Gevoel and Illumination: Bavinck, Augustine, and Bonaventure on Awareness of God

This essay offers a reflection that seeks to clarify and complement Steven Duby’s God in Himself, especially on the natural awareness of God. First, in response to Duby’s assessment of Bavinck’s critique of certain forms of natural theology, I draw particularly from Cory Brock’s recent monograph on Bavinck’s critical appropriation of particular strands of post-Kantian romantic philosophy in order to articulate the affective dimensions of general revelation. This explains Bavinck’s preference for the term “general revelation” over “natural theology,” for the former emphasizes humanity’s pre-categorical dependence on God’s revealing work internal to the human psyche, manifesting as the feeling (gevoel) of dependence. Second, then, following Bavinck’s own connection of Schleiermacher to Augustine’s turn to the subject, I provide a retrieval of Augustine’s and Bonaventure’s accounts of illumination, which escalates the agent’s dependence on God’s revelation to a maximal degree.

The Structure of d.r.e. Degrees

This dissertation is highly motivated by d.r.e. Nondensity Theorem, which is interesting in two perspectives. One is that it contrasts Sacks Density Theorem, and hence shows that the structures of r.e. degrees and d.r.e. degrees are different. The other is to investigate what other properties a maximal degree can have.In Chapter 1, we briefly review the backgrounds of Recursion Theory which motivate the topics of this dissertation.In Chapter 2, we introduce the notion of $(m,n)$ -cupping degree. It is closely related to the notion of maximal d.r.e. degree. In fact, a $(2,2)$ -cupping degree is maximal d.r.e. degree. We then prove that there exists an isolated $(2,\omega )$ -cupping degree by combining strategies for maximality and isolation with some efforts.Chapter 3 is part of a joint project with Steffen Lempp, Yiqun Liu, Keng Meng Ng, Cheng Peng, and Guohua Wu. In this chapter, we prove that any finite boolean algebra can be embedded into d.r.e. degrees as a final segment. We examine the proof of d.r.e. Nondensity Theorem and make developments to the technique to make it work for our theorem. The goal of the project is to see what lattice can be embedded into d.r.e. degrees as a final segment, as we observe that the technique has potential be developed further to produce other interesting results.Abstract prepared by Yong Liu.E-mail: [email protected]

2-distance choice number of planar graphs with maximal degree no more than 4

Regarding the 2-[Formula: see text] coloring of planar graphs, in 1977, Wegner conjectured that for a graph [Formula: see text]: (1) [Formula: see text] if [Formula: see text]. (2) [Formula: see text] if [Formula: see text]. (3) [Formula: see text] if [Formula: see text]. In this paper, we proved that for every planar graph with maximum degree [Formula: see text]: (1) [Formula: see text] if [Formula: see text]. (2) [Formula: see text] if [Formula: see text].

Indecomposable orthogonal invariants of several matrices over a field of positive characteristic

We consider the algebra of invariants of [Formula: see text]-tuples of [Formula: see text] matrices under the action of the orthogonal group by simultaneous conjugation over an infinite field of characteristic [Formula: see text] different from two. It is well known that this algebra is generated by the coefficients of the characteristic polynomial of all products of generic and transpose generic [Formula: see text] matrices. We establish that in case [Formula: see text] the maximal degree of indecomposable invariants tends to infinity as [Formula: see text] tends to infinity. In other words, there does not exist a constant [Formula: see text] such that it only depends on [Formula: see text] and the considered algebra of invariants is generated by elements of degree less than [Formula: see text] for any [Formula: see text]. This result is well-known in case of the action of the general linear group. On the other hand, for the rest of [Formula: see text] the given phenomenon does not hold. We investigate the same problem for the cases of symmetric and skew-symmetric matrices.

The degree of Kummer extensions of number fields

Let [Formula: see text] be a number field, and let [Formula: see text] be elements of [Formula: see text] which generate a subgroup of [Formula: see text] of rank [Formula: see text]. Consider the cyclotomic-Kummer extensions of [Formula: see text] given by [Formula: see text], where [Formula: see text] divides [Formula: see text] for all [Formula: see text]. There is an integer [Formula: see text] such that these extensions have maximal degree over [Formula: see text], where [Formula: see text] and [Formula: see text]. We prove that the constant [Formula: see text] is computable. This result reduces to finitely many cases the computation of the degrees of the extensions [Formula: see text] over [Formula: see text].

Improved Resource State for Verifiable Blind Quantum Computation

Recent advances in theoretical and experimental quantum computing raise the problem of verifying the outcome of these quantum computations. The recent verification protocols using blind quantum computing are fruitful for addressing this problem. Unfortunately, all known schemes have relatively high overhead. Here we present a novel construction for the resource state of verifiable blind quantum computation. This approach achieves a better verifiability of 0.866 in the case of classical output. In addition, the number of required qubits is 2N+4cN, where N and c are the number of vertices and the maximal degree in the original computation graph, respectively. In other words, our overhead is less linear in the size of the computational scale. Finally, we utilize the method of repetition and fault-tolerant code to optimise the verifiability.

Another Approach to Non-Repetitive Colorings of Graphs of Bounded Degree

We propose a new proof technique that applies to the same problems as the Lovász Local Lemma or the entropy-compression method. We present this approach in the context of non-repetitive colorings and we use it to improve upper-bounds relating different non-repetitive chromatic numbers to the maximal degree of a graph. It seems that there should be other interesting applications of the presented approach. In terms of upper-bounds our approach seems to be as strong as entropy-compression, but the proofs are more elementary and shorter. The applications we provide in this paper are upper bounds for graphs of maximal degree at most $\Delta$: a minor improvement on the upper-bound of the non-repetitive chromatic number, a $4.25\Delta +o(\Delta)$ upper-bound on the weak total non-repetitive chromatic number, and a $ \Delta^2+\frac{3}{2^{1/3}}\Delta^{5/3}+ o(\Delta^{5/3})$ upper-bound on the total non-repetitive chromatic number of graphs. This last result implies the same upper-bound for the non-repetitive chromatic index of graphs, which improves the best known bound.