Uniqueness for the signature of a path of bounded variation and the reduced path group

We introduce the notions of tree-like path and tree-like equivalence between paths and prove that the latter is an equivalence relation for paths of finite length. We show that the equivalence classes form a group with some similarity to a free group, and that in each class there is one special tree reduced path. The set of these paths is the Reduced Path Group. It is a continuous analogue to the group of reduced words. The signature of the path is a power series whose coefficients are definite iterated integrals of the path. We identify the paths with trivial signature as the tree-like paths, and prove that two paths are in tree-like equivalence if and only if they have the same signature. In this way, we extend Chen's theorems on the uniqueness of the sequence of iterated integrals associated with a piecewise regular path to finite length paths and identify the appropriate extended meaning for reparameterisation in the general setting. It is suggestive to think of this result as a non-commutative analogue of the result that integrable functions on the circle are determined, up to Lebesgue null sets, by their Fourier coefficients. As a second theme we give quantitative versions of Chen's theorem in the case of lattice paths and paths with continuous derivative, and as a corollary derive results on the triviality of exponential products in the tensor algebra.Comment: 52 pages - considerably extended and revised version of the previous version of the pape

Identifying sensitive marine areas in the South China Sea under international law

At the interface of law, science and policy, this study is centred on the scientific and technical criteria adopted for the identification of sensitive marine and coastal areas, the role of international law and the adequacy of scientific evidence. It uses the South China Sea (SCS) as a case-study the basis that it is (1) a regional sea which is not protected under any binding regional instrument -this makes international law particularly relevant for direct implementation; and (2) a regional sea whose resources are subjected to intense and often competing uses in a politically and ecologically sensitive environment -identification of sensitive marine areas therefore appear opportune. The identification of sensitive area necessarily precedes the choice of and designation of Marine Protected Areas and Area-Based Management Tools (ABMTs). It also precedes the question of the content of states’ obligation to protect and preserve the marine environment, which are outside the scope of this study. This study investigates the legal status of sensitive area criteria adopted under the auspices of intergovernmental processes but in often non-binding instruments and documents. It asks whether and how these sensitive area criteria may contribute to inform implementation of the obligation to protect and preserve the marine environment under international law

Demographic study of craniosynostosis patients in mid-Missouri

Craniosynostosis is a congenital defect in which one or more of the cranial sutures close prematurely, affecting 1 in 2000 to 2500 live births worldwide. Historically, sagittal craniosynostosis has been reported to be the most common form of single-suture craniosynostosis. Our previous retrospective review of craniosynostosis at our institution reported that the incidence of metopic craniosynostosis in mid-Missouri is significantly greater than that of sagittal craniosynostosis, 65% versus 13% (Table 1). Our current aim is to further investigate the demographic characteristics of our institution's craniosynostosis population

The signature of a rough path: uniqueness

In the context of controlled differential equations, the signature is the exponential function on paths. B. Hambly and T. Lyons proved that the signature of a bounded variation path is trivial if and only if the path is tree-like. We extend Hambly–Lyons' result and their notion of tree-like paths to the setting of weakly geometric rough paths in a Banach space. At the heart of our approach is a new definition for reduced path and a lemma identifying the reduced path group with the space of signatures

Walnut Allergy Across Europe: Distribution of Allergen Sensitization Patterns and Prediction of Severity

Background: Walnut allergy is common across the globe, but data on the involvement of individual walnut components are scarce. Objectives: To identify geographical differences in walnut component sensitization across Europe, explore cosensitization and cross-reactivity, and assess associations of clinical and serological determinants with severity of walnut allergy. Methods: As part of the EuroPrevall outpatient surveys in 12 European cities, standardized clinical evaluation was conducted in 531 individuals reporting symptoms to walnut, with sensitization to all known walnut components assessed in 202 subjects. Multivariable Lasso regression was applied to investigate predictors for walnut allergy severity. Results: Birch-pollen-related walnut sensitization (Jug r 5) dominated in Northern and Central Europe and lipid transfer protein sensitization (Jug r 3) in Southern Europe. Profilin sensitization (Jug r 7) was prominent throughout Europe. Sensitization to storage proteins (Jug r 1, 2, 4, and 6) was detected in up to 10% of subjects. The walnut components that showed strong correlations with pollen and other foods differed between centers. The combination of determinants best predicting walnut allergy severity were symptoms upon skin contact with walnut, atopic dermatitis (ever), family history of atopic disease, mugwort pollen allergy, sensitization to cat or dog, positive skin prick test result to walnut, and IgE to Jug r 1, 5, 7, or carbohydrate determinants (area under the curve = 0.81; 95% CI, 0.73-0.89). Conclusions: Walnut-allergic subjects across Europe show clear geographical differences in walnut component sensitization and cosensitization patterns. A predictive model combining results from component-based serology testing with results from extract-based testing and information on clinical background allows for good discrimination between mild to moderate and severe walnut allerg

From constructive field theory to fractional stochastic calculus. (II) Constructive proof of convergence for the L\'evy area of fractional Brownian motion with Hurst index α∈(1/8,1/4)\alpha\in(1/8,1/4)

