Let $\Omega _1, \ldots , \Omega _m$ be probability spaces, let ${\mathbf \Omega }=\Omega _1 \times \cdots \times \Omega _m$ be their product and let $A_1, \ldots , A_n \subset {\mathbf \Omega }$ be events. Suppose that each event $A_i$ depends on $r_i$ coordinates of a point $x \in {\mathbf \Omega }$, $x=\left (\xi _1, \ldots , \xi _m\right )$, and that for each event $A_i$ there are $\Delta _i$ other events $A_j$ that depend on some of the coordinates that $A_i$ depends on. Let $\Delta =\max \{5,\ \Delta _i\,:\, i=1, \ldots , n\}$ and let $\mu _i=\min \{r_i,\ \Delta _i+1\}$ for $i=1, \ldots , n$. We prove that if ${\mathbb P}(A_i) \lt (3\Delta )^{-3\mu _i}$ for all $i$, then for any $0 \lt \epsilon \lt 1$, the probability ${\mathbb P}\left ( \bigcap _{i=1}^n \overline {A}_i\right )$ of the intersection of the complements of all $A_i$ can be computed within relative error $\epsilon$ in polynomial time from the probabilities ${\mathbb P}\left (A_{i_1} \cap \ldots \cap A_{i_k}\right )$ of $k$-wise intersections of the events $A_i$ for $k = e^{O(\Delta )} \ln (n/\epsilon )$.

This paper studies a class of degenerate parabolic partial differential equation models that describe the dynamics of single-species populations with cognitive functions in toxic environments. The core innovation lies in introducing cognitive functions to simulate the ability of species to perceive toxins and adaptively adjust their behaviours, which in turn affects population density. This mechanism is characterized by a nonlinear degenerate Pratial Differential Equation. The degradation of the model is mainly manifested in the diffusion coefficient and response term approaching zero when the population density is zero or the cognitive state reaches the boundary. For this model, we have achieved the following theoretical breakthroughs: First, by using regularization techniques, we constructed a non-degenerate approximate system. Combining prior energy estimates with the fixed point theorem, we proved the existence of local classical solutions for this regularized system; subsequently, we constructed an appropriate Lyapunov function. Based on strict energy dissipation analysis and uniform prior estimates, we proved that under conditions such as bounded toxins and regular initial values, the classical solutions of the regularized system exist globally and remain bounded on the interval $[0, \infty )$ ultimately, based on this, through compactness theory and limit processes, the existence of the global weak solution for the original degenerate system is further derived. Numerical simulations verify the rationality of the theoretical results and visually demonstrate the dynamic regulatory effect of cognitive function on population tolerance.

We consider a smooth fibration equipped with a flat complex vector bundle and a hypersurface cutting the fibration into two pieces. Our main result is a gluing formula relating the Bismut-Lott analytic torsion form of the whole fibration to that of each piece. This result solves a conjecture proposed at a conference in Göttingen in 2003. This result also leads to a higher Cheeger-Müller/Bismut-Zhang theorem. Our approach combines an adiabatic limit along the normal direction of the hypersurface and a Witten-type deformation on the flat vector bundle.

Let $\Omega$ be a lattice in $\mathbb C$ with invariants $g_2,g_3$ and let $\wp(z), \zeta(z)$ be the associated Weierstrass elliptic and zeta functions, respectively. In this paper, we prove that if $\omega$ is any non-zero period of $\wp(z)$ and $u_1,u_2$ complex numbers such that $u_1,u_2, \omega$ are $\mathbb{Q}$-linearly independent with $({\mathbb{Z}} u_1+{\mathbb{Z}} u_2)\cap\Omega=(0),$ then at least two of the numbers

\begin{equation*}g_2,g_3,\omega, \eta,u_1,u_2,\wp(u_i),\zeta(u_i)~~(1\leq i\leq 2)\end{equation*}

$\mathbb{Q},$

$\eta$

$\zeta(z)$

$\omega.$

In this paper, we show that for each sufficiently large integer $H$, there is a real cubic number $\alpha$ and a rational number $r$, both of height smaller than $H$, such that $0\lt \alpha -r\lt 272/H^4$. The exponent $4$ of $H$ in this inequality is best possible. The numbers $\alpha$ and $r$ are both constructed explicitly. We also show that a necessary and sufficient condition on $(u,v)$ under which there is a positive constant $c_1=c_1(d,u,v)$ such that the inequality $|\alpha -\beta | \geq c_1 H(\alpha )^{-u} H(\beta )^{-v}$ holds for all real algebraic numbers $\alpha$ of degree $d \geq 2$ and all real algebraic numbers $\beta$ of degree $d-1$ is $u \geq d-1$ and $v \geq d$.

