Given a Dirichlet form with generator $\mathcal{L}$ and a measure $\mu$, we consider superharmonic functions of the Schrödinger operator $\mathcal{L} + \mu$. We probabilistically prove that the existence of superharmonic functions gives rise to the Hardy inequality. More precisely, the $L^2$-Hardy inequality is derived from Itô's formula applied to the superharmonic function.

The derivatives of Nash functions are Nash functions which are derived algebraically from their minimal polynomial equations. In this paper we show that, for any non-Nash analytic function, it is impossible to derive its derivatives algebraically, i.e., by using linearity and Leibniz rule finite times. In fact we prove the impossibility of such kind of algebraic computations, algebraically by using Kähler differentials. Then the notion of Leibniz complexity of a Nash function is introduced in this paper, as a computational complexity on its derivative, by the minimal number of usages of Leibniz rules to compute the total differential algebraically. We provide general observations and upper estimates on Leibniz complexity of Nash functions, related to the binary expansions, the addition chain complexity, the non-scalar complexity and the complexity of Nash functions in the sense of Ramanakoraisina.

We compare various notions of weak subsolutions to degenerate complex Monge–Ampère equations, showing that they all coincide. This allows us to give an alternative proof of mixed Monge–Ampère inequalities due to Kołodziej and Dinew.

A small cover is a closed smooth manifold of dimension $n$ having a locally standard $\mathbb{Z}_2^n$-action whose orbit space is isomorphic to a simple polytope. In the paper, we classify small covers and real toric manifolds whose orbit space is isomorphic to the dual of the simplicial complex obtainable by a sequence of wedgings from a polygon, using a systematic combinatorial method of puzzles finding toric spaces.

We study the homological properties of random simplicial complexes. In particular, we obtain the asymptotic behavior of lifetime sums for a class of increasing random simplicial complexes; this result is a higher-dimensional counterpart of Frieze's $\zeta(3)$-limit theorem for the Erdős–Rényi graph process. The main results include solutions to questions posed in an earlier study by Hiraoka and Shirai about the Linial–Meshulam complex process and the random clique complex process. One of the key elements of the arguments is a new upper bound on the Betti numbers of general simplicial complexes in terms of the number of small eigenvalues of Laplacians on links. This bound can be regarded as a quantitative version of the cohomology vanishing theorem.

We study the formal neighborhoods at rational non-degenerate arcs of the arc scheme associated with a toric variety. The first main result of this article shows that these formal neighborhoods are generically constant on each Nash component of the variety. Furthermore, using our previous work, we attach to every such formal neighborhood, and in fact to every toric valuation, a minimal formal model (in the class of stable isomorphisms) which can be interpreted as a measure of the singularities of the base-variety. As a second main statement, for a large class of toric valuations, we compute the dimension and the embedding dimension of such minimal formal models, and we relate the latter to the Mather discrepancy. The class includes the strongly essential valuations, that is to say those the center of which is a divisor in the exceptional locus of every resolution of singularities of the variety. We also obtain a similar result for monomial curves.

We derive explicit formulas for the discriminants of classical quasi-orthogonal polynomials, as a full generalization of the result of Dilcher and Stolarsky (2005). We consider a certain system of Diophantine equations, originally designed by Hausdorff (1909) as a simplification of Hilbert's solution (1909) of Waring's problem, and then create the relationship to quadrature formulas and quasi-Hermite polynomials. We reduce these equations to the existence problem of rational points on a hyperelliptic curve associated with discriminants of quasi-Hermite polynomials, and show a nonexistence theorem for solutions of Hausdorff-type equations by applying our discriminant formula.

We study algebro-geometric consequences of the quantised extremal Kähler metrics, introduced in the previous work of the author. We prove that the existence of quantised extremal metrics implies weak relative Chow polystability. As a consequence, we obtain asymptotic weak relative Chow polystability and relative $K$-semistability of extremal manifolds by using quantised extremal metrics; this gives an alternative proof of the results of Mabuchi and Stoppa–Székelyhidi. In proving them, we further provide an explicit local density formula for the equivariant Riemann–Roch theorem.

We consider a $2m$th-order strongly elliptic operator $A$ subject to Dirichlet boundary conditions in a domain $\Omega$ of $\mathbb{R}^{n}$, and show the $L_{p}$ regularity theorem, assuming that the domain has less smooth boundary. We derive the regularity theorem from the following isomorphism theorems in Sobolev spaces. Let $k$ be a nonnegative integer. When $A$ is a divergence form elliptic operator, $A-\lambda$ has a bounded inverse from the Sobolev space $W^{k-m}_{p}(\Omega)$ into $W^{k+m}_{p}(\Omega)$ for $\lambda$ belonging to a suitable sectorial region of the complex plane, if $\Omega$ is a uniformly $C^{k,1}$ domain. When $A$ is a non-divergence form elliptic operator, $A-\lambda$ has a bounded inverse from $W^{k}_{p}(\Omega)$ into $W^{k+2m}_{p}(\Omega)$, if $\Omega$ is a uniformly $C^{k+m,1}$ domain. Compared with the known results, we weaken the smoothness assumption on the boundary of $\Omega$ by $m-1$.

We present an $L^2$-extension theorem with an estimate depending on the weight functions for domains in $\mathbb{C}$. When the Hartogs domain defined by the weight function is strictly pseudoconvex, this estimate is strictly sharper than known optimal estimates. When the weight function is radial, we prove that our estimate provides the $L^2$-minimum extension.

We introduce in this work a concept of rough driver that somehow provides a rough path-like analogue of an enriched object associated with time-dependent vector fields. We use the machinery of approximate flows to build the integration theory of rough drivers and prove well-posedness results for rough differential equations on flows and continuity of the solution flow as a function of the generating rough driver. We show that the theory of semimartingale stochastic flows developed in the 80's and early 90's fits nicely in this framework, and obtain as a consequence some strong approximation results for general semimartingale flows and provide a fresh look at large deviation theorems for ‘Gaussian’ stochastic flows.

A unified family of $P_{\mathrm{J}}$-hierarchies (J=I, II, IV, 34) with a large parameter is introduced and we construct general formal solutions which are called instanton-type solutions for the system.

For an odd prime number $p$, we give an explicit upper bound of $\lambda$-invariants for all $\mathbb{Z}_p$-extensions of an imaginary quadratic field $k$ under several assumptions. We also give an explicit upper bound of $\lambda$-invariants for all $\mathbb{Z}_p$-extensions of $k$ in the case where the $\lambda$-invariant of the cyclotomic $\mathbb{Z}_p$-extension of $k$ is equal to 3.