We show that if an open set in $\mathbb{R}^d$ can be fibered by unit n-spheres, then $d \geq 2n+1$, and if $d = 2n+1$, then the spheres must be pairwise linked, and $n \in \left\{0, 1, 3, 7 \right\}$. For these values of n, we construct unit n-sphere fibrations in $\mathbb{R}^{2n+1}$.

Motivated by new examples of functional Banach spaces over the unit disk, arising as the symbol spaces in the study of random analytic functions, for which the monomials $\{z^n\}_{n\geq 0}$ exhibit features of an unconditional basis yet they often don’t even form a Schauder basis, we introduce a notion called solid basis for Banach spaces and p-Banach spaces and study its properties. Besides justifying the rich existence of solid bases, we study their relationship with unconditional bases, the weak-star convergence of Taylor polynomials, the problem of a solid span and the curious roles played by c0. The two features of this work are as follows: (1) during the process, we are led to revisit the axioms satisfied by a typical Banach space of analytic functions over the unit disk, leading to a notion of $\mathcal{X}^\mathrm{max}$ (and $\mathcal{X}^\mathrm{min}$), as well as a number of related functorial constructions, which are of independent interests; (2) the main interests of solid basis lie in the case of non-separable (p-)Banach spaces, such as BMOA and the Bloch space instead of VMOA and the little Bloch space.

This paper first studies the multiplicity of normalized solutions to the non-autonomous Schrödinger equation with mixed nonlinearities

\begin{equation*}\begin{cases}-\Delta u=\lambda u+h(\epsilon x)|u|^{q-2}u+\eta |u|^{p-2}u,\quad x\in \mathbb{R}^N, \\\int_{\mathbb{R}^N}|u|^2\,\textrm{d}x=a^2,\end{cases}\end{equation*}

$a, \epsilon, \eta \gt 0$

$\lambda\in \mathbb{R}$

$p=2N/(N-2)$

Let $f\,:\,X\,\longrightarrow \,Y$ be a generically smooth morphism between irreducible smooth projective curves over an algebraically closed field of arbitrary characteristic. We prove that the vector bundle $((f_*{\mathcal O}_X)/{\mathcal O}_Y)^*$ is virtually globally generated. Moreover, $((f_*{\mathcal O}_X)/{\mathcal O}_Y)^*$ is ample if and only if f is genuinely ramified.

For $-1\leq B \lt A\leq 1$, let $\mathcal{C}(A,B)$ denote the class of normalized Janowski convex functions defined in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z| \lt 1\}$ that satisfy the subordination relation $1+zf''(z)/f'(z)\prec (1+Az)/(1+Bz)$. In the present article, we determine the sharp estimate of the Schwarzian norm for functions in the class $\mathcal{C}(A,B)$. The Dieudonné’s lemma which gives the exact region of variability for derivatives at a point of bounded functions, plays the key role in this study, and we also use this lemma to construct the extremal functions for the sharpness by a new method.

We obtain a new bound on the second moment of modified shifted convolutions of the generalized threefold divisor function and show that, for applications, the modified version is sufficient.

We show a result on propagation of the anisotropic Gabor wave front set for linear operators with a tempered distribution Schwartz kernel. The anisotropic Gabor wave front set is parametrized by a positive parameter relating the space and frequency variables. The anisotropic Gabor wave front set of the Schwartz kernel is assumed to satisfy a graph type criterion. The result is applied to a class of evolution equations that generalizes the Schrödinger equation for the free particle. The Laplacian is replaced by any partial differential operator with constant coefficients, real symbol and order at least two.

In dimension n = 1, we obtain $L^{p_1}(\mathbb R) \times\dots\times L^{p_m}(\mathbb R)$ to $L^p(\mathbb R)$ boundedness for the multilinear spherical maximal function in the largest possible open set of indices and we provide counterexamples that indicate the optimality of our results.

Given any commutative Noetherian ring R and an element x in R, we consider the full subcategory $\mathsf{C}(x)$ of its singularity category consisting of objects for which the morphism that is given by the multiplication by x is zero. Our main observation is that we can establish a relation between $\mathsf{C}(x), \mathsf{C}(y)$ and $\mathsf{C}(xy)$ for any two ring elements x and y. Utilizing this observation, we obtain a decomposition of the singularity category and consequently an upper bound on the dimension of the singularity category.

