Delta Function Research Articles

A Data Centric Approach Towards Server Time Series Model Compression

Effective model compression plays a pivotal role in mitigating the computational and interpretational challenges inherent in the domain of time series forecasting. In this study, we introduce an innovative data-centric methodology tailored to identify a representative data subset from the entirety of the dataset. This chosen representative segment forms the cornerstone for the training of proficient time series forecasting models. Furthermore, our investigation unveils a compelling outcome of this approach—a substantial reduction in the size of time series forecasting models when trained with this selected representative data segment. This model compression strategy results in a remarkable 56.31% decrease in storage consumption, a discovery of considerable significance for optimizing resources and enhancing scalability in time series forecasting. By distilling the dataset to its fundamental components through our data-centric approach, we aim to enhance both computational efficiency and the interpretability of the resultant models. This paper introduces a pioneering technique to tackle the challenges associated with data volume and model complexity in the field of time series forecasting, offering potential pathways for more efficient and insightful modelling in this domain.

Generalized One-Dimensional Periodic Potential Wells Tending to the Dirac Delta Potential

The solution of a quantum periodic potential in solid state physics is relevant because one can understand how electrons behave in a corresponding crystal. In this paper, the analytical solution of the energy formulation for a one-dimensional periodic potential that meets certain boundary conditions is developed in a didactic and detailed way. In turn, the group speed and effective mass are also calculated from the transcendental energy equation of a potential V=V(x). From this, a comparison is made with periodic potentials with known analytical solutions, such as the Dirac delta, as well as rectangular and triangular potentials. Finally, some limits are proposed in which these periodic potentials of the form V=V(x) approach the periodic Dirac delta potential of positive intensity. Therefore, the calculations described in this paper can be used as the basis for more-complex one-dimensional potentials and related simulations.

A new method for digital processing cardio signals using the wavelet function

Biomedical signal processing algorithms offer a variety of opportunities to improve performance. In this study, a new wavelet delta- function (WDF) method was proposed to effectively detect pathology from ECG signals. In this method, the delta function is determined for each value of the ECG signal. The coefficients 0, 1, and -1 are determined by summing this WDF. The level of detection of the difference between pathology and norm using data in the form of this vector allows changes in the diagnosis to be made more accurately than conventional methods. This is a new way of processing ECG signals by converting graphical data into vectors. The new method created a bar-code - like image. It has become possible to see myocardial infarction (MI), one of the diseases of the cardiovascular system, more effectively in real numbers of invisible peaks on the ECG. This allows for early detection and accurate diagnosis. The model we propose is an effective way to analyze an ECG. The WDF model we proposed can be used for early assessment of disease in biomedical signal processing.

Customized and Optimized Treatment of Extended-Spectrum-Beta-Lactamase Bacteria for Individual Patients

This paper discusses the optimization of customized intravenous-drip therapy regimens for individual patients infected with antibiotic-resistant bacteria, namely Extended-Spectrum-Beta-Lactamase (ESBL) bacteria. By utilizing Ordinary Differential Equations (ODE) to model the system’s dynamics, evaluated the efficiencies of the four most typical types of pulse functions, namely Dirac delta, sine, trapezoid, and normal distribution functions, as well as the time required for each treatment to suppress the pathogen population. The results revealed that a trapezoid pulse function for intravenous delivery of antibiotics is most favorable. Subsequently, infection severity was randomized by altering the bacterial and Beta-Lactamase population. The efficiency of each randomized regimen was evaluated using a predetermined score matrix that assessed the dosing length, dosing interval, maximum rate of antibiotic infusion, net consumption of antibiotics, and the total number of required treatments. Radar charts and box plot models were employed to present the findings. It was discovered that for more severely infected patients with a higher initial population and growth rate, it is important to decrease the time interval between subsequent doses and increase the time of maximum infusion rate. Conversely, for mild infections, the time interval between doses should be increased, and the time of maximum infusion rate should be decreased. It should be noted that maximizing the infusion rate does not enhance treatment efficiency. Furthermore, in comparison to the initial infected bacteria population size, the growth rate, and Bla production rate play more important roles in impacting treatment efficacy.

Dirac Delta as a Useful Technical Tool in Modelling Signals but Hard to Think About It as a Physical Signal Itself

Dirac Delta as a Useful Technical Tool in Modelling Signals but Hard to Think About It as a Physical Signal Itself

Você esqueceu mesmo uma função Delta?

This is the third article in a series dedicated to examining the use of the delta function in the physics literature. Here we discuss the appearance of a delta function in the second derivative of the wave function if its derivative is discontinuous. When this happens it is often claimed in the literature that the proposed wave function must be eigenfunction of an operator that contains a delta function and, in some cases, it is rejected. We study in this paper what we believe are alternatives ways to deal with this situation.

