Infinitely many homoclinic solutions for double phase problem on integers

We consider a discrete double phase problem on integers with an unbounded potential and reaction term, which does not satisfy the Ambrosetti–Rabinowitz condition. A new functional setting was provided for this problem. Using the Fountain and Dual Fountain Theorem with Cerami condition, we obtain some existence of infinitely many solutions. Our results extend some recent findings expressed in the literature.

Functional Settings of Hospital Outdoor Spaces and the Perceptions from Public and Hospital Occupant during COVID-19

Hospital outdoor spaces play an important role for the safety and well-being of users (patients, visitors, and staff), particularly during a pandemic. However, the actual needs of these spaces are often overlooked due to the design and management process. This study investigates the perceptions of the public and occupants on the functional settings of outdoor spaces, and provides evidence for building a safe and resilient hospital during (and after) COVID-19. A multi-method approach of web content analysis (WCA) and a web-based survey was employed. Reports were collected from three mainstream websites; keywords were extracted and then categorized, pertaining to the functional settings of outdoor spaces. Three groups of occupants from Southwest Hospital (staff n = 47, patients n = 64, visitors n = 73) participated in the survey to identify their perceptions of these functional settings. Based on the 657 reports and 33 keywords selected, 7 functional settings were identified: health check (HC), quarantine and observation (QO), food and delivery (FD), healing and restoration (HR), waiting and rest (WR), transportation and parking (TP), load and unload (LU). From all users, HC (4.13) was thought to be the most expected function setting while FD (2.61) was the least. Regarding the satisfaction level, most users were satisfied with HC (3.22) while WR (2.16) was the least satisfying. The users also showed significant differences regarding expectation and satisfaction pertaining to their groups. The results indicate that the current outdoor space could not fully meet the needs of users, regarding the emerging functional setting, due to the pandemic. Users showed significant different perceptions on the functional setting due to their roles. The mismatch between the outdoor space and the users’ needs on emerging functional settings resulted in low satisfaction and high expectation in the survey. Environmental interventions with adaptive and flexible strategies should be adapted for these functional settings. The differences of users should be fully recognized by administrators, decision-makers, and designers.

Generic Ownership: a Practical Approach to Ownership and Confinement in Object-Oriented Programming Languages

<p>Modern object-oriented programming languages support many techniques that simplify the work of a programmer. Among them is generic types: the ability to create generic descriptions of algorithms and object structures that will be automatically specialised by supplying the type information when they are used. At the same time, object-oriented technologies still suffer from aliasing: the case of many objects in a program's memory referring to the same object via different references. Ownership types enforce encapsulation in object-oriented programs by ensuring that objects cannot be referred to from the outside of the object(s) that own them. Existing ownership programming languages either do not support generic types or attempt to add them on top of ownership restrictions. The goal of this work is to bring object ownership into mainstream object-oriented programming languages. This thesis presents Generic Ownership which provides perobject ownership on top of a generic imperative language. Surprisingly, the resulting system not only provides ownership guarantees comparable to the established systems, but also requires few additional language mechanisms to achieve them due to full reuse of generic types. In this thesis I formalise the core of Generic Ownership, highlighting that the restriction of this calls, owner preservation over subtyping, and appropriate owner nesting are the only necessary requirements for ownership. I describe two formalisms: (1) a simple formalism, capturing confinement in a functional setting, and (2) a complete formalism, providing a way for Generic Ownership to support both deep and shallow variations of ownership types. I support the formal work by describing how the Ownership Generic Java (OGJ) language is implemented as a minimal extension to Java 5. OGJ is the first publicly available language implementation that supports ownership, confinement, and generic types at the same time. I demonstrate OGJ in practice: show how to use OGJ to write programs and provide insights into the implementations of Generic Ownership.</p>

Generic Ownership: a Practical Approach to Ownership and Confinement in Object-Oriented Programming Languages

<p>Modern object-oriented programming languages support many techniques that simplify the work of a programmer. Among them is generic types: the ability to create generic descriptions of algorithms and object structures that will be automatically specialised by supplying the type information when they are used. At the same time, object-oriented technologies still suffer from aliasing: the case of many objects in a program's memory referring to the same object via different references. Ownership types enforce encapsulation in object-oriented programs by ensuring that objects cannot be referred to from the outside of the object(s) that own them. Existing ownership programming languages either do not support generic types or attempt to add them on top of ownership restrictions. The goal of this work is to bring object ownership into mainstream object-oriented programming languages. This thesis presents Generic Ownership which provides perobject ownership on top of a generic imperative language. Surprisingly, the resulting system not only provides ownership guarantees comparable to the established systems, but also requires few additional language mechanisms to achieve them due to full reuse of generic types. In this thesis I formalise the core of Generic Ownership, highlighting that the restriction of this calls, owner preservation over subtyping, and appropriate owner nesting are the only necessary requirements for ownership. I describe two formalisms: (1) a simple formalism, capturing confinement in a functional setting, and (2) a complete formalism, providing a way for Generic Ownership to support both deep and shallow variations of ownership types. I support the formal work by describing how the Ownership Generic Java (OGJ) language is implemented as a minimal extension to Java 5. OGJ is the first publicly available language implementation that supports ownership, confinement, and generic types at the same time. I demonstrate OGJ in practice: show how to use OGJ to write programs and provide insights into the implementations of Generic Ownership.</p>

