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.

In this paper, we prove an optimal isoperimetric inequality for spacelike, compact, star-shaped, and $2$-convex hypersurfaces in de Sitter space.

In the goundbreaking paper [BD11] (which opened a wide avenue of research regarding unlikely intersections in arithmetic dynamics), Baker and DeMarco prove that for the family of polynomials $f_\lambda (x):=x^d+\lambda $ (parameterized by $\lambda \in \mathbb {C}$), given two starting points a and b in $\mathbb {C}$, if there exist infinitely many $\lambda \in \mathbb {C}$ such that both a and b are preperiodic under the action of $f_\lambda $, then $a^d=b^d$. In this paper, we study the same question, this time working in a field of characteristic $p>0$. The answer in positive characteristic is more nuanced, as there are three distinct cases: (i) both starting points a and b live in ${\overline {\mathbb F}_p}$; (ii) d is a power of p; and (iii) not both a and b live in ${\overline {\mathbb F}_p}$, while d is not a power of p. Only in case (iii), one derives the same conclusion as in characteristic $0$ (i.e., that $a^d=b^d$). In case (i), one has that for each $\lambda \in {\overline {\mathbb F}_p}$, both a and b are preperiodic under the action of $f_\lambda $, while in case (ii), one obtains that also whenever $a-b\in {\overline {\mathbb F}_p}$, then for each parameter $\lambda $, we have that a is preperiodic under the action of $f_\lambda $ if and only if b is preperiodic under the action of $f_\lambda $.

Alexandrov’s estimate states that if $\Omega $ is a bounded open convex domain in $\mathbb {R}^n$ and $u:\bar \Omega \to \mathbb {R}$ is a convex solution of the Monge-Ampère equation $\det D^2 u = f$ that vanishes on $\partial \Omega $, then

$$\begin{align*}|u(x) - u(y)| \le \omega(|x-y|)(\int_\Omega f)^{1/n} \qquad \text{for }\omega(\delta) = C_n\,\text{diam}(\Omega)^{\frac{n-1}n} \delta^{1/n}. \end{align*}$$

$\partial \Omega $

$0$

$\omega (\delta )$

$C_\Omega \delta ^{\frac 12 + \frac 1{2n}}$

$C_\Omega $

$n=2$

$n\ge 3$

$\partial \Omega $

$C^2$

$\omega _\Omega (\delta )$

$\delta \to 0$

$\omega _\Omega $

$\partial \Omega $

$\omega _\Omega (\delta ) = o(\delta ^{1/n})$

$\delta \to 0$

$\omega _\Omega (\delta ) \le C \delta ^\alpha $

$\alpha>1/n$

Let $p(n)$ and $M(m,L;n)$ be the number of partitions of n and the number of partitions of n with crank congruent to m modulo L, respectively, and let

$$ \begin{align*}M(a,b,c;L;n):= M(a,L;n) + M(b,L;n) + M(c,L;n).\end{align*} $$

In this paper, we focus on some relations between $M(m,L;n)$ and $p(n),$ which Dyson, Andrews, and Garvan etc. have studied previously. By applying a modification of the circle method to estimate the Fourier coefficients of generating functions, we establish the following inequalities between $M(a,b,c;L;n)$ and $p(n):$ for $n \geq 467$,

$$ \begin{align*} M(0,1,1;9;n)> \frac{p(n)}{3} &\;\: \mathrm{when}\;\: n \equiv 0,1,5,8 ~ (\mathrm{mod}~9), \\ M(0,1,1;9;n) < \frac{p(n)}{3} &\;\: \mathrm{when}\;\: n \equiv 2,3,4,6,7 ~ (\mathrm{mod}~9), \\ M(2,3,4;9;n) < \frac{p(n)}{3} &\;\: \mathrm{when}\;\: n \equiv 0,1,5,8 ~ (\mathrm{mod}~9), \\ M(2,3,4;9;n)> \frac{p(n)}{3} &\;\: \mathrm{when}\;\: n \equiv 2,3,4,6,7 ~ (\mathrm{mod}~9). \end{align*} $$

In the proof of these inequalities, an inequality for the logarithm of the generating function for $p(n)$ is derived and applied. Our method reduces the last possible counterexamples to $467 \leq n \leq 22471$, and it will produce more effective estimates when proving inequalities of such types.

In this paper, we give a complete coarse classification of strongly exceptional Legendrian realizations of the connected sum of two Hopf links in contact 3-spheres. This is the first classification result about exceptional Legendrian representatives for connected sums of link families.

In this paper, we establish a definitive result which almost completely closes the problem of bounded elementary generation for Chevalley groups of rank $\ge 2$ over arbitrary Dedekind rings R of arithmetic type, with uniform bounds. Namely, we show that for every reduced irreducible root system $\Phi $ of rank $\ge 2$, there exists a universal bound $L=L(\Phi )$ such that the simply connected Chevalley groups $G(\Phi ,R)$ have elementary width $\le L$ for all Dedekind rings of arithmetic type R.

