence of the methods that are used in them. 14 proofs which are proved from 2012 to 2017 are given in Subsection 2.9, and 18 recent proofs from 2018 to 2022 are presented in Subsection 2.10. In Section 3, we mainly survey elementary proofs of the infinitude of primes in different arithmetic progressions. Presented proofs are special cases of Dirichlet's theorem. In Section 4, we give a new simple ''Euclidean's proof'' of the infinitude of primes.

math.HO↗

arXiv2019

Closed maximal ideals ideals in some Fréchet algebras of holomorphic functions

The space $F^p$ ($1 0}$ defined for $f\in F^q$ as $\Vert f\Vert_{q,c}:=\sum_{n=0}^{\infty}\vert a_n\vert\exp\left(-cn^{1/(q+1)} \right)<\infty$, becomes a countably normed Fréchet algebra. It is known that for every $p>1$, $F^p$ is the Fréchet envelope of the Privalov space $N^p$. In this paper, we extend our study of [32] on the structure of maximal ideals in the algebras $F^p$ ($1<p<\infty$). Namely, the obtained characterization of closed maximal ideals in $F^p$ from [32] is extended here in terms of topology of uniform convergence on compact subsets of $\Bbb D$.

math.NT↗

arXiv2019

A characterization of some prime ideals in certain $F$-algebras of holomorphic functions

The class $M^p$ $(1<p<\infty)$ consists of all holomorphic functions $f$ on the open unit disk $\Bbb D$ for which $$ \int_0^{2π}\left(\log^+Mf(θ)\right)^p\,\frac{dθ}{2π}<\infty, $$ where $Mf(θ)=\sup_{0\leqslant r<1}\big\lvert f\big(re^{iθ}\big)\big \rvert$. The class $M^p$ equipped with the topology given by the metric $ρ_p$ defined by $$ ρ_p(f,g)=\left(\int_0^{2π}\log^p(1+M(f-g)(θ))\, \frac{dθ}{2π}\right)^{1/p}\quad (f,g\in M^p) $$ becomes an $F$-algebra. In this paper, we consider the ideal structure of the classes $M^p$ $(1<p<\infty)$. Our main result gives a complete characterization of prime ideals in $M^p$ which are not dense subsets of $M^p$. As a consequence, we obtaiin a related Mochizuki's result concerning the Privalov classes $N^p$ $(1<p<\infty)$.

math.CV↗

arXiv2019

Goldbach's like conjectures arising from arithmetic progressions whose first two terms are primes

For two odd primes $p$ and $q$ such that $p 1$ there exist a positive integer $n$ and two odd primes $p$ and $q$ such that $a$ can be expressed as a sum of the first $2n$ terms of the arithmetic progression $A(p,q)$. Notice that in the case of even $a$, this conjecture immediately follows from Goldbach's conjecture. We also propose the analogous conjecture for odd positive integers $a>1$ as well as some related Goldbach's like conjectures arising from the previously mentioned arithmetic progressions.

math.NT↗

arXiv2018

A characterization of maximal ideals in the Fréchet algebras of holomorphic functions $F^p$ $(1<p<\infty$

The space $F^p$ ($1 0}$ defined for $f\in F^q$ as $\Vert f\Vert_{q,c}:=\sum_{n=0}^{\infty}\vert a_n\vert\exp\left(-cn^{1/(q+1)} \right)<\infty,$ is a countably normed Fréchet algebra. Notice that for each $p>1$, $F^p$ is the Fréchet envelope of the Privalov space $N^p$. In this paper we study the structure of maximal ideals in the algebras $F^p$ ($1<p<\infty$). In particular, we give a complete characterization of closed maximal ideals in $F^p$. Moreover, we characterize multiplicative linear functionals on $F^p$.

math.NT↗

arXiv2018

On some metric topologies on Privalov spaces on the unit disk

Let $N^p$ $(1 1$, we consider the class $N^p$ as the space $M^p$ equipped with the topology induced by the metric $ρ_p$ defined as $$ ρ_p(f,g) = \Bigg(\int_0^{2π}\log^p(1+M(f-g)(θ))\, \frac{dθ}{2π}\Bigg)^{1/p},\quad f,g\in M^p,\,\, {\mathrm where}\,\, Mf(θ) = \sup_{0\leqslant r<1} \big\vert f \big(re^{iθ})\big\vert.$$ On the other hand, we consider the class $N^p$ with the metric topology introduced by Meštrović, Pavićević and Labudović (1999) which generalizes the Gamelin-Lumer's metric which is generally defined on a measure space $(Ω, Σ, μ)$ with a positive finite measure $μ$. The space $N^p$ with the associated modular in the sense of Musielak and Orlicz becomes the Hardy-Orlicz class. It is noticed that the all considered metrics induce the same topology on the space $N^p$.

