On Fano and weak Fano Bott–Samelson–Demazure–Hansen varieties

Let G be a simple algebraic group over the field of complex numbers. Fix a maximal torus T and a Borel subgroup B of G containing T. Let w be an element of the Weyl group W of G, and let $Z(\stackrel{?}{w})$ be the Bott–Samelson–Demazure–Hansen (BSDH) variety corresponding to a reduced expression $\stackrel{?}{w}$ of w with respect to the data $(G,B,T)$.In this article we give complete characterization of the expressions $\stackrel{?}{w}$ such that the corresponding BSDH variety $Z(\stackrel{?}{w})$ is Fano or weak Fano. As a consequence we prove vanishing theorems of the cohomology of tangent bundle of certain BSDH varieties and hence we get some local rigidity results.     

Upper and lower solutions and quasilinearization for a class of second order singular nonlinear differential equations with nonlinear boundary conditions

We consider the differential equation $-(1/w)({\mathit{pu}}^{\prime }{)}^{\prime }=f(·,u)$, where $f$ is a nonlinear function, with nonlinear boundary conditions. Under appropriate assumptions on $p,w,f$ and the boundary conditions, the existence of solutions is established. If the problem has a lower solution and an upper solution, then we use a quasilinearization method to obtain two monotonic sequences of approximate solutions converging quadratically to a solution of the equation.     

Toeplitz quantization on Fock space

For Toeplitz operators $T_f^{(t)}$ acting on the weighted Fock space $H_t^2$, we consider the semi-commutator $T_f^{(t)}{T}_g^{(t)}?T_{fg}^{(t)}$, where $t>0$ is a certain weight parameter that may be interpreted as Planck's constant ? in Rieffel's deformation quantization. In particular, we are interested in the semi-classical limit

($?$)$\underset{t\to 0}{\lim }?{6T_f^{(t)}{T}_g^{(t)}?T_{fg}^{(t)}6}_t.$

It is well-known that ${6T_f^{(t)}{T}_g^{(t)}?T_{fg}^{(t)}6}_t$ tends to 0 under certain smoothness assumptions imposed on f and g. This result was recently extended to $f,g\in \mathrm{BUC}({\mathbb{C}}^n)$ by Bauer and Coburn. We now further generalize (?) to (not necessarily bounded) uniformly continuous functions and symbols in the algebra $\mathrm{VMO}\cap L^{\infty }$ of bounded functions having vanishing mean oscillation on ${\mathbb{C}}^n$. Our approach is based on the algebraic identity $T_f^{(t)}{T}_g^{(t)}?T_{fg}^{(t)}=?{(H_{\overline{f}}^{(t)})}^{?}{H}_g^{(t)}$, where $H_g^{(t)}$ denotes the Hankel operator corresponding to the symbol g, and norm estimates in terms of the (weighted) heat transform. As a consequence, only f (or likewise only g) has to be contained in one of the above classes for (?) to vanish. For g we only have to impose ${\lim \sup }_{t\to 0}{6H_g^{(t)}6}_t<\infty$, e.g. $g\in L^{\infty }({\mathbb{C}}^n)$. We prove that the set of all symbols $f\in L^{\infty }({\mathbb{C}}^n)$ with the property that ${\lim }_{t\to 0}?{6T_f^{(t)}{T}_g^{(t)}?T_{fg}^{(t)}6}_t={\lim }_{t\to 0}?{6T_g^{(t)}{T}_f^{(t)}?T_{gf}^{(t)}6}_t=0$ for all $g\in L^{\infty }({\mathbb{C}}^n)$ coincides with $\mathrm{VMO}\cap L^{\infty }$. Additionally, we show that $\underset{t\to 0}{\lim }?{6T_f^{(t)}6}_t={6f6}_{\infty }$ holds for all $f\in L^{\infty }({\mathbb{C}}^n)$. Finally, we present new examples, including bounded smooth functions, where (?) does not vanish.     

