Two novel algorithms, which incorporate inertial terms and relaxation effects, are introduced to tackle a monotone inclusion problem. The weak and strong convergence of the algorithms are obtained under certain conditions, and the R-linear convergence for the first algorithm is demonstrated if the set-valued operator involved is strongly monotone in real Hilbert spaces. The proposed algorithms are applied to signal recovery problems and demonstrate improved performance compared to existing algorithms in the literature.

We prove a functional version of the additive kinematic formula as an application of the Hadwiger theorem on convex functions together with a Kubota-type formula for mixed Monge–Ampère measures. As an application, we give a new explanation for the equivalence of the representations of functional intrinsic volumes as singular Hessian valuations and as integrals with respect to mixed Monge–Ampère measures. In addition, we obtain a new integral geometric formula for mixed area measures of convex bodies, where integration on $\operatorname {SO}(n-1)\times \operatorname {O}(1)$ is considered.

Let $\overline {M}(a,c,n)$ denote the number of overpartitions of n with first residual crank congruent to a modulo c with $c\geq 3$ being odd and $0\leq a<c$. The central objective of this paper is twofold: firstly, to establish an asymptotic formula for the crank of overpartitions; and secondly, to establish several inequalities concerning $\overline {M}(a,c,n)$ that encompasses crank differences, positivity, and strict log-subadditivity.

We establish bounds for exponential sums twisted by generalized Möbius functions and their convolutions. As an application, we prove asymptotic formulas for certain weighted chromatic partitions by using the Hardy–Littlewood circle method. Lastly, we provide an explicit formula relating the contributions from the major arcs with a sum over the zeros of the Riemann zeta-function.

In this paper, we investigate locally finitely presented pure semisimple (hereditary) Grothendieck categories. We show that every locally finitely presented pure semisimple (resp., hereditary) Grothendieck category $\mathscr {A}$ is equivalent to the category of left modules over a left pure semisimple (resp., left hereditary) ring when $\mathrm {Mod}(\mathrm {fp}(\mathscr {A}))$ is a QF-3 category, and every representable functor in $\mathrm {Mod}(\mathrm {fp}(\mathscr {A}))$ has finitely generated essential socle. In fact, we show that there exists a bijection between Morita equivalence classes of left pure semisimple (resp., left hereditary) rings $\Lambda $ and equivalence classes of locally finitely presented pure semisimple (resp., hereditary) Grothendieck categories $\mathscr {A}$ that $\mathrm {Mod}(\mathrm {fp}(\mathscr {A}))$ is a QF-3 category, and every representable functor in $\mathrm {Mod}(\mathrm {fp}(\mathscr {A}))$ has finitely generated essential socle. To prove this result, we study left pure semisimple rings by using Auslander’s ideas. We show that there exists, up to equivalence, a bijection between the class of left pure semisimple rings and the class of rings with nice homological properties. These results extend the Auslander and Ringel–Tachikawa correspondence to the class of left pure semisimple rings. As a consequence, we give several equivalent statements to the pure semisimplicity conjecture.

In this paper, we study the ranges of the Schwartz space $\mathcal {S}$ and its dual $\mathcal {S}'$ (space of tempered distributions) under the Bargmann transform. The characterization of these two ranges leads to interesting reproducing kernel Hilbert spaces whose reproducing kernels can be expressed, respectively, in terms of the Touchard polynomials and the hypergeometric functions. We investigate the main properties of some associated operators and introduce two generalized Bargmann transforms in this framework. This can be considered as a continuation of an interesting research path that Neretin started earlier in his book on Gaussian integral operators.

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.

This article explores the notions of $\mathcal {F}$-transitivity and topological $\mathcal {F}$-recurrence for backward shift operators on weighted $\ell ^p$-spaces and $c_0$-spaces on directed trees, where $\mathcal {F}$ represents a Furstenberg family of subsets of $\mathbb {N}_0$. In particular, we establish the equivalence between recurrence and hypercyclicity of these operators on unrooted directed trees. For rooted directed trees, a backward shift operator is hypercyclic if and only if it possesses an orbit of a bounded subset that is weakly dense.

We establish the restricted sumset analog of the celebrated conjecture of Sárközy on additive decompositions of the set of nonzero squares over a finite field. More precisely, we show that if $q>13$ is an odd prime power, then the set of nonzero squares in $\mathbb {F}_q$ cannot be written as a restricted sumset $A \hat {+} A$, extending a result of Shkredov. More generally, we study restricted sumsets in multiplicative subgroups over finite fields as well as restricted sumsets in perfect powers (over integers) motivated by a question of Erdős and Moser. We also prove an analog of van Lint–MacWilliams’ conjecture for restricted sumsets, which appears to be the first analogue of Erdős--Ko–Rado theorem in a family of Cayley sum graphs.

