In this paper we study integral operators of the form Tf(x) = |x − a1y| −α1 . . . |x − amy| mf(y) dy, α1 + . . . + αm = n. We obtain the L (w) boundedness for them, and a weighted (1, 1) inequality for weights w in Ap satisfying that there exists c > 1 such that w(aix) 6 cw(x) for a.e. x ∈ n , 1 6 i 6 m. Moreover, we prove ‖Tf‖BMO 6 c‖f‖∞ for a wide family of functions f ∈ L( ).

A Domain of Positivity D is an open convex cone associated with a nonsingular symmetric matrix S, called the characteristic, such that xÇzD if and only if x'Sy>0 for all y(~D* As such they were introduced by Koecher (1) in generalization of the cone of positive definite matrices studied by Siegel. The automorphisms of D are the nonsingular linear transformations mapping D onto itself. The group...

Bousquet-Mélou and Petkovšek investigated the generating functions of multivariate linear recurrences with constant coefficients. We will give a reinterpretation of their theory by means of division theorems for formal power series, which clarifies the structural background and provides short, conceptual proofs. In addition, extending the division to the context of differential operators, the c...

We show that the only ow solving the stochastic di erential equation (SDE) on R dXt = 1{Xt>0}W+(dt) + 1{Xt<0}dW−(dt), where W and W− are two independent white noises, is a coalescing ow we will denote φ±. The ow φ± is a Wiener solution. Moreover, K = E[δφ±|W+] is the unique solution (it is also a Wiener solution) of the SDE K s,tf(x) = f(x) + ∫ t s Ks,u(1R+f )(x)W+(du) + 1 2 ∫ t s Ks,uf ′′(x)du...

We establish a …rst order asymptotic for the entropy integrals Z I pn log pn W 2 and Z I pn log (pnW ) 2 W 2 where fpngn=0 are the orthonormal polynomials associated with the exponential weight W . 1 The Result Let I = (c; d) be a real interval, where 1 c < 0 < d 1; and let Q : I ! [0;1) be convex. Let W := exp ( Q) and assume that all power moments Z I xW (x)dx; n = 0; 1; 2; 3; ::: are …nite. ...

Let c(n, q) be the number of connected labeled graphs with n vertices and q 5 N = ( ) edges. Let x = q/n and k = q n. We determine functions w k 1 , a(x ) and cp(x) such that c (n , q) w k ( z ) e n r p ( x ) e o ( x ) uniformly for all n and q 2 n. If Q > O is fixed, n+= and 4q > ( 1 + ~ ) n log n, this formula simplifies to c(n, q) ( t ) exp(-ne-zq’n). On the other hand, if k = o(n”’), this f...

suppose $f$ is a map from a non-empty finite set $x$ to a finite group $g$. define the map $zeta^f_g: glongrightarrow mathbb{n}cup {0}$ by $gmapsto |f^{-1}(g)|$. in this article, we show that for a suitable choice of $f$, the map $zeta^f_g$ is a character. we use our results to show that the solution function for the word equation $w(t_1,t_2,dots,t_n)=g$ ($gin g$) is a character, where $w(t_1,t...

Let 1 p 1 and W : R ! (0;1) be continuous. Does W admit a Jackson Theorem in Lp? That is, does there exist a sequence f ng 1 n=1 of positive numbers with limit 0 such that inf deg(P ) n k (f P )W kLp(R) n k f W kLp(R) for all absolutely continuous f with k f 0W kLp(R) …nite? We show that such a theorem is true i¤ lim x!1 W 1 Lq [0;x] kWkLp[x;1) = 0; where q is the conjugate parameter of p. In a...

" r? Let X~= X and Y~= Y for t = 1, 2 . . . . . X, ,= ~ X ~ and !1, = Y~. Let S be any set, cg = {w(. 1" I s ) lseS} t = l t = l n be a set of (a x 2) stochastic matrices w( ' l ' ls) , and St=S , t= 1 . . . . . n. For every sn=(s 1 . . . . . s " ) e H S t n t = l define P (. ]" Is.) by P (y.[x.ls.) = I~ w (yt]x~lst) for every x. = (x a . . . . . x")~ X . and every y. = (yl, ... , y.)e 11.. t =...

Let In be the set of partial one-to-one transformations on chain Xn={1,2, . , n} and, for each α in In, let h(α)=|Imα|, f(α)=|{x∈Xn:xα=x}| and w(α)=max(Imα). this note, we obtain formulae involving binomial coefficients F(n; p, m, k)=|{α ∈ In:h(α)=p∧f(α)=m∧w(α)=k}| F(n;·, In:f(α)=m∧w(α)=k}| analogous results derangements In.