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Quantitative bounds for sets lacking polynomial progressions with shifted prime difference
Authors:
Ben Krause,
Hamed Mousavi,
Terence Tao,
Joni Teräväinen
Abstract:
We prove quantitative polynomial Szemerédi-type theorems involving polynomial progressions with shift parameter restricted to the set of shifted primes $\mathbb{P}-1$. The types of configurations covered are distinct degree progressions and progressions involving integer multiples of a fixed polynomial.
For nonlinear configurations of length at least three, these results provide the first quanti…
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We prove quantitative polynomial Szemerédi-type theorems involving polynomial progressions with shift parameter restricted to the set of shifted primes $\mathbb{P}-1$. The types of configurations covered are distinct degree progressions and progressions involving integer multiples of a fixed polynomial.
For nonlinear configurations of length at least three, these results provide the first quantitative versions of such theorems. In the linear case, our results improve on work by the last two authors. Our density bounds are strongest in the case of distinct degree polynomials, where they give polylogarithmic bounds, of the same shape as recent bounds by Shao and Wang with integer shifts.
The proofs combine recent quantitative results for polynomial configurations in the integers with quantitative Gowers uniformity bounds of the primes. For multiples of a fixed polynomial, we adapt a comparison argument of Altman and Sawhney to obtain uniformity over the polynomial families produced by the $W$-trick. For distinct degree progressions, we establish a comparison between prime-weighted and unweighted polynomial counts that is uniform throughout the density increment argument and accounts for a possible Siegel zero.
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Submitted 19 August, 2026;
originally announced August 2026.
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The Wiener Wintner Theorem Along the Primes
Authors:
Jan Fornal,
Anastasios Fragkos,
Ben Krause,
Michael Lacey,
Hamed Mousavi,
Yu-Chen Sun
Abstract:
We prove the following Wiener-Wintner Theorem along the sequence of prime times, the first extension of the Wiener-Wintner Theorem to arithmetic sequences: for every probability space, $(X, ν),$ equipped with a measure-preserving transformation, $T : X \to X,$ and every $f \in L^p(X), 1 < p \leq \infty$, there exists a set of full probability, $X_f \subset X$ with $ν(X_f) = 1,$ so that for all…
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We prove the following Wiener-Wintner Theorem along the sequence of prime times, the first extension of the Wiener-Wintner Theorem to arithmetic sequences: for every probability space, $(X, ν),$ equipped with a measure-preserving transformation, $T : X \to X,$ and every $f \in L^p(X), 1 < p \leq \infty$, there exists a set of full probability, $X_f \subset X$ with $ν(X_f) = 1,$ so that for all $ω\in X_f$, \[ \frac{1}{N} \sum_{n \leq N} e^{ 2 πi p_n θ} f(T^{p_n} ω) \] converges for all $θ\in [0,1]$; above, $\{2 = p_1 < p_2 < \dots\}$ are an enumeration of the primes.
Our proof lives at the interface of classical Fourier analysis, combinatorial number theory, higher order Fourier analysis, and pointwise ergodic theory, with U^3 theory playing an important role; our $U^3$-estimates for Heath-Brown models of the von Mangoldt function may be of independent interest.
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Submitted 9 July, 2026; v1 submitted 15 January, 2026;
originally announced January 2026.
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An Approach To Endpoint Problems in Oscillatory Singular Integrals
Authors:
Alex Iosevich,
Ben Krause,
Hamed Mousavi
Abstract:
In this note we provide a quick proof that maximal truncations of oscillatory singular integrals are bounded from $L^1(\mathbb{R})$ to $L^{1,\infty}(\mathbb{R})$. The methods we use are entirely elementary, and rely only on pigeonholing and stationary phase considerations.
In this note we provide a quick proof that maximal truncations of oscillatory singular integrals are bounded from $L^1(\mathbb{R})$ to $L^{1,\infty}(\mathbb{R})$. The methods we use are entirely elementary, and rely only on pigeonholing and stationary phase considerations.
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Submitted 26 February, 2025;
originally announced February 2025.
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On the Normalizer-Solubilizer Conjecture_V3
Authors:
Hamid Mousavi
Abstract:
Let $G$ be a finite group and $x$ be an element of $G$. Define $\textrm{Sol}_G(x)$ as the set of all $y \in G$ such that $\langle {x,y}\rangle$ is soluble. We provide an equivalent condition for the normalizer-solubilizer conjecture, namely $|\mathcal{N}_G(\langle x\rangle)| \mid |\textrm{Sol}_G(x)|$, where $\mathcal{N}_G(\langle x\rangle)$ is the normalizer of $\langle x\rangle$. Furthermore, we…
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Let $G$ be a finite group and $x$ be an element of $G$. Define $\textrm{Sol}_G(x)$ as the set of all $y \in G$ such that $\langle {x,y}\rangle$ is soluble. We provide an equivalent condition for the normalizer-solubilizer conjecture, namely $|\mathcal{N}_G(\langle x\rangle)| \mid |\textrm{Sol}_G(x)|$, where $\mathcal{N}_G(\langle x\rangle)$ is the normalizer of $\langle x\rangle$. Furthermore, we demonstrate that the conjecture holds in the special case where $\mathcal{N}_G(\langle x\rangle)$ is a Frobenius group with kernel $\mathcal{C}_G(x)$, the centralizer of $x$, and $|\mathcal{N}_G(\langle x\rangle): \mathcal{C}_G(x)|$ is of prime order. Finally, we will classify all finite simple groups $G$ that contain an element $x$ for which $\textrm{Sol}_G(x)$ is a maximal subgroup of order $pq$, where $p$ and $q$ are prime numbers.
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Submitted 10 June, 2025; v1 submitted 20 January, 2025;
originally announced January 2025.
