An official website of the United States government
Suppose $$f(x,y) + \frac{\kappa}{2} \|x\|^2 - \frac{\sigma}{2}\|y\|^2$$ is convex where $$\kappa\ge 0, \sigma>0$$, and the argmin function $$\gamma(x) = \{ \gamma : \inf_y f(x,y) = f(x,\gamma)\}$$ exists and is single valued. We will prove $$\gamma$$ is differentiable almost everywhere. As an application we deduce a minimum principle for certain semiconcave subsolutions.
more »
« less
Lower bounds for Maass forms on semisimple groups
Let $$G$$ be an anisotropic semisimple group over a totally real number field $$F$$ . Suppose that $$G$$ is compact at all but one infinite place $$v_{0}$$ . In addition, suppose that $$G_{v_{0}}$$ is $$\mathbb{R}$$ -almost simple, not split, and has a Cartan involution defined over $$F$$ . If $$Y$$ is a congruence arithmetic manifold of non-positive curvature associated with $$G$$ , we prove that there exists a sequence of Laplace eigenfunctions on $$Y$$ whose sup norms grow like a power of the eigenvalue.
more »
« less
- Award ID(s):
- 1902173
- PAR ID:
- 10145415
- Date Published:
- Journal Name:
- Compositio Mathematica
- Volume:
- 156
- Issue:
- 5
- ISSN:
- 0010-437X
- Page Range / eLocation ID:
- 959 to 1003
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
null (Ed.)Suppose $$f(x,y) + \frac{\kappa}{2} \|x\|^2 - \frac{\sigma}{2}\|y\|^2$$ is convex where $$\kappa\ge 0, \sigma>0$$, and the argmin function $$\gamma(x) = \{ \gamma : \inf_y f(x,y) = f(x,\gamma)\}$$ exists and is single valued. We will prove $$\gamma$$ is differentiable almost everywhere. As an application we deduce a minimum principle for certain semiconcave subsolutions.more » « less
-
We generalize Hermite interpolation with error correction, which is the methodology for multiplicity algebraic error correction codes, to Hermite interpolation of a rational function over a field K from function and function derivative values. We present an interpolation algorithm that can locate and correct <= E errors at distinct arguments y in K where at least one of the values or values of a derivative is incorrect. The upper bound E for the number of such y is input. Our algorithm sufficiently oversamples the rational function to guarantee a unique interpolant. We sample (f/g)^(j)(y[i]) for 0 <= j <= L[i], 1 <= i <= n, y[i] distinct, where (f/g)^(j) is the j-th derivative of the rational function f/g, f, g in K[x], GCD(f,g)=1, g <= 0, and where N = (L[1]+1)+...+(L[n]+1) >= C + D + 1 + 2(L[1]+1) + ... + 2(L[E]+1) where C is an upper bound for deg(f) and D an upper bound for deg(g), which are input to our algorithm. The arguments y[i] can be poles, which is truly or falsely indicated by a function value infinity with the corresponding L[i]=0. Our results remain valid for fields K of characteristic >= 1 + max L[i]. Our algorithm has the same asymptotic arithmetic complexity as that for classical Hermite interpolation, namely soft-O(N). For polynomials, that is, g=1, and a uniform derivative profile L[1] = ... = L[n], our algorithm specializes to the univariate multiplicity code decoder that is based on the 1986 Welch-Berlekamp algorithm.more » « less
-
Let $$\phi(x,y)$$ be a continuous function, smooth away from the diagonal, such that, for some $$\alpha>0$$, the associated generalized Radon transforms \begin{equation} \label{Radon} R_t^{\phi}f(x)=\int_{\phi(x,y)=t} f(y) \psi(y) d\sigma_{x,t}(y) \end{equation} map $$L^2({\mathbb R}^d) \to H^{\alpha}({\mathbb R}^d)$$ for all $t>0$. Let $$E$$ be a compact subset of $${\mathbb R}^d$$ for some $$d \ge 2$$, and suppose that the Hausdorff dimension of $$E$$ is $$>d-\alpha$$. We show that any tree graph $$T$$ on $k+1$ ($$k \ge 1$$) vertices is realizable in $$E$$, in the sense that there exist distinct $$x^1, x^2, \dots, x^{k+1} \in E$$ and $t>0$ such that the $$\phi$$-distance $$\phi(x^i, x^j)$$ is equal to $$t$$ for all pairs $(i,j)$ corresponding to the edges of the graph $$T$$.more » « less
-
Suppose that $$\mathbf{G}$$ is a connected reductive group over a finite extension $$F/\mathbb{Q}_{p}$$ and that $$C$$ is a field of characteristic $$p$$ . We prove that the group $$\mathbf{G}(F)$$ admits an irreducible admissible supercuspidal, or equivalently supersingular, representation over $$C$$ .more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1112/S0010437X20007125
