- Award ID(s):
- 1716466
- NSF-PAR ID:
- 10359309
- Date Published:
- Journal Name:
- Memoirs of the American Mathematical Society
- Volume:
- 279
- Issue:
- 1377
- ISSN:
- 0065-9266
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
Abstract Let Γ be a Schottky semigroup in {\mathrm{SL}_{2}(\mathbf{Z})} ,and for {q\in\mathbf{N}} , let {\Gamma(q):=\{\gamma\in\Gamma:\gamma=e~{}(\mathrm{mod}~{}q)\}} be its congruence subsemigroupof level q . Let δ denote the Hausdorff dimension of the limit set of Γ.We prove the following uniform congruence counting theoremwith respect to the family of Euclidean norm balls {B_{R}} in {M_{2}(\mathbf{R})} of radius R :for all positive integer q with no small prime factors, \#(\Gamma(q)\cap B_{R})=c_{\Gamma}\frac{R^{2\delta}}{\#(\mathrm{SL}_{2}(%\mathbf{Z}/q\mathbf{Z}))}+O(q^{C}R^{2\delta-\epsilon}) as {R\to\infty} for some {c_{\Gamma}>0,C>0,\epsilon>0} which are independent of q .Our technique also applies to give a similar counting result for the continued fractions semigroup of {\mathrm{SL}_{2}(\mathbf{Z})} ,which arises in the study of Zaremba’s conjecture on continued fractions.more » « less
-
Abstract We study the problem of estimating a $k$-sparse signal ${\boldsymbol \beta }_{0}\in{\mathbb{R}}^{p}$ from a set of noisy observations $\mathbf{y}\in{\mathbb{R}}^{n}$ under the model $\mathbf{y}=\mathbf{X}{\boldsymbol \beta }+w$, where $\mathbf{X}\in{\mathbb{R}}^{n\times p}$ is the measurement matrix the row of which is drawn from distribution $N(0,{\boldsymbol \varSigma })$. We consider the class of $L_{q}$-regularized least squares (LQLS) given by the formulation $\hat{{\boldsymbol \beta }}(\lambda )=\text{argmin}_{{\boldsymbol \beta }\in{\mathbb{R}}^{p}}\frac{1}{2}\|\mathbf{y}-\mathbf{X}{\boldsymbol \beta }\|^{2}_{2}+\lambda \|{\boldsymbol \beta }\|_{q}^{q}$, where $\|\cdot \|_{q}$ $(0\le q\le 2)$ denotes the $L_{q}$-norm. In the setting $p,n,k\rightarrow \infty $ with fixed $k/p=\epsilon $ and $n/p=\delta $, we derive the asymptotic risk of $\hat{{\boldsymbol \beta }}(\lambda )$ for arbitrary covariance matrix ${\boldsymbol \varSigma }$ that generalizes the existing results for standard Gaussian design, i.e. $X_{ij}\overset{i.i.d}{\sim }N(0,1)$. The results were derived from the non-rigorous replica method. We perform a higher-order analysis for LQLS in the small-error regime in which the first dominant term can be used to determine the phase transition behavior of LQLS. Our results show that the first dominant term does not depend on the covariance structure of ${\boldsymbol \varSigma }$ in the cases $0\le q\lt 1$ and $1\lt q\le 2,$ which indicates that the correlations among predictors only affect the phase transition curve in the case $q=1$ a.k.a. LASSO. To study the influence of the covariance structure of ${\boldsymbol \varSigma }$ on the performance of LQLS in the cases $0\le q\lt 1$ and $1\lt q\le 2$, we derive the explicit formulas for the second dominant term in the expansion of the asymptotic risk in terms of small error. Extensive computational experiments confirm that our analytical predictions are consistent with numerical results.
-
Abstract This paper will study almost everywhere behaviors of functions on partition spaces of cardinals possessing suitable partition properties. Almost everywhere continuity and monotonicity properties for functions on partition spaces will be established. These results will be applied to distinguish the cardinality of certain subsets of the power set of partition cardinals.
The following summarizes the main results proved under suitable partition hypotheses.