The rational homology of the IA-automorphism group $\operatorname {IA}_n$ of the free group $F_n$ is still mysterious. We study the quotient of the rational homology of $\operatorname {IA}_n$ that is obtained as the image of the map induced by the abelianization map, which we call the Albanese homology of $\operatorname {IA}_n$. We obtain a representation-stable $\operatorname {GL}(n,\mathbb {Q})$-subquotient of the Albanese homology of $\operatorname {IA}_n$, which conjecturally coincides with the entire Albanese homology of $\operatorname {IA}_n$. In particular, we obtain a lower bound of the dimension of the Albanese homology of $\operatorname {IA}_n$ for each homological degree in a stable range. Moreover, we determine the entire third Albanese homology of $\operatorname {IA}_n$ for $n\ge 9$. We also study the Albanese homology of an analogue of $\operatorname {IA}_n$ for the outer automorphism group of $F_n$ and the Albanese homology of the Torelli groups of surfaces. Moreover, we study the relation between the Albanese homology of $\operatorname {IA}_n$ and the cohomology of $\operatorname {Aut}(F_n)$ with twisted coefficients.

In this paper, we study a nonlinear free boundary problem modelling the growth of radially symmetric tumours. The tumour consists of a central necrotic core, an intermediate quiescent layer and an outer proliferating shell. The evolution of tumour layers and the movement of the tumour boundary are totally governed by external nutrient supply and conservation of mass. The three-layer structure generates three free boundaries with discontinuous nutrient consumption rates and cell growth rates. We develop a nonlinear analysis method to clarify the interactive relationships among free boundaries. By carefully studying the dependence of the critical-state tumour growth rate on the external nutrient concentration, we reveal the evolutionary mechanism in tumour growth and the mutual transformation of its internal structures. The existence and uniqueness of the radial stationary solution is proved, and its globally asymptotic stability towards different dormant tumour states is established.

The goal of this article is to find some geometric characterizations of Kenmotsu and almost Kenmotsu manifolds that admit a non-gradient $m$-quasi-Einstein structure. First, we prove that a Kenmotsu manifold admits a closed $m$-quasi-Einstein structure is an Einstein manifold. Next, we prove that if a three-dimensional Kenmotsu manifold admits a non-trivial $m$-quasi-Einstein structure $(g, V,m,\lambda )$ with $V$ as a conformal vector field, then it is of constant sectional curvature $-1$. Finally, we prove that if a non-Kenmotsu almost Kenmotsu $(k,\mu )^{\prime}$-manifold admits a closed $m$-quasi-Einstein structure $(g, V,m,\lambda )$ with $m\ne 1$, then it is locally isometric to the product space $\mathbb{H}^{n+1}(-4)\times \mathbb{R}^{n}$.

We show that the existence of non-zero tropical forms of degree at least two implies that the tropical Chow group of points of an integral affine manifold is infinite-dimensional. This can be seen as a tropical analogue of classical results of Mumford and Roitman for Chow groups of smooth (complex) projective algebraic varieties. We also show that the existence of tropical 1-forms on integral affine surfaces does not imply infinite dimensionality by considering the case of a tropical Klein bottle.

In the Morel-Voevodsky motivic stable homotopy category of a quasi-compact quasi-separated scheme S, several candidates exist for a motivic spectrum representing hermitian K-theory. This note shows that the cellular absolute motivic spectrum constructed in [9] via the geometry of orthogonal and hyperbolic Grassmannians over S coincides with the motivic ring spectrum constructed in [4].