Delta function of the Fourier transform

In the analysis of the double window Fourier transform, it is found that both components of the direct transformation contain an equal amount of information sufficient to restore the original time signal. A double conversion diagram has been constructed. For any initial signal, there is a conjugate signal having the same frequency spectrum. Analysis of the harmonic signal transformation showed that at an arbitrary phase of the harmonic, the frequency representation contains two functions: conditionally unipolar with a maximum at the initial frequency and bipolar with a zero value at the specified frequency. For classical Fourier transformations with an infinite limit of integration, these functions turn into two delta functions with the same properties. The practical application of the found patterns is currently limited by the lack of an algorithm that allows the use of data from only one component.

Constructions of mismatched binary periodic complementary pairs

Two mismatched binary sequences are called a mismatched binary periodic complementary pair (MB-PCP) if the sum of their periodic cross-correlation functions is a delta function. Such a pair is a generalization of the well-known Golay sequence pair, and has applications in the scenario where the mismatched filter is used. Up to now, there are only MB-PCPs of length $ 3q $ with peak value 4 or $ 2q-6 $, where $ q \equiv 3\; (\bmod \,4 ) $ is a prime. The objective of this paper is to construct four families of MB-PCPs of length $ pq $ via the generalized cyclotomy and the Chinese Remainder Theorem, where $ p $ and $ q $ are two different odd primes.

Horizontal oscillations of the wood sawing support during the cutting process in a wood sawing line using a vertical bandsaw

The wooden trolley (in the sawing line using a vertical bandsaw) during work generally moves by iron wheels on two rails placed on the concrete floor. Due to the effect of noticeable cutting force applied by the saw blade and the vehicle's uneven weight on the wheels, it causes the vehicle to vibrate during the horizontal movement, affecting the stability of the wood sawing circuit. By modeling the problem of vibration of beams located on Winkler elastic foundations, subjected to mobile concentrated loads, the article has built a system of differential equations of vibration of two rails, giving an expression to calculate the vibration amplitude of the swan plane. The influence of the flexural rigidity EI of the rails and the elastic stiffness k of foundation on the vibration amplitude of the cutting plane is studied and analyzed. The use of the Dirac Delta function and the employed analytical solution allow the designers to evaluate the accuracy and reliability of the calculated results.

An immersed boundary velocity correction method combined with virtual body-fitted grid for simulation of incompressible flows

In this work, a virtual body-fitted grid is introduced into the velocity correction-based immersed boundary method (IBM) to simulate incompressible flows. The impact of the immersed boundary is indirectly transmitted to the flow field via a virtual body-fitted grid. In this method, the fractional step technique consisting of the predictor and the corrector is adopted. The prediction step is executed on the Eulerian mesh, and the correction step is done on the virtual grid to fulfill the no-slip boundary condition. After the correction step, the corrected velocity field on the virtual grid is then assigned to that on the Eulerian mesh to update the flow field. Being able to adjust the grid spacing flexibly, the virtual body-fitted grid alleviates the shortcomings of the conventional IBM that uses the smooth Dirac delta function to associate Lagrangian points with their surrounding Eulerian points. As a result, the present method is easy to apply to non-uniform Cartesian grids, which is inapplicable to the conventional IBM with the smooth Dirac delta function. Numerical experiments concerning flow past a circular cylinder and a NACA0012 airfoil demonstrate the advantages of the present method, i.e., fewer Lagrangian points are required to avoid the streamline penetration of boundary and the range of “diffuse interface” can be narrowed by reducing the normal grid spacing of the virtual body-fitted grid to improve numerical results on a coarse mesh. In addition, an accuracy assessment on the decaying vortex problem reveals that the present IBM has a second-order accuracy.

Cesaro summability of Taylor series in higher order weighted Dirichlet-type spaces

For a positive integer \(m\) and a finite non-negative Borel measure \(\mu\) on the unit circle, we study the Hadamard multipliers of higher order weighted Dirichlet-type spaces \(\mathcal H_{\mu, m}\). We show that if \(\alpha\gt\frac{1}{2}\), then for any \(f\) in \(\mathcal H_{\mu, m}\) the sequence of generalized Cesaro sums \(\{\sigma_n^{\alpha}[f]\}\) converges to \(f\). We further show that if \(\alpha=\frac{1}{2}\) then for the Dirac delta measure supported at any point on the unit circle, the previous statement breaks down for every positive integer \(m\).

