Partial Fourier Series Research Articles

The expressivity of classical and quantum neural networks on entanglement entropy

Analytically continuing the von Neumann entropy from Rényi entropies is a challenging task in quantum field theory. While the n-th Rényi entropy can be computed using the replica method in the path integral representation of quantum field theory, the analytic continuation can only be achieved for some simple systems on a case-by-case basis. In this work, we propose a general framework to tackle this problem using classical and quantum neural networks with supervised learning. We begin by studying several examples with known von Neumann entropy, where the input data is generated by representing TrρAn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {Tr}}\\rho _A^n$$\\end{document} with a generating function. We adopt KerasTuner to determine the optimal network architecture and hyperparameters with limited data. In addition, we frame a similar problem in terms of quantum machine learning models, where the expressivity of the quantum models for the entanglement entropy as a partial Fourier series is established. Our proposed methods can accurately predict the von Neumann and Rényi entropies numerically, highlighting the potential of deep learning techniques for solving problems in quantum information theory.

Exploring the Potential of Mixed Fourier Series in Signal Processing Applications Using One-Dimensional Smooth Closed-Form Functions with Compact Support: A Comprehensive Tutorial

This paper studies and analyzes the approximation of one-dimensional smooth closed-form functions with compact support using a mixed Fourier series (i.e., a combination of partial Fourier series and other forms of partial series). To explore the potential of this approach, we discuss and revise its application in signal processing, especially because it allows us to control the decreasing rate of Fourier coefficients and avoids the Gibbs phenomenon. Therefore, this method improves the signal processing performance in a wide range of scenarios, such as function approximation, interpolation, increased convergence with quasi-spectral accuracy using the time domain or the frequency domain, numerical integration, and solutions of inverse problems such as ordinary differential equations. Moreover, the paper provides comprehensive examples of one-dimensional problems to showcase the advantages of this approach.

Effect of data encoding on the expressive power of variational quantum-machine-learning models

Quantum computers can be used for supervised learning by treating parametrized quantum circuits as models that map data inputs to predictions. While a lot of work has been done to investigate the practical implications of this approach, many important theoretical properties of these models remain unknown. Here, we investigate how the strategy with which data are encoded into the model influences the expressive power of parametrized quantum circuits as function approximators. We show that one can naturally write a quantum model as a partial Fourier series in the data, where the accessible frequencies are determined by the nature of the data-encoding gates in the circuit. By repeating simple data-encoding gates multiple times, quantum models can access increasingly rich frequency spectra. We show that there exist quantum models which can realize all possible sets of Fourier coefficients, and therefore, if the accessible frequency spectrum is asymptotically rich enough, such models are universal function approximators.

Partial Fourier series on compact Lie groups

In this note, we investigate the partial Fourier series on a product of two compact Lie groups. We give necessary and sufficient conditions for a sequence of partial Fourier coefficients to define a smooth function or a distribution. As an application, we will study conditions for the global solvability of an evolution equation defined on T1×S3 and we will show that some properties of this evolution equation can be obtained from a conjugated constant-coefficient equation.

Gibbs-Phenomenon-Reduced Digital PWM for Power Amplifiers Using Pulse Modulation

The pulse-modulated polar transmitter (PMPT) uses radio-frequency (RF) phase modulated signal with pulse-width modulated envelope to drive a power amplifier (PA) as a switch to achieve high linearity and high efficiency. However, as the signal bandwidth increases, various non-idealities limit the performance of the PMPT architecture. In this paper, a Gibbs-phenomenon reduction filter is proposed with a partial Fourier series of digital pulse-width modulation (PWM) such that the pulsed signal is more immune to the distortion in nonlinear high-efficiency RF PAs. For validation, a prototype transmitter was tested with a 20-MHz bandwidth 256-QAM 5G new radio (NR) signal at 2.14 GHz. The proposed method achieved a drain efficiency of 28.3%, error vector magnitude (EVM) of 1.67%, and adjacent channel leakage ratio (ACLR) of -45.4 dBc at an output power of 18 dBm using the prototype transmitter. Linearity requirements for base stations were met without the use of any digital predistortion methods.

Meta-analysis of day-of-week variation of acute aortic rupture or dissection.