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A Density Theorem for Higher Order Sums of Prime Numbers
Authors:
Michael T. Lacey,
Hamed Mousavi,
Yaghoub Rahimi,
Manasa N. Vempati
Abstract:
Let $P$ be a subset of the primes of lower density strictly larger than $\frac12$. Then, every sufficiently large even integer is a sum of four primes from the set $P$. We establish similar results for $k$-summands, with $k\geq 4$, and for $k \geq 4$ distinct subsets of primes. This extends the work of H.~Li, H.~Pan, as well as X.~Shao on sums of three primes, and A.~Alsteri and X.~Shao on sums of…
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Let $P$ be a subset of the primes of lower density strictly larger than $\frac12$. Then, every sufficiently large even integer is a sum of four primes from the set $P$. We establish similar results for $k$-summands, with $k\geq 4$, and for $k \geq 4$ distinct subsets of primes. This extends the work of H.~Li, H.~Pan, as well as X.~Shao on sums of three primes, and A.~Alsteri and X.~Shao on sums of two primes. The primary new contributions come from elementary combinatorial lemmas.
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Submitted 2 November, 2024;
originally announced November 2024.
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A note on meta and para-$\mathfrak{Nil}$-Hamiltonian groups_v3
Authors:
Hamid Mousavi
Abstract:
Let $\mathfrak{Nil}$ be the class of nilpotent groups. This article explores the finiteness of meta and para-$\mathfrak{Nil}$-Hamiltonian groups or their derived subgroups when these groups contain a soluble subgroup of finite index or a non-nilpotent (or insoluble) subgroup of finite order respectively.
Let $\mathfrak{Nil}$ be the class of nilpotent groups. This article explores the finiteness of meta and para-$\mathfrak{Nil}$-Hamiltonian groups or their derived subgroups when these groups contain a soluble subgroup of finite index or a non-nilpotent (or insoluble) subgroup of finite order respectively.
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Submitted 9 February, 2025; v1 submitted 1 October, 2024;
originally announced October 2024.
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Pointwise convergence of bilinear polynomial averages over the primes
Authors:
Ben Krause,
Hamed Mousavi,
Terence Tao,
Joni Teräväinen
Abstract:
We show that on a $σ$-finite measure preserving system $X = (X,ν, T)$, the non-conventional ergodic averages $$ \mathbb{E}_{n \in [N]} Λ(n) f(T^n x) g(T^{P(n)} x)$$ converge pointwise almost everywhere for $f \in L^{p_1}(X)$, $g \in L^{p_2}(X)$, and $1/p_1 + 1/p_2 \leq 1$, where $P$ is a polynomial with integer coefficients of degree at least $2$. This had previously been established with the von…
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We show that on a $σ$-finite measure preserving system $X = (X,ν, T)$, the non-conventional ergodic averages $$ \mathbb{E}_{n \in [N]} Λ(n) f(T^n x) g(T^{P(n)} x)$$ converge pointwise almost everywhere for $f \in L^{p_1}(X)$, $g \in L^{p_2}(X)$, and $1/p_1 + 1/p_2 \leq 1$, where $P$ is a polynomial with integer coefficients of degree at least $2$. This had previously been established with the von Mangoldt weight $Λ$ replaced by the constant weight $1$ by the first and third authors with Mirek, and by the Möbius weight $μ$ by the fourth author. The proof is based on combining tools from both of these papers, together with several Gowers norm and polynomial averaging operator estimates on approximants to the von Mangoldt function of ''Cramér'' and ''Heath-Brown'' type.
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Submitted 22 January, 2026; v1 submitted 16 September, 2024;
originally announced September 2024.
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Bayesian Inference for Estimating Heat Sources through Temperature Assimilation
Authors:
Hanieh Mousavi,
Jeff D. Eldredge
Abstract:
This paper introduces a Bayesian inference framework for two-dimensional steady-state heat conduction, focusing on the estimation of unknown distributed heat sources in a thermally-conducting medium with uniform conductivity. The goal is to infer heater locations, strengths, and shapes using temperature assimilation in the Euclidean space, employing a Fourier series to represent each heater's shap…
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This paper introduces a Bayesian inference framework for two-dimensional steady-state heat conduction, focusing on the estimation of unknown distributed heat sources in a thermally-conducting medium with uniform conductivity. The goal is to infer heater locations, strengths, and shapes using temperature assimilation in the Euclidean space, employing a Fourier series to represent each heater's shape. The Markov Chain Monte Carlo (MCMC) method, incorporating the random-walk Metropolis-Hasting algorithm and parallel tempering, is utilized for posterior distribution exploration in both unbounded and wall-bounded domains. Strong correlations between heat strength and heater area prompt caution against simultaneously estimating these two quantities. It is found that multiple solutions arise in cases where the number of temperature sensors is less than the number of unknown states. Moreover, smaller heaters introduce greater uncertainty in estimated strength. The diffusive nature of heat conduction smooths out any deformations in the temperature contours, especially in the presence of multiple heaters positioned near each other, impacting convergence. In wall-bounded domains with Neumann boundary conditions, the inference of heater parameters tends to be more accurate than in unbounded domains.
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Submitted 17 April, 2024;
originally announced May 2024.
