Let $(\pi ,\mathcal{H})$ be a continuous unitary representation of the (infinite-dimen- sional) Lie group $G$, and let $\gamma :\mathbb{R}\to Aut\left(G\right)$ be a group homomorphism which defines a continuous action of $\mathbb{R}$ on $G$ by Lie group automorphisms. Let ${\pi }^{\#}(g,t)=\pi \left(g\right)U_t$ be a continuous unitary representation of the semidirect product group $G{⋊}_{\gamma }\mathbb{R}$ on $\mathcal{H}$. The first main theorem of the present note provides criteria for the invariance of the space ${\mathcal{H}}^{\infty }$ of smooth vectors of $\pi$ under the operators $U_f={\int }_{\mathbb{R}}f\left(t\right)U_{t}dt$ for $f\in L^1\left(\mathbb{R}\right)$ and $f\in \mathcal{S}\left(\mathbb{R}\right)$, respectively. When $\mathfrak{g}$ is complete and the actions of $\mathbb{R}$ on $G$ and $\mathfrak{g}$ are continuous, we use the above theorem to show that, for suitably defined spectral subspaces ${\mathfrak{g}}_{\mathbb{C}}\left(E\right)$, $E\subseteq \mathbb{R}$, in the complexified Lie algebra ${\mathfrak{g}}_{\mathbb{C}}$ and ${\mathcal{H}}^{\infty }\left(F\right)$, $F\subseteq \mathbb{R}$, for $U_t$ in ${\mathcal{H}}^{\infty }$, we have

$$\mathrm{d}\pi \left({\mathfrak{g}}_{\mathbb{C}}\right(E\left)\right){\mathcal{H}}^{\infty }\left(F\right)\subseteq {\mathcal{H}}^{\infty }(E+F).$$

We show that the graded maximal ideal of a graded $K$-algebra $R$ has linear quotients for a suitable choice and order of its generators if the defining ideal of $R$ has a quadratic Gröbner basis with respect to the reverse lexicographic order, and we show that this linear quotient property for algebras defined by binomial edge ideals characterizes closed graphs. Furthermore, for algebras defined by binomial edge ideals attached to a closed graph and for join-meet rings attached to a finite distributive lattice we present explicit Koszul filtrations.

In this paper, we prove a version of the arithmetic Bertini theorem asserting that there exists a strictly small and generically smooth section of a given arithmetically free graded arithmetic linear series.

For topological semigroups $S$, we consider Tannaka-type duality theorems, which are extensions of the notion of weak Tannaka duality theorem for topological groups. In the case of topological semigroups, we must set as the dual object of $S$ all isometric representations of $S$ instead of all unitary representations. We define a property T-type for $S$. After arguments analogous to previous work from the author, we can prove that our Tannaka-type duality theorem is valid if and only if $S$ is a T-type semigroup.

On a general filtered probability space $(\Omega ,\mathcal{F},({\mathcal{F}}_t,t\in [0,1]),P)$, for a given signal $U_t=B_t+{\int }_0^t{\dot{u}}_{s}ds$, where $(B_t,t\in [0,1\left]\right)$ is a Brownian motion and $\dot{u}$ is adapted and in $L^2(dt\times dP)$, we prove that the filtration of $U$, denoted $({\mathcal{U}}_t,t\in [0,1\left]\right)$, is equal to the filtration of its innovation process $Z$, which is defined as $Z_t=U_t-{\int }_0^t{E}_P[{\dot{u}}_s\mid {\mathcal{U}}_s]ds$, $t\in [0,1]$, if and only if $$H\left(Z\right(\nu )\mid \mu )=\frac{1}{2}{E}_{\nu }\left[{\int }_0^1\right|E_P[{\dot{u}}_s\mid {\mathcal{U}}_s]{|}^{2}ds]$$ where $d\nu =exp(-{\int }_0^1{E}_P[{\dot{u}}_s\mid {\mathcal{U}}_s]d{Z}_s-\frac{1}{2}{\int }_0^1|E_P[{\dot{u}}_s\mid {\mathcal{U}}_s]{|}^{2}ds)dP$ in the case in which the density has expectation $1$; otherwise, we give a localized version of the same strength with a sequence of stopping times of the filtration of $U$.

