We give a quick proof of the fact that the relative de Rham cohomology groups (H^i_{{{,textrm{dR},}}}(X/S)) of a smooth and proper map X/S between schemes over ({mathbb {Q}}) are vector bundles on the base, replacing Hodge-theoretic and transcendental methods with ({mathbb {A}}^1)-homotopy theory.
We prove a novel, tight lower bound for the norm in (textrm{L}^2[0,T]) of the Caputo fractional derivative. It is based on continuous linear functionals, Peano kernels, and the Gaussian hypergeometric function.
For a hyperplane H supporting a convex body C in the hyperbolic space (mathbb {H}^d), we define the width of C determined by H as the distance between H and a most distant ultraparallel hyperplane supporting C. The minimum width of C over all supporting H is called the thickness (Delta (C)) of C. A convex body (R subset mathbb {H}^{d}) is said to be reduced if (Delta (Z) < Delta (R)) for every convex body Z properly contained in R. We describe a class of reduced polygons in (mathbb {H}^{2}) and present some properties of them. In particular, we estimate their diameters in terms of their thicknesses.
Let C be a proper, closed subset with nonempty interior in a normed space X. We define four variants of modulus of convexity for C and prove that they all coincide. This result, which is classical and well-known for (C=B_X) (the unit ball of X), requires a less easy proof than the particular case of (B_X.) We also show that if the modulus of convexity of C is not identically null, then C is bounded. This extends a result by M.V. Balashov and D. Repovš.
Building on the work of Brinkmann and Logan, we show that both the Brinkmann problem and the Brinkmann conjugacy problem are decidable for endomorphisms of the free group (mathbb {F}_{n}).
The purpose of this note is to show that a finitely generated graded module M over (S=k[x_1,ldots ,x_n]), k a field, is sequentially Cohen-Macaulay if and only if its arithmetic degree ({text {adeg}}(M)) agrees with ({text {adeg}}(F/{text {gin}}_textrm{revlex}(U))), where F is a graded free S-module and (M cong F/U). This answers positively a conjecture of Lu and Yu from 2016.
In this paper, we give an alternative proof of the main result in Hatano et al. (Tokyo J Math 46(1):125–160, 2023) that the Hardy–Littlewood maximal operator is bounded on the Orlicz–Lorentz space (L^{Phi ,q}({mathbb {R}}^n)) for a Young function (Phi in nabla _2) and (0<q<1.)
In this paper, we investigate the existence and uniqueness of almost periodic solutions for the parabolic-elliptic Keller–Segel system on the whole space (mathbb {R}^n,, (n geqslant 4)). We work in the framework of critical spaces such as on the weak-Lorentz space (L^{frac{n}{2},infty }(mathbb {R}^n)). Our method is based on the dispersive and smoothing estimates of the heat semigroup and fixed point arguments.
We consider groups G such that the set ([G,varphi ]={g^{-1}g^{varphi }|gin G}) is a subgroup for every automorphism (varphi ) of G, and we prove that there exists such a group G that is finite and nilpotent of class n for every (nin mathbb N). Then there exists an infinite not nilpotent group with the above property and the Conjecture 18.14 of Khukhro and Mazurov (The Kourovka Notebook No. 20, 2022) is false.
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