Supporting complex group decisions: A probabilistic multi-dimensional scaling approach

Groups often face complex decisions; decisions in which the decision alternatives are not clearly defined and the criteria for choosing an alternative are subject to dispute within the group. We present a Group Decision Support System that will use judgments from the group to visualize the decision problem in a probabilistic geometric space. In this geometric representation, actual decision alternatives and an ideal alternative—an artificial alternative that identifies the ideal solution to the group's decision dilemma—are portrayed as distributions in a multi-dimensional space. Dispersions of the distributions measure the uncertainties of the decision process. The psychometric theory used to develop the probabilistic geometric representation is described. Preliminary research is presented which demonstrates that geometric representations of this type help groups both to understand better the decision they face and to find better solutions.     

LOOK: A Lazy Object-Oriented Kernel design for geometric computation

In this paper we describe and discuss a new kernel design for geometric computation in the plane. It combines different kinds of floating-point filter techniques and a lazy evaluation scheme with the exact number types provided by LEDA allowing for efficient and exact computation with rational and algebraic geometric objects.

It is the first kernel design which uses floating-point filter techniques on the level of geometric constructions.

The experiments we present—partly using the CGAL framework—show a great improvement in speed and—maybe even more important for practical applications—memory consumption when dealing with more complex geometric computations.     

On a functional equation related to convex bodies with <Emphasis Type="Italic">SU</Emphasis>(2)-congruent projections

In this note, we consider two Riemannian metrics on a moduli space of metric graphs. Each of them could be thought of as an analogue of the Weil–Petersson metric on the moduli space of metric graphs. We discuss and compare geometric features of these two metrics with the “classic” Weil–Petersson metric in Teichmüller theory. This paper is motivated by Pollicott and Sharp’s work (Pollicott and Sharp in Geom Dedic 172(1):229–244, 2014). Moreover, we fix some errors in Pollicott and Sharp (2014).     

An algebraic geometry algorithm for scheduling in presence of setups and correlated demands

We study here a problem of schedulingn job types onm parallel machines, when setups are required and the demands for the products are correlated random variables. We model this problem as a chance constrained integer program.Methods of solution currently available—in integer programming and stochastic programming—are not sufficient to solve this model exactly. We develop and introduce here a new approach, based on a geometric interpretation of some recent results in Gröbner basis theory, to provide a solution method applicable to a general class of chance constrained integer programming problems.Out algorithm is conceptually simple and easy to implement. Starting from a (possibly) infeasible solution, we move from one lattice point to another in a monotone manner regularly querying a membership oracle for feasibility until the optimal solution is found. We illustrate this methodology by solving a problem based on a real system.Corresponding author.     

Ebene absolute geometrie über Euklidischen Körpern

We give a set of seven postulates for plane absolute geometry based on the geometric notions points and lines and show that every absolute plane is — up to isomorphisms — either a Euclidean or a hyperbolic plane over a Euclidean field.     

An extremal problem for Graham-Rothschild parameter words

This paper exposes connections between the theory of Möbius functions and extremal problems, extending ideas of Frankl and Pach [8]. Extremal results concerning the trace of objects in geometric lattices and Graham—Rothschild parameter posets are proved, covering previous results due to Sauer [16] and Perles and Shelah [17].     

Geometric quantization as zero mass limit in the problem of a particle on a sphere having a magnetic monopole at its center

The quantum-mechanical problem of a point particle on a sphere with a magnetic monopole at its center is shown to be equivalent in the zero mass limit to the quantum theory including geometric action related to the Kirillov—Konstant form for the SU(2) group.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Akademii Nauk SSSR, Vol. 180, pp. 3–8, 1990.     

Auslander-Reiten Triangles, Ziegler Spectra and Gorenstein Rings

We investigate (existence of) Auslander—Reiten triangles in a triangulated category in connection with torsion pairs, existence of Serre functors, representability of homological functors and realizability of injective modules. We also develop an Auslander—Reiten theory in a compactly generated triangulated category and we study the connections with the naturally associated Ziegler spectrum. Our analysis is based on the relative homological theory of purity and Brown's Representability Theorem. Our main interest lies in the structure of Auslander—Reiten triangles in the full subcategory of compact objects. We also study the connections and the interplay between Auslander—Reiten theory, pure-semisimplicity and the finite type property, Grothendieck groups, and we give applications to derived categories of Gorenstein rings.     

Geometric Gibbs theory

We extend the classical Gibbs theory for smooth potentials to the geometric Gibbs theory for certain continuous potentials. We study the existence and uniqueness and the compatibility of geometric Gibbs measures associated with these continuous potentials. We introduce a complex Banach manifold structure on the space of these continuous potentials as well as on the space of all geometric Gibbs measures. We prove that with this complex Banach manifold structure, the space is complete and, moreover, is the completion of the space of all smooth potentials as well as the space of all classical Gibbs measures. There is a maximum metric on the space, which is incomplete. We prove that the topology induced by the newly introduced complex Banach manifold structure and the topology induced by the maximal metric are the same. We prove that a geometric Gibbs measure is an equilibrium state, and the infimum of the metric entropy function on the space is zero.

