Hopf algebras in dynamical systems theory

The theory of exact and of approximate solutions for non-autonomous linear differential equations forms a wide field with strong ties to physics and applied problems. This paper is meant as a stepping stone for an exploration of this long-established theme, through the tinted glasses of a (Hopf and Rota-Baxter) algebraic point of view. By reviewing, reformulating and strengthening known results, we give evidence for the claim that the use of Hopf algebra allows for a refined analysis of differential equations. We revisit the renowned Campbell-Baker-Hausdorff-Dynkin formula by the modern approach involving Lie idempotents. Approximate solutions to differential equations involve, on the one hand, series of iterated integrals solving the corresponding integral equations; on the other hand, exponential solutions. Equating those solutions yields identities among products of iterated Riemann integrals. Now, the Riemann integral satisfies the integration-by-parts rule with the Leibniz rule for derivations as its partner; and skewderivations generalize derivations. Thus we seek an algebraic theory of integration, with the Rota-Baxter relation replacing the classical rule. The methods to deal with noncommutativity are especially highlighted. We find new identities, allowing for an extensive embedding of Dyson-Chen series of time- or path-ordered products (of generalized integration operators); of the corresponding Magnus expansion; and of their relations, into the unified algebraic setting of Rota-Baxter maps and their inverse skewderivations. This picture clarifies the approximate solutions to generalized integral equations corresponding to non-autonomous linear (skew)differential equations.Comment: International Journal of Geometric Methods in Modern Physics, in pres

Renormalization : A number theoretical model

We analyse the Dirichlet convolution ring of arithmetic number theoretic functions. It turns out to fail to be a Hopf algebra on the diagonal, due to the lack of complete multiplicativity of the product and coproduct. A related Hopf algebra can be established, which however overcounts the diagonal. We argue that the mechanism of renormalization in quantum field theory is modelled after the same principle. Singularities hence arise as a (now continuously indexed) overcounting on the diagonals. Renormalization is given by the map from the auxiliary Hopf algebra to the weaker multiplicative structure, called Hopf gebra, rescaling the diagonals.Comment: 15 pages, extended version of talks delivered at SLC55 Bertinoro,Sep 2005, and the Bob Delbourgo QFT Fest in Hobart, Dec 200

miR-1: A comprehensive review of its role in normal development and diverse disorders

MicroRNA-1 (miR-1) is a conserved miRNA with high expression in the muscle tissues. In humans, two discrete genes, MIRN1-1 and MIRN1-2 residing on a genomic region on 18q11.2 produce a single mature miRNA which has 21 nucleotides. miR-1 has a regulatory role on a number of genes including heat shock protein 60 (HSP60), Kruppel-like factor 4 (KLF4) and Heart And Neural Crest Derivatives Expressed 2 (HAND2). miR-1 has critical roles in the physiological processes in the smooth and skeletal muscles as well as other tissues, thus being involved in the pathogenesis of a wide range of disorders. Moreover, dysregulation of miR-1 has been noted in diverse types of cancers including gastric, colorectal, breast, prostate and lung cancer. In the current review, we provide the summary of the data regarding the role of this miRNA in the normal development and the pathogenic processes. © 2020 The Author(s

Modeling and optimization of nanoemulsion containing Sorafenib for cancer treatment by response surface methodology

The aim of this study is the development of nanoemulsions for intravenous administration of Sorafenib, which is a poorly soluble drug with no parenteral treatment. The formulation was prepared by a high energy emulsification method and optimized by response surface methodology. The effects of overhead stirring time, high shear rate, high shear time, and cycles of high-pressure homogenizer were studied in the preparation of nanoemulsion loaded with Sorafenib. Most of the particles in nanoemulsion are spherical in shape, the smallest particle size being 82.14nm. The results of the 3-(4,5-Dimethylthiazol-2-yl)-2,5-diphenyltetrazolium bromide, a tetrazole reveal that the optimum formulation does not affect normal cells significantly in low drug concentrations but could remove the cancer cells. Finally, a formulation containing Sorafenib retained its properties over a period of 90days. With characterization, the study of the formulated nanoemulsion has the potential to be used as a parenteral nanoemulsion in the treatment of cancer. Graphical abstract Schematic figure of high pressure homogenizer device

From AKNS to derivative NLS hierarchies via deformations of associative products

Using deformations of associative products, derivative nonlinear Schrodinger (DNLS) hierarchies are recovered as AKNS-type hierarchies. Since the latter can also be formulated as Gelfand-Dickey-type Lax hierarchies, a recently developed method to obtain 'functional representations' can be applied. We actually consider hierarchies with dependent variables in any (possibly noncommutative) associative algebra, e.g., an algebra of matrices of functions. This also covers the case of hierarchies of coupled DNLS equations.Comment: 22 pages, 2nd version: title changed and material organized in a different way, 3rd version: introduction and first part of section 2 rewritten, taking account of previously overlooked references. To appear in J. Physics A: Math. Ge

Surgical Skill Assessment on In-Vivo Clinical Data via the Clearness of Operating Field

Surgical skill assessment is important for surgery training and quality control. Prior works on this task largely focus on basic surgical tasks such as suturing and knot tying performed in simulation settings. In contrast, surgical skill assessment is studied in this paper on a real clinical dataset, which consists of fifty-seven in-vivo laparoscopic surgeries and corresponding skill scores annotated by six surgeons. From analyses on this dataset, the clearness of operating field (COF) is identified as a good proxy for overall surgical skills, given its strong correlation with overall skills and high inter-annotator consistency. Then an objective and automated framework based on neural network is proposed to predict surgical skills through the proxy of COF. The neural network is jointly trained with a supervised regression loss and an unsupervised rank loss. In experiments, the proposed method achieves 0.55 Spearman's correlation with the ground truth of overall technical skill, which is even comparable with the human performance of junior surgeons.Comment: MICCAI 201

A computationally efficient method for hand–eye calibration

Purpose: Surgical robots with cooperative control and semiautonomous features have shown increasing clinical potential, particularly for repetitive tasks under imaging and vision guidance. Effective performance of an autonomous task requires accurate hand–eye calibration so that the transformation between the robot coordinate frame and the camera coordinates is well defined. In practice, due to changes in surgical instruments, online hand–eye calibration must be performed regularly. In order to ensure seamless execution of the surgical procedure without affecting the normal surgical workflow, it is important to derive fast and efficient hand–eye calibration methods. Methods: We present a computationally efficient iterative method for hand–eye calibration. In this method, dual quaternion is introduced to represent the rigid transformation, and a two-step iterative method is proposed to recover the real and dual parts of the dual quaternion simultaneously, and thus the estimation of rotation and translation of the transformation. Results: The proposed method was applied to determine the rigid transformation between the stereo laparoscope and the robot manipulator. Promising experimental and simulation results have shown significant convergence speed improvement to 3 iterations from larger than 30 with regard to standard optimization method, which illustrates the effectiveness and efficiency of the proposed method

Alkaline-earth phosphonate MOFs with reversible hydration-dependent fluorescence

A new rigid tritopic phosphonic ligand, 2,4,6-tris(4-phosphonophenyl)pyridine (H6L), was synthesized and used to assemble isostructural barium (1) and strontium (2) phosphonate metal organic frameworks that exhibit fully reversible and selective water-dependent fluorescence red-shift at room temperature