On the short-distance structure of irrational non-commutative gauge theories

As shown by Hashimoto and Itzhaki in hep-th/9911057, the perturbative degrees of freedom of a non-commutative Yang-Mills theory (NCYM) on a torus are quasi-local only in a finite energy range. Outside that range one may resort to a Morita equivalent (or T-dual) description appropriate for that energy. In this note, we study NCYM on a non-commutative torus with an irrational deformation parameter $\theta$. In that case, an infinite tower of dual descriptions is generically needed in order to describe the UV regime. We construct a hierarchy of dual descriptions in terms of the continued fraction approximations of $\theta$. We encounter different descriptions depending on the level of the irrationality of $\theta$ and the amount of non-locality tolerated. The behavior turns out to be isomorphic to that found for the phase structure of the four-dimensional Villain $Z_N$ lattice gauge theories, which we revisit as a warm-up. At large 't Hooft coupling, using the AdS/CFT correspondance, we find that there are domains of the radial coordinate $U$ where no T-dual description makes the derivative expansion converge. The radial direction obtains multifractal characteristics near the boundary of AdS.Comment: 17 pages, 4 figures, uses JHEP.cl

Comparative analysis of Kolmogorov ANN and process characteristic input-output modes

In the past decades, representation models of dynamical processes have been developed via both traditional math-analytical and less traditional computational-intelligence approaches. This challenge to system sciences goes on because essentially involves the mathematical approximation theory. A comparison study based on cybernetic input-output view in the time domain on complex dynamical processes has been carried out. An analytical decomposition representation of complex multi-input-multi-output thermal processes is set relative to the neural-network approximation representations, and shown that theoretical background of both emanates from Kolmogorov's theorem. The findings provided a new insight as well as highlighted the efficiency and robustness of fairly simple industrial digital controls, designed and implemented in the past, inherited from input-output decomposition model approximation employed

Testing Point Null Hypothesis of a Normal Mean and the Truth: 21st Century Perspective

Testing a point (sharp) null hypothesis is arguably the most widely used statistical inferential procedure in many fields of scientific research, nevertheless, the most controversial, and misapprehended. Since 1935 when Buchanan-Wollaston raised the first criticism against hypothesis testing, this foundational field of statistics has drawn increasingly active and stronger opposition, including draconian suggestions that statistical significance testing should be abandoned or even banned. Statisticians should stop ignoring these accumulated and significant anomalies within the current point-null hypotheses paradigm and rebuild healthy foundations of statistical science. The foundation for a paradigm shift in testing statistical hypotheses is suggested, which is testing interval null hypotheses based on implications of the Zero probability paradox. It states that in a real-world research point-null hypothesis of a normal mean has zero probability. This implies that formulated point-null hypothesis of a mean in the context of the simple normal model is almost surely false. Thus, Zero probability paradox points to the root cause of so-called large n problem in significance testing. It discloses that there is no point in searching for a cure under the current point-null paradigm

The ontology of number

What is a number? Answering this will answer questions about its philosophical foundations - rational numbers, the complex numbers, imaginary numbers. If we are to write or talk about something, it is helpful to know whether it exists, how it exists, and why it exists, just from a common-sense point of view [Quine, 1948, p. 6]. Generally, there does not seem to be any disagreement among mathematicians, scientists, and logicians about numbers existing in some way, but currently, in the mainstream arena only definitions, descriptions of properties, and effects are presented as evidence. Enough historical description of numbers in history provides an empirical basis of number, although a case can be made that numbers do not exist by themselves empirically. Correspondingly, numbers exist as abstractions. All the while, though, these "descriptions" beg the question of what numbers are ontologically. Advocates for numbers being the ultimate reality have the problem of wrestling with the nature of reality. I start on the road to discovering the ontology of number by looking at where people have talked about numbers as already existing: history. Of course, we need to know not only what ontology is but the problems of identifying one, leading to the selection between metaphysics and provisional approaches. While we seem to be dimensionally limited, at least we can identify a more suitable bootstrapping ontology than mere definitions, leading us to the unity of opposites. The rest of the paper details how this is done and modifies Peano's Postulates

Maths, Logic and Language

A work on the philosophy of mathematics (2017) ‘Number’, such a simple idea, and yet it fascinated and absorbed the greatest proportion of human geniuses over centuries, not to mention the likes of Pythagoras, Euclid, Newton, Leibniz, Descartes and countless maths giants like Euler, Gauss and Hilbert, etc.. Einstein thought of pure maths as the poetry of logical ideas, the exactitude of which, although independent of experience, strangely seems to benefit the study of the objects of reality. And, interestingly as well as surprisingly we are nowhere near any clear understandings of numbers despite discoveries of many productive usages of numbers. This is - rightly or wrongly - a humble attempt to approach the subject from an angle hitherto unthought-of

