arXiv Math @arxiv_math@qoto.org

We present a survey of exact and asymptotic formulas on the number of cyclic subgroups and total number of subgroups of the groups ${\Bbb Z}_{n_1} \times \cdots \times {\Bbb Z}_{n_k}$, where $k\ge 2$ and $n_1,\ldots,n_k$ are arbitrary positive integers.

For every group $\{\pm1\}\subseteq Δ\subseteq (\mathbb{Z}/N\mathbb{Z})^\times$, there exists an intermediate modular curve $X_Δ(N)$. In this paper we determine all curves $X_Δ(N)$ whose $\mathbb{Q}$-gonality is equal to $4$, all curves $X_Δ(N)$ whose $\mathbb{C}$-gonality is equal to $4$, and all curves $X_Δ(N)$ whose $\mathbb{Q}$-gonality is equal to $5$. We also determine the $\mathbb{Q}$-gonality of all curves $X_Δ(N)$ for $N\leq 40$ and $\{\pm1\}\subsetneq Δ\subsetneq (\mathbb{Z}/N\mathbb{Z})^\times$.

Khosravi, Drnovšek and Moslehian [\textit{Filomat, 2012}] derived Buzano inequality for Hilbert C*-modules. Using this inequality we derive Deutsch entropic uncertainty principle for Hilbert C*-modules over commutative unital C*-algebras.

In recent times, a wide variety of combinatorics has been introduced in order to solve problems from algebraic geometry. Newton-Okounkov bodies and tropical geometry are two such combinatorial theories. As shown by Kaveh and Manon, there is a certain correspondence between these two. Building on this correspondence, and exploiting the link of both theories to toric degenerations, Harada and Escobar obtained their wall-crossing result for prime cones. This result states that moving between two adjacent prime maximal cones in a tropical variety corresponds to a mutation between the associated Newton-Okounkov bodies of these cones. In this thesis, we provide a method for applying the wall-crossing result to non-prime cones. Our approach uses a procedure developed by Bossinger, Lamboglia, Mincheva and Mohammadi in order to compute an embedding which changes a tropical variety in such a way that a non-prime cone becomes prime. Assuming that adjacent cones stay adjacent, the wall-crossing result can then be applied in this new embedding. Building on computations by Clarke, Mohammadi and Zaffalon, we show that for Gr(3,6), this approach works. We compute the new embedding and its tropicalization in this case, and study the relation between cones in the new embedding and the original embedding.

In this note, we give a group-theoretic condition which is equivalent to the fact that the trivial character is the only complex irreducible character of a finite group G which is contained in the principal p-block for each prime p in a specified set of prime divisors of the order of G. This character-theoretic condition has been studied previously by a number of authors.

The Katz-Sarnak philosophy states that statistics of zeros of $L$-function families near the central point as the conductors tend to infinity agree with those of eigenvalues of random matrix ensembles as the matrix size tends to infinity. While numerous results support this conjecture, S. J. Miller observed that for finite conductors, very different behavior can occur for zeros near the central point in elliptic curve $L$-function families. This led to the creation of the excised model of Dueñez, Huynh, Keating, Miller, and Snaith, whose predictions for quadratic twists of a given elliptic curve are well fit by the data. The key ingredients are relating the discretization of central values of the $L$-functions to excising matrices based on the value of the characteristic polynomials at 1 and using lower order terms (in statistics such as the one-level density and pair-correlation) to adjust the matrix size. We extended this model for a family of twists of an $L$-function associated to a given holomorphic cuspidal newform of odd prime level and arbitrary weight. We derive the corresponding "effective" matrix size for a given form by computing the one-level density and pair-correlation statistics for a chosen family of twists, and we show there is no repulsion for forms with weight greater than 2 and principal nebentype. We experimentally verify the accuracy of the model, and as expected, our model recovers the elliptic curve model.

We provide the detailed construction of the virtual cycles needed for defining the cohomological field theory associated to a gauged linear sigma model in geometric phase.

