62 research outputs found
On the Equivalence Problem for Toric Contact Structures on S^3-bundles over S^2$
We study the contact equivalence problem for toric contact structures on
-bundles over . That is, given two toric contact structures, one can
ask the question: when are they equivalent as contact structures while
inequivalent as toric contact structures? In general this appears to be a
difficult problem. To find inequivalent toric contact structures that are
contact equivalent, we show that the corresponding 3-tori belong to distinct
conjugacy classes in the contactomorphism group. To show that two toric contact
structures with the same first Chern class are contact inequivalent, we use
Morse-Bott contact homology. We treat a subclass of contact structures which
include the Sasaki-Einstein contact structures studied by physicists.
In this subcase we give a complete solution to the contact equivalence problem
by showing that and are inequivalent as contact structures
if and only if .Comment: 61 page
Determinantal Sieving
We introduce determinantal sieving, a new, remarkably powerful tool in the
toolbox of algebraic FPT algorithms. Given a polynomial on a set of
variables and a linear matroid of
rank , both over a field of characteristic 2, in
evaluations we can sieve for those terms in the monomial expansion of which
are multilinear and whose support is a basis for . Alternatively, using
evaluations of we can sieve for those monomials whose odd support
spans . Applying this framework, we improve on a range of algebraic FPT
algorithms, such as:
1. Solving -Matroid Intersection in time and -Matroid
Parity in time , improving on (Brand and Pratt,
ICALP 2021)
2. -Cycle, Colourful -Path, Colourful -Linkage in undirected
graphs, and the more general Rank -Linkage problem, all in
time, improving on respectively (Fomin et al., SODA 2023)
3. Many instances of the Diverse X paradigm, finding a collection of
solutions to a problem with a minimum mutual distance of in time
, improving solutions for -Distinct Branchings from time
to (Bang-Jensen et al., ESA 2021), and for Diverse
Perfect Matchings from to (Fomin et al.,
STACS 2021)
All matroids are assumed to be represented over a field of characteristic 2.
Over general fields, we achieve similar results at the cost of using
exponential space by working over the exterior algebra. For a class of
arithmetic circuits we call strongly monotone, this is even achieved without
any loss of running time. However, the odd support sieving result appears to be
specific to working over characteristic 2
Mutually avoiding Eulerian circuits
Two Eulerian circuits, both starting and ending at the same vertex, are
avoiding if at every other point of the circuits they are at least distance 2
apart. An Eulerian graph which admits two such avoiding circuits starting from
any vertex is said to be doubly Eulerian. The motivation for this definition is
that the extremal Eulerian graphs, i.e. the complete graphs on an odd number of
vertices and the cycles, are not doubly Eulerian. We prove results about doubly
Eulerian graphs and identify those that are the `densest' and `sparsest' in
terms of the number of edges.Comment: 22 pages; 9 figure
Three Essays in Macroeconomics
The scope of the dissertation is: broadly-defined) general macroeconomics. The first essay is on optimal taxation and capital structure, the second essay is on firm dynamics, and the third essay is on financial crises. The first essay clarifies the role of the corporate income tax: as a form of double taxation) for achieving socially optimal allocations in the Mirrlees framework when the government cannot tax unrealized capital income at the individual level. Use of the corporate tax requires changes in the individual capital tax. The novelty of the paper is that the sophisticated tax system is designed to influence the individual agent\u27s portfolio choice of debt and equity, which in turn endogenizes the leverage ratio. The optimum corporate tax is indeterminate, but a minimal level is ecessary. An immediate question is what happens to capital structure if we increase or decrease the level of the corporate tax. Surprisingly, unlike in classical capital structure theories, in this optimal tax mechanism, the firm\u27s leverage ratio is independent of the corporate tax rate. The second essay examines firm dynamics to explain the following empirical facts:: i) The size of a firm and its growth rate are negatively correlated;: ii) but, they are often independent for firms above a certain size. Existing theories of firm dynamics can explain the first fact, but cannot explain the second. This paper studies a dynamic moral hazard problem under an AK-technology. In a first best world, the expected growth rate is strictly decreasing with capital. However, with information asymmetry our theory is consistent with both empirical facts because the optimal contract dictates under-investment in low-level capital states and over-investment in high-level capital states. The reason is that the given convex production technology becomes nonconvex in equilibrium due to the information asymmetry and the degree of the nonconvexity differs by the level of capital. We also fully characterize the agent\u27s incentives. The capital accumulation mechanism induces incentive schemes that are different from optimal contracts in the literature on principal-agent models. Finally, in the third essay - This essay is a joint work with Costas Azariadis - we propose a model of financial crises as transitions from an efficient and unstable state to an inefficient and stable state in a simple economy with sector-specific shocks. The main driving force of this transition is the unwinding of unsecured loans. Introducing public debt increases the volatility of stock prices. We also discuss possible policy interventions
Random combinatorial structures and randomized search heuristics
This thesis is concerned with the probabilistic analysis of random combinatorial structures and the runtime analysis of randomized search heuristics.
