This document summarizes research on the convexity of minimal total dominating functions of Quadratic Residue Cayley graphs. It defines Quadratic Residue Cayley graphs and discusses properties like completeness for certain prime values. It defines concepts like total dominating sets, functions, and minimal total dominating functions. The main results prove that for complete Quadratic Residue Cayley graphs, the characteristic function of a minimal total dominating set is a minimal total dominating function. It also shows that the convex combination of two such functions is a total dominating function but not minimal.
On Twisted Paraproducts and some other Multilinear Singular IntegralsVjekoslavKovac1
Presentation.
9th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial, June 12, 2012.
The 24th International Conference on Operator Theory, Timisoara, July 3, 2012.
Tales on two commuting transformations or flowsVjekoslavKovac1
1) The document summarizes recent work on ergodic averages and flows for commuting transformations. It discusses convergence results for single and double linear ergodic averages in L2 and almost everywhere, as well as providing norm estimates to quantify the rate of convergence.
2) It also considers double polynomial ergodic averages and provides proofs for almost everywhere convergence in the continuous-time setting. Open problems remain for the discrete-time case.
3) An ergodic-martingale paraproduct is introduced, motivated by an open question from 1950. Convergence in Lp norm is shown, while almost everywhere convergence remains open.
This document discusses power spectral density (PSD) of various digitally modulated signals. It begins by deriving the PSD of modulated signals with memory, then discusses the PSD of linearly modulated signals such as ASK, PSK and QAM. It also derives the PSD of continuous phase modulation (CPM) signals such as continuous phase frequency shift keying (CPFSK). Graphs show the PSD of CPFSK varies with modulation index and signal constellation size. The document concludes by comparing the PSD of minimum shift keying (MSK) and offset quadrature phase-shift keying (OQPSK).
Trilinear embedding for divergence-form operatorsVjekoslavKovac1
The document discusses a trilinear embedding theorem for divergence-form operators with complex coefficients. It proves that if matrices A, B, C are appropriately p,q,r-elliptic, then there is a bound on the integral of the product of the gradients of the semigroups associated with the operators. The proof uses a Bellman function technique and shows the relationship to the concept of p-ellipticity. It generalizes previous work on bilinear embeddings to the trilinear case.
Regularity and complexity in dynamical systemsSpringer
This chapter discusses how variational methods have been used to analyze three classes of snakelike robots: 1) hyper-redundant manipulators guided by backbone curves, 2) flexible steerable needles, and 3) concentric tube continuum robots. Variational methods provide a means to determine optimal backbone curves for manipulators, generate optimal plans for needle steering, and model equilibrium conformations for concentric tube robots based on elastic mechanics principles. The chapter reviews how variational formulations using Euler-Lagrange and Euler-Poincare equations are applied in each case.
On Twisted Paraproducts and some other Multilinear Singular IntegralsVjekoslavKovac1
Presentation.
9th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial, June 12, 2012.
The 24th International Conference on Operator Theory, Timisoara, July 3, 2012.
Tales on two commuting transformations or flowsVjekoslavKovac1
1) The document summarizes recent work on ergodic averages and flows for commuting transformations. It discusses convergence results for single and double linear ergodic averages in L2 and almost everywhere, as well as providing norm estimates to quantify the rate of convergence.
2) It also considers double polynomial ergodic averages and provides proofs for almost everywhere convergence in the continuous-time setting. Open problems remain for the discrete-time case.
3) An ergodic-martingale paraproduct is introduced, motivated by an open question from 1950. Convergence in Lp norm is shown, while almost everywhere convergence remains open.
This document discusses power spectral density (PSD) of various digitally modulated signals. It begins by deriving the PSD of modulated signals with memory, then discusses the PSD of linearly modulated signals such as ASK, PSK and QAM. It also derives the PSD of continuous phase modulation (CPM) signals such as continuous phase frequency shift keying (CPFSK). Graphs show the PSD of CPFSK varies with modulation index and signal constellation size. The document concludes by comparing the PSD of minimum shift keying (MSK) and offset quadrature phase-shift keying (OQPSK).
Trilinear embedding for divergence-form operatorsVjekoslavKovac1
The document discusses a trilinear embedding theorem for divergence-form operators with complex coefficients. It proves that if matrices A, B, C are appropriately p,q,r-elliptic, then there is a bound on the integral of the product of the gradients of the semigroups associated with the operators. The proof uses a Bellman function technique and shows the relationship to the concept of p-ellipticity. It generalizes previous work on bilinear embeddings to the trilinear case.