We investigate the convexity property on $(0,1)$ of the function $$f_a(x)=\frac{{\cal K}{(\sqrt x)}}{a-(1/2)\log(1-x)}.$$ We show that $f_a$ is strictly convex on $(0,1)$ if and only if $a\geq a_c$ and $1/f_a$ is strictly convex on $(0,1)$ if and only if $a\leq\log 4$, where $a_c$ is some critical value. The second main result of the paper is to study the log-convexity and log-concavity of the function $$h_p(x)=(1-x)^p{\cal K}(\sqrt x).$$ We prove that $h_p$ is strictly log-concave on $(0,1)$ if and only if $p\geq 7/32$ and strictly log-convex if and only if $p\leq 0$. This solves some problems posed by Yang and Tian and complete their result and a result of Alzer and Richards that $f_a$ is strictly concave on $(0,1)$ if and only if $a=4/3$ and $1/f_a$ is strictly concave on $(0,1)$ if and only if $a\geq 8/5$. As applications of the convexity and concavity, we establish among other inequalities, that for $a\geq a_c$ and all $r\in(0,1)$ $$\frac{2π\sqrtπ}{(2a+\log 2)Γ(3/4)^2}\leq \frac{{\cal K}(\sqrt r)}{a-\frac12\log (r)}+\frac{{\cal K}(\sqrt{1-r})}{a-\frac12\log (1-r)}<1+\fracπ{2a},$$ and for $p\geq 3(2+\sqrt 2)/8$ and all $r\in(0,1)$ $$\sqrt{(r-r^2)^p{\cal K}(\sqrt{1-r}){\cal K}(\sqrt r)}< \frac{π\sqrtπ}{2^{p+1}Γ(3/4)^2}<\frac{r^p{\cal K}(\sqrt{1-r})+(1-r)^p{\cal K}(\sqrt r)}{2}.$$

The study of probability distributions for random variables and their algebraic combinations has been a central focus driving the advancement of probability and statistics. Since the 1920s, the challenge of calculating the probability distributions of sums, differences, products, and quotients of independent random variables have drawn the attention of numerous statisticians and mathematicians who studied the algebraic properties and relationships of random variables. Statistical distributions are highly helpful in data science and machine learning, as they provide a range of possible values for the variables, aiding in the development of a deeper understanding of the underlying problem. In this paper, we have presented a new probability distribution based on the $\hat{I}$-function. Also, we have discussed the applications of the $\hat{I}$ function, particularly in deriving the distributions of product and the quotient involving two independent $\hat{I}$ function variates. Additionally, it has been shown that both the product and quotient of two independent $\hat{I}$-function variates also follow the $\hat{I}$-function distribution. Furthermore, the new distribution, known as the $\hat{I}$-function distribution, includes several well-known classical distributions such as the gamma, beta, exponential, normal H-function, and G-function distributions, among others, as special cases. Therefore, the $\hat{I}$-function distribution can be considered a characterization or generalization of the above-mentioned distributions.

We explore the application of the quasi-Monte Carlo (QMC) method in deep backward dynamic programming (DBDP) (Hure et al. 2020) for numerically solving high-dimensional nonlinear partial differential equations (PDEs). Our study focuses on examining the generalization error as a component of the total error in the DBDP framework, discovering that the rate of convergence for the generalization error is influenced by the choice of sampling methods. Specifically, for a given batch size $m$, the generalization error under QMC methods exhibits a convergence rate of $O(m^{-1+\varepsilon})$, where $\varepsilon$ can be made arbitrarily small. This rate is notably more favorable than that of the traditional Monte Carlo (MC) methods, which is $O(m^{-1/2+\varepsilon})$. Our theoretical analysis shows that the generalization error under QMC methods achieves a higher order of convergence than their MC counterparts. Numerical experiments demonstrate that QMC indeed surpasses MC in delivering solutions that are both more precise and stable.

Motivated by recent research of Krattenthaler and Wang, we propose five new ``Borwein-type'' conjectures modulo $3$ and two new ``Borwein-type'' conejctures modulo $5$.

