Closure properties of parametric subcompleteness

For an ordinal \(\varepsilon \), I introduce a variant of the notion of subcompleteness of a forcing poset, which I call \(\varepsilon \)-subcompleteness, and show that this class of forcings enjoys some closure properties that the original class of subcomplete forcings does not seem to have: factors of \(\varepsilon \)-subcomplete forcings are \(\varepsilon \)-subcomplete, and if \(\mathbb {P}\) and \(\mathbb {Q}\) are forcing-equivalent notions, then \(\mathbb {P}\) is \(\varepsilon \)-subcomplete iff \(\mathbb {Q}\) is. I formulate a Two Step Theorem for \(\varepsilon \)-subcompleteness and prove an RCS iteration theorem for \(\varepsilon \)-subcompleteness which is slightly less restrictive than the original one, in that its formulation is more careful about the amount of collapsing necessary. Finally, I show that an adequate degree of \(\varepsilon \)-subcompleteness follows from the \(\kappa \)-distributivity of a forcing, for \(\kappa >\omega _1\).     

Asymptotic diophantine approximation: the multiplicative case

Let \(\alpha \) and \(\beta \) be irrational real numbers and \(0<\varepsilon <1/30\). We prove a precise estimate for the number of positive integers \(q\le Q\) that satisfy \(\Vert q\alpha \Vert \cdot \Vert q\beta \Vert <\varepsilon \). If we choose \(\varepsilon \) as a function of Q, we get asymptotics as Q gets large, provided \(\varepsilon Q\) grows quickly enough in terms of the (multiplicative) Diophantine type of \((\alpha ,\beta )\), e.g., if \((\alpha ,\beta )\) is a counterexample to Littlewood’s conjecture, then we only need that \(\varepsilon Q\) tends to infinity. Our result yields a new upper bound on sums of reciprocals of products of fractional parts and sheds some light on a recent question of Lê and Vaaler.     

Existence of self-Cheeger sets on Riemannian manifolds

Let \(({\mathcal M},g)\) be a smooth compact Riemannian manifold of dimension \(N\ge 2\). We prove the existence of a family \((\Omega _\varepsilon )_{\varepsilon \in (0,\varepsilon _0)}\) of self-Cheeger sets in \(({\mathcal M},g)\). The domains \(\Omega _\varepsilon \subset {\mathcal M}\) are perturbations of geodesic balls of radius \(\varepsilon \) centered at \(p \in {\mathcal M}\), and in particular, if \(p_0\) is a non-degenerate critical point of the scalar curvature of g, then the family \((\partial \Omega _\varepsilon )_{\varepsilon \in (0,\varepsilon _0)}\) constitutes a smooth foliation of a neighborhood of \(p_0\).     

Construction of sign-changing solutions for a harmonic equation with subcritical exponent

In this paper, we study the harmonic equation involving subcritical exponent \((P_{\varepsilon })\): \( \Delta u = 0 \), in \(\mathbb {B}^n\) and \(\displaystyle \frac{\partial u}{\partial \nu } + \displaystyle \frac{n-2}{2}u = \displaystyle \frac{n-2}{2} K u^{\frac{n}{n-2}-\varepsilon }\) on \( \mathbb {S}^{n-1}\) where \(\mathbb {B}^n \) is the unit ball in \(\mathbb {R}^n\), \(n\ge 5\) with Euclidean metric \(g_0\), \(\partial \mathbb {B}^n = \mathbb {S}^{n-1}\) is its boundary, K is a function on \(\mathbb {S}^{n-1}\) and \(\varepsilon \) is a small positive parameter. We construct solutions of the subcritical equation \((P_{\varepsilon })\) which blow up at two different critical points of K. Furthermore, we construct solutions of \((P_{\varepsilon })\) which have two bubbles and blow up at the same critical point of K.     

General Natural $$(\alpha ,\varepsilon )$$-Structures

We study in a unified way the \((\alpha ,\varepsilon )\)-structures of general natural lift type on the tangent bundle of a Riemannian manifold. We characterize the general natural \(\alpha \)-structures on the total space of the tangent bundle of a Riemannian manifold, and provide their integrability conditions (the base manifold is a space form and some involved coefficients are rational functions of the other ones). Then, we characterize the two classes (with respect to the sign of \(\alpha \varepsilon \)) of \((\alpha ,\varepsilon )\)-structures of general natural type on TM. The class \(\alpha \varepsilon =-1\) is characterized by some proportionality relations between the coefficients of the metric and those of the \(\alpha \)-structure, and in this case, the structure is almost Kählerian if and only if the first proportionality factor is the derivative of the second one. Moreover, the total space of the tangent bundle is a Kähler manifold if and only if it depends on three coefficients only (two coefficients of the integrable \(\alpha \)-structure and a proportionality factor).     

