A path <i>π</i> = [<i>v</i><sub>1</sub>, <i>v</i><sub>2</sub>, …, <i>v</i><sub><em>k</em></sub>] in a graph <i>G&...A path <i>π</i> = [<i>v</i><sub>1</sub>, <i>v</i><sub>2</sub>, …, <i>v</i><sub><em>k</em></sub>] in a graph <i>G</i> = (<i>V</i>, <i>E</i>) is an uphill path if <i>deg</i>(<i>v</i><sub><i>i</i></sub>) ≤ <i>deg</i>(<i>v</i><sub><i>i</i>+1</sub>) for every 1 ≤ <i>i</i> ≤ <i>k</i>. A subset <i>S </i><span style="white-space:nowrap;"><span style="white-space:nowrap;">⊆</span></span> <i>V</i>(<i>G</i>) is an uphill dominating set if every vertex <i>v</i><sub><i>i</i></sub> <span style="white-space:nowrap;"><span style="white-space:nowrap;">∈</span> </span><i>V</i>(<i>G</i>) lies on an uphill path originating from some vertex in <i>S</i>. The uphill domination number of <i>G</i> is denoted by <i><span style="white-space:nowrap;"><i><span style="white-space:nowrap;"><i>γ</i></span></i></span></i><sub><i>up</i></sub>(<i>G</i>) and is the minimum cardinality of the uphill dominating set of <i>G</i>. In this paper, we introduce the uphill domination polynomial of a graph <i>G</i>. The uphill domination polynomial of a graph <i>G</i> of <i>n</i> vertices is the polynomial <img src="Edit_75fb5c37-6ef5-4292-9d3a-4b63343c48ce.bmp" alt="" />, where <em>up</em>(<i>G</i>, <i>i</i>) is the number of uphill dominating sets of size <i>i</i> in <i>G</i>, and <i><span style="white-space:nowrap;"><i><span style="white-space:nowrap;"><i>γ</i></span></i></span></i><i><sub>up</sub></i>(<i>G</i>) is the uphill domination number of <i>G</i>, we compute the uphill domination polynomial and its roots for some families of standard graphs. Also, <i>UP</i>(<i>G</i>, <em>x</em>) for some graph operations is obtained.展开更多
Let G = (V, E) be a simple graph. A set S í V is a dominating set of G, if every vertex in V-S is adjacent to at least one vertex in S. Let be the square of the Path and let denote the family of all dominating se...Let G = (V, E) be a simple graph. A set S í V is a dominating set of G, if every vertex in V-S is adjacent to at least one vertex in S. Let be the square of the Path and let denote the family of all dominating sets of with cardinality i. Let . In this paper, we obtain a recursive formula for . Using this recursive formula, we construct the polynomial, , which we call domination polynomial of and obtain some properties of this polynomial.展开更多
Let G = (V, E) be a simple graph. A set S E(G) is an edge-vertex dominating set of G (or simply an ev-dominating set), if for all vertices v V(G);there exists an edge eS such that e dominates v. Let denote the family ...Let G = (V, E) be a simple graph. A set S E(G) is an edge-vertex dominating set of G (or simply an ev-dominating set), if for all vertices v V(G);there exists an edge eS such that e dominates v. Let denote the family of all ev-dominating sets of with cardinality i. Let . In this paper, we obtain a recursive formula for . Using this recursive formula, we construct the polynomial, , which we call edge-vertex domination polynomial of (or simply an ev-domination polynomial of ) and obtain some properties of this polynomial.展开更多
Fermat’s Last Theorem is a famous theorem in number theory which is difficult to prove.However,it is known that the version of polynomials with one variable of Fermat’s Last Theorem over C can be proved very concisely...Fermat’s Last Theorem is a famous theorem in number theory which is difficult to prove.However,it is known that the version of polynomials with one variable of Fermat’s Last Theorem over C can be proved very concisely.The aim of this paper is to study the similar problems about Fermat’s Last Theorem for multivariate(skew)-polynomials with any characteristic.展开更多
Video watermarking plays a crucial role in protecting intellectual property rights and ensuring content authenticity.This study delves into the integration of Galois Field(GF)multiplication tables,especially GF(2^(4))...Video watermarking plays a crucial role in protecting intellectual property rights and ensuring content authenticity.This study delves into the integration of Galois Field(GF)multiplication tables,especially GF(2^(4)),and their interaction with distinct irreducible polynomials.The primary aim is to enhance watermarking techniques for achieving imperceptibility,robustness,and efficient execution time.The research employs scene selection and adaptive thresholding techniques to streamline the watermarking process.Scene selection is used strategically to embed watermarks in the most vital frames of the video,while adaptive thresholding methods ensure that the watermarking process adheres to imperceptibility criteria,maintaining the video's visual quality.Concurrently,careful consideration is given to execution time,crucial in real-world scenarios,to balance efficiency and efficacy.The Peak Signal-to-Noise Ratio(PSNR)serves