Explicit formulas for 2-characters

Ganter and Kapranov associated a 2-character to 2-representations of a finite group. Elgueta classified 2-representations in the category of 2-vector spaces 2Vectk in terms of cohomological data. We give an explicit formula for the 2-character in terms of this cohomological data and derive some consequences.     

Explicit formulas for the codimension of certain T-ideals

Sofia, Bulgaria. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 29, No. 6, pp. 30–36, November–December, 1988.     

Explicit formulas for the waldspurger and bessel models

This paper studies certain models of irreducible admissible representations of the split special orthogonal group SO(2n+1) over a nonarchimedean local field. Ifn=1, these models were considered by Waldspurger. Ifn=2, they were considered by Novodvorsky and Piatetski-Shapiro, who called them Bessel models. In the works of these authors, uniqueness of the models is established; in this paper functional equations and explicit formulas for them are obtained. As a global application, the Bessel period of the Eisenstein series on SO(2n+1) formed with a cuspidal automorphic representation π on GL(n) is computed—it is shown to be a product of L-series. This generalizes work of Böcherer and Mizumoto forn=2 and base field ?, and puts it in a representation-theoretic context. In an appendix by M. Furusawa, a new Rankin-Selberg integral is given for the standardL-function on SO(2n+1)×GL(n). The local analysis of the integral is carried out using the formulas of the paper.     

Further Pieri-type formulas for the nonsymmetric Macdonald polynomial

The branching coefficients in the expansion of the elementary symmetric function multiplied by a symmetric Macdonald polynomial P _?? (z) are known explicitly. These formulas generalise the known r=1 case of the Pieri-type formulas for the nonsymmetric Macdonald polynomials E _?? (z). In this paper, we extend beyond the case r=1 for the nonsymmetric Macdonald polynomials, giving the full generalisation of the Pieri-type formulas for symmetric Macdonald polynomials. The decomposition also allows the evaluation of the generalised binomial coefficients $\tbinom{\eta }{\nu }_{q,t}$ associated with the nonsymmetric Macdonald polynomials.     

Explicit formulas for matrix-valued Padé approximants

Formulas are derived for calculating the [N + m, N] Padé approximants to an operator-valued function from the first 2N + m + 1 expansion coefficients of the perturbation series for this function.     

Explicit formulas for the estimation of linear regression coefficients

Lithuanian Mathematical Journal -     

A method for proving polynomial enumeration formulas

We present an elementary method for proving enumeration formulas which are polynomials in certain parameters if others are fixed and factorize into distinct linear factors over Z. Roughly speaking the idea is to prove such formulas by “explaining” their zeros using an appropriate combinatorial extension of the objects under consideration to negative integer parameters. We apply this method to prove a new refinement of the Bender-Knuth (ex-)Conjecture, which easily implies the Bender-Knuth (ex-)Conjecture itself. This is probably the most elementary way to prove this result currently known. Furthermore we adapt our method to q-polynomials, which allows us to derive generating function results as well. Finally we use this method to give another proof for the enumeration of semistandard tableaux of a fixed shape which differs from our proof of the Bender-Knuth (ex-)Conjecture in that it is a multivariate application of our method.     

Explicit formulas for repeated games with absorbing states

Explicit formulas are given for the asymptotic value lim_{λ → 0} v(λ) and the asymptotic minmax lim w(λ) of finite λ-discounted absorbing games together with new simple proofs for the existence of the limits as λ goes to zero. Similar characterizations for stationary Nash equilibrium payoffs are obtained. The results may be extended to absorbing games with compact metric action sets and jointly-continuous payoff functions.     

Explicit formulas for wavelet-homogenized coefficients of elliptic operators

Wavelet-based homogenization provides a method for constructing a coarse-grid discretization of a variable–coefficient differential operator that implicitly accounts for the influence of the fine scale medium parameters on the coarse scale of the solution. The method is applied to discretizations of operators of the form in one dimension and μ(x)Δ in one and more dimensions. The resulting homogenized matrices are shown to correspond to differential operators of the same (or closely related) form. In dimension one, results are obtained for periodic two-phase and for arbitrary coefficients μ(x). For periodic two-phase coefficients, the homogenized coefficients may be computed exactly as the harmonic mean of the function μ. For non-periodic coefficients, the “mass-lumping” approximation results in an explicit formula for the homogenized coefficients. In higher dimensions, results are obtained for operators of the form μ(x)Δ, where μ(x) may or may not be periodic; explicit formulae for the homogenized coefficients are also derived. Numerical examples in 1D and 2D are also presented.     

Explicit formulas for Drinfeld modules and their periods

We provide explicit series expansions for the exponential and logarithm functions attached to a rank r Drinfeld module that generalize well-known formulas for the Carlitz exponential and logarithm. Using these results we obtain a procedure and an analytic expression for computing the periods of rank 2 Drinfeld modules and also a criterion for supersingularity.     

Explicit formulas for Vasil'ev invariants of the fourth order for knots

In this work, the integer-valued basic Vasil'ev invariants of the fourth order for the oriented knots V ₄ ¹ (K) and V ₄ ² (K) are presented. The explicit diagram-arrow formulas with rational coefficients defined according to the Gaussian diagram of a knot are obtained for them. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 53, Suzdal Conference-2006, Part 1, 2008.     

Cubic formulas for computing the zeros of certain quaternionic polynomial

In this paper, we derive the explicit formulas for computing the zeros of certain cubic quaternionic polynomial. From these, we obtain a necessary and sufficient condition to quaternionic cubic polynomial have a spherical zero, and some examples are also provided. Moreover, we will discuss some applications of the cubic quaternionic formulas. Copyright © 2017 John Wiley & Sons, Ltd.     

Explicit formulas for biharmonic submanifolds in non-Euclidean 3-spheres

We obtain the parametric equations of all biharmonic Legendre curves and Hopf cylinders in the 3-dimensional unit sphere endowed with the modified Sasakian structure defined byTanno.     

Explicit formulas for Hankel norm approximations of infinite-dimensional systems

Recently Hankel norm approximation for finite dimensional systems has been studied extensively, both for continuous time systems and discrete time systems. One of the approaches is to combine the work of J.A. Ball and J.W. Helton with results on Wiener-Hopf factorization to produce explicit formulas for the solutions. This has been done by J.A. Ball and A.C.M. Ran for the finite-dimensional case and for a class of infinite-dimensional systems by A.C.M. Ran. Here we show that this approach can be extended to a much larger class of infinite-dimensional systems; those with an exponentially stable semigroup and unbounded input and output operators which satisfy a smoothness condition. This class is larger than the nuclear class for which this problem was solved by K. Glover, R.F. Curtain and J.R. Partington by an approximation approach.     

Explicit ruin formulas for models with dependence among risks

We show that a simple mixing idea allows one to establish a number of explicit formulas for ruin probabilities and related quantities in collective risk models with dependence among claim sizes and among claim inter-occurrence times. Examples include compound Poisson risk models with completely monotone marginal claim size distributions that are dependent according to Archimedean survival copulas as well as renewal risk models with dependent inter-occurrence times.     

Explicit formulas for spectral characteristics and solutions of the sh-Gordon equation

To the sh-Gordon equation

Optimal approximate formulas with polynomial precision

For any given linear functional bounded over the Hardy spaceH ², we obtain explicitly optimal approximate formulas, with polynomial precision, based on nodes lying equispaced on |z|=r. In particular, we obtain optimal formulas with polynomial precision for interpolation, numerical differentiation and integration.