Thermal response of nonequilibrium RC-circuits

We analyze experimental data obtained from an electrical circuit having components at different temperatures, showing how to predict its response to temperature variations. This illustrates in detail how to utilize a recent linear response theory for nonequilibrium overdamped stochastic systems. To validate these results, we introduce a reweighting procedure that mimics the actual realization of the perturbation and allows extracting the susceptibility of the system from steady state data. This procedure is closely related to other fluctuation-response relations based on the knowledge of the steady state probability distribution. As an example, we show that the nonequilibrium heat capacity in general does not correspond to the correlation between the energy of the system and the heat flowing into it. Rather, also non-dissipative aspects are relevant in the nonequilbrium fluctuation response relations.Comment: 2 figure

Türkiye'nin Problem Çözen Çocuğu: Matematik Eğitiminin Kültürel Alanlarının Tarihsel Bir Analizi

This paper, as a history of the present, explores the connection between pedagogical practices of teaching and learning mathematics and historical conditions in relation to the production of the problem-solving child as a desired human kind in Turkey’s early republican years (1923–1940). Archival resources are school mathematics curricula, textbooks, and teacher guidelines published during those years. Analysis focuses on the epistemological principles that make, order, classify, normalize, and differentiate the self and the other in curricular and instructional materials. Findings reveal that knowledge and practices that organize mathematics education contain normative principles that mark some as faithful and disciplined bodies and others as needing intervention to be fully recognized as Turkish citizens. The paper further explores how mathematics education contains a set of precautionary pedagogies to help not-yet-fit bodies to secure the social order and how those pedagogical practices reinscribe differences between children. Implications are discussed in terms of mathematics education and contemporary schooling. The analysis contributes to the field by addressing the issues of equality and inequality in education from a historical perspective, highlighting that differences between children are the product of a complex, multifaceted set of historical–cultural–pedagogical processes.Bu çalışma, matematiği öğrenme ve öğretme ile ilgili pedagojik uygulamaların tarihsel koşullar ile nasıl ilişkilendiğini ve Türkiye'nin erken Cumhuriyet yıllarında (1923-1940) belirli bir insan türü olarak problem çözen çocuğun oluşum süreçlerini şimdinin tarihi yöntemi ile araştırmaktadır. Veri kaynakları dönemin matematik öğretim programları, ders kitapları ve öğretmen kılavuz kitaplarıdır. Analiz, program ve öğretim materyallerinde modern özneyi ve ötekini oluşturan, normalleştiren, farklılaştıran ve sınıflandıran epistemolojik ilkelere odaklanmaktadır. Elde edilen bulgular, matematik eğitimini düzenleyen bilgi ve uygulamaların, bazılarını inançlı-disiplinli bedenler ve diğerlerini ulusun vatandaşı olarak tanınmaları için müdahale edilmesi gerekenler olarak belirleyen normatif prensipleri içerdiğini ortaya koymaktadır. Makale ayrıca matematik eğitiminin, sosyal düzeni güvenceye almak için henüz uyum sağlamamış bedenlere sunduğu pedagojik tedbirleri araştırmakta, her çocuğun matematik öğrenebilmesi için önerilen uygulamaların çocukların farklılıklarını yeniden ortaya koyduğunu görünür kılmaktadır. Sonuçlar, matematik eğitiminin ve çağdaş okullaşmanın kültürel politikaları açısından tartışılmıştır. Yapılan analiz, çocuklar arasındaki farklılıkların çok yönlü ve karmaşık tarihsel-kültürel-pedagojik süreçlerin ürünü olarak oluştuğunu belirgin kılarak, eğitimde eşitlik ve eşitsizlik konularını tarihsel bir perspektifte ele almasıyla alana katkıda bulunmaktadır

The R-algebra of Quasiknowledge and Convex Optimization

This article develops a convex description of a classical or quantum learner's or agent's state of knowledge about its environment, presented as a convex subset of a commutative R-algebra. With caveats, this leads to a generalization of certain semidefinite programs in quantum information (such as those describing the universal query algorithm dual to the quantum adversary bound, related to optimal learning or control of the environment) to the classical and faulty-quantum setting, which would not be possible with a naive description via joint probability distributions over environment and internal memory. More philosophically, it also makes an interpretation of the set of reduced density matrices as "states of knowledge" of an observer of its environment, related to these techniques, more explicit. As another example, I describe and solve a formal differential equation of states of knowledge in that algebra, where an agent obtains experimental data in a Poissonian process, and its state of knowledge evolves as an exponential power series. However, this framework currently lacks impressive applications, and I post it in part to solicit feedback and collaboration on those. In particular, it may be possible to develop it into a new framework for the design of experiments, e.g. the problem of finding maximally informative questions to ask human labelers or the environment in machine-learning problems. The parts of the article not related to quantum information don't assume knowledge of it.Comment: 40 page