as a pivotal metric to gauge the watermark's imperceptibility and video quality.The study explores various irreducible polynomials,navigating the trade-offs between computational efficiency and watermark imperceptibility.In parallel,the study pays careful attention to the execution time,a paramount consideration in real-world scenarios,to strike a balance between efficiency and efficacy.This comprehensive analysis provides valuable insights into the interplay of GF multiplication tables,diverse irreducible polynomials,scene selection,adaptive thresholding,imperceptibility,and execution time.The evaluation of the proposed algorithm's robustness was conducted using PSNR and NC metrics,and it was subjected to assessment under the impact of five distinct attack scenarios.These findings contribute to the development of watermarking strategies that balance imperceptibility,robustness,and processing efficiency,enhancing the field's practicality and effectiveness.展开更多
A certain variety of non-switched polynomials provides a uni-figure representation for a wide range of linear functional equations. This is properly adapted for the calculations. We reinterpret from this point of view...A certain variety of non-switched polynomials provides a uni-figure representation for a wide range of linear functional equations. This is properly adapted for the calculations. We reinterpret from this point of view a number of algorithms.展开更多
In this article,we construct the generating functions for new families of special polynomials including two parametric kinds of Bell-based Bernoulli and Euler polynomials.Some fundamental properties of these functions...In this article,we construct the generating functions for new families of special polynomials including two parametric kinds of Bell-based Bernoulli and Euler polynomials.Some fundamental properties of these functions are given.By using these generating functions and some identities,relations among trigonometric functions and two parametric kinds of Bell-based Bernoulli and Euler polynomials,Stirling numbers are presented.Computational formulae for these polynomials are obtained.Applying a partial derivative operator to these generating functions,some derivative formulae and finite combinatorial sums involving the aforementioned polynomials and numbers are also obtained.In addition,some remarks and observations on these polynomials are given.展开更多
It is remarkable that studying degenerate versions of polynomials from algebraic point of view is not limited to only special polynomials but can also be extended to their hybrid polynomials.Indeed for the first time,...It is remarkable that studying degenerate versions of polynomials from algebraic point of view is not limited to only special polynomials but can also be extended to their hybrid polynomials.Indeed for the first time,a closed determinant expression for the degenerate Appell polynomials is derived.The determinant forms for the degenerate Bernoulli and Euler polynomials are also investigated.A new class of the degenerate Hermite-Appell polynomials is investigated and some novel identities for these polynomials are established.The degenerate Hermite-Bernoulli and degenerate Hermite-Euler polynomials are considered as special cases of the degenerate Hermite-Appell polynomials.Further,by using Mathematica,we draw graphs of degenerate Hermite-Bernoulli polynomials for different values of indices.The zeros of these polynomials are also explored and their distribution is presented.展开更多
In expansions of arbitrary functions in Bessel functions or Spherical Bessel functions, a dual partner set of polynomials play a role. For the Bessel functions, these are the Chebyshev polynomials of first kind and fo...In expansions of arbitrary functions in Bessel functions or Spherical Bessel functions, a dual partner set of polynomials play a role. For the Bessel functions, these are the Chebyshev polynomials of first kind and for the Spherical Bessel functions the Legendre polynomials. These two sets of functions appear in many formulas of the expansion and in the completeness and (bi)-orthogonality relations. The analogy to expansions of functions in Taylor series and in moment series and to expansions in Hermite functions is elaborated. Besides other special expansion, we find the expansion of Bessel functions in Spherical Bessel functions and their inversion and of Chebyshev polynomials of first kind in Legendre polynomials and their inversion. For the operators which generate the Spherical Bessel functions from a basic Spherical Bessel function, the normally ordered (or disentangled) form is found.展开更多
In this note, we first derive an exponential generating function of the alternating run polynomials. We then deduce an explicit formula of the alternating run polynomials in terms of the partial Bell polynomials.
