Rational Spectrum Research Articles

On rationality of spectrums for spectral sets in [formula omitted

Let Ω⊂R be a compact measurable set of measure 1 and with null boundary measure. We show that if Ω is a spectral set, then it admits a rational spectrum. The proof relies on the periodicity of spectrums shown in [16], and adopts the technique in [28] for analyzing zeros of exponential sums as well as the technique in [16] that relates the spectrum to the tiling of |1ˆΩ|2. The key technical ingredient we contribute that eventually leads to the result is the periodicity of values from an exponential sum on a certain subgroup of Z, which characterizes the torsion part of the spectrum. An immediate consequence of this result, together with periodicity and rationality results in [16,28], is the equivalence between the Fuglede conjecture in R and in Zn.

Mutual Information for Electromagnetic Information Theory Based on Random Fields

Traditional channel capacity based on the discrete spatial dimensions mismatches the continuous electromagnetic fields. For the wireless communication system in a limited region, the spatial discretization may results in information loss because the continuous field can not be perfectly recovered from the sampling points. Therefore, electromagnetic information theory based on spatially continuous electromagnetic fields becomes necessary to reveal the fundamental theoretical capacity bound of communication systems. In this paper, we propose analyzing schemes for the performance limit between continuous transceivers. Specifically, we model the communication process between two continuous regions by random fields. Then, for the white noise model, we use Mercer expansion to derive the mutual information between the source and the destination. For the close-form expression, an analytic method is introduced based on autocorrelation functions with rational spectrum. Moreover, the Fredholm determinant is used for the general autocorrelation functions to provide the numerical calculation scheme. Further works extend the white noise model to colored noise and discuss the mutual information under it. Finally, we build an ideal model with infinite-length source and destination which shows a strong correpsondence with the time-domain model in classical information theory. The mutual information and the capacity are derived through the spatial spectral density.

Suspension spectra of matrix algebras, the rank filtration, and rational noncommutative CW-spectra

In a companion paper (Arone et al. in Noncommutative CW-spectra as enriched presheaves on matrix algebras, arXiv:2101.09775, 2021) we introduced the stable infty -category of noncommutative CW-spectra, which we denoted mathtt {NSp}. Let {mathcal {M}} denote the full spectrally enriched subcategory of mathtt {NSp} whose objects are the non-commutative suspension spectra of matrix algebras. In Arone et al. (2021) we proved that mathtt {NSp} is equivalent to the infty -category of spectral presheaves on {mathcal {M}}. In this paper we investigate the structure of {mathcal {M}}, and derive some consequences regarding the structure of mathtt {NSp}. To begin with, we introduce a rank filtration of {mathcal {M}}. We show that the mapping spectra of {mathcal {M}} map naturally to the connective K-theory spectrum ku, and that the rank filtration of {mathcal {M}} is a lift of the classical rank filtration of ku. We describe the subquotients of the rank filtration in terms of spaces of direct-sum decompositions which also arose in the study of K-theory and of Weiss’s orthogonal calculus. We prove that the rank filtration stabilizes rationally after the first stage. Using this we give an explicit model of the rationalization of mathtt {NSp} as presheaves of rational spectra on the category of finite-dimensional Hilbert spaces and unitary transformations up to scaling. Our results also have consequences for the p-localization and the chromatic localization of {mathcal {M}}.

Yield-Adjusted Operation for Convolution Filter Denoising.

A perfect denoising for measurement shall remove noise, while keeping signal truth, so it is a dual-objective optimization of the signal yield and the noise residue. The frequency difference between the noise and signal is the basis of band-limited filter denoising. The root cause for the sharp peak denoise distortion is the insufficient spectrum sampling because of the scattered frequency distribution, which makes it hard to achieve dual-objective optimization. Thus, this article proposes a four-step operation of the signal yield adjustment for beyond the band-limited system. The first step is identifying the signal and noise levels in raw data, then adjusting the sampling density of high-signal level areas and enriching it by linear interpolation, then smoothing the reshaped profile, which is friendly to the filter, and finally, restoring the deformed one to its original form. An executable script function has fully achieved the whole operation. Some actual sharp spectra (Raman, NMR, laser-induced breakdown spectroscopy, and X-ray diffraction) make a comparison between the way with the Savitzky-Golay (SG) method and wavelet (multi-scale) denoising. The results show that all the effects are better than those of the SG filter, all estimations of the yield of signals are more than 99%, and the residue of noise is less than 10%. With multi-scale denoising, this operation is more targeted and gets more rational spectrum profiles─noise reduction without spectrum distortion.

