More Like this

1. Zeroes of polynomials on definable hypersurfaces: pathologies exist, but they are rare

   https://doi.org/10.1093/qmath/haz022  

   Basu, Saugata; Lerario, Antonio; Natarajan, Abhiram (October 2019, The Quarterly Journal of Mathematics)

   Abstract Given a sequence $$\{Z_d\}_{d\in \mathbb{N}}$$ of smooth and compact hypersurfaces in $${\mathbb{R}}^{n-1}$$, we prove that (up to extracting subsequences) there exists a regular definable hypersurface $$\Gamma \subset {\mathbb{R}}\textrm{P}^n$$ such that each manifold $$Z_d$$ is diffeomorphic to a component of the zero set on $$\Gamma$$ of some polynomial of degree $$d$$. (This is in sharp contrast with the case when $$\Gamma$$ is semialgebraic, where for example the homological complexity of the zero set of a polynomial $$p$$ on $$\Gamma$$ is bounded by a polynomial in $$\deg (p)$$.) More precisely, given the above sequence of hypersurfaces, we construct a regular, compact, semianalytic hypersurface $$\Gamma \subset {\mathbb{R}}\textrm{P}^{n}$$ containing a subset $$D$$ homeomorphic to a disk, and a family of polynomials $$\{p_m\}_{m\in \mathbb{N}}$$ of degree $$\deg (p_m)=d_m$$ such that $$(D, Z(p_m)\cap D)\sim ({\mathbb{R}}^{n-1}, Z_{d_m}),$$ i.e. the zero set of $$p_m$$ in $$D$$ is isotopic to $$Z_{d_m}$$ in $${\mathbb{R}}^{n-1}$$. This says that, up to extracting subsequences, the intersection of $$\Gamma$$ with a hypersurface of degree $$d$$ can be as complicated as we want. We call these ‘pathological examples’. In particular, we show that for every $$0 \leq k \leq n-2$$ and every sequence of natural numbers $$a=\{a_d\}_{d\in \mathbb{N}}$$ there is a regular, compact semianalytic hypersurface $$\Gamma \subset {\mathbb{R}}\textrm{P}^n$$, a subsequence $$\{a_{d_m}\}_{m\in \mathbb{N}}$$ and homogeneous polynomials $$\{p_{m}\}_{m\in \mathbb{N}}$$ of degree $$\deg (p_m)=d_m$$ such that (0.1)$$\begin{equation}b_k(\Gamma\cap Z(p_m))\geq a_{d_m}.\end{equation}$$ (Here $$b_k$$ denotes the $$k$$th Betti number.) This generalizes a result of Gwoździewicz et al. [13]. On the other hand, for a given definable $$\Gamma$$ we show that the Fubini–Study measure, in the Gaussian probability space of polynomials of degree $$d$$, of the set $$\Sigma _{d_m,a, \Gamma }$$ of polynomials verifying (0.1) is positive, but there exists a constant $$c_\Gamma$$ such that $$\begin{equation*}0<{\mathbb{P}}(\Sigma_{d_m, a, \Gamma})\leq \frac{c_{\Gamma} d_m^{\frac{n-1}{2}}}{a_{d_m}}.\end{equation*}$$ This shows that the set of ‘pathological examples’ has ‘small’ measure (the faster $$a$$ grows, the smaller the measure and pathologies are therefore rare). In fact we show that given $$\Gamma$$, for most polynomials a Bézout-type bound holds for the intersection $$\Gamma \cap Z(p)$$: for every $$0\leq k\leq n-2$$ and $t>0$: $$\begin{equation*}{\mathbb{P}}\left(\{b_k(\Gamma\cap Z(p))\geq t d^{n-1} \}\right)\leq \frac{c_\Gamma}{td^{\frac{n-1}{2}}}.\end{equation*}$$ 

   more » « less
2. Independent dominating sets in graphs of girth five

   https://doi.org/10.1017/s0963548320000279  

   Harutyunyan, Ararat; Horn, Paul; Verstraete, Jacques (May 2021, Combinatorics, Probability and Computing)

   Abstract Let $$\gamma(G)$$ and $${\gamma _ \circ }(G)$$ denote the sizes of a smallest dominating set and smallest independent dominating set in a graph G, respectively. One of the first results in probabilistic combinatorics is that if G is an n -vertex graph of minimum degree at least d , then $$\begin{equation}\gamma(G) \leq \frac{n}{d}(\log d + 1).\end{equation}$$ In this paper the main result is that if G is any n -vertex d -regular graph of girth at least five, then $$\begin{equation}\gamma_(G) \leq \frac{n}{d}(\log d + c)\end{equation}$$ for some constant c independent of d . This result is sharp in the sense that as $$d \rightarrow \infty$$ , almost all d -regular n -vertex graphs G of girth at least five have $$\begin{equation}\gamma_(G) \sim \frac{n}{d}\log d.\end{equation}$$ Furthermore, if G is a disjoint union of $${n}/{(2d)}$$ complete bipartite graphs $$K_{d,d}$$ , then $${\gamma_\circ}(G) = \frac{n}{2}$$ . We also prove that there are n -vertex graphs G of minimum degree d and whose maximum degree grows not much faster than d log d such that $${\gamma_\circ}(G) \sim {n}/{2}$$ as $$d \rightarrow \infty$$ . Therefore both the girth and regularity conditions are required for the main result. 