We performed a meta-analysis to assess the presence of a day-of-week rhythmic variability of acute aortic rupture or dissection (AARD) onset. Eligible studies were observational studies enrolling patients with AARD and reporting day-of-week variation of AARD. Study-specific estimates, i.e. day-of-week incidence of AARD, were combined using the random-effects model. Chronobiological analysis was performed by applying a partial Fourier series to pooled day-of-week incidence by using the inverse-variance weighted least-squares method. We identified 9 eligible studies enrolling a total of 28,036 patients with AARD. Pooled incidence of AARD was 12.8% on Sunday, 15.9% on Monday, 14.8% on Tuesday, 15.1% on Wednesday, 14.7% on Thursday, 14.1% on Friday, and 12.1% on Saturday. Chronobiological analysis identified a significant (P=0.0098) day-of-week pattern in the occurrence of AARD with a peak on Monday and a nadir on Saturday. Pooled analysis demonstrated significantly more incidence on Monday than on Saturday (relative risk: 1.247; 95% CI: 1.131 to 1.374; P=0.012). Incidence of AARD was 12.8%, 15.9%, 14.8%, 15.1%, 14.7%, 14.1%, and 12.1%, on Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, and Saturday, respectively. A significant day-of-week pattern in the occurrence of AARD with a peak on Monday and a nadir on Saturday was identified with significantly more incidence on Monday than on Saturday.

A meta-analysis of monthly variation in occurrence of abdominal aortic aneurysm rupture.

We performed a meta-analysis to assess the presence of an annual rhythmic variability of ruptured abdominal aortic aneurysm (RAAA) onset. Eligible studies were observational studies enrolling patients with RAAA and reporting monthly incidence of RAAA. Study-specific estimates, i. e. monthly incidence of RAAA, were combined using the random-effects model. Chronobiological analysis was performed by applying a partial Fourier series to pooled monthly incidence by using the weighted least-squares method. We identified 14 eligible studies enrolling a total of 3,798 patients with RAAA. Pooled monthly incidence of RAAA was 8.7 % in January, 7.7 % in February, 8.7 % in March, 7.3 % in April, 7.8 % in May, 7.2 % in June, 7.0 % in July, 7.0 % in August, 8.1 % in September, 8.8 % in October, 8.4 % in November, and 8.3 % in December. Chronobiological analysis identified a significant (p = 0.0020) annual pattern in the occurrence of RAAA with a peak in December to January and a nadir in June to July. Pooled analysis demonstrated significantly more incidence in December than in June (p = 0.03) as well as in January than in July (p = 0.05). A significant annual pattern in the occurrence of RAAA with a peak in December to January and a nadir in June to July was identified with significantly more incidence in December than in June and in January than in July.

Meta-Analysis of Circadian Variation in the Onset of Acute Aortic Dissection

Circadian variation in the onset of acute aortic dissection (AAD) has been less investigated than other cardiovascular diseases. We performed a meta-analysis to assess the presence of an circadian rhythmic variability of AAD onset. Eligible studies were observational studies enrolling patients with AAD and reporting a circadian variation in AAD. Study-specific estimates, that is, 2-hour incidence of AAD, were combined using the random-effects model. Chronobiological analysis (analysis of circadian rhythmicity) was performed by applying a partial Fourier series to the pooled 2-hour incidence using the weighted least-squares method. We identified 7 eligible studies enrolling a total of 1,827 patients with AAD. Pooled 2-hour period incidence of AAD was 3.4% in 0:00 to 2:00, 4.8% in 2:00 to 4:00, 5.4% in 4:00 to 6:00, 9.6% in 6:00 to 8:00, 13.8% in 8:00 to 10:00, 11.1% in 10:00 to 12:00, 8.1% in 12:00 to 14:00, 8.9% in 14:00 to 16:00, 8.8% in 16:00 to 18:00, 7.0% in 18:00 to 20:00, 8.1% in 20:00 to 22:00), and 5.5% in 22:00 to 24:00. Chronobiological analysis (nonlinear Fourier rhythm analysis) identified a significant (p = 0.0082) circadian pattern in the occurrence of AAD with a peak in 8:00 to 10:00 and a nadir in 0:00 to 2:00. Pooled analysis demonstrated significantly more incidence in 8:00 to 10:00 than in 0:00 to 2:00 (risk ratio 3.59, 95% confidence interval 2.19 to 5.90, p<0.00001). The incidence of AAD was 8.8%, 15.5%, 25.0%, 17.7%, 16.1%, and 13.8% in 0:00 to 4:00, 4:00 to 8:00, 8:00 to 12:00, 12:00 to 16:00, 16:00 to 20:00, and 20:00 to 24:00, respectively. A significant circadian pattern was found in the occurrence of AAD with a peak in 8:00 to 10:00 and a nadir in 0:00 to 2:00.

