Continuous Family Research Articles

On the dynamics of Richtmyer–Meshkov bubbles in unstable three-dimensional interfacial coherent structures with time-dependent acceleration

Richtmyer–Meshkov instability (RMI) plays an important role in many areas of science and engineering, from supernovae and fusion to scramjets and nano-fabrication. Classical RMI is induced by a steady shock and impulsive acceleration, whereas in realistic environments, the acceleration is usually variable. We focus on RMI induced by acceleration with power-law time-dependence and apply group theory to study the dynamics of regular bubbles. For early time linear dynamics, we find the dependence of the growth rate on the initial conditions and show that it is independent of the acceleration parameters. For late-time nonlinear dynamics, we consider regular asymptotic solutions, find a continuous family of such solutions, including their curvature, velocity, Fourier amplitudes, and interfacial shear, and study their stability. For each solution, the interface dynamics is directly linked to the interfacial shear. The non-equilibrium velocity field has intense fluid motion near the interface and effectively no motion in the bulk. The quasi-invariance of the fastest stable solution suggests that the dynamics of nonlinear RM bubbles is characterized by two macroscopic length scales: the wavelength and the amplitude, in agreement with observations. The properties of a number of special solutions are outlined. These are the flat Atwood bubble, the curved Taylor bubble, the minimum shear bubble, the convergence limit bubble, and the critical bubble. We elaborate new theory benchmarks for future experiments and simulations.

Convergence of Heisenberg modules over quantum 2-tori for the modular Gromov-Hausdorff propinquity

The modular Gromov-Hausdorff propinquity is a distance on clas\-ses of modules endowed with quantum metric information, in the form of a metric form of a connection and a left Hilbert module structure. This paper proves that the family of Heisenberg modules over quantum two tori, when endowed with their canonical connections, form a continuous family for the modular propinquity.

The Higson–Roe sequence for étale groupoids. I. Dual algebras and compatibility with the BC map

We introduce the dual Roe algebras for proper étale groupoid actions and deduce the expected Higson–Roe short exact sequence. When the action is co-compact, we show that the Roe C^* -ideal of locally compact operators is Morita equivalent to the reduced C^* -algebra of our groupoid, and we further identify the boundary map of the associated periodic six-term exact sequence with the Baum–Connes map, via a Paschke–Higson map for groupoids. For proper actions on continuous families of manifolds of bounded geometry, we associate with any G -equivariant Dirac-type family, a coarse index class which generalizes the Paterson index class and also the Moore–Schochet Connes’ index class for laminations.

Dynamics of compact quantum metric spaces

We provide a detailed study of actions of the integers on compact quantum metric spaces, which includes general criteria ensuring that the associated crossed product algebra is again a compact quantum metric space in a natural way. Moreover, we provide a flexible set of assumptions ensuring that a continuous family of$\ast$-automorphisms of a compact quantum metric space yields a field of crossed product algebras which varies continuously in Rieffel’s quantum Gromov–Hausdorff distance. Finally, we show how our results apply to continuous families of Lip-isometric actions on compact quantum metric spaces and to families of diffeomorphisms of compact Riemannian manifolds which vary continuously in the Whitney$C^{1}$-topology.

The designs and deformations of rigidly and flat-foldable quadrilateral mesh origami

Rigidly and flat-foldable quadrilateral mesh origami is the class of quadrilateral mesh crease patterns with one fundamental property: the patterns can be folded from flat to fully-folded flat by a continuous one-parameter family of piecewise affine deformations that do not stretch or bend the mesh-panels. In this work, we explicitly characterize the designs and deformations of all possible rigidly and flat-foldable quadrilateral mesh origami. Our key idea is a rigidity theorem (Theorem 3.1) that characterizes compatible crease patterns surrounding a single panel and enables us to march from panel to panel to compute the pattern and its corresponding deformations explicitly. The marching procedure is computationally efficient. So we use it to formulate the inverse problem: to design a crease pattern to achieve a targeted shape along the path of its rigidly and flat-foldable motion. The initial results on the inverse problem are promising and suggest a broadly useful engineering design strategy for shape-morphing with origami.

