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Journal articles on the topic 'Function extrema'

Author: Grafiati

Published: 4 June 2021

Last updated: 10 February 2022

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1

Rieger, J. H. "Hypersurfaces of extremal slope." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 133, no. 2 (April 2003): 449–65. http://dx.doi.org/10.1017/s030821050000247x.

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The hypersurface of extremal slope consists of extrema of the gradient magnitude along integral curves of the gradient vector field of a smooth function. We study the singularities, and other geometric features, of hypersurfaces of extremal slope associated with generic functions and with one-parameter families of functions representing generic solutions to the heat equation.

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2

Hirschhorn, Michael D. "91.35 Extrema of a symmetric function." Mathematical Gazette 91, no. 521 (July 2007): 264–67. http://dx.doi.org/10.1017/s0025557200181653.

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3

Beaulieu, N. C. "Extrema of sin x/x function." Electronics Letters 31, no. 15 (July 20, 1995): 1215. http://dx.doi.org/10.1049/el:19950850.

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4

Dorosiewicz, Slawomir. "On the extrema of integrable functions." International Journal of Mathematics and Mathematical Sciences 27, no. 9 (2001): 547–53. http://dx.doi.org/10.1155/s0161171201005488.

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This paper contains the definition of the extremum of integrable functions (e.g., the mode of density function). It seems to be a generalization of well-known standard definition and can be applied in estimation theory to extend the maximum likelihood method.

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5

El-Khoury, M., O. D. Crisalle, and R. Longchamp. "Discrete Transfer-function Zeros and Step-response Extrema." IFAC Proceedings Volumes 26, no. 2 (July 1993): 537–42. http://dx.doi.org/10.1016/s1474-6670(17)49000-8.

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6

Galkin, O. E., and S. Yu Galkina. "Global extrema of the Delange function, bounds for digital sums and concave functions." Sbornik: Mathematics 211, no. 3 (March 2020): 336–72. http://dx.doi.org/10.1070/sm9143.

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7

Guilloux, Antonin, and Julien Marché. "Volume function and Mahler measure of exact polynomials." Compositio Mathematica 157, no. 4 (April 2021): 809–34. http://dx.doi.org/10.1112/s0010437x21007016.

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We study a class of two-variable polynomials called exact polynomials which contains $A$ -polynomials of knot complements. The Mahler measure of these polynomials can be computed in terms of a volume function defined on the vanishing set of the polynomial. We prove that the local extrema of the volume function are on the two-dimensional torus and give a closed formula for the Mahler measure in terms of these extremal values. This formula shows that the Mahler measure of an irreducible and exact polynomial divided by $\pi$ is greater than the amplitude of the volume function. We also prove a K-theoretic criterion for a polynomial to be a factor of an $A$ -polynomial and give a topological interpretation of its Mahler measure.

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8

Fernando, José F. "On the set of local extrema of a subanalytic function." Collectanea Mathematica 71, no. 1 (February 22, 2019): 1–24. http://dx.doi.org/10.1007/s13348-019-00245-6.

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9

Cvetkovic, Dragos, Pierre Hansen, and Vera Kovacevic-Vujcic. "On some interconnections between combinatorial optimization and extremal graph theory." Yugoslav Journal of Operations Research 14, no. 2 (2004): 147–54. http://dx.doi.org/10.2298/yjor0402147c.

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The uniting feature of combinatorial optimization and extremal graph theory is that in both areas one should find extrema of a function defined in most cases on a finite set. While in combinatorial optimization the point is in developing efficient algorithms and heuristics for solving specified types of problems, the extremal graph theory deals with finding bounds for various graph invariants under some constraints and with constructing extremal graphs. We analyze by examples some interconnections and interactions of the two theories and propose some conclusions.

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10

Veroy, Boris S. "An optimal algorithm for search of extrema of a bimodal function." Journal of Complexity 2, no. 4 (December 1986): 323–32. http://dx.doi.org/10.1016/0885-064x(86)90010-5.

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Veroy, Boris S. "Optimal search algorithm for extrema of a discrete periodic bimodal function." Journal of Complexity 5, no. 2 (June 1989): 238–50. http://dx.doi.org/10.1016/0885-064x(89)90006-x.