     

Geometric separation and exact solutions for the parameterized independent set problem on disk graphs

We consider the parameterized problem, whether for a given set  of n disks (of bounded radius ratio) in the Euclidean plane there exists a set of k non-intersecting disks. For this problem, we expose an algorithm running in time that is—to our knowledge—the first algorithm with running time bounded by an exponential with a sublinear exponent. For λ-precision disk graphs of bounded radius ratio, we show that the problem is fixed parameter tractable with a running time  . The results are based on problem kernelization and a new “geometric ( -separator) theorem” which holds for all disk graphs of bounded radius ratio. The presented algorithm then performs, in a first step, a “geometric problem kernelization” and, in a second step, uses divide-and-conquer based on our new “geometric separator theorem.”     

Self-similar p-adic fractal strings and their complex dimensions

We develop a geometric theory of self-similar p-adic fractal strings and their complex dimensions. We obtain a closed-form formula for the geometric zeta functions and show that these zeta functions are rational functions in an appropriate variable. We also prove that every self-similar p-adic fractal string is lattice. Finally, we define the notion of a nonarchimedean self-similar set and discuss its relationship with that of a self-similar p-adic fractal string. We illustrate the general theory by two simple examples, the nonarchimedean Cantor and Fibonacci strings. The text was submitted by the authors in English.     

Geometric properties of the solutions of a Hele-Shaw type equation

This article deals with the application of the methods of geometric function theory to the investigation of the free boundary problem for the equation describing flows in an unbounded simply-connected plane domain. We prove the invariance of some geometric properties of a moving boundary.

     

Hamiltonization of Solids of Revolution Through Reduction

In this paper, we study the relation between conserved quantities of nonholonomic systems and the hamiltonization problem employing the geometric methods of Balseiro (Arch Ration Mech Anal 214:453–501, 2014) and Balseiro and Garcia-Naranjo (Arch Ration Mech Anal 205(1):267–310, 2012). We illustrate the theory with classical examples describing the dynamics of solids of revolution rolling without sliding on a plane. In these cases, using the existence of two conserved quantities we obtain, by means of gauge transformations and symmetry reduction, genuine Poisson brackets describing the reduced dynamics.     

Bordism of semi-free <Emphasis Type="Italic">S</Emphasis><Superscript>1</Superscript>-actions

We calculate geometric and homotopical bordism rings associated to semi-free S¹ actions on complex manifolds, giving explicit generators for the geometric theory. The classification of semi-free actions with isolated fixed points up to cobordism complements similar results from symplectic geometry.     

Fold-up derivatives of set-valued functions and the change-set problem: A Survey

We give a survey on fold-up derivatives, a notion which was introduced by Khmaladze (J Math Anal Appl 334:1055–1072, 2007) and extended by Khmaladze and Weil (J Math Anal Appl 413:291–310, 2014) to describe infinitesimal changes in a set-valued function. We summarize the geometric background and discuss in detail applications in statistics, in particular to the change-set problem of spatial statistics, and show how the notion of fold-up derivatives leads to the theory of testing statistical hypotheses about the change-set. We formulate Poisson limit theorems for the log-likelihood ratio in two versions of this problem and present also the route to a central limit theorem.     

A framework for constructing partial geometric difference sets

Partial geometric difference sets (PGDSs) were defined in Olmez (J Combin Des 22(6):252–269, 2014). They are used to construct partial geometric designs. We use the framework of extended building sets to find infinite families of PGDSs in abelian groups. Included in our new families of PGDSs are generalizations of the Hadamard, McFarland, Spence, Davis-Jedwab, and Chen difference sets.     

Automorphisms and Distinguishing Numbers of Geometric Cliques

A geometric automorphism is an automorphism of a geometric graph that preserves crossings and noncrossings of edges. We prove two theorems constraining the action of a geometric automorphism on the boundary of the convex hull of a geometric clique. First, any geometric automorphism that fixes the boundary of the convex hull fixes the entire clique. Second, if the boundary of the convex hull contains at least four vertices, then it is invariant under every geometric automorphism. We use these results, and the theory of determining sets, to prove that every geometric n-clique in which n≥7 and the boundary of the convex hull contains at least four vertices is 2-distinguishable.     

Commensurability of geometric subgroups of mapping class groups

Let M be an orientable surface with punctures and/or boundary components. Paris and Rolfsen (J Reine Angew Math 521:47–83, 2000) studied “geometric subgroups” of the mapping class group of M, that is subgroups corresponding to inclusions of connected subsurfaces. In the present paper we extend this analysis to disconnected subsurfaces and to the nonorientable case. We characterise the subsurfaces which lead to virtually abelian geometric subgroups. We provide algebraic and geometric conditions under which two geometric subgroups are commensurable. We also describe the commensurator of a geometric subgroup in terms of the stabiliser of the underlying subsurface. Finally, following the work of Paris (Math Ann 322:301–315, 2002), we show some applications of our analysis to the theory of irreducible unitary representations of mapping class groups.     

Asymptotic Homomorphisms of C *-Algebras and C *-Extensions

We present a brief introduction to two theories in the category of C ^*-algebras—theory of asymptotic homomorphisms and theory of extensions—and explain how these theories are related to each other.