We give a functional analytical definition of the Maslov index for continuous curves in the Fredholm-Lagrangian Grassmannian. Our definition does not require assumptions either at the endpoints or at the crossings of the curve with the Maslov cycle. We demonstrate an application of our definition by developing the symplectic geometry of self-adjoint extensions of unbounded symmetric operators. We discuss continuous variations of the form $A_D+C_t$, where $A_D$ is a fixed self-adjoint unbounded Fredholm operator and $\{C_t\}$ a family of bounded self-adjoint operators. We extend the definition of the spectral flow to such families of unbounded operators in a purely functional analytical way. We then prove that the spectral flow is equal to the Maslov index of the corresponding family of abstract Cauchy data spaces.

In this paper we introduce a new concept of more weakly quadrant dependence of hitting times of stochastic processes. This concept is weaker than the more positively quadrant dependence and it is closed under some statistical operations of weakly positive quadrant dependence (WPQD) ordering.

In this paper, we show that the moduli space of the Weierstrass data for algebraic minimal surfaces in Euclidean 4-space with fixed topological type, orders of branched points and ends, and total curvature, has the structure of a real analytic variety. We provide the lower bounds of its dimension. We also show that the moduli space of the Weierstrass data for stable algebraic minimal surfaces in Euclidean 4-space has the structure of a complex analytic variety.

We say that a partial elementary map $f$ of a structure $M$ is normal if $f$ can be extended to an elementary map on $M$ whose domain or range is equal to $M$. In this paper, we investigate properties for normal elementary maps.

An elliptic curve $E$ defined over $\bar{\mathbf{Q}}$ is called a $\mathbf{Q}$-curve, if $E$ and $E^\sigma$ are isogenous over $\bar{\mathbf{Q}}$ for any $\sigma$ in $\mathrm{Gal}(\bar{\mathbf{Q}}/\mathbf{Q})$. For a real quadratic field $K$ and a prime number $p$, we consider a $\mathbf{Q}$-curve $E$ with the following properties: 1) $E$ is defined over $K$, 2) $E$ has everywhere good reduction over $K$, 3) there exists a $p$-isogeny between $E$ and its conjugate $E^\sigma$. In this paper, a method to construct such a $\mathbf{Q}$-curve $E$ for some $p$ will be given.

We show that a decomposing 2-sphere which gives an essential tangle decomposition of a doubled knot gives also an essential tangle decomposition of its companion knot.

The values of the elliptic modular $j$-invariant at imaginary quadratic arguments are algebraic integers, known as singular moduli of level one. If $d_1$ and $d_2$ are imaginary quadratic discriminants, then we may consider a generalized resultant of the class polynomials of the orders of discriminant $d_1$ and $d_2$; this is the norm of the differences of singular moduli of the corresponding orders, denoted here by $J(d_1,d_2)$. These resultants are highly factorizable; Gross-Zagier established a closed formula for $J(d_1,d_2)^2$ when $d_1$ and $d_2$ are fundamental discriminants, with $(d_1,d_2)=1$. In this paper we present a conjectural extension of the Gross-Zagier formula to the case when $d_1$ and $d_2$ are not necessarily fundamental, and $(d_1,d_2)=l^e$, where 1 is a prime not dividing the product of the conductors of $d_1$ and $d_2$.