Spaces of Polygonal Triangulations and Monsky Polynomials

Given a combinatorial triangulation of an n-gon, we study (a) the space of all possible drawings in the plane such the edges are straight line segments and the boundary has a fixed shape, and (b) the algebraic variety of possibilities for the areas of the triangles in such drawings. We define a generalized notion of triangulation, and we show that the areas of the triangles in a generalized triangulation ${\mathcal {T}}$ of a square must satisfy a single irreducible homogeneous polynomial relation $p({\mathcal {T}})$ depending only on the combinatorics of ${\mathcal {T}}$ . The invariant $p({\mathcal {T}})$ is called the Monsky polynomial; it captures algebraic, geometric, and combinatorial information about ${\mathcal {T}}$ . We give an algorithm that computes a lower bound on the degree of $p({\mathcal {T}})$ , and we present several examples in which the algorithm is used to compute the degree.     

Measurement of the absolute branching fraction of D0-->K-pi+

We measure the absolute branching fraction for D(0)-->K(-)pi(+) using partial reconstruction of B(0)-->D(*+)Xl(-)nu(l) decays, in which only the charged lepton and the pion from the decay D(*+)-->D(0)pi(+) are used. Based on a data sample of 230 x 10(6) BB pairs collected at the Upsilon(4S) resonance with the BABAR detector at the PEP-II asymmetric-energy B factory at SLAC, we obtain B(D(0)-->K(-)pi(+)) = (4.007+/-0.037+/-0.072)%, where the first uncertainty is statistical and the second is systematic.     

Observation and polarization measurements of B+/- -->phiK1 +/- and B +/- -->phiK2 *+/-

With the full BABAR data sample of 465 x 10(6) B(over)B pairs, we observe the decays B+/- -->phiK_(1)(1270) +/- and B +/- -->phiK*_(2)(1430)+/-. We measure the branching fractions (6.1+/-1.6+/-1.1) x 10(-6) and (8.4+/-1.8+/-1.0) x 10(-6) and the fractions of longitudinal polarization 0.46 (+0.12+0.06) _(-0.13-0.07) and 0.80(+0.09)_(-0.10)+/-0.03, respectively. We also report on the B+/- -->phiK*_(0)(1430)+/- decay branching fraction of (7.0+/-1.3+/-0.9) x 10(-6) and several parameters sensitive to CP violation and interference in the above three decays. Upper limits are placed on the B+/- decay rates to final states with phi and K_1(1400)+/-, K*(1410)+/-, K2(1770)+/-, or K_2(1820)+/-. Understanding the observed polarization pattern requires amplitude contributions from an uncertain source.     

Observation of B;{0}-->K;(*0)K[over ](*0) and Search for B;(0)-->K;(*0)K;(*0)

We report the observation of the b-->d penguin-dominated decay B;{0}-->K;{*0}K[over ];{*0} with a sample of 383.2+/-4.2 million BB[over ] pairs collected with the BABAR detector at the PEP-II asymmetric-energy e;{+}e;{-} collider at the Stanford Linear Accelerator Center. The measured branching fraction is B(B;{0}-->K;{*0}K[over ];{*0})=[1.28_{-0.30};{+0.35}+/-0.11]x10;{-6} and the fraction of longitudinal polarization is f_{L}(B;{0}-->K;{*0}K[over ];{*0})=0.80_{-0.12};{+0.10}+/-0.06. The first error quoted is statistical and the second systematic. We also obtain an upper limit at the 90% confidence level on the branching fraction for B(B;{0}-->K;{*0}K;{*0})<0.41x10;{-6}.     

Measurements of B --> {pi,eta,eta;{'}}lnu_{l} branching fractions and determination of |V_{ub}| with semileptonically tagged B mesons

We report measurements of branching fractions for the decays B-->Plnu_{l}, where P are the pseudoscalar charmless mesons pi;{-}, pi;{0}, eta and eta;{'}, based on 348 fb;{-1} of data collected with the BABAR detector, using B0 and B+ mesons found in the recoil of a second B meson decaying as B-->D;{(*)}lnu_{l}. Assuming isospin symmetry, we combine pionic branching fractions to obtain B(B;{0}-->pi;{-}l;{+}nu_{l})=(1.54+/-0.17_{(stat)}+/-0.09_{(syst)})x10;{-4}; we find 3.2sigma evidence of the decay B;{+}-->etal;{+}nu_{l} and measure its branching fraction to be (0.64+/-0.20_{(stat)}+/-0.03_{(syst)})x10;{-4}, and determine B(B;{+}-->eta;{'}l;{+}nu_{l})<0.47x10;{-4} to 90% confidence level. Using partial branching fractions for the pionic decays in ranges of the momentum transfer and a variety of form factor calculation, we obtain values of the magnitude of the Cabibbo-Kobayashi-Maskawa matrix element |V_{ub}| in ranging from 3.6x10;{-3} to 4.1x10;{-3}.     

Search for CP violation in the decays D0 --> K- K+ and D0 --> pi- pi+

We measure time-integrated CP-violating asymmetries of neutral charmed mesons in the modes D0 --> K(-) K(+) and D0 --> pi(-) pi(+) with the highest precision to date by using D0 --> K(-) pi(+) decays to correct detector asymmetries. An analysis of 385.8 fb(-1) of data collected with the BABAR detector yields values of a(CP)(KK)=(0.00+/-0.34(stat)+/-0.13(syst))% and a(CP)(pipi)=(-0.24+/-0.52(stat)+/-0.22(syst))%, which agree with standard model predictions.     

Copper-Catalyzed C(sp3)-Amination of Ketone-Derived Dihydroquinazolinones by Aromatization-Driven C−C Bond Scission

Herein, we describe the development of a copper-catalyzed C(sp³)-amination of proaromatic dihydroquinazolinones derived from ketones. The reaction is enabled by the intermediacy of open-shell species arising from homolytic C−C bond-cleavage driven by aromatization. The protocol is characterized by its operational simplicity and generality, including chemical diversification of advanced intermediates.