Browsing by Subject "Eigenvalues and eigenfunctions"
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Item Open Access Analysis of cross-correlations between financial markets after the 2008 crisis(Elsevier BV, 2013) Sensoy, A.; Yuksel, S.; Erturk, M.We analyze the cross-correlation matrix C of the index returns of the main financial markets after the 2008 crisis using methods of random matrix theory. We test the eigenvalues of C for universal properties of random matrices and find that the majority of the cross-correlation coefficients arise from randomness. We show that the eigenvector of the largest deviating eigenvalue of C represents a global market itself. We reveal that high volatility of financial markets is observed at the same times with high correlations between them which lowers the risk diversification potential even if one constructs a widely internationally diversified portfolio of stocks. We identify and compare the connection and cluster structure of markets before and after the crisis using minimal spanning and ultrametric hierarchical trees. We find that after the crisis, the co-movement degree of the markets increases. We also highlight the key financial markets of pre and post crisis using main centrality measures and analyze the changes. We repeat the study using rank correlation and compare the differences. Further implications are discussed.Item Open Access Canonical-covariant Wigner function in polar form(OSA - The Optical Society, 2000) Hakioǧlu, T.The two-dimensional Wigner function was investigated in polar canonical coordinates. The covariance properties under the action of affine canonical transformations were derived. The polar canonical phase-space representations were considered important for paraxial optical systems as well as other systems in which a rotational symmetry around a particular axis was present.Item Open Access Characteristic equations for the lasing Modes of infinite periodic chain of quantum wires(IEEE, 2008-06) Byelobrov, V. O.; Benson, T. M.; Altıntaş, Ayhan; Nosich, A.I.In this paper, we study the lasing modes of a periodic open optical resonator. The resonator is an infinite chain of active circular cylindrical quantum wires standing in tree space. Characteristic equations for the frequencies and associated linear thresholds of lasing are derived. These quantities are considered as eigenvalues of specific electromagnetic-field problem with "active" imaginary part of the cylinder material's refractive index - Lasing Eigenvalue Problem (LEP). ©2008 IEEE.Item Open Access Classification of closed and open shell pistachio nuts using principal component analysis of impact acoustics(IEEE, 2004-05) Çetin, A. Enis; Pearson, T. C.; Tewfik, A. H.An algorithm was developed to separate pistachio nuts with closed-shells from those with open-shells. It was observed that upon impact on a steel plate, nuts with closed-shells emit different sounds than nuts with open-shells. Two feature vectors extracted from the sound signals were melcepstrum coefficients and eigenvalues obtained from the principle component analysis of the autocorrelation matrix of the signals. Classification of a sound signal was done by linearly combining feature vectors from both mel-cepstrum and PCA feature vectors. An important property of the algorithm is that it is easily trainable. During the training phase, sounds of the nuts with closed-shells and open-shells were used to obtain a representative vector of each class. The accuracy of closed-shell nuts was more than 99% on the test set.Item Open Access Componentwise bounds for nearly completely decomposable Markov chains using stochastic comparison and reordering(Elsevier, 2005) Pekergin, N.; Dayar T.; Alparslan, D. N.This paper presents an improved version of a componentwise bounding algorithm for the state probability vector of nearly completely decomposable Markov chains, and on an application it provides the first numerical results with the type of algorithm discussed. The given two-level algorithm uses aggregation and stochastic comparison with the strong stochastic (st) order. In order to improve accuracy, it employs reordering of states and a better componentwise probability bounding algorithm given st upper- and lower-bounding probability vectors. Results in sparse storage show that there are cases in which the given algorithm proves to be useful. © 2004 Elsevier B.V. All rights reserved.Item Open Access Continuous and discrete fractional fourier domain decomposition(IEEE, 2000) Yetik, İ. Şamil; Kutay, M. A.; Özaktaş, H.; Özaktaş, Haldun M.We introduce the fractional Fourier domain decomposition for continuous and discrete signals and systems. A procedure called pruning, analogous to truncation of the singular-value decomposition, underlies