Monotonicity formulas play a central role in the study of free boundary problems. In this note, we develop a Weiss-type monotonicity formula for solutions to parabolic free boundary problems on metric measure cones.

The purpose of this paper is to represent the cohomology ring of a moment-angle manifold over an m-gon explicitly in terms of the quotient of an exterior algebra, and to count the Betti numbers of the cohomology groups of a special class of quotients of moment-angle manifolds.

The main purpose of this note is to construct almost complex or complex structures on certain isoparametric hypersurfaces in unit spheres. As a consequence, complex structures on S¹×S⁷×S⁶, and on S¹×S³×S² with vanishing first Chern class, are built.

In this paper, we study the Cauchy problem for the nonlinear Schrödinger equations with Coulomb potential i$?_tu+\Delta u+\frac{K}{|x|}u=\lambda|u|^{p-1}u$ with 1<p≤5 on $\mathbb{R}^3$. Our results reveal the influence of the long range potential $K|x|^{-1}$ on the existence and scattering theories for nonlinear Schrödinger equations. In particular, we prove the global existence when the Coulomb potential is attractive, i.e., when $K>0$, and the scattering theory when the Coulomb potential is repulsive, i.e., when $K\leq0$. The argument is based on the newly-established interaction Morawetz-type inequalities and the equivalence of Sobolev norms for the Laplacian operator with the Coulomb potential.

This paper gives a survey of the progress on the minimal genus problem since Lawson’s [26] survey.

For any scheme M with a perfect obstruction theory, Jiang and Thomas associated a scheme N with a symmetric perfect obstruction theory. The scheme N is a cone over M given by the dual of the obstruction sheaf of M, and contains M as its zero section. Locally, N is the critical locus of a regular function. In this note we prove that N is a d-critical scheme in the sense of Joyce. There exists a global motive for N locally given by the motive of the vanishing cycle of the local regular function. We prove a motivic localization formula under the good and circle compact C^*-action for N. When taking the Euler characteristic, the weighted Euler characteristic of N weighted by the Behrend function is the signed Euler characteristic of M by motivic method. As applications, using the main theorem we study the motivic generating series of the motivic Vafa-Witten invariants for K3 surfaces.

Spin group and screw algebra, as extensions of quaternions and vector algebra, respectively, have important applications in geometry, physics and engineering. In threedimensional projective geometry, when acting on lines, each projective transformation can be decomposed into at most three harmonic projective reflections with respect to projective lines, or equivalently, each projective spinor can be decomposed into at most three orthogonal Minkowski bispinors, each inducing a harmonic projective line reflection. In this paper, we establish the corresponding result for three-dimensional affine geometry: with each affine transformation is found a minimal decomposition into general affine reflections, where the number of general affine reflections is at most three; equivalently, each affine spinor can be decomposed into at most three affine Minkowski bispinors, each inducing a general affine line reflection.

In this paper, we will give a sufficient condition for the self-amalgamation of a handlebody to be strongly irreducible.

We consider a Prandtl model derived from MHD in the Prandtl-Hartmann regime that has a damping term due to the effect of the Hartmann boundary layer. A global-in-time well-posedness is obtained in the Gevrey function space with the optimal index 2. The proof is based on a cancellation mechanism through some auxiliary functions from the study of the Prandtl equation and an observation about the structure of the loss of one order tangential derivatives through twice operations of the Prandtl operator.

By studying the distribution of zeros of combinations of a Dirichlet L-function and its first-order derivative, we prove that every Dirichlet L-function has more than 66.7934% distinct zeros.

Using expectations regarding utilities to make decisions in a risk environment hides a paradox, which is called the expected utility enigma. Moreover, the mystery has not been solved yet; an imagined utility function on the risk-return plane has been applied to establish the mean-variance model, but this hypothetical utility function not only lacks foundation, it also holds an internal contradiction. This paper studies these basic problems. Through risk preference VNM condition is proposed to solve the expected utility enigma. How can a utility function satisfy the VNM condition? This is a basic problem that is hard to deal with. Fortunately, it is found in this paper that the VNM utility function can have some concrete forms when individuals have constant relative risk aversion. Furthermore, in order to explore the basis of mean-variance utility, an MV function is founded that is based on the VNM utility function and rooted in underlying investment activities. It is shown that the MV function is just the investor’s utility function on the risk-return plane and that it has normal properties. Finally, the MV function is used to analyze the laws of investment activities in a systematic risk environment. In doing so, a tool, TRR, is used to measure risk aversion tendencies and to weigh risk and return.