Approximation and prediction of movements of the control points of soil dams using Fourier series

The modeling process of settling of concrete top parapet of soil dam reservoir of Khmelnitsky NPP and forecasting movement of individual points is considered in the article. It is proposed to analyze the results of high-precision geometric leveling in 2 stages: at first is allocated trend component using polynomial approximation and then the remaining deviations from trend curve approximating by partial Fourier series

Fourier decomposition of properly elliptic boundary value problems in a half-plane

For a general class of second-order elliptic boundary value problems in the lower half-plane, we show that the existence and uniqueness of solutions in Lp Sobolev spaces is reduced to the invertibility of the ordinary differential operators obtained by Fourier decomposition. This terminology refers to the partial Fourier series expansion in the case of horizontally periodic solutions and to the partial Fourier transform otherwise. The problem is straightforward when p = 2 and, in the periodic case, the same question on a strip with finite width can also be quickly settled by indirect arguments irrespective of \({p \in (1, \infty )}\). However, in the half-plane, the infinite depth raises serious difficulties when p ≠ 2. These difficulties are overcome by writing the problem as a first-order system and using existing abstract results about operator valued Fourier multipliers. In that approach, the randomized boundedness of the resolvent becomes the central issue.

Seasonal pattern of peptic ulcer hospitalizations: analysis of the hospital discharge data of the Emilia-Romagna region of Italy

BackgroundPrevious studies have reported seasonal variation in peptic ulcer disease (PUD), but few large-scale, population-based studies have been conducted.MethodsTo verify whether a seasonal variation in cases of PUD (either compicated or not complicated) requiring acute hospitalization exists, we assessed the database of hospital admissions of the region Emilia Romagna (RER), Italy, obtained from the Center for Health Statistics, between January 1998 and December 2005. Admissions were categorized by sex, age (<65, 65-74, ≥ 75 yrs), site of PUD lesion (stomach or duodenum), main complication (hemorrhage or perforation), and final outcome (intended as fatal outcome: in-hospital death; nonfatal outcome: patient discharged alive). Temporal patterns in PUD admissions were assessed in two ways, considering a) total counts per single month and season, and b) prevalence proportion, such as the monthly prevalence of PUD admissions divided by the monthly prevalence of total hospital admissions, to assess if the temporal patterns in the raw data might be the consequence of seasonal and annual variations in hospital admissions per se in the region. For statistical analysis, the χ2 test for goodness of fit and inferential chronobiologic method (Cosinor and partial Fourier series) were used.ResultsOf the total sample of PUD patients (26,848 [16,795 males, age 65 ± 16 yrs; 10,053 females, age 72 ± 15 yrs, p < 0.001)], 7,151 were <65 yrs of age, 8,849 between 65 and 74 yrs of age, and 10,848 ≥ 75 yrs of age. There were more cases of duodenal (DU). (89.8%) than gastric ulcer (GU) (3.6%), and there were 1,290 (4.8%) fatal events. Data by season showed a statistically difference with the lowest proportion of PUD hospital admissions in summer (23.3%) (p < 0.001), for total cases and rather all subgroups. Chronobiological analysis identified three major peaks of PUD hospitalizations (September-October, January-February, and April-May) for the whole sample (p = 0.035), and several subgroups, with nadir in July. Finally, analysis of the monthly prevalence proportions yielded a significant (p = 0.025) biphasic pattern with a main peak in August-September-October, and a secondary one in January-February.ConclusionsA seasonal variation in PUD hospitalization, characterized by three peaks of higher incidence (Autumn, Winter, and Spring) is observed. When data corrected by monthly admission proportions are analyzed, late summer-autumn and winter are confirmed as higher risk periods. The underlying pathophysiologic mechanisms are unknown, and need further studies. In subjects at higher risk, certain periods of the year could deserve an appropriate pharmacological protection to reduce the risk of PUD hospitalization.

Monday preference in onset of takotsubo cardiomyopathy

Objective Acute cardiovascular events show definite temporal patterns of occurrence. Takotsubo cardiomyopathy (TTC) has been recently shown to exhibit a seasonal (summer) and circadian (morning) temporal distribution. The aim of this study, based on the database of a multicenter Italian network, was to investigate the possible existence of a weekly pattern of onset of TTC. Methods The study included all cases of TTC admitted to the coronary care unit of 8 referral cardiac centers in Italy (five in Southern Italy and three in Northern Italy, respectively), belonging to the Takotsubo Italian Network (January 2002-December 2008). Day of admission was categorized into seven 1-day intervals according by week, and chronobiological analysis was performed by partial Fourier series. Results The database included 112 patients with TTC (92.9% females). The weekly distribution identified the highest number of cases on Monday and the lowest on Saturday. Chronobiologic analysis yielded a rhythmic pattern with a significant peak on Monday ( P = .036). Conclusions This study confirms a Monday preference for TTC occurrence on Monday, similar to that reported for acute myocardial infarction. Stress of starting weekly day life activities, could play a triggering role.

