Extensions of valuations to rational function fields over completions

Given a valued field $(K,v)$ and its completion $(\widehat{K},v)$, we study the set of all possible extensions of v to $\widehat{K}(X)$. We show that any such extension is closely connected with the underlying subextension $(K(X)|K,v)$. The connections between these extensions are studied via minimal pairs, key polynomials, pseudo-Cauchy sequences, and implicit constant fields. As a consequence, we obtain strong ramification theoretic properties of $(\widehat{K},v)$. We also give necessary and sufficient conditions for $(K(X),v)$ to be dense in $(\widehat{K}(X),v)$.     

The étale open topology over the fraction field of a Henselian local domain

Suppose that R is a local domain with fraction field K. If R is Henselian, then the R-adic topology over K refines the étale open topology. If R is regular, then the étale open topology over K refines the R-adic topology. In particular, the étale open topology over $L((t_1,\text{…},t_n))$ agrees with the $L[[t_1,\text{…},t_n]]$-adic topology for any field L and $n\ge 1$.     

Bounded sets structure of CpX and quasi‐(DF)‐spaces

For wide classes of locally convex spaces, in particular, for the space $C_p(X)$ of continuous real‐valued functions on a Tychonoff space X equipped with the pointwise topology, we characterize the existence of a fundamental bounded resolution (i.e., an increasing family of bounded sets indexed by the irrationals which swallows the bounded sets). These facts together with some results from Grothendieck's theory of $(DF)$‐spaces have led us to introduce quasi‐ $(DF)$‐spaces, a class of locally convex spaces containing $(DF)$‐spaces that preserves subspaces, countable direct sums and countable products. Regular $(LM)$‐spaces as well as their strong duals are quasi‐ $(DF)$‐spaces. Hence the space of distributions $D^{\prime }(\mathrm{\Omega })$ provides a concrete example of a quasi‐ $(DF)$‐space not being a $(DF)$‐space. We show that $C_p(X)$ has a fundamental bounded resolution if and only if $C_p(X)$ is a quasi‐ $(DF)$‐space if and only if the strong dual of $C_p(X)$ is a quasi‐ $(DF)$‐space if and only if X is countable. If X is metrizable, then $C_k(X)$ is a quasi‐ $(DF)$‐space if and only if X is a σ‐compact Polish space.     

Geodesics as products of one-parameter subgroups in compact lie groups and homogeneous spaces

We study the geodesic equation for compact Lie groups G and homogeneous spaces $G/H$, and we prove that the geodesics are orbits of products $\exp (t{X}_1)\cdots \exp (t{X}_N)$ of one-parameter subgroups of G, provided that a simple algebraic condition for the Riemannian metric is satisfied. For the group $SO(3)$, we relate this type of geodesics to the free motion of a symmetric top. Moreover, by using series of Lie subgroups of G, we construct a wealth of metrics having the aforementioned type of geodesics.     

Hausdorff operators on compact abelian groups

Necessary and sufficient conditions are given for the boundedness of Hausdorff operators on the generalized Hardy spaces $H_E^p(G)$, real Hardy space $H_{\mathbb{R}}^1(G)$, $\text{BMO}(G)$, and $\text{BMOA}(G)$ for compact Abelian group G. Surprisingly, these conditions turned out to be the same for all groups and spaces under consideration. Applications to Dirichlet series are given. The case of the space of continuous functions on G and examples are also considered.     

Periods of singular double octic Calabi–Yau threefolds and modular forms

By the modularity theorem, every rigid Calabi–Yau threefold X has associated modular form f such that the equality of L-functions $L(X,s)=L(f,s)$ holds. In this case, period integrals of X are expected to be expressible in terms of the special values $L(f,1)$ and $L(f,2)$. We propose a similar interpretation of period integrals of a nodal model of X. It is given in terms of certain variants of a Mellin transform of f. We provide numerical evidence toward this interpretation based on a case of double octics.     

