I examine some analytical properties of a nonlinear reaction-diffusion system that has been used to model the propagation of a wildfire. I establish global-in-time existence and uniqueness of bounded mild solutions to the Cauchy problem for this system given bounded initial data. In particular, this shows that the model does not allow for thermal blow-up. If the initial temperature and fuel density also satisfy certain integrability conditions, the spatial $L^2$-norms of these global solutions are uniformly bounded in time. Additionally, I use a bootstrap argument to show that small initial temperatures give rise to solutions that decay to zero as time goes to infinity, proving the existence of initial states that do not develop into traveling combustion waves.

In this article, we investigate a free boundary problem for the Lotka–Volterra model consisting of an invasive species with density u and a native species with density v in one dimension. We assume that v undergoes diffusion and growth in $[0,+\infty )$, and u invades into the environment with spreading front $x=h(t)$ satisfying free boundary condition $h'(t)=-u_x(t,h(t))-\alpha $ for some decay rate $\alpha>0$, this is caused by the bad environment at the boundary. When u is an inferior competitor, $u(t,x)$ and $h(t)$ tend to 0 within a finite time, while another specie $v(t,x)$ tends to a stationary $\Lambda (x)$ defined on the half-line. When u is a superior competitor, we have a trichotomy result: spreading of u and vanishing of v (i.e., as $t \to +\infty $, $h(t)$ goes to $+\infty $ and $(u,v)\to (\Lambda ,0)$); the transition case (i.e., as $t \to +\infty $, $(u,v)\to (w_\alpha , \eta _\alpha )$, $h(t)$ tends to a finite point); vanishing of u and spreading of v (i.e., $u(t,x)$ and $h(t)$ tends to 0 within a finite time, $v(t,x)$ converges to $\Lambda (x)$). Additionally, we show that this trichotomy result depends on the initial data $u(0,x)$.

In this article, we address the following question: Which hyperbolic or elliptic PDEs admit functional separable solutions. We shall focus on the study of a sinh-Gordon type equation. We construct solutions to this equation via the method of functional separation. We prove that these are the only families that have the property of functional separation and so we obtain a classification. To this end, we construct new families of solutions for the hyperbolic and elliptic versions of both sine and sinh-Gordon equations in a unified way.

A coset partition of a group G is a set partition of G into finitely many left cosets of one or more subgroups. A driving force in this research area is the Herzog–Schönheim Conjecture [15], which states that any nontrivial coset partition of a group contains at least two cosets with the same index. Although many families of groups have been shown to satisfy the conjecture, it remains open.

A Steiner coset partition of G, with respect to distinct subgroups $H_1,\dots ,H_r$, is a coset partition of G that contains exactly one coset of each $H_i$. In the quest of a more structural version of the Herzog–Schönheim Conjecture, it was shown that there is no Steiner coset partition of G with respect to any $r\geq 2$ subgroups $H_i$ that mutually commute [1]. In this article, we show that this result holds for $r=4$ mutually commuting subgroups provided that G does not have $C_2\times C_2\times C_2$ as a quotient, where $C_2$ is the cyclic group of order $2$. We further give an explicit construction of Steiner coset partitions of the n-fold direct product $G^*=C_p\times \ldots \times C_p$ for p prime and $n\geq 3$. This construction lifts to every group extension of $G^*$.

We give necessary and sufficient conditions for certain pushouts of topological spaces in the category of Čech’s closure spaces to agree with their pushout in the category of topological spaces. We prove that in these two categories, the constructions of cell complexes by a finite sequence of closed cell attachments, which attach arbitrarily many cells at a time, agree. Likewise, the constructions of CW complexes relative to a compactly generated weak Hausdorff space that attach only finitely many cells, also agree. On the other hand, we give examples showing that the constructions of finite-dimensional CW complexes, CW complexes of finite type, and relative CW complexes that attach only finitely many cells, need not agree.

We study the class of composition operators acting on the Fréchet space $\mathrm {Hol}(\mathbb {B}_N)$ of all holomorphic maps on the unit ball of $\mathbb {C}^N$. We describe the conditions to make these operators continuous, invertible and compact. We also do the spectral study of these operators, depending on the nature of its symbol.

We analyze generating functions for trees and for connected subgraphs on the complete graph, and identify a single scaling profile which applies for both generating functions in a critical window. Our motivation comes from the analysis of the finite-size scaling of lattice trees and lattice animals on a high-dimensional discrete torus, for which we conjecture that the identical profile applies in dimensions $d \ge 8$.

We explore the relationship between (3-isogeny induced) Selmer group of an elliptic curve and the (3 part of) the ideal class group, over certain non-abelian number fields.

We explore the regularity theory of optimal transport maps for costs satisfying a Ma–Trudinger–Wang condition, by viewing the graphs of the transport maps as maximal Lagrangian surfaces with respect to an appropriate pseudo-Riemannian metric on the product space. We recover the local regularity theory in two-dimensional manifolds.

For a wide class of integer linear recurrence sequences $(u(n))_{n=1}^\infty $, we give an upper bound on the number of s-tuples $\left (n_1, \ldots , n_s\right ) \in \left ({\mathbb Z}\cap [M+1,M+ N]\right )^s$ such that the corresponding elements $u(n_1), \ldots , u(n_s)$ in the sequence are multiplicatively dependent.

In this note, we prove that quadratic algebraic integers, except for trivial cases, are not Mahler measures of algebraic integers and we also answer in negative the question of A. Schinzel [9] whether $1+\sqrt {17}$ is a Mahler measure of an algebraic number.

We present a counterexample to [Kan08, Conjecture 14.1.6], regarding Borel equivalence relations on product spaces.