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12

Grin?, A. A., and L. A. Cherkas. "Extrema of the Andronov-Hopf function of a polynomial Lienard system." Differential Equations 41, no. 1 (January 2005): 50–60. http://dx.doi.org/10.1007/s10625-005-0134-1.

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13

Le Bihan, J. "Locations and amplitudes of the extrema of the sinx/x function." Circuits, Systems and Signal Processing 16, no. 2 (March 1997): 241–45. http://dx.doi.org/10.1007/bf01183277.

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14

van der Hofstad, Remco, and Harsha Honnappa. "Large deviations of bivariate Gaussian extrema." Queueing Systems 93, no. 3-4 (October 15, 2019): 333–49. http://dx.doi.org/10.1007/s11134-019-09632-z.

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Abstract We establish sharp tail asymptotics for componentwise extreme values of bivariate Gaussian random vectors with arbitrary correlation between the components. We consider two scaling regimes for the tail event in which we demonstrate the existence of a restricted large deviations principle and identify the unique rate function associated with these asymptotics. Our results identify when the maxima of both coordinates are typically attained by two different versus the same index, and how this depends on the correlation between the coordinates of the bivariate Gaussian random vectors. Our results complement a growing body of work on the extremes of Gaussian processes. The results are also relevant for steady-state performance and simulation analysis of networks of infinite server queues.

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15

Aksomaitis, A., and A. Jokimaitis. "Asymptotics of the Joint Distribution of Multivariate Extrema." Nonlinear Analysis: Modelling and Control 6, no. 1 (June 5, 2001): 3–8. http://dx.doi.org/10.15388/na.2001.6.1.15220.

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Let Wn and Zn be a bivariate extrema of independent identically distributed bivariate random variables with a distribution function F. in this paper the nonuniform estimate of convergence rate of the joint distribution of the normalized and centralized minima and maxima is obtained.

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16

LINDERHED, ANNA. "VARIABLE SAMPLING OF THE EMPIRICAL MODE DECOMPOSITION OF TWO-DIMENSIONAL SIGNALS." International Journal of Wavelets, Multiresolution and Information Processing 03, no. 03 (September 2005): 435–52. http://dx.doi.org/10.1142/s0219691305000932.

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Previous work on empirical mode decomposition in two dimensions typically generates a residue with many extrema points. In this paper we propose an improved method to decompose an image into a number of intrinsic mode functions and a residue image with a minimum number of extrema points. We further propose a method for the variable sampling of the two-dimensional empirical mode decomposition. Since traditional frequency concept is not applicable in this work, we introduce the concept of empiquency, shortform for empirical mode frequency, to describe the signal oscillations. The very special properties of the intrinsic mode functions are used for variable sampling in order to reduce the number of parameters to represent the image. This is done blockwise using the occurrence of extrema points of the intrinsic mode function to steer the sampling rate of the block. A method of using overlapping 7 × 7 blocks is introduced to overcome blocking artifacts and to further reduce the number of parameters required to represent the image. The results presented here shows that an image can be successfully decomposed into a number of intrinsic mode functions and a residue image with a minimum number of extrema points. The results also show that subsampling offers a way to keep the total number of samples generated by empirical mode decomposition approximately equal to the number of pixels of the original image.

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17

Caban, Paweł, and Jakub Rembieliński. "Relativistic Einstein-Podolsky-Rosen Correlations." Open Systems & Information Dynamics 18, no. 02 (June 2011): 165–73. http://dx.doi.org/10.1142/s123016121100011x.

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We analyse the correlation function in Einstein-Podolsky-Rosen experiment with relativistic massive particles. We show that in the wide range of parameters the correlation function has local extrema as a function of momentum of the EPR particles.

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18

Ponomareva, Inna D. "Computing the Correlation Function of Random Analog Signals Given by Their Extrema." Journal of Automation and Information Sciences 29, no. 2-3 (1997): 13–16. http://dx.doi.org/10.1615/jautomatinfscien.v29.i2-3.30.

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19

Kavvadias, Dimitris J., and Michael N. Vrahatis. "Locating and Computing All the Simple Roots and Extrema of a Function." SIAM Journal on Scientific Computing 17, no. 5 (September 1996): 1232–48. http://dx.doi.org/10.1137/s1064827594265666.

