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We obtain an asymptotic expansion for the solution of the Cauchy problem for the Korteweg-de Vries (KdV) equation ut + 6uux + ǫ uxxx = 0, u(x, t = 0, ǫ) = u0(x), for ǫ small, near the point of gradient catastrophe (xc, tc) for the solution of the dispersionless equation ut + 6uux = 0. The sub-leading term in this expansion is described by the smooth(More)
OBJECTIVE To study age-related metabolic changes in N-acetylaspartate (NAA), total creatine (tCr), choline (Cho) and myo-inositol (Ins). MATERIALS AND METHODS Proton magnetic resonance spectroscopy (1H-MRS) was performed in the posterior cingulate cortex (PCC) and the left hippocampus (HC) of 90 healthy subjects (42 women and 48 men aged 18-76 years,(More)
PURPOSE To assess the outcome of prostate cancer (PCa) patients diagnosed with oligometastatic disease at recurrence and treated with stereotactic body radiotherapy (SBRT). METHODS Non-castrate patients with up to 3 synchronous metastases (bone and/or lymph nodes) diagnosed on positron emission tomography - computed tomography, following biochemical(More)
We establish the existence of a real solution y(x, T ) with no poles on the real line of the following fourth order analogue of the Painlevé I equation, x = Ty − ( 1 6 y + 1 24 (y x + 2yyxx) + 1 240 yxxxx ) . This proves the existence part of a conjecture posed by Dubrovin. We obtain our result by proving the solvability of an associated Riemann-Hilbert(More)
We study unitary random matrix ensembles in the critical regime where a new cut arises away from the original spectrum. We perform a double scaling limit where the size of the matrices tends to infinity, but in such a way that only a bounded number of eigenvalues is expected in the newborn cut. It turns out that limits of the eigenvalue correlation kernel(More)
We study unitary random matrix ensembles of the form Z n,N | detM |2αe−N Tr V dM , where α > −1/2 and V is such that the limiting mean eigenvalue density for n,N → ∞ and n/N → 1 vanishes quadratically at the origin. In order to compute the double scaling limits of the eigenvalue correlation kernel near the origin, we use the Deift/Zhou steepest descent(More)
Unitary random matrix ensembles Z n,N (detM) α exp(−N TrV (M)) dM defined on positive definite matrices M , where α > −1 and V is real analytic, have a hard edge at 0. The equilibrium measure associated with V typically vanishes like a square root at soft edges of the spectrum. For the case that the equilibrium measure vanishes like a square root at 0, we(More)
OBJECTIVES To describe the effects of cytoreductive nephrectomy (CN) on the natural course of metastatic renal cell carcinoma (mRCC). CN appears to stabilize metastatic lesions in mRCC in a subgroup of patients and we hypothesize that systemic treatment might be deferred in these patients with stable disease after CN. SUBJECTS AND METHODS Overall, 45(More)