A class of elliptic equations involving nonlocal integrodifferential operators with sign-changing weight functions

In this paper, we investigate the existence of nontrivial weak solutions to a class of elliptic equations involving a general nonlocal integrodifferential operator LAK, two real parameters, and two weight functions, which can be sign-changing. Considering different situations concerning the growth of the nonlinearities involved in the problem (P), we prove the existence of two nontrivial distinct solutions and the existence of a continuous family of eigenvalues. The proofs of the main results are based on ground state solutions using the Nehari method, Ekeland’s variational principle, and the direct method of the calculus of variations. The difficulties arise from the fact that the operator LAK is nonhomogeneous and the nonlinear term is undefined.

The Fermat–Torricelli theorem in convex geometry

This paper studies a generalization of the Euclidean triangle, the generalized deltoid, which we believe to be the right one for convex geometry. To illustrate the process, our main result shows that the generalized deltoid satisfies a convex generalization of the Fermat–Torricelli theorem. A point that minimizes the sum of distances to the vertices of a triangle (Fermat–Torricelli point) is the same as one through which pass three equiangular affine diameters (Fermat–Ceder point). A generalized deltoid is a triangle whose sides are disjoint, outwardly-looking arcs of convex curves. The Fermat–Torricelli theorem in convex geometry extends the Fermat–Ceder point of a triangle to a Fermat–Ceder point of a generalized deltoid. As an application, we show that the Fermat–Ceder points for the continuous families of affine diameters, area-bisecting lines, and perimeter-bisecting lines are unique for every triangle, and non-unique for every pentagon. In the case of quadrilaterals, the uniqueness of the Fermat–Ceder point for affine diameters holds precisely for all non-trapezoids, the one for the Fermat–Ceder point for area-bisecting lines holds for all quadrilaterals, and the one for the Fermat–Ceder point for perimeter-bisecting lines is open.

The chaotic Black–Scholes equation with time-dependent coefficients

In Emamirad et al. (in: Semigroups of Operators - Theory and Applications, Springer, 2014; and Proc. Amer. Math. Soc. 140:2043–2052, 2012), the Black–Scholes semigroup was studied in various Banach spaces of continuous functions regarding chaotic behavior and the null volatility limit. Here we let the volatility and the interest rates be continuous functions on the half line $$[0,\infty )$$ , and we consider the generalized Black–Scholes equation as a linear nonautonomous abstract Cauchy problem. It is shown that this problem is wellposed and an explicit formula for the corresponding strongly continuous evolution family is given. Furthermore, a hypercyclicity criterion for strongly continuous evolution families on separable Banach spaces is proved and applied to the Black–Scholes evolution family. Finally, the chaotic behavior is defined for strongly continuous evolution families and it is shown that the Black–Scholes evolution family is chaotic when the volatility and the interest rates are periodic functions.

Remarks on equilibria of two-dimensional uniform vortices with polygonal symmetry

Two-dimensional uniform vortices that rotate without deforming in an isochoric, inviscid fluid are investigated. They have polygonal symmetry and are usually considered of the same kind of the famous Kirchhoff elliptical one. By following this idea, their shapes and rotation speeds are calculated through a new integral approach. It is heuristically proved, by examining the co-rotating streamlines, that the similitude with the elliptical vortex is not complete, due to the appearance of external singularities in the Schwarz functions of the polygonal boundaries. Once the effects of these singularities have been accounted for, exactly steady vortices are obtained. In this way, for any number of “sides”, a continuous family of rotating, steady polygonal vortices is obtained, parametrized in the ratio between minimum and maximum radial coordinates of the boundary. Solutions exist above a critical value of this ratio, at which the boundary ceases to be differentiable.

