An update on the Hirsch conjecture

The Hirsch conjecture was posed in 1957 in a letter from Warren M. Hirsch to George Dantzig. It states that the graph of a d-dimensional polytope with n facets cannot have diameter greater than n - d. Despite being one of the most fundamental, basic and old problems in polytope theory, what we know is quite scarce. Most notably, no polynomial upper bound is known for the diameters that are conjectured to be linear. In contrast, very few polytopes are known where the bound $n-d$ is attained. This paper collects known results and remarks both on the positive and on the negative side of the conjecture. Some proofs are included, but only those that we hope are accessible to a general mathematical audience without introducing too many technicalities.Comment: 28 pages, 6 figures. Many proofs have been taken out from version 2 and put into the appendix arXiv:0912.423

A generalization of the concept of distance based on the simplex inequality

We introduce and discuss the concept of n-distance, a generalization to n elements of the classical notion of distance obtained by replacing the triangle inequality with the so-called simplex inequality d(x1,…,xn)≤K∑i=1nd(x1,…,xn)zi,x1,…,xn,z∈X, where K=1. Here d(x1,…,xn)zi is obtained from the function d(x1,…,xn) by setting its ith variable to z. We provide several examples of n-distances, and for each of them we investigate the infimum of the set of real numbers K∈]0,1] for which the inequality above holds. We also introduce a generalization of the concept of n-distance obtained by replacing in the simplex inequality the sum function with an arbitrary symmetric function

SARS-CoV-2 infection induces a dual response in liver function tests: Association with mortality during hospitalization

Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is associated with abnormal liver function tests. We hypothesized that early altered liver biochemistries at admission might have different clinical relevance than subsequent changes during hospitalization. A single-center retrospective study was conducted on 540 consecutive hospitalized patients, PCR-diagnosed with SARS-CoV-2. Liver test abnormalities were defined as the elevation of either gamma-glutamyltransferase (GGT), alanine aminotransferase (ALT), or aspartate aminotransferase (AST), above the upper limit of normality set by our laboratory. Linear mixed models (LMM) evaluated longitudinal associations, incorporating all available follow-up laboratory chemistries. By the end of the follow-up period, 502 patients (94.5%) were discharged (109 (20.5%) died). A total of 319 (64.3%) had at least one abnormal liver test result at admission. More prevalent were elevated AST (40.9%) and GGT (47.3%). Abnormalities were not associated with survival but with respiratory complications at admission. Conversely, LMM models adjusted for age and sex showed that longitudinal increases during hospitalization in ferritin, GGT, and alkaline phosphatase (ALP), as well as a decreased albumin levels, were associated with reduced survival. This dual pattern of liver damage might reconcile previous conflicting reports. GGT and ALP trajectories could be useful to determine who might need more surveillance and intensive care

Изменения сетей покоя фМРТ у пациентов с тяжелой черепно-мозговой травмой при терапевтической ритмической транскраниальной магнитной стимуляции (клиническое наблюдение)

