Let $u(t,x)$ be a solution of the heat equation in ${\mathbb{R}}^n$. Then its kth derivative also solves the heat equation and satisfies a maximum principle: the largest kth derivative of $u(t,x)$ cannot be larger than the largest kth derivative of $u(0,x)$. We prove an analogous statement for the solution of the heat equation on bounded domains $\mathrm{\Omega }\subset {\mathbb{R}}^n$ with Dirichlet boundary conditions. As an application, we give a new and fairly elementary proof of the sharp growth of the second derivatives of Laplacian eigenfunction $-\mathrm{\Delta }{\mathit{\varphi }}_k={\mathit{\lambda }}_k{\mathit{\varphi }}_k$ with Dirichlet conditions on smooth domains $\mathrm{\Omega }\subset {\mathbb{R}}^n$.

Fix a standard measurable space $(X,\mathcal{X})$ and an internal, hyperfinite $H{\subset }^{\ast }X$ that contains each $x\in X$. All finitely additive probabilities on $(X,\mathcal{X})$ can be represented by setting $p(B)$ equal to the standard part of $Q{(}^{\ast }B\cap H)$ for some internal probability Q supported on H. From this starting point, we have the following: a decomposition of two ways in which a probability can fail to be countably additive, a nonstandard characterization of countable additivity for probabilities on complete separable metic spaces, and results on the multiplicity of hyperfinitely supported probabilities that represent a given finitely additive p, from which we have set-valued integrals with respect to products of finitely additive probabilities that respect statistical independence, and for subfields or sub-σ-fields of $\mathcal{X}$, a proper ${}^{\ast }X$-based disintegration of finitely additive probabilities as a countably additive integral over the set of finitely additive probabilities. We end with several applications for which the alternative Stone space approach to representing finitely additive probabilities is an impediment to the analyses.

We give an expression of the Lefschetz number of iterates of the Verdier monodormy associated to an ideal on a complex algebraic variety in terms of the Euler characteristic of a space of truncated arcs. This extends the result of Denef and Loeser in the case of principal ideal on a smooth variety, and motivates a definition of motivic Milnor fiber for ideals. A discussion of the monodromy conjecture for motivic zeta functions of torus-invariant ideals on toric varieties is also included.

Given a convex domain and a convex subdomain, we prove a variant of domain monotonicity for the Neumann eigenvalues of the Laplacian. As an application of our method, we also obtain an upper bound for Neumann eigenvalues of the Laplacian of a convex domain.

In this paper, we prove a limiting absorption principle for high-order Schrödinger operators with a large class of potentials that generalize some results by Ionescu and Schlag. Two key tools we use in this paper are the Stein–Tomas theorem in Lorentz spaces and a sharp trace lemma given by Agmon and Hörmander.

In this paper, we prove that a critical metric of the volume functional with pinched Weyl curvature is isometric to a geodesic ball in ${\mathbb{S}}^n$. Moreover, we provide a necessary and sufficient condition on the norm of the gradient of the potential function in order to classify such critical metrics.

For a left action $S\stackrel{\mathit{\lambda }}{\curvearrowright }X$ of a cancellative right amenable monoid S on a discrete Abelian group X, we construct its Ore localization $G\stackrel{{\mathit{\lambda }}^{\ast }}{\curvearrowright }{X}^{\ast }$, where G is the group of left fractions of S; analogously, for a right action $K\stackrel{\mathit{\rho }}{\curvearrowleft }S$ on a compact space K, we construct its Ore colocalization $K^{\ast }\stackrel{{\mathit{\rho }}^{\ast }}{\curvearrowleft }G$. Both constructions preserve entropy—that is, for the algebraic entropy $h_{\mathrm{alg}}$ and for the topological entropy $h_{\mathrm{top}}$ one has, respectively, $h_{\mathrm{alg}}(\mathit{\lambda })=h_{\mathrm{alg}}({\mathit{\lambda }}^{\ast })$ and $h_{\mathrm{top}}(\mathit{\rho })=h_{\mathrm{top}}({\mathit{\rho }}^{\ast })$. Exploiting these constructions and the theory of quasi-tilings, we extend the addition theorem for $h_{\mathrm{top}}$, known for right actions of countable amenable groups on compact metrizable groups, to right actions $K\stackrel{\mathit{\rho }}{\curvearrowleft }S$ of cancellative right amenable monoids S (with no restrictions on the cardinality) on arbitrary compact groups K.

When the compact group K is Abelian, we prove that $h_{\mathrm{top}}(\mathit{\rho })$ coincides with $h_{\mathrm{alg}}({\mathit{\rho }}^{\wedge })$, where $S\stackrel{{\mathit{\rho }}^{\wedge }}{\curvearrowright }X$ is the dual left action on the discrete Pontryagin dual $X=K^{\wedge }$ (that is, a so-called bridge theorem. From the addition theorem for $h_{\mathrm{top}}$ and the bridge theorem, we obtain an addition theorem for $h_{\mathrm{alg}}$ for left actions $S\stackrel{\mathit{\lambda }}{\curvearrowright }X$ on discrete Abelian groups, so far known only under the hypotheses that either X is torsion or S is locally monotileable. The proofs substantially use the unified approach toward entropy based on the entropy of actions of cancellative right amenable monoids on appropriately defined normed monoids.

In this paper, we study the variance of the number of squarefull numbers in short intervals. As a result, we are able to prove that, for any $0<\mathit{\theta }<1∕2$, almost all short intervals $(x,x+x^{1∕2+\mathit{\theta }}]$ contain about $\frac{\mathit{\zeta }(3∕2)}{2\mathit{\zeta }(3)}{x}^{\mathit{\theta }}$ squarefull numbers.