ANOMALOUS TRANSPORT, QUASIPERIODICITY, AND MEASUREMENT INDUCED PHASE TRANSITIONS

With the advent of the noisy-intermediate scale quantum (NISQ) era quantum computers are increasingly becoming a reality of the near future. Though universal computation still seems daunting, a great part of the excitement is about using quantum simulators to solve fundamental problems in fields ranging from quantum gravity to quantum many-body systems. This so-called second quantum revolution rests on two pillars. First, the ability to have precise control over experimental degrees of freedom is crucial for the realization of NISQ devices. Significant progress in the control and manipulation of qubits, atoms, and ions, as well as their interactions, has not only allowed for emulation of diverse range of physical systems but has also led to realization of quantum systems in non-conventional settings such as systems out-of-equilibrium, driven by oscillating fields, and with quasiperiodic (QP) modulation. These systems often show novel properties which not only provide an interesting testbed for NISQ devices but also an opportunity to exploit them for further development of quantum computing devices. Second, the study of dynamics of quantum information in quantum systems is essential for understanding and designing better quantum computers. In addition to their practical application as resource for quantum computation, quantum information has also become an essential element for our understanding of various physical problems, such as thermalization of isolated quantum many-body systems. This interplay between quantum information and computation, and quantum many-body systems is only expected to increase with time. In this thesis, we explore these topics in two parts, corresponding respectively to the two pillars mentioned above. In the first part, we study effects of quasiperiodicity on many-body quantum systems in low dimensions. QP systems are aperiodic but deterministic, so their behavior differs from that of clean systems and disordered ones as well. Moreover, these systems can be conveniently realized in an experimental setting where it is easier to isolate them from external decoherence. %Recent advancement in experimental techniques has made it easier to design and probe quantum systems with quasi-periodic modulations. We start with the easy-plane regime of the XXZ spin chain and show that the well-known fractal behavior of the spin Drude weight implies the divergence of the low-frequency conductivity for generic values of anisotropy. We tie this to the quasi-periodic structure in the Bethe ansatz solution resulting in different species of quasiparticles getting activated along the time evolution in a quasi-periodic pattern. We then study quantum critical systems under generic quasi-periodic modulations using real-space renormalization group (RSRG) procedure. In 1d, we show that the system flows to a new fixed point with the couplings following a discrete aperiodic sequence which allows us to analytically calculate the critical properties. We dub these new classes of quasi-periodic fixed points infinite-quasiperiodicity fixed points in line with the infinite-randomness fixed point observed in random quantum systems. We use this approach to analyze the quasiperiodic Heisenberg, Ising, and Potts spin chains. The RSRG is not analytically tractable in 2d, but numerically implementing it for the 2d quasi-periodic $q$-state quantum Potts model, we find that it is well controlled and becomes exact in the asymptotic limit. The critical behavior is shown to be largely independent of $q$ and is controlled by an infinite-quasiperiodicity fixed point. We also provide a heuristic argument for the correlation length exponent and the scaling of the energy gap. Moving on to the second part, we study monitored quantum circuits which have recently emerged as a powerful platform for exploring the dynamics of quantum information and errors in quantum systems. Unitary evolution generates entanglement between distant particles of the system. The dynamics of entanglement has been successfully studied by replacing the Hamiltonian evolution with random quantum circuits. Recently, the robustness of unitary evolution\u27s ability to protect the entanglement against external projective measurements has received much attention. Entanglement is also a resource for quantum information, so its stability is directly related to the stability of a quantum computer against external noises. It has been observed that, in absence of any symmetry, there is a measurement induced phase transition (MIPT) in the behavior of bipartite entanglement that goes from volume law to area law as we tune the rate of measurements. Here we focus on monitored quantum circuits with U(1) symmetry which leads to the presence of a conserved charge density. These diffusive hydrodynamic modes scramble very differently than non-symmetric modes and we find that in addition to the entanglement transition, there is another transition \textit{inside} the volume phase which we call a ``charge sharpening\u27\u27 transition. The sharpening transition is a transition in the ability/inability of the measurements to detect the global charge of the system. We study this sharpening transition in a variety of settings, including an effective field theory that predicts the transition to be in a modified Kosterlitz-Thouless universality class. We provide various numerical evidence to back our predictions