On Rayleigh-Taylor and Richtmyer-Meshkov Dynamics With Inverse-Quadratic Power-Law Acceleration

Rayleigh-Taylor (RT) and Richtmyer-Meshkov (RM) instabilities occur in many situations in Nature and technology from astrophysical to atomic scales, including stellar evolution, oceanic flows, plasma fusion, and scramjets. While RT and RM instabilities are sister phenomena, a link of RT-to-RM dynamics requires better understanding. This work focuses on the long-standing problem of RTI/RMI induced by accelerations, which vary as inverse-quadratic power-laws in time, and on RT/RM flows, which are three-dimensional, spatially extended and periodic in the plane normal to the acceleration direction. We apply group theory to obtain solutions for the early-time linear and late-time nonlinear dynamics of RT/RM coherent structure of bubbles and spikes, and investigate the dependence of the solutions on the acceleration’s parameters and initial conditions. We find that the dynamics is of RT type for strong accelerations and is of RM type for weak accelerations, and identify the effects of the acceleration’s strength and the fluid density ratio on RT-to-RM transition. While for given problem parameters the early-time dynamics is uniquely defined, the solutions for the late-time dynamics form a continuous family parameterised by the interfacial shear and include special solutions for RT/RM bubbles/spikes. Our theory achieves good agreement with available observations. We elaborate benchmarks that can be used in future research and in design of experiments and simulations, and that can serve for better understanding of RT/RM relevant processes in Nature and technology.

On the multifractal spectrum of weighted Birkhoff averages

<p style='text-indent:20px;'>In this paper, we study the topological spectrum of weighted Birk–hoff averages over aperiodic and irreducible subshifts of finite type. We show that for a uniformly continuous family of potentials, the spectrum is continuous and concave over its domain. In case of typical weights with respect to some ergodic quasi-Bernoulli measure, we determine the spectrum. Moreover, in case of full shift and under the assumption that the potentials depend only on the first coordinate, we show that our result is applicable for regular weights, like Möbius sequence.</p>

Prevalence of crack cocaine use and its associated factors in patients treated in a specialized outpatient service

Background Despite the advance in studies addressing the use of crack cocaine, knowledge about the characteristics of users that seek treatment in the different modalities of care for substance use disorders is important to plan the operationalization of these services. Objective To analyze the prevalence and factors associated with the use of crack cocaine in outpatients. Method Cross-sectional study consisting in the analysis of the medical records of outpatients of a chemical dependency clinic located in the south of Brazil from 1999 to 2015. The Fisher’s exact test and the Poisson regression model were used to analyze the data. Results Medical records from 1,253 patients were analyzed, and 1,196 (95.5%) of them contained information on the use of crack cocaine. Use of this substance was reported by 47% (95% CI [44, 50]) of the outpatients. The risk group was composed of adults aged 20-39 years, with no income, who had three or more children, did not consume alcohol or marijuana, had continuous family assistance, spontaneously looked for the service, and had already been hospitalized or assisted at a therapeutic community or psychosocial center. Conclusion There is great demand for the outpatient care of crack cocaine users. It is crucial that the risk factors guide treatment planning.

Update on Neurodegenerative Diseases and Glutamate: A PMHNP Single Case Study Report of a Diagnostically Complex Adult Patient, Interventions, and Unexpected Outcomes

OBJECTIVE: While dysfunction of serotonin and dopamine neurotransmitters has been studied in depth, in regard to the etiology of mental illness, the neurotransmitter glutamate and its dysfunction is now being explored as contributing to neurodegenerative psychiatric diseases, schizophrenia, autism, depression, and Alzheimer’s disease. This article explains its synthesis, neurotransmission, and metabolism within the brain and subsequent dysfunction that is responsible for neurocognitive loss associated with several psychiatric disorders. METHOD: The case study will report on the screening for pseudobulbar affective (PBA) disorder in a 29-year-old male with bipolar disorder, autism spectrum disorder, and intellectual developmental disability who was experiencing extreme, uncontrolled emotional outbursts requiring continuous family isolation (pre-COVID-19) for safety. With the positive screen for PBA, the patient was subsequently treated with a glutamatergic drug, dextromethorphan/quinidine. RESULTS: The patient’s unexpected response to this treatment including the acquisition of language, increased cognition, and improved executive functioning is presented. At 2 years post the initiation of treatment, his PBA screening score is reduced, uncontrolled outbursts and aggression have subsided, and the family can spend time outside of their home. CONCLUSIONS: Neurodegeneration and its impact is being researched and treated with medications affecting glutamate. The addition of a glutamatergic medication to this young man’s medication regimen has improved both his and his family’s quality of life. The psychiatric diagnoses, medications, and treatments associated with glutamate are explained in depth. The importance of nurses’ understanding of glutamate, its synthesis, transmission, and dysfunction causing excitotoxicity and brain cell death and its impact on patients’ behavior and safety is explained.