   Severe traumatic brain injury (TBI) accompanied by impaired consciousness manifesting as prolonged postcoma unawareness (PCU) is one of the current medical and social problems causing high morbidity and mortality worldwide. Difficult recovery of such patients necessitates the development of additional neurorehabilitation approaches, including neuromodulation methods, as well as the search for objective markers of treatment efficacy.   Aim of the study: to evaluate the effect of therapeutic rhythmic transcranial magnetic stimulation (rTMS) on fMRI resting state networks (RSN) in PCU after severe TBI.   Materials and Methods. We analyzed individual fMRI RSN in three patients with PCU before and after a course of rTMS performed at different timepoints after severe TBI and with different efficacy of treatment. We assessed the topography and quantitative characteristics of the networks (DMN, sensorimotor, control functions, left and right fronto-parietal, auditory, and speech) known to be most significant for recovery of consciousness.   Results. We found a trend toward normalization of RSN topography as well as an increase in the integral index of network intensity in two of three patients with a distinct increase in consciousness after a course of rTMS.   Conclusion. Using case observations, we have demonstrated the therapeutic efficacy of rTMS and feasibility of using fMRI RSN as a reliable diagnostic approach in PCU following severe TBI.   Тяжелая черепно-мозговая травма (ТЧМТ), сопровождающаяся нарушением сознания в форме затяжных посткоматозных бессознательных состояний (ПКБС), относится к числу актуальных медико-социальных проблем, обусловливающих высокую инвалидизацию или летальность пациентов во всем мире. Трудности восстановления пациентов обосновывают необходимость разработки дополнительных нейрореабилитационных подходов, включая методы нейромодуляции, а также поиск объективных маркеров эффективности лечения.   Цель исследования. Оценить влияние терапевтической ритмической транскраниальной стимуляции (рТМС) на сети покоя фМРТ (resing state networks — RSN) при ПКБС после ТЧМТ.   Материалы и методы. Проанализировали индивидуальные RSN фМРТ у трех пациентов с ПКБС до и после курсовой рТМС, выполненной в разные сроки после ТЧМТ и с разной эффективностью лечения. Оценивали топографию и количественные характеристики сетей (сеть DMN, сенсомоторная, управляющих функций, лобно-теменные (левая и правая), аудиторная, речевая), наиболее значимых для восстановления сознания по данным литературы.   Результаты. Выявили тенденцию к нормализации топографии RSN, а также увеличение интегрального показателя сетевой интенсивности у двух из трех пациентов с выраженным повышением уровня сознания после курсовой рТМС.   Заключение. На примере индивидуальных наблюдений показали перспективность рТМС в качестве лечебного воздействия, а также оценки RSN фМРТ — как объективного диагностического подхода при ПКБС вследствие ТЧМТ

Визуализация сетей покоя (resting state) фМРТ у пациентов с тяжелой черепно-мозговой травмой