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Averages with the Gaussian divisor: Weighted Inequalities and the Pointwise Ergodic Theorem
Authors:
Christina Giannitsi,
Nazar Miheisi,
Hamed Mousavi
Abstract:
We discuss the Pointwise Ergodic Theorem for the Gaussian divisor function $d(n)$, that is, for a measure preserving $\mathbb Z[i]$ action $T$, the limit
$$\lim_{N\rightarrow \infty} \frac{1}{D(N)} \sum _{\mathscr{N} (n) \leq N} d(n) \,f(T^n x) $$ converges for every $f\in L^p$, where $\mathscr{N} (n) = n \bar{n}$, and $D(N) = \sum _{\mathscr{N} (n) \leq N} d(n) $, and $1<p\leq \infty$. To do so…
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We discuss the Pointwise Ergodic Theorem for the Gaussian divisor function $d(n)$, that is, for a measure preserving $\mathbb Z[i]$ action $T$, the limit
$$\lim_{N\rightarrow \infty} \frac{1}{D(N)} \sum _{\mathscr{N} (n) \leq N} d(n) \,f(T^n x) $$ converges for every $f\in L^p$, where $\mathscr{N} (n) = n \bar{n}$, and $D(N) = \sum _{\mathscr{N} (n) \leq N} d(n) $, and $1<p\leq \infty$. To do so we study the averages
$$ A_N f (x) = \frac{1}{D(N)} \sum _{\mathscr{N} (n) \leq N} d(n) \,f(x-n) ,$$ and obtain improving and weighted maximal inequalities for our operator, in the process.
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Submitted 19 February, 2024;
originally announced February 2024.
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Groups whose non-normal subgroups are either nilpotent or minimal non-nilpotent
Authors:
Nasrin Dastborhan,
Hamid Mousavi
Abstract:
Let $\mathfrak{Nil}$ be the class of nilpotent groups and $G$ be a group. We call $G$ a meta-$\mathfrak{Nil}$-Hamiltonian group if any of its non-$\mathfrak{Nil}$ subgroups is normal. Also, we call $G$ a para-$\mathfrak{Nil}$-Hamiltonian group if $G$ is a non-$\mathfrak{Nil}$ group and every non-normal subgroup of $G$ is either a $\mathfrak{Nil}$-group or a minimal non-$\mathfrak{Nil}$ group. In t…
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Let $\mathfrak{Nil}$ be the class of nilpotent groups and $G$ be a group. We call $G$ a meta-$\mathfrak{Nil}$-Hamiltonian group if any of its non-$\mathfrak{Nil}$ subgroups is normal. Also, we call $G$ a para-$\mathfrak{Nil}$-Hamiltonian group if $G$ is a non-$\mathfrak{Nil}$ group and every non-normal subgroup of $G$ is either a $\mathfrak{Nil}$-group or a minimal non-$\mathfrak{Nil}$ group. In this paper we investigate the class of finitely generated meta-$\mathfrak{Nil}$-Hamiltonian and para-$\mathfrak{Nil}$-Hamiltonian groups.
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Submitted 20 February, 2024; v1 submitted 1 November, 2023;
originally announced November 2023.
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Averages over the Gaussian Primes: Goldbach's Conjecture and Improving Estimates
Authors:
Christina Giannitsi,
Ben Krause,
Michael Lacey,
Hamed Mousavi,
Yaghoub Rahimi
Abstract:
We prove versions of Goldbach conjectures for Gaussian primes in arbitrary sectors. Fix an interval $ω\subset \mathbb{T}$. There is an integer $N_ω$, so that every odd integer $n$ with $N(n)>N_ω$ and $\text{dist}( \text{arg}(n) , \mathbb{T}\setminus ω) > (\log N(n)) ^{-B}$, is a sum of three Gaussian primes $n=p_1+p_2+p_3$, with $\text{arg}(p_j) \in ω$, for $j=1,2,3$. A density version of the bina…
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We prove versions of Goldbach conjectures for Gaussian primes in arbitrary sectors. Fix an interval $ω\subset \mathbb{T}$. There is an integer $N_ω$, so that every odd integer $n$ with $N(n)>N_ω$ and $\text{dist}( \text{arg}(n) , \mathbb{T}\setminus ω) > (\log N(n)) ^{-B}$, is a sum of three Gaussian primes $n=p_1+p_2+p_3$, with $\text{arg}(p_j) \in ω$, for $j=1,2,3$. A density version of the binary Goldbach conjecture in a sector is also proved.
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Submitted 20 March, 2024; v1 submitted 25 September, 2023;
originally announced September 2023.
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The impact of the solubilizer of an element on the structure of a finite group
Authors:
Hamid Mousavi,
Mina Poozesh,
Yousef Zamani
Abstract:
Let $G$ be a finite group, and let $x$ be an element of $G$. Denote by $\Sol_G(x)$ the set of all $y \in G$ such that the group generated by $x$ and $y$ is soluble. We investigate the influence of $\Sol_G(x)$ on the structure of $G$.
Let $G$ be a finite group, and let $x$ be an element of $G$. Denote by $\Sol_G(x)$ the set of all $y \in G$ such that the group generated by $x$ and $y$ is soluble. We investigate the influence of $\Sol_G(x)$ on the structure of $G$.
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Submitted 2 April, 2023; v1 submitted 20 October, 2022;
originally announced October 2022.
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On a conjecture of Graham on the p-divisibility of central binomial coefficients
Authors:
Ernie Croot,
Hamed Mousavi,
Maxie Schmidt
Abstract:
We show that for every $r \geq 1$, and all $r$ distinct (sufficiently large) primes $p_1,..., p_r > p_0(r)$, there exist infinitely many integers $n$ such that ${2n \choose n}$ is divisible by these primes to only low multiplicity. From a theorem of Kummer, an upper bound for the number of times that a prime $p_j$ can divide ${2n \choose n}$ is $1+\log n / \log p_j$; and our theorem shows that for…
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We show that for every $r \geq 1$, and all $r$ distinct (sufficiently large) primes $p_1,..., p_r > p_0(r)$, there exist infinitely many integers $n$ such that ${2n \choose n}$ is divisible by these primes to only low multiplicity. From a theorem of Kummer, an upper bound for the number of times that a prime $p_j$ can divide ${2n \choose n}$ is $1+\log n / \log p_j$; and our theorem shows that for every $\varepsilon > 0$, $r \geq 1$, and any sufficiently large primes $p_1,...,p_r > p_0(\varepsilon,r)$, we can find integers $n$ where for $j=1,...,r$, $p_j$ divides ${2n \choose n}$ with multiplicity at most $\varepsilon \log n/\log p_j$. We connect this result to a famous conjecture by R. L. Graham on whether there are infinitely many integers $n$ such that ${2n \choose n}$ is coprime to $105$.
