Modular forms, Schwarzian conditions, and symmetries of differential equations in physics

We give examples of infinite order rational transformations that leave linear differential equations covariant. These examples are non-trivial yet simple enough illustrations of exact representations of the renormalization group. We first illustrate covariance properties on order-two linear differential operators associated with identities relating the same $_2F_1$ hypergeometric function with different rational pullbacks. We provide two new and more general results of the previous covariance by rational functions: a new Heun function example and a higher genus $_2F_1$ hypergeometric function example. We then focus on identities relating the same hypergeometric function with two different algebraic pullback transformations: such remarkable identities correspond to modular forms, the algebraic transformations being solution of another differentially algebraic Schwarzian equation that emerged in a paper by Casale. Further, we show that the first differentially algebraic equation can be seen as a subcase of the last Schwarzian differential condition, the restriction corresponding to a factorization condition of some associated order-two linear differential operator. Finally, we also explore generalizations of these results, for instance, to $_3F_2$, hypergeometric functions, and show that one just reduces to the previous $_2F_1$ cases through a Clausen identity. In a $_2F_1$ hypergeometric framework the Schwarzian condition encapsulates all the modular forms and modular equations of the theory of elliptic curves, but these two conditions are actually richer than elliptic curves or $_2F_1$ hypergeometric functions, as can be seen on the Heun and higher genus example. This work is a strong incentive to develop more differentially algebraic symmetry analysis in physics.Comment: 43 page

Diagonals of rational functions, pullbacked 2F1 hypergeometric functions and modular forms (unabrigded version)

We recall that diagonals of rational functions naturally occur in lattice statistical mechanics and enumerative combinatorics. We find that a seven-parameter rational function of three variables with a numerator equal to one (reciprocal of a polynomial of degree two at most) can be expressed as a pullbacked 2F1 hypergeometric function. This result can be seen as the simplest non-trivial family of diagonals of rational functions. We focus on some subcases such that the diagonals of the corresponding rational functions can be written as a pullbacked 2F1 hypergeometric function with two possible rational functions pullbacks algebraically related by modular equations, thus showing explicitely that the diagonal is a modular form. We then generalise this result to eight, nine and ten parameters families adding some selected cubic terms at the denominator of the rational function defining the diagonal. We finally show that each of these previous rational functions yields an infinite number of rational functions whose diagonals are also pullbacked 2F1 hypergeometric functions and modular forms.Comment: 39 page

Time to failure prediction in rubber components subjected to thermal ageing: A combined approach based upon the intrinsic defect concept and the fracture mechanics

In this contribution, we attempt to derive a tool allowing the prediction of the stretch ratioat failure in rubber components subjected to thermal ageing. To achieve this goal, the mainidea is to combine the fracture mechanics approach and the intrinsic defect concept. Using an accelerated ageing procedure for an Ethylene–Propylene–Diene Monomer (EPDM), it is first shown that the average molar mass of the elastically active chains (i.e. between crosslinks) can be used as the main indicator of the macromolecular network degradation. Byintroducing the time–temperature equivalence principle, a shift factor obeying to an Arrhenius law is derived, and master curves are built as well for the average molar mass as for the ultimate mechanical properties. Fracture mechanics tests are also achieved and the square root dependence of the fracture energy with the average molar mass is pointed out. Moreover, it is shown that the mechanical response could be approximated by the phantom network theory, which allows to relate the strain energy density function to the average molar mass. Assuming that the fracture of a smooth specimen is the consequence of a virtual intrinsic defect whose the size can be easily estimated, the stretch ratio at break can be therefore computed for any thermal ageing condition. The estimated values are found in a very nice agreement with EPDM experimental data, making this approach a useful tool when designing rubber components for moderate to high temperature environments

A Direct Algorithm for Pole Placement by State-derivative Feedback for Single-input Linear Systems