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20

Vafaei Sadr, A., and S. M. S. Movahed. "Clustering of local extrema in Planck CMB maps." Monthly Notices of the Royal Astronomical Society 503, no. 1 (March 16, 2021): 815–29. http://dx.doi.org/10.1093/mnras/stab368.

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ABSTRACT The clustering of local extrema will be exploited to examine Gaussianity, asymmetry, and the footprint of the cosmic-string network on the CMB observed by Planck. The number density of local extrema (npk for peak and ntr for trough) and sharp clipping (npix) statistics support the Gaussianity hypothesis for all component separations. However, the pixel at the threshold reveals a more consistent treatment with respect to end-to-end simulations. A very tiny deviation from associated simulations in the context of trough density, in the threshold range ϑ ∈ [−2–0] for NILC and CR component separations, are detected. The unweighted two-point correlation function, Ψ, of the local extrema illustrates good consistency between different component separations and corresponding Gaussian simulations for almost all available thresholds. However, for high thresholds, a small deficit in the clustering of peaks is observed with respect to the Planck fiducial ΛCDM model. To put a significant constraint on the amplitude of the mass function based on the value of Ψ around the Doppler peak (θ ≈ 70–75 arcmin), we should consider ϑ ≲ 0.0. The scale-independent bias factors for the peak above a threshold for large separation angle and high threshold level are in agreement with the value expected for a pure Gaussian CMB. Applying the npk, ntr, Ψpk − pk and Ψtr − tr measures on the tessellated CMB map with patches of 7.52 deg2 size prove statistical isotropy in the Planck maps. The peak clustering analysis puts the upper bound on the cosmic-string tension, Gμ(up) ≲ 5.59 × 10−7, in SMICA.

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21

Galkin, O. E., and S. Yu Galkina. "Global extrema of the Gray Takagi function of Kobayashi and binary digital sums." Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki 27, no. 1 (March 2017): 17–25. http://dx.doi.org/10.20537/vm170102.

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22

Hassani, M., M. Bayat, and H. Teimoori. "Proof without Words: Extrema of the Function a cos t + b sin t." Mathematics Magazine 77, no. 4 (October 1, 2004): 259. http://dx.doi.org/10.2307/3219283.

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23

Hassani, M., M. Bayat, and H. Teimoori. "Proof Without Words: Extrema of the Function a cos t + b sin t." Mathematics Magazine 77, no. 4 (October 2004): 259. http://dx.doi.org/10.1080/0025570x.2004.11953262.

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24

Newman, D. J., and T. J. Rivlin. "A continuous function whose divided differences at the Chebyshev extrema are all zero." Constructive Approximation 2, no. 1 (December 1986): 221–23. http://dx.doi.org/10.1007/bf01893428.

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25

Gomes, Abel, and José Morgado. "A Generalized Regula Falsi Method for Finding Zeros and Extrema of Real Functions." Mathematical Problems in Engineering 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/394654.

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Many zero-finding numerical methods are based on the Intermediate Value Theorem, which states that a zero of a real function is bracketed in a given interval if and have opposite signs; that is, . But, some zeros cannot be bracketed this way because they do not satisfy the precondition . For example, local minima and maxima that annihilate may not be bracketed by the Intermediate Value Theorem. In this case, we can always use a numerical method for bracketing extrema, checking then whether it is a zero of or not. Instead, this paper introduces a single numerical method, calledgeneralized regula falsi(GRF) method to determine both zeros and extrema of a function. Consequently, it differs from the standardregula falsi methodin that it is capable of finding any function zero in a given interval even when the Intermediate Value Theorem is not satisfied.

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26

Elsinga, Gerrit E., Takashi Ishihara, and Julian C. R. Hunt. "Extreme dissipation and intermittency in turbulence at very high Reynolds numbers." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476, no. 2243 (November 2020): 20200591. http://dx.doi.org/10.1098/rspa.2020.0591.