Let $f(z)=\sum\limits_{j=0}^{\infty} a_j z^j$ be a transcendental entire function and let $f_\omega(z)=\sum\limits_{j=0}^{\infty}\chi_j(\omega) a_j z^j$ be a random entire function, where $\chi_j(\omega)$ are independent and identically distributed random variables defined on a probability space $(\Omega, \mathcal{F}, \mu)$. In this paper, we first define a family of random entire functions, which includes Gaussian, Rademacher and Steinhaus entire functions. We prove that, for almost all functions in the family and for any constant C > 1, there exist a constant $r_0=r_0(\omega)$ and a set $E\subset [e, \infty)$ of finite logarithmic measure such that, for $r \gt r_0$ and $r\notin E$,

\begin{equation*}|\log M(r, f)- N(r,0, f_\omega)|\le (C/A)^{\frac1{B}}\,\log^{\frac1{B}}\,\log M(r,f) +\log\,\log M(r, f), \qquad a.s.\end{equation*}

where $A, B$ are constants, $M(r, f)$ is the maximum modulus and $N(r, 0, f)$ is the integrated zero-counting function of f. As a by-product of our main results, we prove Nevanlinna’s second main theorem for random entire functions. Thus, the characteristic function of almost all functions in the family is bounded above by an integrated counting function, rather than by two integrated counting functions as in the classical Nevanlinna theory. For instance, we show that, for almost all Gaussian entire functions fω and for any ϵ > 0, there is r0 such that, for $r \gt r_0$,

\begin{equation*}T(r, f) \le N(r,0, f_\omega)+\left(\tfrac12+\epsilon\right) \log T(r, f).\end{equation*}

In this note, we establish a boundary maximum principle for a class of stationary pairs of varifolds satisfying a fixed contact angle condition in any compact Riemannian manifold with smooth boundary.

Let Mn and Tn denote the nth Motzkin number and the nth central trinomial coefficient respectively. We prove that for any prime $p\ge 5$,

\begin{equation*}\begin{aligned}\\[-22pt]&\sum_{k=0}^{p-1}M_k^2\equiv \left(\frac{p}{3}\right)\left(2-6p\right)\pmod{p^2},\\&\sum_{k=0}^{p-1}kM_k^2\equiv \left(\frac{p}{3}\right)\left(9p-1\right)\pmod{p^2},\\&\sum_{k=0}^{p-1}T_kM_k\equiv \frac{4}{3}\left(\frac{p}{3}\right)+\frac{p}{6}\left(1-9\left(\frac{p}{3}\right)\right)\pmod{p^2},\\[-6pt]\end{aligned}\end{equation*}

where $\left(-\right)$ is the Legendre symbol. These results confirm three supercongruences conjectured by Z.-W. Sun in 2010.

Beardon and Minda gave a characterization of normal families of holomorphic and meromorphic functions in terms of a locally uniform Lipschitz condition. Here, we generalize this viewpoint to families of mappings in higher dimensions that are locally uniformly continuous with respect to a given modulus of continuity. Our main application is to the normality of families of quasiregular mappings through a locally uniform Hölder condition. This provides a unified framework in which to consider families of quasiregular mappings, both recovering known results of Miniowitz, Vuorinen and others and yielding new results. In particular, normal quasimeromorphic mappings, Yosida quasiregular mappings and Bloch quasiregular mappings can be viewed as classes of quasiregular mappings which arise through consideration of various metric spaces for the domain and range. We give several characterizations of these classes and obtain upper bounds on the rate of growth in each class.

In Communications in Contemporary Mathematics 24 3, (2022),the authors have developed a method for constructing G-invariant partial differential equations (PDEs) imposed on hypersurfaces of an $(n+1)$-dimensional homogeneous space $G/H$, under mild assumptions on the Lie group G. In the present paper, the method is applied to the case when $G=\mathsf{PGL}(n+1)$ (respectively, $G=\mathsf{Aff}(n+1)$) and the homogeneous space $G/H$ is the $(n+1)$-dimensional projective $\mathbb{P}^{n+1}$ (respectively, affine $\mathbb{A}^{n+1}$) space, respectively. The main result of the paper is that projectively or affinely invariant PDEs with n independent and one unknown variables are in one-to-one correspondence with invariant hypersurfaces of the space of trace-free cubic forms in n variables with respect to the group $\mathsf{CO}(d,n-d)$ of conformal transformations of $\mathbb{R}^{d,n-d}$.

We consider the broadest concept of the gradient Yamabe soliton, the conformal gradient soliton. In this paper, we elucidate the structure of complete gradient conformal solitons under some assumption, and provide some applications to gradient Yamabe solitons. These results enhance the understanding gained from previous research. Furthermore, we give an affirmative partial answer to the Yamabe soliton version of Perelman’s conjecture.

