Diagonability of idempotent matrices over noncommutative rings

AbstractLet R be an arbitrary ring. In this paper, the following statements are proved: (a) Each idempotent matrix over R can be diagonalized if and only if each idempotent matrix over R has a characteristic vector. (b) An idempotent matrix over R can be diagonalized under a similarity transformation if and only if it is equivalent to a diagonal matrix. (a) and (b) generalize Foster's and Steger's theorems to arbitrary rings. We give some new results about 0-similarity of idempotent matrices over R

Lower bound estimates for the rank of universal quadratic forms in some families of real cubic fields with density one

In this paper, we establish the explicit lower bound estimates for the rank of universal quadratic forms in some certain families of real cubic fields under the condition of density one. The more general results that represent all multiples of a given rational integer are obtained for totally positive definite quadratic lattices. Our main tools are some properties of indecomposable integers with trace in these fields and short vectors in quadratic lattices

The rank of 2-Selmer group associate to θ\theta-congruent numbers

We study the parity of rank of $2$-${\rm Selmer}$ groups associated to $\pi/3$ and $2\pi/3$-congruent numbers. Our second result gives some positive densities about $\pi/3$ and $2\pi/3$ non-congruent numbers which can support the even part of Goldfeld's conjecture. We give some necessary conditions such that $n$ is non $\pi/3$-congruent number for elliptic curves $E_n$ whose Shafarevich-Tate group is non-trivial. In the last section, we show that for $n=pq\equiv 5(resp. \ 11)\pmod{24}$, the density of non $\pi/3$($resp.$ $2\pi/3$)-congruent numbers is at least 75\%, where $p,q$ are primes

CM points, class numbers, and the Mahler measures of x3+y3+1−kxyx^3+y^3+1-kxy

We study the Mahler measures of the polynomial family $Q_k(x,y) = x^3+y^3+1-kxy$ using the method previously developed by the authors. An algorithm is implemented to search for CM points with class numbers $\leqslant 3$, we employ these points to derive interesting formulas that link the Mahler measures of $Q_k(x,y)$ to $L$-values of modular forms. As a by-product, three conjectural identities of Samart are confirmed. For $k=\sqrt[3]{729\pm405\sqrt{3}}$, we also prove an equality that expresses a $2\times 2$ determinant with entries the Mahler measures of $Q_k(x,y)$ as some multiple of the $L$-value of two isogenous elliptic curves over $\mathbb{Q}(\sqrt{3})$.Comment: 17 pages, 2 tables. Comments are welcom

Distribution of the k-regular partition function modulo composite integers M

Let $b_k(n)$ denote the $k-$regular partitons of a natural number $n$. In this paper, we study the behavior of $b_k(n)$ modulo composite integers $M$ which are coprime to $6$. Specially, we prove that for arbitrary $k-$regular partiton function $b_k(n)$ and integer $M$ coprime to $6$, there are infinitely many Ramanujan-type congruences of $b_k(n)$ modulo $M$

An improvement on the parity of Schur's partition function

We improve S.-C. Chen's result on the parity of Schur's partition function. Let $A(n)$ be the number of Schur's partitions of $n$, i.e., the number of partitions of $n$ into distinct parts congruent to $1, 2 \mod{3}$. S.-C. Chen \cite{MR3959837} shows $\small \frac{x}{(\log{x})^{\frac{47}{48}}} \ll \sharp \{0\le n\le x:A(2n+1)\; \text{is odd}\}\ll \frac{x}{(\log{x})^{\frac{1}{2}}}$. In this paper, we improve Chen's result to $\frac{x}{(\log{x})^{\frac{11}{12}}} \ll \sharp \{0\le n\le x:A(2n+1)\; \text{is odd}\}\ll \frac{x}{(\log{x})^{\frac{1}{2}}}.

Several Integral Estimates and Some Applications

In this paper, the authors first consider the bidirectional estimates of several typical integrals. As some applications of these integral estimates, the authors investigate the pointwise multipliers from the normal weight general function space $F(p,\mu,s)$ to the normal weight Bloch type space $\mathcal{B_{\nu}}(B_{n})$ on the unit ball $B_{n}$ of $\mathbb{C}^{n}$, where $\mu$ and $\nu$ are two normal functions on $[0,1)$. For the special normal function $\displaystyle{\mu(r)=(1-r^{2})^{\alpha}\log^{\beta}\frac{e}{1-r^{2}}}$ ($\alpha>0$, $-\infty<\beta<\infty$), the authors give the necessary and sufficient conditions of pointwise multipliers from $F(p,\mu,s)$ to $\mathcal{B_{\nu}}(B_{n})$ for all cases

Efficacy and safety of small-incision corneal intrastromal lenticule implantation for hyperopia correction: a systematic review and meta-analysis

PurposeTo assess the efficacy and safety of intrastromal lenticule implantation for the treatment of hyperopia.MethodsA systematic search of PubMed, Web of Science, Embase, Cochrane Library, China National Knowledge Internet, and Wan Fang Database identified studies on small-incision intrastromal lenticule implantation for hyperopia correction until January 2023. The Joanna Briggs Institute (JBI) critical appraisal tool was used to assess the quality of the retrospective research, and the Methodological Index for Non-randomized Studies (MINORS) was used to assess the quality of the prospective research. This study included postoperative visual outcomes, corneal morphology, and biomechanical outcomes.ResultsA total of 456 articles were identified, of which 10 were included in the meta-analysis. Ten single-arm studies involving 190 eyes were included. A meta-analysis demonstrated that corneal intrastromal lenticule implantation treatment significantly improved hyperopia. Uncorrected distance visual acuity (UDVA) significantly improved compared to the preoperative value (p = 0.027), corrected distance visual acuity showed no difference compared to the preoperative value (p = 0.27), and 87% eyes have no loss of one or more lines in the Snellen lines of CDVA (p &lt; 0.00001). There was a significant difference between the spherical equivalent refractive (SE) and preoperative examination (p &lt; 0.00001), 52% of eyes had ±0.5 diopters (D) postoperative SE (p &lt; 0.00001), and 74% eyes had ±1.0 D postoperative SE (p &lt; 0.00001). The central corneal thickness (CCT) increased by 72.68 μm compared to that preoperatively (p &lt; 0.00001), and corneal curvature increased by 4.18D (p &lt; 0.00001). The Q-value decreased by 0.82 (p &lt; 0.00001), and higher-order aberration (HOA) decreased by 0.66 (p &lt; 0.00001).ConclusionSmall-incision intrastromal lenticule implantation may be an effective solution for correcting hyperopia. The effect of improved vision is significant, but further exploration is needed for changes in corneal biomechanics and long-term safety.Systematic review registration: https://www.crd.york.ac.uk/PROSPERO/, identifier: CRD42023432343