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Extreme dissipation events in turbulent flows are rare, but they can be orders of magnitude stronger than the mean dissipation rate. Despite its importance in many small-scale physical processes, there is presently no accurate theory or model for predicting the extrema as a function of the Reynolds number. Here, we introduce a new model for the dissipation probability density function (PDF) based on the concept of significant shear layers, which are thin regions of elevated local mean dissipation. At very high Reynolds numbers, these significant shear layers develop layered substructures. The flow domain is divided into the different layer regions and a background region, each with their own PDF of dissipation. The volume-weighted regional PDFs are combined to obtain the overall PDF, which is subsequently used to determine the dissipation variance and maximum. The model yields Reynolds number scalings for the dissipation maximum and variance, which are in agreement with the available data. Moreover, the power law scaling exponent is found to increase gradually with the Reynolds numbers, which is also consistent with the data. The increasing exponent is shown to have profound implications for turbulence at atmospheric and astrophysical Reynolds numbers. The present results strongly suggest that intermittent significant shear layer structures are key to understanding and quantifying the dissipation extremes, and, more generally, extreme velocity gradients.

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27

Shah, Syed Ghoos Ali, Saqib Hussain, Akhter Rasheed, Zahid Shareef, and Maslina Darus. "Application of Quasisubordination to Certain Classes of Meromorphic Functions." Journal of Function Spaces 2020 (October 14, 2020): 1–8. http://dx.doi.org/10.1155/2020/4581926.

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Inequalities play a fundamental role in many branches of mathematics and particularly in real analysis. By using inequalities, we can find extrema, point of inflection, and monotonic behavior of real functions. Subordination and quasisubordination are important tools used in complex analysis as an alternate of inequalities. In this article, we introduce and systematically study certain new classes of meromorphic functions using quasisubordination and Bessel function. We explore various inequalities related with the famous Fekete-Szego inequality. We also point out a number of important corollaries.

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28

Milinovich, Micah B. "Moments of the Riemann zeta-function at its relative extrema on the critical line." Bulletin of the London Mathematical Society 43, no. 6 (July 1, 2011): 1119–29. http://dx.doi.org/10.1112/blms/bdr047.

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29

Truten, I. V., and A. Yu. Korchin. "The top-quark polarization beyond the Standard Model in electron–positron annihilation." International Journal of Modern Physics A 34, no. 12 (April 30, 2019): 1950067. http://dx.doi.org/10.1142/s0217751x19500672.

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Electron–positron annihilation into a pair of top quarks is considered at the energy of the future collider CLIC. Polarization components of the top quark are calculated with the [Formula: see text] and [Formula: see text] interactions which follow from the Lagrangian of the effective field theory of the Standard Model. The polarization vector as a function of the scattering angle is calculated and averaged components of polarization are analyzed as functions of anomalous coupling constants. Extrema of these observables (maxima, minima and saddle points) are studied as functions of the [Formula: see text] energy.

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30

Sahin, Ismet. "Minimization over randomly selected lines." An International Journal of Optimization and Control: Theories & Applications (IJOCTA) 3, no. 2 (June 17, 2013): 111–19. http://dx.doi.org/10.11121/ijocta.01.2013.00167.

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This paper presents a population-based evolutionary optimization method for minimizing a given cost function. The mutation operator of this method selects randomly oriented lines in the cost function domain, constructs quadratic functions interpolating the cost function at three different points over each line, and uses extrema of the quadratics as mutated points. The crossover operator modifies each mutated point based on components of two points in population, instead of one point as is usually performed in other evolutionary algorithms. The stopping criterion of this method depends on the number of almost degenerate quadratics. We demonstrate that the proposed method with these mutation and crossover operations achieves faster and more robust convergence than the well-known Differential Evolution and Particle Swarm algorithms.

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31

LIU, HUI, and JIA-RONG LI. "THE DIELECTRIC FUNCTION EXCITED BY ρNN TENSOR COUPLING IN NUCLEAR MATTER." Modern Physics Letters A 19, no. 11 (April 10, 2004): 855–62. http://dx.doi.org/10.1142/s0217732304013659.

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The dielectric function of nuclear matter excited by ρNN tensor coupling has been studied in the framework of finite temperature field theory. The induced current mechanism has been introduced to explain the three extrema on the dielectric function curve, of which one is in the space-like region and the other two are in the time-like region. It points out that the tensor coupling contributes much more large amplitude than the vector coupling and plays a more important role on the time-like region compared with its effect on the space-like region.

