Large-Scale Regularities of Lattice Embeddings of Posets

Let N denote the set of natural numbers and let P =(N ^k , ) be a countably infinite poset on the k-dimensional lattice N ^k . Given x N ^k , we write max(x) (min(x)) for the maximum (minimum) coordinate of x. Let be the directed-incomparability graph of P which is defined to be the graph with vertex set equal to N ^k and edge set equal to the set of all (x, y) such that max(x) max(y) and x and y not comparable in P. For any subset D N ^k , we let P _D and _D denote the restrictions of P and to D. Points x N ^k with min(x) = 0 will be called boundary points. We define a geometrically natural notion of when a point is interior to P or relative to the lattice N ^k , and an analogous notion of monotone interior with respect to or _D . We wish to identify situations where most of these interior points are exposed to the boundary of the lattice or, in the case of monotone interior points, not concealed very much from the boundary. All of these ideas restrict to finite sublattices F ^k and/or infinite sublattices E ^k of N ^k . Our main result shows that for any poset P and any arbitarily large integer M > 0, there is an F E with F = M where, relative to the sublattices F ^k E ^k , the ideal situation of total exposure of interior points and very little concealment of monotone interior points must occur. Precisely, we prove that for any P =(N ^k , ) and any integer M > 0, there is an infinite E N and a finite D F ^k with F E and F = M such that (1) every interior vertex of P _{E k} or _{E k} is exposed and (2) there is a fixed set C E, C k ^k , such that every monotone-interior point of _D belonging to F ^k has its monotone concealment in the set C. In addition, we show that if P ¹ =(N ^k , ₁),..., P ^r =(N ^k , _r ) is any sequence of posets, then we can find E,D, and F so that the properties (1) and (2) described above hold simultaneously for each P ⁱ . We note that the main point of (2) is that the bound k ^k depends only on the dimension of the lattice and not on the poset P. Statement (1) is derived from classical Ramsey theory while (2) is derived from a recent powerful extension of Ramsey theory due to H. Friedman and shown by Friedman to be independent of ZFC, the usual axioms of set theory. The fact that our result is proved as a corollary to a combinatorial theorem that is known to be independent of the usual axioms of mathematics does not, of course, mean that it cannot be proved using ZFC (we just couldn"t find such a proof). This puts our geometrically natural combinatorial result in a somewhat unusual position with regard to the axioms of mathematics.     

Fermilab top mass and modified Fritzsch mass matrices

Fritzsch like mass matrices with non-zero 22-elements both in U sector and D sector have been investigated in the context of latest data regardingm _t ^phys , |V _ub|, |V _cb|, |V _td| and |V _ts|. Unlike several other phenomenological models, the present model not only accommodates the value ofm _t ^phys in the range 150–240 GeV, encompassing the CDF and D0 values, but is also able to reproduce |V _cb| ≊0.040 and |V _ub/V_cb| = 0.08±0.02 and |V _td| is predicted to lie in the range 0.005–0.014. Further, the angles of the unitarity triangle, related to the CP-violating asymmetries, are calculated to be in the ranges −1.0⩽sin2α⩽−0.1, 0.6 ⩽sin2α⩽1.0 and 0.48⩽sin2β⩽0.56, which are in agreement with other recent calculations.     

Studies in metal sulphite chemistry—III

Manganese(II) iodide, iron(II) iodide and copper(I) iodide each react with tetramethylammonium disulphite to form anhydrous manganese(II) sulphite, iron(II) sulphite and copper(I) disulphite respectively. Iron(II) sulphite and copper(I) disulphite react with dimethylsulphoxide-sulphur dioxide to form iron(II) disulphate and copper(II) disulphate respectively. Hydrated sulphites of manganese(II), iron(II), magnesium(II) and calcium(II) were also prepared. The properties of the sulphites were investigated using thermogravimetric and IR measurements.     

Measurement of mass and lifetime of hadronically produced charmed F mesons

We have observed three unique decays of F mesons into KKπ and five unique decays of D mesons into KKπ using the NA11 spectrometer together with a telescope of high-resolution silicon microstrip detectors. The mass value obtained for the F meson is (1975 ± 4) MeV, the lifetime (3.2±_1.3^3..0)×10^?13 s.     

