 Award ID(s):
 1802119
 Publication Date:
 NSFPAR ID:
 10143896
 Journal Name:
 Journal fĂŒr die reine und angewandte Mathematik (Crelles Journal)
 Volume:
 2019
 Issue:
 753
 Page Range or eLocationID:
 89 to 135
 ISSN:
 00754102
 Sponsoring Org:
 National Science Foundation
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This is the second in a pair of works which study small disturbances to the plane, periodic 3D Couette flow in the incompressible NavierStokes equations at high Reynolds number Re . In this work, we show that there is constant 0 > c 0 âȘ 1 0 > c_0 \ll 1 , independent of R e \mathbf {Re} , such that sufficiently regular disturbances of size Ï” âČ R e â 2 / 3 â ÎŽ \epsilon \lesssim \mathbf {Re}^{2/3\delta } for any ÎŽ > 0 \delta > 0 exist at least until t = c 0 Ï” â 1 t = c_0\epsilon ^{1} and in general evolve to be O ( c 0 ) O(c_0) due to the liftup effect. Further, after times t âł R e 1 / 3 t \gtrsim \mathbf {Re}^{1/3} , the streamwise dependence of the solution is rapidly diminished by a mixingenhanced dissipation effect and the solution is attracted back to the class of â2.5 dimensionalâ streamwiseindependent solutions (sometimes referred to as âstreaksâ). The largest of these streaks are expected to eventually undergo a secondary instability at t â Ï” â 1 t \approx \epsilon ^{1} . Hence, our work strongly suggests, for allmore »

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 logmore »

Let $F$ be a totally real field in which $p$ is unramified. Let $\overline{r}:G_{F}\rightarrow \text{GL}_{2}(\overline{\mathbf{F}}_{p})$ be a modular Galois representation that satisfies the TaylorâWiles hypotheses and is tamely ramified and generic at a place $v$ above $p$ . Let $\mathfrak{m}$ be the corresponding Hecke eigensystem. We describe the $\mathfrak{m}$ torsion in the $\text{mod}\,p$ cohomology of Shimura curves with full congruence level at $v$ as a $\text{GL}_{2}(k_{v})$ representation. In particular, it only depends on $\overline{r}_{I_{F_{v}}}$ and its JordanâHĂ¶lder factors appear with multiplicity one. The main ingredients are a description of the submodule structure for generic $\text{GL}_{2}(\mathbf{F}_{q})$ projective envelopes and the multiplicity one results of Emerton, Gee and Savitt [Lattices in the cohomology of Shimura curves, Invent. Math. Â 200 (1) (2015), 1â96].

Abstract A measurement of the $$ B_{s}^{0} \rightarrow J/\psi \phi $$ B s 0 â J / Ï Ï decay parameters using $$ 80.5\, \mathrm {fb^{1}} $$ 80.5 fb  1 of integrated luminosity collected with the ATLAS detector from 13Â $$\text {Te}\text {V}$$ Te protonâproton collisions at the LHC is presented. The measured parameters include the CP violating phase $$\phi _{s} $$ Ï s , the width difference $$ \Delta \Gamma _{s}$$ Î Î s between the $$B_{s}^{0}$$ B s 0 meson mass eigenstates and the average decay width $$ \Gamma _{s}$$ Î s . The values measured for the physical parameters are combined with those from $$ 19.2\, \mathrm {fb^{1}} $$ 19.2 fb  1 of 7 and 8Â $$\text {Te}\text {V}$$ Te data, leading to the following: $$\begin{aligned} \phi _{s}= & {} 0.087 \pm 0.036 ~\mathrm {(stat.)} \pm 0.021 ~\mathrm {(syst.)~rad} \\ \Delta \Gamma _{s}= & {} 0.0657 \pm 0.0043 ~\mathrm {(stat.)}\pm 0.0037 ~\mathrm {(syst.)~ps}^{1} \\ \Gamma _{s}= & {} 0.6703 \pm 0.0014 ~\mathrm {(stat.)}\pm 0.0018 ~\mathrm {(syst.)~ps}^{1} \end{aligned}$$ Ï s =  0.087 Â± 0.036 ( stat . ) Â± 0.021 ( syst . ) rad Î Î s = 0.0657 Â± 0.0043 ( stat .more »

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