We study scaled topological entropy, scaled measure entropy, and scaled local entropy in the context of amenable group actions. In particular, a variational principle is established.

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.

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.

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.

Inspired by the work of Bourgain and Garaev (2013), we provide new bounds for certain weighted bilinear Kloosterman sums in polynomial rings over a finite field. As an application, we build upon and extend some results of Sawin and Shusterman (2022). These results include bounds for exponential sums weighted by the Möbius function and a level of distribution for irreducible polynomials beyond 1/2, with arbitrary composite modulus. Additionally, we can do better when averaging over the modulus, to give an analogue of the Bombieri-Vinogradov Theorem with a level of distribution even further beyond 1/2.

We give a precise classification, in terms of Shimura data, of all $1$-dimensional Shimura subvarieties of a moduli space of polarized abelian varieties.

By employing the external Kasparov product, in [18], Hawkins, Skalski, White, and Zacharias constructed spectral triples on crossed product C$^\ast $-algebras by equicontinuous actions of discrete groups. They further raised the question for whether their construction turns the respective crossed product into a compact quantum metric space in the sense of Rieffel. By introducing the concept of groups separated with respect to a given length function, we give an affirmative answer in the case of virtually Abelian groups equipped with certain orbit metric length functions. We further complement our results with a discussion of natural examples such as generalized Bunce-Deddens algebras and higher-dimensional noncommutative tori.

Given a Fell bundle $\mathcal {B}=\{B_t\}_{t\in G}$ over a locally compact group G and a closed subgroup $H\subset G,$ we construct quotients $C^{*}_{H\uparrow \mathcal {B}}(\mathcal {B})$ and $C^{*}_{H\uparrow G}(\mathcal {B})$ of the full cross-sectional C*-algebra $C^{*}(\mathcal {B})$ analogous to Exel–Ng’s reduced algebras $C^{*}_{\mathop {\mathrm {r}}}(\mathcal {B})\equiv C^{*}_{\{e\}\uparrow \mathcal {B}}(\mathcal {B})$ and $C^{*}_R(\mathcal {B})\equiv C^{*}_{\{e\}\uparrow G}(\mathcal {B}).$ An absorption principle, similar to Fell’s one, is used to give conditions on $\mathcal {B}$ and H (e.g., G discrete and $\mathcal {B}$ saturated, or H normal) ensuring $C^{*}_{H\uparrow \mathcal {B}}(\mathcal {B})=C^{*}_{H\uparrow G}(\mathcal {B}).$ The tools developed here enable us to show that if the normalizer of H is open in G and $\mathcal {B}_H:=\{B_t\}_{t\in H}$ is the reduction of $\mathcal {B}$ to $H,$ then $C^{*}(\mathcal {B}_H)=C^{*}_{\mathop {\mathrm {r}}}(\mathcal {B}_H)$ if and only if $C^{*}_{H\uparrow \mathcal {B}}(\mathcal {B})=C^{*}_{\mathop {\mathrm {r}}}(\mathcal {B});$ the last identification being implied by $C^{*}(\mathcal {B})=C^{*}_{\mathop {\mathrm {r}}}(\mathcal {B}).$ We also prove that if G is inner amenable and $C^{*}_{\mathop {\mathrm {r}}}(\mathcal {B})\otimes _{\max } C^{*}_{\mathop {\mathrm {r}}}(G)=C^{*}_{\mathop {\mathrm { r}}}(\mathcal {B})\otimes C^{*}_{\mathop {\mathrm {r}}}(G),$ then $C^{*}(\mathcal {B})=C^{*}_{\mathop {\mathrm {r}}}(\mathcal {B}).$

We establish a Central Limit Theorem for tensor product random variables $c_k:=a_k \otimes a_k$, where $(a_k)_{k \in \mathbb {N}}$ is a free family of variables. We show that if the variables $a_k$ are centered, the limiting law is the semi-circle. Otherwise, the limiting law depends on the mean and variance of the variables $a_k$ and corresponds to a free interpolation between the semi-circle law and the classical convolution of two semi-circle laws.

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 construct a reflexive Banach space $X_{\mathcal {D}}$ with an unconditional basis such that all spreading models admitted by normalized block sequences in $X_{\mathcal {D}}$ are uniformly equivalent to the unit vector basis of $\ell _1$, yet every infinite-dimensional closed subspace of $X_{\mathcal {D}}$ fails the Lebesgue property. This is a new result in a program initiated by Odell in 2002 concerning the strong separation of asymptotic properties in Banach spaces.