We study a nonlinear parabolic equation for which the classical Aubin–Lions compactness theorem fails due to critical temporal integrability. We show that this failure is structural and originates from the lack of control over temporal translations rather than from insufficient spatial regularity. A global compactness lemma based on uniform time-translation estimates is introduced, providing a minimal and optimal substitute for the Aubin–Lions framework. As an application, weestablish the existence of global weak solutions for a parabolic equation with unilateral absorption, where strong compactness is essential to identify the nonlinear term.

The intrinsic timed-Hausdorff distance between timed-metric-spaces, first introduced by Sakovich–Sormani, yields a weak notion of convergence for space-times. In this paper we prove a compactness theorem for the intrinsic timed-Hausdorff convergence of timed-metric-spaces using timed-Fréchet maps. Our proof introduces the notion of “addresses” and provides a new way of stating Gromov's original compactness theorem for Gromov–Hausdorff convergence of metric spaces. We also obtain a new Arzelà–Ascoli theorem for real valued uniformly bounded Lipschitz functions on Gromov–Hausdorff converging compact metric spaces. Moreover, we establish the triangle inequality for the intrinsic timed-Hausdorff distance.

A graph $H$ is said to be \emph{common} if the number of monochromatic labelled copies of $H$ in a $2$-colouring of the edges of a large complete graph is asymptotically minimized by a random colouring. It is well known that the disjoint union of two common graphs may be uncommon; e.g., $K_2$ and $K_3$ are common, but their disjoint union is not. We investigate the commonality of disjoint unions of multiple copies of $K_3$ and $K_2$. As a consequence of our results, we obtain an example of a pair of uncommon graphs whose disjoint union is common. Our approach is to reduce the problem of showing that certain disconnected graphs are common to a constrained optimization problem in which the constraints are derived from supersaturation bounds related to Razborov's Triangle Density Theorem. We also improve bounds on the Ramsey multiplicity constant of a triangle with a pendant edge and the disjoint union of $K_3$ and $K_2$.

Abstract Let M be a compact three-dimensional Riemannian manifold with non-negative Ricci curvature and a non-empty boundary $\partial M$ . Fraser and Li [2] established a compactness theorem for the space of compact, properly embedded minimal surfaces of fixed topological type in M with a free boundary on $\partial M$ , assuming that $\partial M$ is strictly convex with respect to the inward unit normal. In this paper, we show that the strict convexity condition on $\partial M$ cannot be relaxed.

Let [Formula: see text] and [Formula: see text], where [Formula: see text] is Euler’s function and [Formula: see text] is Dedekind’s arithmetic function. We obtain the maximal order of [Formula: see text], as well as the average orders of [Formula: see text] and [Formula: see text]. Furthermore, we prove a density theorem for both [Formula: see text] and [Formula: see text].

In two dimensions, the l-level Sierpinski gasket SG(l) is obtained by splitting an equilateral triangle into a collection of l2 equilateral triangles of equal size and with the same total area, retaining only the l(l+1)∕2 triangles with the same orientation as the original triangle, and then iterating this procedure indefinitely. We show that the canonical diffusions on the spaces SG(l), l≥2, can be rescaled to yield Brownian motion on the initial triangle. Our argument also applies to the analogous higher-dimensional Sierpinski gaskets. Moreover, we prove a local central limit theorem for the associated transition densities. Key to this is the derivation of a Poincaré inequality, in the proof of which we exploit the Euclidean-type mixing that occurs between the bottlenecks present at each scale of the fractal.

In this paper, we are interested in studying a pseudo-parabolic equation with a memory term. This type of equation has lots of appearance in many fields, such as: physics, mechanical engineering, and heat conduction theory. For the homogeneous case, we prove well-posedness and convergence as the memory parameter $ b \to 0 $. In the global Lipschitz case, we establish the global existence and uniqueness of mild solution by using the Banach fixed point theorem and show convergence as $ b \to 0 $. Furthermore, when the nonlinear source is locally Lipschitz, we extend the solutions to the interval $ [0, \infty) $ and establish the existence of a global strong solution under specific conditions. In this context, the primary techniques for obtaining mild solutions involve first proving the global existence of solutions to the truncated problem and then analyzing the convergence of these truncated solutions using Aubin - Lions' compactness theorem to achieve a global solution to our problem. To the best of our knowledge, this paper is one of the first result to consider the global strong solution to pseudo-parabolic equations with a memory term.

