The tetrahedron A4 group has been widely used in studying neutrino mixing matrix. It provides a natural framework of model building for the tri-bimaximal mixing matrix. In this class of models, it is necessary to have two Higgs fields, χ and χ′, transforming under A4 as 3 with one of them having vacuum expectation values for the three components to be equal and another having only one of the co...

An Euler characteristic χ is definable if for every definable function f : X → Y and every r ∈ R, the set {y ∈ Y : χ(f−1(y)) = r} is definable. If R = Z/nZ and M is a finite structure, there is a (unique) Euler characteristic χ : Def(M) → Z/nZ assigning every set its size mod n. This χ is always strong and ∅definable. If M is an ultraproduct of finite structures, then there is a canonical stron...

We obtain a number of new bounds for exponential sums of the type S(χ, f) = Pp−1 x=1 χ(x)ep(f(x)), with p a prime, f(x) = Pr i=1 aix ki , ai, ki ∈ Z, 1 ≤ i ≤ r and χ a multiplicative character (mod p). The bounds refine earlier Mordell-type estimates and are particularly effective for polynomials in which a certain number of the ki have a large gcd with p − 1. For instance, if f(x) = Pm i=1 aix...

Let A 11 be the moduli space of (1, 11)-polarized abelian surfaces with a canonical level structure. Let χ be a primitive character of order 5 with conductor 11. In this paper we construct five endoscopic lifts Πi, 0 6 i 6 4 from two elliptic modular forms f ⊗ χ of weight 2 and g ⊗ χ of weight 4 with complex multiplication by Q( √ −11) such that Πi∞ gives a non-holomorphic differential form on ...

A theory of pump-probe spectroscopy is developed in which optical fields drive two-photon Raman transitions between ground states of an ensemble of three-level atoms. Effects related to the recoil the atoms undergo as a result of their interactions with the fields are fully accounted for in this theory. The linear absorption coefficient of a weak probe field in the presence of two pump fields o...

We investigate hierarchies of semidefinite approximations for the chromatic number χ(G) of a graph G. We introduce an operator Ψ mapping any graph parameter β(G), nested between the stability number α(G) and χ(G), to a new graph parameter Ψ β (G), nested between α(G) and χ(G); Ψ β (G) is polynomial time computable if β(G) is. As an application, there is no polynomial time computable graph param...

We present a higher-twist production mechanism for η c and χ J charmonium at large momentum fraction x F in pion-nucleon collisions. The higher-twist contribution is essentially independent of x F and therefore becomes dominant at large x F , where the leading-twist contribution falls off as (1 − x F) 3. We show that the higher-twist mechanism produces longitudinally polarized χ 1 and χ 2. For ...

In this paper, we introduce, for a multiplier χ, a notion of generalized power function x 7→ ∆χ(x), defined on the homogeneous cone P of a Vinberg algebra A. We then extend to A the famous Gindikin result, that is we determine the set of multipliers χ such that the map θ 7→ ∆χ(θ), defined on P∗, is the Laplace transform of a positive measure Rχ. We also determine the set of χ such that Rχ gener...

By Cayley’s theorem, any finite group G of order n can be regarded as a subgroup of the symmetric group $n. Let χ be any irreducible complex character of G and let V n χ (G) denote the symmetry classes of tensors associated with G and χ. In this paper assuming the Cayley representation of G, we obtain a formula for the dimension of V n χ (G) and discuss its non-vanishing in general. A necessary...

The hybrid Chi (χ) language is a formalism for modeling, simulation and verification of hybrid systems. One of the most widely known hybrid system formalisms is that of hybrid automata. The formal translation of χ to hybrid automata enables verification of χ specifications using existing hybrid automata based verification tools. In this paper, we describe the translation from χ to hybrid automa...