a number of potential applications, among which we discuss fast implementation of space-variant linear systems.Item Open Access Counting surrounding nodes using DS-SS signals and de Bruijn sequences in blind environment(IEEE, 2013-03) Warty, C.; Seçer, Görkem; Yu, R.W.; Spinsante, S.In recent years the technological development has encouraged several applications based on node to node communications without any fixed infrastructure. This paper presents preliminary evaluation of popular estimating techniques to populate active nodes in the neighborhood using De Bruijn sequences. They have much higher cardinality compared to any other family of binary sequences at a parity of length. This characteristic of De Bruijn sequences can be exploited to identify the presence of an active node in a dense surrounding, to assist the primary node in making intelligent decisions in a blind or foggy environment. The simulation model in this paper evaluates the use of eigenvalue estimation to estimate the spreading sequence among noisy signals, based on eigenvalues analysis techniques. The received signal is divided into windows, from which a covariance matrix is computed; the sequence can be reconstructed from the two first eigenvectors of this matrix, and that useful information, such as the desynchronization time, can be extracted from the eigenvalues. © 2013 IEEE.Item Open Access Coupled optical microcavities in one-dimensional photonic bandgap structures(Institute of Physics Publishing, 2001) Bayındır, Mehmet; Kural, C.; Özbay, EkmelWe present a detailed theoretical and experimental study of the evanescent coupled optical microcavity modes in one-dimensional photonic bandgap structures. The coupled-cavity samples are fabricated by depositing alternating hydrogenated amorphous silicon nitride and silicon oxide layers. Splitting of the eigenmodes and formation of a defect band due to interaction between the neighbouring localized cavity modes are experimentally observed. Corresponding field patterns and the transmission spectra are obtained by using transfer matrix method (TMM) simulations. A theoretical model based on the classical wave analogue of the tight-binding (TB) picture is developed and applied to these structures. Experimental results are in good agreement with the predictions of the TB approximation and the TMM simulations.Item Open Access Digital computation of the fractional Fourier transform(Institute of Electrical and Electronics Engineers, 1996-09) Özaktaş, Haldun M.; Arıkan, Orhan; Kutay, M. A.; Bozdağı, G.An algorithm for efficient and accurate computation of the fractional Fourier transform is given. For signals with time-bandwidth product N, the presented algorithm computes the fractional transform in O(NlogN) time. A definition for the discrete fractional Fourier transform that emerges from our analysis is also discussed.Item Open Access Disorder and localization in the lowest Landau level in the presence of dilute point scatterers(Pergamon Press, 1999) Gedik, Z.; Bayındır, MehmetWe study the localization properties of a two-dimensional noninteracting electron gas in the presence of randomly distributed short-range scatterers in very high magnetic fields. We evaluate the participation number of the eigenstates obtained by exact diagonalization technique. At low impurity concentrations we obtain self-averaged values showing that all states, except those exactly at the Landau level, are localized with finite localization length. We conclude that in this dilute regime the localization length does not diverge. We also find that the maximum localization length increases exponentially with impurity concentration. Our calculations suggest that scaling behavior may be absent even for higher concentrations of scatterers.Item Open Access An empirical eigenvalue-threshold test for sparsity level estimation from compressed measurements(IEEE, 2014) Lavrenko, A.; Römer, F.; Del Galdo, G.; Thoma, R.; Arıkan, OrhanCompressed sensing allows for a significant reduction of the number of measurements when the signal of interest is of a sparse nature. Most computationally efficient algorithms for signal recovery rely on some knowledge of the sparsity level, i.e., the number of non-zero elements. However, the sparsity level is often not known a priori and can even vary with time. In this contribution we show that it is possible to estimate the sparsity level directly in the compressed domain, provided that multiple independent observations are available. In fact, one can use classical model order selection algorithms for this purpose. Nevertheless, due to the influence of the measurement process they may not perform satisfactorily in the compressed sensing setup. To overcome this drawback, we propose an approach which