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Submitted 6 January, 2023; v1 submitted 26 January, 2022;
originally announced January 2022.
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Improving and Maximal Inequalities for Primes in Progressions
Authors:
Christina Giannitsi,
Michael T. Lacey,
Hamed Mousavi,
Yaghoub Rahimi
Abstract:
Assume that $ y < N$ are integers, and that $ (b,y) =1$. Define an average along the primes in a progression of diameter $ y$, given by integer $ (b,y)=1 $. \begin{align*} A_{N,y,b} := \frac{φ(y)}{N} \sum _{\substack{n <N\\n\equiv b\pmod{y}}} Λ(n) f(x-n) \end{align*} Above, $Λ$ is the von Mangoldt function and $φ$ is the totient function. We establish improving and maximal inequalities for these a…
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Assume that $ y < N$ are integers, and that $ (b,y) =1$. Define an average along the primes in a progression of diameter $ y$, given by integer $ (b,y)=1 $. \begin{align*} A_{N,y,b} := \frac{φ(y)}{N} \sum _{\substack{n <N\\n\equiv b\pmod{y}}} Λ(n) f(x-n) \end{align*} Above, $Λ$ is the von Mangoldt function and $φ$ is the totient function. We establish improving and maximal inequalities for these averages. These bounds are uniform in the choice of progression. For instance, for $ 1< r < \infty $ there is an integer $N _{y, r}$ so that \begin{align*} \lVert \sup _{N>N _{y,r}} \lvert A_{N,y,b} f \rvert \rVert_{r}\ll \lVert f\rVert_{r}. \end{align*} The implied constant is only a function of $ r$. The uniformity over progressions imposes several novel elements on the proof.
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Submitted 17 April, 2022; v1 submitted 14 December, 2021;
originally announced December 2021.
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Nonlocal Games, Compression Theorems, and the Arithmetical Hierarchy
Authors:
Hamoon Mousavi,
Seyed Sajjad Nezhadi,
Henry Yuen
Abstract:
We investigate the connection between the complexity of nonlocal games and the arithmetical hierarchy, a classification of languages according to the complexity of arithmetical formulas defining them. It was recently shown by Ji, Natarajan, Vidick, Wright and Yuen that deciding whether the (finite-dimensional) quantum value of a nonlocal game is $1$ or at most $\frac{1}{2}$ is complete for the cla…
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We investigate the connection between the complexity of nonlocal games and the arithmetical hierarchy, a classification of languages according to the complexity of arithmetical formulas defining them. It was recently shown by Ji, Natarajan, Vidick, Wright and Yuen that deciding whether the (finite-dimensional) quantum value of a nonlocal game is $1$ or at most $\frac{1}{2}$ is complete for the class $Σ_1$ (i.e., $\mathsf{RE}$). A result of Slofstra implies that deciding whether the commuting operator value of a nonlocal game is equal to $1$ is complete for the class $Π_1$ (i.e., $\mathsf{coRE}$). We prove that deciding whether the quantum value of a two-player nonlocal game is exactly equal to $1$ is complete for $Π_2$; this class is in the second level of the arithmetical hierarchy and corresponds to formulas of the form "$\forall x \, \exists y \, φ(x,y)$". This shows that exactly computing the quantum value is strictly harder than approximating it, and also strictly harder than computing the commuting operator value (either exactly or approximately). We explain how results about the complexity of nonlocal games all follow in a unified manner from a technique known as compression. At the core of our $Π_2$-completeness result is a new "gapless" compression theorem that holds for both quantum and commuting operator strategies. Our compression theorem yields as a byproduct an alternative proof of Slofstra's result that the set of quantum correlations is not closed. We also show how a "gap-preserving" compression theorem for commuting operator strategies would imply that approximating the commuting operator value is complete for $Π_1$.
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Submitted 11 October, 2021; v1 submitted 9 October, 2021;
originally announced October 2021.
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Synchronous Values of Games
Authors:
J. William Helton,
Hamoon Mousavi,
Seyed Sajjad Nezhadi,
Vern I. Paulsen,
Travis B. Russell
Abstract:
We study synchronous values of games, especially synchronous games. It is known that a synchronous game has a perfect strategy if and only if it has a perfect synchronous strategy. However, we give examples of synchronous games, in particular graph colouring games, with synchronous value that is strictly smaller than their ordinary value. Thus, the optimal strategy for a synchronous game need not…
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We study synchronous values of games, especially synchronous games. It is known that a synchronous game has a perfect strategy if and only if it has a perfect synchronous strategy. However, we give examples of synchronous games, in particular graph colouring games, with synchronous value that is strictly smaller than their ordinary value. Thus, the optimal strategy for a synchronous game need not be synchronous. We derive a formula for the synchronous value of an XOR game as an optimization problem over a spectrahedron involving a matrix related to the cost matrix. We give an example of a game such that the synchronous value of repeated products of the game is strictly increasing. We show that the synchronous quantum bias of the XOR of two XOR games is not multiplicative. Finally, we derive geometric and algebraic conditions that a set of projections that yields the synchronous value of a game must satisfy.
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Submitted 22 August, 2023; v1 submitted 29 September, 2021;
originally announced September 2021.
