Scale calculus and the Schrodinger equation

We introduce the scale calculus, which generalizes the classical differential calculus to non differentiable functions. The new derivative is called the scale difference operator. We also introduce the notions of fractal functions, minimal resolution, and quantum representation of a non differentiable function. We then define a scale quantization procedure for classical Lagrangian systems inspired by the Scale relativity theory developped by Nottale. We prove that the scale quantization of Newtionian mechanics is a non linear Schrodinger equation. Under some specific assumptions, we obtain the classical linear Schrodinger equation.Comment: 49 page

Scale relativity and fractal space-time: theory and applications

In the first part of this contribution, we review the development of the theory of scale relativity and its geometric framework constructed in terms of a fractal and nondifferentiable continuous space-time. This theory leads (i) to a generalization of possible physically relevant fractal laws, written as partial differential equation acting in the space of scales, and (ii) to a new geometric foundation of quantum mechanics and gauge field theories and their possible generalisations. In the second part, we discuss some examples of application of the theory to various sciences, in particular in cases when the theoretical predictions have been validated by new or updated observational and experimental data. This includes predictions in physics and cosmology (value of the QCD coupling and of the cosmological constant), to astrophysics and gravitational structure formation (distances of extrasolar planets to their stars, of Kuiper belt objects, value of solar and solar-like star cycles), to sciences of life (log-periodic law for species punctuated evolution, human development and society evolution), to Earth sciences (log-periodic deceleration of the rate of California earthquakes and of Sichuan earthquake replicas, critical law for the arctic sea ice extent) and tentative applications to system biology.Comment: 63 pages, 14 figures. In : First International Conference on the Evolution and Development of the Universe,8th - 9th October 2008, Paris, Franc

Geometry and field theory in multi-fractional spacetime

We construct a theory of fields living on continuous geometries with fractional Hausdorff and spectral dimensions, focussing on a flat background analogous to Minkowski spacetime. After reviewing the properties of fractional spaces with fixed dimension, presented in a companion paper, we generalize to a multi-fractional scenario inspired by multi-fractal geometry, where the dimension changes with the scale. This is related to the renormalization group properties of fractional field theories, illustrated by the example of a scalar field. Depending on the symmetries of the Lagrangian, one can define two models. In one of them, the effective dimension flows from 2 in the ultraviolet (UV) and geometry constrains the infrared limit to be four-dimensional. At the UV critical value, the model is rendered power-counting renormalizable. However, this is not the most fundamental regime. Compelling arguments of fractal geometry require an extension of the fractional action measure to complex order. In doing so, we obtain a hierarchy of scales characterizing different geometric regimes. At very small scales, discrete symmetries emerge and the notion of a continuous spacetime begins to blur, until one reaches a fundamental scale and an ultra-microscopic fractal structure. This fine hierarchy of geometries has implications for non-commutative theories and discrete quantum gravity. In the latter case, the present model can be viewed as a top-down realization of a quantum-discrete to classical-continuum transition.Comment: 1+82 pages, 1 figure, 2 tables. v2-3: discussions clarified and improved (especially section 4.5), typos corrected, references added; v4: further typos correcte

A922 Sequential measurement of 1 hour creatinine clearance (1-CRCL) in critically ill patients at risk of acute kidney injury (AKI)

Meeting abstrac

Mechanical Stress Induces Remodeling of Vascular Networks in Growing Leaves

International audienceDifferentiation into well-defined patterns and tissue growth are recognized as key processes in organismal development. However, it is unclear whether patterns are passively, homogeneously dilated by growth or whether they remodel during tissue expansion. Leaf vascu-lar networks are well-fitted to investigate this issue, since leaves are approximately two-dimensional and grow manyfold in size. Here we study experimentally and computationally how vein patterns affect growth. We first model the growing vasculature as a network of viscoelastic rods and consider its response to external mechanical stress. We use the so-called texture tensor to quantify the local network geometry and reveal that growth is heterogeneous , resembling non-affine deformations in composite materials. We then apply mechanical forces to growing leaves after veins have differentiated, which respond by anisotropic growth and reorientation of the network in the direction of external stress. External mechanical stress appears to make growth more homogeneous, in contrast with the model with viscoelastic rods. However, we reconcile the model with experimental data by incorporating randomness in rod thickness and a threshold in the rod growth law, making the rods viscoelastoplastic. Altogether, we show that the higher stiffness of veins leads to their reorientation along external forces, along with a reduction in growth heterogeneity. This process may lead to the reinforcement of leaves against mechanical stress. More generally , our work contributes to a framework whereby growth and patterns are coordinated through the differences in mechanical properties between cell types

Neuronal Antibody Biomarkers for Sydenham’s Chorea Identify a New Group of Children with Chronic Recurrent Episodic Acute Exacerbations of Tic and Obsessive Compulsive Symptoms Following a Streptococcal Infection

<div><p>Several autoantibodies (anti-dopamine 1 (D1R) and 2 (D2R) receptors, anti-tubulin, anti-lysoganglioside-GM1) and antibody-mediated activation of calcium calmodulin dependent protein kinase II (CaMKII) signaling activity are elevated in children with Sydenham’s chorea (SC). Recognizing proposed clinical and autoimmune similarities between SC and PANDAS (pediatric autoimmune neuropsychiatric disorder associated with a streptococcal infection), we sought to identify serial biomarker changes in a slightly different population. Antineuronal antibodies were measured in eight children (mean 11.3 years) with chronic, dramatic, recurrent tics and obsessive-compulsive disorder (OCD) associated with a group A β-hemolytic streptococcal (GABHS) respiratory tract infection, but differing because they lacked choreiform movements. Longitudinal serum samples in most subjects included two pre-exacerbation samples, Exac), one midst Exac (abrupt recurrence of tic/OCD; temporally association with a GABHS infection in six of eight subjects), and two post-Exac. Controls included four groups of unaffected children (n = 70; mean 10.8 years) obtained at four different institutions and published controls. Clinical exacerbations were not associated with a significant rise in antineuronal antibody titers. CaMKII activation was increased at the GABHS exacerbation point in 5/6 subjects, exceeded combined and published control’s 95th percentile at least once in 7/8 subjects, and median values were elevated at each time point. Anti-tubulin and anti-D2R titers did not differ from published or combined control group’s 95th percentile or median values. Differences in anti-lysoganglioside-GM1 and anti-D1R titers were dependent on the selected control. Variances in antibody titers and CaMKII activation were identified among the institutional control groups. Based on comparisons to published studies, results identify two groups of PANDAS: 1) a cohort, represented by this study, which lacks choreiform movements and elevated antibodies against D2R; 2) the originally reported group with choreiform movements and elevated anti-D2R antibodies, similar to SC. Increased antibody mediated CaMKII activation was found in both groups and requires further study as a potential biomarker.</p></div