Regularization of Planar Piecewise Smooth Systems with a Heteroclinic Loop

In this paper, we study bifurcations of the regularized systems of planar piecewise smooth systems, which have a visible fold-regular point and a sliding or grazing heteroclinic loop. Our results show that if the planar piecewise smooth system with a sliding heteroclinic loop undergoes sliding heteroclinic bifurcation, then the regularized system can bifurcate with a stable limit cycle passing through the regularized region and at most two limit cycles outside the regularized region. The regularized system can have at most three periodic orbits. When the upper subsystem is a Hamiltonian system, the regularized system can bifurcate with a semi-stable periodic orbit. Finally, we discuss two cases when the heteroclinic loop of a piecewise smooth system remains unbroken under a small perturbation. Our results show that the regularized system can bifurcate at most two limit cycles from an inner unstable grazing heteroclinic loop.

Criterion for the occurrence of the macro destruction and formation of breaking during metal deformation

When generalizing the geometrically nonlinear law of Murnaghan elasticity to plasticity, a formally mathematical criterion was introduced for deformational macrofracture (macrocrack appearance) associated with an increase in elastic and plastic anisotropy as a failure cause. The use of the double potentiality of the governing equations in stresses and their velocities made it possible to obtain the reliable information on the structure of the deviatory section of the yield surface, the existence of which is a classical hypothesis in solid mechanics. The normal vector to the surface of the deviatory section is selected from two mutually orthogonal eigenvectors of the constructed operator. There are two families of regular concave surfaces, and a section surface is formed by joining the parts of two representatives of the families at singular points. To select normal vectors, the obtained ratio for them is used for isotropy. In connection with the considered problem of a double simple shift, it is established that multiple eigenvalues appear for the both normal vectors. To unambiguously determine the normal vector at a regular point, it is necessary to exclude the presence of multiple eigenvalues for the both normal vectors at the same time. At a singular point, the appearance of a multiple eigenvalue of one of the normal vectors is still unacceptable. These two conditions are necessary and sufficient to validate the governing equations of the generalized Murnaghan model. Otherwise, a macrocrack occurs. The theoretical construction is supported by the developed software complexes.

Some results on the filter method for nonlinear complementary problems

Recent studies show that the filter method has good numerical performance for nonlinear complementary problems (NCPs). Their approach is to reformulate an NCP as a constrained optimization solved by filter algorithms. However, they can only prove that the iterative sequence converges to the KKT point of the constrained optimization. In this paper, we investigate the relation between the KKT point of the constrained optimization and the solution of the NCP. First, we give several sufficient conditions under which the KKT point of the constrained optimization is the solution of the NCP; second, we define regular conditions and regular point which include and generalize the previous results; third, we prove that the level sets of the objective function of the constrained optimization are bounded for a strongly monotone function or a uniform P-function; finally, we present some examples to verify the previous results.

Optimal Łojasiewicz–Simon inequalities and Morse–Bott Yang–Mills energy functions

For any compact Lie group 𝐺 and closed, smooth Riemannian manifold ( X , g ) (X,g) of dimension d ≥ 2 d\geq 2 , we extend a result due to Uhlenbeck (1985) that gives existence of a flat connection on a principal 𝐺-bundle over 𝑋 supporting a connection with L p L^{p} -small curvature, when p > d / 2 p>d/2 , to the case of a connection with L d / 2 L^{d/2} -small curvature. We prove an optimal Łojasiewicz–Simon gradient inequality for abstract Morse–Bott functions on Banach manifolds, generalizing an earlier result due to the author and Maridakis (2019), principally by removing the hypothesis that the Hessian operator be Fredholm with index zero. We apply this result to prove the optimal Łojasiewicz–Simon gradient inequality for the self-dual Yang–Mills energy function near regular anti-self-dual connections over closed Riemannian four-manifolds and for the full Yang–Mills energy function over closed Riemannian manifolds of dimension d ≥ 2 d\geq 2 , when known to be Morse–Bott at a given Yang–Mills connection. We also prove the optimal Łojasiewicz–Simon gradient inequality by direct analysis near a given flat connection that is a regular point of the curvature map.

