Interpolation of Hilbert and Sobolev Spaces:\ud Quantitative Estimates and Counterexamples

This paper provides an overview of interpolation of Banach and Hilbert spaces, with a focus on establishing when equivalence of norms is in fact equality of norms in the key results of the theory. (In brief, our conclusion for the Hilbert space case is that, with the right normalisations, all the key results hold with equality of norms.) In the final section we apply the Hilbert space results to the Sobolev spaces $H^s(\Omega)$ and $\tilde{H}^s(\Omega)$, for $s\in \mathbb{R}$ and an open $\Omega\subset \mathbb{R}^n$. We exhibit examples in one and two dimensions of sets $\Omega$ for which these scales of Sobolev spaces are not interpolation scales. In the cases when they are interpolation scales (in particular, if $\Omega$ is Lipschitz) we exhibit examples that show that, in general, the interpolation norm does not coincide with the intrinsic Sobolev norm and, in fact, the ratio of these two norms can be arbitrarily large

Hybrid Methods for Studyng Stability and Bifurcations in Delayed Feedback Systems

Fil: Itovich, Griselda R. Universidad Nacional de Río Negro. Escuela de Producción, Tecnología y Medio Ambiente. Río Negro, Argentina.Fil: Gentile, Franco S. Universidad Nacional del Sur e Instituto de Investigaciones en Ingeniería Eléctrica, (IIIE). Bahia Blanca, Argentina.Fil: Moiola, Jorge L. Universidad Nacional del Sur e Instituto de Investigaciones en Ingeniería Eléctrica, (IIIE). Bahia Blanca, Argentina.The dynamics of two related models of second order delay differential equations with four bifurcating parameters is analyzed. Through a classical technique in the time-domain which involves the location of the roots of a exponential polynomial equation, the areas of stability of the equilibrium are set. Moreover, the frequency domain methodology allows to study the Hopf bifurcation phenomena and to describe the behavior of the emerging cycles completely. Certain type of singularities, which provoke fold bifurcations of cycles are detected precisely. Also, a complete picture of resonant parameter configurations is established for both models. The whole results are checked with the software DDE-Biftool .Se analiza la dinámica de dos modelos de ecuaciones diferenciales de segundo orden con cuatro parámetros de bifurcación. A través de una técnica clásica en el dominio tiempo que busca la ubicación de las raíces de una ecuación que es un polinomio exponencial, se establecen las áreas de estabilidad. La metodología en el dominio frecuencia se aplica para estudiar el fenómeno de la bifurcación de Hopf y para describir el comportamiento de los ciclos emergentes, por medio de un enfoque de sistemas realimentados. También se detectaron ciertos tipos de singularidades que provocan bifurcaciones de ciclos de tipo silla nodo. Además, también se establece para los dos modelos, una descripción completa de configuraciones de parámetros que produce resonancias. Todos los resultados obtenidos fueron contrastados con el programa DDE-Biftool

Interpolation of Hilbert and Sobolev Spaces: Quantitative Estimates and Counterexamples

This paper provides an overview of interpolation of Banach and Hilbert spaces, with a focus on establishing when equivalence of norms is in fact equality of norms in the key results of the theory. (In brief, our conclusion for the Hilbert space case is that, with the right normalisations, all the key results hold with equality of norms.) In the final section we apply the Hilbert space results to the Sobolev spaces $H^s(\Omega)$ and $\widetilde{H}^s(\Omega)$, for $s\in \mathbb{R}$ and an open $\Omega\subset \mathbb{R}^n$. We exhibit examples in one and two dimensions of sets $\Omega$ for which these scales of Sobolev spaces are not interpolation scales. In the cases when they are interpolation scales (in particular, if $\Omega$ is Lipschitz) we exhibit examples that show that, in general, the interpolation norm does not coincide with the intrinsic Sobolev norm and, in fact, the ratio of these two norms can be arbitrarily large

Melatonin and its derivatives in red wine: contribution of fermenting microorganisms

