We show that {\mathbf{B}}_{\mathrm{dR}}^{+} is the universal thickening of {\mathbf{C}}_{p} . More generally, we show that, if S is a reduced affinoid algebra, {\mathcal{O}}{\mathbb{B}}_{\mathrm{dR}}^{+}({\bar S}) is the universal S -thickening of the completion of {\bar S} . Nous montrons que {\mathbf{B}}_{\mathrm{dR}}^{+} est l’épaississement universel de {\mathbf{C}}_{p} . Plus généralement, nous montrons que, si S est une algèbre affinoïde réduite, {\mathcal{O}}{\mathbb{B}}_{\mathrm{dR}}^{+}({\bar S}) est le S -épaississement universel du complété de {\bar S} .
In this paper we investigate the optimal problem for the L_{p} mixed chord integral. The existence of the L_{p} chord integral–Petty bodies is established, and the L_{p} geominimal chord integral is proposed and its properties, such as invariance under orthogonal matrices, homogeneity, isoperimetric-type inequalities and cyclic-type inequalities, which are provided as well.
Let A(n) denote the largest absolute value of the coefficients of the n-th cyclotomic polynomial Phi(n)(x). In this paper, for odd primes 7 < q < r with r equivalent to +/- 3 (mod 7q), we show that A(7qr) = 1 if q equivalent to 1 (mod 21), 2 if q equivalent to 2, 10, 11, 19, 20 (mod 21), 3 if q equivalent to 16 (mod 21), 2 or 3 if q equivalent to 4, 5, 8, 13,17 (mod 21).
In the theory of shape optimization, the rearrangements of sets are a key concept, because they allow us to keep some properties of the original set while improving other aspects. This paper is devoted to the proof of an isoperimetric property of the double spherical cap rearrangement of planar sets. In particular, we prove that, under the assumption of disconnection of non-trivial spherical slices, the rearranged set has a lower perimeter than the original one. In the general case, the symmetrized set does not decrease the perimeter, but we show that the "excess" is bounded above by 2H(1)(F/, where P denotes the set of radii such that the spherical slice is a non-trivial arc of a circle. Additionally, the higher-dimensional case is briefly discussed; in particular, an explicit counterexample is given, thus explaining why an analogous result cannot hold. The main reason for this is that, in dimension N = 3 or higher, the union of two spherical caps of equal size does not minimize the (N-2/-dimensional measure of the boundary.
T is a minimal homeomorphism of the Cantor set X, and [[T]](0) is the kernel of the index map, we give a short proof of the characterization of the maximal subgroups G of [[T]](0) such that G does not act minimally on X. They are stabilizers of nonempty closed subsets Y of X, and we present a classification when Y is finite.
We undertake a study of the asymptotic stability of the Moore-Gibson-Thompson equation with memory effect, which is one of the nonlinear acoustic equations describing the propagation of sound waves in gas and liquid. The effects of viscosity, thermal conductivity, and thermal radiation on acoustic wave propagation are considered in this model. The current work improves the previously related results in that it removes the convexity assumption on the memory kernels and obtains the uniform decay rate t(-1) of solution energy to the Moore-Gibson-Thompson equation, with only basic conditions on the memory kernels (without any decay conditions).
Let alpha is an element of R\Q and beta is an element of R be given. Suppose that a(1),& mldr;,a(s) are distinct positive integers that do not contain a reduced residue system modulo p(2 )for any prime p. We prove that there exist infinitely many primes p such that the inequality parallel to alpha p+beta parallel to
Let G be a finite group, and let p be an odd prime of |G| and Sylow p-subgroups of G be non-abelian. In this paper, we prove that any two p-subgroups of equal order in G are conjugate if and only if G/O-p'(G) is isomorphic to F-2(4)(2) and p = 3, Ru and p = 3, J(4) and p = 3, Th and p = 5, or an almost simple group of the socle F-2(4)(2(2n+1)) with n > 0, n equivalent to/ 1 mod 3 and p = 3. This solved a problem of Brandl's.
We show that B-dR(+) is the universal thickening of C-p. More generally, we show that, if S is a reduced affinoid algebra, OBdR+(S-) is the universal S-thickening of the completion of S-.
In this paper we investigate the optimal problem for the L(p )mixed chord integral. The existence of the L-p chord integral-Petty bodies is established, and the L-p geominimal chord integral is proposed and its properties, such as invariance under orthogonal matrices, homogeneity, isoperimetric-type inequalities and cyclic-type inequalities, which are provided as well.
We give equivalences between given properties of a commutative ring, and other properties on its ring of Witt vectors. Amongst them, we characterise all commutative rings whose rings of Witt vectors are Noetherian. We define a new category of commutative rings called preduced rings, and explain how it is the category of rings whose ring of Witt vectors has no p-torsion. We then extend this characterisation to the torsion of the de Rham-Witt complex.
We construct a new Weil cohomology for smooth projective varieties over a field, universal among Weil cohomologies with values in rigid additive tensor categories. A similar universal problem for Weil cohomologies with values in rigid abelian tensor categories also has a solution. We give a variant for Weil cohomologies satisfying more axioms, like Weak and Hard Lefschetz. As a consequence, we get a different construction of André's category of motives for motivated correspondences and show that it has a universal property. This theory extends over suitable bases.