On the subject of random structures, we investigate two classes of combinatorial objects. The first is the class of planar maps and the second is the class of generalized parking functions. We identify typical properties of these structures and show strong concentration results on the probabilities that these properties hold. To this end, we develop and apply techniques based on exact enumeration by generating functions. For several types of random planar maps, this culminates in concentration results for the degree sequence. For parking functions, we determine the distribution of the defect, the most characteristic parameter. On the subject of randomized search heuristics, we present, improve, and unify different probabilistic methods and their applications. In this, special focus is given to potential functions and the analysis of the drift of stochastic processes. We apply these techniques to investigate the runtimes of evolutionary algorithms. In particular, we show for several classical problems in combinatorial optimization how drift analysis can be used in a uniform way to give bounds on the expected runtimes of evolutionary algorithms.Diese Dissertationsschrift beschĂ€ftigt sich mit der wahrscheinlichkeitstheoretischen Analyse von zufĂ€lligen kombinatorischen Strukturen und der Laufzeitanalyse randomisierter Suchheuristiken. Im Bereich der zufĂ€lligen Strukturen untersuchen wir zwei Klassen kombinatorischer Objekte. Dies sind zum einen die Klasse aller kombinatorischen Einbettungen planarer Graphen und zum anderen eine Klasse diskreter Funktionen mit bestimmten kombinatorischen Restriktionen (generalized parking functions). FĂŒr das Studium dieser Klassen entwickeln und verwenden wir zĂ€hlkombinatorische Methoden die auf erzeugenden Funktionen basieren. Dies erlaubt uns, Konzentrationsresultate fĂŒr die Gradsequenzen verschiedener Typen zufĂ€lliger kombinatorischer Einbettungen planarer Graphen zu erzielen. DarĂŒber hinaus erhalten wir Konzentrationsresultate fĂŒr den charakteristischen Parameter, den Defekt, zufĂ€lliger Instanzen der untersuchten diskreten Funktionen.
Im Bereich der randomisierten Suchheuristiken prÀsentieren und erweitern wir verschiedene wahrscheinlichkeitstheoretische Methoden der Analyse. Ein besonderer Fokus liegt dabei auf der Analyse der Drift stochastischer Prozesse. Wir wenden diese Methoden in der Laufzeitanalyse evolutionÀrer Algorithmen an. Insbesondere zeigen wir, wie mit Hilfe von Driftanalyse die erwarteten Laufzeiten evolutionÀrer Algorithmen auf verschiedenen klassischen Problemen der kombinatorischen Optimierung auf einheitliche Weise abgeschÀtzt werden können
Developing and Testing an Anguilliform Robot Swimming with Theoretically High Hydrodynamic Efficiency
An anguilliform swimming robot replicating an idealized motion is a complex marine vehicle necessitating both a theoretical and experimental analysis to completely understand its propulsion characteristics. The ideal anguilliform motion within is theorized to produce ``wakeless\u27\u27 swimming (Vorus, 2011), a reactive swimming technique that produces thrust by accelerations of the added mass in the vicinity of the body. The net circulation for the unsteady motion is theorized to be eliminated.
The robot was designed to replicate the desired, theoretical motion by applying control theory methods. Independent joint control was used due to hardware limitations. The fluid velocity vectors in the propulsive wake downstream of the tethered, swimming robot were measured using Particle Image Velocimetry (PIV). Simultaneously, a load cell measured the thrust (or drag) forces of the robot via a hydrodynamic tether. The measured field velocities and thrust forces were compared to the theoretical predictions for each.
The desired, ideal motion was not replicated consistently during PIV testing, producing off-design scenarios. The thrust-computing method for the ideal motion was applied to the actual, recorded motion and compared to the load cell results. The theoretical field velocities were computed differently by accounting for shed vortices due to a different shape than ideal. The theoretical thrust shows trends similar to the measured thrust over time. Similarly promising comparisons are found between the theoretical and measured flow-field velocities with respect to qualitative trends and velocity magnitudes. The initial thrust coefficient prediction was deemed insufficient, and a new one was determined from an iterative process. The off-design cases shed flow structures into the downstream wake of the robot. The first is a residual disturbance of the shed boundary layer, which is to be expected for the ideal case, and dissipates within one motion cycle. The second are larger-order vortices that are being shed at two distinct times during a half-cycle.
These qualitative and quantitative comparisons were used to confirm the possibility of the original hypothesis of ``wakeless\u27\u27 swimming. While the ideal motion could not be tested consistently, the results of the off-design cases agree significantly with the adjusted theoretical computations. This shows that the boundary conditions derived from slender-body constraints and the assumptions of ideal flow theory are sufficient enough to predict the propulsion characteristics of an anguilliform robot undergoing this specific motion
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
LIPIcs, Volume 248, ISAAC 2022, Complete Volum
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