Compact $$\lambda $$-translating Solitons with Boundary

A \(\lambda \)-translating soliton with density vector \(\mathbf {v}\) is a surface \(\varSigma \) in Euclidean space \(\mathbb {R}^3\) whose mean curvature H satisfies \(2H=2\lambda +\langle N,\mathbf {v}\rangle \), where N is the Gauss map of \(\varSigma \). In this article, we study the shape of a compact \(\lambda \)-translating soliton in terms of its boundary. If \(\varGamma \) is a given closed curve, we deduce under what conditions on \(\lambda \) there exists a compact \(\lambda \)-translating soliton \(\varSigma \) with boundary \(\varGamma \) and we provide estimates of the surface area depending on the height of \(\varSigma \). Finally, we study the shape of \(\varSigma \) related with the geometry of \(\varGamma \), in particular, we give conditions that assert that \(\varSigma \) inherits the symmetries of its boundary \(\varGamma \).     

Edge Flips in Surface Meshes

Little theoretical work has been done on edge flips in surface meshes despite their popular usage in graphics and solid modeling to improve mesh equality. We propose the class of \((\varepsilon ,\alpha )\)-meshes of a surface that satisfy several properties: the vertex set is an \(\varepsilon \)-sample of the surface, the triangle angles are no smaller than a constant \(\alpha \), some triangle has a good normal, and the mesh is homeomorphic to the surface. We believe that many surface meshes encountered in practice are \((\varepsilon ,\alpha )\)-meshes or close to being one. We prove that flipping the appropriate edges can smooth a dense \((\varepsilon ,\alpha )\)-mesh by making the triangle normals better approximations of the surface normals and the dihedral angles closer to \(\pi \). Moreover, the edge flips can be performed in time linear in the number of vertices. This helps to explain the effectiveness of edge flips as observed in practice and in our experiments. A corollary of our techniques is that, in \(\mathbb {R}^2\), every triangulation with a constant lower bound on the angles can be flipped in linear time to the Delaunay triangulation.     

The Sobolev Stability Threshold for 2D Shear Flows Near Couette

We consider the 2D Navier–Stokes equation on \(\mathbb T \times \mathbb R\), with initial datum that is \(\varepsilon \)-close in \(H^N\) to a shear flow (U(y), 0), where \(\Vert U(y) - y\Vert _{H^{N+4}} \ll 1\) and \(N>1\). We prove that if \(\varepsilon \ll \nu ^{1/2}\), where \(\nu \) denotes the inverse Reynolds number, then the solution of the Navier–Stokes equation remains \(\varepsilon \)-close in \(H^1\) to \((e^{t \nu \partial _{yy}}U(y),0)\) for all \(t>0\). Moreover, the solution converges to a decaying shear flow for times \(t \gg \nu ^{-1/3}\) by a mixing-enhanced dissipation effect, and experiences a transient growth of gradients. In particular, this shows that the stability threshold in finite regularity scales no worse than \(\nu ^{1/2}\) for 2D shear flows close to the Couette flow.     

rthogonality Property $$\mathcal {}$$ and Compact Perturbations

A bounded linear operator T acting on a Hilbert space is said to have orthogonality property \(\mathcal {O}\) if the subspaces \(\ker (T-\alpha )\) and \(\ker (T-\beta )\) are orthogonal for all \(\alpha , \beta \in \sigma _p(T)\) with \(\alpha \ne \beta \). In this paper, the authors investigate the compact perturbations of operators with orthogonality property \(\mathcal {O}\). We give a sufficient and necessary condition to determine when an operator T has the following property: for each \(\varepsilon >0\), there exists \(K\in \mathcal {K(H)}\) with \(\Vert K\Vert <\varepsilon \) such that \(T+K\) has orthogonality property \(\mathcal {O}\). Also, we study the stability of orthogonality property \(\mathcal {O}\) under small compact perturbations and analytic functional calculus.     

Approximation,solution operators and quantale-valued metrics

A generalized solution operator is a mapping abstractly describing a computational problem and its approximate solutions. It assigns a set of \(\varepsilon \)-approximations of a solution to the problem instance f and accuracy of approximation \(\varepsilon \). In this paper we study generalized solution operators for which the accuracy of approximation is described by elements of a complete lattice equipped with a compatible monoid structure, namely, a quantale. We provide examples of computational problems for which the accuracy of approximation of a solution is measured by such objects. We show that the sets of \(\varepsilon \)-approximations are, roughly, closed balls with radii \(\varepsilon \) with respect to a certain family of quantale-valued generalized metrics induced by a generalized solution operator.     

