Optimal Monotone Drawings of Trees

A monotone drawing of a graph G is a straight-line drawing of G such that, for every pair of vertices u,w in G, there exists abpath P_{uw} in G that is monotone in some direction l_{uw}. (Namely, the order of the orthogonal projections of the vertices of P_{uw} on l_{uw} is the same as the order they appear in P_{uw}.) The problem of finding monotone drawings for trees has been studied in several recent papers. The main focus is to reduce the size of the drawing. Currently, the smallest drawing size is O(n^{1.205}) x O(n^{1.205}). In this paper, we present an algorithm for constructing monotone drawings of trees on a grid of size at most 12n x 12n. The smaller drawing size is achieved by a new simple Path Draw algorithm, and a procedure that carefully assigns primitive vectors to the paths of the input tree T. We also show that there exists a tree T_0 such that any monotone drawing of T_0 must use a grid of size Omega(n) x Omega(n). So the size of our monotone drawing of trees is asymptotically optimal

On W_2 lifting of Frobenius of Algebraic Surfaces

We completely decide which minimal algebraic surfaces in positive characteristics allow a lifting of their Frobenius over the trucated witt rings of lengh 2.Comment: 10 pages. Comments are welcom

Local convergence of critical random trees and continuous-state branching processes

We study the local convergence of critical Galton-Watson trees and Levy trees under various conditionings. Assuming a very general monotonicity property on the functional of random trees, we show that random trees conditioned to have large functional values always converge locally to immortal trees. We also derive a very general ratio limit property for functionals of random trees satisfying the monotonicity property. Then we move on to study the local convergence of critical continuous-state branching processes, and prove a similar result. Finally we give a definition of continuum condensation trees, which should be the correct local limits for certain subcritical Levy trees under suitable conditionings.Comment: 22 page

Frobenius splitting of projective toric bundles

We give several mild conditions on a toric bundle on a nonsingular toric variety under which the projectivization of the toric bundle is Frobenius split.Comment: 9 pages. Comments are welcom

Identical Relations among Transverse Parts of Variant Green Functions and the Full Vertices in Gauge Theories

The identical relations among the transverse parts of variant vertex functions are derived by computing the curl of the time-ordered products of three-point Green functions involving the vector, the axial-vector and the tensor current operators, respectively. These transverse relations are coupled each other. Combining these transverse relations with the normal (longitudinal) Ward-Takahashi identities forms a complete set of constraint relations for three-point vertex functions. As a consequence, the full vector, the full axial-vector and the full tensor vertex functions in the momentum space are exactly obtained.Comment: 12 pages, revte

Particle Physics Inflation Model Constrained from Astrophysics Observations

The early Universe inflation is well known as a promising theory to explain the origin of large scale structure of the Universe, a causal theory for the origin of primordial density fluctuations which may explain the observed density inhomogeneities and cosmic microwave fluctuations in the very early Universe, and to solve the early universe pressing problems for the standard hot big bang theory. For a resonable inflation model, the potential during inflation must be very flat in, at least, the direction of the inflaton. To construct a resonable inflation model, or the inflaton potential, all the known related astrophysics observations should be included. For a general tree-level hybrid inflation potential, which is not discussed fully so far for the quartic term, the parameters in it are shown how to be constrained via the astrophysics data observed and to be obtained to the expected accuracy by the soon lauched MAP and PLANCK satellite missions, as well as the consistent cosmology requirements. We find the effective inflaton mass parameter is in the TeV range, and the quartic term's self-coupling constant tiny, needs fine-tunning.Comment: 10 page

Transverse Ward-Takahashi Relation for the Fermion-Boson Vertex Function in 4-dimensional QED

I present a general expression of the transverse Ward-Takahashi relation for the fermion-boson vertex function in momentum space in 4-dimensional QED, from which the corresponding one-loop expression is derived straightforwardly. Then I deduce carefully this transverse Ward-Takahashi relation to one-loop order in d-dimensions, with $d = 4 + \epsilon$. The result shows that this relation in d-dimensions has the same form as one given in 4-dimensions and there is no need for an additional piece proportional to $(d-4)$ to include for this relation to hold in 4-dimensions. This result is confirmed by an explicit computation of terms in this transverse WT relation to one-loop order. I also make some comments on the paper given by Pennington and Williams who checked the transverse Ward-Takahashi relation at one loop order in d-dimensions.Comment: 15 page

Nonperturbative Fermion-Boson Vertex Function in Gauge Theories

The nonperturbative fermion-boson vertex function in four-dimensional Abelian gauge theories is self-consistently and exactly derived in terms of a complete set of normal (longitudinal) and transverse Ward-Takahashi relations for the The nonperturbative fermion-boson vertex function in four-dimensional Abelian gauge theories is self-consistently and exactly derived in terms of a complete set of normal(longitudinal) and transverse Ward-Takahashi relations for the fermion-boson and the axial-vector vertices in the case of massless fermion, in which the possible quantum anomalies and perturbative corrections are taken into account simultaneously. We find that this nonperturbative fermion-boson vertex function is expressed nonperturbatively in terms of the full fermion propagator and contains the contributions of the transverse axial anomaly and perturbative corrections. The result that the transverse axial anomaly contributes to the nonperturbative fermion-boson vertex arises from the coupling between the fermion-boson and the axial-vector vertices through the transverse Ward-Takahashi relations for them and is a consequence of gauge invariance.Comment: 11 pages, RevTa

Transverse Symmetry Transformations and the Quark-Gluon Vertex Function in QCD

The transverse symmetry transformations associated with the normal symmetry transformations in gauge theories are introduced, which at first are used to reproduce the transverse Ward-Takahashi identities in the Abelian theory QED. Then the transverse symmetry transformations associated with the BRST symmetry and chiral transformations in the non-Abelian theory QCD are used to derive the transverse Slavnov-Taylor identities for the vector and axial-vector quark-gluon vertices, respectively. Based on the set of normal and transverse Slavnov-Taylor identities, an expression of the quark-gluon vertex function is derived, which describes the constraints on the structure of the quark-gluon vertex imposed from the underlying gauge symmetry of QCD alone. Its role in the study of the Dyson-Schwinger equation for the quark propagator in QCD is discussed.Comment: 13 pages, no figur

Angular Momentum-Free of the Entropy Relations for Rotating Kaluza-Klein Black Holes

Based on a mathematical lemma related to the Vandermonde determinant and two theorems derived from the first law of black hole thermodynamics, we investigate the angular momentum independence of the entropy sum as well as the entropy product of general rotating Kaluza-Klein black holes in higher dimensions. We show that for both non-charged rotating Kaluza-Klein black holes and non-charged rotating Kaluza-Klein-AdS black holes, the angular momentum of the black holes will not be present in entropy sum relation in dimensions $d\geq4$, while the independence of angular momentum of the entropy product holds provided that the black holes possess at least one zero rotation parameter $a_j$ = 0 in higher dimensions $d\geq5$, which means that the cosmological constant does not affect the angular momentum-free property of entropy sum and entropy product under the circumstances that charge $\delta=0$. For the reason that the entropy relations of charged rotating Kaluza-Klein black holes as well as the non-charged rotating Kaluza-Klein black holes in asymptotically flat spacetime act the same way, it is found that the charge has no effect in the angular momentum-independence of entropy sum and product in asymptotically flat spactime.Comment: Final version, accepted for publication by International Journal of Theoretical Physic