Utilization of the shift operator to represent Euler polynomials as polynomials of Appell type leads directly to its algebraic properties, its relations with powers sums;may be all its relations with Bernoulli polynom...Utilization of the shift operator to represent Euler polynomials as polynomials of Appell type leads directly to its algebraic properties, its relations with powers sums;may be all its relations with Bernoulli polynomials, Bernoulli numbers;its recurrence formulae and a very simple formula for calculating simultaneously Euler numbers and Euler polynomials. The expansions of Euler polynomials into Fourier series are also obtained;the formulae for obtaining all π<sup>m</sup> as series on k<sup>-m</sup> and for expanding functions into series of Euler polynomials.展开更多
By looking at the situation when the coefficients Pj(z)(j=1,2,…,n-1)(or most of them) are exponential polynomials,we investigate the fact that all nontrivial solutions to higher order differential equations f((n))+Pn...By looking at the situation when the coefficients Pj(z)(j=1,2,…,n-1)(or most of them) are exponential polynomials,we investigate the fact that all nontrivial solutions to higher order differential equations f((n))+Pn-1(z)f((n-1))+…+P0(z)f=0 are of infinite order.An exponential polynomial coefficient plays a key role in these results.展开更多
图 G 的一个 Domination 染色是使得图 G 的每个顶点 v 控制至少一个色类(可能是自身的色类), 并且每一个色类至少被 G 中一个顶点控制的一个正常染色。 图 G 的 Domination 色数是图 G 的 Domination 染色所需最小的颜色数目,用 χdd(...图 G 的一个 Domination 染色是使得图 G 的每个顶点 v 控制至少一个色类(可能是自身的色类), 并且每一个色类至少被 G 中一个顶点控制的一个正常染色。 图 G 的 Domination 色数是图 G 的 Domination 染色所需最小的颜色数目,用 χdd(G) 表示。 本文研究了图 G 的 Domination 色数与图 G 通过某种操作得到图 G"的 Domination 色数之间的关系。展开更多
In this manuscript,an algorithm for the computation of numerical solutions to some variable order fractional differential equations(FDEs)subject to the boundary and initial conditions is developed.We use shifted Legen...In this manuscript,an algorithm for the computation of numerical solutions to some variable order fractional differential equations(FDEs)subject to the boundary and initial conditions is developed.We use shifted Legendre polynomials for the required numerical algorithm to develop some operational matrices.Further,operational matrices are constructed using variable order differentiation and integration.We are finding the operationalmatrices of variable order differentiation and integration by omitting the discretization of data.With the help of aforesaid matrices,considered FDEs are converted to algebraic equations of Sylvester type.Finally,the algebraic equations we get are solved with the help of mathematical software like Matlab or Mathematica to compute numerical solutions.Some examples are given to check the proposed method’s accuracy and graphical representations.Exact and numerical solutions are also compared in the paper for some examples.The efficiency of the method can be enhanced further by increasing the scale level.展开更多
In this paper,suppose that a,c∈C{0},c_(j)∈C(j=1,2,···,n) are not all zeros and n≥2,and f (z) is a finite order transcendental entire function with Borel finite exceptional value or with infinitely ma...In this paper,suppose that a,c∈C{0},c_(j)∈C(j=1,2,···,n) are not all zeros and n≥2,and f (z) is a finite order transcendental entire function with Borel finite exceptional value or with infinitely many multiple zeros,the zero distribution of difference polynomials of f (z+c)-af^(n)(z) and f (z)f (z+c_1)···f (z+c_n) are investigated.A number of examples are also presented to show that our results are best possible in a certain sense.展开更多
This paper presents a novel approach to improve aliasing rejection in comb-based decimation filters. The method is established on certain palindromic polynomials with all zeros on the unit circle and the sharpening te...This paper presents a novel approach to improve aliasing rejection in comb-based decimation filters. The method is established on certain palindromic polynomials with all zeros on the unit circle and the sharpening technique. As a result, aliasing rejection and the passband characteristic are improved. The method is illustrated with various examples and compared with the methods from the literature. .展开更多
