Generalized Rayleigh-quotient Formulas for the Real Parts, Imaginary Parts, and Moduli of the Eigenvalues of Diagonalizable Matrices

In the present paper, generalized Rayleigh-quotient formulas for the real parts, imaginary parts, and moduli of the eigenvalues of diagonalizable matrices are derived. These formulas are new and correspond to similar formulas for the eigenvalues of self-adjoint matrices obtained recently. Numerical examples underpin the theoretical findings

Differential calculus for p-norms of complex-valued vector functions with applications

AbstractFor complex-valued n-dimensional vector functions t↦s(t), supposed to be sufficiently smooth, the differentiability properties of the mapping t↦∥s(t)∥p at every point t=t0∈R0+≔{t∈R|t⩾0} are investigated, where ∥·∥p is the usual vector norm in Cn resp. Rn, for p∈[1,∞]. Moreover, formulae for the first three right derivatives D+k∥s(t)∥p,k=1,2,3 are determined. These formulae are applied to vibration problems by computing the best upper bounds on ∥s(t)∥p in certain classes of bounds. These results cannot be obtained by the methods used so far. The systematic use of the differential calculus for vector norms, as done here for the first time, could lead to major advances also in other branches of mathematics and other sciences

Rayleigh-Quotient Representation of the Real Parts, Imaginary Parts, and Moduli of the Eigenvalues of General Matrices

In the present paper, formulas for the Rayleigh-quotient representation of the real parts, imaginary parts, and moduli of the eigenvalues of general matrices are obtained that resemble corresponding formulas for the eigenvalues of self-adjoint matrices. These formulas are of interest in Linear Algebra and in the theory of linear dynamical systems. The key point is that a weighted scalar product is used that is defined by means of a special positive definite matrix. As applications, one obtains convexity properties of newly-defined numerical ranges of a matrix. A numerical example underpins the theoretical findings

NT-proBNP or Self-Reported Functional Capacity in Estimating Risk of Cardiovascular Events After Noncardiac Surgery

ImportanceNearly 16 million surgical procedures are conducted in North America yearly, and postoperative cardiovascular events are frequent. Guidelines suggest functional capacity or B-type natriuretic peptides (BNP) to guide perioperative management. Data comparing the performance of these approaches are scarce.ObjectiveTo compare the addition of either N-terminal pro-BNP (NT-proBNP) or self-reported functional capacity to clinical scores to estimate the risk of major adverse cardiac events (MACE).Design, Setting, and ParticipantsThis cohort study included patients undergoing inpatient, elective, noncardiac surgery at 25 tertiary care hospitals in Europe between June 2017 and April 2020. Analysis was conducted in January 2023. Eligible patients were either aged 45 years or older with a Revised Cardiac Risk Index (RCRI) of 2 or higher or a National Surgical Quality Improvement Program, Risk Calculator for Myocardial Infarction and Cardiac (NSQIP MICA) above 1%, or they were aged 65 years or older and underwent intermediate or high-risk procedures.ExposuresPreoperative NT-proBNP and the following self-reported measures of functional capacity were the exposures: (1) questionnaire-estimated metabolic equivalents (METs), (2) ability to climb 1 floor, and (3) level of regular physical activity.Main Outcome and MeasuresMACE was defined as a composite end point of in-hospital cardiovascular mortality, cardiac arrest, myocardial infarction, stroke, and congestive heart failure requiring transfer to a higher unit of care.ResultsA total of 3731 eligible patients undergoing noncardiac surgery were analyzed; 3597 patients had complete data (1258 women [35.0%]; 1463 (40.7%) aged 75 years or older; 86 [2.4%] experienced a MACE). Discrimination of NT-proBNP or functional capacity measures added to clinical scores did not significantly differ (Area under the receiver operating curve: RCRI, age, and 4MET, 0.704; 95% CI, 0.646-0.763; RCRI, age, and 4MET plus floor climbing, 0.702; 95% CI, 0.645-0.760; RCRI, age, and 4MET plus physical activity, 0.724; 95% CI, 0.672-0.775; RCRI, age, and 4MET plus NT-proBNP, 0.736; 95% CI, 0.682-0.790). Benefit analysis favored NT-proBNP at a threshold of 5% or below, ie, if true positives were valued 20 times or more compared with false positives. The findings were similar for NSQIP MICA as baseline clinical scores.Conclusions and relevanceIn this cohort study of nearly 3600 patients with elevated cardiovascular risk undergoing noncardiac surgery, there was no conclusive evidence of a difference between a NT-proBNP–based and a self-reported functional capacity–based estimate of MACE risk.Trial RegistrationClinicalTrials.gov Identifier: NCT0301693

Introduction to the discrete Fourier series considering both mathematical and engineering aspects -A linear algebra approach

Abstract: The discrete Fourier series is a valuable tool developed and used by mathematicians and engineers alike. One of the most prominent applications is signal processing. Usually, it is important that the signals be transmitted fast, for example, when transmitting images over large distances such as between the moon and the earth or when generating images in computer tomography. In order to achieve this, appropriate algorithms are necessary. In this context, the fast Fourier transform (FFT) plays a key role which is an algorithm for calculating the discrete Fourier transform (DFT); this, in turn, is tightly connected with the discrete Fourier series. The last one itself is the discrete analog of the common (continuous-time) Fourier series and is usually learned by mathematics students from a theoretical point of view. The aim of this expository/pedagogical paper is to give an introduction to the discrete Fourier series for both mathematics and engineering students. It is intended to expand the purely mathematical view; the engineering aspect is taken into account by applying the FFT to an example from signal processing that is small enough to be used in class-room teaching and elementary enough to be understood also by mathematics students. The MATLAB program is employed to do the computations

Two-sided bounds on some output-related quantities in linear stochastically excited vibration systems with application of the differential calculus of norms

A linear stochastic vibration model in state-space form, $\dot{x}(t) = A x(t)+b(t), \; x(0)=x_0,$ with output equation $x_S(t)=S x(t)$ is investigated, where A is the system matrix and b(t) is the white noise excitation. The output equation $x_S(t)=S x(t)$ can be viewed as a transformation of the state vector x(t) that is mapped by the rectangular matrix S into the output vector x(t). It is known that, under certain conditions, the solution x(t) is a random vector that can be completely described by its mean vector, $m_x(t):=m_{x(t)}$, and its covariance matrix, $P_x(t):=P_{x(t)}$. If matrix A is asymptotically stable, then $m_x(t) \rightarrow 0 \; (t \rightarrow \infty )$ and $P_{x_S}(t) \rightarrow P_S \; (t \rightarrow \infty )$, where $P_S$ is a positive (semi-)definite matrix. Similar results will be derived for some output-related quantities. The obtained results are of special interest to applied mathematicians and engineers