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The Hill estimator is often used to infer the power behavior in tails of experimental distribution functions. This estimator is known to produce bad results in certain situations which have lead to the so-called Hill horror plots. In this brief note, we propose an improved estimator which is simple and coherent and often provides an efficient remedy in the bad situations, especially when the distribution is decreasing slowly, when the data is restricted by external cuts to lie within a finite domain, or even when the distribution is increasing., Comment: 22 pages, 4 figures

Ternary algebras, constructed from ternary commutators, or as we call them, ternutators, defined as the alternating sum of products of three operators, have been shown to satisfy cubic identities as necessary conditions for their existence. Here we examine the situation where we permit identities not solely constructed from ternutators or nested ternutators and we find that in general, these impose additional restrictions; for example, the anti-commutators or commutators of the operators must obey some linear relations among themselves., Comment: 10 pages

We consider a free real vector field propagating in a five dimensional flat space with its fifth dimension compactified either on a strip or on a circle and perform a Kalaza Klein reduction which breaks SO(4,1) invariance while reserving SO(3,1) invariance. Taking into account the Lorenz gauge condition, we obtain from the most general hermiticity conditions for the relevant operators all the allowed boundary conditions which have to be imposed on the fields in the extra-dimension. The physical Kaluza-Klein mass towers, which result in a four-dimensional brane, are determined in the different distinct allowed cases. They depend on the bulk mass, on the parameters of the boundary conditions and on the extra parameter present in the Lagrangian. In general, they involve vector states together with accompanying scalar states., Comment: 28 pages, 4 independent tables

The ternary commutator or ternutator, defined as the alternating sum of the product of three operators, has recently drawn much attention as an interesting structure generalising the commutator. The ternutator satisfies cubic identities analogous to the quadratic Jacobi identity for the commutator. We present various forms of these identities and discuss the possibility of using them to define ternary algebras., Comment: 12 pages, citation added

All the boundary conditions compatible with the reduction of a five dimensional spinor field of bulk mass $M$ in a compactified warped space to a four dimensional brane are derived from the hermiticity conditions of the relevant operator. The possible presence of metric singularities is taken into account. Examples of resulting Kaluza-Klein spinor towers are given for a representative set of values for the basic parameters of the model and of the parameters describing the allowed boundary conditions, within the hypothesis that there exists one-mass-scale-only, the Planck mass. In many cases, the lowest mass in the tower is small and very sensitive to the parameters while the other masses are much higher and become more regularly spaced. In these cases, if a basic fermion of the standard model (lepton or quark) happens to be the lowest mass of a Kaluza-Klein tower, the other masses would be much larger and weakly dependent on the fermion which defines the tower., Comment: 39 pages

Special p-forms are forms which have components \phi_{\mu_1...\mu_p} equal to +1,-1 or 0 in some orthonormal basis. A p-form \phi\in \Lambda^p R^d is called democratic if the set of nonzero components {\phi_{\mu_1...\mu_p}} is symmetric under the transitive action of a subgroup of O(d,Z) on the indices {1,...,d}. Knowledge of these symmetry groups allows us to define mappings of special democratic p-forms in d dimensions to special democratic P-forms in D dimensions for successively higher P \geq p and D \geq d. In particular, we display a remarkable nested stucture of special forms including a U(3)-invariant 2-form in six dimensions, a G_2-invariant 3-form in seven dimensions, a Spin(7)-invariant 4-form in eight dimensions and a special democratic 6-form \Omega in ten dimensions. The latter has the remarkable property that its contraction with one of five distinct bivectors, yields, in the orthogonal eight dimensions, the Spin(7)-invariant 4-form. We discuss various properties of this ten dimensional form., Comment: 25 pages

Considering a massive or massless free spinor field propagating in a flat five dimensional space with its fifth dimension compactified either on a strip or on a circle, we analyse the procedure of generation of the four dimensional Kaluza-Klein spinor mass towers. Requiring the five dimensional Dirac operator to be symmetric, the set of all the allowed boundary conditions is obtained. In the determination of the boundary conditions and in the Kaluza-Klein reduction equations, the SO(3,1) and parity invariances in the space-time subspace are carefully taken into account. The equations determining the mass towers are written in full generality. A few numerical examples are given., Comment: 17 pages