Novel Fuzzy and Game Theory Based Clustering and Decision Making for VANETs

Different studies have recently emphasized the importance of deploying clustering schemes in Vehicular ad hoc Network (VANET) to overcome challenging problems related to scalability, frequent topology changes, scarcity of spectrum resources, maintaining clusters stability, and rational spectrum management. However, most of these studies addressed the clustering problem using conventional performance metrics while spectrum shortage, and the combination of spectrum trading and VANET architecture have not been tackled so far. Thus, this paper presents a new fuzzy logic based clustering control scheme to support scalability, enhance the stability of the network topology, motivate spectrum owners to share spectrum and provide efficient and cost-effective use of spectrum. Unlike existing studies, our context-aware scheme is based on multi-criteria decision making where fuzzy logic is adopted to rank the multi-attribute candidate nodes for optimizing the selection of cluster heads (CH)s. Criteria related to each candidate node include: received signal strength, speed of vehicle, vehicle location, spectrum price, reachability, and stability of node. Our model performs efficiently, exhibits faster recovery in response to topology changes and enhances the network efficiency life time.

A commutative P1-spectrum representing motivic cohomology over dedekind domains

We construct a motivic Eilenberg-MacLane spectrum with a highly structured multiplication over smooth schemes over Dedekind domains which represents Levine's motivic cohomology. The latter is defined via Bloch's cycle complexes. Our method is by gluing p-completed and rational parts along an arithmetic square. Hereby the finite coefficient spectra are obtained by truncated \'etale sheaves (relying on the now proven Bloch-Kato conjecture) and a variant of Geisser's version of syntomic cohomology, and the rational spectra are the ones which represent Beilinson motivic cohomology. As an application the arithmetic motivic cohomology groups can be realized as Ext-groups in a triangulated category of Tate sheaves with integral coefficients. These can be modelled as representations of derived fundamental groups. Our spectrum is compatible with base change giving rise to a formalism of six functors for triangulated categories of motivic sheaves over general base schemes including the localization triangle. Further applications include a generalization of the Hopkins-Morel isomorphism and a structure result for the dual motivic Steenrod algebra in the case where the coefficient characteristic is invertible on the base scheme.

Rational orthogonal calculus

We show that one can use model categories to construct rational orthogonal calculus. That is, given a continuous functor from vector spaces to based spaces one can construct a tower of approximations to this functor depending only on the rational homology type of the input functor, whose layers are given by rational spectra with an action of O(n). By work of Greenlees and Shipley, we see that these layers are classified by torsion {{mathrm{H}}}^*({{mathrm{B}}}SO(n))[O(n)/SO(n)]-modules.

Dynamic factor models with infinite-dimensional factor spaces: One-sided representations

Factor model methods recently have become extremely popular in the theory and practice of large panels of time series data. Those methods rely on various factor models which all are particular cases of the Generalized Dynamic Factor Model (GDFM) introduced in Forniet al. (2000). That paper, however, rests on Brillinger’s dynamic principal components. The corresponding estimators are two-sided filters whose performance at the end of the observation period or for forecasting purposes is rather poor. No such problem arises with estimators based on standard principal components, which have been dominant in this literature. On the other hand, those estimators require the assumption that the space spanned by the factors has finite dimension. In the present paper, we argue that such an assumption is extremely restrictive and potentially quite harmful. Elaborating upon recent results by Anderson and Deistler (2008a, b) on singular stationary processes with rational spectrum, we obtain one-sided representations for the GDFM without assuming finite dimension of the factor space. Construction of the corresponding estimators is also briefly outlined. In a companion paper, we establish consistency and rates for such estimators, and provide Monte Carlo results further motivating our approach.

Some reductions of the spectral set conjecture to integers

AbstractThe spectral set conjecture, also known as the Fuglede conjecture, asserts that every bounded spectral set is a tile and vice versa. While this conjecture remains open on${\mathbb R}^1$, there are many results in the literature that discuss the relations among various forms of the Fuglede conjecture on${\mathbb Z}_n$,${\mathbb Z}$and${\mathbb R}^1$and also the seemingly stronger universal tiling (spectrum) conjectures on the respective groups. In this paper, we clarify the equivalences between these statements in dimension one. In addition, we show that if the Fuglede conjecture on${\mathbb R}^1$is true, then every spectral set with rational measure must have a rational spectrum. We then investigate the Coven–Meyerowitz property for finite sets of integers, introduced in [1], and we show that if the spectral sets and the tiles in${\mathbb Z}$satisfy the Coven–Meyerowitz property, then both sides of the Fuglede conjecture on${\mathbb R}^1$are true.