exploits the empirical distributions of the noise eigenvalues. We demonstrate its superior performance compared to state-of-the-art model order estimation algorithms numerically.Item Open Access Equivalence of linear canonical transform domains to fractional Fourier domains and the bicanonical width product: a generalization of the space-bandwidth product(Optical Society of America, 2010-07-30) Oktem, F. S.; Özaktaş, Haldun M.Linear canonical transforms (LCTs) form a three-parameter family of integral transforms with wide application in optics. We show that LCT domains correspond to scaled fractional Fourier domains and thus to scaled oblique axes in the space-frequency plane. This allows LCT domains to be labeled and ordered by the corresponding fractional order parameter and provides insight into the evolution of light through an optical system modeled by LCTs. If a set of signals is highly confined to finite intervals in two arbitrary LCT domains, the space-frequency (phase space) support is a parallelogram. The number of degrees of freedom of this set of signals is given by the area of this parallelogram, which is equal to the bicanonical width product but usually smaller than the conventional space-bandwidth product. The bicanonical width product, which is a generalization of the space-bandwidth product, can provide a tighter measure of the actual number of degrees of freedom, and allows us to represent and process signals with fewer samples.Item Open Access Exact diffraction calculation from fields specified over arbitrary curved surfaces(Elsevier, 2011-07-30) Esmer, G. B.; Onural, L.; Özaktaş, Haldun M.Calculation of the scalar diffraction field over the entire space from a given field over a surface is an important problem in computer generated holography. A straightforward approach to compute the diffraction field from field samples given on a surface is to superpose the emanated fields from each such sample. In this approach, possible mutual interactions between the fields at these samples are omitted and the calculated field may be significantly in error. In the proposed diffraction calculation algorithm, mutual interactions are taken into consideration, and thus the exact diffraction field can be calculated. The algorithm is based on posing the problem as the inverse of a problem whose formulation is straightforward. The problem is then solved by a signal decomposition approach. The computational cost of the proposed method is high, but it yields the exact scalar diffraction field over the entire space from the data on a surface.Item Open Access Fast and accurate algorithm for the computation of complex linear canonical transforms(Optical Society of America, 2010-08-05) Koç A.; Özaktaş, Haldun M.; Hesselink, L.A fast and accurate algorithm is developed for the numerical computation of the family of complex linear canonical transforms (CLCTs), which represent the input-output relationship of complex quadratic-phase systems. Allowing the linear canonical transform parameters to be complex numbers makes it possible to represent paraxial optical systems that involve complex parameters. These include lossy systems such as Gaussian apertures, Gaussian ducts, or complex graded-index media, as well as lossless thin lenses and sections of free space and any arbitrary combinations of them. Complex-ordered fractional Fourier transforms (CFRTs) are a special case of CLCTs, and therefore a fast and accurate algorithm to compute CFRTs is included as a special case of the presented algorithm. The algorithm is based on decomposition of an arbitrary CLCT matrix into real and complex chirp multiplications and Fourier transforms. The samples of the output are obtained from the samples of the input in ∼N log N time, where N is the number of input samples. A space-bandwidth product tracking formalism is developed to ensure that the number of samples is information-theoretically sufficient to reconstruct the continuous transform, but not unnecessarily redundant.Item Open Access Fast and accurate linear canonical transform algorithms(IEEE, 2015) Özaktaş, Haldun M.; Koç, A.Linear canonical transforms are encountered in many areas of science and engineering. Important transformations such as the fractional Fourier transform and the ordinary Fourier transform are special cases of this transform family. This family of transforms is especially important for the modelling of wave propagation. It has many applications such as noise removal, image encryption, and analysis of optical systems. Here we discuss algorithms for fast and accurate computation of these transforms. These algorithms can achieve the same accuracy and speed as fast Fourier transform algorithms, so that they can be viewed as optimal algorithms. Efficient