Abstract In this paper we consider the classes of all continuous $\mathcal {L}$ -(pre-)structures for a continuous first-order signature $\mathcal {L}$ . We characterize the moduli of continuity for which the classes of finite, countable, or all continuous $\mathcal {L}$ -(pre-)structures have the amalgamation property. We also characterize when Urysohn continuous $\mathcal {L}$ -(pre)-structures exist, establish that certain classes of finite continuous $\mathcal {L}$ -structures are countable Fraïssé classes, prove the coherent EPPA for these classes of finite continuous $\mathcal {L}$ -structures, and show that actions by automorphisms on finite $\mathcal {L}$ -structures also form a Fraïssé class. As consequences, we have that the automorphism group of the Urysohn continuous $\mathcal {L}$ -structure is a universal Polish group and that Hall’s universal locally finite group is contained in the automorphism group of the Urysohn continuous $\mathcal {L}$ -structure as a dense subgroup.

Abstract In this paper we study a notion of HL-extension (HL standing for Herwig–Lascar) for a structure in a finite relational language $\mathcal {L}$ . We give a description of all finite minimal HL-extensions of a given finite $\mathcal {L}$ -structure. In addition, we study a group-theoretic property considered by Herwig–Lascar and show that it is closed under taking free products. We also introduce notions of coherent extensions and ultraextensive $\mathcal {L}$ -structures and show that every countable $\mathcal {L}$ -structure can be extended to a countable ultraextensive structure. Finally, it follows from our results that the automorphism group of any countable ultraextensive $\mathcal {L}$ -structure has a dense locally finite subgroup.

Abstract We prove a number of results about countable Borel equivalence relations with forcing constructions and arguments. These results reveal hidden regularity properties of Borel complete sections on certain orbits. As consequences they imply the nonexistence of Borel complete sections with certain features.

The marching cube algorithm is currently one of the most popular three-dimensional (3D) reconstruction surface rendering algorithms. It forms cube voxels based on an input image and then uses 15 basic topological configurations to extract isosurfaces from the voxels. The algorithm processes each cube voxel in a traversal-based manner, but it does not consider the relationship between the isosurfaces in adjacent cubes. Owing to ambiguity, the final reconstructed model may have holes. In this paper, we propose a marching cube algorithm based on edge growth. The algorithm first extracts seed triangles, grows these seed triangles, and then reconstructs the entire 3D model. According to the position of the growth edge, we propose 17 topological configurations with isosurfaces. The reconstruction results showed that the algorithm can reconstruct the 3D model well. When only the main contour of the 3D model is required, the algorithm performs well. In addition, when there are multiple scattered parts in the data, the algorithm can extract only the 3D contours of the parts connected to the seed by setting the region selected based on the seed.

This paper explores the realization of viable, scalable, automated, and authentic alternatives to paper-only-based testing within Engineering disciplines. Currently, manual delivery and grading of paper-based exams incurs vast logistical burdens that have low impact to learning achievement, especially as enrollments increase. Meanwhile, Engineering's design-oriented and problem-solving emphases pose substantial challenges to the digitized delivery of assessments, and thus they warrant a substantive evaluation of their validity. To address this research need, novel Computer-Based Assessment (CBA) infrastructures and delivery protocols were launched via an IRB-approved crossover study to investigate the impact of lockdown-proctored digitized quiz and exam delivery in terms of test score validity, and learning achievement within a large-size undergraduate Mechanical and Aerospace Engineering (MAE) course. Results indicate that well-formed CBAs can determine scores differing as little as 0.6% from Paper-Based Assessments (PBA). Student achievement was de-correlated by technical topic of the assessment delivery mode during crossover and results revealed that the CBA delivery and remediation cohort attained up to 16.9% higher learning outcomes during summative assessment. The encouraging results are discussed in detail along with lessons learned, and suggestions for transportability of CBA approaches to other Engineering courses and institutions.

Distributed source coding addresses the compression of correlated sources without communication links among them. This paper is concerned with the Wyner-Ziv problem: coding of an information source with side information available only at the decoder in the form of a noisy version of the source. Both the problems of theoretical analysis and code design are addressed in the framework of multi-dimensional nested lattice coding. For theoretical analysis, accurate computation of the rate-distortion function is given under the high-resolution assumption, and a new upper bound using the derivative of the theta series is derived. For practical code design, several low-complexity techniques are proposed. Compared to the existing Slepian-Wolf coded nested quantization for Wyner-Ziv coding based on one or two-dimensional lattices, our proposed multi-dimensional lattice coding can offer better performance at arguably lower complexity, since it does not require the second stage of Slepian-Wolf coding.

Abstract Let E be a coanalytic equivalence relation on a Polish space X and (An)n∈ω a sequence of analytic subsets of X. We prove that if lim supn∈kAn meets uncountably many E-equivalence classes for every K ∈ [ω]ω, then there exists K ∈ [ω]ω such that ∩n∈kAn contains a perfect set of pairwise E-inequivalent elements.

