Let $L$ be an ample line bundle over a smooth complex projective $n$-fold $X$ such that $K_X+L$ is nef. We prove that $H^0(X, \mathcal{O}_X(m(K_X+L)))\ne 0$ holds for every $m\geq n(n+1)/2+2$.

We consider the leafwise Brownian motion on a mapping torus of a homeomorphism on a compact metric space. We prove a central limit theorem for a class of additive functionals of the leafwise Brownian motion starting at any quasi-regular point and apply it to the leafwise Brownian motions on generalized Kronecker foliations. As an auxiliary result, we also give an elementary proof of the characterization of harmonic measures for the leafwise Brownian motion on the mapping torus.

Many researchers have investigated sums of products of Bernoulli numbers and related types of sums of products. Recently, the concept of poly-Cauchy numbers has been introduced by the author as a generalization of the classical Cauchy number. In this paper, we investigate sums of products of Cauchy numbers including generalized poly-Cauchy numbers. We give a relation among these sums and explicit expressions of sums of any arbitrary products. In addition, we also show the similar results in the case of poly-Cauchy numbers of the second kind and those of mixed ones of both kinds.

We consider second order linear elliptic equations $ - \div (A(x) \nabla u) + \mathbf{b}(x) \cdot \nabla u = 0$ with a singular vector field $\mathbf{b}$. We prove a refined subsolution estimate, which contains a precise dependence of the quantities of $\mathbf{b}$, for weak subsolutions and a weak Harnack inequality for weak supersolutions under certain assumptions on $\mathbf{b}$.

We define relatively quasiconvex subgroups of relatively hyperbolic groups in the sense of Osin and show that such subgroups have expected properties. Also we state several definitions equivalent to the definition of relatively hyperbolic groups in the sense of Osin.

This paper purposes to characterize Noetherian local rings $(A,\mathfrak{m})$ of positive dimension such that the first Hilbert coefficients of $\mathfrak{m}$-primary ideals in $A$ range among only finitely many values. Examples are explored to illustrate our theorems.

We establish a characterization of dualizing modules among semidualizing modules. Let $R$ be a finite dimensional commutative Noetherian ring with identity and $C$ a semidualizing $R$-module. We show that $C$ is a dualizing $R$-module if and only if $\mathrm{Tor}_i^R(E,E')$ is $C$-injective for all $C$-injective $R$-modules $E$ and $E'$ and all $i\geq 0$.

We generalize Toponogov's maximal diameter sphere theorem from the radial curvature geometry's standpoint. As a corollary to our main theorem, we prove that for a complete connected Riemannian $n$-manifold $M$ having radial sectional curvature at a point bounded from below by the radial curvature function of an ellipsoid of prolate type, the diameter of $M$ does not exceed the diameter of the ellipsoid. Furthermore if the diameter of such an $M$ equals that of the ellipsoid, then $M$ is isometric to the $n$-dimensional ellipsoid of revolution.

In this paper, we consider the combinatorial formula for the Schur coefficients of the integral form of the Macdonald polynomials. As an attempt to prove Haglund's conjecture that $\Biggl<\frac{J_{\mu}(X;q,q^k)}{(1-q)^{|\mu|}},s_{\lambda}(X)\Biggr> \in \mathbb{N}[q]$, we have found explicit combinatorial formulas for the Schur coefficients in one row case, two column case and certain hook shape cases [Yoo12]. A result of Egge-Loehr-Warrington [ELW] gives a combinatorial way of getting Schur expansion of symmetric functions when the expansion of the function in terms of Gessel's fundamental quasi symmetric functions is known. We apply this result to the combinatorial formula for the integral form Macdonald polynomials of Haglund [Hag] in quasi symmetric functions to prove the Haglund's conjecture in more general cases.

We shall give a characterization of the discrete Radon transform of functions in the Schwartz space on $\mathbb{Z}^n$ and obtain various inversion formulas for the discrete Radon transform on $\mathbb{Z}^2$.

We study the product Hausdorff operator $H_{\Phi}$ on the product Hardy spaces, and prove that, for a nonnegative valued function $\Phi$, $H_{\Phi}$ is bounded on the product Hardy space $H^{1}(\mathbb{R}\times \mathbb{R})$ if and only if $\Phi$ is a Lebesgue integrable function on $(0,\infty)\times (0,\infty)$. As an application, we know that the product Hardy operator $\mathcal{H}$ is not bounded on $H^{1}(\mathbb{R}\times \mathbb{R})$. On the other hand, we prove that $\left\Vert \mathcal{H}f\ \right\Vert_{H^{1}(\mathbb{R}\times \mathbb{R} )}$\ $\preceq \left\Vert f\ \right\Vert_{H^{1}(\mathbb{R}\times \mathbb{R} )}$ if $f$ is an even function. Furthermore, using the $H^{p}(\mathbb{R}\times \mathbb{R})$ boundedness criterion of Fefferman, we prove that the $k$-th order Hardy operator is bounded on $H^{p}(\mathbb{R}\times \mathbb{R})$ whenever $k>1/p-1$.

Let $R$ be an $n$-dimensional Cohen-Macaulay local ring and $Q$ a parameter ideal of $R$. Suppose that an acyclic complex $(F_{\bullet}, \varphi_{\bullet})$ of length $n$ of finitely generated free $R$-modules is given. We put $M = {\rm Im} \, \varphi_{1}$, which is an $R$-submodule of $F_{0}$. Then $F_{\bullet}$ is an $R$-free resolution of $F_{0}/M$. In this paper, we describe a concrete procedure to get an acyclic complex $^\ast\!{F_{\bullet}}$ of length $n$ that resolves $F_{0}/(M :_{F_{0}} Q)$.

We prove that braided semigroups with suitable conditions can produce solutions to the quantum Yang-Baxter equation in every tensor category. As an application, some dynamical Yang-Baxter maps, set-theoretic solutions to a version of the quantum dynamical Yang-Baxter equation, are constructed.

We generalize Toponogov's maximal diameter sphere theorem from the radial curvature geometry's standpoint. As a corollary to our main theorem, we prove that for a complete connected Riemannian $n$-manifold $M$ having radial sectional curvature at a point bounded from below by the radial curvature function of an ellipsoid of prolate type, the diameter of $M$ does not exceed the diameter of the ellipsoid. Furthermore if the diameter of such an $M$ equals that of the ellipsoid, then $M$ is isometric to the $n$-dimensional ellipsoid of revolution.

A one-dimensional diffusion process with a Brownian potential including a zero potential part is studied. The maximum process and the minimum process of the diffusion process are also investigated.

Based on the fact that projective monomial curves in the plane are complete intersections, we give an effective inductive method for creating infinitely many monomial curves in the projective $n$-space that are set theoretic complete intersections. We illustrate our main result by giving different infinite families of examples. Our proof is constructive and provides one binomial and $(n-2)$ polynomial explicit equations for the hypersurfaces cutting out the curve in question.

In this paper we characterize behavior of the sequence of norm of primitives of functions in Orlicz spaces by its spectrum (the support of its Fourier transform).