The purpose of adversarial deep learning is to train robust DNNs against adversarial attacks, and this is one of the major research focuses of deep learning. Game theory has been used to answer some of the basic questions about adversarial deep learning, such as those regarding the existence of a classifier with optimal robustness and the existence of optimal adversarial samples for a given class of classifiers. In most previous works, adversarial deep learning was formulated as a simultaneous game and the strategy spaces were assumed to be certain probability distributions in order for the Nash equilibrium to exist. However, this assumption is not applicable to practical situations. In this paper, we give answers to these basic questions for the practical case where the classifiers are DNNs with a given structure; we do that by formulating adversarial deep learning in the form of Stackelberg games. The existence of Stackelberg equilibria for these games is proven. Furthermore, it is shown that the equilibrium DNN has the largest adversarial accuracy among all DNNs with the same structure, when Carlini-Wagner’s margin loss is used. The trade-off between robustness and accuracy in adversarial deep learning is also studied from a game theoretical perspective.

The Hodge bound for the Newton polygon of L-functions of T-adic exponential sums associated to a Laurent polynomial is established. We improve the lower bound and study the properties of this new bound. We also study when this new bound is reached with large p arbitrarily, and hence the generic Newton polygon is determined.

A Heegaard splitting is a type of combinatorial structure on an orientable compact 3-manifold. We will give a survey on Heegaard spliting and its applications, including those pertaining to the classification and stabilization problem, reducibilities, minimal Heegaard splitting and Heegaard distance.

The two-center problem, also known as Euler’s three-body problem, is a classic example of integrable systems. Among its periodic solutions, planetary type solutions are periodic solutions which enclose both centers. Inspired by advances on n-body and n-center problems via variational techniques developed during the past two decades, a recent paper (Arch. Rat. Mech. Ana. 2022) shows the minimizing property of planetary type solutions for any given masses of centers at fixed positions, as long as the period is above a mass-dependent threshold value. In this paper, we provide further discussions regarding this minimizing approach. In particular, we improve the above-mentioned mass-dependent threshold value by refining estimates for action values.

Opinion dynamics has recently attracted much attention, and there have been a lot of achievements in this area. This paper first gives an overview of the development of opinion dynamics on social networks. We introduce some classical models of opinion dynamics in detail, including the DeGroot model, the Krause model, 0-1 models, sign networks and models related to Gossip algorithms. Inspired by some real life cases, we choose the unit circle as the range of the individuals’ opinion values. We prove that the individuals’ opinions of the randomized gossip algorithm in which the individuals’ opinion values are on the unit circle reaches consensus almost surely.

This paper provides a survey on symbolic computational approaches for the analysis of qualitative behaviors of systems of ordinary differential equations, focusing on symbolic and algebraic analysis for the local stability and bifurcation of limit cycles in the neighborhoods of equilibria and periodic orbits of the systems, with a highlight on applications to computational biology.

In this paper, we are concerned with the asymptotic behavior of $L^{\infty}$ weak-entropy solutions to the compressible Euler equations with a vacuum and time-dependent damping $-\frac{m}{(1+t)^{\lambda}}$. As $\lambda \in (0,\frac17]$, we prove that the $L^{\infty}$ weak-entropy solution converges to the nonlinear diffusion wave of the generalized porous media equation (GPME) in $L^2(\mathbb R)$. As $\lambda \in (\frac17,1)$, we prove that the $L^{\infty}$ weak-entropy solution converges to an expansion around the nonlinear diffusion wave in $L^2(\mathbb R)$, which is the best asymptotic profile. The proof is based on intensive entropy analysis and an energy method.