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32

Beucher, Serge. "GEODESIC RECONSTRUCTION, SADDLE ZONES & HIERARCHICAL SEGMENTATION." Image Analysis & Stereology 20, no. 3 (May 3, 2011): 137. http://dx.doi.org/10.5566/ias.v20.p137-141.

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The morphological reconstruction based on geodesic operators, is a powerful tool in mathematical morphology. The general definition of this reconstruction supposes the use of a marker function f which is not necessarily related to the function g to be built. However, this paper deals with operations where the marker function is defined from given characteristic regions of the initial function f, as it is the case, for instance, for the extrema (maxima or minima) but also for the saddle zones. Firstly, we show that the intuitive definition of a saddle zone is not easy to handle, especially when digitised images are involved. However, some of these saddle zones (regional ones also called overflow zones) can be defined, this definition providing a simple algorithm to extract them. The second part of the paper is devoted to the use of these overflow zones as markers in image reconstruction. This reconstruction provides a new function which exhibits a new hierarchy of extrema. This hierarchy is equivalent to the hierarchy produced by the so-called waterfall algorithm. We explain why the waterfall algorithm can be achieved by performing a watershed transform of the function reconstructed by its initial watershed lines. Finally, some examples of use of this hierarchical segmentation are described.

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33

Bakirov, N. K. "The Extrema of the Distribution Function for the Ratio of Quadratic Forms in Gaussian Variables." Theory of Probability & Its Applications 40, no. 3 (January 1996): 542–46. http://dx.doi.org/10.1137/1140058.

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34

Le Bihan, J. "Straightforward recursive relations for computing the locations and amplitudes of extrema of sinx/x function." Electronics Letters 34, no. 25 (1998): 2385. http://dx.doi.org/10.1049/el:19981655.

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35

Norkin, Vladimir. "A Stochastic Smoothing Method for Nonsmooth Global Optimization." Cybernetics and Computer Technologies, no. 1 (March 31, 2020): 5–14. http://dx.doi.org/10.34229/2707-451x.20.1.1.

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The paper presents the results of testing the stochastic smoothing method for global optimization of a multiextremal function in a convex feasible subset of Euclidean space. Preliminarily, the objective function is extended outside the admissible region so that its global minimum does not change, and it becomes coercive. The smoothing of a function at any point is carried out by averaging the values of the function over some neighborhood of this point. The size of the neighborhood is a smoothing parameter. Smoothing eliminates small local extrema of the original function. With a sufficiently large value of the smoothing parameter, the averaged function can have only one minimum. The smoothing method consists in replacing the original function with a sequence of smoothed approximations with vanishing to zero smoothing parameter and optimization of the latter functions by contemporary stochastic optimization methods. Passing from the minimum of one smoothed function to a close minimum of the next smoothed function, we can gradually come to the region of the global minimum of the original function. The smoothing method is also applicable for the optimization of nonsmooth nonconvex functions. It is shown that the smoothing method steadily solves test global optimization problems of small dimensions from the literature.

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36

HU, XIYUAN, SILONG PENG, and WEN-LIANG HWANG. "ESTIMATION OF INSTANTANEOUS FREQUENCY PARAMETERS OF THE OPERATOR-BASED SIGNAL SEPARATION METHOD." Advances in Adaptive Data Analysis 01, no. 04 (October 2009): 573–86. http://dx.doi.org/10.1142/s1793536909000308.

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The main function of the operator-based signal separation approach is to construct an operator from a signal and use it to decompose the signal into two subcomponents. The procedure involves two steps: estimating the operator, and decomposing the signal into two subcomponents. Existing approaches estimate the operator's parameters from the local extrema of a signal. In contrast, we show that the parameters can be estimated by adopting a variational approach. Because the proposed approach imposes a global constraint on the operator's parameter, the estimated parameters are more robust than those derived from the local extrema of the signal. We also compare the signal separation results with those obtained by using the empirical mode decomposition (EMD) method.

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37

Bettencourt, João H., and Frédéric Dias. "Wall pressure and vorticity in the intermittently turbulent regime of the Stokes boundary layer." Journal of Fluid Mechanics 851 (July 25, 2018): 479–506. http://dx.doi.org/10.1017/jfm.2018.520.

