On Yau's problem of evolving one curve to another: Convex case

A revised Yau's Curvature Difference Flow is considered to deform one convex curve $X_0$ to another one $\stackrel{?}{X}$. It is proved that this flow exists globally on time interval $[0,+\infty )$ and the evolving curve, preserving its convexity and bounded area A, converges to a fixed limiting curve $X_{\infty }$ (congruent to $\sqrt{A/\stackrel{?}{A}}\stackrel{?}{X}$) as time tends to infinity, where $\stackrel{?}{A}$ is the area bounded by the target curve $\stackrel{?}{X}$.     

Singular p-Laplacian equations with superlinear perturbation

We consider a nonlinear Dirichlet problem driven by the p-Laplace operator and with a right-hand side which has a singular term and a parametric superlinear perturbation. We are interested in positive solutions and prove a bifurcation-type theorem describing the changes in the set of positive solutions as the parameter $\lambda >0$ varies. In addition, we show that for every admissible parameter $\lambda >0$ the problem has a smallest positive solution ${\stackrel{￣}{u}}_{\lambda }$ and we establish the monotonicity and continuity properties of the map $\lambda \to {\stackrel{￣}{u}}_{\lambda }$.     

A time-periodic reaction–diffusion–advection equation with a free boundary and sign-changing coefficients

We consider a reaction–diffusion–advection equation of the form: $u_t=u_{xx}-\beta \left(t\right)u_x+f(t,u)$ for $x\in [0,h\left(t\right))$, where $\beta \left(t\right)$ is a $T$-periodic function, $f(t,u)$ is a $T$-periodic Fisher–KPP type of nonlinearity with $a\left(t\right)≔f_u(t,0)$ changing sign, $h\left(t\right)$ is a free boundary satisfying the Stefan condition. We study the long time behavior of solutions and find that there are two critical numbers $\bar{c}$ and $B\left(\tilde{\beta }\right)$ with $B\left(\tilde{\beta }\right)>\bar{c}>0$, $\bar{\beta }≔\frac{1}{T}{\int }_0^T\beta \left(t\right)dt$ and $\tilde{\beta }\left(t\right)≔\beta \left(t\right)-\bar{\beta }$, such that a vanishing–spreading dichotomy result holds when $\left|\bar{\beta }\right|<\bar{c}$; a vanishing–transition–virtual spreading trichotomy result holds when $\bar{\beta }\in [\bar{c},B\left(\tilde{\beta }\right))$; all solutions vanish when $\bar{\beta }⩾B\left(\tilde{\beta }\right)$ or $\bar{\beta }⩽-\bar{c}$.     

Positive solutions to delayed differential equations of the second-order

A new criterion for the existence of positive solutions of the second-order delayed differential equation $\ddot{y}\left(t\right)=f(t,y_t,{\dot{y}}_t)$, $t\in [t_0,\infty )$ is given with applications to linear equations. Open problems for future research are formulated.     

Non-uniqueness for a critical heat equation in two dimensions with singular data

Nonlinear heat equations in two dimensions with singular initial data are studied. In recent works nonlinearities with exponential growth of Trudinger-Moser type have been shown to manifest critical behavior: well-posedness in the subcritical case and non-existence for certain supercritical data. In this article we propose a specific model nonlinearity with Trudinger-Moser growth for which we obtain surprisingly complete results: a) for initial data strictly below a certain singular threshold function $\stackrel{?}{u}$ the problem is well-posed, b) for initial data above this threshold function $\stackrel{?}{u}$, there exists no solution, c) for the singular initial datum $\stackrel{?}{u}$ there is non-uniqueness. The function $\stackrel{?}{u}$ is a weak stationary singular solution of the problem, and we show that there exists also a regularizing classical solution with the same initial datum $\stackrel{?}{u}$.     

Relation between the H-rank of a mixed graph and the rank of its underlying graph

Given a simple graph $G=(V_G,E_G)$ with vertex set $V_G$ and edge set $E_G$, the mixed graph $\stackrel{?}{G}$ is obtained from $G$ by orienting some of its edges. Let $H\left(\stackrel{?}{G}\right)$ denote the Hermitian adjacency matrix of $\stackrel{?}{G}$ and $A\left(G\right)$ be the adjacency matrix of $G$. The $H$-rank (resp. rank) of $\stackrel{?}{G}$ (resp. $G$), written as $rk\left(\stackrel{?}{G}\right)$ (resp. $r\left(G\right)$), is the rank of $H\left(\stackrel{?}{G}\right)$ (resp. $A\left(G\right)$). Denote by $d\left(G\right)$ the dimension of cycle space of $G$, that is $d\left(G\right)=\left|E_G\right|?\left|V_G\right|+\omega \left(G\right)$, where $\omega \left(G\right)$ denotes the number of connected components of $G$. In this paper, we concentrate on the relation between the $H$-rank of $\stackrel{?}{G}$ and the rank of $G$. We first show that $?2d\left(G\right)?rk\left(\stackrel{?}{G}\right)?r\left(G\right)?2d\left(G\right)$ for every mixed graph $\stackrel{?}{G}$. Then we characterize all the mixed graphs that attain the above lower (resp. upper) bound. By these obtained results in the current paper, all the main results obtained in Luo et al. (2018); Wong et al. (2016) may be deduced consequently.     