Pairs of quadratic forms over a quadratic field extension

Let F be a field of characteristic distinct from 2, $L=F(\sqrt{d})$ a quadratic field extension. Let further f and g be quadratic forms over L considered as polynomials in n variables, $M_f$, $M_g$ their matrices. We say that the pair $(f,g)$ is a k-pair if there exist $S\in G{L}_n(L)$ such that all the entries of the $k\times k$ upper-left corner of the matrices $S{M}_f{S}^t$ and $S{M}_g{S}^t$ are in F. We give certain criteria to determine whether a given pair $(f,g)$ is a k-pair. We consider the transfer ${\mathrm{cor}}_{L(t)/F(t)}$ determined by the $F(t)$-linear map $s:L(t)\to F(t)$ with $s(1)=0$, $s(\sqrt{d})=1$, and prove that if $\dim {\mathrm{cor}}_{L(t)/F(t)}{(f+tg)}_{an}\le 2(n?k)$, then $(f,g)$ is a $[\frac{k+1}{2}]$-pair. If, additionally, the form $f+tg$ does not have a totally isotropic subspace of dimension $p+1$ over $L(t)$, we show that $(f,g)$ is a $(k?2p)$-pair. In particular, if the form $f+tg$ is anisotropic, and $\dim {\mathrm{cor}}_{L(t)/F(t)}{(f+tg)}_{an}\le 2(n?k)$, then $(f,g)$ is a k-pair.     

Sampling in a weighted Sobolev space

We show that functions f in some weighted Sobolev space are completely determined by time-frequency samples ${\{f(t_n)\}}_{n\in \mathbb{Z}}\cup {\{\stackrel{?}{f}({\lambda }_k)\}}_{k\in \mathbb{Z}}$ along appropriate slowly increasing sequences ${\{t_n\}}_{n\in \mathbb{Z}}$ and ${\{{\lambda }_n\}}_{n\in \mathbb{Z}}$ tending to ±∞ as $n\to \pm \infty$.     

Dedekind–Mertens lemma and content formulas in power series rings

Let $R?X?$ be the power series ring over a commutative ring R with identity. For $f\in R?X?$, let $A_f$ denote the content ideal of f, i.e., the ideal of R generated by the coefficients of f. We show that if R is a Prüfer domain and if $g\in R?X?$ such that $A_g$ is locally finitely generated (or equivalently locally principal), then a Dedekind–Mertens type formula holds for g, namely $A_f^2{A}_g=A_f{A}_{fg}$ for all $f\in R?X?$. More generally for a Prüfer domain R, we prove the content formula ${(A_f{A}_g)}^2=(A_f{A}_g)A_{fg}$ for all $f,g\in R?X?$. As a consequence it is shown that an integral domain R is completely integrally closed if and only if ${(A_f{A}_g)}_v={(A_{fg})}_v$ for all nonzero $f,g\in R?X?$, which is a beautiful result corresponding to the well-known fact that an integral domain R is integrally closed if and only if ${(A_f{A}_g)}_v={(A_{fg})}_v$ for all nonzero $f,g\in R[X]$, where $R[X]$ is the polynomial ring over R.For a ring R and $g\in R?X?$, if $A_g$ is not locally finitely generated, then there may be no positive integer k such that $A_f^{k+1}{A}_g=A_f^k{A}_{fg}$ for all $f\in R?X?$. Assuming that the locally minimal number of generators of $A_g$ is $k+1$, Epstein and Shapiro posed a question about the validation of the formula $A_f^{k+1}{A}_g=A_f^k{A}_{fg}$ for all $f\in R?X?$. We give a negative answer to this question and show that the finiteness of the locally minimal number of special generators of $A_g$ is in fact a more suitable assumption. More precisely we prove that if the locally minimal number of special generators of $A_g$ is $k+1$, then $A_f^{k+1}{A}_g=A_f^k{A}_{fg}$ for all $f\in R?X?$. As a consequence we show that if $A_g$ is finitely generated (in particular if $g\in R[X]$), then there exists a nonnegative integer k such that $A_f^{k+1}{A}_g=A_f^k{A}_{fg}$ for all $f\in R?X?$.