According to the literature, fMRI analysis at resting state (RS) is an informative methodological approach to the study of the basic level of a healthy and diseased brain’s functional activity. Averaging data over observation groups for various forms of cerebral pathology is often unacceptable. Previously, we mastered and applied the Independent Component Algorithm (ICA) in FSL software to visualize and analyze individual fMRI resting networks of healthy people.Objective: to analyze individual fMRI resting networks associated with the state of motor activity and consciousness in patients with severe traumatic brain injury (STBI).Materials and methods. Observation groups: 23 patients with SТBI (main) and 17 healthy volunteers (control). 3T fMRI recorded at rest. Individual (norm and STBI) and group (norm) analysis of RS networks was carried out by FSL software (ICA algorithm) and SPM8 in MATLAB.For the DMN and Sensorimotor networks, topography and total volume and intensity of their activation of their activation were determined.Results. The topography features reproduced in the group and individual analysis of fMRI of healthy people, as well as the averaged quantitative indicators of the rest networks were used as reference for pathology.In the context of motor activity, the RS Sensorimotor network was considered. Its topography is close to normal in most patients without or with mild hemiparesis. The growth of this defect is accompanied by a decrease in the integral quantitative indicators of the network, combined with asymmetric reduction (lack of activation in the contralateral motor cortex) in rough hemiparesis.In the context of consciousness, the expression and characteristics of the DMN network were compared in healthy people and in patients with STBI at its various levels: from clear to chronic vegetative state. It was revealed that a decrease in the level of consciousness is accompanied by a reduction in the cortical components of DMN, primarily the frontal (anterior DMN), not pronounced in the vegetative state. Activation of the caudal component of DMN (in particular, the posterior cingular cortex) persists in patients with depressed consciousness: distinct and even somewhat enhanced compared to the norm with its reversible form, less pronounced with chronicConclusion. The data obtained indicate the informative value of fMRI analysis of individual resting networks in the context of studying cerebral structural and functional foundations of consciousness and motor activity, as well as diagnosing the state of these functions in STBI.По данным литературы, анализ функциональной манитно-резонансной томографии (фМРТ) в состоянии покоя (RS) является информативным методическим подходом к исследованию базового уровня функциональной активности здорового и больного мозга. Усреднение данных по группам наблюдений при разных формах церебральной патологии зачастую неприемлемо. Ранее нами был освоен и применен алгоритм независимых компонент (ICA) в программном обеспечении FSL для визуализации и анализа индивидуальных сетей покоя фМРТ здоровых людей.Цель исследования: анализ индивидуальных сетей покоя фМРТ, сопряженных с состоянием двигательной активности и сознания, у пациентов с тяжелой черепно-мозговой травмой (ТЧМТ).Материал и методы. Группы наблюдений: 23 пациента с ТЧМТ, сопровождавшейся разной степенью угнетения сознания и двигательных нарушений в форме гемипареза (основная), и 17 здоровых испытуемых (контроль). У каждого записывали фМРТ 3 Тл в состоянии покоя с закрытыми глазами. Проводили групповой (в норме) и индивидуальный (в норме и при патологии) фМРТ-анализ RS с использованием программных средств FSL (алгоритм ICA) и SPM8 в среде MATLAB. Для сетей DMN и Sensorimotor определяли топографию активированных зон мозга, представленность сетей в группах наблюдений, общий объем и интенсивность их активации.Результаты. Воспроизводимые при групповом и индивидуальном анализе фМРТ здоровых людей особенности топографии, а также усредненные количественные показатели сетей покоя были использованы в качестве эталонных для патологии.В контексте двигательной активности рассматривали сенсомоторную сеть RS. Ее топография близка к норме у большинства пациентов без или с легким гемипарезом. Нарастание этого дефекта сопровождается  уменьшением интегральных количественных показателей сети, сочетающимся с асимметричной редукцией (отсутствием активации в контралатеральной моторной коре) при грубом гемипарезе.В контексте сознания анализировали особенности сети DMN у пациентов с ТЧМТ при разном его уровне: от ясного до хронического вегетативного состояния. Выявлено, что снижение уровня сознания сопровождается редукцией корковых составляющих DMN, в первую очередь лобных (anterior DMN), не выраженной при вегетативном состоянии. Активациия каудального компонента DMN (в частности, задней цингулярной коры) у пострадавших с угнетенным сознанием сохраняется: отчетливая и даже несколько усиленная по сравнению с нормой при обратимой его форме, менее выраженная – при хронической.Заключение. Полученные данные свидетельствуют об информативности фМРТ-анализа индивидуальных сетей покоя в контексте изучения церебральных структурно-функциональных основ сознания и двигательной активности, а также диагностики состояния этих функций при ТЧМТ

Sonidegib as a Locally Advanced Basal Cell Carcinoma Therapy in Real-life Clinical Setting: A National Multicentre Study

Background: Basal cell carcinoma (BCC) is the most prevalent cancer. A minority of BCCs have an aggressive behaviour (laBCC) and may require hedgehog pathway inhibitors such as sonidegib as its treatment. Objective: To describe the use of sonidegib in a large number of patients and provide more data on its real-life efficacy and safety profile.Methods: We conducted a retrospective and multicentric study that included patients treated with sonidegib. Epidemiological, effectiveness and safety data were collected.Results: A total of 82 patients with a mean age of 73.9 years were included. Ten patients had Gorlin syndrome. Median treatment duration was 6 months. Median follow-up duration was 34.2 months. Globally, 81.7% of the patients showed clinical improvement (52.4% partial response and 29.3% complete response), 12.2% clinical stability and 6.1% disease progression. There was no statistically significant difference in clinical improvement between the 24 h and 48 h sonidegib posology. After 6 months of treatment, 48.8% of the patients discontinued sonidegib. Prior vismodegib treatment and recurrent primary BCC were associated with a poorer response to sonidegib. At 6 months of treatment, 68.3% of the patients experienced at least one adverse effect. Conclusion: Sonidegib shows good effectiveness and acceptable safety profile in usual clinical practice.& COPY; 2023 AEDV. Published by Elsevier Espan & SIM;a, S.L.U.This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)

