Global asymptotic stability of bifurcating, positive equilibria of p-Laplacian boundary value problems with p-concave nonlinearities

We consider the parabolic, initial value problem $$v_t =\Delta_p(v)+\lambda g(x,v)\phi_p(v), \quad \text{in $\Omega \times (0,\infty),$}$$ $$v =0, \text{in $\partial\Omega \times (0,\infty),$}\tag{IVP} v =v_0\ge0, \text{in $\Omega \times \{0\},$}$$ where $\Omega$ is a bounded domain in ${\mathbb R}^N$, for some integer $N\ge1$, with smooth boundary $\partial\Omega$, $\phi_p(s):=|s|^{p-1} {\rm sgn}s$, $s\in{\mathbb R}$, $\Delta_p$ denotes the $p$-Laplacian, with $p>\max\{2,N\}$, $v_0\in C^0(\overline{\Omega})$, and $\lambda>0$. The function $g:\overline{\Omega } \times [0,\infty)\to(0,\infty)$ is $C^0$ and, for each $x\in\overline{\Omega }$, the function $g(x,\cdot):[0,\infty)\to(0,\infty)$ is Lipschitz continuous and strictly decreasing. Clearly, (IVP) has the trivial solution $v\equiv0$, for all $\lambda>0$. In addition, there exists $0<\lambda_{\rm min}(g)<\lambda_{\rm max}(g)$ ($\lambda_{\rm max}(g)$ may be $\infty$) such that: $(a)$ if $\lambda\not\in(\lambda_{\rm min}(g),\lambda_{\rm max}(g))$ then (IVP) has no non-trivial, positive equilibrium; $(b)$ if $\lambda\in(\lambda_{\rm min}(g),\lambda_{\rm max}(g))$ then (IVP) has a unique, non-trivial, positive equilibrium $e_\lambda\in W_0^{1,p}(\Omega)$. We prove the following results on the positive solutions of (IVP): $(a)$ if $0<\lambda<\lambda_{\rm min}(g)$ then the trivial solution is globally asymptotically stable; $(b)$ if $\lambda_{\rm min}(g)<\lambda<\lambda_{\rm max}(g)$ then $e_\lambda$ is globally asymptotically stable; $(c)$ if $\lambda_{\rm max}(g)<\lambda$ then any non-trivial solution blows up in finite time

The impact of Indigenous community sports programs: the case of surfing

The objective of this research project was to consider the social impact of sport and physical activity on the lives of Indigenous Australians and their communities. There has been strong research interest in the links between sport and recreation programs and various health and social outcomes and a well-established body of literature exists on the use of sport to address social issues in mainstream society. The consensus is that physical activity is an important contributor to health for all people. While there is strong research interest, what remains unclear is the value and impact of sport and physical activity on Indigenous communities.Indigenous groups cannot be considered to be homogenous as there is much diversity between and within groups. It is therefore important this report is not viewed as taking an essentialist view of who Indigenous people are and how they develop. Rather, this paper attempts to describe and discuss the experiences of some individuals and their communities in site-specific surfing programs.The results of this project indicate that surfing should continue to be considered as an appropriate sport for use with Indigenous communities.It should be noted, however, that any program needs to take into consideration the diversity of Indigenous culture, how communities operate and should seek appropriate guidance.While there was great variety in terms of how surf programs in this study operated, there were some common elements across all programs including: strict surfer-to-coach ratios, opt-in opt-out structures and avoidance of controlling coach behaviours.In addition to these characteristics, quality programs also had consistency in personnel (small groups rather than single providers).This research project provided experience and information to develop recommendations relating to future similar programs including:Programs should be supported in longer term allotments Programs should collect meaningful data (well-constructed interview and survey protocols) over and above simple participation statistics To achieve meaningful outcomes, programs need continuity and should not be one-offs As transport costs represent the greatest barrier to participation in programs and surfing as a lifestyle pursuit, programs should consider the transport options most suitable for their area. Individuals who participated in the research highlighted that it can take a long time for the effects of programs to be felt. As a result, longitudinal tracking by program providers is required to provide ongoing evidence in support of programs. Funding to programs should also be sought across longer time frames to provide a greater platform to generate positive outcomes.Through this research it was possible to identify the development of individual and organisational capacity. In both areas, however, these capacities were not automatically enhanced. Careful planning and strong encouragement and support from program providers were required