   more » « less
3. On local well-posedness of logarithmic inviscid regularizations of generalized SQG equations in borderline Sobolev spaces

   https://doi.org/10.3934/cpaa.2021169  

   Jolly, Michael S.; Kumar, Anuj; Martinez, Vincent R. (January 2021, Communications on Pure & Applied Analysis)

   This paper studies a family of generalized surface quasi-geostrophic (SQG) equations for an active scalar \begin{document}$$ \theta $$\end{document} on the whole plane whose velocities have been mildly regularized, for instance, logarithmically. The well-posedness of these regularized models in borderline Sobolev regularity have previously been studied by D. Chae and J. Wu when the velocity \begin{document}$ u $$\end{document} is of lower singularity, i.e., \begin{document}$$ u = -\nabla^{\perp} \Lambda^{ \beta-2}p( \Lambda) \theta $$\end{document}, where \begin{document}$$ p $$\end{document} is a logarithmic smoothing operator and \begin{document}$$ \beta \in [0, 1] $$\end{document}. We complete this study by considering the more singular regime \begin{document}$$ \beta\in(1, 2) $$\end{document}$. The main tool is the identification of a suitable linearized system that preserves the underlying commutator structure for the original equation. We observe that this structure is ultimately crucial for obtaining continuity of the flow map. In particular, straightforward applications of previous methods for active transport equations fail to capture the more nuanced commutator structure of the equation in this more singular regime. The proposed linearized system nontrivially modifies the flux of the original system in such a way that it coincides with the original flux when evaluated along solutions of the original system. The requisite estimates are developed for this modified linear system to ensure its well-posedness. 

   more » « less
4. Dimensional Reduction Guides Electronic Structure Evolution in the A _n Cu _4–n SnS ₄ Semiconductor Series

   https://doi.org/10.1021/jacs.5c07366  

   Viti, Michael A; Li, Zhi; Wolverton, Christopher; Kanatzidis, Mercouri G (August 2025, Journal of the American Chemical Society)

   The search for new functional materials with tunable properties remains a central challenge in chemistry, particularly for applications in energy and electronics. In this work, we present a framework for predictive crystal design in alkali metal chalcogenides that enables controlled dimensional reduction of a parent covalent motif, yielding a broad range of electronic structures, which systematically evolve from one parent to the other. We present 11 new members of the AnCu4–nSnS4 family (A = alkali metal; n = 0–4), which reduce the three-dimensional (3D) covalent network of Cu4SnS4 into various 3D, 2D, 1D, and 0D [Cu4–nSnS4]n− motifs through the substitution of Cu with alkali metals of various radii. The end members of the family set the range in achievable band gaps at 0.99 eV for fully covalent Cu4SnS4 (n = 0) and 3.38 eV for K4SnS4 (n = 4) with 0D [SnS4]n− tetrahedra. As the dimensionality of [Cu4–nSnS4]n− systematically reduces within AnCu4–nSnS4 (n = 1–3), a stepwise increase in band gap energy occurs through a gradual decrease in the energy of the valence band maximum and an increase in the conduction band minimum, with an increase in the effective masses of charge carriers. Furthermore, irrespective of the alkali metal, the thermal stability decreases with decreasing [Cu4–nSnS4]n− dimensionality within the quaternary members. Most importantly, we demonstrate that predictable crystal structure and property evolution for a given composition space is possible by deriving a general formula based on substituting the covalent metals of a parent structure with alkali metals. 

   more » « less
5. On the nonlinear stability and the existence of selective decay states of 3D quasi‐geostrophic potential vorticity equation

   https://doi.org/10.1002/mma.5962  

   Esen, Oğul; Han, Daozhi; Şengül, Taylan; Wang, Quan (November 2019, Mathematical Methods in the Applied Sciences)

   In this article, we study the dynamics of large‐scale motion in atmosphere and ocean governed by the 3D quasi‐geostrophic potential vorticity (QGPV) equation with a constant stratification. It is shown that for a Kolmogorov forcing on the first energy shell, there exist a family of exact solutions that are dissipative Rossby waves. The nonlinear stability of these exact solutions are analyzed based on the assumptions on the growth rate of the forcing. In the absence of forcing, we show the existence of selective decay states for the 3D QGPV equation. The selective decay states are the 3D Rossby waves traveling horizontally at a constant speed. All these results can be regarded as the expansion of that of the 2D QGPV system and in the case of 3D QGPV system with isotropic viscosity. Finally, we present a geometric foundation for the model as a general equation for nonequilibrium reversible‐irreversible coupling. 

   more » « less