Discovery of potent, novel, non-toxic anti-malarial compounds via quantum modelling, virtual screening and in vitro experimental validation

<p>Abstract</p> <p>Background</p> <p>Developing resistance towards existing anti-malarial therapies emphasize the urgent need for new therapeutic options. Additionally, many malaria drugs in use today have high toxicity and low therapeutic indices. Gradient Biomodeling, LLC has developed a quantum-model search technology that uses quantum similarity and does not depend explicitly on chemical structure, as molecules are rigorously described in fundamental quantum attributes related to individual pharmacological properties. Therapeutic activity, as well as toxicity and other essential properties can be analysed and optimized simultaneously, independently of one another. Such methodology is suitable for a search of novel, non-toxic, active anti-malarial compounds.</p> <p>Methods</p> <p>A set of innovative algorithms is used for the fast calculation and interpretation of electron-density attributes of molecular structures at the quantum level for rapid discovery of prospective pharmaceuticals. Potency and efficacy, as well as additional physicochemical, metabolic, pharmacokinetic, safety, permeability and other properties were characterized by the procedure. Once quantum models are developed and experimentally validated, the methodology provides a straightforward implementation for lead discovery, compound optimizzation and <it>de novo </it>molecular design.</p> <p>Results</p> <p>Starting with a diverse training set of 26 well-known anti-malarial agents combined with 1730 moderately active and inactive molecules, novel compounds that have strong anti-malarial activity, low cytotoxicity and structural dissimilarity from the training set were discovered and experimentally validated. Twelve compounds were identified <it>in silico </it>and tested <it>in vitro</it>; eight of them showed anti-malarial activity (IC50 ≤ 10 μM), with six being very effective (IC50 ≤ 1 μM), and four exhibiting low nanomolar potency. The most active compounds were also tested for mammalian cytotoxicity and found to be non-toxic, with a therapeutic index of more than 6,900 for the most active compound.</p> <p>Conclusions</p> <p>Gradient's metric modelling approach and electron-density molecular representations can be powerful tools in the discovery and design of novel anti-malarial compounds. Since the quantum models are agnostic of the particular biological target, the technology can account for different mechanisms of action and be used for <it>de novo </it>design of small molecules with activity against not only the asexual phase of the malaria parasite, but also against the liver stage of the parasite development, which may lead to true causal prophylaxis.</p

Selected Open Problems in Discrete Geometry and Optimization

A list of questions and problems posed and discussed in September 2011 at the following consecutive events held at the Fields Institute, Toronto: Workshop on Discrete Geometry, Conference on Discrete Geometry and Optimization, and Workshop on Optimization. We hope these questions and problems will contribute to further stimulate the interaction between geometers and optimizers

A dynamic inequality generation scheme for polynomial programming

Hierarchies of semidefinite programs have been used to approximate or even solve polynomial programs. This approach rapidly becomes computationally expensive and is often tractable only for problems of small size. In this paper, we propose a dynamic inequality generation scheme to generate valid polynomial inequalities for general polynomial programs. When used iteratively, this scheme improves the bounds without incurring an exponential growth in the size of the relaxation. As a result, the proposed scheme is in principle scalable to large general polynomial programming problems. When all the variables of the problem are non-negative or when all the variables are binary, the general algorithm is specialized to a more efficient algorithm. In the case of binary polynomial programs, we show special cases for which the proposed scheme converges to the global optimal solution. We also present several examples illustrating the computational behavior of the scheme and provide comparisons with Lasserre’s approach and, for the binary linear case, with the lift-and-project method of Balas, Ceria, and Cornuejols