In this article, I consider projection groups on function spaces, more specifically the space of polynomials P<sub>n</sub>[x]. I will show that a very similar construct of projection operators allows us to...In this article, I consider projection groups on function spaces, more specifically the space of polynomials P<sub>n</sub>[x]. I will show that a very similar construct of projection operators allows us to project into the subspaces of P<sub>n</sub>[x] where the function h ∈P<sub>n</sub>[x] represents the closets function to f ∈P<sub>n</sub>[x] in the least square sense. I also demonstrate that we can generalise projections by constructing operators i.e. in R<sup>n+1</sup> using the metric tensor on P<sub>n</sub>[x]. This allows one to project a polynomial function onto another by mapping it to its coefficient vector in R<sup>n+1</sup>. This can be also achieved with the Kronecker Product as detailed in this paper.展开更多
Sums of convergent series for any desired number of terms, which may be infinite, are estimated very accurately by establishing definite rational polynomials. For infinite number of terms the sum infinite is obtained ...Sums of convergent series for any desired number of terms, which may be infinite, are estimated very accurately by establishing definite rational polynomials. For infinite number of terms the sum infinite is obtained by taking the asymptotic limit of the rational polynomial. A rational function with second-degree polynomials both in the numerator and denominator is found to produce excellent results. Sums of series with different characteristics such as alternating signs are considered for testing the performance of the proposed approach.展开更多
Many graph domination applications can be expanded to achieve complete cototal domination.If every node in a dominating set is regarded as a record server for a PC organization,then each PC affiliated with the organiz...Many graph domination applications can be expanded to achieve complete cototal domination.If every node in a dominating set is regarded as a record server for a PC organization,then each PC affiliated with the organization has direct access to a document server.It is occasionally reasonable to believe that this gateway will remain available even if one of the scrape servers fails.Because every PC has direct access to at least two documents’servers,a complete cototal dominating set provides the required adaptability to non-critical failure in such scenarios.In this paper,we presented a method for calculating a graph’s complete cototal roman domination number.We also examined the properties and determined the bounds for a graph’s complete cototal roman domination number,and its applications are presented.It has been observed that one’s interest fluctuate over time,therefore inferring them just from one’s own behaviour may be inconclusive.However,it may be able to deduce a user’s constant interest to some level if a user’s networking is also watched for similar or related actions.This research proposes a method that considers a user’s and his channel’s activity,as well as common tags,persons,and organizations from their social media posts in order to establish a solid foundation for the required conclusion.展开更多
文摘A path <i>π</i> = [<i>v</i><sub>1</sub>, <i>v</i><sub>2</sub>, …, <i>v</i><sub><em>k</em></sub>] in a graph <i>G</i> = (<i>V</i>, <i>E</i>) is an uphill path if <i>deg</i>(<i>v</i><sub><i>i</i></sub>) ≤ <i>deg</i>(<i>v</i><sub><i>i</i>+1</sub>) for every 1 ≤ <i>i</i> ≤ <i>k</i>. A subset <i>S </i><span style="white-space:nowrap;"><span style="white-space:nowrap;">⊆</span></span> <i>V</i>(<i>G</i>) is an uphill dominating set if every vertex <i>v</i><sub><i>i</i></sub> <span style="white-space:nowrap;"><span style="white-space:nowrap;">∈</span> </span><i>V</i>(<i>G</i>) lies on an uphill path originating from some vertex in <i>S</i>. The uphill domination number of <i>G</i> is denoted by <i><span style="white-space:nowrap;"><i><span style="white-space:nowrap;"><i>γ</i></span></i></span></i><sub><i>up</i></sub>(<i>G</i>) and