Even (s̄,t̄)-core partitions and self-associate characters of S̃n

We extend a method of Olsson and Bessenrodt to determine the number of even partitions that are simultaneously $\stackrel{?}{s}$-core and $\stackrel{?}{t}$-core. When $p$ and $q$ are distinct primes, this also determines the number of self-associate characters of ${\stackrel{?}{S}}_n$ that are simultaneously defect 0 for $p$ and $q$.     

Concentrated steady vorticities of the Euler equation on 2-d domains and their linear stability

We consider concentrated vorticities for the Euler equation on a smooth domain $\mathrm{\Omega }?{\mathbf{R}}^2$ in the form of

$\omega =\sum _{j=1}^N{\omega }_j{\chi }_{{\mathrm{\Omega }}_j},|{\mathrm{\Omega }}_j|=\pi r_j^2,\underset{{\mathrm{\Omega }}_j}{\int }{\omega }_{j}d\mu ={\mu }_j\ne 0,$

supported on well-separated vortical domains ${\mathrm{\Omega }}_j$, $j=1,\dots ,N$, of small diameters $O(r_j)$. A conformal mapping framework is set up to study this free boundary problem with ${\mathrm{\Omega }}_j$ being part of unknowns. For any given vorticities ${\mu }_1,\dots ,{\mu }_N$ and small $r_1,\dots ,r_N\in {\mathbf{R}}^{+}$, through a perturbation approach, we obtain such piecewise constant steady vortex patches as well as piecewise smooth Lipschitz steady vorticities, both concentrated near non-degenerate critical configurations of the Kirchhoff–Routh Hamiltonian function. When vortex patch evolution is considered as the boundary dynamics of $?{\mathrm{\Omega }}_j$, through an invariant subspace decomposition, it is also proved that the spectral/linear stability of such steady vortex patches is largely determined by that of the 2N-dimensional linearized point vortex dynamics, while the motion is highly oscillatory in the 2N-codim directions corresponding to the vortical domain shapes.     

A generalization of lifting non-proper tropical intersections

Let X and $X^{\prime }$ be closed subschemes of an algebraic torus T over a non-archimedean field. We prove the rational equivalence as tropical cycles in the sense of [11, §2] between the tropicalization of the intersection product $X?X^{\prime }$ and the stable intersection $\mathrm{trop}(X)?\mathrm{trop}(X^{\prime })$, when restricted to (the inverse image under the tropicalization map of) a connected component C of $\mathrm{trop}(X)\cap \mathrm{trop}(X^{\prime })$. This requires possibly passing to a (partial) compactification of T with respect to a suitable fan. We define the compactified stable intersection in a toric tropical variety, and check that this definition is compatible with the intersection product in [11, §2]. As a result we get a numerical equivalence between $\stackrel{￣}{X}?{\stackrel{￣}{X}}^{\prime }{|}_{\stackrel{￣}{C}}$ and $\stackrel{￣}{\mathrm{trop}(X)?\mathrm{trop}(X^{\prime }){|}_C}$ via the compactified stable intersection, where the closures are taken inside the compactifications of T and ${\mathbb{R}}^n$. In particular, when X and $X^{\prime }$ have complementary codimensions, this equivalence generalizes [15, Theorem 6.4], in the sense that $X\cap X^{\prime }$ is allowed to be of positive dimension. Moreover, if $\stackrel{￣}{X}\cap {\stackrel{￣}{X}}^{\prime }$ has finitely many points which tropicalize to $\stackrel{￣}{C}$, we prove a similar equation as in [15, Theorem 6.4] when the ambient space is a reduced subscheme of T (instead of T itself).     

On Nordhaus–Gaddum type inequalities for the game chromatic and game coloring numbers

A seminal result by Nordhaus and Gaddum states that $2\sqrt{n}\le \chi \left(G\right)+\chi \left(\overline{G}\right)\le n+1$ for every graph $G$ of order $n$, where $\overline{G}$ is the complement of $G$ and $\chi$ is the chromatic number. We study similar inequalities for ${\chi }_g\left(G\right)$ and $co{l}_g\left(G\right)$, which denote, respectively, the game chromatic number and the game coloring number of $G$. Those graph invariants give the score for, respectively, the coloring and marking games on $G$ when both players use their best strategies.