math.FA↗

arXiv2018

Several generalizations and variations of Chu-Vandermonde identity

In this paper we prove some combinatorial identities which can be considered as generalizations and variations of remarkable Chu-Vandermonde identity. These identities are proved by using an elementary combinatorial-probabilistic approach to the expressions for the $k$-th moments ($k=1,2,3$) of some particular cases of recently investigated discrete random variables. Using one of these Chu-Vandermonde-type identities, two combinatorial congruences are established.

math.CO↗

arXiv2018

A generalization of some random variables involving in certain compressive sensing problems

In this paper we give a generalization of the discrete complex-valued random variable defined and investigated in \cite{ssa} and \cite{m8}. We prove the statements concerning the expressions for the excepted value and the variance of this random variable. In partucular, such a random variable here is defined for each of $m$ rows of any $m\times N$ complex or real matrix ${\rm{\bf A}}$ with $1\le m\le N$. We consider the arithmetic mean $\bar{X}(m)$ of these $m$ random variables and we deduce the expressions for the expected value $\Bbb E[\bar{X}(m)]$ and the variance ${\rm{ Var}}[\bar{X}(m)]$ of $\bar{X}(m)$. Using the expression for ${\rm{ Var}}[\bar{X}(m)]$, we establish some equalities and inequalities involving ${\rm{Var}}[\bar{X}(m)]$, the Frobenius norm, the largest eigenvalue, the largest singular value and the coherence of a matrix ${\rm{\bf A}}$. It is showed that some of these estimates are closely related to the Welch bound of the coherence of a $m\times N$ complex or real matrix ${\rm{\bf A}}$ with $1\le m\le N$. Taking into account that the value of coherence of the measurement matrix in the theory of compressive sensing has a significant role, we believe that our results should be useful for some topics of this theory.

eess.SP↗

arXiv2018

On the distribution of primes in the alternating sums of concecutive primes

Quite recently, in [8] the authoor of this paper considered the distribution of primes in the sequence $(S_n)$ whose $n$th term is defined as $S_n=\sum_{k=1}^{2n}p_k$, where $p_k$ is the $k$th prime. Some heuristic arguments and the numerical evidence lead to the conjecture that the primes are distributed among sequence $(S_n)$ in the same way that they are distributed among positive integers. More precisely, Conjecture 3.3 in [8] asserts that $π_n\sim \frac{n}{\log n}$ as $n\to \infty$, where $π_n$ denotes the number of primes in the set $\{S_1,S_2,\ldots, S_n\}$. Motivated by this, here we consider the distribution of primes in aletrnating sums of first $2n$ primes, i.e., in the sequences $(A_n)$ and $(T_n)$ defined by $A_n:=\sum_{i=1}^{2n}(-1)^ip_i$ and $T_n:=A_n-2=\sum_{i=2}^{2n}(-1)^ip_i$ ($n=1,2,\ldots$). Heuristic arguments and computational results suggest the conjecture that (Conjecture 2.5) $$ π_{(A_k)}(A_n)\sim π_{(T_k)}(T_n)\sim \frac{2n}{\log n} \quad {\rm as}\,\, n\to \infty, $$ where $π_{(A_k)}(A_n)$ (respectively, $π_{(T_k)}(T_n)$) denotes the number of primes in the set $\{A_1,A_2,\ldots, A_n\}$(respectively, $\{T_1,T_2,\ldots, T_n\}$). Under Conjecture 2.5 and Pillai's conjecture, we establish two results concerning the expressions for the $k$th prime in the sequences $(A_n)$ and $(T_n)$. Furthermore, we propose some other related conjectures and we deduce some their consequences.

math.NT↗

arXiv2018

Ideals in some Rings of Nevanlinna-Smirnov Type

Let $N^p$ $(1<p<\infty)$ denote the algebra of holomorphic functions in the open unit disk, introduced by I.~I.~Privalov with the notation $A_q$ in [8]. Since $N^p$ becomes a ring of Nevanlinna--Smirnov type in the sense of Mortini [7], the results from [7] can be applied to the ideal structure of the ring $N^p$. In particular, we observe that $N^p$ has the Corona Property. Finally, we prove the $N^p$-analogue of the Theorem 6 in [7], which gives sufficient conditions for an ideal in $N^p$, generated by a finite number of inner functions, to be equal to the whole algebra $N^p$.