Convergence and Precise Asymptotics for Series Involving Self-normalized Sums

Let \(X, X_{1}, X_{2}, \ldots \) be i.i.d. random variables, and set \(S_{n}=X_{1}+\cdots +X_{n}\) and \( V_{n}^{2}=X_{1}^{2}+\cdots +X_{n}^{2}.\) Without any moment conditions on \(X\), assuming that \(\{S_{n}/V_{n}\}\) is tight, we establish convergence of series of the type (*) \(\sum \nolimits _{n}w_{n}P(\left| S_{n}\right| /V_{n}\ge \varepsilon b_{n}),\) \(\varepsilon >0.\) Then, assuming that \(X\) is symmetric and belongs to the domain of attraction of a stable law, and choosing \(w_{n}\) and \(b_{n}\) suitably\(,\) we derive the precise asymptotic behavior of the series (*) as \(\varepsilon \searrow 0. \)     

Associate space with respect to a locally $$\sigma $$-finite measure on a $$\delta $$δ-ring and applications to spaces of integrable functions defined by a vector measure

We show that for a locally \(\sigma \)-finite measure \(\mu \) defined on a \(\delta \)-ring, the associate space theory can be developed as in the \(\sigma \)-finite case, and corresponding properties are obtained. Given a saturated \(\sigma \)-order continuous \(\mu \)-Banach function space E, we prove that its dual space can be identified with the associate space \(E ^\times \) if, and only if, \(E^\times \) has the Fatou property. Applying the theory to the spaces \(L^p (\nu )\) and \(L_w^p (\nu )\), where \(\nu \) is a vector measure defined on a \(\delta \)-ring \(\mathcal {R}\) and \(1 \le p < \infty \), we establish results corresponding to those of the case when the vector measure is defined on a \(\sigma \)-algebra.     

Reducibility of operator semigroups and values of vector states

Let \(\mathcal S\) be a multiplicative semigroup of bounded linear operators on a complex Hilbert space \(\mathcal H\), and let \(\Omega \) be the range of a vector state on \(\mathcal S\) so that \(\Omega = \{ \langle S \xi , \xi \rangle \,{:}\,S \in \mathcal S\}\) for some fixed unit vector \(\xi \in \mathcal H\). We study the structure of sets \(\Omega \) of cardinality two coming from irreducible semigroups \(\mathcal S\). This leads us to sufficient conditions for reducibility and, in some cases, for the existence of common fixed points for \(\mathcal S\). This is made possible by a thorough investigation of the structure of maximal families \(\mathcal F\) of unit vectors in \(\mathcal H\) with the property that there exists a fixed constant \(\rho \in \mathbb C\) for which \(\langle x, y \rangle = \rho \) for all distinct pairs x and y in \(\mathcal F\).     

Quasi-Linear Equations with a Small Diffusion Term and the Evolution of Hierarchies of Cycles

We study the long-time behavior (at times of order \(\exp (\lambda /\varepsilon ^2\))) of solutions to quasi-linear parabolic equations with a small parameter \(\varepsilon ^2\) at the diffusion term. The solution to a PDE can be expressed in terms of diffusion processes, whose coefficients, in turn, depend on the unknown solution. The notion of a hierarchy of cycles for diffusion processes was introduced by Freidlin and Wentzell and applied to the study of the corresponding linear equations. In the quasi-linear case, it is not a single hierarchy that corresponds to an equation, but rather a family of hierarchies that depend on the timescale \(\lambda \). We describe the evolution of the hierarchies with respect to \(\lambda \) in order to gain information on the limiting behavior of the solution of the PDE.     

The Allen–Cahn equation on closed manifolds

We study global variational properties of the space of solutions to \(-\varepsilon ^2\Delta u + W'(u)=0\) on any closed Riemannian manifold M. Our techniques are inspired by recent advances in the variational theory of minimal hypersurfaces and extend a well-known analogy with the theory of phase transitions. First, we show that solutions at the lowest positive energy level are either stable or obtained by min–max and have index 1. We show that if \(\varepsilon \) is not small enough, in terms of the Cheeger constant of M, then there are no interesting solutions. However, we show that the number of min–max solutions to the equation above goes to infinity as \(\varepsilon \rightarrow 0\) and their energies have sublinear growth. This result is sharp in the sense that for generic metrics the number of solutions is finite, for fixed \(\varepsilon \), as shown recently by G. Smith. We also show that the energy of the min–max solutions accumulate, as \(\varepsilon \rightarrow 0\), around limit-interfaces which are smooth embedded minimal hypersurfaces whose area with multiplicity grows sublinearly. For generic metrics with \(\mathrm{Ric}_M>0\), the limit-interface of the solutions at the lowest positive energy level is an embedded minimal hypersurface of least area in the sense of Mazet–Rosenberg. Finally, we prove that the min–max energy values are bounded from below by the widths of the area functional as defined by Marques–Neves.     

