In the following we obtain the explicit conversion formulas between $v_i$’s and ${\lambda }_i$’s, the right unit images of $v_i$’s, and the $BP$-cohomology operation of $v_i$’s.

We describe a characterization for the Maaß space associated with the paramodular group of degree $2$ and square-free level $N$. As an application we show that the Maaß space is invariant under all Hecke operators. As a consequence, we conclude that the associated Siegel–Eisenstein series belongs to the Maaß space.

We study $(2,2)$ divisors in ${\mathbb{P}}^2\times {\mathbb{P}}^2$ giving rise to pairs of nonisomorphic, derived equivalent, and $\mathbb{L}$-equivalent K3 surfaces of degree $2$. In particular, we confirm the existence of such fourfolds as predicted recently by Kuznetsov and Shinder.

We study the stochastic nonlinear Schrödinger equations with additive stochastic forcing. By using the dispersive estimate, we present a simple argument, constructing a unique local-in-time solution with rougher stochastic forcing than those considered in the literature.

We give explicit descriptions of the adjoint group $Ad\left(Q_W\right)$ of the Coxeter quandle $Q_W$ associated with an arbitrary Coxeter group $W$. The adjoint group $Ad\left(Q_W\right)$ turns out to be an intermediate group between $W$ and the corresponding Artin group $A_W$, and it fits into a central extension of $W$ by a finitely generated free abelian group. We construct $2$-cocycles of $W$ corresponding to the central extension. In addition, we prove that the commutator subgroup of the adjoint group $Ad\left(Q_W\right)$ is isomorphic to the commutator subgroup of $W$. Finally, the root system ${\Phi }_W$ associated with a Coxeter group $W$ turns out to be a rack. We prove that the adjoint group $Ad\left({\Phi }_W\right)$ of ${\Phi }_W$ is isomorphic to the adjoint group of $Q_W$.

We analyze a family of models for a qubit interacting with a bosonic field. This family of models is very large and contains models where higher-order perturbations of field operators are added to the Hamiltonian. The Hamiltonian has a special symmetry, called spin-parity symmetry, which plays a central role in our analysis. Using this symmetry, we find the domain of self-adjointness and we decompose the Hamiltonian into two fiber operators each defined on Fock space. We then prove the Hunziker–van Winter–Zhislin (HVZ) theorem for the fiber operators, and we single out a particular fiber operator which has a ground state if and only if the full Hamiltonian has a ground state. From these results, we deduce a simple criterion for the existence of an excited state.

We consider the nonrelativistic limit of a semilinear field equation in a homogeneous and isotropic space, where the scale function of the space is constructed based on the Einstein equations. Furthermore, we examine the Cauchy problem for the limit equation and show the existence of global and blowup solutions in Sobolev spaces. Finally, we study the effects of spatial variance on the problem and remark on some dissipative and antidissipative properties of the limit equation.

Consider $E$ a CM elliptic curve over $\mathbb{Q}$. Assume that ${rank}_{\mathbb{Q}}E\ge 1$, and let $a\in E\left(\mathbb{Q}\right)$ be a point of infinite order. For $p$ a rational prime, we denote by ${\mathbb{F}}_p$ the residue field at $p$. If $E$ has good reduction at $p$, let $\overline{E}$ be the reduction of $E$ at $p$, let $\overline{a}$ be the reduction of $a(modulop$), and let $\langle \overline{a}\rangle$ be the subgroup of $\overline{E}\left({\mathbb{F}}_p\right)$ generated by $\overline{a}$. Assume that $\mathbb{Q}\left(E\right[2\left]\right)=\mathbb{Q}$ and $\mathbb{Q}\left(E\right[2],2^{-1}a)\ne \mathbb{Q}$. Then in this article we obtain an asymptotic formula for the number of rational primes $p$, with $p\le x$, for which $\overline{E}\left({\mathbb{F}}_p\right)/\langle \overline{a}\rangle$ is cyclic, and we prove that the number of primes $p$, for which $\overline{E}\left({\mathbb{F}}_p\right)/\langle \overline{a}\rangle$ is cyclic, is infinite. This result is a generalization of the classical Artin’s primitive root conjecture, in the context of CM elliptic curves; that is, this result is an unconditional proof of Artin’s primitive root conjecture for CM elliptic curves. Artin’s conjecture states that, for any integer $a\ne \pm 1$ or a perfect square (or equivalently $a\ne \pm 1$, and $\mathbb{Q}(\pm 1,\sqrt{a})=\mathbb{Q}\left(1\right[2],2^{-1}a)\ne \mathbb{Q}$), there are infinitely many primes $p$ for which $a$ is a primitive root (mod $p$), and an asymptotic formula for such primes is satisfied (this conjecture is not known for any specific $a$).

We study $K$-types of degenerate principal series of $Sp(n,\mathbb{C})$ by using two realizations of these infinite-dimensional representations. The first model we use is the classical compact picture; the second model is conjugate to the noncompact picture via an appropriate partial Fourier transform. In the first case, we find a family of $K$-finite vectors that can be expressed as solutions of specific hypergeometric differential equations; the second case leads to a family of $K$-finite vectors whose expressions involve Bessel functions.

We consider the Cauchy problem for first-order systems. Assuming that the set $\Sigma$ of singular points of the characteristic variety is a smooth manifold and the characteristic values are real and semisimple, we introduce a new class which is strictly hyperbolic in the directions transverse to $\Sigma$. If the propagation cone and $\Sigma$ are compatible, we prove, under some additional conditions, that transversally strictly hyperbolic systems are strongly hyperbolic. On the other hand, if the propagation cone is incompatible with $\Sigma$, then transversally strictly hyperbolic systems are much more involved, which is discussed utilizing an interesting example.

In this article, we construct a family of genus $2$ Lefschetz fibrations $f_n:X_{{\theta }_n}\to {\mathbb{S}}^2$ with $e\left(X_{{\theta }_n}\right)=11$, $b_2^{+}\left(X_{{\theta }_n}\right)=1$, and $c_1^2\left(X_{{\theta }_n}\right)=1$ by applying a single lantern substitution to the twisted fiber sums of Matsumoto’s genus $2$ Lefschetz fibration over ${\mathbb{S}}^2$. Moreover, we compute the fundamental group of $X_{{\theta }_n}$ and show that it is isomorphic to the trivial group if $n=-3$ or $-1$, $\mathbb{Z}$ if $n=-2$, and ${\mathbb{Z}}_{|n+2|}$ for all integers $n\ne -3,-2,-1$. Also, we prove that our fibrations admit $-2$ section, that their total spaces are symplectically minimal, and that they have symplectic Kodaira dimension $\kappa =2$. In addition, using techniques developed over the past decade with other authors, we also construct the genus $2$ Lefschetz fibrations over ${\mathbb{S}}^2$ with $c_1^2=1,2$ and $\chi =1$ via the fiber sums of Matsumoto’s and Xiao’s genus $2$ Lefschetz fibrations, and present some applications in constructing exotic smooth structures on small $4$-manifolds with $b_2^{+}=1$ and $b_2^{+}=3$.

We prove Fujita-type basepoint-freeness for projective quasilog canonical curves and surfaces.