In this paper, we consider blowup of solutions to the Cauchy problem for the following biharmonic nonlinear Schrödinger equation (NLS),

\begin{equation*}\text{i } \partial_t u=\Delta^2 u-\mu \Delta u-|u|^{2 \sigma} u \quad \text{in} \,\, \mathbb{R} \times \mathbb{R}^d,\end{equation*}

where $d \geq 1$, $\mu \in \mathbb{R}$ and $0 \lt \sigma \lt \infty$ if $1 \leq d \leq 4$ and $0 \lt \sigma \lt 4/(d-4)$ if $d \geq 5$. In the mass critical and supercritical cases, we establish the existence of blowup solutions to the problem for cylindrically symmetric data. The result extends the known ones with respect to blowup of solutions to the problem for radially symmetric data.

A suitable notion of weak amenability for dual Banach algebras, which we call weak Connes amenability, is defined and studied. Among other things, it is proved that the measure algebra M(G) of a locally compact group G is always weakly Connes amenable. It can be a complement to Johnson’s theorem that $L^1(G)$ is always weakly amenable [10].

Let $X, Y$ be two locally compact Hausdorff spaces and $T:C_0(X)\rightarrow C_0(Y)$ be a standard surjective ɛ-norm-additive map, i.e.

\begin{equation*}\big|\|T(f)+T(g)\|-\|f+g\|\big|\leq \varepsilon,\;{\rm for\;all}\; f, g\in C_0(X).\end{equation*}

Then there exist a homeomorphism $\varphi:Y\rightarrow X$ and a continuous function $\lambda:Y\rightarrow\lbrace\pm1\rbrace$ such that

\begin{equation*}|T(f)(y)-\lambda(y)f(\varphi(y))|\leq\frac{3}{2}\varepsilon,\;{\rm for\;all}\;y\in Y,\;f\in C_0(X).\end{equation*}

The estimate ‘$\frac{3}{2}\varepsilon$’ is optimal. And this result can be regarded as a new nonlinear extension of the Banach–Stone theorem.

Let $H^{\infty}(\Omega,X)$ be the space of bounded analytic functions $f(z)=\sum_{n=0}^{\infty} x_{n}z^{n}$ from a proper simply connected domain Ω containing the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z| \lt 1\}$ into a complex Banach space X with $\left\lVert f\right\rVert_{H^{\infty}(\Omega,X)} \leq 1$. Let $\phi=\{\phi_{n}(r)\}_{n=0}^{\infty}$ with $\phi_{0}(r)\leq 1$ such that $\sum_{n=0}^{\infty} \phi_{n}(r)$ converges locally uniformly with respect to $r \in [0,1)$. For $1\leq p,q \lt \infty$, we denote

\begin{equation*}R_{p,q,\phi}(f,\Omega,X)= \sup \left\{r \geq 0: \left\lVert x_{0}\right\rVert^p \phi_{0}(r) + \left(\sum_{n=1}^{\infty} \left\lVert x_{n}\right\rVert\phi_{n}(r)\right)^q \leq \phi_{0}(r)\right\}\end{equation*}

\begin{equation*}R_{p,q,\phi}(\Omega,X)=\inf \left\{R_{p,q,\phi}(f,\Omega,X): \left\lVert f\right\rVert_{H^{\infty}(\Omega,X)} \leq 1\right\}.\end{equation*}

$R_{p,q,\phi}(\Omega,X)$

$X=\mathcal{B}(\mathcal{H})$

$\mathcal{H}$

We consider the radially symmetric positive solutions to quasilinear problem

\begin{equation*}-\triangle u-u\triangle u^{2}+\lambda u=f(u),\quad{\rm in} \ \mathbb{R}^{N},\end{equation*}

having prescribed mass $\int_{\mathbb{R}^{N}}|u|^2 =a^2,$ where a > 0 is a constant, λ appears as a Lagrange multiplier. We focus on the pure L2-supercritical case and combination case of L2-subcritical and L2-supercritical nonlinearities

\begin{equation*}f(u)=\tau |u|^{q-2}u+|u|^{p-2}u,\quad \tau \gt 0,\qquad{\rm where}\ \ 2 \lt q \lt 2+\frac{4}{N} \ {\rm and} \quad \ p \gt \bar{p},\end{equation*}

where $\bar{p}:=4+\frac{4}{N}$ is the L2-critical exponent. Our work extends and develops some recent results in the literature.