Support needs of Canadian adult siblings of brothers and sisters with intellectual/developmental disabilities

AbstractBackgroundIt is becoming more common for siblings to fulfill a caregiving role for their brother or sister, particularly because people with intellectual/developmental disabilities (IDD) are often living longer and outliving their parents. However, most of what we know about siblings of people with IDD is based on research with children, and limited studies on the adult sibling experiences in Canada have been published. To meet the support needs of Canadian adult siblings, “The Sibling Collaborative”, a grass‐roots initiative, conducted a needs assessment.Specific AimsThe purpose of this study was to better understand the current challenges siblings experience and how requested resources may differ across three age groups: adults between the ages of 20 and 29 years, 30 to 49 years, and 50 years of age and older.MethodA total of 260 siblings of individuals with IDD from across Ontario completed an online survey.FindingsSiblings endorsed a relatively low rating of intensity of support that they provided for their brothers or sisters with IDD at the time of survey completion; however, the majority indicated that they intended to take a greater caregiving role in the future. Ratings of support differed by sibling age groups, as did challenges related to supporting brothers or sisters with IDD. Overall, participants reported a range of desired resources and preferred methods of accessing resources.DiscussionSiblings' caregiving relationships with their brothers and sisters with IDD differed across age groups. Results from the current study indicate different supports may be needed for different age groups. As policies around the world continue to encourage continued family involvement in caregiving for adults with IDD, it is important to understand how systems can better support sibling caregivers.

Parameters of a class of complex continuous wavelet family

In this paper, we have established the relationship between the wavelet scale and equivalent Fourier wavelength for the family of the new complex continuous wavelet. The same relation is used to relate the maximum of the Scalogram of each coefficient obtained from the wavelet transform of the signal to its amplitude. The relationship was applied to a synthetic signal and the simulated results show the effectiveness of using the proper wavelet transform definition to this proposed family of wavelets.

Richtmyer–Meshkov dynamics with variable acceleration by group theory approach

Richtmyer–Meshkov instability (RMI) plays an important role in nature and technology, including supernovae, fusion, scramjets and nano-fabrication. Canonical RMI is induced by a steady shock and impulsive acceleration. In realistic environments the acceleration is often variable. We focus on the accelerations with power-law time-dependence, identify the values of the power-law exponent for which the dynamics is Richtmyer–Meshkov type, and apply group theory to solve the long-standing problem of RMI with variable acceleration. For early-time dynamics, we find that the RMI growth-rate depends on the initial conditions and not on the acceleration parameters. For late-time dynamics, we find a continuous family of regular asymptotic solutions, including their curvature, velocity, Fourier amplitudes, and interfacial shear function, and we study stability of these solutions. For each solution, the interface dynamics is set by the interfacial shear; the non-equilibrium velocity field has effectively no motion in the bulk and intense motion near the interface. The fastest stable solution in the family has the flattened front and the quasi-invariance property suggesting the multi-scale character of nonlinear dynamics in RMI. Our results agree with available observations and elaborate new benchmarks for future studies.

Generalized trapezoidal ogive curves for fatality rate modeling

The construction of a continuous family of distributions on a compact set is demonstrated by concatenating, in a continuous manner, three probability density functions with bounded support using a modified mixture technique. The construction technique is similar to that of generalized trapezoidal (GT) distributions, but contrary to GT distributions, the resulting density function is smooth within its bounded domain. The construction of Generalized Trapezoidal Ogive (GTO) distributions was motivated by the COVID-19 epidemic, where smoothness of an infection rate curve may be a desirable property combined with the ability to separately model three stages and their durations as the epidemic progresses, being: (1) an increasing infection rate stage, (2) an infection rate stage of some stability and (3) a decreasing infection rate stage. The resulting model allows for asymmetry of the infection rate curve opposite to, for example, the Gaussian Error Infection (GEI) rate curve utilized early on for COVID-19 epidemic projections by the Institute for Health Metrics and Evaluation (IHME). While other asymmetric distributions too allow for the modeling of asymmetry, the ability to separately model the above three stages of an epidemic’s progression is a distinct feature of the model proposed. The latter avoids unrealistic projections of an epidemic’s right-tail in the absence of right tail data, which is an artifact of any fatality rate model where a left-tail fit determines its right-tail behavior.

