Loss functions with a large number of saddle points are one of the major obstacles for training modern machine learning (ML) models efficiently. First-order methods such as gradient descent (GD) are usually the methods of choice for training ML models. However, these methods converge to saddle points for certain choices of initial guesses. In this paper, we propose a modification of the recently proposed Laplacian smoothing gradient descent (LSGD) [Osher et al., arXiv:1806.06317], called modified LSGD (mLSGD), and demonstrate its potential to avoid saddle points without sacrificing the convergence rate. Our analysis is based on the attraction region, formed by all starting points for which the considered numerical scheme converges to a saddle point. We investigate the attraction region’s dimension both analytically and numerically. For a canonical class of quadratic functions, we show that the dimension of the attraction region for mLSGD is $\lfloor (n-1)/2\rfloor$, and hence it is significantly smaller than that of GD whose dimension is $n-1$.

In this paper, we study the quadratic perturbations of a one-parameter family of reversible quadratic systems whose first integral contains the logarithmic function. By the criterion function for determining the lowest upper bound of the number of zeros of Abelian integrals, we obtain that the cyclicity of either period annulus is two. To the best of our knowledge, this is the first result for the cyclicity of period annulus of the one-parameter family of reversible quadratic systems whose first integral contains the logarithmic function. Moreover, the simultaneous bifurcation and distribution of limit cycles from two-period annuli are considered.

We classify all $\mathbb{Q}$-factorial Fano intrinsic quadrics of dimension three and Picard number one having at most canonical singularities.

We study approximations for the Lévy area of Brownian motion which are based on the Fourier series expansion and a polynomial expansion of the associated Brownian bridge. Comparing the asymptotic convergence rates of the Lévy area approximations, we see that the approximation resulting from the polynomial expansion of the Brownian bridge is more accurate than the Kloeden–Platen–Wright approximation, whilst still only using independent normal random vectors. We then link the asymptotic convergence rates of these approximations to the limiting fluctuations for the corresponding series expansions of the Brownian bridge. Moreover, and of interest in its own right, the analysis we use to identify the fluctuation processes for the Karhunen–Loève and Fourier series expansions of the Brownian bridge is extended to give a stand-alone derivation of the values of the Riemann zeta function at even positive integers.

In the present note, we establish a finiteness theorem for $L^p$ harmonic 1-forms on hypersurfaces with finite index, which is an extension of the result of Choi and Seo (J. Geom. Phys. 129 (2018), 125–132).

An associative ring R is called potent provided that for every $x\in R$, there is an integer $n(x)>1$ such that $x^{n(x)}=x$. A celebrated result of N. Jacobson is that every potent ring is commutative. In this note, we show that a ring R is potent if and only if every nonzero subring S of R contains a nonzero idempotent. We use this result to give a generalization of a recent result of Anderson and Danchev for reduced rings, which in turn generalizes Jacobson’s theorem.

In this paper, we investigate the $H^{p}(G) \rightarrow L^{p}(G)$, $0< p \leq 1$, boundedness of multiplier operators defined via group Fourier transform on a graded Lie group $G$, where $H^{p}(G)$ is the Hardy space on $G$. Our main result extends those obtained in [Colloq. Math. 165 (2021), 1–30], where the $L^{1}(G)\rightarrow L^{1,\infty }(G)$ and $L^{p}(G) \rightarrow L^{p}(G)$, $1< p <\infty$, boundedness of such Fourier multiplier operators were proved.

In this paper, we consider the following Hardy–Littlewood–Sobolev inequality with extended kernel(0.1)

\begin{equation} \int_{\mathbb{R}_+^{n}}\int_{\partial\mathbb{R}^{n}_+} \frac{x_n^{\beta}}{|x-y|^{n-\alpha}}f(y)g(x) {\rm d}y{\rm d}x\leq C_{n,\alpha,\beta,p}\|f\|_{L^{p}(\partial\mathbb{R}_+^{n})} \|g\|_{L^{q'}(\mathbb{R}_+^{n})}, \end{equation}

$f\in L^{p}(\partial \mathbb {R}_+^{n})$

$g\in L^{q'}(\mathbb {R}_+^{n})$

$p,\,\ q'\in (1,\,\infty )$

$\beta \geq 0$

$\alpha +\beta >1$

$\frac {n-1}{n}\frac {1}{p}+\frac {1}{q'}-\frac {\alpha +\beta -1}{n}=1$

We prove the existence of all extremal functions for (0.1). We show that if $f$ and $g$ are extremal functions for (0.1) then both of $f$ and $g$ are radially decreasing. Moreover, we apply the regularity lifting method to obtain the smoothness of extremal functions. Finally, we derive the sufficient and necessary condition of the existence of any nonnegative nontrivial solutions for the Euler–Lagrange equations by using Pohozaev identity.

A subset E of a metric space X is said to be starlike-equivalent if it has a neighbourhood which is mapped homeomorphically into $\mathbb{R}^n$ for some n, sending E to a starlike set. A subset $E\subset X$ is said to be recursively starlike-equivalent if it can be expressed as a finite nested union of closed subsets $\{E_i\}_{i=0}^{N+1}$ such that $E_{i}/E_{i+1}\subset X/E_{i+1}$ is starlike-equivalent for each i and $E_{N+1}$ is a point. A decomposition $\mathcal{D}$ of a metric space X is said to be recursively starlike-equivalent, if there exists $N\geq 0$ such that each element of $\mathcal{D}$ is recursively starlike-equivalent of filtration length N. We prove that any null, recursively starlike-equivalent decomposition $\mathcal{D}$ of a compact metric space X shrinks, that is, the quotient map $X\to X/\mathcal{D}$ is the limit of a sequence of homeomorphisms. This is a strong generalisation of results of Denman–Starbird and Freedman and is applicable to the proof of Freedman’s celebrated disc embedding theorem. The latter leads to a multitude of foundational results for topological 4-manifolds, including the four-dimensional Poincaré conjecture.