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Endpoint $ \ell ^{r}$ improving estimates for Prime averages
Authors:
Michael T. Lacey,
Hamed Mousavi,
Yaghoub Rahimi
Abstract:
Let $ Λ$ denote von Mangoldt's function, and consider the averages \begin{align*} A_N f (x) &=\frac{1}{N}\sum_{1\leq n \leq N}f(x-n)Λ(n) . \end{align*} We prove sharp $ \ell ^{p}$-improving for these averages, and sparse bounds for the maximal function. The simplest inequality is that for sets $ F, G\subset [0,N]$ there holds \begin{equation*} N ^{-1} \langle A_N \mathbf 1_{F} , \mathbf 1_{G} \ran…
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Let $ Λ$ denote von Mangoldt's function, and consider the averages \begin{align*} A_N f (x) &=\frac{1}{N}\sum_{1\leq n \leq N}f(x-n)Λ(n) . \end{align*} We prove sharp $ \ell ^{p}$-improving for these averages, and sparse bounds for the maximal function. The simplest inequality is that for sets $ F, G\subset [0,N]$ there holds \begin{equation*} N ^{-1} \langle A_N \mathbf 1_{F} , \mathbf 1_{G} \rangle \ll \frac{\lvert F\rvert \cdot \lvert G\rvert} { N ^2 } \Bigl( \operatorname {Log} \frac{\lvert F\rvert \cdot \lvert G\rvert} { N ^2 } \Bigr) ^{t}, \end{equation*} where $ t=2$, or assuming the Generalized Riemann Hypothesis, $ t=1$. The corresponding sparse bound is proved for the maximal function $ \sup_N A_N \mathbf 1_{F}$. The inequalities for $ t=1$ are sharp. The proof depends upon the Circle Method, and an interpolation argument of Bourgain.
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Submitted 1 May, 2023; v1 submitted 25 January, 2021;
originally announced January 2021.
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On Reduced archimedean skew power series rings
Authors:
Hamed Mousavi,
Farzad Padashnik,
Ayesha Asloob Qureshi
Abstract:
In this paper, we prove that if $R$ is an Archimedean reduced ring and satisfy ACC on annihilators, then $R[[x]]$ is also an Archimedean reduced ring. More generally we prove that if $R$ is a right Archimedean ring satisfying the \emph{ACC} on annihilators and $α$ is a rigid automorphism of $R$, then the skew power series ring $R[[x;α]]$ is right Archimedean reduced ring. We also provide some exam…
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In this paper, we prove that if $R$ is an Archimedean reduced ring and satisfy ACC on annihilators, then $R[[x]]$ is also an Archimedean reduced ring. More generally we prove that if $R$ is a right Archimedean ring satisfying the \emph{ACC} on annihilators and $α$ is a rigid automorphism of $R$, then the skew power series ring $R[[x;α]]$ is right Archimedean reduced ring. We also provide some examples to justify the assumptions we made to obtain the required result.
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Submitted 3 September, 2020;
originally announced September 2020.
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On a Class of Sums with Unexpectedly High Cancellation, and its Applications
Authors:
Ernie Croot,
Hamed Mousavi
Abstract:
Following attempts at an analytic proof of the Pentagonal Number Theorem, we report on the discovery of a general principle leading to an unexpected cancellation of oscillating sums. After stating the motivation, and our theorem, we apply it to prove several results on the Prouhet-Tarry-Escott Problem, integer partitions, and the distribution of prime numbers. Regarding the Prouhet-Tarry-Escott pr…
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Following attempts at an analytic proof of the Pentagonal Number Theorem, we report on the discovery of a general principle leading to an unexpected cancellation of oscillating sums. After stating the motivation, and our theorem, we apply it to prove several results on the Prouhet-Tarry-Escott Problem, integer partitions, and the distribution of prime numbers. Regarding the Prouhet-Tarry-Escott problem, we show that \begin{align*} \sum_{|\ell|\leq x}(4x^2-4\ell^2)^{2r}-\sum_{|\ell|<x}(4x^2-(2\ell+1)^2)^{2r}=\text{polynomial w.r.t. } x \text{ with degree }2r-1. \end{align*} This can perhaps be proved using properties of Bernoulli polynomials, but the claim fell out of our method in a more natural and motivated way. Using this result, we solve an approximate version of the PTE Problem, and in doing so our work in the approximate case exceeds the bounds one can prove using a pigeonhole argument, which seems remarkable. Also, we prove that $$ \sum_{\ell^2 < n} (-1)^\ell p(n-\ell^2)\ \sim\ (-1)^n 2^{-3/4} n^{-1/4} \sqrt{p(n)}, $$ where $p(n)$ is the usual partition function. We get the following "Weak pentagonal number theorem", in which we can replace the partition function $p(n)$ with Chebyshev $Ψ$ function: $$ \sum_{0 < \ell < \sqrt{xT}/2} Ψ([e^{\sqrt{x - \frac{(2\ell)^2}{T}}},\ e^{\sqrt{x - \frac{(2\ell-1)^2}{T}}}])\ =Ψ(e^{\sqrt{x}})\left(\frac{1}{2} + O\left (e^{-0.196\sqrt{x}}\right)\right), $$ where $T=e^{0.786\sqrt{x}}$, where $Ψ([a,b]) := \sum_{n\in [a,b]} Λ(n)$ and $Ψ(x) = Ψ([1,x])$, where $Λ$ is the von Mangoldt function. Note that this last equation (sum over $\ell$) is stronger than one would get using a strong form of the Prime Number Theorem and also a naive use of the Riemann Hypothesis in each interval, since the widths of the intervals are smaller than $e^{\frac{1}{2} \sqrt{x}}$, making the RH estimate ``trivial".
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Submitted 22 June, 2022; v1 submitted 26 September, 2019;
originally announced September 2019.