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In this paper we study the wall pressure and vorticity fields of the Stokes boundary layer in the intermittently turbulent regime through direct numerical simulation (DNS). The DNS results are compared to experimental measurements and a good agreement is found for the mean and fluctuating velocity fields. We observe maxima of the turbulent kinetic energy and wall shear stress in the early deceleration stage and minima in the late acceleration stage. The wall pressure field is characterized by large fluctuations with respect to the root mean square level, while the skewness and kurtosis of the wall pressure show significant deviations from their Gaussian values. The wall vorticity components show different behaviours during the cycle: for the streamwise component, positive and negative fluctuations have the same probability of occurrence throughout the cycle while the spanwise fluctuations favour negative extrema in the acceleration stage and positive extrema in the deceleration stage. The wall vorticity flux is a function of the wall pressure gradients. Vorticity creation at the wall reaches a maximum at the beginning of the deceleration stage due to the increase of uncorrelated wall pressure signals. The spanwise vorticity component is the most affected by the oscillations of the outer flow. These findings have consequences for the design of wave energy converters. In extreme seas, wave induced fluid velocities can be very high and extreme wall pressure fluctuations may occur. Moreover, the spanwise vortical fields oscillate violently in a wave cycle, inducing strong interactions between vortices and the device that can enhance the device motion.

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38

Heydari, Mohammad Hossein, Mohammad Reza Mahmoudi, Zakieh Avazzadeh, and Dumitru Baleanu. "Chebyshev cardinal functions for a new class of nonlinear optimal control problems with dynamical systems of weakly singular variable-order fractional integral equations." Journal of Vibration and Control 26, no. 9-10 (January 8, 2020): 713–23. http://dx.doi.org/10.1177/1077546319889862.

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The main objectives of this study are to introduce a new class of optimal control problems governed by a dynamical system of weakly singular variable-order fractional integral equations and to establish a computational method by utilizing the Chebyshev cardinal functions for their numerical solutions. In this way, a new operational matrix of variable-order fractional integration is generated for the Chebyshev cardinal functions. In the established method, first the control and state variables are approximated by the introduced basis functions. Then, the interpolation property of these basis functions together with their mentioned operational matrix is applied to derive an algebraic equation instead of the objective function and an algebraic system of equations instead of the dynamical system. Eventually, the constrained extrema technique is applied by adjoining the constraints generated from the dynamical system to the objective function using a set of Lagrange multipliers. The accuracy of the established approach is examined through several test problems. The obtained results confirm the high accuracy of the presented method.

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39

El-Daou, M. K., and E. L. Ortiz. "The number of extrema of the error function of a class of methods for differential equations." Computers & Mathematics with Applications 33, no. 1-2 (January 1997): 69–79. http://dx.doi.org/10.1016/s0898-1221(96)00220-9.

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40

Lahrech, S., J. Hlal, A. Ouahab, A. Jaddar, and A. Mbarki. "Characterization of the extrema of a pseudoconvex function in terms of limiting and strong limiting subdifferential." International Mathematical Forum 2 (2007): 2711–18. http://dx.doi.org/10.12988/imf.2007.07241.

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41

Tamrazyan, Ashot, and Ekaterina Filimonova. "Searching Method of Optimization of Bending Reinforced Concrete Slabs with Simultaneous Assessment of Criterion Function and the Boundary Conditions." Applied Mechanics and Materials 467 (December 2013): 404–9. http://dx.doi.org/10.4028/www.scientific.net/amm.467.404.

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Problems of optimal design of reinforced concrete structures are characterized by existence of nonlinear objective function and a set of local extrema. For the solution of similar tasks for their optimization its offered to use an algorithm based on the method of casual search. A prerequisite for this is the requirement for simultaneous consideration of changes in the objective function and the boundary conditions in the motion search for the optimum. This procedure provides a guaranteed search of optimal parameters plate.

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42

WEBER, MICHEL J. G. "LOCAL SUPREMA OF DIRICHLET POLYNOMIALS AND ZEROFREE REGIONS OF THE RIEMANN ZETA-FUNCTION." Glasgow Mathematical Journal 56, no. 3 (August 22, 2014): 643–55. http://dx.doi.org/10.1017/s001708951400007x.