Unexpected quadratic behaviors for the small-time local null controllability of scalar-input parabolic equations

We consider scalar-input control systems in the vicinity of an equilibrium, at which the linearized systems are not controllable. For finite dimensional control systems, the authors recently classified the possible quadratic behaviors. Quadratic terms introduce coercive drifts in the dynamics, quantified by integer negative Sobolev norms, which are linked to Lie brackets and which prevent smooth small-time local controllability for the full nonlinear system.In the context of nonlinear parabolic equations, we prove that the same obstructions persist. More importantly, we prove that two new behaviors occur, which are impossible in finite dimension. First, there exists a continuous family of quadratic obstructions quantified by fractional negative Sobolev norms or by weighted variations of them. Second, and more strikingly, small-time local null controllability can sometimes be recovered from the quadratic expansion. We also construct a system for which an infinite number of directions are recovered using a quadratic expansion.As in the finite dimensional case, the relation between the regularity of the controls and the strength of the possible quadratic obstructions plays a key role in our analysis.

Turkish delight a public affairs study on family business: The influence of owners in the entrepreneurship orientation of family‐owned businesses

Family‐owned businesses (FOBs) are as unique as the families that own and control them. As reported by Miller, Steier, and Le Breton‐Miller (2003, p.513), the founders of many of these businesses try to continue their legacy and ensure continued family control via intergenerational succession, as when they hand over leadership to their children. The initial statistics suggest only approximately one third of FOBs survive into the second generation, with just 12% remaining “viable” by the third, and only about 3% operating into the fourth generation or beyond. Thus, one of the central problems for FOBs is this inability to ensure competent cross‐generational family leadership through successful transfer of ownership and management to the next family generation. This is a core issue for the modern public affairs practitioner and policy maker, nationally and internationally, and the Turkish case is a good example of the multicomplex issues evident in succession planning and leadership for business founders and leaders in these organisations.A firm's strategic orientation is an indicator of the processes developed to integrate new information, to coordinate decisions, to examine the evolution of environmental factors, and to assess new projects (Escriba‐Esteve, Sanchez‐Peinado & Sanchez‐Peinado, 2009). However, few studies have provided a framework that jointly analyses the FOB owner characteristics, the mediating processes and attitudes by which owners shape the direction of their family firms, and the effect of these postures on firm performance.This paper addresses the influence of family business owner, over the behaviour of FOBs. By treating FOB owners' characteristics as predictors of a firm's strategic orientation, we seek to provide a deeper understanding of how the characteristics of FOB owners shape decision making process and FOBs' behaviours in order to successfully survive in generations. This study introduced the concept of FOB's entrepreneurship orientation (EO) as a variable that mediates between FOB owners' characteristics and business performance. The objective of this paper is twofold: (a) to identify the demographic predictors FOBs' EO and (b) to analyse the role of EO as a mediator of the relationship between FOB owners' characteristics and FOBs' performance.

Scheme of multiplicative linearization of a random process

In this paper, we consider a stochastic differential equation in a Hilbert space. This is a natural situation when modeling a physical system having disturbances such as white noise. The equation describing such a process is a nonlinear stochastic equation in the form of Ito. In this paper, we consider the method of multiplicative linearization of such an equation. To this end, the time interval for the existence of a solution is divided into elementary ones in such a way that the diameter of the maximum partition tends to zero. At each of these intervals, the nonlinear coefficients of the equation are expanded in a Taylor series. This becomes possible when additional conditions for the coefficients of the equation are satisfied. It is required that the coefficients have Lipschitz uniformly bounded first and second order derivatives, and also that the fourth moment of the initial value be bounded. The conditions of the equivalence lemma of two multiplicative representations are verified. One of these multiplicative families is a stochastic strongly continuous evolutionary family of resolving operators of the original equation. Another family is the product of the resolving operators of linear equations on the corresponding elementary time intervals. Under the additional conditions on the coefficients of the equation, there exists a unique, up to stochastic equivalence, solution of a linear inhomogeneous equation. Using the Lagrange form for the remainder terms when expanding in a Taylor series and the Gronwall lemma, the conditions of the lemma are verified and the solution is proved to be a product of linear processes on each elementary interval. This multiplicative product can be interpreted as an approximate solution.