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Factorization Theorems for Relatively Prime Divisor Sums, GCD Sums and Generalized Ramanujan Sums
Authors:
Hamed Mousavi,
Maxie D. Schmidt
Abstract:
We generalize recent matrix-based factorization theorems for Lambert series generating functions generating the coefficients $(f \ast 1)(n)$ for some arithmetic function $f$. Our new factorization theorems provide analogs to these established expansions generating sums of the form $\sum_{d: (d,n)=1} f(d)$ (type I) and the Anderson-Apostol sums $\sum_{d|(m,n)} f(d) g(n/d)$ (type II) for any arithme…
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We generalize recent matrix-based factorization theorems for Lambert series generating functions generating the coefficients $(f \ast 1)(n)$ for some arithmetic function $f$. Our new factorization theorems provide analogs to these established expansions generating sums of the form $\sum_{d: (d,n)=1} f(d)$ (type I) and the Anderson-Apostol sums $\sum_{d|(m,n)} f(d) g(n/d)$ (type II) for any arithmetic functions $f$ and $g$. Our treatment of the type II sums includes a matrix-based factorization method relating the partition function $p(n)$ to arbitrary arithmetic functions $f$. We also conclude the last section of the article by directly expanding new formulas for an arithmetic function $g$ by the type II sums using discrete Fourier transforms for functions over inputs of greatest common divisors and by suitably defined orthogonal polynomial sequences whose weight function we can define by a discrete time Fourier transform (DTFT) involving the partition function $p(n)$. There are numerous applications and special cases of our new results which we are able to cite as examples in the article. Particular cases of the applications we give in the article include new identities for Euler's totient function, the Ramanujan sums $c_q(n)$, the generalized sum-of-divisors functions, the Mertens function which is the summatory function of the Möbius function, and the cyclotomic polynomials.
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Submitted 19 September, 2019; v1 submitted 19 October, 2018;
originally announced October 2018.
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Stability analysis of networked control systems with not necessarily UGES protocols
Authors:
Seyed Hossein Mousavi,
Navid Noroozi,
Anton H. J. de Ruiter,
Roman Geiselhart
Abstract:
This note studies (practical) asymptotic stability of nonlinear networked control systems whose protocols are not necessarily uniformly globally exponentially stable. In particular, we propose a Lyapunov-based approach to establish (practical) asymptotic stability of the networked control systems. Considering so-called modified Round Robin and Try-Once-Discard protocols, which are only uniformly g…
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This note studies (practical) asymptotic stability of nonlinear networked control systems whose protocols are not necessarily uniformly globally exponentially stable. In particular, we propose a Lyapunov-based approach to establish (practical) asymptotic stability of the networked control systems. Considering so-called modified Round Robin and Try-Once-Discard protocols, which are only uniformly globally asymptotically stable, we explicitly construct Lyapunov functions for these two protocols, which fit our proposed setting. In order to optimize the usage of communication resource, we exploit the following transmission policy: wait for a certain minimum amount of time after the last sampling instant and then check a state-dependent criterion. When the latter condition is violated, a transmission occurs. In that way, the existence of the minimum amount of time between two consecutive transmission is established and so-called Zeno phenomenon, therefore, is avoided. Finally, illustrative examples are given to verify the effectiveness of our results.
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Submitted 9 October, 2018; v1 submitted 5 October, 2018;
originally announced October 2018.
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Integral versions of input-to-state stability for dual-rate nonlinear sampled-data systems
Authors:
Navid Noroozi,
Seyed Hossein Mousavi,
Horacio J. Marquez
Abstract:
This paper presents versions of integral input-to-state stability and integral input-to-integral-state stability for nonlinear sampled-data systems, under the low measurement rate constraint. In particular, we compensate the lack of measurements using an estimator approximately reconstructing the current state. Interestingly, under certain checkable conditions, we establish that a controller that…
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This paper presents versions of integral input-to-state stability and integral input-to-integral-state stability for nonlinear sampled-data systems, under the low measurement rate constraint. In particular, we compensate the lack of measurements using an estimator approximately reconstructing the current state. Interestingly, under certain checkable conditions, we establish that a controller that semiglobally practically integral input-to-(integral-) state stabilizes an approximate discrete-time model of a single-rate nonlinear sampled-data system, also stabilizes the exact discrete-time model of the nonlinear sampled-data system in the same sense implemented in a dual-rate setting. Numerical simulations are given to illustrate the effectiveness of our results.
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Submitted 22 April, 2018;
originally announced April 2018.
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Solving the L1 regularized least square problem via a box-constrained smooth minimization
Authors:
Majid Mohammadi,
Wout Hofman,
Yaohua Tan,
S. Hamid Mousavi
Abstract:
In this paper, an equivalent smooth minimization for the L1 regularized least square problem is proposed. The proposed problem is a convex box-constrained smooth minimization which allows applying fast optimization methods to find its solution. Further, it is investigated that the property "the dual of dual is primal" holds for the L1 regularized least square problem. A solver for the smooth probl…
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In this paper, an equivalent smooth minimization for the L1 regularized least square problem is proposed. The proposed problem is a convex box-constrained smooth minimization which allows applying fast optimization methods to find its solution. Further, it is investigated that the property "the dual of dual is primal" holds for the L1 regularized least square problem. A solver for the smooth problem is proposed, and its affinity to the proximal gradient is shown. Finally, the experiments on L1 and total variation regularized problems are performed, and the corresponding results are reported.
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Submitted 20 October, 2021; v1 submitted 11 April, 2017;
originally announced April 2017.
