Being and Ground

This chapter focuses on grounding. In what sense might grounding be primitive? Perhaps conceptually or methodologically, but not metaphysically. Several ways of defining up a relation of grounding in terms of some kind of ontological superiority plus other connecting relations are explored. The chapter argues that the grounding pluralist, who accepts many metaphysically important grounding relations, and the grounding monist have reasons to believe in an additional relation of ontological superiority. The pluralist does because she needs to account for the unity of the generic relation of ground; it is not a mere disjunction, and so it is either a determinable or an analogous property. But these distinctions were accounted for in terms of naturalness, which is a kind of ontological superiority. The monist about grounding needs some way to defuse grounding variantism, a view analogous to quantifier variantism, and here again appealing to naturalness does the job.

A Return to the Analogy of Being

This chapter explores the metaphysics of what the author calls “analogous properties.” An analogous property is a non-specific property that is less natural than its specifications (called “analogue instances”) but is more natural as a merely disjunctive property. The author discusses and then applies two tests for being an analogous property: a property is analogous provided that it has more unity than a mere disjunction but yet systematically varies with respect to either its logical form or the axioms that govern its behavior. The notion of an analogous property is used to formulate several more versions of ontological pluralism. One kind of ontological pluralism appeals to a distinction between absolute and relative modes of existence. This distinction between modes of being is then used to articulate one kind of ontological superiority, which the author calls “orders of being.”

Skewhermitian and mixed pencils

This chapter turns to matrix pencils of the form A + tB, where one of the matrices A or B is skewhermitian and the other may be hermitian or skewhermitian. Canonical forms of such matrix pencils are given under strict equivalence and under simultaneous congruence, with full detailed proofs, again based on the Kronecker forms. Comparisons with real and complex matrix pencils are presented. In contrast to hermitian matrix pencils, two complex skewhermitian matrix pencils that are simultaneously congruent under quaternions need not be simultaneously congruent under the complex field, although an analogous property is valid for pencils of real skewsymmetric matrices. Similar results hold for real or complex matrix pencils A + tB, where A is real symmetric or complex hermitian and B is real skewsymmetric or complex skewhermitian.

ON THE RELATIONSHIP BETWEEN THE CLASS OF A LIE ALGEBRA AND THE CLASSES OF ITS SUBALGEBRAS

A classical nilpotency result considers finite p-groups whose proper subgroups all have class bounded by a fixed number n. We consider the analogous property in nilpotent Lie algebras. In particular, we investigate whether this condition puts a bound on the class of the Lie algebra. Some p-group results and proofs carry over directly to the Lie algebra case, some carry over with modified proofs and some fail. For the final of these cases, a certain metabelian Lie algebra is constructed to show a case when the p-groups and Lie algebra cases differ.

Subcritical pattern languages for and/or trees

International audience Let $P_k(f)$ denote the density of and/or trees defining a boolean function $f$ within the set of and/or trees with fixed number of variables $k$. We prove that there exists constant $B_f$ such that $P_k(f) \sim B_f \cdot k^{-L(f)-1}$ when $k \to \infty$, where $L(f)$ denote the complexity of $f$ (i.e. the size of a minimal and/or tree defining $f$). This theorem has been conjectured by Danièle Gardy and Alan Woods together with its counterpart for distribution $\pi$ defined by some critical Galton-Watson process. Methods presented in this paper can be also applied to prove the analogous property for $\pi$.

A homogeneity property for Besov spaces

A homogeneity property for some Besov spacesBp,qsis proved. An analogous property for someFp,qsspaces is already known.

The WA/GA distinction and switch-reference for ellipted subject identification in Japanese complex sentences

This paper elucidates one interacting aspect of two problematic issues in Japanese grammar: the functional differences between the nominative marker GA and the topic marker WA, and the referential identity of ellipted subjects. My analysis shows that the interaction of WA and GA in complex sentences has an analogous property to switch-reference systems which determine the identity of ellipted subjects; GA in subordinate clauses signals Different Subject, and WA in all loci Same Subject. This syntactic finding is further evaluated in the light of non-syntactic features, some of which can override the interpretation created by these particles. Statistically in examining texts, however, over 90% of ellipted subjects in complex sentences are successfully identified solely by WA and GA.

Analytic cell decomposition and the closure of p-adic semianalytic sets

The p-adic semianalytic sets are defined, locally, as boolean combinations of sets of the form over the p-adic fields ℚp, where f is an analytic function. A well-know example due to Osgood showed the projection of a semianalytic set need not be a semianalytic set. We call those sets that are, locally, the projections of p-adic semianalytic sets p-adic subanalytic sets. The theory of p-adic subanalytic sets was presented by Denef and Van den Dries in [5]. The basic tools are the quantifier elimination techniques together with the ultrametric Weierstrass Preparation Theorem. Simultaneously with their developments of the p-adic subanalytic sets, they established some basic properties of p-adic semianalytic sets.In this paper, we prove that the closure of any p-adic semianalytic set is also a semianalytic set. The analogous property for real semianalytic sets was proved in [12] and that for rigid semianalytic sets, informed by the referee, has been proved recently by a quite different method in [14] (cf. [9]). The keys to the proof are a separation lemma (Lemma 2) and an analytic cell decomposition theorem (Theorem 2) which is an analytic version of Denef's cell decomposition theorem (see [3, 4]; A total different form of anayltic cells appeared in [13]). The analytic cell decomposition theorem allows us to partition certain kinds of basic subsets into analytic cells that possess the closure property (see §1 for the definition).

Frames and Stable Bases for Shift-Invariant Subspaces of L2(ℝd)

Let X be a countable fundamental set in a Hilbert space H, and let T be the operator Whenever T is well-defined and bounded, X is said to be a Bessel sequence. If, in addition, ran T is closed, then X is a frame. Finally, a frame whose corresponding T is injective is a stable basis (also known as a Riesz basis). This paper considers the above three properties for subspaces H of L2(ℝd), and for sets X of the form with Φ either a singleton, a finite set, or, more generally, a countable set. The analysis is performed on the Fourier domain, where the two operators TT* and T* T are decomposed into a collection of simpler "fiber" operators. The main theme of the entire analysis is the characterization of each of the above three properties in terms of the analogous property of these simpler operators.