Melatonin (MEL) is an indoleamine produced mainly by the pineal gland in vertebrates and it has a significant role in regulation of circadian rhythm, mitigation of sleeping disorder and jet lag. It has been found in medicinal plants, plant foods including seeds, fruits and fermented beverages, and, therefore, its occurrence in plants is now ascertained. In grapes and wines, MEL ranges from sub-ng/g to \ub5g/g and from sub-ng/mL to ng/mL, respectively, with varying levels according to both endogenous and exogenous factors. In addition, MEL isomers and tryptophan-ethylester (TEE, a compound with the same molecular weight of MEL) have recently been detected in wine and the fermenting yeast plays an important role for their production [1]. It has been suggested that, in synergy with polyphenols, MEL in wine may contribute to maximize the health-promoting effects of Mediterranean diet. The research aimed to validate an analytical method for the simultaneous detection of MEL, TEE and tryptophan. The sample preparation was developed by means of SPE purification. Purified samples of synthetic wine solution (tartaric acid 5 g/L, ethanol 12% [v/v], pH 3.2) and red wine spiked with the analytes of interest were analyzed by liquid chromatography coupled with both fluorescence and mass spectrometry detectors. MEL, TEE and tryptophan were successfully detected and quantified by both the analytical conditions adopted. The response was linear for all the investigated compounds and it was comparable between synthetic wine solution and red wine. The recovery was higher than 85% and the relative standard deviation was lower than 10%. The developed method was applied for the analysis of red wine samples produced in a cold area of North of Italy. Preliminary results showed an increase of TEE concentrations suggesting the possible positive influence of secondary fermentations other than the alcoholic fermentation

A high frequency boundary element method for scattering by a class of multiple obstacles

Engineering and Physical Sciences Research Council (EP/K000012/1)https://arxiv.org/abs/1903.0444

Integral equation methods for acoustic scattering by fractals

We study sound-soft time-harmonic acoustic scattering by general scatterers, including fractal scatterers, in 2D and 3D space. For an arbitrary compact scatterer $\Gamma$ we reformulate the Dirichlet boundary value problem for the Helmholtz equation as a first kind integral equation (IE) on $\Gamma$ involving the Newton potential. The IE is well-posed, except possibly at a countable set of frequencies, and reduces to existing single-layer boundary IEs when $\Gamma$ is the boundary of a bounded Lipschitz open set, a screen, or a multi-screen. When $\Gamma$ is uniformly of $d$-dimensional Hausdorff dimension in a sense we make precise (a $d$-set), the operator in our equation is an integral operator on $\Gamma$ with respect to $d$-dimensional Hausdorff measure, with kernel the Helmholtz fundamental solution, and we propose a piecewise-constant Galerkin discretization of the IE, which converges in the limit of vanishing mesh width. When $\Gamma$ is the fractal attractor of an iterated function system of contracting similarities we prove convergence rates under assumptions on $\Gamma$ and the IE solution, and describe a fully discrete implementation using recently proposed quadrature rules for singular integrals on fractals. We present numerical results for a range of examples and make our software available as a Julia code

Endometrial cancer PDX-derived organoids (PDXOs) and PDXs with FGFR2c isoform expression are sensitive to FGFR inhibition

Endometrial cancer; Immunohistochemistry; Predictive markersCàncer d'endometri; Immunohistoquímica; Marcadors predictiusCáncer de endometrio; Inmunohistoquímica; Marcadores predictivosEndometrial cancer (EC) patients with metastatic/recurrent disease have limited treatment options and poor survival outcomes. Recently, we discovered the FGFR2c splice isoform is associated with poor prognosis in EC patients. Here we report the establishment of 16 EC patient-derived xenografts (PDX)-derived organoids (PDXOs) with or without FGFR2c expression. In vitro treatment of 5 EC PDXOs with BGJ398 showed significant cell death in 3 models with FGFR2c expression. PDXs with high/moderate FGFR2c expression showed significant tumour growth inhibition (TGI) following 21-day treatment with FGFR inhibitors (BGJ398 or pemigatinib) and significantly prolonged survival in 4/5 models. Pemigatinib + cisplatin combination therapy (n = 5) resulted in significant TGI and prolonged survival in one of two p53abn PDXs. All five models treated with cisplatin alone showed de novo resistance and no survival benefit. Seven-day treatment with BGJ398 revealed a significant reduction in angiogenesis and CD206 + M2 macrophages. These data collectively support the evaluation of FGFR inhibitors in a clinical trial

The amount of preoperative endometrial tissue surface in relation to final endometrial cancer classification