Seasonal and weekly patterns of hospital admissions for nonfatal and fatal myocardial infarction

Objective This retrospective study, based on the database of hospital admissions of the region Emilia-Romagna [RER], Italy, was aimed to confirm the existence of a seasonal or weekly pattern of hospital admission of acute myocardial infarction (AMI) and to verify possible differences between nonfatal or fatal cases. Methods The study included all cases of patients with AMI hospitalized between 1998 and 2006. Day of admission was categorized, respectively, into four 3-month intervals, into twelve 1-month intervals, and into seven 1-day intervals for statistical analysis, performed by χ 2 test goodness of fit and partial Fourier series on total cases, males, females, and nonfatal and fatal cases. Results The database included 64 191 cases of AMI (62.9% males, 12.3% fatal). Acute myocardial infarction was most frequent in winter and least in summer ( P < .0001). The highest number of cases was recorded in January and the lowest in July ( P < .0001). Chronobiologic analysis showed winter peaks for total cases (January, P = .035), females (December, P = .009), and fatal cases (January, P < .001). Acute myocardial infarction was most frequent on Monday and least on Sunday ( P < .0001). Comparing observed vs expected events, there was a significantly higher frequency of cases on weekdays and reduced on weekends, for total ( P < .0001), nonfatal ( P < .0001), and fatal cases ( P = .0001). Conclusions This study confirms a significantly higher frequency of AMI admissions in winter and on a Monday. No difference in the frequency of nonfatal vs fatal events, depending of patients' admissions on weekdays or weekends, was found.

Disturbance of Circadian Rhythms in Analgosedated Intensive Care Unit Patients with and without Craniocerebral Injury

Melatonin, cortisol, heart rate, blood pressure, spontaneous motor activity, and body temperature follow stable circadian rhythms in healthy individuals. These circadian rhythms may be influenced or impaired by the loss of external zeitgebers during analgosedation, critical illness, continuous therapeutic intervention in the intensive care unit (ICU), and cerebral injury. This prospective, observational, clinical study examined 24 critically ill analgo‐sedated patients, 13 patients following surgery, trauma, or acute respiratory distress (ICU), and 11 patients with acute severe brain injury following trauma or cerebral hemorrhage (CCI). Blood samples for the determination of melatonin and cortisol were obtained from each patient at 2 h intervals for 24 h beginning at 18:00 h on day 1 and ending 16:00 h on day 2. Blood pressure, heart rate, body temperature, and spontaneous motor activity were monitored continuously. Level of sedation was assessed using the Ramsey Sedation Scale. The severity of illness was assessed using the APACHE‐II‐score. The time series data were analyzed by rhythm analysis with the Chronos‐Fit program, using partial Fourier series with up to six harmonics. The 24 h profiles of all parameters from both groups of patients were greatly disturbed/abolished compared to the well‐known rhythmic 24 h patterns in healthy controls. These rhythm disturbances were more pronounced in patients with brain injury. The results of this study provide evidence for a pronounced disturbance of the physiological temporal organization in ICU patients. The relative contribution of analgosedation and/or brain injury, however, is a point of future investigation.

Free Vibration Analysis of Spherical Caps Using a G.D.Q. Numerical Solution

In this paper we present the frequency evaluation of spherical shells by means of the generalized differential quadrature method (G.D.Q.M.), an effective numerical procedure which pertains to the class of generalized collocation methods. The shell theory used in this study is a first-order shear deformation theory with transverse shearing deformations and rotatory inertia included. The shell governing equations in terms of mid-surface displacements are obtained and, after expansion in partial Fourier series of the circumferential coordinate, solved with the G.D.Q.M. Several comparisons are made with available results, showing the reliability and modeling capability of the numerical scheme in argument.

The G. D. Q. method for the harmonic dynamic analysis of rotational shell structural elements

This paper deals with the modal analysis of rotational shell structures by means of the numerical solution technique known as the Generalized Differential Quadrature (G. D. Q.) method. The treatment is conducted within the Reissner first order shear deformation theory (F. S. D. T.) for linearly elastic isotropic shells. Starting from a non-linear formulation, the compatibility equations via Principle of Virtual Works are obtained, for the general shell structure, given the internal equilibrium equations in terms of stress resultants and couples. These equations are subsequently linearized and specialized for the rotational geometry, expanding all problem variables in a partial Fourier series, with respect to the longitudinal coordinate. The procedure leads to the fundamental system of dynamic equilibrium equations in terms of the reference surface kinematic harmonic components. Finally, a one-dimensional problem, by means of a set of five ordinary differential equations, in which the only spatial coordinate appearing is the one along meridians, is obtained. This can be conveniently solved using an appropriate G. D. Q. method in meridional direction, yielding accurate results with an extremely low computational cost and not using the so-called "delta-point" technique.