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Characterization of the skew cyclic codes over Fp+vFp
Authors:
Reza Dastbasteh,
Seyyed Hamed Mousavi,
Javad Haghighat
Abstract:
We study cyclic codes with arbitrary length over Fp+vFp where theta(v)=av, a in Fp and v^2=0. We characterize all existing codes in case of O(theta)|n by using certain projections from (Fp+vFp)[x;theta] to Fp[x]. We provide an explicit expression for the ensemble of all possible codes. We also prove useful properties of these codes in the case where O(theta)|n does not hold. We provide results and…
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We study cyclic codes with arbitrary length over Fp+vFp where theta(v)=av, a in Fp and v^2=0. We characterize all existing codes in case of O(theta)|n by using certain projections from (Fp+vFp)[x;theta] to Fp[x]. We provide an explicit expression for the ensemble of all possible codes. We also prove useful properties of these codes in the case where O(theta)|n does not hold. We provide results and examples to illustrate how the codes are constructed and how their encoding and decoding are realized.
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Submitted 5 July, 2016;
originally announced July 2016.
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S-Noetherian generalized power series rings
Authors:
F. Padashnik,
A. Moussavi,
H. Mousavi
Abstract:
Let R be a ring with identity, (M;\leq) a commutative positive strictly ordered monoid and w_m an automorphism for each m \in M . The skew generalized power series ring R[[M,w]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal'cev Neumann Laurent series rings. If S\subset R is a multiplicative set, then R…
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Let R be a ring with identity, (M;\leq) a commutative positive strictly ordered monoid and w_m an automorphism for each m \in M . The skew generalized power series ring R[[M,w]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal'cev Neumann Laurent series rings. If S\subset R is a multiplicative set, then R is called right S-Noetherian, if for each ideal I of R, Is \subseteq J\subseteq I for some s\in S and some finitely generated right ideal J . Unifying and generalizing a number of known results, we study transfers of S-Noetherian property to the ring R[[M,w]]. We also show that the ring R[[M,w]] is left Noetherian if and only if R is left Noetherian and M is finitely generated. Generalizing a result of Anderson and Dumitrescu, we show that,when S\subset R is a-anti-Archimedean multiplicative set with a an automorphism of R, then R is right S-Noetherian if and only if the skew polynomial ring R[x,a] is right S-Noetherian.
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Submitted 30 May, 2016;
originally announced May 2016.
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Automatic Theorem Proving in Walnut
Authors:
Hamoon Mousavi
Abstract:
Walnut is a software package that implements a mechanical decision procedure for deciding certain combinatorial properties of some special words referred to as automatic words or automatic sequences. Walnut is written in Java and is open source. It is licensed under GNU General Public License.
Walnut is a software package that implements a mechanical decision procedure for deciding certain combinatorial properties of some special words referred to as automatic words or automatic sequences. Walnut is written in Java and is open source. It is licensed under GNU General Public License.
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Submitted 25 May, 2021; v1 submitted 18 March, 2016;
originally announced March 2016.
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The ascending chain condition for principal left or right ideals of skew generalized power series rings
Authors:
F. Padashnik,
A. Moussavi,
H. Mousavi
Abstract:
Let $R$ be a ring, $(S,\leq)$ a strictly ordered monoid and $ω: S\rightarrow End(R)$ a monoid homomorphism. In this paper we study the ascending chain conditions on principal left (resp. right) ideals of the skew generalized power series ring $R[[S,ω]]$. Among other results, it is shown that $R[[S,ω]]$ is a right archimedean reduced ring if $S$ is an Artinian strictly totally ordered monoid, $R$ i…
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Let $R$ be a ring, $(S,\leq)$ a strictly ordered monoid and $ω: S\rightarrow End(R)$ a monoid homomorphism. In this paper we study the ascending chain conditions on principal left (resp. right) ideals of the skew generalized power series ring $R[[S,ω]]$. Among other results, it is shown that $R[[S,ω]]$ is a right archimedean reduced ring if $S$ is an Artinian strictly totally ordered monoid, $R$ is a right archimedean and $S$-rigid ring which satisfies the ACC on annihilators and $ω_s$ preserves nonunits of $R$ for each $s\in S$. As a consequence we deduce that the power series rings, Laurent series rings, skew power series rings, skew Laurent series rings and generalized power series rings are reduced satisfying the ascending chain condition on principal left (or right) ideals. It is also proved that, the skew Laurent polynomial ring $R[x,x^{-1};α]$ satisfies \emph{ACCPL(R)}, if $R$ is $α$-rigid and satisfies \emph{ACCPL(R)} and the $ACC$ on left(resp. right) annihilators. Examples are provided to illustrate and delimit our results.
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Submitted 21 January, 2016;
originally announced January 2016.
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ICR: Iterative Convex Refinement for Sparse Signal Recovery Using Spike and Slab Priors
Authors:
Hojjat S. Mousavi,
Vishal Monga,
Trac D. Tran
Abstract:
In this letter, we address sparse signal recovery using spike and slab priors. In particular, we focus on a Bayesian framework where sparsity is enforced on reconstruction coefficients via probabilistic priors. The optimization resulting from spike and slab prior maximization is known to be a hard non-convex problem, and existing solutions involve simplifying assumptions and/or relaxations. We pro…
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In this letter, we address sparse signal recovery using spike and slab priors. In particular, we focus on a Bayesian framework where sparsity is enforced on reconstruction coefficients via probabilistic priors. The optimization resulting from spike and slab prior maximization is known to be a hard non-convex problem, and existing solutions involve simplifying assumptions and/or relaxations. We propose an approach called Iterative Convex Refinement (ICR) that aims to solve the aforementioned optimization problem directly allowing for greater generality in the sparse structure. Essentially, ICR solves a sequence of convex optimization problems such that sequence of solutions converges to a sub-optimal solution of the original hard optimization problem. We propose two versions of our algorithm: a.) an unconstrained version, and b.) with a non-negativity constraint on sparse coefficients, which may be required in some real-world problems. Experimental validation is performed on both synthetic data and for a real-world image recovery problem, which illustrates merits of ICR over state of the art alternatives.