Diagnosis; Endometrial carcinoma; Endometrial samplingDiagnóstico; Carcinoma de endometrio; Muestreo endometrialDiagnòstic; Carcinoma endometrial; Mostreig endometrialObjective To evaluate whether the amount of preoperative endometrial tissue surface is related to the degree of concordance with final low- and high-grade endometrial cancer (EC). In addition, to determine whether discordance is influenced by sampling method and impacts outcome. Methods A retrospective cohort study within the European Network for Individualized Treatment of Endometrial Cancer (ENITEC). Surface of preoperative endometrial tissue samples was digitally calculated using ImageJ. Tumor samples were classified into low-grade (grade 1–2 endometrioid EC (EEC)) and high-grade (grade 3 EEC + non-endometroid EC). Results The study cohort included 573 tumor samples. Overall concordance between pre- and postoperative diagnosis was 60.0%, and 88.8% when classified into low- and high-grade EC. Upgrading (preoperative low-grade, postoperative high-grade EC) was found in 7.8% and downgrading (preoperative high-grade, postoperative low-grade EC) in 26.7%. The median endometrial tissue surface was significantly lower in concordant diagnoses when compared to discordant diagnoses, respectively 18.7 mm2 and 23.5 mm2 (P = 0.022). Sampling method did not influence the concordance in tumor classification. Patients with preoperative high-grade and postoperative low-grade showed significant lower DSS compared to patients with concordant low-grade EC (P = 0.039). Conclusion The amount of preoperative endometrial tissue surface was inversely related to the degree of concordance with final tumor low- and high-grade. Obtaining higher amount of preoperative endometrial tissue surface does not increase the concordance between pre- and postoperative low- and high-grade diagnosis in EC. Awareness of clinically relevant down- and upgrading is crucial to reduce subsequent over- or undertreatment with impact on outcome

Is the Helmholtz equation really sign-indefinite?

The usual variational (or weak) formulations of the Helmholtz equation are sign-indefinite in the sense that the bilinear forms cannot be bounded below by a positive multiple of the appropriate norm squared. This is often for a good reason, since in bounded domains under certain boundary conditions the solution of the Helmholtz equation is not unique at wavenumbers that correspond to eigenvalues of the Laplacian, and thus the variational problem cannot be sign-definite. However, even in cases where the solution is unique for all wavenumbers, the standard variational formulations of the Helmholtz equation are still indefinite when the wavenumber is large. This indefiniteness has implications for both the analysis and the practical implementation of finite element methods. In this paper we introduce new sign-definite (also called coercive or elliptic) formulations of the Helmholtz equation posed in either the interior of a star-shaped domain with impedance boundary conditions, or the exterior of a star-shaped domain with Dirichlet boundary conditions. Like the standard variational formulations, these new formulations arise just by multiplying the Helmholtz equation by particular test functions and integrating by parts

Intratumor genetic heterogeneity and clonal evolution to decode endometrial cancer progression

Endometrial cancer; Clonal evolution; MutationCáncer endometrial; Evolución clonal; MutaciónCàncer d'endometri; Evolució clonal; MutacióAnalyzing different tumor regions by next generation sequencing allows the assessment of intratumor genetic heterogeneity (ITGH), a phenomenon that has been studied widely in some tumor types but has been less well explored in endometrial carcinoma (EC). In this study, we sought to characterize the spatial and temporal heterogeneity of 9 different ECs using whole-exome sequencing, and by performing targeted sequencing validation of the 42 primary tumor regions and 30 metastatic samples analyzed. In addition, copy number alterations of serous carcinomas were assessed by comparative genomic hybridization arrays. From the somatic mutations, identified by whole-exome sequencing, 532 were validated by targeted sequencing. Based on these data, the phylogenetic tree reconstructed for each case allowed us to establish the tumors’ evolution and correlate this to tumor progression, prognosis, and the presence of recurrent disease. Moreover, we studied the genetic landscape of an ambiguous EC and the molecular profile obtained was used to guide the selection of a potential personalized therapy for this patient, which was subsequently validated by preclinical testing in patient-derived xenograft models. Overall, our study reveals the impact of analyzing different tumor regions to decipher the ITGH in ECs, which could help make the best treatment decision.We thank all those at the Translational Research Laboratory of the MD Anderson Cancer Center Madrid for their invaluable help with this study. Tissue samples were obtained with the support of the MD Anderson Foundation Biobank (record number B.0000745, ISCIII National Biobank Record), the “Xarxa Catalana de Bancs de Tumors” and “Plataforma de Biobancos” ISCIII (PT13/0010/0014, B.000609). This study has been supported by the Spanish Ministry of Economy and Innovation (PID2019-104644RB-I00 (GMB), the Instituto de Salud Carlos III (ISCIII, CIBERONC, CB16/12/00295 - GMB-, CB16/12/00328 -EC, AGM- and CB16/12/00231 -XMG- [all partly supported by FEDER funds]) and by the AECC Scientific Foundation (FC_AECC PROYE19036MOR -GMB- and Coordinated groups 2018 -XMG, AGM, GMB-). SO is funded by an AECC-postdoctoral grant (2020). JSR-F and BW are funded in part by the Breast Cancer Research Foundation and in part by the NIH/NCI P50 CA247749 01 grant. Research reported in this publication was supported in part by a Cancer Center Support Grant of the NIH/NCI (Grant No. P30CA008748; MSK). We thank the Eurofins Megalab laboratory for helping us to perform the analysis of DNA HPV detection