http://web.stanford.edu/group/brunet/Cell1999

Akt Promotes Cell Survival by Phosphorylating and Inhibiting a Forkhead Transcription Factor

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Mitogen-activated protein kinases (MAPKs) are important signal transducing enzymes, unique to eukaryotes, that are involved in many facets of cellular regulation. Initial research concentrated on defining the components and organization of MAPK signalling cascades, but recent studies have begun to shed light on the physiological functions of these cascades in the control of gene expression, cell proliferation and programmed cell death.

/pdf/mammalian-map-kinase-signalling-cascades-2wtn4qrixr.pdf

Mammalian MAP kinase signalling cascades

Cells undergo a variety of biological responses when placed in hypoxic conditions, including activation of signalling pathways that regulate proliferation, angiogenesis and death. Cancer cells have adapted these pathways, allowing tumours to survive and even grow under hypoxic conditions, and tumour hypoxia is associated with poor prognosis and resistance to radiation therapy. Many elements of the hypoxia-response pathway are therefore good candidates for therapeutic targeting.

Hypoxia — a key regulatory factor in tumour growth

Specificity of receptor tyrosine kinase signaling: Transient versus sustained extracellular signal-regulated kinase activation

The sharp asymptotics for the L^2-quantization errors of Gaussian measures on a Hilbert space and, in particular, for Gaussian processes is derived. 
The condition imposed is regular variation of the eigenvalues.

Sharp asymptotics of the functional quantization problem for Gaussian processes

We establish conditions to characterize probability measures by their $L^{p}$-quantization error functions in both $\mathbb{R}^{d}$ and Hilbert settings. This characterization is two-fold: static (identity of two distributions) and dynamic (convergence for the $L^{p}$-Wasserstein distance). We first propose a criterion on the quantization level $N$, valid for any norm on $\mathbb{R}^{d}$ and any order $p$ based on a geometrical approach involving the Voronoi diagram. Then, we prove that in the $L^{2}$-case on a (separable) Hilbert space, the condition on the level $N$ can be reduced to $N=2$, which is optimal. More quantization based characterization cases in dimension 1 and a discussion of the completeness of a distance defined by the quantization error function can be found at the end of this paper.

Characterization of probability distribution convergence in Wasserstein distance by $L^{p}$-quantization error function

https://www.scientificarchives.com/admin/assets/articles/pdf/relevance-of-neuropilin-1-and-neuropilin-2-targeting-for-cancer-treatment-20210603060628.pdf

Relevance of Neuropilin 1 and Neuropilin 2 Targeting for Cancer Treatment

Abstract We investigate the functional quantization problem for stochastic processes with respect to Lp(IRd, μ)-norms, where μ is a fractal measure namely, μ is self-similar or a homogeneous Cantor measure. The derived functional quantization upper rate bounds are universal depending only on the mean-regularity index of the process and the quantization dimension of μ and as universal rates they are optimal. Furthermore, for arbitrary Borel probability measures μ we establish a (nonconstructive) link between the quantization errors of μ and the functional quantization errors of the process in the space Lp(IRd, μ).

Fractal functional quantization of mean-regular stochastic processes

We propose a new approach to quantize the marginals of the discrete Euler diffusion process. The method is built recursively and involes the conditional distribution of the marginals of the discrete Euler process. Analytically, the method raises several questions like the analysis of the induced quadratic quantization error between the marginals of the Euler process and the proposed quantizations. We show in particular that at every discretization step $t_k$ of the Euler scheme, this error is bounded by the cumulative quantization errors induced by the Euler operator, from times $t_0=0$ to time $t_k$. For numerics, we restrict our analysis to the one dimensional setting and show how to compute the optimal grids using a Newton-Raphson algorithm. We then propose a close formula for the companion weights and the transition probabilities associated to the proposed quantizations. This allows us to quantize in particular diffusion processes in local volatility models by reducing dramatically the computational complexity of the search of optimal quantizers while increasing their computational precision with respect to the algorithms commonly proposed in this framework. Numerical tests are carried out for the Brownian motion and for the pricing of European options in a local volatility model. A comparison with the Monte Carlo simulations shows that the proposed method is more efficient (w.r.t. both computational precision and time complexity) than the Monte Carlo method.

Gilles Pagès

Papers

Sharp asymptotics of the functional quantization problem for Gaussian processes

Characterization of probability distribution convergence in Wasserstein distance by $L^{p}$-quantization error function

Relevance of Neuropilin 1 and Neuropilin 2 Targeting for Cancer Treatment

Fractal functional quantization of mean-regular stochastic processes

Recursive marginal quantization of an Euler scheme with applications to local volatility models