Seasonal variation in the onset of acute pancreatitis.

A circannual variation in the onset of several acute diseases, mostly dealing with cardiovascular system, has been reported. The present study was to verify the possible existence of a seasonal variability in the onset of acute pancreatitis. All patients consecutively admitted to the Hospital of Ferrara, Italy, between January 1998 to December 2002, whose discharge diagnosis was acute pancreatitis, were considered. According to the time of admission, cases were categorized into twelve 1-mo intervals and in four periods by season. chi (2) test for goodness of fit and partial Fourier series were used for statistical analysis. During the study period, 549 cases of acute pancreatitis were observed. A significant peak of higher incidence was found in March-May, both for total population, males and subgroups with and without cholelithiasis or alcoholism. Fourier analysis showed the existence of a circannual rhythmic pattern with its main peak in March (95% C.L.: February-April, P = 0.005), and a secondary one in September. Death occurred more frequently in December-February, compared to the other periods (P = 0.029), and chronobiologic analysis yielded a seasonal peak in November-December (P<0.001). This study shows the existence of a circannual variation in the onset of acute pancreatitis, with a significantly higher frequency of events in the spring, especially for patients with cholelithiasis or alcoholism. Moreover, events occurring during the colder months seem to be characterized by a higher mortality rate.

Seasonal variation in onset of pulmonary embolism is independent of patients' underlying risk comorbid conditions.

As for many cardiovascular events, pulmonary embolism (PE) is not randomly distributed over time, but shows rhythmic patterns. The purpose of this study was to investigate whether such temporal pattern of occurrence varied in subgroups of patients according to different risk comorbid conditions. All cases of PE observed at the Hospital of Ferrara, Italy, from 1998 to 2001, were considered. After determination of the day of onset, the population was grouped by gender and the most common underlying risk comorbid conditions, e.g., deep vein thrombosis (DVT), neoplasms, cardiomyopathies, traumas/surgical operations, diabetes mellitus, pulmonary diseases, hypertension, cerebrovascular diseases, heart failure, hematologic diseases. For statistical analysis, chi-square test for goodness of fit and partial Fourier series were used. A total of 784 cases (mean age 71 +/- 14 years) were included. Frequency of onset was higher in winter for total population (p = 0.002), men (p = 0.004), DVT (p = 0.001), pulmonary disease (p = 0.008), cardiomyopathies (p = 0.011), and major traumas/surgical operations (p = 0.049). Chronobiologic analysis identified a winter peak for total population (p = 0.008), men (p < 0.001), DVT (p = 0.006), pulmonary diseases (p = 0.017), and hypertension (p = 0.026). This study confirms the winter peak of PE and provides evidence that it is not influenced by the underlying clinical conditions, but probably by endogenous variations.

Classical Limit of Expectation Values in a Wave Packet Involving Few Stationary States

Expectation values of physical quantities in a wave packet involving few stationary states in an infinite square well are calculated. Explicit results show that the expectation values in the classical limit go over to the corresponding classical quantity in the form of the arithmetic mean (in mathematical term, the Fejer's average) of the partial Fourier series converging to the classical quantity. The number of the stationary states is that of the partial Fourer series in the Fejer's average. The quantum uncertainty is then demonstrated to have a classical counterpart.

Circadian pattern of emergency calls: Implications for ED organization

A circadian variation has been shown in the onset of acute medical diseases and we postulate that there is a circadian variation in emergency calls. We reviewed the 20,858 emergency calls addressed to the Emergency Coordinating Unit of the Hospital of Ferrara, Italy, from January 1 to December 31, 1998. Precise determination of the time of calls was available from the recordings. Total calls and subgroups by different diseases were categorized into 24 one-hour increments and analyzed for circadian rhythmicity by applying a partial Fourier series. A circadian variation was found for all subgroups, except for alcoholic intoxication. There was a peak frequency of calls in the morning hours for cardiologic, respiratory, and neurologic disease. There was a peak frequency of calls in the afternoon for trauma, neoplastic diseases, and acute poisoning. Organization of quantity and quality of Emergency Department (ED) staff should take into account the increased demand of specific facilities during certain hours of the day.(Am J Emerg Med 2002;20:282-286. Copyright 2002, Elsevier Science (USA). All rights reserved.)