{Let $B=(B_1(t),...,B_d(t))$ be a $d$-dimensional fractional Brownian motion with Hurst index $\alpha<1/4$, or more generally a Gaussian process whose paths have the same local regularity. Defining properly iterated integrals of $B$ is a difficult task because of the low H\"older regularity index of its paths. Yet rough path theory shows it is the key to the construction of a stochastic calculus with respect to $B$, or to solving differential equations driven by $B$. We intend to show in a series of papers how to desingularize iterated integrals by a weak, singular non-Gaussian perturbation of the Gaussian measure defined by a limit in law procedure. Convergence is proved by using "standard" tools of constructive field theory, in particular cluster expansions and renormalization. These powerful tools allow optimal estimates, and call for an extension of Gaussian tools such as for instance the Malliavin calculus. After a first introductory paper \cite{MagUnt1}, this one concentrates on the details of the constructive proof of convergence for second-order iterated integrals, also known as L\'evy area

Differential segregation in a cell-cell contact interface: the dynamics of the immunological synapse

Receptor-ligand couples in the cell-cell contact interface between a T cell and an antigen-presenting cell form distinct geometric patterns and undergo spatial rearrangement within the contact interface. Spatial segregation of the antigen and adhesion receptors occurs within seconds of contact, central aggregation of the antigen receptor then occurring over 1-5 min. This structure, called the immunological synapse, is becoming a paradigm for localized signaling. However, the mechanisms driving its formation, in particular spatial segregation, are currently not understood. With a reaction diffusion model incorporating thermodynamics, elasticity, and reaction kinetics, we examine the hypothesis that differing bond lengths (extracellular domain size) is the driving force behind molecular segregation. We derive two key conditions necessary for segregation: a thermodynamic criterion on the effective bond elasticity and a requirement for the seeding/nucleation of domains. Domains have a minimum length scale and will only spontaneously coalesce/aggregate if the contact area is small or the membrane relaxation distance large. Otherwise, differential attachment of receptors to the cytoskeleton is required for central aggregation. Our analysis indicates that differential bond lengths have a significant effect on synapse dynamics, i.e., there is a significant contribution to the free energy of the interaction, suggesting that segregation by differential bond length is important in cell-cell contact interfaces and the immunological synapse

Wavelength-scale stationary-wave integrated Fourier-transform spectrometry

Spectrometry is a general physical-analysis approach for investigating light-matter interactions. However, the complex designs of existing spectrometers render them resistant to simplification and miniaturization, both of which are vital for applications in micro- and nanotechnology and which are now undergoing intensive research. Stationary-wave integrated Fourier-transform spectrometry (SWIFTS)-an approach based on direct intensity detection of a standing wave resulting from either reflection (as in the principle of colour photography by Gabriel Lippmann) or counterpropagative interference phenomenon-is expected to be able to overcome this drawback. Here, we present a SWIFTS-based spectrometer relying on an original optical near-field detection method in which optical nanoprobes are used to sample directly the evanescent standing wave in the waveguide. Combined with integrated optics, we report a way of reducing the volume of the spectrometer to a few hundreds of cubic wavelengths. This is the first attempt, using SWIFTS, to produce a very small integrated one-dimensional spectrometer suitable for applications where microspectrometers are essential

Factors Influencing the Incidence of Severe Complications in Head and Neck Free Flap Reconstructions.

Complications after head and neck free-flap reconstructions are detrimental and prolong hospital stay. In an effort to identify related variables in a tertiary regional head and neck unit, the microvascular reconstruction activity over the last 5 years was captured in a database along with patient-, provider-, and volume-outcome-related parameters. Retrospective cohort study (level of evidence 3), a modified Clavien-Dindo classification, was used to assess severe complications. A database of 217 patients was created with consecutively reconstructed patients from 2009 to 2014. In the univariate analysis of severe complications, we found significant associations (P &lt; 0.05) between type of flap used, American Society of Anesthesiologists classification, T-stage, microscope use, surgeon, flap frequency, and surgeon volume. Within a binomial logistic regression model, less frequently versus frequently performed flap (odds ratio [OR] = 3.2; confidence interval [CI] = 2.9-3.5; P = 0.000), high-volume versus low-volume surgeon (OR = 0.52; CI = -0.22 to 0.82; P = 0.007), and ASA classification (OR = 2.9; CI = 2.4-3.4; P = 0.033) were retained as independent predictors of severe complications. In a Cox-regression model, surgeon (P = 0.011), site of reconstruction (P = 0.000), T-stage (P = 0.001), and presence of severe complications (P = 0.015) correlated with a prolonged hospitalization. In this study, we identified a correlation of patient-related factors with severe complications (ASA score) and prolonged hospital stay (T-stage, site). More importantly, we identified several provider- (surgeon) and volume-related (frequency with which a flap was performed and high-volume surgeon) factors as predictors of severe complications. Our data indicate that provider- and volume-related parameters play an important role in the outcome of microvascular free-flap procedures in the head and neck region