math.CV↗

arXiv2018

Curious conjectures on the distribution of primes among the sums of the first $2n$ primes

Let $p_n$ be $n$th prime, and let $(S_n)_{n=1}^\infty:=(S_n)$ be the sequence of the sums of the first $2n$ consecutive primes, that is, $S_n=\sum_{k=1}^{2n}p_k$ with $n=1,2,\ldots$. Heuristic arguments supported by the corresponding computational results suggest that the primes are distributed among sequence $(S_n)$ in the same way that they are distributed among positive integers. In other words, taking into account the Prime Number Theorem, this assertion is equivalent to \begin{equation*}\begin{split} &\# \{p:\, p\,\,{\rm is\,\,a\,\, prime\,\, and}\,\, p=S_k \,\,{\rm for\,\,some\,\,} k \,\,{\rm with\,\,} 1\le k\le n\} \sim & \# \{p:\, p\,\,{\rm is\,\,a\,\, prime\,\, and}\,\, p=k \,\,{\rm for\,\,some\,\,} k \,\,{\rm with\,\,} 1\le k\le n\}\sim\frac{\log n}{n}\,\, {\rm as}\,\, n\to\infty, \end{split}\end{equation*} where $|S|$ denotes the cardinality of a set $S$. Under the assumption that this assertion is true (Conjecture 3.3), we say that $(S_n)$ satisfies the Restricted Prime Number Theorem. Motivated by this, in Sections 1 and 2 we give some definitions, results and examples concerning the generalization of the prime counting function $π(x)$ to increasing positive integer sequences. The remainder of the paper (Sections 3-7) is devoted to the study of mentioned sequence $(S_n)$. Namely, we propose several conjectures and we prove their consequences concerning the distribution of primes in the sequence $(S_n)$. These conjectures are mainly motivated by the Prime Number Theorem, some heuristic arguments and related computational results. Several consequences of these conjectures are also established.

math.NT↗

arXiv2018

Remarks on Generalized Hardy Algebras

For a measure space $(Ω, Σ, μ)$ with a positive finite measure $μ$, and a positive real number $p$, we define the space $L_p^{+}(μ)=L_p^{+}$ of all (equivalence classes of) $Σ$-measurable complex functions $f$ defined on $Ω$ such that the function $\left(\log^+|f|\right)^p$ is integrable with respect to $μ$.We define the metric $d_p$ on $L^{+}_p$ which generalizes the metric introduced by Gamelin and Lumer in [G] for the case $p=1$. It is shown that the space $L^{+}_p$ is a topological algebra. On the other hand, one can define on the space $L_p^{+}$ an equivalent $F$-norm $| \cdot|_p$ that makes $L_p^{+}$ into an Orlicz space. For the case of the normalized Lebesgue's measure $dt/2π$ on $[0,2π)$, it follows that the class $N^p(1<p<\infty)$ introduced by I. I. Privalov in [P], may be considered as a generalization of the Smirnov class $N^+$. Furthermore, $N^p(1<p<\infty)$ with the associated modular becomes an Hardy-Orlicz class. Finally, for a strictly positive and measurable on $[0,2π)$ function $w$, we define the generalized Orlicz space $L_p^{w}(\mathrm{d}t/2π)=L^w_p$ with the modular $ρ^w_p$ given by the function $ψ_w(t,u)=\big(\log(1+uw(t))\big)^p$, with a "weight" $w$. We observe that the space $L^w_p$ is a generalized Orlicz space with respect to the modular $ρ^w_p$. We examine and compare different topologies induced on $L^w_p$ by corresponding "weights" $w$.

math.FA↗

arXiv2018

Different classes of binary necklaces and a combinatorial method for their enumerations

In this paper we investigate enumeration of some classes of $n$-character strings and binary necklaces. Recall that binary necklaces are necklaces in two colors with length $n$. We prove three results (Theorems 1, 1' and 2) concerning the numbers of three classes of $j$-character strings (closely related to some classes of binary necklaces or Lyndon words). Using these results, we deduce Moreau's necklace-counting function for binary aperiodic necklaces of length $k$ \cite{mo} (Theorem 3), and we prove the binary case of MacMahon's formula from 1892 \cite{ma} (also called Witt's formula) for the number of necklaces (Theorem 4). Notice that we give proofs of Theorems 3 and 4 without use of Burnside's lemma and Pólya enumeration theorem. Namely, the methods used in our proofs of auxiliary and main results presented in Sections 3 and 4 are combinatorial in spirit and they are based on counting method and some facts from elementary number theory.

