Non-commutative Donaldson-Thomas theory and the conifold

Given a quiver algebra A with relations defined by a superpotential, this paper defines a set of invariants of A counting framed cyclic A-modules, analogous to rank-1 Donaldson-Thomas invariants of Calabi-Yau threefolds. For the special case when A is the non-commutative crepant resolution of the threefold ordinary double point, it is proved using torus localization that the invariants count certain pyramid-shaped partition-like configurations, or equivalently infinite dimer configurations in the square dimer model with a fixed boundary condition. The resulting partition function admits an infinite product expansion, which factorizes into the rank-1 Donaldson-Thomas partition functions of the commutative crepant resolution of the singularity and its flop. The different partition functions are speculatively interpreted as counting stable objects in the derived category of A-modules under different stability conditions; their relationship should then be an instance of wall crossing in the space of stability conditions on this triangulated category.Comment: Infinite product form, conjectured in v1, now a theorem of Ben Young. Additional discussion of small-volume expansion related to Eisenstein-like serie

A Class of Logistic Functions for Approximating State-Inclusive Koopman Operators

An outstanding challenge in nonlinear systems theory is identification or learning of a given nonlinear system's Koopman operator directly from data or models. Advances in extended dynamic mode decomposition approaches and machine learning methods have enabled data-driven discovery of Koopman operators, for both continuous and discrete-time systems. Since Koopman operators are often infinite-dimensional, they are approximated in practice using finite-dimensional systems. The fidelity and convergence of a given finite-dimensional Koopman approximation is a subject of ongoing research. In this paper we introduce a class of Koopman observable functions that confer an approximate closure property on their corresponding finite-dimensional approximations of the Koopman operator. We derive error bounds for the fidelity of this class of observable functions, as well as identify two key learning parameters which can be used to tune performance. We illustrate our approach on two classical nonlinear system models: the Van Der Pol oscillator and the bistable toggle switch.Comment: 8 page

Classical versus Quantum Time Evolution of Densities at Limited Phase-Space Resolution

We study the interrelations between the classical (Frobenius-Perron) and the quantum (Husimi) propagator for phase-space (quasi-)probability densities in a Hamiltonian system displaying a mix of regular and chaotic behavior. We focus on common resonances of these operators which we determine by blurring phase-space resolution. We demonstrate that classical and quantum time evolution look alike if observed with a resolution much coarser than a Planck cell and explain how this similarity arises for the propagators as well as their spectra. The indistinguishability of blurred quantum and classical evolution implies that classical resonances can conveniently be determined from quantum mechanics and in turn become effective for decay rates of quantum correlations.Comment: 10 pages, 3 figure

Data Acquisition and Control System of Hydroelectric Power Plant Using Internet Techniques

