When Is σ (A(t)) ⊂ {z ∈ ℂ; ℜz ≤ −α < 0} the Sufficient Condition for Uniform Asymptotic Stability of LTV System ẋ = A(t)x?

In this paper, the class of matrix functions A(t) is determined for which the condition that the pointwise spectrum σ(A(t))⊂z∈C;ℜz≤−α for all t≥t0 and some α>0 is sufficient for uniform asymptotic stability of the linear time-varying system x˙=A(t)x. We prove that this class contains as a proper subset the matrix functions with the values in the special orthogonal group SO(n).

Difference between Riesz derivative and fractional Laplacian on the proper subset of ℝ

In general, the Riesz derivative and the fractional Laplacian are equivalent on ℝ. But they generally are not equivalent with each other on any proper subset of ℝ. In this paper, we focus on the difference between them on the proper subset of ℝ.

Disciplines in the Scientonomic Ontology

The role of categories of knowledge, or disciplines, in science has not previously been explored in scientonomy. While disciplinary communities devoted to the production of knowledge are a modern phenomenon, the practice of dividing knowledge into categories is a universal feature of science. Although at any moment of time, many questions and theories can be part of a given discipline, not all of these are essential to the discipline. We show that two components are essential to a discipline: the discipline’s core questions as well as the discipline’s delineating theory, a second-order theory that identifies these questions as essential to the discipline. If the questions of one discipline are a proper subset of the questions of another discipline, the former discipline is a subdiscipline of the latter. Since a discipline is a complex entity consisting of questions and a theory, epistemic agents can take epistemic stances towards disciplines. A discipline is said to be accepted if its core questions and its delineating theory are all accepted. To illustrate the applicability of these new concepts, the transition from physical to biological anthropology is discussed.

ON THE HEIGHT AND RELATIONAL COMPLEXITY OF A FINITE PERMUTATION GROUP

Let G be a permutation group on a set $\Omega $ of size t. We say that $\Lambda \subseteq \Omega $ is an independent set if its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset of $\Lambda $ . We define the height of G to be the maximum size of an independent set, and we denote this quantity $\textrm{H}(G)$ . In this paper, we study $\textrm{H}(G)$ for the case when G is primitive. Our main result asserts that either $\textrm{H}(G)< 9\log t$ or else G is in a particular well-studied family (the primitive large–base groups). An immediate corollary of this result is a characterization of primitive permutation groups with large relational complexity, the latter quantity being a statistic introduced by Cherlin in his study of the model theory of permutation groups. We also study $\textrm{I}(G)$ , the maximum length of an irredundant base of G, in which case we prove that if G is primitive, then either $\textrm{I}(G)<7\log t$ or else, again, G is in a particular family (which includes the primitive large–base groups as well as some others).

Sudoku Ripeto

We present a variant of Sudoku called Sudoku Ripeto. It seems to be the first to admit any combination of repeated symbols, and includes Sudoku as a proper subset. We present other Sudoku Ripeto families, each with a different repetition pattern. We define Sudoku Ripeto squares and puzzles, prove several solving rules that generalize those for Sudoku, and give sufficient conditions to flexibly solve puzzles with rules only, without search.

A Special One-Point Connectification that Characterizes $T_{0}$ Spaces

We show that every $T_{0}$ space $X$ has some $T_{0}$ "special" one-point connectification $X_{\infty}$, unique up to a homeomorphism, such that $X$ is a closed subspace of $X_{\infty}$ and a closed subset of $X$ is precisely a closed proper subset of $X_{\infty}$; moreover, having such a one-point connectification characterizes $T_{0}$ spaces. As an application, it is also shown that our one-point connectification of every given topological $n$-manifold is a space more general than but "close to" a topological $n$-manifold with boundary.

