The quantum measurement is one of the key elements of quantum theory. In his book the <italic>Mathematical Foundations of Quantum Mechanics</italic>, von Neumann delved into the 问题 of measurements. He concluded that the entire physical world could be governed by a universal wave function. The collapse of the wave function is caused by the experimenter's consciousness due to the need for “something other than computation". He argued that the mathematics of quantum mechanics allows the collapse of the wave function to be located anywhere in the causal chain from the measuring device to the human observer's “subjective consciousness". Measurements lead to the inevitable collapse of the system under tests. For this reason, Aharonov et al. (1987) put forward the theory of weak measurements and the concept of weak values. In this paper, the causes of the weak value anomalies, and the uncertainty and complementarity caused by the weak measurements are analyzed from the basic level.

In this paper, we consider the commutators $[b,\mathcal{I}_\alpha]_i$ $(i=1,2)$, where $\mathcal{I}_\alpha$ is a bilinear fractional integral operator, and $b$ is a function in generalized Campanato spaces with the variable growth condition. We give necessary and sufficient 条件 for the boundedness of $~[b,\mathcal{I}_\alpha]_i$ $(i=1,2)$ on generalized Morrey spaces with the variable growth condition.

In this paper, we first introduce the history, developing process, recent progress, andworks contributed by Chinese researchers in integral and convex geometry.Then we show our recent works in invariant kinematicformulas and chord power integrals. The affine isoperimetric inequalities on the set of convex bodies, closely related to integral geometry, are also addressed.

A famous Camina-Gagen 定理 is that if $G$ acts as a block-transitive group of automorphisms of a $2$-$(v,k,1)$ design with $k\mid~v$, then $G$ is flag-transitive.In this paper, we generalize this 定理 to $5$-$(v,k,1)$ designs.We prove that if $G$ acts block-transitively on a non-trivial $5$-$(v,k,1)$ design ${D}$ with $k\mid~v$, then ${D}$ is flag-transitive.

Breast cancer is one of the most common cancers for females. In order to propose efficient screening strategies and evaluate the effect, one fundamental and important step is to choose proper parameters, i.e., transition rates, in the natural history model of breast cancer in China. There are two challenges to choosing reasonable transition rates. Firstly, the transition rates used in other countries may be not applicable to China, due to the cancer epidemiology. Secondly, little screening sample data are available, making the traditional statistical based approaches such as maximum likelihood methods fail. In this paper, we propose an optimization approach for parameter selection in the natural history of breast cancer in China. A mathematical optimization model is established based on the published statistical data. A coordinate descent algorithm is proposed to solve the resulting box-constrained and highly nonlinear problem, whose objective function does not have an explicit form. Finally, the best matching method is proposed to estimate the distributions of each transition rate. Our approach provides solid foundations to further propose reasonable screening strategies for breast cancer in China and analyze the economic burden of breast cancer.

Let $({\mathcal~X},~d,~\mu)$ be a metric measurespace satisfying the geometrically doubling condition. We establish some weak type weighted norm inequalitiesfor the multilinear fractional integral operatorand the multilinear fractional maximal function.As an application, we also establish a weak type weighted norm inequalityon Morrey spaces for the multilinear fractional integral.

We study the size of the largest component of the randomintersection graph $G(n,m,p)$ with $m=[n^r]$ in the subcritical regime, i.e., the case where the expected vertex degree is smaller than one. If $r&gt;1$, then the largest component and the largest tree component have the same type of weak law of large numbers, and both of them are of order $\Theta(\log~n)$; if $r=1$, then the largest component is not a tree, but the largest component and the largest tree component are also of order $\Theta(\log~n)$; if $0&lt;r&lt;1$, then the largest tree component is of order $o(\log~n)$while the largest component is of order $\Theta(np\log~n)$.

Distributed statistical learning for massive data has attracted enormous attention recently.There are two noteworthy issues in the existing methods. First, they all require that the massive data are randomly distributed on different machines, which is seldom the case in practice. Second, they are usually built upon the least squares, which are sensitive to the heavy-tailed noise and outliers.To fix these problems, we propose a new robust distributed modal regression and also apply it to nonconvex penalized learning problems.The new method can overcome the non-randomly distributed nature of the big data, and the theoretical results also guarantee this statement. Simulation studies and the real world data evaluation are also used to illustrate the proposed methods.

Suppose that $q$ is a prime power such that $q\equiv~2\pmod{3}$. By using Kantor's model of $Q^+(7,q)$, we study the orbit structure of the action of $\textup{PSU}_3(q)$ on the polar space $Q^+(7,q)$ and obtain new $(q^2+q)$- and $q^3$-ovoids with an automorphism group $\textup{PSU}_3(q)$. In this way, we determine all $\textup{PSU}_3(q)$-invariant intriguing sets of $Q^+(7,q)$. It turns out that such an intriguing set is either the unitary ovoid, one of the two new $m$-ovoids, or their complements.

Let $p(\cdot)$ be a measurable function on $\mathbb{R}^n$ satisfyingthe globally log-Hölder continuous condition.Its essential supremum $p_+$ and infimum $p_-$ satisfy$0&lt;p_-\le~p_+\le1$.Let $H_A^{p(\cdot),q}(\mathbb{R}^n)$ be the variable anisotropicHardy-Lorentz spaces defined via the radial grand maximal function,where $q\in(0,1]$ and $A$ is an expansive matrix.In this article, byusing the atomic decomposition of $H_A^{p(\cdot),q}(\mathbb{R}^n)$,we prove that the Fourier transform of $f\in~H_A^{p(\cdot),q}(\mathbb{R}^n)$equals a continuous function $F$ on $\mathbb{R}^n$in the sense of tempered distributions.Moreover, the function $F$ can be pointwisely controlled bythe product of the $H_A^{p(\cdot),q}(\mathbb{R}^n)$ norm of $f$and the homogeneous quasi-norm associated with the transpose matrix of $A$.As applications, we obtain a higher order of convergence for the function$F$ at the origin, and an analogue of Hardy-Littlewood inequalities in the present setting of $H_A^{p(\cdot),q}(\mathbb{R}^n)$. All theseresults are new even for the isotropic Hardy-Lorentz spaces$H^{p,q}(\mathbb{R}^n)$ and they generalize the corresponding conclusionsof Taibleson and Weiss on classical Hardy spaces $H^{p}(\mathbb{R}^n)$.