Концептуальна рамка резильєнтної поведінки інформаційних систем

This study focuses on the development of a conceptual framework that delineates the resilience of information systems through the analysis of their key operational stages in the context of countering prevalent threats. These stages encompass preliminary security measures, active counteraction against threats, as well as strategies for adaptation and recovery following security breaches. The realization of this objective necessitates the identification and formulation of fundamental goals and objectives associated with resilience, which play a crucial role in the neutralization of potential threats and significantly influence risk management policies, emphasizing the importance of a comprehensive analysis of the key functions of organizations. Throughout the research, special attention was devoted to the creation of models for resilient and non-resilient behavioral scenarios of systems and their responses to threats. The employment of the delta function facilitated a detailed examination of the stages of resilient response to threats, including the processes of adaptation and subsequent recovery. The work contributes significantly to the field of information security, presenting a multifaceted approach to advancing the resilience of information systems.

Stabilizability for nonautonomous linear parabolic equations with actuators as distributions

The stabilizability of a general class of abstract parabolic-like equations is investigated, with a finite number of actuators. This class includes the case of actuators given as delta distributions located at given points in the spatial domain of concrete parabolic equations. A stabilizing feedback control operator is constructed and given in explicit form. Then, an associated optimal control is considered and the corresponding Riccati feedback is investigated. Results of simulations are presented showing the stabilizing performance of both explicit and Riccati feedbacks.

Friedel Oscillations in a One-dimensional Non-interacting Electron Gas in the Presence of Two Impuritites

Using the linear response theory, we analyze Friedel oscillations in a one-dimensional non-interacting electron gas in the presence of two impurities with different potential strengths. The impurities potentials are modeled using Dirac delta function, as well as Lorentzian and Gaussian distribution functions. Our findings show that density oscillations are strongly sensitive to both the distance between the impurities and their respective potential strengths. Keywords: one-dimensional electron gas, one impurity, two impurities, Friedel oscillations.

Paradoxes of Competition in Periodic Environments: Delta Functions in Ecological Models

We demonstrate a basic technique for simplifying time-periodic competition models, which is based on the utilization of periodic delta functions as population growth rates. We show that the Poincare mapping splits into a sequence of one-dimensional mappings. The study of the corresponding stable equilibria allows us to make conclusions concerning the coexistence and selection of the family of competitors. In particular, in “all vs. all” systems, for one of the populations to dominate, it is enough to surpass the others with a certain margin, and the correspondent stock constant does not depend on the number of competitors. We present paradoxical examples, where (1) a low-productive population can displace a highly productive one, (2) the displacement is non-transitive, (3) the coexistence is non-transitive. We also show how the delta functions can be utilized for the analysis of a “predator–prey” system.

STATIONARY MODE OF FLUID FLOW IN THE PIPELINE IN THE PRESENCE OF ASSOCIATED SAMPLING AND PUMPING

Pipeline transportation of oil and petroleum products plays a significant role in the transport system of the Russian Federation. When operating oil pipelines, reliability and energy efficiency are the most important criteria. To achieve these indicators, it is necessary to perform optimal technological modes. When constructing models of technological regimes, a methodology is used in which the movement of oil and petroleum products is considered as steady. Based on this assumption, the throughput, pump operating modes, strength characteristics and other parameters are determined. During the operation of the equipment, almost every day it is necessary to switch from one mode to another, which entails unsteady fluid movement for some time. At the same time, the unsteady mode makes up a small part of the total operating time of the system.The beginning of studies of unsteady fluid processes in pipes dates back to the last quarter of the 19th century. N. E. Zhukovsky made a fundamental contribution to the study of this problem. His classic work on water hammer in water pipes was the beginning for the creation of a large number of works in pipe hydraulics. The problem was solved for an ideal elastic fluid, i.e. without taking into account the friction forces against the pipe walls. The resulting formula, according to which the pressure increment is proportional to the velocity, has received numerous experimental confirmations for cases when it is possible to neglect pressure losses to overcome hydraulic resistances. N. E. Zhukovsky also obtained a formula for determining the sound velocity of a liquid droplet in a pipe with elastic walls. It is proved that this speed depends on the pipe material and its thickness.In the future, a large number of works were carried out to study the flow of liquid and gas in round pipes. The fundamental works of I. A. Charny, who obtained a mathematical model in the form of a system of partial differential equations of hyperbolic type, proved fruitful in this direction. Linearization of the equation, taking into account viscosity and hydraulic resistance allowed the researcher to obtain solutions to a number of engineering problems.The introduction of the Dirac function in the description of selectionpumping at specified points made it possible to describe processes in complex pipelines with one partial differential equation.This article discusses a horizontal section of the pipeline with sampling and pumping points. A stationary process for a drip liquid is considered. The problem is solved using the sine-Fourier transform.