Upper triangle free detour number of a graph

For any two vertices [Formula: see text] and [Formula: see text] in a connected graph [Formula: see text], the [Formula: see text] path [Formula: see text] is called a [Formula: see text] triangle free path if no three vertices of [Formula: see text] induce a triangle. The triangle free detour distance [Formula: see text] is the length of a longest [Formula: see text] triangle free path in [Formula: see text]. A [Formula: see text] path of length [Formula: see text] is called a [Formula: see text] triangle free detour. A set [Formula: see text] is called a triangle free detour set of [Formula: see text] if every vertex of [Formula: see text] lies on a [Formula: see text] triangle free detour joining a pair of vertices of [Formula: see text]. The triangle free detour number [Formula: see text] of [Formula: see text] is the minimum order of its triangle free detour sets and any triangle free detour set of order [Formula: see text] is a triangle free detour basis of [Formula: see text]. A triangle free detour set [Formula: see text] of [Formula: see text] is called a minimal triangle free detour set if no proper subset of [Formula: see text] is a triangle free detour set of [Formula: see text]. The upper triangle free detour number [Formula: see text] of [Formula: see text] is the maximum order of its minimal triangle free detour sets and any minimal triangle free detour set of order [Formula: see text] is an upper triangle free detour basis of [Formula: see text]. We determine bounds for it and characterize graphs which realize these bounds. For any connected graph [Formula: see text] of order [Formula: see text], [Formula: see text]. Also, for any four positive integers [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] with [Formula: see text], it is shown that there exists a connected graph [Formula: see text] such that [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], where [Formula: see text] is the upper detour number, [Formula: see text] is the upper detour monophonic number and [Formula: see text] is the upper geodetic number of a graph [Formula: see text].

Free will

Wilson considers whether free will is either Weakly or Strongly emergent. She starts by drawing on Bernstein and Wilson (2016) to present a framework for connecting positions on the problem of free will with positions on the problem of mental (higher-level) causation. Bernstein and Wilson argue that compatibilist accounts implement a ‘proper subset’ strategy relevantly similar to that implemented by nonreductive physicalists/Weak emergentists; here Wilson extends this result to establish that the compatibilist strategy entails satisfaction of the conditions in Weak emergence. Wilson then argues that libertarian accounts implement a ‘new power’ strategy entailing satisfaction of the conditions on Strong emergence. Wilson goes on to suggest that free will of the compatibilist/Weakly emergent variety is plausibly widespread, and to present a novel argument for taking some instances of seemingly free choice to be Strongly emergent.

Two schemas for metaphysical emergence

Wilson presents the problem of higher-level causation (Kim 1989, 1993, 1998), according to which metaphysical emergence gives rise to problematic causal overdetermination. She argues that there are two and only two strategies of response to this problem of making sense of metaphysical emergence. One strategy provides a schematic basis for ‘Weak’ (physically acceptable) emergence; core and crucial here is that a macro-entity or feature has a proper subset of the powers of its base-level configuration. The other strategy provides a schematic basis for ‘Strong’ (physically unacceptable) emergence; core and crucial here is that a macro-entity or feature has a new power as compared to its base-level configuration. Wilson shows that a range of seemingly diverse accounts of metaphysical emergence are plausibly seen as satisfying the conditions in one or the other schema, and thus are more unified than they appear.

Unconcealed Questions

Which price does John know? This sentence exemplifies what I call an unconcealed question (UQ): a sentence with a structure and meaning analogous to those of an ordinary concealed question (CQ), but where the sentence is interrogative in form and interpretation, with the relevant DP headed by which. Such examples are almost completely unstudied in the otherwise wide-ranging CQ literature. As I show, UQs exhibit a proper subset of the ambiguities that have been observed for ordinary CQs. In particular, UQs lack what is known as Reading B, where a relative clause (or other modifying adjunct) containing the relevant sort of predicate is interpreted in the scope of the higher CQ-selecting predicate (e.g. know). I survey the properties of UQs and evaluate the CQ theories currently on the market in light of the UQ data, concluding tentatively that the absence of Reading B is the result of syntactic factors whose description is straightforward but whose explanation remains murky.