In this paper, we establish the sharp fractional subelliptic Sobolev inequalities and Gagliardo-Nirenberg inequalities on stratified Lie groups. The best constants are given in terms of a ground state solution of a fractional subelliptic equation involving the fractional p-sublaplacian (1<p<infty ) on stratified Lie groups. We also prove the existence of ground state (least energy) solutions to nonlinear subelliptic fractional Schrödinger equation on stratified Lie groups. Different from the proofs of analogous results in the setting of classical Sobolev spaces on Euclidean spaces given by Weinstein (Comm. Math. Phys. 87(4):576-676, 1982/1983) using the rearrangement inequality which is not available in stratified Lie groups, we apply a subelliptic version of vanishing lemma due to Lions extended in the setting of stratified Lie groups combining it with the compact embedding theorem for subelliptic fractional Sobolev spaces obtained in our previous paper (Math. Ann. 388(4):4201-4249, 2024). We also present subelliptic fractional logarithmic Sobolev inequalities with explicit constants on stratified Lie groups. The main results are new for p=2 even in the context of the Heisenberg group.

Abstract We prove some results concerning the finitely additive, vector integrals of Bochner and Pettis and their representation over a countably additive probability space. An application to the non compact Choquet theorem is also provided.

ABSTRACT This paper deals with a coupled system of nonlinear partial differential equations describing unsteady 3D flows of a viscoelastic fluid (with the constitutive law of differential type) through a porous medium. Using a modified Faedo‐Galerkin approximation procedure with special basis eigenfunctions and the Aubin compactness theorem, we prove a theorem about the existence of a global‐in‐time weak solution in a bounded Lipschitz domain under natural assumptions on model data. The proof schema of this theorem can be used for semi‐analytical and numerical solving of viscoelastic flow problems in tubes and channels filled with a porous medium. Moreover, in our paper, a new Serrin‐type criterion for regularity and uniqueness of a weak solution is established in terms of the velocity field and the elastic part of the extra stress tensor. Our results hold for both the no‐slip and perfect‐slip boundary conditions.

Abstract Let K / k be a finite Galois extension of number fields, and let $$H_K$$ H K be the Hilbert class field of K . We find a way to verify the nonsplitting of the short exact sequence $$\begin{aligned} 1\rightarrow Cl_K\rightarrow \textrm{Gal}(H_K/k){\rightarrow }\textrm{Gal}(K/k)\rightarrow 1 \end{aligned}$$ 1 → C l K → Gal ( H K / k ) → Gal ( K / k ) → 1 by finite calculation. Our method is based on the study of the principal version of the Chebotarev density theorem, which represents the density of the prime ideals of k that factor into the product of principal prime ideals in K . We also find explicit equations to express the principal density in terms of the invariants of K / k . In particular, we prove that the group structure of the ideal class group of K can be determined by reading the principal densities.

In this work, we reformulated the incompressible Navier–Stokes equations in four spatial dimensions as a gauge theory. We identified the velocity field with a connection one-form on a principal bundle so that we established higher-order energy estimates and invoked Uhlenbeck’s compactness theorem to analyze the limiting behavior of solutions. Under suitable curvature–dissipation conditions, the global existence and smoothness of solutions were proven.

Abstract We prove old and new convergence statements for fixed-points statistics and characters of symmetric groups using tensor envelope categories, such as the Deligne–Knop category of representations of the “symmetric group” $$S_t$$ S t for an indeterminate t . We also speculate on a generalization of Chebotarev’s density theorem to pseudopolynomials.

In this paper, we investigate the continuity of weak pullback mean random attractors for mean random dynamical systems defined in Bochner spaces. We first introduce the concept of weak pullback mean random attractor with respect to the weak topology of reflexive Bochner spaces, and then discuss the abstract criteria on the continuity of weak pullback mean random attractors using Baire residual Theorem and Baire density Theorem. As an application, we establish the residual dense continuity and full upper semi-continuity of weak pullback mean random attractors for non-autonomous stochastic wave equations with nonlinear noise as the coefficient of the white noise term tends to zero.