If
is a cardinal,$\kappa $ ,$\epsilon < \kappa $ ,${\mathrm {cof}}(\epsilon ) = \omega $ and$\kappa \rightarrow _* (\kappa )^{\epsilon \cdot \epsilon }_2$ , then$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$ satisfies the almost everywhere short length continuity property: There is a club$\Phi $ and a$C \subseteq \kappa $ so that for all$\delta < \epsilon $ , if$f,g \in [C]^\epsilon _*$ and$f \upharpoonright \delta = g \upharpoonright \delta $ , then$\sup (f) = \sup (g)$ .$\Phi (f) = \Phi (g)$ If
is a cardinal,$\kappa $ is countable,$\epsilon $ holds and$\kappa \rightarrow _* (\kappa )^{\epsilon \cdot \epsilon }_2$ , then$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$ satisfies the strong almost everywhere short length continuity property: There is a club$\Phi $ and finitely many ordinals$C \subseteq \kappa $ so that for all$\delta _0, ..., \delta _k \leq \epsilon $ , if for all$f,g \in [C]^\epsilon _*$ ,$0 \leq i \leq k$ , then$\sup (f \upharpoonright \delta _i) = \sup (g \upharpoonright \delta _i)$ .$\Phi (f) = \Phi (g)$ If
satisfies$\kappa $ ,$\kappa \rightarrow _* (\kappa )^\kappa _2$ and$\epsilon \leq \kappa $ , then$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$ satisfies the almost everywhere monotonicity property: There is a club$\Phi $ so that for all$C \subseteq \kappa $ , if for all$f,g \in [C]^\epsilon _*$ ,$\alpha < \epsilon $ , then$f(\alpha ) \leq g(\alpha )$ .$\Phi (f) \leq \Phi (g)$ Suppose dependent choice (
),$\mathsf {DC}$ and the almost everywhere short length club uniformization principle for${\omega _1} \rightarrow _* ({\omega _1})^{\omega _1}_2$ hold. Then every function${\omega _1}$ satisfies a finite continuity property with respect to closure points: Let$\Phi : [{\omega _1}]^{\omega _1}_* \rightarrow {\omega _1}$ be the club of$\mathfrak {C}_f$ so that$\alpha < {\omega _1}$ . There is a club$\sup (f \upharpoonright \alpha ) = \alpha $ and finitely many functions$C \subseteq {\omega _1}$ so that for all$\Upsilon _0, ..., \Upsilon _{n - 1} : [C]^{\omega _1}_* \rightarrow {\omega _1}$ , for all$f \in [C]^{\omega _1}_*$ , if$g \in [C]^{\omega _1}_*$ and for all$\mathfrak {C}_g = \mathfrak {C}_f$ ,$i < n$ , then$\sup (g \upharpoonright \Upsilon _i(f)) = \sup (f \upharpoonright \Upsilon _i(f))$ .$\Phi (g) = \Phi (f)$ Suppose
satisfies$\kappa $ for all$\kappa \rightarrow _* (\kappa )^\epsilon _2$ . For all$\epsilon < \kappa $ ,$\chi < \kappa $ does not inject into$[\kappa ]^{<\kappa }$ , the class of${}^\chi \mathrm {ON}$ -length sequences of ordinals, and therefore,$\chi $ . As a consequence, under the axiom of determinacy$|[\kappa ]^\chi | < |[\kappa ]^{<\kappa }|$ , these two cardinality results hold when$(\mathsf {AD})$ is one of the following weak or strong partition cardinals of determinacy:$\kappa $ ,${\omega _1}$ ,$\omega _2$ (for all$\boldsymbol {\delta }_n^1$ ) and$1 \leq n < \omega $ (assuming in addition$\boldsymbol {\delta }^2_1$ ).$\mathsf {DC}_{\mathbb {R}}$ -
Abstract A flavour-tagged time-dependent angular analysis of $${{B} ^0_{s}} \!\rightarrow {{J /\psi }} \phi $$ B s 0 → J / ψ ϕ decays is presented where the $${J /\psi }$$ J / ψ meson is reconstructed through its decay to an $$e ^+e ^-$$ e + e - pair. The analysis uses a sample of pp collision data recorded with the LHCb experiment at centre-of-mass energies of 7 and $$8\text {\,Te V} $$ 8 \,Te V , corresponding to an integrated luminosity of $$3 \text {\,fb} ^{-1} $$ 3 \,fb - 1 . The $$C\!P$$ C P -violating phase and lifetime parameters of the $${B} ^0_{s} $$ B s 0 system are measured to be $${\phi _{{s}}} =0.00\pm 0.28\pm 0.07\text {\,rad}$$ ϕ s = 0.00 ± 0.28 ± 0.07 \,rad , $${\Delta \Gamma _{{s}}} =0.115\pm 0.045\pm 0.011\text {\,ps} ^{-1} $$ Δ Γ s = 0.115 ± 0.045 ± 0.011 \,ps - 1 and $${\Gamma _{{s}}} =0.608\pm 0.018\pm 0.012\text {\,ps} ^{-1} $$ Γ s = 0.608 ± 0.018 ± 0.012 \,ps - 1 where the first uncertainty is statistical and the second systematic. This is the first time that $$C\!P$$ C P -violating parameters are measured in the $${{B} ^0_{s}} \!\rightarrow {{J /\psi }} \phi $$ B s 0 → J / ψ ϕ decay with an $$e ^+e ^-$$ e + e - pair in the final state. The results are consistent with previous measurements in other channels and with the Standard Model predictions.more » « less
-
Abstract We consider integral area-minimizing 2-dimensional currents
in$T$ with$U\subset \mathbf {R}^{2+n}$ , where$\partial T = Q\left [\!\![{\Gamma }\right ]\!\!]$ and$Q\in \mathbf {N} \setminus \{0\}$ is sufficiently smooth. We prove that, if$\Gamma $ is a point where the density of$q\in \Gamma $ is strictly below$T$ , then the current is regular at$\frac{Q+1}{2}$ . The regularity is understood in the following sense: there is a neighborhood of$q$ in which$q$ consists of a finite number of regular minimal submanifolds meeting transversally at$T$ (and counted with the appropriate integer multiplicity). In view of well-known examples, our result is optimal, and it is the first nontrivial generalization of a classical theorem of Allard for$\Gamma $ . As a corollary, if$Q=1$ is a bounded uniformly convex set and$\Omega \subset \mathbf {R}^{2+n}$ a smooth 1-dimensional closed submanifold, then any area-minimizing current$\Gamma \subset \partial \Omega $ with$T$ is regular in a neighborhood of$\partial T = Q \left [\!\![{\Gamma }\right ]\!\!]$ .$\Gamma $