Landesman-Lazer conditions at half-eigenvalues of the p-Laplacian

We study the existence of solutions of the Dirichlet problem {gather} -\phi_p(u')' -a_+ \phi_p(u^+) + a_- \phi_p(u^-) -\lambda \phi_p(u) = f(x,u), \quad x \in (0,1), \label{pb.eq} \tag{1} u(0)=u(1)=0,\label{pb_bc.eq} \tag{2} {gather} where $p>1$, \phi_p(s):=|s|^{p-1}\sgn s for $s \in \mathbb{R}$, the coefficients $a_\pm \in C^0[0,1]$, $\lambda \in \mathbb{R}$, and $u^\pm := \max\{\pm u,0\}$. We suppose that $f\in C^1([0,1]\times\mathbb{R})$ and that there exists $f_\pm \in C^0[0,1]$ such that $\lim_{\xi\to\pm\infty} f(x,\xi) = f_\pm(x)$, for all $x \in [0,1]$. With these conditions the problem \eqref{pb.eq}-\eqref{pb_bc.eq} is said to have a `jumping nonlinearity'. We also suppose that the problem {gather} -\phi_p(u')' = a_+ \phi_p(u^+) - a_- \phi_p(u^-) + \lambda \phi_p(u) \quad\text{on} \ (0,1), \tag{3} \label{heval_pb.eq} {gather} together with \eqref{pb_bc.eq}, has a non-trivial solution $u$. That is, $\lambda$ is a `half-eigenvalue' of \eqref{pb_bc.eq}-\eqref{heval_pb.eq}, and the problem \eqref{pb.eq}-\eqref{pb_bc.eq} is said to be `resonant'. Combining a shooting method with so called `Landesman-Lazer' conditions, we show that the problem \eqref{pb.eq}-\eqref{pb_bc.eq} has a solution. Most previous existence results for jumping nonlinearity problems at resonance have considered the case where the coefficients $a_\pm$ are constants, and the resonance has been at a point in the `Fucik spectrum'. Even in this constant coefficient case our result extends previous results. In particular, previous variational approaches have required strong conditions on the location of the resonant point, whereas our result applies to any point in the Fucik spectrum.Comment: 14 page

Half eigenvalues and the Fucik spectrum of multi-point, boundary value problems

We consider the nonlinear boundary value problem consisting of the equation \tag{1} -u" = f(u) + h, \quad \text{a.e. on $(-1,1)$,} where $h \in L^1(-1,1)$, together with the multi-point, Dirichlet-type boundary conditions \tag{2} u(\pm 1) = \sum^{m^\pm}_{i=1}\alpha^\pm_i u(\eta^\pm_i) where $m^\pm \ge 1$ are integers, $\alpha^\pm = (\alpha_1^\pm, ...,\alpha_m^\pm) \in [0,1)^{m^\pm}$, $\eta^\pm \in (-1,1)^{m^\pm}$, and we suppose that $$\sum_{i=1}^{m^\pm} \alpha_i^\pm < 1 .$$ We also suppose that $f : \mathbb{R} \to \mathbb{R}$ is continuous, and $$0 < f_{\pm\infty}:=\lim_{s \to \pm\infty} \frac{f(s)}{s} < \infty.$$ We allow $f_{\infty} \ne f_{-\infty}$ --- such a nonlinearity $f$ is {\em jumping}. Related to (1) is the equation \tag{3} -u" = \lambda(a u^+ - b u^-), \quad \text{on $(-1,1)$,} where $\lambda,\,a,\,b > 0$, and $u^{\pm}(x) =\max\{\pm u(x),0\}$ for $x \in [-1,1]$. The problem (2)-(3) is `positively-homogeneous' and jumping. Regarding $a,\,b$ as fixed, values of $\lambda = \lambda(a,b)$ for which (2)-(3) has a non-trivial solution $u$ will be called {\em half-eigenvalues}, while the corresponding solutions $u$ will be called {\em half-eigenfunctions}. We show that a sequence of half-eigenvalues exists, the corresponding half-eigenfunctions having specified nodal properties, and we obtain certain spectral and degree theoretic properties of the set of half-eigenvalues. These properties lead to solvability and non-solvability results for the problem (1)-(2). The set of half-eigenvalues is closely related to the `Fucik spectrum' of the problem, which we briefly describe. Equivalent solvability and non-solvability results for (1)-(2) are obtained from either the half-eigenvalue or the Fucik spectrum approach

Self-adjoint boundary-value problems on time-scales

In this paper we consider a second order, Sturm-Liouville-type boundary-value operator of the form $$L u := -[p u^{ abla}]^{Delta} + qu,$$ on an arbitrary, bounded time-scale $mathbb{T}$, for suitable functions $p,q$, together with suitable boundary conditions. We show that, with a suitable choice of domain, this operator can be formulated in the Hilbert space $L^2(mathbb{T}_kappa)$, in such a way that the resulting operator is self-adjoint, with compact resolvent (here, "self-adjoint" means in the standard functional analytic meaning of this term). Previous discussions of operators of this, and similar, form have described them as self-adjoint, but have not demonstrated self-adjointness in the standard functional analytic sense

Hybridity, Style and Identity, the Court Art and Architecture of Lucknow 1770-1850