Normes cyclotomiques naïves et unités logarithmiques

We compute the \({\mathbb {Z}}\)-rank of the subgroup \(\widetilde{E}_K =\bigcap _{n\in {\mathbb {N}}} N_{K_n/K}(K_n^\times )\) of elements of the multiplicative group of a number field K that are norms from every finite level of the cyclotomic \({\mathbb {Z}}_\ell \)-extension \(K^c\) of K. Thus we compare its \(\ell \)-adification \({\mathbb {Z}}_\ell \otimes _{\mathbb {Z}}\widetilde{E}_K\) with the group of logarithmic units \(\widetilde{\varepsilon }_K\). By the way we point out an easy proof of the Gross–Kuz’min conjecture for \(\ell \)-undecomposed extensions of abelian fields.     

Almost band preservers

We study the stability of band preserving operators on Banach lattices. To this end the notion of \(\varepsilon \)-band preserving mapping is introduced. It is shown that, under quite general assumptions, a \(\varepsilon \)-band preserving operator is in fact a small perturbation of a band preserving one. However, a counterexample can be produced in some circumstances. Some results on automatic continuity of \(\varepsilon \)-band preserving maps are also obtained.     

Spectral density estimates with partial symmetries and an application to Bahri–Lions-type results

We are concerned with the existence of infinitely many solutions for the problem \(-\Delta u=|u|^{p-2}u+f\) in \(\Omega \), \(u=u_0\) on \(\partial \Omega \), where \(\Omega \) is a bounded domain in \(\mathbb {R}^N\), \(N\ge 3\). This can be seen as a perturbation of the problem with \(f=0\) and \(u_0=0\), which is odd in u. If \(\Omega \) is invariant with respect to a closed strict subgroup of O(N), then we prove infinite existence for all functions f and \(u_0\) in certain spaces of invariant functions for a larger range of exponents p than known before. In order to achieve this, we prove Lieb–Cwikel–Rosenbljum-type bounds for invariant potentials on \(\Omega \), employing improved Sobolev embeddings for spaces of invariant functions.     

On Hecke groups,Schwarzian triangle functions and a class of hyper-elliptic functions

Let m be a positive integer \(\ge \)3 and \(\lambda =2\cos \frac{\pi }{m}\). The Hecke group \(\mathfrak {G}(\lambda )\) is generated by the fractional linear transformations \(\tau + \lambda \) and \(-\frac{1}{\tau }\) for \(\tau \) in the upper half plane \(\mathbb H\) of the complex plane \(\mathbb C\). We consider a set of functions \(\mathfrak {f}_0, \mathfrak {f}_i\) and \(\mathfrak {f}_{\infty }\) automorphic with respect to \(\mathfrak {G}(\lambda )\), constructed from the conformal mapping of the fundamental domain of \(\mathfrak {G}(\lambda )\) to the upper half plane \(\mathbb H\), and establish their connection with the Legendre functions and a class of hyper-elliptic functions. Many well-known classical identities associated with the cases of \(\lambda =1\) and 2 are preserved. As an application, we will establish a set of identities expressing the reciprocal of \(\pi \) in terms of the hypergeometric series.     

Eichler’s commutation relation and some other invariant subspaces of Hecke operators

Let k be an odd positive integer, L a lattice on a regular positive definite k-dimensional quadratic space over \(\mathbb {Q}\), \(N_L\) the level of L, and \(\mathscr {M}(L)\)  be the linear space of \(\theta \)-series attached to the distinct classes in the genus of L. We prove that, for an odd prime \(p|N_L\), if \(L_p=L_{p,1}\,\bot \, L_{p,2}\), where \(L_{p,1}\) is unimodular, \(L_{p,2}\) is (p)-modular, and \(\mathbb {Q}_pL_{p,2}\) is anisotropic, then \(\mathscr {M}(L;p):=\) \(\mathscr {M}(L)\) \(+T_{p^2}.\) \(\mathscr {M}(L)\)  is stable under the Hecke operator \(T_{p^2}\). If \(L_2\) is isometric to \(\left( \begin{array}{ll}0&{}\frac{1}{2}\\ \frac{1}{2}&{}0\end{array}\right) ^{\kappa }\,\bot \, \langle \varepsilon \rangle \) or \(\left( \begin{array}{ll}0&{}\frac{1}{2}\\ \frac{1}{2}&{}0\end{array}\right) ^{\kappa }\,\bot \, \langle 2\varepsilon \rangle \) or \(\left( \begin{array}{ll}0&{}1\\ 1&{}0\end{array}\right) ^{\kappa }\,\bot \, \langle \varepsilon \rangle \) with \(\varepsilon \in \mathbb {Z}_2^{\times }\) and \(\kappa :=\frac{k-1}{2}\), then \(\mathscr {M}(L;2):=T_{2^2}.\mathscr {M}(L)+T_{2^2}^2.\,\mathscr {M}(L)\) is stable under the Hecke operator \(T_{2^2}\). Furthermore, we determine some invariant subspaces of the cusp forms for the Hecke operators.