In this paper, we consider the coding of an independent and identically distributed (i.i.d.) Gaussian source with side information available only at the decoder in the form of a noisy version of the source to be encoded. This problem is known as Wyner-Ziv coding in literature. In this paper, we propose concrete implementation by using the strategy of multi-dimensional nested lattice quantization (NLQ). By investigating various lattices in the dimensions considered, we give some analysis on how lattice properties affect performance. We also propose a method on choosing good coarse lattices in multiple dimensions. By introducing scale factors, we examine the relationship between distortion and scale factor for various rates. As dimension increases to eight and twenty-four, we obtain distortion performance close to the Wyner-Ziv limit. Meanwhile, our scheme is simple without causing long delay and large storage, which is suitable for sensor networks.

In response to a question of Farah, “How many Boolean algebras are there?” [Far04], one of us (Oliver) proved that there are continuum-many nonisomorphic Boolean algebras of the form with I a Borel ideal on the natural numbers, and in fact that this result could be improved simultaneously in two directions: (i) “Borel ideal” may be improved to “analytic P-ideal” (ii) “continuum-many” may be improved to “E0-many”; that is, E0 is Borel reducible to the isomorphism relation on quotients by analytic P-ideals. See [Oli04]. In [AdKechOO], Adams and Kechris showed that the relation of equality on Borel sets (and therefore, any Borel equivalence relation whatsoever) is Borel reducible to the equivalence relation of Borel bireducibility. (In somewhat finer terms, they showed that the partial order of inclusion on Borel sets is Borel reducible to the quasi-order of Borel reducibility.) Their technique was to find a collection of, in some sense, strongly mutually ergodic equivalence relations, indexed by reals, and then assign to each Borel set B a sort of “direct sum” of the equivalence relations corresponding to the reals in B. Then if B1, ⊆ B2 it was easy to see that the equivalence relation thus induced by B1 was Borel reducible to the one induced by B2, whereas in the opposite case, taking x to be some element of B / B2, it was possible to show that the equivalence relation corresponding to x, which was part of the equivalence relation induced by B1, was not Borel reducible to the equivalence relation corresponding to B2.

Abstract We investigate diagonal actions of Polish groups and the related intersection operator on closed subgroups of the acting group. The Borelness of the diagonal orbit equivalence relation is characterized and is shown to be connected with the Borelness of the intersection operator. We also consider relatively tame Polish groups and give a characterization of them in the class of countable products of countable abelian groups. Finally an example of a logic action is considered and its complexity in the Borel reducbility hierarchy determined.

The maze routing problem is to find an optimal path between a given pair of cells on a grid plane. Lee's algorithm and its variants, probably the most widely used maze routing method, fails to work in the 4-geometry of the grid plane. Our algorithm solves this problem by using a suitable data structure for uniform wave propagation in the 4-geometry, 8-geometry, etc. The algorithm guarantees finding an optimal path if it exists and has linear time and space complexities. Next, to solve the obstacle-avoiding rectilinear and 4-geometry Steiner tree problems, a heuristic algorithm is presented. The algorithm utilizes a cost accumulation scheme based on the maze router to determine the Torricelli vertices (points) for improving the quality of multiterminal nets. Our experimental results show that the algorithm works well in practice. Furthermore, using the 4-geometry router, path lengths can be significantly reduced up to 12% compared to those in the rectilinear router.

Abstract We prove that the strong Martin conjecture is false. The counterexample is the first-order theory of infinite atomic Boolean algebras. We show that for this class of Boolean algebras, the classification of their (ω + ω)-elementary theories can be reduced to the classification of the elementary theories of their quotient algrbras modulo the Frechét ideals.

Abstract Strengthening known instances of Vaught Conjecture, we prove the Glimm-Effros dichotomy theorems for countable linear orderings and for simple trees. Corollaries of the theorems answer some open questions of Friedman and Stanley in an Lω1ω-interpretability theory. We also give a survey of this theory.

Abstract Strengthening a theorem of D. W. Kueker, this paper completely charaterizes which countable structures do not admit uncountable Lω1ω-elementarily equivalent models. In particular, it is shown that if the automorphism group of a countable structure M is abelian, or even just solvable, then there is no uncountable model of the Scott sentence of M. These results arise as part of a study of Polish groups with compatible left-invariant complete metrics.

Abstract In this paper we define and study conditional problems and their degrees. The main result is that the class of conditional degrees is a lattice extending the ordinary Turing degrees and it is dense. These properties are not shared by ordinary Turing degrees. We show that the class of conditional many-one degrees is a distributive lattice. We also consider properties of semidecidable problems and their degrees, which are analogous to r.e. sets and degrees.