Vodní energie se nyní stala nejlepším zdrojem elektrické energie na zemi. Vyrábí se pomocí energie poskytované pohybem nebo pádem vody. Historie dokazuje, že náklady na tuto elektrickou energii zůstávají konstantní v průběhu celého roku. Vzhledem k mnoha výhodám, většina zemí nyní využívá vodní energie jako hlavní zdroj pro výrobu elektrické energie.Nejdůležitější výhodou je, že vodní energie je zelená energie, což znamená, že žádné vzdušné nebo vodní znečišťující látky nejsou vyráběny, také žádné skleníkové plyny jako oxid uhličitý nejsou vyráběny, což činí tento zdroj energie šetrný k životnímu prostředí. A tak brání nebezpečí globálního oteplování. Použití internetové techniky k ovladání několika vodních elektráren má velmi významné výhody, jako snížení provozních nákladů a flexibilitu uspokojení změny poptávky po energii na straně spotřeby. Také velmi efektivně čelí velkým narušením elektrické sítě, jako je například přidání nebo odebrání velké zátěže, a poruch. Na druhou stranu, systém získávání dat poskytuje velmi užitečné informace pro typické i vědecké analýzy, jako jsou ekonomické náklady, predikce poruchy systémů, predikce poptávky, plány údržby, systémů pro podporu rozhodování a mnoho dalších výhod. Tato práce popisuje všeobecný model, který může být použit k simulaci pro sběr dat a kontrolní systémy pro vodní elektrárny v prostředí Matlab / Simulink a TrueTime Simulink knihovnu. Uvažovaná elektrárna sestává z vodní turbíny připojené k synchronnímu generátoru s budicí soustavou, generátor je připojen k veřejné elektrické síti. Simulací vodní turbíny a synchronního generátoru lze provést pomocí různých simulačních nástrojů. V této práci je upřednostňován SIMULINK / MATLAB před jinými nástroji k modelování dynamik vodní turbíny a synchronního stroje. Program s prostředím MATLAB SIMULINK využívá k řešení schematický model vodní elektrárny sestavený ze základních funkčních bloků. Tento přístup je pedagogicky lepší než komplikované kódy jiných softwarových programů. Knihovna programu Simulink obsahuje funkční bloky, které mohou být spojovány, upravovány a modelovány. K vytvoření a simulování internetových a Real Time systémů je možné použít bud‘ knihovnu simulinku Real-Time nebo TRUETIME, v práci byla použita knihovna TRUETIME.Hydropower has now become the best source of electricity on earth. It is produced due to the energy provided by moving or falling water. History proves that the cost of this electricity remains constant over the year. Because of the many advantages, most of the countries now have hydropower as the source of major electricity producer. The most important advantage of hydropower is that it is green energy, which mean that no air or water pollutants are produced, also no greenhouse gases like carbon dioxide are produced which makes this source of energy environment-friendly. It prevents us from the danger of global warming. Using internet techniques to control several hydroelectric plants has very important advantages, as reducing operating costs and the flexibility of meeting changes of energy demand occurred in consumption side. Also it is very effective to confront large disturbances of electrical grid, such as adding or removing large loads, and faults. In the other hand, data acquisition systems provides very useful information for both typical and scientific analysis, such as economical costs reducing, fault prediction systems, demand prediction, maintenance schedules, decision support systems and many other benefits. This thesis describes a generalized model which can be used to simulate a data acquisition and control system of hydroelectric power plant using MATLAB/SIMULINK and TrueTime simulink library. The plant considered consists of hydro turbine connected to synchronous generator with excitation system, and the generator is connected to public grid. Simulation of hydro turbine and synchronous generator can be done using various simulation tools, In this work, SIMULINK/MATLAB is favored over other tools in modeling the dynamics of a hydro turbine and synchronous machine. The SIMULINK program in MATLAB is used to obtain a schematic model of the hydro plant by means of basic function blocks. This approach is pedagogically better than using a compilation of program code as in other software programs .The library of SIMULINK software programs includes function blocks which can be linked and edited to model. Either Simulink Real-Time library or TrueTime library can be used to build and simulate internet and real time systems, in this thesis the TrueTime library was used.

Reducible family of height three level algebras

Let $R=k[x_1,..., x_r]$ be the polynomial ring in $r$ variables over an infinite field $k$, and let $M$ be the maximal ideal of $R$. Here a \emph{level algebra} will be a graded Artinian quotient $A$ of $R$ having socle $Soc(A)=0:M$ in a single degree $j$. The Hilbert function $H(A)=(h_0,h_1,... ,h_j)$ gives the dimension $h_i=\dim_k A_i$ of each degree-$i$ graded piece of $A$ for $0\le i\le j$. The embedding dimension of $A$ is $h_1$, and the \emph{type} of $A$ is \dim_k \Soc (A), here $h_j$. The family \Levalg (H) of level algebra quotients of $R$ having Hilbert function $H$ forms an open subscheme of the family of graded algebras or, via Macaulay duality, of a Grassmannian. We show that for each of the Hilbert functions $H=H_1=(1,3,4,4)$ and $H=H_2=(1,3,6,8,9,3)$ the family $LevAlg (H)$ parametrizing level Artinian algebras of Hilbert function $H$ has several irreducible components. We show also that these examples each lift to points. However, in the first example, an irreducible Betti stratum for Artinian algebras becomes reducible when lifted to points. These were the first examples we obtained of multiple components for \Levalg(H) in embedding dimension three. We also show that the second example is the first in an infinite sequence of examples of type three Hilbert functions $H(c)$ in which also the number of components of LevAlg(H) gets arbitrarily large. The first case where the phenomenon of multiple components can occur (i.e. the lowest embedding dimension and then the lowest type) is that of dimension three and type two. Examples of this first case have been obtained by the authors and also by J.-O. Kleppe.Comment: 20 pages. Minor revisio