Let C be a convex body (i.e., a proper closed convex subset with nonempty interior) in a normed space X. We consider four moduli of local uniform rotundity for C at a given point $x\in \partial C$, that extend in a natural way the corresponding notions for the unit ball $B_X$, and we prove that they all coincide. This extends a known result of J. Daneš from 1976 concerning the particular case when $C=B_X$.

Let $\mu $ be a finite positive Borel measure on $[0,1)$ and $f(z)=\sum _{n=0}^{\infty }a_{n}z^{n} \in H(\mathbb {D})$. For $0<\alpha <\infty $, the generalized Cesàro-like operator $\mathcal {C}_{\mu ,\alpha }$ is defined by

$$ \begin{align*}\mathcal {C}_{\mu,\alpha}(f)(z)=\sum^\infty_{n=0}\left(\mu_n\sum^n_{k=0}\frac{\Gamma(n-k+\alpha)}{\Gamma(\alpha)(n-k)!}a_k\right)z^n, \ z\in \mathbb{D}, \end{align*} $$

where, for $n\geq 0$, $\mu _n$ denotes the nth moment of the measure $\mu $, that is, $\mu _n=\int _{0}^{1} t^{n}d\mu (t)$.

For $s>1$, let X be a Banach subspace of $H(\mathbb {D})$ with $\Lambda ^{s}_{\frac {1}{s}}\subset X\subset \mathcal {B}$. In this article, for $1\leq p <\infty $, we characterize the measure $\mu $ for which $\mathcal {C}_{\mu ,\alpha }$ is bounded (resp. compact) from X into the analytic Besov space $B_{p}$.

Let $H_n$ be the minimal number such that any n-dimensional convex body can be covered by $H_n$ translates of the interior of that body. Similarly $H_n^s$ is the corresponding quantity for symmetric bodies. It is possible to define $H_n$ and $H_n^s$ in terms of illumination of the boundary of the body using external light sources, and the famous Hadwiger’s covering conjecture (illumination conjecture) states that $H_n=H_{n}^s=2^n$. In this note, we obtain new upper bounds on $H_n$ and $H_{n}^s$ for small dimensions n. Our main idea is to cover the body by translates of John’s ellipsoid (the inscribed ellipsoid of the largest volume). Using specific lattice coverings, estimates of quermassintegrals for convex bodies in John’s position, and calculations of mean widths of regular simplexes, we prove the following new upper bounds on $H_n$ and $H_n^s$: $H_5\le 933$, $H_6\le 6137$, $H_7\le 41377$, $H_8\le 284096$, $H_4^s\le 72$, $H_5^s\le 305$, and $H_6^s\le 1292$. For larger n, we describe how the general asymptotic bounds $H_n\le \binom {2n}{n}n(\ln n+\ln \ln n+5)$ and $H_n^s\le 2^n n(\ln n+\ln \ln n+5)$ due to Rogers, Shephard and Roger, Zong, respectively, can be improved for specific values of n.

We study $\ell $-isogeny graphs of ordinary elliptic curves defined over $\mathbb {F}_q$ with an added level structure. Given an integer N coprime to p and $\ell ,$ we look at the graphs obtained by adding $\Gamma _0(N)$, $\Gamma _1(N),$ and $\Gamma (N)$-level structures to volcanoes. Given an order $\mathcal {O}$ in an imaginary quadratic field $K,$ we look at the action of generalized ideal class groups of $\mathcal {O}$ on the set of elliptic curves whose endomorphism rings are $\mathcal {O}$ along with a given level structure. We show how the structure of the craters of these graphs is determined by the choice of parameters.

We establish Trudinger-type inequalities for variable Riesz potentials $J_{\alpha (\cdot ), \tau }f$ of functions in Musielak–Orlicz spaces $L^{\Phi }(X)$ over bounded metric measure spaces X equipped with lower Ahlfors $Q(x)$-regular measures under conditions on $\Phi $ which are weaker than conditions in the previous paper (Houston J. Math. 48 (2022), no. 3, 479–497). We also deal with the case $\Phi $ is the double phase functional with variable exponents. As an application, Trudinger-type inequalities are discussed for Sobolev functions.

In this note, we give a new necessary condition for the boundedness of the composition operator on the Dirichlet-type space on the disc, via a two dimensional change of variables formula. With the same formula, we characterize the bounded composition operators on the anisotropic Dirichlet-type spaces $\mathfrak {D}_{\vec {a}}(\mathbb {D}^2)$ induced by holomorphic self maps of the bidisc $\mathbb {D}^2$ of the form $\Phi (z_1,z_2)=(\phi _1(z_1),\phi _2(z_2))$. We also consider the problem of boundedness of composition operators $C_{\Phi }:\mathfrak {D}(\mathbb {D}^2)\to A^2(\mathbb {D}^2)$ for general self maps of the bidisc, applying some recent results about Carleson measures on the Dirichlet space of the bidisc.

In this article, we describe meromorphic solutions of certain partial differential equations, which are originated from the algebraic equation $P(f,g)=0$, where P is a polynomial on $\mathbb {C}^2$. As an application, with the theorem of Coman–Poletsky, we give a proof of the classic theorem: Every meromorphic solution $u(s)$ on $\mathbb {C}$ of $P(u,u')=0$ belongs to W, which is the class of meromorphic functions on $\mathbb {C}$ that consists of elliptic functions, rational functions and functions of the form $R(e^{a s})$, where R is rational and $a\in \mathbb {C}$. In addition, we consider the factorization of meromorphic solutions on $\mathbb {C}^n$ of some well-known PDEs, such as Inviscid Burgers’ equation, Riccati equation, Malmquist–Yosida equation, PDEs of Fermat type.

We establish sharp upper bounds for shifted moments of quadratic Dirichlet L-function under the generalized Riemann hypothesis. Our result is then used to prove bounds for moments of quadratic Dirichlet character sums.