Whitney’s jets for Sobolev functions

We present two fundamental facts of the jet theory for Sobolev spaces Wmp. One of them is that the formal differentiation of k-jets theory is compatible with the pointwise definition of Sobolev (m − 1)-jet spaces on regular subsets of Euclidean spaces Rⁿ. The second result describes the Sobolev embedding operator of Sobolev jet spaces increasing the order of integrability of Sobolev functions up to the critical Sobolev exponent.Встановлено два фундаментальних факти теорії струменів для просторів Соболева Wm,p. Перший із них полягає в тому, що формальне диференціювання k-струменів є сумісним з поточковим визначенням соболевських просторів (m−1)-струменя на регулярних підмножинах евклідових просторів Rⁿ. Другий результат описує соболевські оператори вкладення соболевських просторів струменів, що покращують порядок сумовності соболевських функцій аж до критичного показника

Conformational constraints on side chains in protein residues increase their information content

Like all other complex biological systems, proteins exhibit properties not seen in free amino acids (i.e., emergent properties). The present investigation arose from the deduction that proteins should offer a good model to approach the reverse phenomenon, namely top-down constraints experienced by protein residues compared to free amino acids. The crystalline structure of profilin Ib, a contractile protein of Acanthamoeba castellanii, was chosen as the object of study and submitted to 2-ns molecular dynamics simulation. The results revealed strong conformational constraints on the side chain of residues compared to the respective free amino acids. A Shannon entropy (SE) analysis of the conformational behavior of the side chains showed in most cases a strong decrease in the SE of the χ1 and χ2 dihedral angles compared to free amino acids. This is equivalent to stating that conformational constraints on the side chain of residues increase their information content and hence recognition specificity compared to free amino acids. In other words, the vastly increased information content of a protein relative to its free monomers is embedded not only in the tertiary structure of the backbone, but also in the conformational behavior of the side chains. The postulated implication is that both backbone and side chains, by virtue of being conformationally constrained, contribute to the recognition specificity of the protein toward other macromolecules and ligand

The age of data-driven proteomics : how machine learning enables novel workflows

A lot of energy in the field of proteomics is dedicated to the application of challenging experimental workflows, which include metaproteomics, proteogenomics, data independent acquisition (DIA), non-specific proteolysis, immunopeptidomics, and open modification searches. These workflows are all challenging because of ambiguity in the identification stage; they either expand the search space and thus increase the ambiguity of identifications, or, in the case of DIA, they generate data that is inherently more ambiguous. In this context, machine learning-based predictive models are now generating considerable excitement in the field of proteomics because these predictive models hold great potential to drastically reduce the ambiguity in the identification process of the above-mentioned workflows. Indeed, the field has already produced classical machine learning and deep learning models to predict almost every aspect of a liquid chromatography-mass spectrometry (LC-MS) experiment. Yet despite all the excitement, thorough integration of predictive models in these challenging LC-MS workflows is still limited, and further improvements to the modeling and validation procedures can still be made. In this viewpoint we therefore point out highly promising recent machine learning developments in proteomics, alongside some of the remaining challenges

dbar-equations, integrable deformations of quasiconformal mappings and Whitham hierarchy

It is shown that the dispersionless scalar integrable hierarchies and, in general, the universal Whitham hierarchy are nothing but classes of integrable deformations of quasiconformal mappings on the plane. Examples of deformations of quasiconformal mappings associated with explicit solutions of the dispersionless KP hierarchy are presented.Comment: 13 pages, LaTe

General Spectral Flow Formula for Fixed Maximal Domain

We consider a continuous curve of linear elliptic formally self-adjoint differential operators of first order with smooth coefficients over a compact Riemannian manifold with boundary together with a continuous curve of global elliptic boundary value problems. We express the spectral flow of the resulting continuous family of (unbounded) self-adjoint Fredholm operators in terms of the Maslov index of two related curves of Lagrangian spaces. One curve is given by the varying domains, the other by the Cauchy data spaces. We provide rigorous definitions of the underlying concepts of spectral theory and symplectic analysis and give a full (and surprisingly short) proof of our General Spectral Flow Formula for the case of fixed maximal domain. As a side result, we establish local stability of weak inner unique continuation property (UCP) and explain its role for parameter dependent spectral theory.Comment: 22 page

Mammotome HH biopsy : the future of minimal invasive breast surgery?

Background: Vacuum-assisted breast biopsy / Mammotome HH ® Breast Biopsy System/ is the milestone in the diagnosis of breast lesions. This system has proven to be as diagnostically reliable as open surgery, but without scarring, deformations and hospitalizations associated with an open procedure. The aim of our study was to assess the role and possibilities of using this biopsy in treatment of benign breast lesions like fibroadenoma. Material/Methods: From 2001 to 2004, about 1118 Mammotome biopsies were performed in our Department. Among 445 Mammotome biopsies performed under US control there were 211 cases of fibroadenomas. Follow-up was performed in 156 patients with this result at 6 and 12 months after biopsy. In our study we took into considerations the size, localizations as well as performers. Results: In 2002 there were 70.8% patients with total lesion excision, 16.7% with residual lesion and 12.5% women with hematomas or scars. In 2003-2004 there were more women with total lesion excision (84.3%), fewer residual tumors and other lesions. Conclusions: In future, Mammotome breast biopsy can replace scalpel, and will become an alternative method to open surgical excision of fibroadenomas. It is important especially in the cases of young women to prevent cosmetic deformations and scars

HIGHER-ORDER REGULARITY FOR THE SOLUTIONS OF SOME DEGENERATE QUASI-LINEAR ELLIPTIC-EQUATIONS IN THE PLANE

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