Improved Measurement of the Cabibbo-Kobayashi-Maskawa angle alpha using B0(B) --> rho+rho- decays

We present results from an analysis of B(0)B(0)--> rho(+)rho(-) using 232 x 10(6) Gamma (4S) --> BB decays collected with the BABAR detector at the PEP-II asymmetric-energy B factory at SLAC. We measure the longitudinal polarization fraction f(L) = 0.978 +/- 0.014(stat) + 0.021 / -0.029(syst) and the CP-violating parameters S(L)= -0.33 +/- 0.24(stat) + 0.08 / -0.14(syst) and C(L)= -0.03 +/- 0.18(stat) +/- 0.09(syst). Using an isospin analysis of B --> rhorho decays, we determine the unitarity triangle parameter alpha. The solution compatible with the standard model is alpha = (100 +/- 13) degrees.     

Measurement of the branching fraction of Gamma(4S) --> B0B0

We report the first measurement of the branching fraction f(00) for Gamma(4S) --> B(0)B(0). The data sample consists of 81.7 fb(-1) collected at the Gamma(4S) resonance with the BABAR detector at the SLAC PEP-II asymmetric-energy e(+)e(-) storage ring. Using partial reconstruction of the decay B(0) --> D(*+) l(-)nu(l) in which only the charged lepton and the soft pion from the decay D(*+) --> D(0)pi(+) are reconstructed, we obtain f(00) = 0.487 +/- 0.010(stat) +/- 0.008(syst). Our result does not depend on the branching fractions of B(0) --> D(*+)l(-)nu(l) and D(*+) --> D(0)pi(+) decays, on the ratio of the charged and neutral B meson lifetimes, nor on the assumption of isospin symmetry.     

Search for the rare leptonic decay B--->tau-nutau

We present a search for the decay B(-)--> tau(-)nu(tau) in a sample of 88.9 x 10(6) BB pairs recorded with the BABAR detector at the Stanford Linear Accelerator Center B factory. One of the two B mesons from the Gamma(4S) is reconstructed in a hadronic or a semileptonic final state, and the decay products of the other B in the event are analyzed for consistency with a B(-) --> tau(-)nu(tau) decay. We find no evidence of a signal and set an upper limit on the branching fraction of B(B(-) --> tau(-) nu(tau)) < 4.2 x 10(-4) at the 90% confidence level.     

Search for lepton flavor violation in the decay tau+/- --> mu+ gamma

A search for the nonconservation of lepton flavor number in the decay tau(+/-) --> mu(+/-) has been performed using 2.07 x 10(8) e(+ )e(-) tau(+) tau(-) events produced at a center-of-mass energy near 10.58 GeV with the BABAR detector at the PEP-II storage ring. We find no evidence for a signal and set an upper limit on the branching ratio of Beta(tau(+/-) --> mu(+/-) gamma) < 6.8 x 10(-8) at 90% confidence level.     

Limit on the B0-->rho0rho0 branching fraction and implications for the Cabibbo-Kobayashi-Maskawa angle alpha

We search for the decay B0-->rho(0)rho(0) in a data sample of about 227x10(6) Upsilon(4S)-->BB decays collected with the BABAR detector at the PEP-II asymmetric-energy e(+)e(-) collider at SLAC. We find no significant signal and set an upper limit of 1.1x10(-6) at 90% C.L. on the branching fraction. As a result, the uncertainty due to penguin contributions on the Cabibbo-Kobayashi-Maskawa unitarity angle alpha measured in B-->rhorho decays is decreased to 11 degrees at 68% C.L.     

Measurements of the branching fraction and CP-violation asymmetries in B0-->f0(980)K0S

We present measurements of the branching fraction and CP-violating asymmetries in the decay B0-->f0(980)K0S. The results are obtained from a data sample of 123 x 10(6) Upsilon(4S)-->BB decays. From a time-dependent maximum likelihood fit, we measure the branching fraction B(B0-->f0(980)(-->pi+pi-)K0)=(6.0+/-0.9+/-0.6+/-1.2)x10(-6), the mixing-induced CP violation parameter S=-1.62(+0.56)(-0.51)+/-0.09+/-0.04, and the direct CP violation parameter C=0.27+/-0.36+/-0.10+/-0.07, where the first errors are statistical, the second systematic, and the third due to model uncertainties. We measure the f0(980) mass and width to be mf0(980)=(980.6+/-4.1+/-0.5+/-4.0) MeV/c2 and Gammaf0(980)=(43(+12)(-9)+/-3+/-9) MeV/c2, respectively.