The version of the warp model that we proposed to explain the mass scale hierarchy has been extended by the introduction of one or more singularities in the metric. We restricted ourselves to a real massless scalar field supposed to propagate in a five dimensional bulk with the extradimension being compactified on a strip or on a circle. With the same emphasis on the hermiticity and commutativity properties of the Kakuza Klein operators, we have established all the allowed boundary conditions to be imposed on the fields. From them, for given positions of the singularities, one can deduce either mass eigenvalues building up a Kaluza Klein tower, or a tachyon, or a zero mass state. Assuming the Planck mass to be the high mass scale and by a choice, unique for all boundary conditions, of the major warp parameters, the low lying mass eigenvalues are of the order of the TeV, in this way explaining the mass scale hierarchy. In our model, the physical masses are related to the Kaluza Klein eigenvalues, depending on the location of the physical brane which is an arbitrary parameter of the model. Illustrative numerical calculations are given to visualize the structure of Kaluza Klein mass eigenvalue towers. Observation at high energy colliders like LHC of a mass tower with its characteristic structure would be the fingerprint of the model., Comment: 33 pages, 1 figure

Inspired by the warped Randall Sundrum scenario proposed to solve the mass scale hierarchy problem with a compactified fifth extra dimension, a similar model with no metric singularities has been elaborated. In this framework, the Kaluza-Klein reduction equations for a real massless scalar field propagating in the bulk have been studied carefully from the point of view of hermiticity so as to formulate in a mathematically rigorous way all the possible boundary conditions and corresponding mass eigenvalue towers and tachyon states. The physical masses as observable in our four-dimensional brane are deduced from these mass eigenvalues depending on the location of the brane on the extra dimension axis. Examples of mass towers and tachyons and related field probability densities are presented from numerical computations performed for some arbitrary choices of the parameters of the model., Comment: 34 pages, 5 figures

Considering that the momentum squared in the extra dimensions is the physically relevant quantity for the generation of the Kaluza-Klein mass states, we have reanalyzed mathematically the procedure for five dimensional scalar fields within the Arkhani-Ahmed, Dimopoulos and Dvali scenario. We find new sets of physically allowed boundary conditions. Beside the usual results, they lead to new towers with non regular mass spacing, to lonely mass states and to tachyons. We remark that, since the SO(1,4) symmetry is to be broken due to the compactification of the extra dimensions, the speed of light could be different in the fifth dimension. This would lead to the possible appearance of a new universal constant besides $\hbar$ and $c$., Comment: 20 pages, 1 figure

A special p-form is a p-form which, in some orthonormal basis {e_\mu}, has components \phi_{\mu_1...\mu_p} = \phi(e_{\mu_1},..., e_{\mu_p}) taking values in {-1,0,1}. We discuss graphs which characterise such forms., Comment: 8 pages, V2: a texing error corrected

The requirement of Hermiticity of a Quantum Mechanical Hamiltonian, for the description of physical processes with real eigenvalues which has been challenged notably by Carl Bender, is examined for the case of a Fock space Hamilitonian which is bilinear in two creation and destruction operators. An interpretation of this model as a Schr\"odinger operator leads to an identification of the Hermitian form of the Hamiltonian as the Landau model of a charged particle in a plane, interacting with a constant magnetic field at right angles to the plane. When the parameters of the Hamiltonian are suitably adjusted to make it non-Hermitian, the model represents two harmonic oscillators at right angles interacting with a constant magnetic field in the third direction, but with a pure imaginary coupling, and real energy eigenvalues. It is now ${\cal PT}$ symmetric. Multiparticle states are investigated., Comment: LaTeX2e, 18 pages

The Convolution and Master equations governing the time behavior of the term structure of Interest Rates are set up both for continuous variables and for their discretised forms. The notion of Seed is introduced. The discretised theoretical distributions matching the empirical data from the Federal Reserve System (FRS) are deduced from a discretised seed which enjoys remarkable scaling laws. In particular the tails of the distributions are very well reproduced. These results may be used to develop new methods for the computation of the value-at-risk and fixed-income derivative pricing., Comment: 27 pages, 12 figures

The technique of Pad\'e Approximants, introduced in a previous work, is applied to extended recent data on the distribution of variations of interest rates compiled by the Federal Reserve System in the US. It is shown that new power laws and new scaling laws emerge for any maturity not only as a function of the Lag but also as a function of the average inital rate. This is especially true for the one year maturity where critical forms and critical exponents are obtained. This suggests future work in the direction of constructing a theory of variations of interest rates at a more ``microscopic'' level.}, Comment: 22 pages, 9 figures and 2 tables