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AbstractA new family of zerofree region of the Riemann Zeta-function ζ is identified by using Turán's (P. Turán, Eine neue Methode inter Analysis und deren Anwendungen (Akadémiai Kiadó, Budapest, Hungary, 1953); Analytic number theory, Proc. Symp. Pure Math., vol. XXIV (Amer. Math. Soc. Providence, RI, 1972)) localization criterion linking zeros of ζ with uniform local suprema of sets of Dirichlet polynomials expanded over the primes. The proof is based on a randomization argument. An estimate for local extrema for some finite families of shifted Dirichlet polynomials is established by preliminary considering their local increment properties by means of Montgomery-Vaughan's variant of Hilbert's inequality. A covering argument combined with Turán's localization criterion allows to conclude.

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43

WANG, GANG, XIAN-YAO CHEN, FANG-LI QIAO, ZHAOHUA WU, and NORDEN E. HUANG. "ON INTRINSIC MODE FUNCTION." Advances in Adaptive Data Analysis 02, no. 03 (July 2010): 277–93. http://dx.doi.org/10.1142/s1793536910000549.

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Empirical Mode Decomposition (EMD) has been widely used to analyze non-stationary and nonlinear signal by decomposing data into a series of intrinsic mode functions (IMFs) and a trend function through sifting processes. For lack of a firm mathematical foundation, the implementation of EMD is still empirical and ad hoc. In this paper, we prove mathematically that EMD, as practiced now, only gives an approximation to the true envelope. As a result, there is a potential conflict between the strict definition of IMF and its empirical implementation through natural cubic spline. It is found that the amplitude of IMF is closely connected with the interpolation function defining the upper and lower envelopes: adopting the cubic spline function, the upper (lower) envelope of the resulting IMF is proved to be a unitary cubic spline line as long as the extrema are sparsely distributed compared with the sampling data. Furthermore, when natural spline boundary condition is adopted, the unitary cubic spline line degenerates into a straight line. Unless the amplitude of the IMF is a strictly monotonic function, the slope of the straight line will be zero. It explains why the amplitude of IMF tends to be a constant with the number of sifting increasing ad infinitum. Therefore, to get physically meaningful IMFs the sifting times for each IMF should be kept low as in the practice of EMD. Strictly speaking, the resolution of these difficulties should be either to change the EMD implementation method and eschew the spline, or to define the stoppage criterion more objectively and leniently. Short of the full resolution of the conflict, we should realize that the EMD as implemented now yields an approximation with respect to cubic spline basis. We further concluded that a fixed low number of iterations would be the best option at this time, for it delivers the best approximation.

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44

Прохорова, Ольга Михайловна. "Ж.-Л. ЛАГРАНЖ КАК ОДИН ИЗ ОСНОВОПОЛОЖНИКОВ ТЕОРИИ ЭКСТРЕМУМОВ ФУНКЦИЙ МНОГИХ ПЕРЕМЕННЫХ." RADIOELECTRONIC AND COMPUTER SYSTEMS, no. 1 (January 28, 2020): 103–11. http://dx.doi.org/10.32620/reks.2020.1.10.

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In the article we carried out a detailed analysis of the results obtained by J.-L. Lagrange in his first. The theory of extrema of functions of many variables, as part of mathematical analysis, refers to the mathematical foundations of the study of operations. In turn, many optimization problems are actually problems on the conditional extremum of the function of many variables. The relevance of this topic is determined by the fact that the methods for solving problems on the extremum of the function of many variables obtained in the mid 18th - early 20th centuries are used in solving modern problems. A special place here is occupied by L. Euler and J.-L. Lagrange. The aim of the article is to study the conditions for the maximum and minimum functions of many variables obtained by J.-L. Lagrange, and a comparison of its results with the presentation of this topic in modern textbooks on higher mathematics and mathematical analysis. It was established that in his first printed work he first formulated and proved sufficient conditions for the existence of an extremum of the function of many variables by actually establishing a criterion for the positive (negative) definiteness of quadratic forms, long before it appeared in J. Sylvester in the mid-19th century. A comparative analysis of the results of L. Euler and J.-L. Lagrange. It was found that sufficient conditions for the existence of an extremum of functions of many variables obtained in the first printed work of the young Lagrange are included in all modern textbooks in virtually the same form. The examples shown illustrate his theory. These are tasks of geometric and physical content. Special cases are considered in detail: functions of two and three variables. It is noted that this article became programmatic for the young Lagrange, although it remained unnoticed by his contemporaries. Subsequently, based on the method he obtained, he created the variational calculus, using the principle of least action and the theory of extrema, derived the basic laws of mechanics, the rule of factors for finding the conditional extremum of functions of many variables, which is named after him.

