For microscale heterogeneous partial differential equations (PDEs), this article further develops novel theory and methodology for their macroscale mathematical/asymptotic homogenization. This article specifically encompasses the case of quasi-periodic heterogeneity with finite scale separation: no scale separation limit is required. A key innovation herein is to analyse the ensemble of all phase-shifts of the heterogeneity. Dynamical systems theory then frames the homogenization as a slow manifold of the ensemble. Depending upon any perceived scale separation within the quasi-periodic heterogeneity, the homogenization may be done in either one step or two sequential steps: the results are equivalent. The theory not only assures us of the existence and emergence of an exact homogenization at finite scale separation, it also provides a practical systematic method to construct the homogenization to any specified order. For a class of heterogeneities, we show that the macroscale homogenization is potentially valid down to lengths which are just twice that of the microscale heterogeneity! This methodology complements existing well-established results by providing a new rigorous and flexible approach to homogenization that potentially also provides correct macroscale initial and boundary conditions, treatment of forcing and control, and analysis of uncertainty.

In this paper, we study the singular boundary value problem

$$ \begin{align*} \begin{cases} \Delta_\infty^h u=\lambda f(x,u,Du) \quad &\mathrm{in}\; \Omega, \\ u>0\quad &\mathrm{in}\; \Omega,\\ u=0 \quad &\mathrm{on} \;\partial\Omega, \end{cases} \end{align*} $$

where $\lambda>0$ is a parameter, $h>1$ and $\Delta _\infty ^h u=|Du|^{h-3} \langle D^2uDu,Du \rangle $ is the highly degenerate and h-homogeneous operator related to the infinity Laplacian. The nonlinear term $f(x,t,p):\Omega \times (0,\infty )\times \mathbb {R}^{n}\rightarrow \mathbb {R}$ is a continuous function and may exhibit singularity at $t\rightarrow 0^{+}$. We establish the comparison principle by the double variables method for the general equation $\Delta _\infty ^h u=F(x,u,Du)$ under some conditions on the term $F(x,t,p)$. Then, we establish the existence of viscosity solutions to the singular boundary value problem in a bounded domain based on Perron’s method and the comparison principle. Finally, we obtain the existence result in the entire Euclidean space by the approximation procedure. In this procedure, we also establish the local Lipschitz continuity of the viscosity solution.

In a number of texts throughout his career, Thomas Aquinas identifies different senses of the term ‘esse’. Most notably, he notes that according to one sense, the term signifies the act of existence (actus essendi), which he famously holds is really distinct from essence in all beings other than God. Perhaps surprisingly, he also notes on a number of occasions that according to another sense, the term ‘esse’ can signify that very principle that he says is distinct from the act of existence, namely, essence. In light of Aquinas's semantic theory, this paper investigates how he coherently holds within his metaphysical system that this term ‘esse’ can signify in different ways both essence and the act of existence. More broadly, what it shows is how, for Aquinas, the metaphysician can look to the modes of signification (modi significandi) of terms and as well as their modes of predication (modi praedicandi) to draw careful conclusions about the modes of existence (modi essendi) of real beings. These considerations reveal that in Aquinas's view, although the grammarian and logician in their way are also concerned with these semantic modes, it is not their job to employ them to discern the various senses of the term ‘being’ or the fundamental modes of being. In the end, this is a task for the metaphysician.

We will present the proof of existence and uniqueness of renormalized solutions to a broad family of strongly non-linear elliptic equations with lower order terms and data of low integrability. The leading part of the operator satisfies general growth conditions settling the problem in the framework of fully anisotropic and inhomogeneous Musielak–Orlicz spaces. The setting considered in this paper generalized known results in the variable exponents, anisotropic polynomial, double phase and classical Orlicz setting.

In this article we define a logical system called Hybrid Partial Type Theory ($\mathcal {HPTT}$). The system is obtained by combining William Farmer’s partial type theory with a strong form of hybrid logic. William Farmer’s system is a version of Church’s theory of types which allows terms to be non-denoting; hybrid logic is a version of modal logic in which it is possible to name worlds and evaluate expressions with respect to particular worlds. We motivate this combination of ideas in the introduction, and devote the rest of the article to defining, axiomatising, and proving a completeness result for $\mathcal {HPTT}$.

According to Carolyne Larrington, legends of the past ‘offer particular kinds of answers – beautiful and mysterious answers. . . – to very large questions through a kind of metaphorical thinking . . . which, in their stripped-down clarity, show us what's really important in an unfamiliar light’. The claim that ‘what is really important [is disclosed] by casting it in an unfamiliar light’ I take into a philosophical engagement with the figure of the ghost. Far from being of dubious interest for the philosopher of religion, the continuing fascination with ghosts and hauntings offers promising ground for the discussion of religion, for the study of ghosts holds out the possibility of engaging with the wonder and terror of the human condition. The figure of something that is dead yet alive is a creative representation of the fact that we who are alive are also mortal, destined to die. The resulting confrontation with death arouses anxiety, but also has the potential to enrich life. The wisdom of the ghost thus enables the possibility of returning philosophy of religion to the great themes of human existence – birth, suffering, loss, and death – which provide rich resources for understanding religion and its relation to the experience of being human.

