Approximate Capacities of Two-Dimensional Codes by Spatial Mixing

We apply several state-of-the-art techniques developed in recent advances of counting algorithms and statistical physics to study the spatial mixing property of the two-dimensional codes arising from local hard (independent set) constraints, including: hard-square, hard-hexagon, read/write isolated memory (RWIM), and non-attacking kings (NAK). For these constraints, the strong spatial mixing would imply the existence of polynomial-time approximation scheme (PTAS) for computing the capacity. It was previously known for the hard-square constraint the existence of strong spatial mixing and PTAS. We show the existence of strong spatial mixing for hard-hexagon and RWIM constraints by establishing the strong spatial mixing along self-avoiding walks, and consequently we give PTAS for computing the capacities of these codes. We also show that for the NAK constraint, the strong spatial mixing does not hold along self-avoiding walks

Aeromechanical stability analysis of COPTER

A plan was formed for developing a comprehensive, second-generation system with analytical capabilities for predicting performance, loads and vibration, handling qualities, aeromechanical stability, and acoustics. This second-generation system named COPTER (COmprehensive Program for Theoretical Evaluation of Rotorcraft) is designed for operational efficiency, user friendliness, coding readability, maintainability, transportability, modularity, and expandability for future growth. The system is divided into an executive, a data deck validator, and a technology complex. At present a simple executive, the data deck validator, and the aeromechanical stability module of the technology complex were implemented. The system is described briefly, the implementation of the technology module is discussed, and correlation data presented. The correlation includes hingeless-rotor isolated stability, hingeless-rotor ground-resonance stability, and air-resonance stability of an advanced bearingless-rotor in forward flight

Engineered Porous Media for Groundwater Contaminant Remediation: Applications of Nano-TiO2

Nonmetallic Arsenic and hydrocarbon BTEX are the most common and harmful contaminants in groundwater. Many oxide, biological and metal based adsorbent materials have been investigated in the laboratory and in the field environment. These materials, especially for nano-TiO2, have been extensively studied in the laboratory due to its high efficiency and remediation outcomes. This research compared two different synthetic nano-TiO2, aggregated dendritic nano-scale anatase TiO2 and the polymorphed rutile nano-TiO2 attached to the surface of SiO2, to study the feasibility of nano-TiO2 in geoenvironmental engineering applications. Characterization of crystal structure, purity, and morphologic, microscopic features of the materials were examed by XRD, XRF and SEM, respectively. The BTEX and arsenic removal rate/efficiency were compared between the two materials. Finally, via analyzing the materials and solutions before and after the experiment, nano-TiO2 could be used in the environmental engineering site