On pointwise decay rates of time‐periodic solutions to the Navier–Stokes equation

We study the existence of a time‐periodic solution with pointwise decay properties to the Navier–Stokes equation in the whole space. We show that if the time‐periodic external force is sufficiently small in an appropriate sense, then there exists a time‐periodic solution $\{u,p\}$ of the Navier–Stokes equation such that $|{?}^j{u(t,x)|=O(|x|}^{1?n?j})$ and $|{?}^j{p(t,x)|=O(|x|}^{?n?j})(j=0,1,\dots )$ uniformly in $t\in \mathbb{R}$ as $|x|\to \infty$. Our solution decays faster than the time‐periodic Stokes fundamental solution and the faster decay of its spatial derivatives of higher order is also described.     

On the average degree of some irreducible characters of a finite group

Let G be a finite group and N be a non-trivial normal subgroup of G, such that the average degree of irreducible characters in $\mathrm{Irr}(G|N)$ is less than or equal to 16/5. Then, we prove that N is solvable. Also, we prove the solvability of G, by assuming that the average degree of irreducible characters in $\mathrm{Irr}(G|N)$ is strictly less than 16/5. We show that the bounds are sharp.     

Gauss-Prym maps on Enriques surfaces

We prove that the kth Gaussian map ${\gamma }_H^k$ is surjective on a polarized unnodal Enriques surface $(S,H)$ with $\phi (H)>2k+4$. In particular, as a consequence, when $\phi (H)>4(k+2)$, we obtain the surjectivity of the kth Gauss-Prym map ${\gamma }_{{\omega }_C\otimes \alpha }^k$, with $\alpha :={\omega }_{{S|}_C}$, on smooth hyperplane sections $C\in |H|$. In case $k=1$, it is sufficient to ask $\phi (H)>6$.     

Resolvent family for the evolution process with memory

In this paper, we investigate a class of the linear evolution process with memory in Banach space by a different approach. Suppose that the linear evolution process is well posed, we introduce a family pair of bounded linear operators, $\{(G(t),F(t)),t\ge 0\}$, that is, called the resolvent family for the linear evolution process with memory, the $F(t)$ is called the memory effect family. In this paper, we prove that the families $G(t)$ and $F(t)$ are exponentially bounded, and the family $(G(t),F(t))$ associate with an operator pair $(A,L)$ that is called generator of the resolvent family. Using $(A,L)$, we derive associated differential equation with memory and representation of $F(t)$ via L. These results give necessary conditions of the well-posed linear evolution process with memory. To apply the resolvent family to differential equation with memory, we present a generation theorem of the resolvent family under some restrictions on $(A,L)$. The obtained results can be directly applied to linear delay differential equation, integro-differential equation and functional differential equations.     

Rubio de Francía's weighted extrapolation in mixed-norm spaces and applications

Weighted extrapolation for pairs of functions in mixed-norm Banach function spaces defined on the product of quasi-metric measure spaces $(X,d,\mu )$ and $(Y,\rho ,\nu )$ are derived. As special cases, we have appropriate results for mixed-norm Lebesgue, Lorentz, and Orlicz spaces. Some of the derived results are applied to get weighted extrapolation in mixed-norm grand Lebesgue spaces.     

Weight generalization of the space of continuous functions vanishing at infinity

In this paper, we characterize the weighted generalization of the space of continuous functions vanishing at infinity and correct some wrong results in the paper. Let X be a locally compact space and ν is an arbitrary weight (non-negative function) on X. We give a correct and comprehensive definition of the weighted generalization $C_0^{\nu }(X)$ of $C_0(X)$, and show that it is a seminormed space with respect to the canonical seminorm ${\parallel f\parallel }_{\nu }={sup}_{x\in X}|f(x)|$, where $f\in C_0^{\nu }(X)$. We find conditions on ν under which $C_0^{\nu }(X)$, with respect to ${\parallel .\parallel }_{\nu }$, becomes a normed space or a Banach space or an algebra, or a topological algebra, respectively.