is the minimum cardinality of the uphill dominating set of <i>G</i>. In this paper, we introduce the uphill domination polynomial of a graph <i>G</i>. The uphill domination polynomial of a graph <i>G</i> of <i>n</i> vertices is the polynomial <img src="Edit_75fb5c37-6ef5-4292-9d3a-4b63343c48ce.bmp" alt="" />, where <em>up</em>(<i>G</i>, <i>i</i>) is the number of uphill dominating sets of size <i>i</i> in <i>G</i>, and <i><span style="white-space:nowrap;"><i><span style="white-space:nowrap;"><i>γ</i></span></i></span></i><i><sub>up</sub></i>(<i>G</i>) is the uphill domination number of <i>G</i>, we compute the uphill domination polynomial and its roots for some families of standard graphs. Also, <i>UP</i>(<i>G</i>, <em>x</em>) for some graph operations is obtained.
文摘Let G = (V, E) be a simple graph. A set S í V is a dominating set of G, if every vertex in V-S is adjacent to at least one vertex in S. Let be the square of the Path and let denote the family of all dominating sets of with cardinality i. Let . In this paper, we obtain a recursive formula for . Using this recursive formula, we construct the polynomial, , which we call domination polynomial of and obtain some properties of this polynomial.
文摘Let G = (V, E) be a simple graph. A set S E(G) is an edge-vertex dominating set of G (or simply an ev-dominating set), if for all vertices v V(G);there exists an edge eS such that e dominates v. Let denote the family of all ev-dominating sets of with cardinality i. Let . In this paper, we obtain a recursive formula for . Using this recursive formula, we construct the polynomial, , which we call edge-vertex domination polynomial of (or simply an ev-domination polynomial of ) and obtain some properties of this polynomial.
基金supported by the National Natural Science Foundation of China(12131015,12071422).
文摘Fermat’s Last Theorem is a famous theorem in number theory which is difficult to prove.However,it is known that the version of polynomials with one variable of Fermat’s Last Theorem over C can be proved very concisely.The aim of this paper is to study the similar problems about Fermat’s Last Theorem for multivariate(skew)-polynomials with any characteristic.
文摘Video watermarking plays a crucial role in protecting intellectual property rights and ensuring content authenticity.This study delves into the integration of Galois Field(GF)multiplication tables,especially GF(2^(4)),and their interaction with distinct irreducible polynomials.The primary aim is to enhance watermarking techniques for achieving imperceptibility,robustness,and efficient execution time.The research employs scene selection and adaptive thresholding techniques to streamline the watermarking process.Scene selection is used strategically to embed watermarks in the most vital frames of the video,while adaptive thresholding methods ensure that the watermarking process adheres to imperceptibility criteria,maintaining the video's visual quality.Concurrently,careful consideration is given to execution time,crucial in real-world scenarios,to balance efficiency and efficacy.The Peak Signal-to-Noise Ratio(PSNR)serves as a pivotal metric to gauge the watermark's imperceptibility and video quality.The study explores various irreducible polynomials,navigating the trade-offs between computational efficiency and watermark imperceptibility.In parallel,the study pays careful attention to the execution time,a paramount consideration in real-world scenarios,to strike a balance between efficiency and efficacy.This comprehensive analysis provides valuable insights into the interplay of GF multiplication tables,diverse irreducible polynomials,scene selection,adaptive thresholding,imperceptibility,and execution time.The evaluation of the proposed algorithm's robustness was conducted using PSNR and NC metrics,and it was subjected to assessment under the impact of five distinct attack scenarios.These findings contribute to the development of watermarking strategies that balance imperceptibility,robustness,and processing efficiency,enhancing the field's practicality and effectiveness.