New preconditioners for the Laplace and Helmholtz integral equations on open curves: analytical framework and numerical results

Helmholtz wave scattering by open screens in 2D can be formulated as first-kind integral equations which lead to ill-conditioned linear systems after discretization. We introduce two new preconditioners in the form of square-roots of on-curve differential operators both for the Dirichlet and Neumann boundary conditions on the screen. They generalize the so-called “analytical” preconditioners available for Lipschitz scatterers. We introduce a functional setting adapted to the singularity of the problem and enabling the analysis of those preconditioners. The efficiency of the method is demonstrated on several numerical examples.

High-dimensional regimes of non-stationary Gaussian correlated Wishart matrices

We study the high-dimensional asymptotic regimes of correlated Wishart matrices [Formula: see text], where [Formula: see text] is a [Formula: see text] Gaussian random matrix with correlated and non-stationary entries. We prove that under different normalizations, two distinct regimes emerge as both [Formula: see text] and [Formula: see text] grow to infinity. The first regime is the one of central convergence, where the law of the properly renormalized Wishart matrices becomes close in Wasserstein distance to that of a Gaussian orthogonal ensemble matrix. In the second regime, a non-central convergence happens, and the law of the normalized Wishart matrices becomes close in Wasserstein distance to that of the so-called Rosenblatt–Wishart matrix recently introduced by Nourdin and Zheng. We then proceed to show that the convergences stated above also hold in a functional setting, namely as weak convergence in [Formula: see text]. As an application of our main result (in the central convergence regime), we show that it can be used to prove convergence in expectation of the empirical spectral distributions of the Wishart matrices to the semicircular law. Our findings complement and extend a rich collection of results on the study of the fluctuations of Gaussian Wishart matrices, and we provide explicit examples based on Gaussian entries given by normalized increments of a bi-fractional or a sub-fractional Brownian motion.

On Time-Periodic Bifurcation of a Sphere Moving under Gravity in a Navier-Stokes Liquid

We provide sufficient conditions for the occurrence of time-periodic Hopf bifurcation for the coupled system constituted by a rigid sphere, S, freely moving under gravity in a Navier-Stokes liquid. Since the region of flow is unbounded (namely, the whole space outside S), the main difficulty consists in finding the appropriate functional setting where general theory may apply. In this regard, we are able to show that the problem can be formulated as a suitable system of coupled operator equations in Banach spaces, where the relevant operators are Fredholm of index 0. In such a way, we can use the theory recently introduced by the author and give sufficient conditions for time-periodic bifurcation to take place.

Quasilinear Dirichlet Problems with Degenerated p-Laplacian and Convection Term

The paper develops a sub-supersolution approach for quasilinear elliptic equations driven by degenerated p-Laplacian and containing a convection term. The presence of the degenerated operator forces a substantial change to the functional setting of previous works. The existence and location of solutions through a sub-supersolution is established. The abstract result is applied to find nontrivial, nonnegative and bounded solutions.

Sparse Functional Linear Discriminant Analysis

Functional linear discriminant analysis offers a simple yet efficient method for classification, with the possibility of achieving a perfect classification. Several methods are proposed in the literature that mostly address the dimensionality of the problem. On the other hand, there is a growing interest in interpretability of the analysis, which favors a simple and sparse solution. In this work, we propose a new approach that incorporates a type of sparsity that identifies nonzero sub-domains in the functional setting, offering a solution that is easier to interpret without compromising performance. With the need to embed additional constraints in the solution, we reformulate the functional linear discriminant analysis as a regularization problem with an appropriate penalty. Inspired by the success of ℓ1-type regularization at inducing zero coefficients for scalar variables, we develop a new regularization method for functional linear discriminant analysis that incorporates an L1-type penalty, ∫ |f|, to induce zero regions. We demonstrate that our formulation has a well-defined solution that contains zero regions, achieving a functional sparsity in the sense of domain selection. In addition, the misclassification probability of the regularized solution is shown to converge to the Bayes error if the data are Gaussian. Our method does not presume that the underlying function has zero regions in the domain, but produces a sparse estimator that consistently estimates the true function whether or not the latter is sparse. Numerical comparisons with existing methods demonstrate this property in finite samples with both simulated and real data examples.