On a Steklov eigenvalue problem associated with the (p,q)-Laplacian

"Consider in a bounded domain \Omega \subset \mathbb{R}^N, N\ge 2, with smooth boundary \partial \Omega, the following eigenvalue problem (1) \begin{eqnarray*} &~&\mathcal{A} u:=-\Delta_p u-\Delta_q u=\lambda a(x) \mid u\mid ^{r-2}u\ \ \mbox{ in} ~ \Omega, \nonumber \\ &~&\big(\mid \nabla u\mid ^{p-2}+\mid \nabla u\mid ^{q-2}\big)\frac{\partial u}{\partial\nu}=\lambda b(x) \mid u\mid ^ {r-2}u ~ \mbox{ on} ~ \partial \Omega, \nonumber \end{eqnarray*} where 1<r<q<p<\infty or 1<q<p<r<\infty; r\in \Big(1, \frac{p(N-1)}{N-p}\Big) if p<N and r\in (1, \infty) if p\ge N; a\in L^{\infty}(\Omega),~ b\in L^{\infty}(\partial\Omega) are given nonnegative functions satisfying \[ \int_\Omega a~dx+\int_{\partial\Omega} b~d\sigma >0. \] Under these assumptions we prove that the set of all eigenvalues of the above problem is the interval [0, \infty). Our result complements those previously obtained by Abreu, J. and Madeira, G., [Generalized eigenvalues of the (p, 2)-Laplacian under a parametric boundary condition, Proc. Edinburgh Math. Soc., 63 (2020), No. 1, 287–303], Barbu, L. and Moroşanu, G., [Full description of the eigenvalue set of the (p,q)-Laplacian with a Steklov-like boundary condition, J. Differential Equations, in press], Barbu, L. and Moroşanu, G., [Eigenvalues of the negative (p,q)– Laplacian under a Steklov-like boundary condition, Complex Var. Elliptic Equations, 64 (2019), No. 4, 685–700], Fărcăşeanu, M., Mihăilescu, M. and Stancu-Dumitru, D., [On the set of eigen-values of some PDEs with homogeneous Neumann boundary condition, Nonlinear Anal. Theory Methods Appl., 116 (2015), 19–25], Mihăilescu, M., [An eigenvalue problem possesing a continuous family of eigenvalues plus an isolated eigenvale, Commun. Pure Appl. Anal., 10 (2011), 701–708], Mihăilescu, M. and Moroşanu, G., [Eigenvalues of -\triangle_p-\triangle_q under Neumann boundary condition, Canadian Math. Bull., 59 (2016), No. 3, 606–616]."

Superconformal index of low-rank gauge theories via the Bethe Ansatz

We study the Bethe Ansatz formula for the superconformal index, in the case of 4d $$ \mathcal{N} $$ N = 4 super-Yang-Mills with gauge group SU(N). We observe that not all solutions to the Bethe Ansatz Equations (BAEs) contribute to the index, and thus formulate “reduced BAEs” such that all and only their solutions contribute. We then propose, sharpening a conjecture of Arabi Ardehali et al. [1], that there is a one-to-one correspondence between branches of solutions to the reduced BAEs and vacua of the 4d $$ \mathcal{N} $$ N = 1* theory. We test the proposal in the case of SU(2) and SU(3). In the case of SU(3), we confirm that there is a continuous family of solutions, whose contribution to the index is non-vanishing.

The Theoretical and experimental study of the bending of a thin substrate during electrolytic deposition

The present paper is aimed at the theoretical and experimental study of the shape distortion of thin substrates during electrolytic deposition and gaccumulation of residual stresses in them. The theoretical modeling is provided in the framework of the theory of solids with variable material composition. The result of the deposition process is modeled with a continuous family of elastic bodies, which local deformations are incompatible. These deformations act as internal sources for stresses. Formally they are equivalent to the field of distributed defects. Unlike the classical approach adopted in nonlinear elasticity, the elements of the family which present a body with a variable material composition don’t have a global reference natural (free of stresses) form. Instead we used the continuous family being only locally free from stresses. To formulate the boundary value problem, continuous families of reference, intermediate and actual forms and corresponding families of deformations are defined. The deformations, belonging to these families, locally represent implants (local deformations of reference forms into intermediate ones) and deformations that bring intermediate forms into actual ones. Relations for stresses and strains in such bodies are obtained under the assumption that the displacement gradients are small with respect to unity and satisfy the kinematic hypothesis of the technical plate theory. Under these assumptions the equilibrium equations are derived. They include specific terms which determine formal loading that is caused by incompatible deformations. Axisymmetric problems for a circular substrate under various types of fixing and tension on the boundary, which characterize the conditions of the experiment, are obtained. The theoretical distribution for displacements of the substrate surface is formulated upon the obtained solution. They are intended to identify incompatible deformations that cause bending during the deposition process. The experimental measuring setup is constructed according to a holographic scheme of displacement measurements in real time. The deposition process is carried out in a cylindrical chamber with flange fastening of the cathode. The electrochemical process is implemented in sulphate electrolyte. As a result of comparing the theoretically obtained relations for bending surfaces of the substrate with the experimental results, the parameters that characterize the substrate shrinkage and tension are estimated.

Renormalization of Galilean electrodynamics

We study the quantum properties of a Galilean-invariant abelian gauge theory coupled to a Schrödinger scalar in 2+1 dimensions. At the classical level, the theory with minimal coupling is obtained from a null-reduction of relativistic Maxwell theory coupled to a complex scalar field in 3+1 dimensions and is closely related to the Galilean electromagnetism of Le-Bellac and Lévy-Leblond. Due to the presence of a dimensionless, gauge-invariant scalar field in the Galilean multiplet of the gauge-field, we find that at the quantum level an infinite number of couplings is generated. We explain how to handle the quantum corrections systematically using the background field method. Due to a non-renormalization theorem, the beta function of the gauge coupling is found to vanish to all orders in perturbation theory, leading to a continuous family of fixed points where the non-relativistic conformal symmetry is preserved.