For semisimple groups, possibly multiplied by U(1)'s, the number of Yang-Mills gauge fields is equal to the number of generators of the group. In this paper, it is shown that, for non-semisimple groups, the number of Yang-Mills fields can be larger. These additional Yang-Mills fields are not irrelevant because they appear in the gauge transformations of the original Yang-Mills fields. Such non-semisimple Yang-Mills theories may lead to physical consequences worth studying. The non-semisimple group with only two generators that do not commute is studied in detail., Comment: 16 pages, no figures, prepared with ReVTeX4

Self-duality equations for Yang-Mills fields in d-dimensional Euclidean spaces consist of linear algebraic relations amongst the components of the curvature tensor which imply the Yang-Mills equations. For the extension to superspace gauge fields, the super self-duality equations are investigated, namely, systems of linear algebraic relations on the components of the supercurvature, which imply the self-duality equations on the even part of superspace. A group theory based algorithm for finding such systems is developed. Representative examples in various dimensions are provided, including the Spin(7) and G(2) invariant systems in d=8 and 7, respectively., Comment: 51 pages, latex

We present generalisations of N-extended supersymmetry algebras in four dimensions, using Lorentz covariance and invariance under permutation of the N supercharges as selection criteria., Comment: 26 pages, latex file

The classical approach in finance attempts to model the term structure of interest rates using specified stochastic processes and the no arbitrage argument. Up to now, no universally accepted theory has been obtained for the description of experimental data. We have chosen a more phenomenological approach. It is based on results obtained some twenty years ago by physicists, results which show that Pad\'e Approximants are very suitable for approximating large classes of functions in a very precise and coherent way. In this paper, we have chosen to compare Pad\'e Approximants with very low indices with the experimental densities of interest rates variations. We have shown that the data published by the Federal Reserve System in the United States are very well reproduced with two parameters only. These parameters are rather simple functions of the lag and of the maturity and are directly related to the moments of the distributions., Comment: LaTeX, 28 pages, 13 figures

Lorentz covariant generalisations of the notions of supersymmetry, superspace and self-duality are discussed. The essential idea is to extend standard constructions by allowing tangent vectors and coordinates which transform according to more general Lorentz representations than solely the spinorial and vectorial ones of standard lore. Such superspaces provide model configuration spaces for theories of arbitrary spin fields. Our framework is an elegant one for handling higher-dimensional theories in a manifestly SO(3,1) covariant fashion. A further application is the construction of a hierarchy of solvable Lorentz covariant systems generalising four-dimensional self-duality., Comment: 6 pages, latex file

Superalgebras including generators having spins up to two and realisable as tangent vector fields on Lorentz covariant generalised superspaces are considered. The latter have a representation content reminiscent of configuration spaces of (super)gravity theories. The most general canonical supercommutation relations for the corresponding phase space coordinates allowed by Lorentz covariance are discussed. By including generators transforming according to every Lorentz representation having spin up to two, we obtain, from the super Jacobi identities, the complete set of quadratic equations for the Lorentz covariant structure constants. These defining equations for spin two Heisenberg superalgebras are highly overdetermined. Nevertheless, non-trivial solutions can indeed be found. By making some simplifying assumptions, we explicitly construct several classes of these superalgebras., Comment: 20 pages, latex, typos corrected

We generalise the notions of supersymmetry and superspace by allowing generators and coordinates transforming according to more general Lorentz representations than the spinorial and vectorial ones of standard lore. This yields novel SO(3,1)-covariant superspaces, which we call hyperspaces, having dimensionality greater than (4|4) of traditional super-Minkowski space. As an application, we consider gauge fields on complexifications of these superspaces; and extending the concept of self-duality, we obtain classes of completely solvable equations analogous to the four-dimensional self-duality equations., Comment: 29 pages, latex

Let $P(N,V)$ denote the vector space of polynomials of maximal degree less than or equal to $N$ in $V$ independent variables. This space is preserved by the enveloping algebra generated by a set of linear, differential operators representing the Lie algebra $gl(V+1)$. We establish the counterpart of this property for the vector space $P(M,V) \oplus P(N,V)$ for any values of the integers $M,N,V$. We show that the operators preserving $P(M,V) \oplus P(N,V)$ generate an abstract superalgebra (non linear if $\Delta=\mid M-N\mid\geq 2$). A family of algebras is also constructed, extending this particular algebra by $\Delta -1$ arbitrary complex parameters., Comment: 19 pages, latex

The superconformal algebras of Ademollo et al are generalised to a multi-index form. The structure obtained is similar to the Moyal Bracket analogue of the Neveu-Schwarz Algebra., Comment: 8 pages, LaTeX, no figures