This article examines the essence-existence distinction in Spinoza's theory of modes. This distinction is commonly made in two ways. First, essence and existence are separated by cause. Essences are understood to follow vertically from the essence of God, while existence follows horizontally from other modes. I present textual and systematic arguments against such a causal bifurcation. Second, essence and existence are distinguished by their temporal nature. Essence is eternal. Existence is durational. However, in several passages, Spinoza writes that eternity and duration constitute two ways of understanding nature rather than two really distinct aspects of nature.

Much of historic Christian philosophical theology has affirmed that God not only exists, but is Existence itself. Nowadays, this claim is widely rejected as unintelligible by theists and non-theists alike. I argue in contrast that if there is such a thing as Existence itself, that thing must be a maximally excellent being, which is what many philosophers call God. This is because Existence would itself need to exist, which is only possible if Existence exists in a paradigmatic way, that is, as a perfect instance of existence. My argument thus offers both a defence of the coherence of the claim that God is Existence itself, and a new way of arguing for theism.

To what extent can Aquinas' philosophy of truth accommodate ambiguity? If an ambiguous object is that which exhibits multiple conflicting meanings, and truth, as ‘the conformity of thing and intellect', has its source and purpose in the divine, does the ambiguous lead us away from God? If so, how do we square this with the experience of the ambiguous, such as in art, that appears to draw us towards the divine? The paper explores this aporia by an analysis of the first two questions of De Veritate in conversation with Feser's Scholastic Metaphysics and Pickstock's Truth in Aquinas. Drawing on these three sources, truth is posited as a translation of being. However, it becomes clear that any translation is imperfect, given the difference between the medium of the existence of the thing and the medium of truth in the intellect. Hence, multiple, sometimes contradictory, propositions are needed in order to express the being of the thing. Moreover, it is shown how the ambiguous can prompt recursive returning to the singular, drawing us beyond merely identifying ‘what' a thing is, and beyond propositions, to share in the divine actualization of existence.

We establish some existence results for a class of critical $N$-Laplacian problems in a bounded domain in $\mathbb {R}^{N}$. In the absence of a suitable direct sum decomposition of the underlying Sobolev space to which the classical linking theorem can be applied, we use an abstract linking theorem based on the $\mathbb {Z}_2$-cohomological index to obtain a non-trivial critical point.

In some languages assertions about ‘somebody’ or ‘nobody’ are existential in a strong sense, i.e. they need or prominently allow an explicit syntactic marker of existence (‘there is’, ‘exist’). This paper presents a state-of-the-art typology of existential indefinite constructions and finds the typological understanding to be inconclusive in many respects. The paper responds to this inconclusiveness with a study of the existential indefinite constructions in four mainland Southeast Asian languages, namely Thai, Lao, Vietnamese, and Khmer. These are languages in which existential indefinite constructions take pride of place, although the typological literature has not acknowledged this. The paper then sketches the implications of the study of the aforementioned languages for typology.

In this paper, we concern with a backward problem for a nonlinear time fractional wave equation in a bounded domain. By applying the properties of Mittag-Leffler functions and the method of eigenvalue expansion, we establish some results about the existence and uniqueness of the mild solutions of the proposed problem based on the compact technique. Due to the ill-posedness of backward problem in the sense of Hadamard, a general filter regularization method is utilized to approximate the solution and further we prove the convergence rate for the regularized solutions.

We will give a new proof of the existence of hypercylinder expander of the inverse mean curvature flow which is a radially symmetric homothetic soliton of the inverse mean curvature flow in $\mathbb {R}^{n}\times \mathbb {R}$ , $n\ge 2$ , of the form $(r,y(r))$ or $(r(y),y)$ , where $r=|x|$ , $x\in \mathbb {R}^{n}$ , is the radially symmetric coordinate and $y\in \mathbb {R}$ . More precisely, for any $\lambda>\frac {1}{n-1}$ and $\mu>0$ , we will give a new proof of the existence of a unique even solution $r(y)$ of the equation $\frac {r^{\prime \prime }(y)}{1+r^{\prime }(y)^{2}}=\frac {n-1}{r(y)}-\frac {1+r^{\prime }(y)^{2}}{\lambda (r(y)-yr^{\prime }(y))}$ in $\mathbb {R}$ which satisfies $r(0)=\mu $ , $r^{\prime }(0)=0$ and $r(y)>yr^{\prime }(y)>0$ for any $y\in \mathbb {R}$ . We will prove that $\lim _{y\to \infty }r(y)=\infty $ and $a_{1}:=\lim _{y\to \infty }r^{\prime }(y)$ exists with $0\le a_{1}<\infty $ . We will also give a new proof of the existence of a constant $y_{1}>0$ such that $r^{\prime \prime }(y_{1})=0$ , $r^{\prime \prime }(y)>0$ for any $0<y<y_{1}$ , and $r^{\prime \prime }(y)<0$ for any $y>y_{1}$ .