On the rational functions in non-commutative random variables

This thesis is devoted to some problems on non-commutative rational functions in non-commutative random variables that come from free probability theory and from random matrix theory. First, we will consider the non-commutative random variables in tracial W*-probability spaces, such as freely independent semicircular and Haar unitary random variables. A natural question on rational functions in these random variables is the well-definedness question. Namely, how large is the family of rational functions that have well-defined evaluations for a given tuple X of random variables? Note that for a fixed rational function r, the well-definedness of its evaluation r(X) depends on the interpretation of the invertibility of random variables. This is because the invertibility of a random variable in a tracial W*-probability space (or an operator in a finite von Neumann algebra) can be also considered in a larger algebra, i.e., the *-algebra of affiliated operators. One of our goals in this thesis to show some criteria that characterize the well-definedness of all rational functions in the framework of affiliated operators. In particular, one of these criteria is given by a homological-algebraic quantity on non-commutative random variables. We will also show that some notions provided by free probability are related to this quantity. So we can finally answer the well-definedness question via these related notions from free probability. Those criteria for the well-definedness of rational functions are actually intrinsic connected to the Atiyah conjecture or Atiyah property. We will explore these connections between the Atiyah property and our question on the well-definedness of rational functions. In particular, we will present a result to show a connection between the so-called strong Atiyah property and the invertibility of evaluations of rational functions. In this result, the evaluation of a rational function at a tuple of random variables may not be well-defined, but it is always invertible as an affiliated operator once it is well-defined. In the last part of this thesis, we will turn to the questions on rational functions in random matrices. Besides the well-definedness problem for rational functions in random matrices, we will also address the convergence problem for rational functions in random matrices. Due to the unstableness of the convergence in distribution, we will limit our random matrices to the ones that strongly converge in distribution and our rational functions to the ones that have bounded evaluations. We will show that both the well-defineness and the convergence problem have an affirmative answer under such conditions.Diese Doktorarbeit widmet sich Problemen aus der freien Wahrscheinlichkeitstheorie und der Zufallsmatrizentheorie über nicht-kommutative rationale Funktionen in nichtkommutativen Zufallsvariablen. Zunächst betrachten wir nicht-kommutative Zufallsvariable in endlichen (d.h. tracial) W*-Wahrscheinlichkeitsräumen, wie z.B. freie Halbkreiselemente oder freie Haar unitäre Zufallsvariable. Eine natürliche Frage über rationale Funktionen in solchen Zufallsvariablen ist die nach der Wohldefiniertheit. Genauer gesagt, wie groß ist die Familie von rationalen Funktionen, die eine wohldefinierte Auswertung für ein gegebenes Tupel X von Zufallsvariablen haben? Man muss beachten, dass für eine feste rationale Funktion r die Wohldefiniertheit der Auswertung r(X) von der Interpretation der Invertierbarkeit von Zufallsvariablen abhängt. Dies liegt daran, dass die Invertierbarkeit einer Zufallsvariablen in einem endlichen W*-Wahrscheinlichkeitsraum (oder eines Operators in einer endlichen von Neumann Algebra) in einer gr oßeren Algebra, nämlich der *-Algebra der affiliierten Operatoren, betrachtet werden kann. Eines der Ziele dieser Doktorarbeit ist es Kriterien zu finden, welche die Wohldefiniertheit von allen rationalen Funktionen im Rahmen von affiliierten Operatoren charakterisieren. Insbesondere wird eines dieser Kriterien durch eine homologisch-algebraische Gr oße von nicht-kommutativen Zufallsvariablen gegeben sein. Wir werden auch zeigen, dass diese Gr oße mit verschiedenen Gr oßen aus der freien Wahrscheinlichkeitstheorie zusammenhängt. So werden wir schließlich die Wohldefiniertheitsfrage durch diese Gr oßen aus der freien Wahrscheinlichheitstheorie beschreiben. Diese Kriterien für die Wohldefiniertheit von rationalen Funktionen hängen inhärent mit der Atiyah Vermutung/Eigenschaft zusammen. Wir werden dieser Verbindung zwischen der Atiyah Eigenschaft und unserer Frage nach der Wohldefiniertheit von rationalen Funktionen auf den Grund gehen. Insbesondere werden wir den Zusammenhang zwischen der sogennanten starken Atiyah Eigenschaft und der Wohldefiniertheitsfrage klären. Dabei mag die Auswertung einer rationalen Funktion an einem Tupel von Zufallsvariablen nicht wohldefiniert sein, aber sofern sie wohldefiniert ist, ist sie immer invertierbar als affiliierter Operator. Im letzten Teil der Doktorarbeit wenden wir uns Fragen zu rationalen Funktionen in Zufallsmatrizen zu. Neben dem Wohldefiniertheitsproblem für rationale Funktionen in Zufallsmatrizen werden wir auch das Konvergenzproblem in dem Rahmen ansprechen. Wegen der Instabilität der Konvergenz in Verteilung schränken wir uns dabei auf Zufallsmatrizen ein, welche stark in Verteilung konvergieren, und betrachen nur rationale Funktionen, welche beschränkte Auswertungen besitzen. Wir werden zeigen, dass under solchen Voraussetzungen sowohl die Wohldefiniertheitsfrage als auch die Konvergenzfrage eine positive Antwort hat

Ab initio modeling of the energy landscape for screw dislocations in body-centered cubic high-entropy alloys

In traditional body-centered cubic (bcc) metals, the core properties of screw dislocations play a critical role in plastic deformation at low temperatures. Recently, much attention has been focused on refractory high-entropy alloys (RHEAs), which also possess bcc crystal structures. However, unlike face-centered cubic high-entropy alloys (HEAs), there have been far fewer investigations on bcc HEAs, specifically on the possible effects of chemical short-range order (SRO) in these multiple principal element alloys on dislocation mobility. Here, using density functional theory, we investigate the distribution of dislocation core properties in MoNbTaW RHEAs alloys, and how they are influenced by SRO. The average values of the core energies in the RHEA are found to be larger than those in the corresponding pure constituent bcc metals, and are relatively insensitive to the degree of SRO. However, the presence of SRO is shown to have a large effect on narrowing the distribution of dislocation core energies and decreasing the spatial heterogeneity of dislocation core energies in the RHEA. It is argued that the consequences for the mechanical behavior of HEAs is a change in the energy landscape of the dislocations which would likely heterogeneously inhibit their motion

Chiral magnetic currents with QGP medium response in heavy ion collisions at RHIC and LHC energies

We calculate the electromagnetic current with a more realistic approach in the RHIC and LHC energy regions in the article. We take the partons formation time as the initial time of the magnetic field response of QGP medium. The maximum electromagnetic current and the time-integrated current are two important characteristics of the chiral magnetic effect (CME), which can characterize the intensity and duration of fluctuations of CME. We consider the finite frequency response of CME to a time-varying magnetic field, find a significant impact from QGP medium feedback, and estimate the generated electromagnetic current as a function of time, beam energy and impact parameter.Comment: 10 pages, 12 figur