Quantum extensions of ordinary maps

We define a loop to be quantum nullhomotopic if and only if it admits a nonempty quantum set of extensions to the unit disk. We show that the canonical loop in the unit circle is not quantum nullhomotopic, but that every loop in the real projective plane is quantum nullhomotopic. Furthermore, we apply Kuiper’s theorem to show that the canonical loop admits a continuous family of extensions to the unit disk that is indexed by an infinite quantum space. We obtain these results using a purely topological condition that we show to be equivalent to the existence of a quantum family of extensions of a given map.

On orientations for gauge-theoretic moduli spaces

Let X be a compact manifold, D:Γ∞(E0)→Γ∞(E1) a real elliptic operator on X, G a Lie group, P→X a principal G-bundle, and BP the infinite-dimensional moduli space of all connections ∇P on P modulo gauge, as a topological stack. For each [∇P]∈BP, we can consider the twisted elliptic operator D∇Ad(P):Γ∞(Ad(P)⊗E0)→Γ∞(Ad(P)⊗E1) on X. This is a continuous family of elliptic operators over the base BP, and so has an orientation bundle OPE•→BP, a principal Z2-bundle parametrizing orientations of KerD∇Ad(P)⊕CokerD∇Ad(P) at each [∇P]. An orientation on (BP,E•) is a trivialization OPE•≅BP×Z2.In gauge theory one studies moduli spaces MPga of connections ∇P on P satisfying some curvature condition, such as anti-self-dual instantons on Riemannian 4-manifolds (X,g). Under good conditions MPga is a smooth manifold, and orientations on (BP,E•) pull back to orientations on MPga in the usual sense of differential geometry under the inclusion MPga↪BP. This is important in areas such as Donaldson theory, where one needs an orientation on MPga to define enumerative invariants.We explain a package of techniques, some known and some new, for proving orientability and constructing canonical orientations on (BP,E•), after fixing some algebro-topological information on X. We use these to construct canonical orientations on gauge theory moduli spaces, including new results for moduli spaces of flat connections on 2- and 3-manifolds, instantons, the Kapustin–Witten equations, and the Vafa–Witten equations on 4-manifolds, and the Haydys–Witten equations on 5-manifolds.

Optimal controls for second‐order stochastic differential equations driven by mixed‐fractional Brownian motion with impulses

We study optimal control problems for a class of second-order stochastic differential equation driven by mixed-fractional Brownian motion with non-instantaneous impulses. By using stochastic analysis theory, strongly continuous cosine family, and a fixed point approach, we establish the existence of mild solutions for the stochastic system. Moreover, the optimal control results are derived without uniqueness of mild solutions of the stochastic system. Finally, the main results are validated with the aid of an example.

The need for renovating patient education in kidney transplantation: A qualitative study.

BACKGROUND:Many kidney transplant recipients lack the knowledge, abilities, and support they need for self-care. On the other hand, most kidney transplant centers do not have a well-planned and specific training program for them, and educational interventions for kidney transplant recipients have not been adequately effective. This study aimed to describe strategies for improving patient education in kidney transplantation.MATERIALS AND METHODS:Data were collected through semi-structured individual and group interviews with 24 patients, family members, and health-care staff in one of the main kidney transplant centers in Tehran. Participants were selected purposefully, and qualitative content analysis was used to analyze the data.RESULTS:The main finding emerged from the data was the shift from current patient education program to patient- and family-centered education (PFCE). The strategies to achieve this goal were categorized into four main categories including “continuous patient and family education” (pre- and posttransplant patient education), “facilitating the process” (using new technologies, teamwork education, and patient and family accessibility), “strengthening human resources” (empowerment health-care team, allocation of human resources, promoting staffs' motivation, and updating educational content and materials), and “monitoring and evaluation” (correcting patient education recording, supervising the patient education, and appropriate educational evaluation).CONCLUSIONS:Transforming from the current patient education program to PFCE seems to be essential to increase the effectiveness of patient education in kidney transplant process. To this end, providing continuous patient and family education, facilitating the processes, strengthening human resources, and monitoring and evaluation in health-care organizations conducting the kidney transplantation is necessary.