The solution of Apollonius' problem on constructing a circle (line), tangent to three given circles (lines), is presented in terms of oriented circles and inversive invariants. Tangency is understood as the coincidence of tangent vectors at the common point, in contrast to counter-tangency. The problem has 0, 1 or 2 solutions. By reversing each of the given circles one by one, we obtain the remaining solutions of the classical non-oriented problem.
Preference cycles are prevalent in problems of decision-making, and are contradictory when preferences are assumed to be transitive. This contradiction underlies Condorcet's Paradox, a pioneering result of Social Choice Theory, wherein intuitive and seemingly desirable constraints on decision-making necessarily lead to contradictory preference cycles. Topological methods have since broadened Social Choice Theory and elucidated existing results. However, characterisations of preference cycles in Topological Social Choice Theory are lacking. In this paper, we address this gap by introducing a framework for topologically modelling preference cycles that generalises Baryshnikov's existing topological model of strict, ordinal preferences on 3 alternatives. In our framework, the contradiction underlying Condorcet's Paradox topologically corresponds to the non-orientability of a surface homeomorphic to either the Klein Bottle or Real Projective Plane, depending on how preference cycles are represented. These findings allow us to reduce Arrow's Impossibility Theorem to a statement about the orientability of a surface. Furthermore, these results contribute to existing wide-ranging interest in the relationship between non-orientability, impossibility phenomena in Economics, and logical paradoxes more broadly.
Bernardo N. B. de Lima, Daniel Ungaretti, Maria Eulália Vares
This note was motivated by natural questions related to oriented percolation on a layered environment that introduces long range dependence. As a convenient tool, we are led to deal with questions on the strict decrease of the percolation parameter in the oriented setup when an extra dimension is added.
We show that the order of the cardinality of maximal complete $1$-systems of loops on non-orientable surfaces is $\sim |χ|^{2}$. In particular, we determine the exact cardinality of maximal complete $1$-systems of loops on punctured projective planes. To prove these results, we show that the cardinality of maximal systems of arcs pairwise-intersecting at most once on a non-orientable surface is $2|χ|(|χ|+1)$.
Péter Ágoston, Gábor Damásdi, Balázs Keszegh
et al.
We study systems of orientations on triples that satisfy the following so-called interiority condition: $\circlearrowleft(ABD)=~\circlearrowleft(BCD)=~\circlearrowleft(CAD)=1$ implies $\circlearrowleft(ABC)=1$ for any $A,B,C,D$. We call such an orientation a P3O (partial 3-order), a natural generalization of a poset, that has several interesting special cases. For example, the order type of a planar point set (that can have collinear triples) is a P3O; we denote a P3O realizable by points as p-P3O. If we do not allow $\circlearrowleft(ABC)=0$, we obtain a T3O (total 3-order). Contrary to linear orders, a T3O can have a rich structure. A T3O realizable by points, a p-T3O, is the order type of a point set in general position. In our paper "Orientation of convex sets" we defined a 3-order on pairwise intersecting convex sets; such a P3O is called a C-P3O. In this paper we extend this 3-order to pairwise intersecting good covers; such a P3O is called a GC-P3O. If we do not allow $\circlearrowleft(ABC)=0$, we obtain a C-T3O and a GC-T3O, respectively. The main result of this paper is that there is a p-T3O that is not a GC-T3O, implying also that it is not a C-T3O -- this latter problem was left open in our earlier paper. Our proof involves several combinatorial and geometric observations that can be of independent interest. Along the way, we define several further special families of GC-T3O's.
We show the Teichmüller space of a non-orientable surface with marked points (considered as a Klein surface) can be identified with a subspace of the Teichmüller space of its orientable double cover. Also, it is well known that the mapping class group $\text{Mod} (N_g; k)$ of a non-orientable surface can be identified with a subgroup of $\text{Mod} (S_{g-1}; 2k)$, the mapping class group of its orientable double cover. These facts together with the classical Nielsen realization theorem are used to prove that every finite subgroup of $\text{Mod}(N_g; k)$ can be lifted isomorphically to a subgroup of the group of diffeomorphisms $\text{Diff}(N_g; k)$. In contrast, we show the projection $\text{Diff}(N_g) \to \text{Mod}(N_g)$ does not admit a section for large $g$.
Upon the arrival of emerging devices, including Extended Reality (XR) and Unmanned Aerial Vehicles (UAVs), the traditional communication framework is approaching Shannon's physical capacity limit and fails to guarantee the massive amount of transmission within latency requirements. By jointly exploiting the context of data and its importance to the task, an emerging communication paradigm shift to semantic level and effectiveness level is envisioned to be a key revolution in Sixth Generation (6G) networks. However, an explicit and systematic communication framework incorporating both semantic level and effectiveness level has not been proposed yet. In this article, we propose a generic goal-oriented semantic communication framework for various tasks with diverse data types, which incorporates both semantic level information and effectiveness-aware performance metrics. We first analyze the unique characteristics of all data types, and summarise the semantic information, along with corresponding extraction methods. We then propose a detailed goal-oriented semantic communication framework for different time-critical and non-critical tasks. In the goal-oriented semantic communication framework, we present the goal-oriented semantic information, extraction methods, recovery methods, and effectiveness-aware performance metrics. Last but not least, we present a goal-oriented semantic communication framework tailored for Unmanned Aerial Vehicle (UAV) control task to validate the effectiveness of the proposed goal-oriented semantic communication framework.
