Scale exhaustivity and the Modification Condition

This paper pursues a Vector Space Semantics (VSS) analysis of evaluative and extreme adjectives in absolute and comparative constructions, with a particular emphasis on the licensing of measure phrases (MPs) in these environments. I show that the Modification Condition (Winter 2005), which restricts the distribution of MPs with locative/directional PPs and dimensional adjectives, can be extended to account for MP licensing with evaluative and extreme adjectives as well. Importantly, the non-satisfaction of the Modification Condition is entailed when a set of vectors does not exhaust the range of possible values on a particular scale. This observation thus allows us to link a long-standing generalization that scale exhaustivity and MP licensing are crucially related (Bierwisch 1989) with the formal denotational properties of certain linguistic expressions

Promiscuous Modification and Cross-Categorial Scale Structure

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Sequence of optional tense

In English, sentences containing a past tense in the complement clause of a past-marked propositional attitude predicate (e.g, John said Mary was sick) are ambiguous between a simultaneous and a back-shifted reading, in a phenomenon known as Sequence of Tense. In languages like Japanese, only a shifted reading is available for such past-under-past sentences. Two families of theories have been proposed in the literature to account for this variation: structural and pragmatic. Structural accounts rely on a syntactic rule or licensing condition to derive simultaneous readings of embedded clauses. Pragmatic accounts rely on competition between past and present tense in embedded clauses to derive the readings. In this paper, I provide new data from Washo, an optional tense language, to weigh in on these theories. In Washo, both tensed and (past-oriented) tenseless embedded clauses can have simultaneous and back-shifted readings. I argue that structural approaches can account for the Washo generalizations fairly straightforwardly, while pragmatic approaches encounter difficulties. The result is that the distribution of simultaneous readings cross-linguistically is more fruitfully viewed as a syntactic phenomenon rather than a pragmatic one

Quantity and gradability across categories

This paper proposes a unified analysis of scalar modifiers across the adjec- tival, nominal and verbal domains, with a special focus on the proportional modifier half in English. I claim that half has a scalar meaning in all the environments in which it appears. Specifically, I show that in partitive and event-modifying uses, half targets a quantity-based scale whose scale structure crucially depends on the part structure of a nominal argument, just like many adjectival cases. To formalize the analysis, I extend the degree-based analysis of Kennedy & McNally (2005) for gradable adjectives to partitives and VPs headed by incremental theme verbs, with some discussion of compounding uses of half with both nouns and verbs

Detecting multipartite entanglement

We discuss the problem of determining whether the state of several quantum mechanical subsystems is entangled. As in previous work on two subsystems we introduce a procedure for checking separability that is based on finding state extensions with appropriate properties and may be implemented as a semidefinite program. The main result of this work is to show that there is a series of tests of this kind such that if a multiparty state is entangled this will eventually be detected by one of the tests. The procedure also provides a means of constructing entanglement witnesses that could in principle be measured in order to demonstrate that the state is entangled.Comment: 9 pages, REVTE

Monotone functions and maps

In [S. Basu, A. Gabrielov, N. Vorobjov, Semi-monotone sets. arXiv:1004.5047v2 (2011)] we defined semi-monotone sets, as open bounded sets, definable in an o-minimal structure over the reals, and having connected intersections with all translated coordinate cones in R^n. In this paper we develop this theory further by defining monotone functions and maps, and studying their fundamental geometric properties. We prove several equivalent conditions for a bounded continuous definable function or map to be monotone. We show that the class of graphs of monotone maps is closed under intersections with affine coordinate subspaces and projections to coordinate subspaces. We prove that the graph of a monotone map is a topologically regular cell. These results generalize and expand the corresponding results obtained in Basu et al. for semi-monotone sets.Comment: 30 pages. Version 2 appeared in RACSAM. In version 3 Corollaries 1 and 2 were corrected. In version 4 Theorem 3 is correcte

The Łojasiewicz exponent over a field of arbitrary characteristic

Let K be an algebraically closed field and let K((XQ)) denote the field of generalized series with coefficients in K. We propose definitions of the local Łojasiewicz exponent of F = ( f1, . . . , fm) ∈ K[[X, Y ]]m as well as of the Łojasiewicz exponent at infinity of F = ( f1, . . . , fm) ∈ K[X, Y ]m, which generalize the familiar case of K = C and F ∈ C{X, Y }m (resp. F ∈ C[X, Y ]m), see Cha˛dzy´nski and Krasi´nski (In: Singularities, 1988; In: Singularities, 1988; Ann Polon Math 67(3):297–301, 1997; Ann Polon Math 67(2):191–197, 1997), and prove some basic properties of such numbers. Namely, we show that in both cases the exponent is attained on a parametrization of a component of F (Theorems 6 and 7), thus being a rational number. To this end, we define the notion of the Łojasiewicz pseudoexponent of F ∈ (K((XQ))[Y ])m for which we give a description of all the generalized series that extract the pseudoexponent, in terms of their jets. In particular, we show that there exist only finitely many jets of generalized series giving the pseudoexponent of F (Theorem 5). The main tool in the proofs is the algebraic version of Newton’s Polygon Method. The results are illustrated with some explicit examples

Towards Physical Hybrid Systems

Some hybrid systems models are unsafe for mathematically correct but physically unrealistic reasons. For example, mathematical models can classify a system as being unsafe on a set that is too small to have physical importance. In particular, differences in measure zero sets in models of cyber-physical systems (CPS) have significant mathematical impact on the mathematical safety of these models even though differences on measure zero sets have no tangible physical effect in a real system. We develop the concept of "physical hybrid systems" (PHS) to help reunite mathematical models with physical reality. We modify a hybrid systems logic (differential temporal dynamic logic) by adding a first-class operator to elide distinctions on measure zero sets of time within CPS models. This approach facilitates modeling since it admits the verification of a wider class of models, including some physically realistic models that would otherwise be classified as mathematically unsafe. We also develop a proof calculus to help with the verification of PHS.Comment: CADE 201

Topological complexity of the relative closure of a semi-Pfaffian couple

Gabrielov introduced the notion of relative closure of a Pfaffian couple as an alternative construction of the o-minimal structure generated by Khovanskii's Pfaffian functions. In this paper, use the notion of format (or complexity) of a Pfaffian couple to derive explicit upper-bounds for the homology of its relative closure. Keywords: Pfaffian functions, fewnomials, o-minimal structures, Betti numbers.Comment: 12 pages, 1 figure. v3: Proofs and bounds have been slightly improve