Lucknow, the capital city of present day Uttar Pradesh, and previously of the Kingdom of Oudh, was the site of an extraordinary cultural and artistic milieu in the late eighteenth and early nineteenth centuries. Oudh was an semi-independent state within the declining mughal empire, and close contact with Europeans provoked an exchange of architectural ideas which developed into a hybrid of Indian and European prototypes. In addition, the publication of Indian scenes in England initiated a response to Indian architectural styles with some fascinating results. Adopting elements of neo-classical architecture. combined with the local late mughal style, the nawabs of Lucknow initiated an extraordinary program of palatial and religious building. Drawing extensively on local prototypes, and using neo-classical forms and motifs. a hybrid architectural milieu developed which functioned as a multivalent sign of the nawabs' aspirations and identity statements. The buildings provoked, and continue to provoke, comment and debate, and are examined here in terms of their use as signs of identity, gendered spaces, and ritual spaces, both social and religious. The converse; British uses of Indian architectural forms; is discussed as being equally significant as a demonstration of how exotic forms are adopted and assimilated into an existing style in the west. In late eighteenth and early nineteenth century England, this took place within a larger debate about styles and values, and a commitment to notions of primacy and purity in architecture. One result of this thinking is the denigration of hybridity and deviation from style which informed the art historical attitudes to Lucknavi architecture, and this is discussed with particular reference to the western art historical discourse. The use of the other in the formation of diverse identity statements is examined and developed in three specific cases: the little-known culture. architecture and rituals of the Lucknavi Shias, the employment of a European artist in the king of Lucknow's inner circle, and his influence on the courtly style, and the use of the first visual mass-medium, the Panorama, to display Lucknow to the British public. These significant examples are used to illustrate the essentially tripartite nature of the hybrid culture of Lucknow: Indian (mughal), European and Shia; and subverting the polarity that is usually assumed in the contacts between east and west

‘Fast track’ and ‘traditional path’ coaches: affordances, agency and social capital

A recent development in large-scale coach accreditation (certification) structures has been the ‘fast tracking’ of former elite athletes. Former elite athletes are often exempted from entry-level qualifications and are generally granted access to fast track courses that are shortened versions of the accreditation courses undertaken by ‘traditional path’ coaches. While formal coach accreditation is not the focus of this research note, it does provide the context for the two coaching case studies. The aim of this article is to consider and contrast the experiences of a former elite athlete and a traditional pathway coach with respect to their development and their trajectory towards employment in high performance coaching settings. The notion of relational interdependence (Billett, 2006) is used to consider the characteristics that particular coaches may bring to their work. In examining the social nature of coaching work and coaching appointments further, it is possible to connect with the notion of social capital (Field, 2006). Informed by accreditation course information (coaching history, aspirations and educational achievements) and three days of in-course observations by the author, the interpretivist case study design incorporated a semi-structured interview with one former elite athlete and one traditional pathway coach during the top level coach accreditation course of one of Australia's most popular team sports. Interviews were transcribed verbatim and coded via a hierarchical content analysis. From this study it was possible to identify a range of affordances that are available to former elite athletes that are not readily accessible for traditional pathway coaches and vice versa. Regarding social capital, former athletes appear to possess greater amounts and are better able to leverage that capital for development and employment. Recommendations are offered and implications discussed for coaches and those individuals and organisations charged with employing high performance coaches

Young Ireland and Irish Revolutions

This paper gives an overview of the development of the republican armed force tradition in Irish politics from the 1790s. It concludes that while Wolfe Tone and Emmet may have been inspirational, it was the experiences in politics and developments in political theory stemming from the 1840s Young Ireland movement that had the greatest impact. Though the 1848 rebellion led by William Smith O’Brien has often been derided by historians, it was a pivotal event which led directly to the foundation of Fenianism, which in turn led directly to the Land League revolution 1879-82 and indeed the 1916 Rising. The influence of James Fintan Lalor is highlighted as it was Lalor who came up an alternative formula to constitutional agitation arguing that England’s treatment of Ireland had given the Irish a moral right to a legal tabula rasa over both land ownership and constitutional claims.Cette étude propose un bilan de la tradition de la force armée républicaine dans la politique irlandaise depuis les années 1790. Elle conclut que même si Wolfe Tone et Emmet furent des sources d’inspiration indéniables, ce sont les expériences politiques et les développements des théories politiques à partir du mouvement de la Jeune Irlande des années 1840 qui ont eu l’impact le plus important. Alors que la révolution de 1848 dirigée par William Smith O’Brien a souvent été minimisée par les historiens, ce fut en réalité un événement charnière qui eut pour conséquence directe la création du fenianisme qui a à son tour mené directement à la révolution de la Land League de 1879-1882 et par la suite à l’insurrection de 1916. L’influence de James Fintan Lalor est soulignée car celui-ci proposa une alternative à l’agitation constitutionnelle en affirmant que le traitement de l’Irlande par l’Angleterre avait donné aux Irlandais un droit moral à repartir sur de nouvelles bases juridiques, à la fois du point de vue de la propriété de la terre et des revendications constitutionnels

Diophantine approximation and the solubility of the Schroedinger equation

We characterise the set of periods for which number theoretical obstructions prevent us from finding periodic solutions of the Schroedinger equation on a two dimensional torus as well as the asymptotic occurrence of possible resonances.Comment: 7 pages, revised versio