文摘A certain variety of non-switched polynomials provides a uni-figure representation for a wide range of linear functional equations. This is properly adapted for the calculations. We reinterpret from this point of view a number of algorithms.
基金funded by Research Deanship at the University of Ha’il,Saudi Arabia,through Project No.RG-21144.
文摘In this article,we construct the generating functions for new families of special polynomials including two parametric kinds of Bell-based Bernoulli and Euler polynomials.Some fundamental properties of these functions are given.By using these generating functions and some identities,relations among trigonometric functions and two parametric kinds of Bell-based Bernoulli and Euler polynomials,Stirling numbers are presented.Computational formulae for these polynomials are obtained.Applying a partial derivative operator to these generating functions,some derivative formulae and finite combinatorial sums involving the aforementioned polynomials and numbers are also obtained.In addition,some remarks and observations on these polynomials are given.
文摘It is remarkable that studying degenerate versions of polynomials from algebraic point of view is not limited to only special polynomials but can also be extended to their hybrid polynomials.Indeed for the first time,a closed determinant expression for the degenerate Appell polynomials is derived.The determinant forms for the degenerate Bernoulli and Euler polynomials are also investigated.A new class of the degenerate Hermite-Appell polynomials is investigated and some novel identities for these polynomials are established.The degenerate Hermite-Bernoulli and degenerate Hermite-Euler polynomials are considered as special cases of the degenerate Hermite-Appell polynomials.Further,by using Mathematica,we draw graphs of degenerate Hermite-Bernoulli polynomials for different values of indices.The zeros of these polynomials are also explored and their distribution is presented.
文摘In expansions of arbitrary functions in Bessel functions or Spherical Bessel functions, a dual partner set of polynomials play a role. For the Bessel functions, these are the Chebyshev polynomials of first kind and for the Spherical Bessel functions the Legendre polynomials. These two sets of functions appear in many formulas of the expansion and in the completeness and (bi)-orthogonality relations. The analogy to expansions of functions in Taylor series and in moment series and to expansions in Hermite functions is elaborated. Besides other special expansion, we find the expansion of Bessel functions in Spherical Bessel functions and their inversion and of Chebyshev polynomials of first kind in Legendre polynomials and their inversion. For the operators which generate the Spherical Bessel functions from a basic Spherical Bessel function, the normally ordered (or disentangled) form is found.
文摘In this note, we first derive an exponential generating function of the alternating run polynomials. We then deduce an explicit formula of the alternating run polynomials in terms of the partial Bell polynomials.
文摘Utilization of the shift operator to represent Euler polynomials as polynomials of Appell type leads directly to its algebraic properties, its relations with powers sums;may be all its relations with Bernoulli polynomials, Bernoulli numbers;its recurrence formulae and a very simple formula for calculating simultaneously Euler numbers and Euler polynomials. The expansions of Euler polynomials into Fourier series are also obtained;the formulae for obtaining all π<sup>m</sup> as series on k<sup>-m</sup> and for expanding functions into series of Euler polynomials.
基金supported partly by the National Natural Science Foundation of China(12171050,11871260)National Science Foundation of Guangdong Province(2018A030313508)。
文摘By looking at the situation when the coefficients Pj(z)(j=1,2,…,n-1)(or most of them) are exponential polynomials,we investigate the fact that all nontrivial solutions to higher order differential equations f((n))+Pn-1(z)f((n-1))+…+P0(z)f=0 are of infinite order.An exponential polynomial coefficient plays a key role in these results.