Dilatations of Linear Operators

The article is devoted to building various dilatations of linear operators. The explicit construction of a unitary dilation of a compression operator is considered. Then the J -unitary dilatation of a bounded operator is constructed by means of the operator knot concept of a bounded linear operator. Using the Pavlov method, we construct the self-adjoint dilatation of a bounded dissipative operator. We consider spectral and translational representations of the self-adjoint dilatation of a densely defined dissipative operator with nonempty set of regular points. Using the concept of an operator knot for a bounded operator and the Cayley transform, we introduce an operator knot for a linear operator. By means of this concept, we construct the J -self-adjoint dilatation of a densely defined operator with a regular point. We obtain conditions of isomorphism of extraneous dilations and their minimality.

Hypertension Management in Emergency Departments

BACKGROUND Elevated blood pressure (BP) is pervasive among patients that visit emergency departments (EDs) for their care. METHODS In this review article, we outline the current approach to the management of these individuals and highlight the crucial role emergency medicine clinicians play in reducing the morbidity associated with elevated BP. RESULTS We highlight the critical importance of immediate treatment when elevated BP contributes to new or worsening end-organ injury but emphasize that such hypertensive emergencies are rare. For the vast majority of patients with elevated BP in the ED who do not have new or worsening end-organ injury from elevated BP, immediate BP reduction within the ED is not recommended or safe. Nonetheless, within weeks after an ED visit, there is a pressing need to improve the care of patients with elevated or previously undiagnosed hypertension. For many, it may be their only regular point of engagement with the healthcare system. To address this, we present novel perspectives that envision a new role for emergency medicine in chronic hypertension management—one that acknowledges the significant population-level gaps in BP control that contribute to disparities in cardiovascular disease and sets the stage for future changes in systems-based practice. CONCLUSIONS Emergency medicine plays a key and evolving role in reducing morbidity associated with elevated BP.

Sustained Reduction in Third-generation Cephalosporin Usage in Adult Inpatients Following Introduction of an Antimicrobial Stewardship Program in a Large, Urban Hospital in Malawi

Background Third-generation cephalosporins (3GC) remain the first-choice empiric antibiotic for severe infection in many sub-Saharan African hospitals. In Malawi, the limited availability of alternatives means that strategies to prevent the spread of 3GC resistance are imperative; however, suitable approaches to antimicrobial stewardship (AMS) in low-income settings are not well studied. Methods We introduced an AMS intervention to Queen Elizabeth Central Hospital in Blantyre. The intervention consisted of a prescribing application for smartphones and regular point-prevalence surveys with prescriber feedback. We evaluate the effects of the intervention on 3GC usage and on the cost of providing antibiotics. Using a thematic analysis of semi-structured interviews and participant observations, we additionally evaluate the acceptability of the stewardship program. Results The proportion of antibiotic prescriptions for a 3GC reduced from 193/241 (80.1%) to 177/330 (53.6%; percentage decrease, 26.5%; 95% confidence interval, 18.7–34.1) with no change in the case-fatality rate. The cost analysis estimated an annual savings of US$15 000. Qualitative research revealed trust in the guideline and found that its accessibility through smartphones helpful to guide clinical decisions. Operational health-system barriers and hierarchal clinical relationships lead to continued reliance on 3GC. Conclusions We report the successful introduction of an antimicrobial stewardship approach in Malawi. By focusing on pragmatic interventions and simple aims, we demonstrate the feasibility, acceptability, and cost savings of a stewardship program where resources are limited. In doing so, we provide a suitable starting point for expansions of AMS interventions in this and other low-income settings.

On generalized quasi-Einstein manifolds

We study generalized quasi-Einstein manifolds, or briefly, GQE manifolds. Here, we present relations between the Bach, Cotton and D tensors on GQE manifolds. Next, a 3-tensor E which measures the deviation of m-quasi-Einstein manifolds from GQE manifolds is introduced. Among others in dimension 3, it is shown that Bach-flatness implies locally conformally flatness. Furthermore, it is proved that, around a regular point of the fourth-order divergence free Weyl tensor, a GQE manifold is a locally warped product manifold with {(n-1)} -dimensional Einstein fibers in suitable cases.