We establish a new class of weighted L^2 Poincaré and elliptic functional inequalities on smooth two-manifolds with explicit constants, for a family of weights satisfying a differential equation. This family includes, in particular, weights comparable to products of positive powers of the geodesic distance to finitely many points. Our primary motivation is the derivation of estimates associated with a weighted Hodge decomposition for one-forms.
Let G be a reductive group over a field k, and let mu be a cocharacter of G. We prove that Viehmann's double coset spaces associated with (G, mu) are represented by certain Lusztig varieties, and establish a similar result for the mixed characteristic case. This representability enables a comparison between moduli stacks of truncated local shtukas and zip stacks. Over a perfect field of positive characteristic, we establish a homeomorphism between the coarse moduli stack of 1-1-truncated local G-shtukas and that of G-zips, enriching our understanding of zip period maps in the context of Shimura varieties.
In this paper we study the existence of solution to the problem \begin{equation*} \left\{\begin{array}{l} u\in H_{0}^{1}(\Omega), \\[4pt] -\textrm{div}\,(A(x)Du)=H(x,u,Du)+f(x)+a_{0}(x)\, u\quad \text{in} \quad\mathcal{D}'(\Omega), \end{array} \right. \end{equation*} where $\Omega$ is an open bounded set of $\mathbb{R}^{2}$, $A(x)$ a coercive matrix with coefficients in $L^\infty(\Omega)$, $H(x,s,\xi)$ a Carath\'eodory function satisfying, for some $\gamma >0$, $$ -c_{0}\, A(x)\, \xi\xi\leq H(x,s,\xi)\,{\rm sign}(s)\leq \gamma\,A(x)\,\xi\xi \;\;\; {\rm a.e. }\; x\in \Omega,\;\;\;\forall s\in\mathbb{R},\;\;\; \forall\xi \in \mathbb{R}^{2}. $$ Here $f$ belongs to $L^1(\log L^1)(\Omega)$ and $a_{0} \geq 0$ to $L^{q}(\Omega )$, $q>1$. For $f$ and $a_{0}$ sufficiently small, we prove the existence of at least one solution $u$ of this problem which is such that $e^{\delta_0 |u|} -1$ belongs to $H_{0}^{1}(\Omega)$ for some $\delta_0\geq\gamma$ and satisfies an \textit{a priori} estimate.
Let R be an arbitrary ring and \mathcal{E} an injectively resolving class of left R -modules. We prove that the class of \mathcal{E} -Gorenstein flat right R -modules is closed under extensions, and hence projectively resolving. This answers an open question in Gao and Zhong [Rocky Mountain J. Math. 54 (2024), 143–160] affirmatively. As a consequence, we get that this class is covering. In addition, we introduce the notion of \mathcal{E} -projectively coresolved Gorenstein flat modules, and prove that the class of \mathcal{E} -projectively coresolved Gorenstein flat right R -modules is projectively resolving and closed under transfinite extensions.
In this work, we study a bound of the form |\zeta(1+it)|\leq v\log t for t\geq t_{0} . We show that the exponential sum method with second-order derivatives can achieve any v >\frac{1}{2} as long as t_{0} is sufficiently large. Using the Riemann–Siegel formula and numerical computations, we show that when t\geq e , |\zeta(1+it)|\leq\frac{1}{2}\log t+0.6633 . This allows us to show that |\zeta(1+it)|\leq 0.6443 \log t when t\geq e . This is the best possible result of the form |\zeta(1+it)|\leq v\log t that holds for all t\geq e , as the equality is achieved when t=17.7477 .
Let R be an arbitrary ring and & an injectively resolving class of left R-modules. We prove that the class of E-Gorenstein flat right R-modules is closed under extensions, and hence projectively resolving. This answers an open question in Gao and Zhong [Rocky Mountain J. Math. 54 (2024), 143-160] affirmatively. As a consequence, we get that this class is covering. In addition, we introduce the notion of E-projectively coresolved Gorenstein flat modules, and prove that the class of E-projectively coresolved Gorenstein flat right R-modules is projectively resolving and closed under transfinite extensions.
We deal with pseudo-spherical surfaces admitting certain singularities and their caustics. In particular, we give characterizations of cuspidal butterfly, cuspidal lip and cuspidal beak singularities on a pseudo-spherical surface. Moreover, characterizations of certain singularities on the caustics of a pseudo-spherical surface are given. Furthermore, when the caustic has a cuspidal edge singularity, we investigate geometric invariants defined at that point.
In this work, we study a bound of the form divided by zeta(1 + it)divided by <= v log t for t >= t(0). We show that the exponential sum method with second-order derivatives can achieve any v > 1/2 as long as t(0) is sufficiently large. Using the Riemann-Siegel formula and numerical computations, we show that when t >= e, divided by zeta(1 + it)divided by <= 1/2 log t + 0.6633. This allows us to show that divided by zeta(1 + it)divided by <= 0.6443 log t when t >= e. This is the best possible result of the form divided by zeta(1 + it)divided by <= v log t that holds for all t >= e, as the equality is achieved when t = 17.7477.
