In this paper, we present a characterization of strong subdifferentiability of the norm of bounded linear operators on $\ell _p$ spaces, $1\leq p<\infty $. Furthermore, we prove that the set of all bounded linear operators in ${B}(\ell _p, \ell _q)$ for which the norm of ${B}(\ell _p, \ell _q)$ is strongly subdifferentiable is dense in ${B}(\ell _p, \ell _q)$. Additionally, we present a characterization of Fréchet differentiability of the norm of bounded linear operators from $\ell _p$ to $\ell _q$, where $1 < p, q < \infty $. Applying this result, we will show that the Fréchet differentiability and the Gateaux differentiability of the norm of bounded linear operators on $\ell _p$ spaces coincide, extending a known theorem regarding the operator norm on Hilbert spaces.

We introduce the L-series of weakly holomorphic modular forms using Laplace transforms and give their functional equations. We then determine converse theorems for vector-valued harmonic weak Maass forms, Jacobi forms, and elliptic modular forms of half-integral weight in Kohnen plus space.

We provide a characterization of differentially large fields in arbitrary characteristic and a single derivation in the spirit of Blum axioms for differentially closed fields. In the case of characteristic zero, we use these axioms to characterize differential largeness in terms of being existentially closed in the differential algebraic Laurent series ring, and we prove that any large field of infinite transcendence degree can be expanded to a differentially large field even under certain prescribed constant fields. As an application, we show that the theory of proper dense pairs of models of a complete and model-complete theory of large fields, is a complete theory. As a further consequence of the expansion result we show that there is no real closed and differential field that has a prime model extension in closed ordered differential fields, unless it is itself a closed ordered differential field.

A graph G is called chromatic-choosable if $\chi (G)=ch(G)$. A natural problem is to determine the minimum number of vertices in a non-chromatic-choosable graph with given chromatic number. It was conjectured by Ohba, and proved by Noel, Reed, and Wu that k-chromatic graphs G with $|V(G)| \le 2k+1$ are chromatic-choosable. This upper bound on $|V(G)|$ is tight. It is known that if k is even, then $G=K_{3 \star (k/2+1), 1 \star (k/2-1)}$ and $G=K_{4, 2 \star (k-1)}$ are non-chromatic-choosable k-chromatic graphs with $|V(G)| =2 k+2$. Some subgraphs of these two graphs are also non-chromatic-choosable. The main result of this paper is that all other k-chromatic graphs G with $|V(G)| =2 k+2$ are chromatic-choosable. In particular, if $\chi (G)$ is odd and $|V(G)| \le 2\chi (G)+2$, then G is chromatic-choosable, which was conjectured by Noel.

In this article, we study a class of convective diffusive elliptic problem with Dirichlet boundary condition and measure data in variable exponent spaces. We begin by introducing an approximate problem via a truncation approach and Yosida’s regularization. Then, we apply the technique of maximal monotone operators in Banach spaces to obtain a sequence of approximate solutions. Finally, we pass to the limit and prove that this sequence of solutions converges to at least one weak or entropy solution of the original problem. Furthermore, under some additional assumptions on the convective diffusive term, we prove the uniqueness of the entropy solution.

Let P be a pointed, closed convex cone in $\mathbb {R}^d$. We prove that for two pure isometric representations $V^{(1)}$ and $V^{(2)}$ of P, the associated CAR flows $\beta ^{V^{(1)}}$ and $\beta ^{V^{(2)}}$ are cocycle conjugate if and only if $V^{(1)}$ and $V^{(2)}$ are unitarily equivalent. We also give a complete description of pure isometric representations of P with commuting range projections that give rise to type I CAR flows. We show that such an isometric representation is completely reducible with each irreducible component being a pullback of the shift semigroup $\{S_t\}_{t \geq 0}$ on $L^2[0,\infty )$. We also compute the index and the gauge group of the associated CAR flows and show that the action of the gauge group on the set of normalized units need not be transitive.

Let $f^{(r)}(n;s,k)$ denote the maximum number of edges in an n-vertex r-uniform hypergraph containing no subgraph with k edges and at most s vertices. Brown, Erdős, and Sós [New directions in the theory of graphs (Proc. Third Ann Arbor Conf., Univ. Michigan 1971), pp. 53–63, Academic Press 1973] conjectured that the limit $\lim _{n\rightarrow \infty }n^{-2}f^{(3)}(n;k+2,k)$ exists for all k. The value of the limit was previously determined for $k=2$ in the original paper of Brown, Erdős, and Sós, for $k=3$ by Glock [Bull. Lond. Math. Soc., 51 (2019) 230–236] and for $k=4$ by Glock, Joos, Kim, Kühn, Lichev, and Pikhurko [Proc. Amer. Math. Soc., Series B, 11 (2024) 173–186] while Delcourt and Postle [Proc. Amer. Math. Soc., 152 (2024), 1881–1891] proved the conjecture (without determining the limiting value).

In this article, we determine the value of the limit in the Brown–Erdős–Sós problem for $k\in \{5,6,7\}$. More generally, we obtain the value of $\lim _{n\rightarrow \infty }n^{-2}f^{(r)}(n;rk-2k+2,k)$ for all $r\geqslant 3$ and $k\in \{5,6,7\}$. In addition, by combining these new values with recent results of Bennett, Cushman, and Dudek [arxiv:2309.00182, 2023] we obtain new asymptotic values for several generalized Ramsey numbers.

We compute the class groups of full rank upper cluster algebras in terms of the exchange polynomials. This characterizes the UFDs among these algebras. Our results simultaneously generalize theorems of Garcia Elsener, Lampe, and Smertnig from 2019 and of Cao, Keller, and Qin from 2023. Furthermore, we show that every (upper) cluster algebra is a finite factorization domain.

In this article, it is shown that the lattice of C$^*$-covers of an operator algebra does not contain enough information to distinguish operator algebras up to completely isometric isomorphism. In addition, four natural equivalences of the lattice of C$^*$-covers are developed and proven to be distinct. The lattice of C$^*$-covers of direct sums and tensor products are studied. Along the way key examples are found of operator algebras, each of which generates exactly n C$^*$-algebras up to $*$-isomorphism, and a simple operator algebra that is not similar to a C$^*$-algebra.

We establish hyperweak boundedness of area functions, square functions, maximal operators, and Calderón–Zygmund operators on products of two stratified Lie groups.

We introduce the concept of pseudo-trisections of smooth oriented compact 4-manifolds with boundary. The main feature of pseudo-trisections is that they have lower complexity than relative trisections for given 4-manifolds. We prove existence and uniqueness of pseudo-trisections, and further establish a one-to-one correspondence between pseudo-trisections and their diagrammatic representations. We next introduce the concept of pseudo-bridge trisections of neatly embedded surfaces in smooth oriented compact 4-manifolds. We develop a diagrammatic theory of pseudo-bridge trisections and provide examples of computations of invariants of neatly embedded surfaces in 4-manifolds using said diagrams.

A pebble tree is an ordered tree where each node receives some colored pebbles, in such a way that each unary node receives at least one pebble, and each subtree has either one more or as many leaves as pebbles of each color. We show that the contraction poset on pebble trees is isomorphic to the face poset of a convex polytope called pebble tree polytope. Beside providing intriguing generalizations of the classical permutahedra and associahedra, our motivation is that the faces of the pebble tree polytopes provide realizations as convex polytopes of all assocoipahedra constructed by K. Poirier and T. Tradler only as polytopal complexes.