This paper deals with the direct solution of the pole placement problem for single-input linear systems using state-derivative feedback. This pole placement problem is always solvable for any controllable systems if all eigenvalues of the original system are nonzero. Then any arbitrary closed-loop poles can be placed in order to achieve the desired system performance. The solving procedure results in a formula similar to the Ackermann formula. Its derivation is based on the transformation of a linear single-input system into Frobenius canonical form by a special coordinate transformation, then solving the pole placement problem by state derivative feedback. Finally the solution is extended also for single-input time-varying control systems. The simulation results are included to show the effectiveness of the proposed approach

Familial Facial Palsy: A Case Series of Six Families from the Northern State, Sudan

Familial facial palsy is uncommon, accounting only for 4–14% of Bell’s palsy cases. We report six families with single or recurrent episodes of familial facial palsy from Northern State, Sudan. The first family had two brothers with single episodes of Bell’s palsy. The index case of the second family was a 19-year-old female who and nine other members of her family had a single or recurrent episodes of Bell’s palsy. The third, fourth, fifth, and sixth families had eight, five, four, and five members, respectively, who developed either single or recurrent episodes of Bell’s palsy. None of the index cases or other members of the six families who were examined showed evidence of facial swelling or fissured tongue suggestive of Melkersson-Rosenthal syndrome. Literature review revealed two studies on Bell’s palsy from Sudan but no studies on familial facial palsy. The mode of inheritance was either autosomal dominant with variable penetrance or autosomal recessive. In the second family, there could be a possibility of autosomal recessive  inheritance due to increased number of cases after consanguineous marriage. Steroids remain the mainstay of treatment together with protective eye regimens. The role of physiotherapy, although widely used, is controversial. Genetic analysis is recommended and family history should be considered in patients with Bell’s palsy. Keywords: Bell’s palsy, familial facial palsy, Northern State, Suda

The Relationship between Trauma due to Winter Storm Alexa, PTSD, Mental Health of Palestinians in the Gaza Strip

Aim: This study investigated the relationship between trauma due to winter storm Alexa, PTSD and other mental health problems of Palestinian in Gaza Strip. Method: The sample consisted of 105 males (50%) and 105 females (50%) selected from three of the most affected areas by flooding in 2014 due to Alexa storm in Gaza Strip. Participants age range was 20-65 years, with a mean age 40.88 (SD = 9.8), with a mean age of years. Mental health status was assessed by a sociodemographic scale, the Trauma Due to Flood Scale, PTSD scale, and General Health Questionnaire (28 items). Results: Mean traumatic events experienced were 7.8. There were no statistically significant differences between males and females in reporting traumatic events. Mean post-traumatic stress disorder was 18.65, re-experiences symptoms was 6.4, avoidance symptoms was 5.7 and mean arousal symptoms was 5.73. The study showed that 34.8% reported full criteria of PTSD. There were no statistically significant differences in PTSD total scores and subscales and sex of participants. Mean GHQ-28 was 12.12, somatization mean was 3.21, anxiety was 3.31, social dysfunction was 3.34, and depression was 2.27, 91% of the participants were rated as psychiatric morbidity cases and need further investigation. Males significantly scored more in social dysfunction than females. Traumatic events were significantly correlated with PTSD and general mental health and all subscales. Conclusion and implications: This study has important implications for need of establishing and implementing psychosocial intervention programs for in the Gaza Strip not only for those victims of political violence but also for people exposed to other types of traumatic events such as natural disasters

Eigenstructure Assignment by State-derivative and Partial Output-derivative Feedback for Linear Time-invariant Control Systems

This paper introduces a parametric approach for solving the problem of eigenstructure assignment via state-derivative feedback for linear control time-invariant systems. This problem is always solvable for any controllable systems if the open-loop system matrix is nonsingular. In this work, the parametric solution to the feedback gain matrix is introduced that describes the available degrees of freedom offered by the state-derivative feedback in selecting the associated eigenvectors from an admissible class. These freedoms can be utilized to improve the robustness of the closed-loop system. Finally, the eigenstructure assignment problem via partial output-derivative feedback is introduced. Numerical examples are included to show the effectiveness of the proposed approach