math.CO↗

arXiv2018

Two curious inequalities involving different means of two arguments

For two positive real numbers $x$ and $y$ let $H$, $G$, $A$ and $Q$ be the harmonic mean, the geometric mean, the arithmetic mean and the quadratic mean of $x$ and $y$, respectively. In this note, we prove that \begin{equation*} A\cdot G\ge Q\cdot H, \end{equation*} and that for each integer $n$ \begin{equation*} A^n+G^n\le Q^n+H^n.\end{equation*} We also discuss and compare the first and the second above inequality for $n=1$ with some known inequalities involving the mentioned classical means, the Seiffert mean $P$, the logarithmic mean $L$ and the identric mean $I$ of two positive real numbers $x$ and $y$.

math.NT↗

arXiv2018

On some random variables involving Bernoulli random variable

Motivated by the recent investigations given in [25] and the fact that Bernoulli probability-type models were often used in the study on some problems in theory of compressive sensing, here we define and study the complex-valued discrete random variables $\widetilde{X}_l(m,N)$ ($0\le l\le N-1$, $1\le m\le N$). Each of these random variables is defined as a linear combination of $N$ independent identically distributed $0-1$ Bernoulli random variables. We prove that for $l\not=0$, $\widetilde{X}_l(m,N)$ is the zero-mean random variable, and we also determine the variance of $\widetilde{X}_l(m,N)$ and its real and imaginary parts. Notice that $\widetilde{X}_l(m,N)$ belongs to the class of sub-Gaussian random variables that are significant in some areas of theory of compressive sensing. In particular, we prove some probability estimates for the mentioned random variables. These estimates are used to establish the upper bounds of the sub-Gaussian norm of their real and imaginary parts. We believe that our results should be implemented in certain applications of sub-Gaussian random variables for solving some problems in compressive sensing of sparse signals.

math.PR↗

arXiv2018

A note on some sub-Gaussian random variables

In [8] the author of this paper continued the research on the complex-valued discrete random variables $X_l(m,N)$ ($0\le l\le N-1$, $1\le M\le N)$ recently introduced and studied in [24]. Here we extend our results by considering $X_l(m,N)$ as sub-Gaussian random variables. Our investigation is motivated by the known fact thatthe so-called Restricted Isometry Property (RIP) introduced in [4] holds with high probability for any matrix generated by a sub-Gaussian random variable. Notice that sensing matrices with the RIP play a crucial role in Theory of compressive sensing. Our main results concern the proofs of the lower and upper bound estimates of the expected values of the random variables $|X_l(m,N)|$, $|U_l(m,N)|$ and $|V_l(m,N)|$, where $U_l(m,N)$ and $U_l(m,N)$ are the real and the imaginary part of $X_l(m,N)$, respectively. These estimates are also given in terms of related sub-Gaussian norm $\Vert \cdot\Vert_{ψ_2}$ considered in [28]. Moreover, we prove a refinement of the mentioned upper bound estimates for the real and the imaginary part of $X_l(m,N)$.

math.PR↗

arXiv2018

On some discrete random variables arising from recent study on statistical analysis of compressive sensing

The recent paper [27] provides a statistical analysis for efficient detection of signal components when missing data samples are present. Here we focus our attention to some complex-valued discrete random variables $X_l(m,N)$ ($0\le l\le N-1$, $1\le M\le N$), which are closely related to the random variables investigated by LJ. Stanković, S. Stanković and M. Amin in \cite{ssa}. In particular, by using a combinatorial approach, we prove that for $l\not=0$ the expected value of $X_l(m,N)$ is equal to zero, and we deduce the expression for the variance of the random variables $X_l(m,N)$. The same results are also deduced for the real part $U_l(m,N)$ and the imaginary part $V_l(m,N)$ of $X_l(m,N)$, as well as the facts that the $k$th moments of $U_l(m,N)$ and $V_l(m,N)$ are equal to zero for every positive integer $k$ which is not divisible by $N/\gcd(N,l)$. Moreover, some additional assertions and examples concerning the random variables $X_l(m,N)$, $U_l(m,N)$ and $V_l(m,N)$ are also presented.

math.ST↗