Uniqueness result for the BSDE whose generator is monotonic in y and uniformly continuous in z

In this Note, we prove that if g is continuous, monotonic and has a general growth in y, g is uniformly continuous in z, and $(g{(t,0,0))}_{t\in [0,T]}$ is square integrable, then for each square integrable terminal condition ξ, the one-dimensional backward stochastic differential equation (BSDE) with the generator g has a unique solution. This generalizes some corresponding (one-dimensional) results.     

Pairs of quadratic forms over a quadratic field extension

Let F be a field of characteristic distinct from 2, $L=F(\sqrt{d})$ a quadratic field extension. Let further f and g be quadratic forms over L considered as polynomials in n variables, $M_f$, $M_g$ their matrices. We say that the pair $(f,g)$ is a k-pair if there exist $S\in G{L}_n(L)$ such that all the entries of the $k\times k$ upper-left corner of the matrices $S{M}_f{S}^t$ and $S{M}_g{S}^t$ are in F. We give certain criteria to determine whether a given pair $(f,g)$ is a k-pair. We consider the transfer ${\mathrm{cor}}_{L(t)/F(t)}$ determined by the $F(t)$-linear map $s:L(t)\to F(t)$ with $s(1)=0$, $s(\sqrt{d})=1$, and prove that if $\dim {\mathrm{cor}}_{L(t)/F(t)}{(f+tg)}_{an}\le 2(n?k)$, then $(f,g)$ is a $[\frac{k+1}{2}]$-pair. If, additionally, the form $f+tg$ does not have a totally isotropic subspace of dimension $p+1$ over $L(t)$, we show that $(f,g)$ is a $(k?2p)$-pair. In particular, if the form $f+tg$ is anisotropic, and $\dim {\mathrm{cor}}_{L(t)/F(t)}{(f+tg)}_{an}\le 2(n?k)$, then $(f,g)$ is a k-pair.     

Odd values of the Rogers–Ramanujan functions

Let $g(n)$ and $h(n)$ be the coefficients of the Rogers–Ramanujan identities. We obtain asymptotic formulas for the number of odd values of $g(n)$ for odd n, and $h(n)$ for even n, which improve Gordon's results. We also obtain lower bounds for the number of odd values of $g(n)$ for even n, and $h(n)$ for odd n.     

On the Erdös–Lax inequality concerning polynomials

Let $P(z)$ be a polynomial of degree n and for any complex number α, let $D_{\alpha }P(z):=nP(z)+(\alpha ?z)P^{\prime }(z)$ denote the polar derivative of $P(z)$ with respect to α. In this paper, we present an integral inequality for the polar derivative of a polynomial. Our theorem includes as special cases several interesting generalisations and refinements of Erdöx–Lax theorem.