A q-analog of the exponential formula

A q-analog of functional composition for Eulerian generating functions is introduced and applied to the enumeration of permutations by inversions and distribution of left-right maxima.     

A q-analog of the exponential formula

A q-analog of functional composition for Eulerian generating functions is introduced and applied to the enumeration of permutations by inversions and distribution of left-right maxima.     

Convergence domains for the campbell-baker-hausdorff formula

Three neighbourhoods of zero within which the Campbell-Baker-Hausdorff series for log(e^xe^y) converges are compared. Two are large enough to include the unit sphere under a norm compatible with Lie multiplication.     

Irreducible weight modules for the Virasoro-like algebra and its q-analog

In this paper, using Larsson’s functor with irreducible 𝔰𝔩₂-modules V, we construct a class of ?²-graded modules for the Virasoro-like algebra and its q-analogs. We determine the irreducibility of these modules for finite-dimensional or infinite-dimensional V using a unified method. In particular, these modules provide new irreducible weight modules with infinite-dimensional weight spaces for the corresponding algebras.     

q-类似Virasoro-like代数模的导子

记Lq为两个变量的量子环面上的斜导子李代数,当0≠q∈C为非单位时,Lq就是q-类似Virasoro-like代数.本文给出了文中构造的Lq的模上的导子及一上同调群H^1（Lq,M）.     

A simplification of the Kovarik formula

Let P,Q be two idempotents on a Hilbert space. Z.V. Kovarik (Z.V. Kovarik, Similarity and interpolation between projectors, Acta Sci. Math. (Szeged) 39 (1977) 341-351) showed that when P+Q−I is invertible, the formula K(P,Q)=P⁻²(P+Q−I)Q gives the only idempotent such that R(K)=R(P), N(K)=N(Q), where N(T) and R(T) denote the nullspace and the range of a bounded linear operator T on a Hilbert space, respectively. This formula was later extended to the context of Banach algebras and used in 1983 by J. Esterle to show that two homotopic idempotents may always be connected by a polynomial idempotent valued path. In the present paper, we give a simplification of Kovarik's original formula and one natural generalization of it.     

A generalization of the Christoffel-Schwartz formula

As a generalization of the well-known Christoffel-Schwartz formula, a formula is obtained for mapping the interior of the unit disk onto a domain whose boundary consists of n arcs of curves, each of which, under the choice of some branch of the transformation ζ=w^m, m>0, passes through a rectilinear segment of the ζ-plane. It is shown that the class B_m of Bazilevich functions coincides with the class ¯L_m of functions representable by means of the obtained formula of the special type.     

A trace formula

To my Father, William F. Eberlein.     

A collection formula

A table of partially ordered sets, corresponding to the basis commutators (in Hall's sense) of weight 9, is given. On the basis of this table, the exponents in the collection formula can be found by elementary combinatorial methods. An example is given.Translated from Zapiski Nauchnykh Seminarov Leningardskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 46, pp. 59–63, 1974.     

The derivation algebra and the universal central extension of the q-analog of the virasoro-like algebra *

In this paper, the derivation algebra and the universal central extension of the q-analog of the Virasoro-like algebra are given.     

A combinatorial proof of the Mehler formula

A combinatorial proof of the Mehler formula on Hermite polynomials is given that is based upon the techniques of the partitional complex.     

A simple proof of the kinematic formula

We give a short proof of the kinematic fundamental formula in geometric probability theory, using a few basic properties of the Haar measure and some convexity.     

A functional extension of the Ito formula

We develop a non-anticipative pathwise calculus for functionals of a Brownian semimartingale and its quadratic variation. A functional Ito formula is obtained for locally Lipschitz functionals of a Brownian semimartingale and its quadratic variation. As a result we obtain a constructive martingale representation theorem for Brownian martingales verifying a regularity property.     

A variational proof of the Gauss-Bonnet formula

The classical Gauss-Bonnet formula has the form I(g_ij)=2, where I(g_ij) represents a sum of three terms each of which depends on the metric tensor g_ij. It is shown that the first variation I of I(g_ij) with respect to the metric g_ij vanishes and that for the Euclidean metric _ij we have I(_ij)=2. From this the formula I(g_ij)=2 follows. In the process, explicit expressions are obtained for the first variation of each of the three terms which comprise I(g_ij). Furthermore, a general expression for the first variation of a multiple integral whose integrand is a scalar density depending on the metric tensor g_ij and its derivatives up to the second order is obtained with the aid of results of Rund [1].