Finding the maximum and minimum

AbstractWe consider the problem of finding the maximum out of a list of n ordered items with binary comparisons where the pth fraction of the answers may be false. It is shown that the maximum can be determined iff p < 12 and that a successful strategy needs Θ(11−p)n questions. A few similar problems are also discussed, including the problem of finding the maximum and minimum simultaneously with lies and in the nuts and bolts model

Realizability and uniqueness in graphs

AbstractConsider a finite graph G(V,E). Let us associate to G a finite list P(G) of invariants. To any P the following two natural problems arise: (R) Realizability. Given P, when is P=P(G) for some graph G?, (U) Uniqueness. Suppose P(G)=P(H) for graphs G and H. When does this imply G ≅ H? The best studied questions in this context are the degree realization problem for (R) and the reconstruction conjecture for (U). We discuss the problems (R) and (U) for the degree sequence and the size sequence of induced subgraphs for undirected and directed graphs, concentrating on the complexity of the corresponding decision problems and their connection to a natural search problem on graphs

Interlace polynomials

AbstractIn a recent paper Arratia, Bollobás and Sorkin discuss a graph polynomial defined recursively, which they call the interlace polynomial q(G,x). They present several interesting results with applications to the Alexander polynomial and state the conjecture that |q(G,−1)| is always a power of 2. In this paper we use a matrix approach to study q(G,x). We derive evaluations of q(G,x) for various x, which are difficult to obtain (if at all) by the defining recursion. Among other results we prove the conjecture for x=−1. A related interlace polynomial Q(G,x) is introduced. Finally, we show how these polynomials arise as the Martin polynomials of a certain isotropic system as introduced by Bouchet

The Relationship between Personality Organization and Psychiatric Classification in Chronic Pain Patients

The assessment of PO is a crucial issue for diagnosis and treatment planning in CPPs, since it represents a measure of structural impairment that is to a considerable extent independent of axis I and II diagnoses. Moreover, the STIPO dimensional rating focuses on the most salient dysfunctions at a given time. Copyright (C) 2010 S. Karger AG, BaselBackground: The present study investigated the relationship between psychiatric classification and personality organization (PO) in a secondary/tertiary clinical sample of chronic pain patients (CPPs). Sampling and Methods: Forty-three patients were administered the Structured Clinical Interview for DSM-IV (SCID I+II) and the Structured Interview of Personality Organization (STIPO). The prevalence of axis I and axis II disorders was correlated with the STIPO level of PO. The STIPO dimensional ratings of patients without personality disorder (PD) were compared to those of patients diagnosed with one or more PDs. Results: Axis I comorbidity was high (93%), and 63% of the patients met the criteria for at least one axis II diagnosis. Twenty-five patients (58%) were diagnosed as borderline PO, with high-level impairments in the dimensions `coping/rigidity', `primitive defenses' and `identity'. Higher axis I and axis II comorbidity corresponded with greater severity of PO impairment. No difference was found between the dimensional ratings of patients without PD and those of patients with one or more PDs. Conclusions

Recent Advances and the Potential for Clinical Use of Autofluorescence Detection of Extra-Ophthalmic Tissues

The autofluorescence (AF) characteristics of endogenous fluorophores allow the label-free assessment and visualization of cells and tissues of the human body. While AF imaging (AFI) is well-established in ophthalmology, its clinical applications are steadily expanding to other disciplines. This review summarizes clinical advances of AF techniques published during the past decade. A systematic search of the MEDLINE database and Cochrane Library databases was performed to identify clinical AF studies in extra-ophthalmic tissues. In total, 1097 articles were identified, of which 113 from internal medicine, surgery, oral medicine, and dermatology were reviewed. While comparable technological standards exist in diabetology and cardiology, in all other disciplines, comparability between studies is limited due to the number of differing AF techniques and non-standardized imaging and data analysis. Clear evidence was found for skin AF as a surrogate for blood glucose homeostasis or cardiovascular risk grading. In thyroid surgery, foremost, less experienced surgeons may benefit from the AF-guided intraoperative separation of parathyroid from thyroid tissue. There is a growing interest in AF techniques in clinical disciplines, and promising advances have been made during the past decade. However, further research and development are mandatory to overcome the existing limitations and to maximize the clinical benefits