Efficient and rational spectrum utilization in opportunistic OFDMA networks with imperfect CSI: a utility‐based top‐down approach

AbstractIn this paper, we consider the Orthogonal Frequency Division Multiple Access (OFDMA)‐based Opportunistic Radio (OR) networks, in which the nodes are able to adopt a subset of subcarriers (SCs) available in the considered network area, as well as the transmission rate and power associated with these SCs either for their uplink or downlink transmission. The idea of our top‐down approach to the efficient and rational spectrum utilization is to allow each node to first optimize the resources acquisition at OSI‐layer 2 (top), where the competition of nodes for available common resources is observed, and then to autonomously optimize the transmission parameters accounting for the radio channel conditions at the physical (down) layer. We assume that the nodes act independently based on their local channel quality information, which contains an error. In these optimization tasks, the efficiency and rationality mechanisms have been introduced in the objective (utility) functions, which reflect the throughput achieved, the network potential to serve multiple nodes, as well as pricing of resources and the power economy. Copyright © 2010 John Wiley & Sons, Ltd.

Automorphism Groups of Rational Circulant Graphs

The paper concerns the automorphism groups of Cayley graphs over cyclic groups which have a rational spectrum (rational circulant graphs for short). With the aid of the techniques of Schur rings it is shown that the problem is equivalent to consider the automorphism groups of orthogonal group block structures of cyclic groups. Using this observation, the required groups are expressed in terms of generalized wreath products of symmetric groups.

Rational Use of Antimicrobials in Patients with Severe Acute Pancreatitis

Infectious complications in severe acute pancreatitis are associated with considerable morbidity and mortality. The course of the disease is often protracted, and patients often stay in the hospital for several weeks. Diagnosis of infected pancreatic necrosis is difficult, and the treatment consists of source control and antibiotic treatment. Antibiotic use should be rational in terms of a rational indication, a rational spectrum, and a rational duration. Prophylactic antibiotics are not effective in reducing the incidence of (peri)-pancreatic infection in patients with severe disease (or even documented necrotizing pancreatitis). The only rational indication for antibiotics is documented infection. The spectrum of empirical antibiotics should include both aerobic and anaerobic gram-negative and gram-positive microorganisms. Also, fungal infections are often present in these patients, and antifungal coverage or even prophylaxis should be considered, especially if multiple risk factors for invasive candidiasis are present. Although initiation of antibiotics may be a difficult decision, stopping antibiotic therapy often proves to be even more difficult. Currently, no tools are available to guide antimicrobial treatment. Antibiotic use is only effective if proper source control has been established.

Quasioptimal estimates of the center frequency of a narrow-band normal random process with a Gaussian spectrum

Based on the well-known algorithm of the most likely estimates of the center frequencies of a normal narrow-band random process with a fractional rational spectrum, a similar algorithm for a process with a Gaussian spectrum is proposed and investigated by computer simulation. The practical importance of the development of algorithms for efficiently estimating the parameters of such processes is that they are a sufficiently close model of the output signal of laser Doppler systems used to study hydrodynamic and aerodynamic flows, measure the velocity of diffusely reflecting solid objects, and perform laser sensing of the atmosphere. The results of the studies show that the proposed algorithm does not shift estimates of the Doppler frequency (velocity), gives fairly accurate estimates (for a Doppler system with 25 interference fringes in the measuring volume at a measurement time equal to 100 Doppler periods, the error is only 0.12%), has high noise immunity, and evaluates the resulting errors.

An algebraic model for free rational G-spectra for connected compact Lie groups G

We show that the category of free rational G-spectra for a connected compact Lie group G is Quillen equivalent to the category of torsion differential graded modules over the polynomial ring H*(BG). The ingredients are the enriched Morita equivalences of Schwede and Shipley (Topology 42(1):103–153, 2003), the functors of Shipley (Am J Math 129:351–379, 2007) making rational spectra algebraic, Koszul duality and thick subcategory arguments based on the simplicity of the derived category of a polynomial ring.

Accuracy analysis of time domain maximum likelihood method and sample maximum likelihood method for errors-in-variables and output error identification

For identifying errors-in-variables models, the time domain maximum likelihood (TML) method and the sample maximum likelihood (SML) method are two approaches. Both methods give optimal estimation accuracy but under different assumptions. In the TML method, an important assumption is that the noise-free input signal is modelled as a stationary process with rational spectrum. For SML, the noise-free input needs to be periodic. It is interesting to know which of these assumptions contain more information to boost the estimation performance. In this paper, the estimation accuracy of the two methods is analyzed statistically for both errors-in-variables (EIV) and output error models (OEM). Numerical comparisons between these two estimates are also done under different signal-to-noise ratios (SNRs). The results suggest that TML and SML have similar estimation accuracy at moderate or high SNR for EIV. For OEM identification, these two methods have the same accuracy at any SNR.