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Ghosh, Aurobrata, Elias Tsigaridas, Bernard Mourrain, and Rachid Deriche. "A polynomial approach for extracting the extrema of a spherical function and its application in diffusion MRI." Medical Image Analysis 17, no. 5 (July 2013): 503–14. http://dx.doi.org/10.1016/j.media.2013.03.004.

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Conrey, J. B., and A. Ghosh. "A Mean Value Theorem for the Riemann Zeta-Function at its Relative Extrema on the Critical Line." Journal of the London Mathematical Society s2-32, no. 2 (October 1985): 193–202. http://dx.doi.org/10.1112/jlms/s2-32.2.193.

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Neymeyr, Klaus, Azadeh Golshan, Konrad Engel, Romà Tauler, and Mathias Sawall. "Does the signal contribution function attain its extrema on the boundary of the area of feasible solutions?" Chemometrics and Intelligent Laboratory Systems 196 (January 2020): 103887. http://dx.doi.org/10.1016/j.chemolab.2019.103887.

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Angeles, J., and A. Bernier. "A General Method of Four-Bar Linkage Mobility Analysis." Journal of Mechanisms, Transmissions, and Automation in Design 109, no. 2 (June 1, 1987): 197–203. http://dx.doi.org/10.1115/1.3267438.

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The problem of mobility associated with four-bar linkages is addressed in this paper. The mobility analysis is reduced to finding the global extrema of a quadratic function on a cylinder, which then leads to the geometric problem of finding the intersections of a circle and a hyperbola. The method proposed here produces an efficient mobility analysis that can be readily integrated into any suitable optimization algorithm.

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Ben Slimane, Mourad, and Clothilde Mélot. "Analysis of a Fractal Boundary: The Graph of the Knopp Function." Abstract and Applied Analysis 2015 (2015): 1–14. http://dx.doi.org/10.1155/2015/587347.

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A usual classification tool to study a fractal interface is the computation of its fractal dimension. But a recent method developed by Y. Heurteaux and S. Jaffard proposes to compute either weak and strong accessibility exponents or localLpregularity exponents (the so-calledp-exponent). These exponents describe locally the behavior of the interface. We apply this method to the graph of the Knopp function which is defined forx∈0, 1asFx=∑j=0∞2-αjϕ2jx, where0<α<1andϕx=distx, z. The Knopp function itself has everywhere the samep-exponentα. Nevertheless, using the characterization of the maxima and minima done by B. Dubuc and S. Dubuc, we will compute thep-exponent of the characteristic function of the domain under the graph ofFat each point(x, F(x))and show thatp-exponents, weak and strong accessibility exponents, change from point to point. Furthermore we will derive a characterization of the local extrema of the function according to the values of these exponents.

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Treiman, Jay S. "Generalized Gradients, Lipschitz behavior and Directional Derivatives." Canadian Journal of Mathematics 37, no. 6 (December 1, 1985): 1074–84. http://dx.doi.org/10.4153/cjm-1985-058-1.

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In the study of optimization problems it is necessary to consider functions that are not differentiable. This has led to the consideration of generalized gradients and a corresponding calculus for certain classes of functions. Rockafellar [16] and others have developed a very strong and elegant theory of subgradients for convex functions. This convex theory gives point-wise criteria for the existence of extrema in optimization problems.There are however many optimization problems that involve functions which are neither differentiable nor convex. Such functions arise in many settings including optimal value functions [15]. In order to deal with such problems Clarke [3] defined a type of subgradient for nonconvex functions. This definition was initially for Lipschitz functions on R”. Clarke extended this definition to include lower semicontinuous (l.s.c.) functions on Banach spaces through the use of a directional derivative, the distance function from a closed set and tangent and normal cones to closed sets.

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