We associate to a sufficiently generic oriented matroid program and choice of linear system of parameters a finite dimensional algebra, whose representation theory is analogous to blocks of Bernstein--Gelfand--Gelfand category $\mathcal O$. When the data above comes from a generic linear program for a hyperplane arrangement, we recover the algebra defined by Braden--Licata--Proudfoot--Webster. Applying our construction to nonlinear oriented matroid programs provides a large new class of algebras. For Euclidean oriented matroid programs, the resulting algebras are quasi-hereditary and Koszul, as in the linear setting. In the non-Euclidean case, we obtain algebras that are not quasi-hereditary and not known to be Koszul, but still have a natural class of standard modules and satisfy numerical analogues of quasi-heredity and Koszulity on the level of graded Grothendieck groups.
We introduce the notion of real phase structure on rational polyhedral fans in Euclidean space. Such a structure consists of an assignment of affine spaces over $\mathbb{Z}/2\mathbb{Z}$ to each top dimensional face of the fan subject to two conditions. Given an oriented matroid we can construct a real phase structure on the fan of the underlying matroid. Conversely, we show that from a real phase structure on a matroid fan we can produce an orientation of the underlying matroid. Thus real phase structures are cryptomorphic to matroid orientations. The topes of the orientated matroid are recovered immediately from the real phase structure. We also provide a direct way to recover the signed circuits of the oriented matroid from the real phase structure.
We present preliminary results from measuring Galactic orientation angles of 800 elliptical and bipolar Planetary Nebulae (PNe) in the Hong Kong/Australian Astronomical Observatory/Strasbourg Observatory H-alpha Planetary Nebula (HASH) research platform and database (Parker et al. [1], [2]). For elliptical PNe the distribution of orientation angles is found to be essentially uniform. However, for bipolar PNe there is statistically significant evidence for preferred orientation angles (as tentatively reported before with smaller samples) across the whole Galaxy.
Repeated application of machine-learning, eigen-centric methods to an evolving dataset reveals that eigenvectors calculated by well-established computer implementations are not stable along an evolving sequence. This is because the sign of any one eigenvector may point along either the positive or negative direction of its associated eigenaxis, and for any one eigen call the sign does not matter when calculating a solution. This work reports an algorithm that creates a consistently oriented basis of eigenvectors. The algorithm postprocesses any well-established eigen call and is therefore agnostic to the particular implementation of the latter. Once consistently oriented, directional statistics can be applied to the eigenvectors in order to track their motion and summarize their dispersion. When a consistently oriented eigensystem is applied to methods of machine-learning, the time series of training weights becomes interpretable in the context of the machine-learning model. Ordinary linear regression is used to demonstrate such interpretability. A reference implementation of the algorithm reported herein has been written in Python and is freely available, both as source code and through the thucyd Python package.
Object-oriented programming is often regarded as too inefficient for high-performance computing (HPC), despite the fact that many important HPC problems have an inherent object structure. Our goal is to bring efficient, object-oriented programming to massively parallel SIMD architectures, especially GPUs. In this thesis, we develop various techniques for optimizing object-oriented GPU code. Most notably, we identify the object-oriented Single-Method Multiple-Objects (SMMO) programming model. We first develop an embedded C++ Structure of Arrays (SOA) data layout DSL for SMMO applications. We then design a lock-free, dynamic memory allocator that stores allocations in SOA layout. Finally, we show how to further optimize the memory access of SMMO applications with memory defragmentation.
We define an abstract framework for object-oriented programming and show that object-oriented languages, such as C++, can be interpreted as parallel programming languages. Parallel C++ code is typically more than ten times shorter than the equivalent C++ code with MPI. The large reduction in the number of lines of code in parallel C++ is primarily due to the fact that coordination of concurrency, and the communications instructions, including packing and unpacking of messages, are automatically generated in the implementation of object operations. We implemented a prototype of a compiler and a runtime system for parallel C++ and used them to create complex data-intensive and HPC applications. These results indicate that adoption of the parallel object-oriented framework has the potential to drastically reduce the cost of parallel programming. We also show that standard sequential object-oriented programs can be ported to parallel architecture, parallelized automatically, and potentially sped up. The parallel object-oriented framework enables an implementation of a compiler with a dedicated backend for the interconnect fabric, which exposes the network hardware features directly to the application. We discuss the potential implications for computer architecture.
We call that a vector field has the oriented shadowing property if for any $\varepsilon>0$ there is $d>0$ such that each $d$-pseudo orbit is $\varepsilon$-oriented shadowed by some real orbit. In this paper, we show that the $C^1$-interior of the set of vector fields with the oriented shadowing property is contained in the set of vector fields with the $Ω$-stability.
It seems that the ancient Roman towns were oriented with the sunrise. Here I propose a discussion on the orientation of Torino, the Julia Augusta Taurinorum, which has the ancient Roman structure perfectly preserved. According to this ancient ritual, we can use the sunrise amplitude to determine the Turin's birthday. The use of the hour angle is also proposed, in this case the day of the foundation of Turin could be the winter solstice.