We introduce a generalization of sequential compactness using barriers on $\omega $ extending naturally the notion introduced in [W. Kubiś and P. Szeptycki, On a topological Ramsey theorem, Canad. Math. Bull., 66 (2023), 156–165]. We improve results from [C. Corral and O. Guzmán and C. López-Callejas, High dimensional sequential compactness, Fund. Math.] by building spaces that are ${\mathcal {B}}$-sequentially compact but not ${\mathcal {C}}$-sequentially compact when the barriers ${\mathcal {B}}$ and ${\mathcal {C}}$ satisfy certain rank assumption which turns out to be equivalent to a Katětov-order assumption. Such examples are constructed under the assumption ${\mathfrak {b}} ={\mathfrak {c}}$. We also exhibit some classes of spaces that are ${\mathcal {B}}$-sequentially compact for every barrier ${\mathcal {B}}$, including some classical classes of compact spaces from functional analysis, and as a byproduct, we obtain some results on angelic spaces. Finally, we introduce and compute some cardinal invariants naturally associated to barriers.

Let $\hat {\mathbb {R}}^n$ be the one-point compactification of $\mathbb {R}^n$ obtained by adding a point at infinity. We say that a subset $A\subseteq \hat {\mathbb {R}}^n$ is $\mathbf {u}$-convex if for every pair of points $\mathbf {z}_1, \mathbf {z}_2 \in A$, the arc of the unique circle through $\mathbf {u}, \mathbf {z}_1$ and $\mathbf {z}_2$, from $\mathbf {z}_1$ to $\mathbf {z}_2$ and not containing $\mathbf {u}$, is contained in A. In this case, we call $\mathbf {u}$ a pole of A. When the pole $\mathbf {u}$ approaches infinity, $\mathbf {u}$-convex sets become convex in the classical sense.

The notion of polar convexity in the complex plane has been used to analyze the behavior of critical points of polynomials. In this paper, we extend the notion to finite-dimensional Euclidean spaces. The goal of this paper is to start building the theory of polar convexity and to show that the introduction of a pole creates a richer theory. For example, polar convexity enjoys a beautiful duality (see Theorem 4.3) that does not exist in classical convexity. We formulate polar analogues of several classical results of the alternatives, such as Gordan’s and Farkas’ lemmas; see Section 5. Finally, we give a full description of the convex hull of finitely many points with respect to finitely many poles; see Theorem 6.7.

We improve known estimates for the number of points of bounded height in semigroup orbits of polarized dynamical systems. In particular, we give exact asymptotics for generic semigroups acting on the projective line. The main new ingredient is the Wiener-Ikehara Tauberian theorem, which we use to count functions in semigroups of bounded degree.

We analyze the limit of stable solutions to the Ginzburg-Landau (GL) equations when ${\varepsilon }$, the inverse of the GL parameter, goes to zero and in a regime where the applied magnetic field is of order $|\log {\varepsilon } |$ whereas the total energy is of order $|\log {\varepsilon }|^2$. In order to do that, we pass to the limit in the second inner variation of the GL energy. The main difficulty is to understand the convergence of quadratic terms involving derivatives of functions converging only weakly in $H^1$. We use an assumption of convergence of energies, the limiting criticality conditions obtained by Sandier-Serfaty by passing to the limit in the first inner variation, and properties of limiting vorticities to find the limit of all the desired quadratic terms. At last, we investigate the limiting stability condition we have obtained. In the case with magnetic field, we study an example of an admissible limiting vorticity supported on a line in a square ${{\Omega }}=(-L,L)^2$ and show that if L is small enough, this vorticiy satisfies the limiting stability condition, whereas when L is large enough, it stops verifying that condition. In the case without magnetic field, we use a result of Iwaniec-Onninen to prove that every measure in $H^{-1}({{\Omega }})$ satisfying the first-order limiting criticality condition also verifies the second-order limiting stability condition.

In this paper, we prove the global exstence of weak solutions for a porous medium dynamics of m species moving between two domains separated by a zero-thickness membrane. On this membrane, Kedem–Katchalsky conditions are considered, and the study is characterized by natural structural conditions applied to the nonlinear reactive terms. The global existence is established under the assumption that these reactive terms are bounded in $L^1$. This problem has already been analyzed in the linear diffusion case by Ciavolella and Perthame in Ciavolella and Perthame (2021, Journal of Evolution Equations 21, 1513–1540). The present work constitutes an extension for nonlinear diffusion, particularly of the porous medium type, in the form $\partial _t v_i - \Delta v_i^{r_i} = R_i$, for an exponent $r_i < 2$. The case $r_i \geq 2$ remains an open problem. This paper is an adaptation of the ideas from Ciavolella and Perthame (2021, Journal of Evolution Equations 21, 1513–1540), with new strategies to overcome the appearance of nonlinearity and degeneracy in the diffusion term.

We use a special tiling for the hyperbolic d-space $\mathbb {H}^d$ for $d=2,3,4$ to construct an (almost) explicit isomorphism between the Lipschitz-free space $\mathcal {F}(\mathbb {H}^d)$ and $\mathcal {F}(P)\oplus \mathcal {F}(\mathcal {N})$, where P is a polytope in $\mathbb {R}^d$ and $\mathcal {N}$ a net in $\mathbb {H}^d$ coming from the tiling. This implies that the spaces $\mathcal {F}(\mathbb {H}^d)$ and $\mathcal {F}(\mathbb {R}^d)\oplus \mathcal {F}(\mathcal {M})$ are isomorphic for every net $\mathcal {M}$ in $\mathbb {H}^d$. In particular, we obtain that, for $d=2,3,4$, $\mathcal {F}(\mathbb {H}^d)$ has a Schauder basis. Moreover, using a similar method, we also give an explicit isomorphism between $\mathrm {Lip}(\mathbb {H}^d)$ and $\mathrm {Lip}(\mathbb {R}^d)$.