Spectral decomposition of fractional operators and a reflected stable semigroup

In this paper, we provide the spectral decomposition in Hilbert space of the $\mathcal{C}_0$-semigroup $P$ and its adjoint \hatP having as generator, respectively, the Caputo and the right-sided Riemann-Liouville fractional derivatives of index $1<\alpha<2$. These linear operators, which are non-local and non-self-adjoint, appear in many recent studies in applied mathematics and also arise as the infinitesimal generators of some substantial processes such as the reflected spectrally negative $\alpha$-stable process. Our approach relies on intertwining relations that we establish between these semigroups and the semigroup of a Bessel type process whose generator is a self-adjoint second order differential operator. In particular, from this commutation relation, we characterize the positive real axis as the continuous point spectrum of $P$ and provide a power series representation of the corresponding eigenfunctions. We also identify the positive real axis as the residual spectrum of the adjoint operator \hatP and elucidates its role in the spectral decomposition of these operators. By resorting to the concept of continuous frames, we proceed by investigating the domain of the spectral operators and derive two representations for the heat kernels of these semigroups. As a by-product, we also obtain regularity properties for these latter and also for the solution of the associated Cauchy problem.Comment: in pres

Intertwining, Excursion Theory and Krein Theory of Strings for Non-self-adjoint Markov Semigroups

In this paper, we start by showing that the intertwining relationship between two minimal Markov semigroups acting on Hilbert spaces implies that any recurrent extensions, in the sense of It\^o, of these semigroups satisfy the same intertwining identity. Under mild additional assumptions on the intertwining operator, we prove that the converse also holds. This connection, which relies on the representation of excursion quantities as developed by Fitzsimmons and Getoor, enables us to give an interesting probabilistic interpretation of intertwining relationships between Markov semigroups via excursion theory: two such recurrent extensions that intertwine share, under an appropriate normalization, the same local time at the boundary point. Moreover, in the case when one of the (non-self-adjoint) semigroup intertwines with the one of a quasi-diffusion, we obtain an extension of Krein's theory of strings byshowing that its densely defined spectral measure is absolutely continuous with respect to the measure appearing in the Stieltjes representation of the Laplace exponent of the inverse local time. Finally, we illustrate our results with the class of positive self-similar Markov semigroups and also the reflected generalized Laguerre semigroups. For the latter, we obtain their spectral decomposition and provide, under some conditions, a perturbed spectral gap estimate for its convergence to equilibrium

Research on Community Building in China, from the Perspective of Community Empowerment

The official document of the 19th National Congress of the ruling party highly stresses the strategy of rural revitalization for the first time, calling for more attention to the crisis of rural communities. This provides a political agenda to promote rural planning lacking local extensive participation. While most research concerning the rural community involvement in governance and building is focused on the improvement of professional planning methods and techniques spatially, little attention is on the motivations and abilities to participate in the community. The empowerment theory, rooted in the American environment in the 1970s, provides a new perspective for inspiring the enthusiasm of vulnerable groups to participate in the community and improving their ability to participate. The purpose of this dissertation is to discuss how to improve community construction in China from the perspective of empowerment theory

Out-of-Distribution Knowledge Distillation via Confidence Amendment

Out-of-distribution (OOD) detection is essential in identifying test samples that deviate from the in-distribution (ID) data upon which a standard network is trained, ensuring network robustness and reliability. This paper introduces OOD knowledge distillation, a pioneering learning framework applicable whether or not training ID data is available, given a standard network. This framework harnesses OOD-sensitive knowledge from the standard network to craft a binary classifier adept at distinguishing between ID and OOD samples. To accomplish this, we introduce Confidence Amendment (CA), an innovative methodology that transforms an OOD sample into an ID one while progressively amending prediction confidence derived from the standard network. This approach enables the simultaneous synthesis of both ID and OOD samples, each accompanied by an adjusted prediction confidence, thereby facilitating the training of a binary classifier sensitive to OOD. Theoretical analysis provides bounds on the generalization error of the binary classifier, demonstrating the pivotal role of confidence amendment in enhancing OOD sensitivity. Extensive experiments spanning various datasets and network architectures confirm the efficacy of the proposed method in detecting OOD samples