Abstract We develop an alternative approach to the study of Fourier series, based on the Short-Time-Fourier Transform (STFT) acting on $$L_{\nu }^{2}(0,1)$$ L ν 2 ( 0 , 1 ) , the space of measurable functions f in $$\mathbb {R}$$ R , square-integrable in (0, 1), and time-periodic up to a phase factor: for fixed $$\nu \in \mathbb {R}$$ ν ∈ R , $$\begin{aligned} f(t+k)=e^{2\pi ik\nu }f(t){, \ }k\in \mathbb {Z}\text {.} \end{aligned}$$ f ( t + k ) = e 2 π i k ν f ( t ) , k ∈ Z . The resulting phase space is the vertical strip $$\mathbb {C}/\mathbb {Z}=[0,1)\times \mathbb {R}$$ C / Z = [ 0 , 1 ) × R , a flat model of an infinite cylinder, which leads to Gabor frames with an interesting structure theory, allowing for a Janssen-type representation. As expected, a Gaussian window leads to a Fock space of entire functions, studied in the companion paper by the same authors [Beurling-type density theorems for sampling and interpolation on the flat cylinder]. When g is a Hermite function, we are lead to true Fock spaces of polyanalytic functions (Landau level eigenspaces) on the vertical strip $$[0,1)\times \mathbb {R}$$ [ 0 , 1 ) × R . We first prove a density condition for a lattice to be interpolating in this space. Furthermore, an analogue of the sufficient Wexler-Raz conditions is obtained which leads to new criteria for Gabor frames in $$L^{2}(\mathbb {R})$$ L 2 ( R ) , and to sufficient conditions for Gabor frames in $$L_{\nu }^{2}(0,1)$$ L ν 2 ( 0 , 1 ) with Hermite windows (an analogue of a theorem of Gröchenig and Lyubarskii about Gabor frames with Hermite windows) and with totally positive windows in the Feichtinger algebra (an analogue of a recent theorem of Gröchenig). We also consider a vectorial STFT in $$L_{\nu }^{2}(0,1)$$ L ν 2 ( 0 , 1 ) and, using the vector with the first Hermite functions as window, we introduce the (full) Fock spaces of polyanalytic functions on $$[0,1)\times \mathbb {R}$$ [ 0 , 1 ) × R and their associated Bargmann-type transforms, and prove an analogue of Vasilevski’s orthogonal decomposition into true polyanalytic Fock spaces (Landau level eigenspaces on $$[0,1)\times \mathbb {R}$$ [ 0 , 1 ) × R ). We conclude the paper with an analogue of Gröchenig-Lyubarskii’s sufficient condition for Gabor super-frames with Hermite functions, which is equivalent to a sufficient sampling condition on the full Fock space of polyanalytic functions on $$[0,1)\times \mathbb {R}$$ [ 0 , 1 ) × R . The proofs of the results about Gabor frames, involving some of Gröchenig’s most significant results of the past 25 years, are a clear indication of his influence on the field during this period.

Purpose This paper investigates the topological and geometric properties of complete shrinking Riemann solitons (Mm, g, µ, V), extending classical results from Ricci solitons to the more general Riemann soliton setting. Design/methodology/approach We employ techniques from Riemannian geometry, including the analysis of the Riemann curvature tensor, Lie derivatives along vector fields and comparison theorems for geodesics. By establishing suitable inequalities on the divergence and norm of the soliton vector field V, we derive diameter bounds and compactness criteria. Findings We prove that any complete shrinking Riemann soliton is compact if the divergence of V satisfies divV ≤ −K1 and ||V || ≤ K2 for positive constants K1 and K2. Moreover, we show that the fundamental group of such a manifold is finite. Explicit diameter estimates in terms of K1, K2 and the soliton constant µ are provided. Originality/value These results generalize known compactness and finiteness theorems for Ricci solitons to the framework of Riemann solitons, offering new insights into their geometric structure and topological constraints.

Abstract The Rarita‐Schwinger–Seiberg‐Witten (RS–SW) equations are defined similarly to the classical Seiberg–Witten equations, where a geometric non–Dirac‐type operator replaces the Dirac operator called the Rarita–Schwinger operator. In dimension 4, the RS–SW equation was first considered by the second author (Nguyen [J. Geom. Anal. 33(2023), no. 10, 336]). The variational approach will also give us a three‐dimensional version of the equations. The RS–SW equations share some features with the multiple‐spinor Seiberg–Witten equations, where the moduli space of solutions could be noncompact. In this paper, we prove a compactness theorem regarding the moduli space of solutions of the RS–SW equations defined on 3‐manifolds.

We prove that if a polynomial with rational coefficients has a root mod p for every large prime p, then it has a real root. We show with examples how to use Chebotarev’s density theorem to study roots of polynomials mod p, leading up to our proof. As an application, we show that the primes can’t be covered by finitely many positive definite binary quadratic forms.

Some Density Theorems in Neural Network with Variable Exponent

Abstract Let E be an elliptic curve over the rationals which does not have complex multiplication. Serre showed that the adelic representation attached to $E/\mathbb {Q}$ has open image, and in particular, there is a minimal natural number $C_E$ such that the mod $\ell $ representation ${\bar {\rho }}_{E,\ell }$ is surjective for any prime $\ell&gt; C_E$ . Assuming the Generalized Riemann Hypothesis, Mayle–Wang gave explicit bounds for $C_E$ which are logarithmic in the conductor of E and have explicit constants. The method is based on using effective forms of the Chebotarev Density Theorem together with the Faltings–Serre method, in particular, using the “deviation group” of the $2$ -adic representations attached to two elliptic curves. By considering quotients of the deviation group and a characterization of the images of the $2$ -adic representation $\rho _{E,2}$ by Rouse and Zureick–Brown, we show in this article how to further reduce the constants in Mayle–Wang’s results. Another result of independent interest are improved effective isogeny theorems for elliptic curves over the rationals.