Celestial holography and AdS3/CFT2 from a scaling reduction of twistor space

Celestial amplitudes obtained from Mellin transforming 4d momentum space scattering amplitudes contain distributional delta functions, hindering the application of conventional CFT techniques. In this paper, we propose to bypass this problem by recognizing Mellin transforms as integral transforms projectivizing certain components of the angular momentum. It turns out that the Mellin transformed wavefunctions in the conformal primary basis can be regarded as representatives of certain cohomology classes on the minitwistor space of the hyperbolic slices of 4d Minkowski space. Geometrically, this amounts to treating 4d Minkowski space as the embedding space of AdS3. By considering scattering of such on-shell wavefunctions on the projective spinor bundle ℙ\U0001d54a of Euclidean AdS3, we bypass the difficulty of the distributional properties of celestial correlators using the traditional AdS3/CFT2 dictionary and find conventional 2d CFT correlators for the scaling reduced Yang-Mills theory living on the hyperbolic slices. In the meantime, however, one is required to consider action functionals on the auxiliary space ℙ\U0001d54a, which introduces additional difficulties. Here we provide a framework to work on the projective spinor bundle of hyperbolic slices, obtained from a careful scaling reduction of the twistor space of 4d Minkowski spacetime.

Стационарная вращательно-симметричная конвекция при изотахофорезе в круговой цилиндрической области с недеформируемой свободной непроницаемой боковой поверхностью

The results of a study of the isotachophoresis process convective stability are presented. Isotachophoresis is the method of separation of a multicomponent mixture to individual components by an external electric field. It is assumed that the process of isotachophoresis occur in an infinite vertical circular cylinder with free side boundaries. The problem of the stationary (monotonic, neutral) concentration gravitational convection occurrence is transform to the study of the solution stability for the problem linearized on the solution of the original problem corresponding to the mechanical equilibrium of the final stage of the isotachophoresis process, in which the zones of the individual components of the mixture move with the identical constant velocity in a stationary flow. It is indicated that the main contribution to the stability/instability of the equilibrium is made by concentration gradients in the vicinity of the boundaries between the zones of individual components. Due to the complexity of constructing an exact solution corresponding to mechanical equilibrium, the exact solution replaced by an asymptotic solution (piecewise constant concentration distribution), which, in turn, is an exact solution of the asymptotic version of the transport under action of an electric field component equations (diffusion-free model). The choice of piecewise constant mechanical equilibrium leads to concentration gradients in the form of delta functions and a linear boundary value problem with delta-like coefficients to determine the critical parameters of instability. An exact solution of the problem (a system of ordinary differential equations with boundary conditions) and, using the method of one-dimensional perturbations, an exact dispersion relation for determining critical parameters (in the case of vanishingly small diffusion) is constructed.

Justification of the exact asymptotics of the fundamental solution for a degenerate parabolic equation with a small parameter

AbstractIn this paper, we propose a scheme for constructing the asymptotics of the fundamental solution of a linear degenerate parabolic equation with a small parameter. The asymptotics is constructed using the operator representation of the Dirac delta function and the non‐oscillating Wentzel‐Kramers‐Brillouin (WKB) method. In addition, a method to justify the obtained asymptotics by proving the convergence of the formal series that arises when using the WKB method is shown. Several examples of degenerate parabolic equations are considered, for which a fundamental solution is constructed, and its main properties are indicated.

On torsion contribution to chiral anomaly via Nieh–Yan term

In this note we present a solution to the question of whether or not, in the presence of torsion, the topological Nieh–Yan term contributes to chiral anomaly. The integral of Nieh–Yan term is non-zero if topology is non-trivial; the manifold has a boundary or vierbeins have singularities. Noting that singular Nieh–Yan term could be written as a sum of delta functions, we argue that the heat kernel expansion cannot end at finite steps. This leads to a sinusoidal dependence on the Nieh–Yan term and the UV cut-off of the theory (or alternatively the minimum length of spacetime). We show this ill-behaved dependence can be removed if a quantization condition on length scales is applied. It is expected as the Nieh–Yan term can be derived as the difference of two Chern class integrals (i.e. Pontryagin terms). On the other hand, in the presence of a cosmological constant, we find that indeed the Nieh–Yan term contributes to the index with a dimensionful anomaly coefficient that depends on the de Sitter length or equivalently inverse Hubble rate. We find similar result in thermal field theory where the anomaly coefficient depends on temperature. In both examples, the anomaly coefficient depends on IR cut-off of the theory. Without singularities, the Nieh–Yan term can be smoothly rotated away, does not contribute to topological structure and consequently does not contribute to chiral anomaly.