Self-dual Embeddings of K_{4m,4n} in Different Orientable and Nonorientable Pseudosurfaces with the Same Euler Characteristic

A proper embedding of a graph G in a pseudosurface P is an embedding in which the regions of the complement of G in P are homeomorphic to discs and a vertex of G appears at each pinchpoint in P; we say that a proper embedding of G in P is self dual if there exists an isomorphism from G to its dual graph. We give an explicit construction of a self-dual embedding of the complete bipartite graph K_{4m,4n} in an orientable pseudosurface for all $m, n\ge 1$; we show that this embedding maximizes the number of umbrellas of each vertex and has the property that for any vertex v of K_{4m,4n}, there are two faces of the constructed embedding that intersect all umbrellas of v. Leveraging these properties and applying a lemma of Bruhn and Diestel, we apply a surgery introduced here or a different known surgery of Edmonds to each of our constructed embeddings for which at least one of m or n is at least 2. The result of these surgeries is that there exist distinct orientable and nonorientable pseudosurfaces with the same Euler characteristic that feature a self-dual embedding of K_{4m,4n}

Dissipative systems: uncontrollability, observability and RLC realizability

The theory of dissipativity has been primarily developed for controllable systems/behaviors. For various reasons, in the context of uncontrollable systems/behaviors, a more appropriate definition of dissipativity is in terms of the dissipation inequality, namely the {\em existence} of a storage function. A storage function is a function such that along every system trajectory, the rate of increase of the storage function is at most the power supplied. While the power supplied is always expressed in terms of only the external variables, whether or not the storage function should be allowed to depend on unobservable/hidden variables also has various consequences on the notion of dissipativity: this paper thoroughly investigates the key aspects of both cases, and also proposes another intuitive definition of dissipativity. We first assume that the storage function can be expressed in terms of the external variables and their derivatives only and prove our first main result that, assuming the uncontrollable poles are unmixed, i.e. no pair of uncontrollable poles add to zero, and assuming a strictness of dissipativity at the infinity frequency, the dissipativities of a system and its controllable part are equivalent. We also show that the storage function in this case is a static state function. We then investigate the utility of unobservable/hidden variables in the definition of storage function: we prove that lossless autonomous behaviors require storage function to be unobservable from external variables. We next propose another intuitive definition: a behavior is called dissipative if it can be embedded in a controllable dissipative {\em super-behavior}. We show that this definition imposes a constraint on the number of inputs and thus explains unintuitive examples from the literature in the context of lossless/orthogonal behaviors.Comment: 26 pages, one figure. Partial results appeared in an IFAC conference (World Congress, Milan, Italy, 2011

Computational Topology and the Unique Games Conjecture

Covering spaces of graphs have long been useful for studying expanders (as "graph lifts") and unique games (as the "label-extended graph"). In this paper we advocate for the thesis that there is a much deeper relationship between computational topology and the Unique Games Conjecture. Our starting point is Linial\u27s 2005 observation that the only known problems whose inapproximability is equivalent to the Unique Games Conjecture - Unique Games and Max-2Lin - are instances of Maximum Section of a Covering Space on graphs. We then observe that the reduction between these two problems (Khot-Kindler-Mossel-O\u27Donnell, FOCS \u2704; SICOMP \u2707) gives a well-defined map of covering spaces. We further prove that inapproximability for Maximum Section of a Covering Space on (cell decompositions of) closed 2-manifolds is also equivalent to the Unique Games Conjecture. This gives the first new "Unique Games-complete" problem in over a decade. Our results partially settle an open question of Chen and Freedman (SODA, 2010; Disc. Comput. Geom., 2011) from computational topology, by showing that their question is almost equivalent to the Unique Games Conjecture. (The main difference is that they ask for inapproximability over Z_2, and we show Unique Games-completeness over Z_k for large k.) This equivalence comes from the fact that when the structure group G of the covering space is Abelian - or more generally for principal G-bundles - Maximum Section of a G-Covering Space is the same as the well-studied problem of 1-Homology Localization. Although our most technically demanding result is an application of Unique Games to computational topology, we hope that our observations on the topological nature of the Unique Games Conjecture will lead to applications of algebraic topology to the Unique Games Conjecture in the future