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Submitted 16 February, 2015;
originally announced February 2015.
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Mechanical Proofs of Properties of the Tribonacci Word
Authors:
Hamoon Mousavi,
Jeffrey Shallit
Abstract:
We implement a decision procedure for answering questions about a class of infinite words that might be called (for lack of a better name) "Tribonacci-automatic". This class includes, for example, the famous Tribonacci word T = 0102010010202 ..., the fixed point of the morphism 0 -> 01, 1 -> 02, 2 -> 0. We use it to reprove some old results about the Tribonacci word from the literature, such as as…
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We implement a decision procedure for answering questions about a class of infinite words that might be called (for lack of a better name) "Tribonacci-automatic". This class includes, for example, the famous Tribonacci word T = 0102010010202 ..., the fixed point of the morphism 0 -> 01, 1 -> 02, 2 -> 0. We use it to reprove some old results about the Tribonacci word from the literature, such as assertions about the occurrences in T of squares, cubes, palindromes, and so forth. We also obtain some new results.
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Submitted 27 July, 2014; v1 submitted 22 July, 2014;
originally announced July 2014.
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Decision Algorithms for Fibonacci-Automatic Words, with Applications to Pattern Avoidance
Authors:
Chen Fei Du,
Hamoon Mousavi,
Luke Schaeffer,
Jeffrey Shallit
Abstract:
We implement a decision procedure for answering questions about a class of infinite words that might be called (for lack of a better name) "Fibonacci-automatic". This class includes, for example, the famous Fibonacci word f = 01001010..., the fixed point of the morphism 0 -> 01 and 1 -> 0. We then recover many results about the Fibonacci word from the literature (and improve some of them), such as…
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We implement a decision procedure for answering questions about a class of infinite words that might be called (for lack of a better name) "Fibonacci-automatic". This class includes, for example, the famous Fibonacci word f = 01001010..., the fixed point of the morphism 0 -> 01 and 1 -> 0. We then recover many results about the Fibonacci word from the literature (and improve some of them), such as assertions about the occurrences in f of squares, cubes, palindromes, and so forth. As an application of our method we prove a new result: there exists an aperiodic infinite binary word avoiding the pattern x x x^R. This is the first avoidability result concerning a nonuniform morphism proven purely mechanically.
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Submitted 27 July, 2014; v1 submitted 3 June, 2014;
originally announced June 2014.
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Shortest Repetition-Free Words Accepted by Automata
Authors:
Hamoon Mousavi,
Jeffrey Shallit
Abstract:
We consider the following problem: given that a finite automaton $M$ of $N$ states accepts at least one $k$-power-free (resp., overlap-free) word, what is the length of the shortest such word accepted? We give upper and lower bounds which, unfortunately, are widely separated.
We consider the following problem: given that a finite automaton $M$ of $N$ states accepts at least one $k$-power-free (resp., overlap-free) word, what is the length of the shortest such word accepted? We give upper and lower bounds which, unfortunately, are widely separated.
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Submitted 10 April, 2013;
originally announced April 2013.
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Repetition Avoidance in Circular Factors
Authors:
Hamoon Mousavi,
Jeffrey Shallit
Abstract:
We consider the following novel variation on a classical avoidance problem from combinatorics on words: instead of avoiding repetitions in all factors of a word, we avoid repetitions in all factors where each individual factor is considered as a "circular word", i.e., the end of the word wraps around to the beginning. We determine the best possible avoidance exponent for alphabet size 2 and 3, and…
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We consider the following novel variation on a classical avoidance problem from combinatorics on words: instead of avoiding repetitions in all factors of a word, we avoid repetitions in all factors where each individual factor is considered as a "circular word", i.e., the end of the word wraps around to the beginning. We determine the best possible avoidance exponent for alphabet size 2 and 3, and provide a lower bound for larger alphabets.
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Submitted 17 March, 2013; v1 submitted 30 November, 2012;
originally announced December 2012.
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On the Number of Unbordered Factors
Authors:
Daniel Goc,
Hamoon Mousavi,
Jeffrey Shallit
Abstract:
We illustrate a general technique for enumerating factors of k-automatic sequences by proving a conjecture on the number f(n) of unbordered factors of the Thue-Morse sequence. We show that f(n) <= n for n >= 4 and that f(n) = n infinitely often. We also give examples of automatic sequences having exactly 2 unbordered factors of every length.
We illustrate a general technique for enumerating factors of k-automatic sequences by proving a conjecture on the number f(n) of unbordered factors of the Thue-Morse sequence. We show that f(n) <= n for n >= 4 and that f(n) = n infinitely often. We also give examples of automatic sequences having exactly 2 unbordered factors of every length.
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Submitted 6 November, 2012;
originally announced November 2012.
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The Structure of $G/Φ(G)$ in Terms of $σ(G)$
Authors:
Alireza Jamali,
Hamid Mousavi
Abstract:
Let $G$ be a finite group. We let $\f{m}(G)$ and $\sig(G)$ denote the number of maximal subgroups of $G$ and the least positive integer $n$ such that $G$ is written as the union of $n$ proper subgroups, respectively. In this paper we determine the structure of $G/Φ(G)$ when $G$ is a finite soluble group with $\f{m}(G)\leq 2\sig(G)$.
Let $G$ be a finite group. We let $\f{m}(G)$ and $\sig(G)$ denote the number of maximal subgroups of $G$ and the least positive integer $n$ such that $G$ is written as the union of $n$ proper subgroups, respectively. In this paper we determine the structure of $G/Φ(G)$ when $G$ is a finite soluble group with $\f{m}(G)\leq 2\sig(G)$.
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Submitted 19 July, 2005;
originally announced July 2005.