On the Finiteness Problem for classes of modular lattices
The Finiteness Problem is shown to be unsolvable for any sufficiently large class of modular lattices.
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arXiv subjects
Christian Herrmann
Publications and source records attributed to Christian Herrmann.
On linear representation of $\ast$-regular rings having representable ortholattice of projections
This paper aims at the following results: \begin{enumerate} \item The class of all $*$-regular rings forms a variety. \item A subdirectly irreducible $*$-regular ring $R$ is faithfully representable (i.e. isomorphic to a subring of an endomorphisms ring of vector spaces, where the involution is given by adjunction with respect to a scalar product on the vector space) if so is its ortholattice of projections. \end{enumerate} This is a short version of the second author's PhD thesis.
math.RA↗
Unconstrained Face Detection and Open-Set Face Recognition Challenge
Face detection and recognition benchmarks have shifted toward more difficult environments. The challenge presented in this paper addresses the next step in the direction of automatic detection and identification of people from outdoor surveillance cameras. While face detection has shown remarkable success in images collected from the web, surveillance cameras include more diverse occlusions, poses, weather conditions and image blur. Although face verification or closed-set face identification have surpassed human capabilities on some datasets, open-set identification is much more complex as it needs to reject both unknown identities and false accepts from the face detector. We show that unconstrained face detection can approach high detection rates albeit with moderate false accept rates. By contrast, open-set face recognition is currently weak and requires much more attention.
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Open Set Logo Detection and Retrieval
Current logo retrieval research focuses on closed set scenarios. We argue that the logo domain is too large for this strategy and requires an open set approach. To foster research in this direction, a large-scale logo dataset, called Logos in the Wild, is collected and released to the public. A typical open set logo retrieval application is, for example, assessing the effectiveness of advertisement in sports event broadcasts. Given a query sample in shape of a logo image, the task is to find all further occurrences of this logo in a set of images or videos. Currently, common logo retrieval approaches are unsuitable for this task because of their closed world assumption. Thus, an open set logo retrieval method is proposed in this work which allows searching for previously unseen logos by a single query sample. A two stage concept with separate logo detection and comparison is proposed where both modules are based on task specific CNNs. If trained with the Logos in the Wild data, significant performance improvements are observed, especially compared with state-of-the-art closed set approaches.
cs.CV↗
On the consistency problem for modular lattices and related structures
The consistency problem for a class of algebraic structures asks for an algorithm to decide for any given conjunction of equations whether it admits a non-trivial satisfying assignment within some member of the class. By Adyan (1955) and Rabin (1958) it is known unsolvable for (the class of) groups and, recently, by Bridson and Wilton (2015) for finite groups. We derive unsolvability for (finite) modular lattices and various subclasses; in particular, the class of all subspace lattices of finite dimensional vector spaces over a fixed or arbitrary field of characteristic $0$. The lattice results are used to prove unsolvability of the consistency problem for (finite) rings and (finite) representable relation algebras. These results in turn apply to equations between simple expressions in Grassmann-Cayley algebra and to functional and embedded multivalued dependencies in databases.
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Linear representations of regular rings and complemented modular lattices with involution
Faithful representations of regular $\ast$-rings and modular complemented lattices with involution within orthosymmetric sesquilinear spaces are studied within the framework of Universal Algebra. In particular, the correspondence between classes of spaces and classes of representables is analyzed; for a class of spaces which is closed under ultraproducts and non-degenerate finite dimensional subspaces, the latter are shown to be closed under complemented [regular] subalgebras, homomorphic images, and ultraproducts and being generated by those members which are associated with finite dimensional spaces. Under natural restrictions, this is refined to a $1$-$1$-correspondence between the two types of classes.
math.RA↗
Proatomic modular ortholattices: Representation and equational theory
We study modular ortholattices in the variety generated by the finite dimensional ones from an equational and geometric point of view. We relate this to coordinatization results.
math.RA↗
Satisfiability of cross product terms is complete for real nondeterministic polytime Blum-Shub-Smale machines
Nondeterministic polynomial-time Blum-Shub-Smale Machines over the reals give rise to a discrete complexity class between NP and PSPACE. Several problems, mostly from real algebraic geometry / polynomial systems, have been shown complete (under many-one reduction by polynomial-time Turing machines) for this class. We exhibit a new one based on questions about expressions built from cross products only.
cs.CC↗
Computational Complexity of Quantum Satisfiability
Quantum logic was introduced in 1936 by Garrett Birkhoff and John von Neumann as a framework for capturing the logical peculiarities of quantum observables. It generalizes, and on 1-dimensional Hilbert space coincides with, Boolean propositional logic. We introduce the weak and strong satisfiability problem for quantum logic terms. It turns out that in dimension two both are also NP-complete. For higher-dimensional spaces R^d and C^d with d>2 fixed, on the other hand, we show both problems to be complete for the nondeterministic Blum-Shub-Smale model of real computation. This provides a unified view on both Turing and real BSS complexity theory; and extends the still relatively scarce family of NP_R-complete problems with one perhaps closest in spirit to the classical Cook-Levin Theorem. Our investigations on the dimensions a term is weakly/strongly satisfiable in lead to satisfiability problems in indefinite finite and finally in infinite dimension. Here, strong satisfiability turns out as polynomial-time equivalent to the feasibility of noncommutative integer polynomial equations
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