SMARANDACHE MULTI-SPACE THEORY, Second Edition

We are used to the idea that our space has three dimensions: length, breadth and height with time providing the fourth dimension of spacetime by Einstein. In the string or superstring theories, we encounter 10 dimensions. However, we do not even know what the right degree of freedom is, as Witten said. Today, we have known two heartening notions for sciences. One is the Smarandache multi-space came into being by purely logic. Another is the mathematical combinatorics motivated by a combinatorial speculation, i.e., a mathematical science can be reconstructed from or made by combinatorialization. Both of them contribute sciences for consistency of research with that human progress in 21st century

Diffusion problems in lead-tin overlay bearings

The corrosion of lead-tin overlay bearings in hot lubricating oils, through the depletion of tin in the overlay, has been recognised as a problem of some severity in the automotive bearing industry. The inter-diffusion of tin in the overlay and the copper in the underlying substrate results in the formation of copper-tin intermetallic compounds, consequently depleting tin in the overlay. This renders the bearing subject to corrosion in degraded oil. Attempts have been made to slow down this diffusion process by placing electro- deposited barriers between the bonding material and the overlay. The most common barrier is a thin electro-deposit of nickel. Claims have been made that this nickel barrier retards the growth of intermetallic compounds. However, no evidence to this effect has been found in the present work. The nickel barrier, as well as other barriers, forms intermetallic compounds with tin at almost the same rate as when a barrier is not present. This work was carried out to investigate the problem of diffusion in detail and to search for possible solutions. The kinetics of the diffusion processes involved was studied. After experimenting on a number of barriers, two successful barriers were discovered. These are electrodeposited alloys of Cu-P and Cu-B. A mechanism of prevention of intermetallic compound growth and tin depletion by the two barriers is discussed. Another important outcome of the investigation is that it has been established that the diffusion of copper and nickel are the rate controlling elements in the process of formation of the respective intermetallic compounds with tin. This disproves the frequently held view that the diffusion of tin, commonly associated with its low melting point, is the rate controlling step

A Humean account of laws and causation

The thesis proposes a new account of laws of nature and token causation within the Humean tradition. After a brief introduction in §1, I specify and defend in §2 a Humean approach to the question of laws and causation. In §3 I defend the view that laws are conditional generalisations which concern 'systems' and detail further issues concerning the scope, content and universality of laws. On the basis of the discussion concerning laws' logical form, I argue in §4 against a view of laws as mirroring the structure of causal relations. Moreover, I show how this conception is implicit in the best system account of laws, thereby giving us reason to reject that account too. §5 presents an alternative `causal-junctions conception' of laws in terms of four causal features often associated with laws: component-level and law-level dispositionality, and variable-level and law-level causal asymmetry. These causal features combine to demarcate a central class of laws called `robust causal junction laws' from which other laws can be accounted for. §6 provides a Humean analysis of the causal features used to characterise robust causal junction laws. This is done first by providing an analysis of dispositions in terms of systems and laws, and second, by providing an analysis of causal asymmetry in terms of relations of probabilistic independence. In §7, I then provide a nomological analysis of token causation by showing how the causal junctions described by robust causal junction laws can be chained together in a particular context

Control Theory: Mathematical Perspectives on Complex Networked Systems

Control theory is an interdisciplinary field that is located at the crossroads of pure and applied mathematics with systems engineering and the sciences. Its range of applicability and its techniques evolve rapidly with new developments in communication systems and electronic data processing. Thus, in recent years networked control systems emerged as a new fundamental topic, which combines complex communication structures with classical control methods and requires new mathematical methods. A substantial number of contributions to this workshop was devoted to the control of networks of systems. This was complemented by a series of lectures on other current topics like fundamentals of nonlinear control systems, model reduction and identification, algorithmic aspects in control, as well as open problems in control