sampling of signals plays an important part in the development of these algorithms.Item Open Access The fractional Fourier domain decomposition(Elsevier, 1999) Kutay, M. A.; Özaktaş, H.; Özaktaş, Haldun M.; Arıkan, OrhanWe introduce the fractional Fourier domain decomposition. A procedure called pruning, analogous to truncation of the singular-value decomposition, underlies a number of potential applications, among which we discuss fast implementation of space-variant linear systems.Item Open Access The Fractional Fourier transform and harmonic oscillation(Springer, 2002) Kutay, M. A.; Özaktaş, Haldun M.The ath-order fractional Fourier transform is a generalization of the ordinary Fourier transform such that the zeroth-order fractional Fourier transform operation is equal to the identity operation and the first-order fractional Fourier transform is equal to the ordinary Fourier transform. This paper discusses the relationship of the fractional Fourier transform to harmonic oscillation; both correspond to rotation in phase space. Various important properties of the transform are discussed along with examples of common transforms. Some of the applications of the transform are briefly reviewed.Item Open Access The fractional fourier transform and its applications to image representation and beamforming(ASME, 2003-09) Yetik, I. Ş; Kutay, M. A.; Özaktaş, Haldun. M.The ath order fractional Fourier transform operator is the ath power of the ordinary Fourier transform operator. We provide a brief introduction to the fractional Fourier transform, discuss some of its more important properties, and concentrate on its applications to image representation and compression, and beamforming. We show that improved performance can be obtained by employing the fractional Fourier transform instead of the ordinary Fourier transform in these applications.Item Open Access A general theory on spectral properties of state-homogeneous finite-state quasi-birth-death processes(Elsevier, 2001) Fadıloğlu, M. M.; Yeralan, S.In this paper a spectral theory pertaining to Quasi-Birth–Death Processes (QBDs) is presented. The QBD, which is a generalization of the birth–death process, is a powerful tool that can be utilized in modeling many stochastic phenomena. Our theory is based on the application of a matrix polynomial method to obtain the steady-state probabilities in state-homogeneous finite-state QBDs. The method is based on finding the eigenvalue–eigenvector pairs that solve a matrix polynomial equation. Since the computational effort in the solution procedure is independent of the cardinality of the counting set, it has an immediate advantage over other solution procedures. We present and prove different properties relating the quantities that arise in the solution procedure. By also compiling and formalizing the previously known properties, we present a formal unified theory on the spectral properties of QBDs, which furnishes a formal framework to embody much of the previous work. This framework carries the prospect of furthering our understanding of the behavior the modeled systems manifest.Item Open Access Impurity coupled to an artificial magnetic field in a Fermi gas in a ring trap(American Physical Society, 2015) Ünal, F. N.; Hetényi, B.; Oktel, M. Ö.The dynamics of a single impurity interacting with a many-particle background is one of the central problems of condensed-matter physics. Recent progress in ultracold-atom experiments makes it possible to control this dynamics by coupling an artificial gauge field specifically to the impurity. In this paper, we consider a narrow toroidal trap in which a Fermi gas is interacting with a single atom. We show that an external magnetic field coupled to the impurity is a versatile tool to probe the impurity dynamics. Using a Bethe ansatz, we calculate the eigenstates and corresponding energies exactly as a function of the flux through the trap. Adiabatic change of flux connects the ground state to excited states due to flux quantization. For repulsive interactions, the impurity disturbs the Fermi sea by dragging the fermions whose momentum matches the flux. This drag transfers momentum from the impurity to the background and increases the effective mass. The effective mass saturates to the total mass of the system for infinitely repulsive interactions. For attractive interactions, the drag again increases the effective mass which quickly saturates to twice the mass of a single particle as a dimer of the impurity and one fermion is formed. For excited states with momentum comparable to number of particles, effective mass shows a resonant behavior. We argue that standard tools in cold-atom experiments can be used to test these predictions.
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