文摘图 G 的一个 Domination 染色是使得图 G 的每个顶点 v 控制至少一个色类(可能是自身的色类), 并且每一个色类至少被 G 中一个顶点控制的一个正常染色。 图 G 的 Domination 色数是图 G 的 Domination 染色所需最小的颜色数目,用 χdd(G) 表示。 本文研究了图 G 的 Domination 色数与图 G 通过某种操作得到图 G"的 Domination 色数之间的关系。
基金Supporting Project No.(PNURSP2022R 14),Princess Nourah bint A bdurahman University,Riyadh,Saudi Arabia.
文摘In this manuscript,an algorithm for the computation of numerical solutions to some variable order fractional differential equations(FDEs)subject to the boundary and initial conditions is developed.We use shifted Legendre polynomials for the required numerical algorithm to develop some operational matrices.Further,operational matrices are constructed using variable order differentiation and integration.We are finding the operationalmatrices of variable order differentiation and integration by omitting the discretization of data.With the help of aforesaid matrices,considered FDEs are converted to algebraic equations of Sylvester type.Finally,the algebraic equations we get are solved with the help of mathematical software like Matlab or Mathematica to compute numerical solutions.Some examples are given to check the proposed method’s accuracy and graphical representations.Exact and numerical solutions are also compared in the paper for some examples.The efficiency of the method can be enhanced further by increasing the scale level.
基金Supported by the National Natural Science Foundation of China (11926201)Natural Science Foundation of Guangdong Province (2018A030313508)。
文摘In this paper,suppose that a,c∈C{0},c_(j)∈C(j=1,2,···,n) are not all zeros and n≥2,and f (z) is a finite order transcendental entire function with Borel finite exceptional value or with infinitely many multiple zeros,the zero distribution of difference polynomials of f (z+c)-af^(n)(z) and f (z)f (z+c_1)···f (z+c_n) are investigated.A number of examples are also presented to show that our results are best possible in a certain sense.
文摘This paper presents a novel approach to improve aliasing rejection in comb-based decimation filters. The method is established on certain palindromic polynomials with all zeros on the unit circle and the sharpening technique. As a result, aliasing rejection and the passband characteristic are improved. The method is illustrated with various examples and compared with the methods from the literature. .
文摘In this article, I consider projection groups on function spaces, more specifically the space of polynomials P<sub>n</sub>[x]. I will show that a very similar construct of projection operators allows us to project into the subspaces of P<sub>n</sub>[x] where the function h ∈P<sub>n</sub>[x] represents the closets function to f ∈P<sub>n</sub>[x] in the least square sense. I also demonstrate that we can generalise projections by constructing operators i.e. in R<sup>n+1</sup> using the metric tensor on P<sub>n</sub>[x]. This allows one to project a polynomial function onto another by mapping it to its coefficient vector in R<sup>n+1</sup>. This can be also achieved with the Kronecker Product as detailed in this paper.
文摘Sums of convergent series for any desired number of terms, which may be infinite, are estimated very accurately by establishing definite rational polynomials. For infinite number of terms the sum infinite is obtained by taking the asymptotic limit of the rational polynomial. A rational function with second-degree polynomials both in the numerator and denominator is found to produce excellent results. Sums of series with different characteristics such as alternating signs are considered for testing the performance of the proposed approach.
文摘Many graph domination applications can be expanded to achieve complete cototal domination.If every node in a dominating set is regarded as a record server for a PC organization,then each PC affiliated with the organization has direct access to a document server.It is occasionally reasonable to believe that this gateway will remain available even if one of the scrape servers fails.Because every PC has direct access to at least two documents’servers,a complete cototal dominating set provides the required adaptability to non-critical failure in such scenarios.In this paper,we presented a method for calculating a graph’s complete cototal roman domination number.We also examined the properties and determined the bounds for a graph’s complete cototal roman domination number,and its applications are presented.It has been observed that one’s interest fluctuate over time,therefore inferring them just from one’s own behaviour may be inconclusive.However,it may be able to deduce a user’s constant interest to some level if a user’s networking is also watched for similar or related actions.This research proposes a method that considers a user’s and his channel’s activity,as well as common tags,persons,and organizations from their social media posts in order to establish a solid foundation for the required conclusion.