Research on Consumption Upgrading of Rural Residents in Hebei Province

China has achieved significant economic growth in the past 40 years, however, there is still great difference between urban and rural areas on social and economic development. This paper took rural area of Hebei province as an example, introduced the current situation and potential of rural consumption in Hebei Province, analyzed the factors affecting the consumption upgrading, and proposed suggestions on how to promote the consumption upgrading in rural areas of Hebei Province

Design and performance analysis of radio frequency meandered-line microstrips

In the design of miniature microstrip transmission lines, a meandered line is often used as an effective method to insert proper electrical delay while reducing the feature size. However, this design technique also introduces undesirable noise to the applied network. In order to provide detailed analysis regarding the electrical response from variously shaped meandered lines, eight different structures of meandered line microstrip, all of the same physical length of 125 mm, were designed, modeled and characterized in terms of their electrical properties and footprints. Both the simulations and measurements of these microstrip lines were carried out in a frequency range of 50 MHz to 10 GHz. In addition, this thesis compares the eight designs’ insertion losses and discusses the possible relationship between particular structures of meandered lines and their corresponding frequency responses.Ope

Living with nonstandard work schedules -- Wellbeing challenges for Australian parents

Shift work is prevalent and has the potential to impact workers' mental health and wellbeing. However, existing research regarding an association between shift work and wellbeing is mixed, and there is a paucity of research examining the wellbeing impacts specifically for working parents. The overarching goal of this thesis is to clarify to what extent shift work is associated with wellbeing specifically for parents (mothers and fathers), focusing on the challenges shift work might bring to negotiating work and family responsibilities. Parental wellbeing is explored using three indicators - psychological distress, work-family conflict and couple relationship quality. While shift work is primarily positioned as an 'adverse exposure', the dynamic, bi-directional nature of the relationships is explored. Three inter-related studies were conducted: Study 1 is a cross-sectional study of 1208 employed parents participating in the "Families at Work" nationwide survey. Using moderated-mediation analyses, study 1 investigated the association between shift work and psychological distress, whether work-family conflict explained this association, and gender differences. The findings show some evidence that working irregular shifts was associated with higher work-family conflict in connection with higher distress, with fathers at greater risk. Working regular shifts was associated with greater distress (largely for mothers), but was not related to work-family conflict, suggesting other mechanisms are likely involved. Study 2 investigated a possible causal link between working shifts and workers' parental wellbeing (i.e. psychological distress, work-family conflict and couple relationship quality). The study used five waves of longitudinal data (n=1915 couples) from a nationally representative sample of dual-earner parents (from The Longitudinal Study of Australian Children - LSAC) - positioning shift work as an occupational exposure. This study adopted a hybrid analysis model to identify within-individual changes in wellbeing in association with changes in shift work (separate from time-invariant differences between individuals). Changes in shift work were found to be linked to changes in fathers' (but not mothers') relationship quality, and this association could not be attributed to shift working fathers' personal or work-related characteristics. Study 3 was conducted as a complement to study 2, but the reverse causal direction was tested to explore whether health selection might drive any association between shift work and wellbeing. Using a sample from the LSAC largely comparable to study 2, population-average logit models investigated whether parents' wellbeing predicted a subsequent transition into or out of shift work. The results showed that mothers with higher psychological distress were more likely to move into shift work. No other significant associations were found. The findings suggest caution is needed when assuming the causal direction of associations found between shift work and mental health, especially in cross-sectional studies. Overall, the findings from this thesis provide some evidence that shift working parents in Australia have poorer wellbeing than non-shift working parents. However, the findings differ somewhat for mothers and fathers, may depend on the type of shift work undertake and the wellbeing indicator measured. Further research is needed to clarify the nature of these relationships to inform family-friendly policy and practices to support shift working parents