Datenbasierte Optimierung unter Unsicherheit für Stromnetzwerke

This cumulative doctoral thesis presents a comprehensive exploration of data-driven optimization under uncertainty in the context of power system analysis. Critical mathematical challenges related to the operation of power grids are addressed, accompanied by innovative solution approaches. The primary focus is on several extensions of the optimal power flow problem which is the predominantly used model in the literature to optimize the power distribution in an electricity network. We study nonconvex mixed-integer nonlinear programs arising in power system analysis, which we solve by the construction and successive refinement of piecewise linear relaxations. Our work introduces various problem-specific and generally applicable algorithmic enhancements to obtain an efficient implementation that outperforms state-of-the-art solvers. Another focus are stochastic mixed-integer linear optimal power flow problems with probabilistic constraints. The solution approach is based on the robust safe approximation of the computationally intractable chance constraints. To construct the approximative problems, suitably defined confidence sets from historical data are computed. We derive a tractable reformulation of the resulting problems and prove quality guarantees about the robustness of the calculated solutions. Numerical experiments on benchmark instances with real weather and network data demonstrate the quality of our solutions. Further improvements are achieved by combining stochastic programming with a model-based prediction of uncertainties. Finally, we present a novel algorithmic framework for optimization under uncertainty over time. The approach uses online learning and scenario observations arriving as a data stream to learn more about the uncertain parameters. We provide a dynamic regret bound for our solutions and illustrate the broad applicability of our approach.Die vorliegende Arbeit widmet sich der datenbasierten Optimierung unter Unsicherheit im Kontext der optimalen Lastflussberechnung in Stromnetzwerken. Wir thematisieren zentrale mathematische Herausforderungen und präsentieren innovative Lösungsansätze. Besonderes Augenmerk liegt auf verschiedene Erweiterungen des klassischen Optimal Power Flow Problems, das als das am häufigsten verwendete Modell zur Stromnetzoptimierung gilt. Wir befassen uns mit nichtkonvexen gemischt-ganzzahligen nichtlinearen Optimierungsproblemen, die in der Steuerung von Stromnetzwerken auftreten. Diese werden durch die Konstruktion und sukzessive Verfeinerung von stückweise linearen Relaxierungen gelöst. Um eine performante Implementierung zu erreichen, präsentieren wir verschiedene, sowohl problemspezifische als auch allgemein anwendbare, algorithmische Erweiterungen. Das resultierende Lösungsverfahren übertrifft existierende Ansätze an Schnelligkeit und Lösungsqualität für die betrachteten Anwendungsprobleme. Wir beschäftigen uns auch mit stochastischen gemischt-ganzzahligen linearen Lastflussoptimierungsproblemen, die probabilistische Nebenbedingungen enthalten. Unser Lösungsansatz basiert auf der robust sicheren Approximation der anspruchsvollen stochastischen Nebenbedingungen. Um die approximativen Modelle aufzustellen werden geeignete Konfidenzmengen mithilfe von historischen Daten berechnet. Wir präsentieren für die resultierenden Probleme eine algorithmisch behandelbare Reformulierung und können Qualitätsaussagen für die berechneten Lösungen treffen. Numerische Experimente an Benchmark-Instanzen unter Verwendung realer Wetter- und Netzwerkdaten verdeutlichen die Performanz unseres Ansatzes. Eine zusätzliche Verbesserung wird durch die innovative Verknüpfung von stochastischer Optimierung mit einer modellbasierten Vorhersage von Unsicherheiten erreicht. Schließlich präsentieren wir einen neuartigen algorithmischen Ansatz für die zeitexpandierte Optimierung unter Unsicherheiten. Unsere Methode nutzt Online-Lernen und die Beobachtung von Szenarien, die als fortlaufender Datenstrom eintreffen, um eine bessere Schätzung von Wahrscheinlichkeitsverteilungen der unsicheren Parameter zu erhalten. Wir beweisen eine dynamische Qualitätsgüte unserer Lösungen und veranschaulichen die vielseitige Anwendbarkeit des Ansatzes

Strongly walk-regular graphs

We study a generalization of strongly regular graphs. We call a graph strongly walk-regular if there is an $\ell >1$ such that the number of walks of length $\ell$ from a vertex to another vertex depends only on whether the two vertices are the same, adjacent, or not adjacent. We will show that a strongly walk-regular graph must be an empty graph, a complete graph, a strongly regular graph, a disjoint union of complete bipartite graphs of the same size and isolated vertices, or a regular graph with four eigenvalues. Graphs from the first three families in this list are indeed strongly $\ell$-walk-regular for all $\ell$, whereas the graphs from the fourth family are $\ell$-walk-regular for every odd $\ell$. The case of regular graphs with four eigenvalues is the most interesting (and complicated) one. Such graphs cannot be strongly $\ell$-walk-regular for even $\ell$. We will characterize the case that regular four-eigenvalue graphs are strongly $\ell$-walk-regular for every odd $\ell$, in terms of the eigenvalues. There are several examples of infinite families of such graphs. We will show that every other regular four-eigenvalue graph can be strongly $\ell$-walk-regular for at most one $\ell$. There are several examples of infinite families of such graphs that are strongly 3-walk-regular. It however remains open whether there are any graphs that are strongly $\ell$-walk-regular for only one particular $\ell$ different from 3