Let $K$ be a simplicial complex, and let $Δ_i^{up}(K)$ be the $i$-th up normalized Laplacian of $K$. Horak and Jost showed that the largest eigenvalue of $Δ_i^{up}(K)$ is at most $i+2$, and characterized the equality case by the orientable or non-orientable circuits. In this paper, by using the balancedness of signed graphs, we show that $Δ_i^{up}(K)$ has an eigenvalue $i+2$ if and only if $K$ has an $(i+1)$-path connected component $K'$ such that the $i$-th signed incidence graph $B_i(K')$ is balanced, which implies Horak and Jost's characterization. We also characterize the multiplicity of $i+2$ as an eigenvalue of $Δ_i^{up}(K)$, which generalizes the corresponding result in graph case. Finally we gave some classes of infinitely many simplicial complexes $K$ with $Δ_i^{up}(K)$ having an eigenvalue $i+2$ by using wedge, Cartesian product and duplication of motifs.

The two-dimensional Jacobian Conjecture says that a Keller map $f: (x,y) \mapsto (p,q) \in k[x,y]^2$ having an invertible Jacobian is an automorphism of $k[x,y]$. We prove that there is no Keller map with $[k(x,y): k(p,q)]$ prime.

We make use of the Laplace transform and gamma function to construct a new integral transform having the property of mapping a derivative to the backward difference, whence we derive a method for solving difference equations and, relying on classical orthogonal polynomials, for obtaining combinatorial identities. A table of some basic functions is given in the Appendix.

We give a criterion for separability of subgroups of certain outer automorphism groups. This answers questions of Hagen and Sisto, by strengthening and generalizing a result of theirs on mapping class groups.

We investigate Riemannian manifolds $(M^n,g)$ whose curvature operator of the second kind $\mathring{R}$ satisfies the condition \begin{equation*} α^{-1} (λ_1 +\cdots +λ_α) > - θ\barλ, \end{equation*} where $λ_1 \leq \cdots \leq λ_{(n-1)(n+2)/2}$ are the eigenvalues of $\mathring{R}$, $\barλ$ is their average, and $θ> -1$. Under such conditions with optimal $θ$ depending on $n$ and $α$, we prove two differentiable sphere theorems in dimensions three and four, a homological sphere theorem in higher dimensions, and a curvature characterization of Kähler space forms. These results generalize recent works corresponding to $θ=0$ of Cao-Gursky-Tran, Nienhaus-Petersen-Wink, and the author. Moreover, examples are provided to demonstrate the sharpness of all results.

We study the commuting graph of $n\times n$ matrices over the field of $p$-adics $\mathbb{Q}_p$, whose vertices are non-scalar $n\times n$ matrices with entries in $\mathbb{Q}_p$ and whose edges connect pairs of matrices that commute under matrix multiplication. We prove that this graph is connected if and only if $n\geq 3$, with $n$ neither prime nor a power of $p$. We also prove that in the case of $p=2$ and $n=2q$ for $q$ a prime with $q\geq 7$, the commuting graph has the maximum possible diameter of $6$; these are the first known such examples independent of the axiom of choice. We also find choices of $p$ and $n$ yielding diameter $4$ and diameter $5$ commuting graphs, and prove general bounds depending on $p$ and $n$.

We demonstrate that almost all elliptic curves over $\mathbb{Q}$ with prescribed torsion subgroup, when ordered by naive height, have Szpiro ratio arbitrarily close to the expected value. We also provide upper and lower bounds for the Szpiro ratio that hold for almost all elliptic curves in certain one-parameter families. The results are achieved by proving that, given any multivariate polynomial within a general class, the absolute value of the polynomial over an expanding box is typically bounded by a fixed power of its radical. The proof adapts work of Fouvry--Nair--Tenenbaum, which shows that almost all elliptic curves have Szpiro ratio close to $1$.

We study the dynamics of the exponential maps $E_λ: \mathbb{C} \longrightarrow \mathbb{C}$ defined by $E_λ(z) = λe^z$, where $λ> \frac{1}{e}$. We prove that for itineraries of a certain form, the set of all points sharing the given itinerary, together with the point at infinity, is an indecomposable continuum in the Riemann sphere. These itineraries contain infinitely many blocks of zeros whose lengths increase, and they may be unbounded. We prove that in every such continuum, there exists exactly one point whose $ω$-limit set contains the repelling fixed point of $E_λ$. For every other point, the $ω$-limit set is equal either to the point at infinity, or to the forward orbit of $0$ together with the point at infinity. Thus, we generalize the results of R. Devaney and X. Jarque concerning indecomposable continua for bounded itineraries.