This paper is devoted to the study of the relation between Osserman algebraic curvature tensors and algebraic curvature tensors which satisfy the duality principle. We give a short overview of the duality principle in Osserman manifolds and extend this notion to null vectors. Here, it is proved that a Lorentzian totally Jacobi-dual curvature tensor is a real space form. Also, we find out that a Clifford curvature tensor is Jacobi-dual. We provide a few examples of Osserman manifolds which are totally Jacobi-dual and an example of an Osserman manifold which is not totally Jacobi-dual.

In this paper, we consider terminal reflexive polytopes arising from finite directed graphs and study the problem of deciding which directed graphs yield smooth Fano polytopes (SFPs). We show that any centrally symmetric or pseudosymmetric SFPs can be obtained from directed graphs. Moreover, by using directed graphs, we provide new examples of SFPs whose corresponding varieties admit Kähler–Einstein metrics.

We consider a semilinear Dirichlet problem driven by the Laplacian and with an indefinite (that is, sign-changing) weight and a nonlinearity which is asymptotically linear near $\pm \infty$. Using variational methods together with truncation techniques and Morse theory, we show that the problem has at least three nontrivial solutions, two of which have constant sign (one positive and the other negative).

We apply Ueda theory to a study of singular Hermitian metrics of a (strictly) nef line bundle $L$. Especially we study minimal singular metrics of $L$, metrics of $L$ with the mildest singularities among singular Hermitian metrics of $L$ whose local weights are plurisubharmonic. In some situations, we determine a minimal singular metric of $L$. As an application, we give new examples of (strictly) nef line bundles which admit no smooth Hermitian metric with semipositive curvature.

In this short note we prove that a saturated fusion system admitting some special type of automorphism is nilpotent. This generalizes classical results by J. G. Thompson.

We shall prove that the map $H_i\left({\mathrm{SO}}_n\right(\mathbb{K}),\mathbb{Z})\to H_i\left({\mathrm{SO}}_{n+1}\right(\mathbb{K}),\mathbb{Z})$ is bijective for $2i<n$ and surjective for $2i\le n$. Here $\mathbb{K}$ is an arbitrary Pythagorean field and the special orthogonal group ${\mathrm{SO}}_n\left(\mathbb{K}\right)$ is the subgroup of $\mathbb{K}$-linear automorphisms over ${\mathbb{K}}^n$ with determinant one which preserve the Euclidean quadratic form $\mathbf{q}\left(x\right)=x_1^2+\cdots +x_n^2$. It is derived from the homological stability of the orthogonal groups ${\mathrm{O}}_n\left(\mathbb{K}\right)$ with twisted coefficients ${\mathbb{Z}}^t$.

Let $f\in S_{2\kappa -2}\left({\Gamma }_0\right(M\left)\right)$ be a Hecke eigenform with $\kappa \ge 2$ even and $M\ge 1$ and odd and square-free. In this paper we survey the construction of the Saito–Kurokawa lifting from the classical point of view. We also provide some arithmetic results on the Fourier coefficients of Saito–Kurokawa liftings. We then calculate the norm of the Saito–Kurokawa lift.

Let $R$ be an affine $k$-domain over the field $k$. The paper’s main result is that if $R$ admits a nontrivial embedding in a polynomial ring $K\left[s\right]$ for some field $K$ containing $k$, then $R$ can be embedded in a polynomial ring $F\left[t\right]$ which extends $R$ algebraically. This theorem can be applied to subrings of a ring which admits a nonzero locally nilpotent derivation. In this way, we obtain a concise new proof of the cancellation theorem for rings of transcendence degree one for fields of characteristic $0$.

In this paper, using arcsine variables, we get a new elementary proof of $\zeta \left(2\right)={\pi }^2/6$, known as the Basel problem, and the Euler formula. Using exponential variables, we get an Euler-like formula. We can also solve the Basel problem by using Wigner’s semicircle law and the Legendre generating function.