Chains on suspension spectra

Some background is needed to understand why this functor deserves attention. There is a much simpler functor called N (normalised simplicial chains) from simplicial sets to integral chain complexes that computes integral homology, and one can just tensor with Q to compute rational homology. There is a dual complex N that calculates integral cohomology. This is equipped with a natural product N .X / N .X /! N .X / which is commutative up to homotopy but not on the nose. The theory of Steenrod operations shows that if we work integrally then neither N .X / nor any reasonable replacement can be given a strictly commutative product (even with the usual signs). Rationally, however, the situation is better: in [10] Sullivan developed a rational and simplicial version of de Rham theory giving a cochain complex  .X / with a strictly commutative product that computes the ordinary rational cohomology of X . This can be used as a starting point for the rich and powerful theory of rational homotopy (originally introduced by Quillen [8] using slightly different machinery). One can then stabilise and consider the category SQ of rational spectra, which makes things considerably simpler: it is well-known that the homotopy category of SQ is equivalent to the category of graded rational vector spaces. However, we can make things harder again by considering rational spectra with a ring structure or a group action. To handle these, we need to improve the homotopy classification of rational spectra to some kind of monoidal Quillen equivalence of SQ with a suitable model category ChQ of rational chain complexes.

Group actions and rational ideals

We develop the theory of rational ideals for arbitrary associative algebras R without assuming the standard finiteness conditions, noetherianness or the Goldie property. The Amitsur-Martindale ring of quotients replaces the classical ring of quotients which underlies the previous definition of rational ideals but is not available in a general setting. Our main result concerns rational actions of an affine algebraic group G on R. Working over an algebraically closed base field, we prove an existence and uniqueness result for generic rational ideals in the sense of Dixmier: for every G-rational ideal I of R, the closed subset of the rational spectrum RatR that is defined by I is the closure of a unique G-orbit in RatR. Under additional Goldie hypotheses, this was established earlier by Mœglin and Rentschler (in characteristic 0) and by Vonessen (in arbitrary characteristic), answering a question of Dixmier.

Accuracy Analysis of Time-domain Maximum Likelihood Method and Sample Maximum Likelihood Method for Errors-in-Variables Identification

The time domain maximum likelihood (TML) method and the sample maximum Likelihood (SML) method are two approaches for identifying errors-in-variables models. Both methods may give the optimal estimation accuracy (achieve Cramér-Rao lower bound) but in different senses. In the TML method, an important assumption is that the noise-free input signal is modeled as a stationary process with rational spectrum. For SML, the noise-free input needs to be periodic. It is interesting to know which of these assumptions contain more information to boost the estimation performance. In this paper, the estimation accuracy of the two methods is analyzed statistically. Numerical comparisons between the two estimates are also done under different signal-to-noise ratios (SNRs). The results suggest that TML and SML have similar estimation accuracy at moderate or high SNR.

Carbohydrate Structural Isomers Analyzed by Sequential Mass Spectrometry

Consistent with the goals of a comprehensive carbohydrate sequencing strategy, we extend earlier reports to include the characterization of structural (constitutional) isomers. Protocols were developed around ion trap instrumentation providing sequential mass spectrometry (MSn) and supported with automation and related computational tools. These strategies have been built on the principle that for a single structure all product spectra upon sequential fragmentation are reproducible and each stage represents a rational spectrum of its precursor; i.e., all major fragments should be accounted for. Anomalous ions at any stage are clues indicating the presence of structural isomers. Gas-phase isolation and subsequent fragmentation of such ions provide an opportunity to specifically resolve selected structures for their detailed characterization. Importantly, some isomers were not detected following MS2 and required multiple (MSn>2) stages for their characterization. Derivatization remains critical to position substructures in a glycan array since product ions carry fragmentation "scars" throughout the MSn tree. Equally as important are the pathway relationships between each stage and the greater yield of fragments with the smaller number of oscillators. Applications were directed to the structural isomers in ovalbumin and IgG, where, in the latter case, several previously unreported glycans were detected. Procedures were supported with bioinformatics tools for assimilating structure from the MSn data sets.

Spectral Analysis of Signals [Book Review

This book is a revised and expanded version of Introduction to Spectral Analysis by the same authors, a popular text and reference on the subject since its publication in 1997. The updated book consists of six chapters and four appendices. The main text of each chapter discusses the principles and fundamentals, while the complements included at the end of each chapter examine more advanced topics. Much of the background material need for the book, such as linear algebra, Cramer-Rao bound analysis, and model order selection, is provided in the appendices. Some of the topics covered include: Fourier-based nonparametric methods; parametric methods for rational spectra; parametric methods for line spectra; the filter-bank approach; and beamforming and direction-of-arrival estimation problems. This timely book is suitable for senior undergraduate and graduate students as well as engineers and other professionals. The material can easily fit in a one-semester spectral analysis course.