 Award ID(s):
 1716466
 NSFPAR ID:
 10359309
 Date Published:
 Journal Name:
 Memoirs of the American Mathematical Society
 Volume:
 279
 Issue:
 1377
 ISSN:
 00659266
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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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 nonrigorous replica method. We perform a higherorder analysis for LQLS in the smallerror 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 A flavourtagged timedependent 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 centreofmass 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

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We present a weighted approach to compute a maximum cardinality matching in an arbitrary bipartite graph. Our main result is a new algorithm that takes as input a weighted bipartite graph G(A cup B,E) with edge weights of 0 or 1. Let w <= n be an upper bound on the weight of any matching in G. Consider the subgraph induced by all the edges of G with a weight 0. Suppose every connected component in this subgraph has O(r) vertices and O(mr/n) edges. We present an algorithm to compute a maximum cardinality matching in G in O~(m(sqrt{w} + sqrt{r} + wr/n)) time. When all the edge weights are 1 (symmetrically when all weights are 0), our algorithm will be identical to the wellknown HopcroftKarp (HK) algorithm, which runs in O(m sqrt{n}) time. However, if we can carefully assign weights of 0 and 1 on its edges such that both w and r are sublinear in n and wr=O(n^{gamma}) for gamma < 3/2, then we can compute maximum cardinality matching in G in o(m sqrt{n}) time. Using our algorithm, we obtain a new O~(n^{4/3}/epsilon^4) time algorithm to compute an epsilonapproximate bottleneck matching of A,B subsetR^2 and an 1/(epsilon^{O(d)}}n^{1+(d1)/(2d1)}) poly log n time algorithm for computing epsilonapproximate bottleneck matching in ddimensions. All previous algorithms take Omega(n^{3/2}) time. Given any graph G(A cup B,E) that has an easily computable balanced vertex separator for every subgraph G'(V',E') of size V'^{delta}, for delta in [1/2,1), we can apply our algorithm to compute a maximum matching in O~(mn^{delta/1+delta}) time improving upon the O(m sqrt{n}) time taken by the HKAlgorithm.more » « less