Parabolic systems and an underlying Lagrangian

In this thesis, we extend De Giorgi's interpolation method to a class of parabolic equations which are not gradient flows but possess an entropy functional and an underlying Lagrangian. The new fact in the study is that not only the Lagrangian may depend on spatial variables, but also it does not induce a metric. Assuming the initial condition is a density function, not necessarily smooth, but solely of bounded first moments and finite "entropy", we use a variational scheme to discretize the equation in time and construct approximate solutions. Moreover, De Giorgi's interpolation method is revealed to be a powerful tool for proving convergence of our algorithm. Finally, we analyze uniqueness and stability of our solution in L¹.Ph.D.Committee Chair: Gangbo, Wilfrid; Committee Member: Chow, Shui-Nee; Committee Member: Harrell, Evans; Committee Member: Swiech, Andrzej; Committee Member: Yezzi, Anthony Josep

Eigenvalue inequalities for relativistic Hamiltonians and fractional Laplacian

Some eigenvalue inequalities for Klein-Gordon operators and fractional Laplacians restricted to a bounded domain are proved. Such operators became very popular recently as they arise in many problems ranging from mathematical finance to crystal dislocations, especially relativistic quantum mechanics and symmetric stable stochastic processes. Many of the results obtained here are concerned with finding bounds for some functions of the spectrum of these operators. The subject, which is well developed for the Laplacian, is examined from the spectral theory perspective through some of the tools used to prove analogous results for the Laplacian. This work highlights some important results, sparking interest in constructing a similar theory for Klein-Gordon operators. For instance, the Weyl asymptotics and semiclassical bounds for the Klein-Gordon operator are developed. As a result, a Berezin-Li-Yau type inequality is derived and an improvement of the bound is proved in a separate chapter. Other results involving some universal bounds for the Klein-Gordon Hamiltonian with an external interaction are also obtained.Ph.D.Committee Chair: Harrell, Evans; Committee Member: Chow, Shui-Nee; Committee Member: Geronimo, Jeffrey; Committee Member: Kennedy, Brian; Committee Member: Loss, Michae

A general fluctuation-response relation for noise variations and its application to driven hydrodynamic experiments

The effect of a change of noise amplitudes in overdamped diffusive systems is linked to their unperturbed behavior by means of a nonequilibrium fluctuation-response relation. This formula holds also for systems with state-independent nontrivial diffusivity matrices, as we show with an application to an experiment of two trapped and hydrodynamically coupled colloids, one of which is subject to an external random forcing that mimics an effective temperature. The nonequilibrium susceptibility of the energy to a variation of this driving is an example of our formulation, which improves an earlier version, as it does not depend on the time-discretization of the stochastic dynamics. This scheme holds for generic systems with additive noise and can be easily implemented numerically, thanks to matrix operations

The initial-boundary value problem for the biharmonic Schr\"odinger equation on the half-line

We study the local and global wellposedness of the initial-boundary value problem for the biharmonic Schr\"odinger equation on the half-line with inhomogeneous Dirichlet-Neumann boundary data. First, we obtain a representation formula for the solution of the linear nonhomogenenous problem by using the Fokas method (also known as the \emph{unified transform method}). We use this representation formula to prove space and time estimates on the solutions of the linear model in fractional Sobolev spaces by using Fourier analysis. Secondly, we consider the nonlinear model with a power type nonlinearity and prove the local wellposedness by means of a classical contraction argument. We obtain Strichartz estimates to treat the low regularity case by using the oscillatory integral theory directly on the representation formula provided by the Fokas method. Global wellposedness of the defocusing model is established up to cubic nonlinearities by using the multiplier technique and proving hidden trace regularities.Comment: 35 pages, 3 figure

The Forecasting of Labour Force Participation and the Unemployment Rate in Poland and Turkey Using Fuzzy Time Series Methods

Fuzzy time series methods based on the fuzzy set theory proposed by Zadeh (1965) was first introduced by Song and Chissom (1993). Since fuzzy time series methods do not have the assumptions that traditional time series do and have effective forecasting performance, the interest on fuzzy time series approaches is increasing rapidly. Fuzzy time series methods have been used in almost all areas, such as environmental science, economy and finance. The concepts of labour force participation and unemployment have great importance in terms of both the economy and sociology of countries. For this reason there are many studies on their forecasting. In this study, we aim to forecast the labour force participation and unemployment rate in Poland and Turkey using different fuzzy time series methods