Regularity and complexity in dynamical systemsSpringer
This chapter discusses how variational methods have been used to analyze three classes of snakelike robots: 1) hyper-redundant manipulators guided by backbone curves, 2) flexible steerable needles, and 3) concentric tube continuum robots. Variational methods provide a means to determine optimal backbone curves for manipulators, generate optimal plans for needle steering, and model equilibrium conformations for concentric tube robots based on elastic mechanics principles. The chapter reviews how variational formulations using Euler-Lagrange and Euler-Poincare equations are applied in each case.
Quantitative norm convergence of some ergodic averagesVjekoslavKovac1
The document summarizes quantitative estimates for the convergence of multiple ergodic averages of commuting transformations. Specifically, it presents a theorem that provides an explicit bound on the number of jumps in the Lp norm for double averages over commuting Aω actions on a probability space. The proof transfers the structure of the Cantor group AZ to R+ and establishes norm estimates for bilinear averages of functions on R2+. This allows bounding the variation of the double averages and proving the theorem.
A T(1)-type theorem for entangled multilinear Calderon-Zygmund operatorsVjekoslavKovac1
This document summarizes a talk given by Vjekoslav Kovač at a joint mathematics conference. The talk concerned establishing T(1)-type theorems for entangled multilinear Calderón-Zygmund operators. Specifically, Kovač discussed studying multilinear singular integral forms where the functions partially share variables, known as an "entangled structure." He outlined establishing generalized modulation invariance and Lp estimates for such operators. The talk motivated further studying related problems involving bilinear ergodic averages and forms with more complex graph structures. Kovač specialized his techniques to bipartite graphs, multilinear Calderón-Zygmund kernels, and "perfect" dyadic models.
This document discusses multilinear twisted paraproducts, which are generalizations of classical paraproduct operators to higher dimensions. It begins by reviewing classical paraproducts on the real line and their generalization to higher dimensions using dyadic squares. It then discusses complications that arise, such as twisted paraproducts. The document presents a unified framework for studying such operators using bipartite graphs and selections of vertices. It proves a main boundedness result and discusses special cases like classical dyadic paraproducts and dyadic twisted paraproducts. It introduces tools like Bellman functions and calculus of finite differences to analyze estimates for paraproduct-like operators on finite trees of dyadic squares.
Variants of the Christ-Kiselev lemma and an application to the maximal Fourie...VjekoslavKovac1
1. The document discusses variants of the Christ-Kiselev lemma and its application to maximal Fourier restriction estimates.
2. The Christ-Kiselev lemma allows block-diagonal and block-triangular truncations of operators while controlling their operator norms.
3. These lemmas can be used to prove maximal and variational estimates for the restriction of the Fourier transform to surfaces, which has applications in harmonic analysis.
This document summarizes research on norm-variation estimates for ergodic bilinear and multiple averages. It begins by motivating the study of ergodic averages and their convergence properties. Previous results are discussed that provide pointwise convergence and norm estimates for certain cases. The document then presents new norm-variation estimates obtained by the authors for bilinear and multiple ergodic averages over general measure-preserving systems. These estimates bound the number of jumps in the L2 norm as the averages converge. Finally, analogous results are discussed for bilinear averages on R2 and Z2, linking the estimates to established bounds for singular integrals.
Scattering theory analogues of several classical estimates in Fourier analysisVjekoslavKovac1
This document summarizes some classical estimates in Fourier analysis and their analogues in nonlinear Fourier analysis. It discusses Carleson's theorem on convergence of Fourier series and Fourier transforms, Hausdorff-Young inequalities bounding Lp norms, and results on lacunary trigonometric series and products. Open questions are presented about extending these classical estimates to the nonlinear setting of the SU(1,1) Fourier transform and lacunary SU(1,1) trigonometric products.
The document discusses scaling sets and MRA wavelet sets, which are measurable sets associated with multiresolution analyses and wavelets. It provides definitions and theorems characterizing scaling sets and MRA wavelet sets. Some simple examples of scaling sets and MRA wavelet sets are given as finite unions of intervals. The document then poses questions about the properties of general wavelet sets and provides counterexamples to ideas about possible restrictions on their structure. Finally, more complex examples of scaling sets and MRA wavelet sets are constructed using Rademacher functions.
This document summarizes paraproduct operators with general dilations. It defines paraproducts and provides classical examples. It then introduces a non-classical example of paraproducts with respect to general dilations defined by groups of dilations on Cartesian product spaces. The author and co-author establish Lp estimates for such paraproduct operators by applying martingale estimates to dyadic structures and using square functions. The estimates depend only on the dilation structure and hold for certain exponent ranges.
Fourier Series for Continuous Time & Discrete Time SignalsJayanshu Gundaniya
- Fourier introduced Fourier series in 1807 to solve the heat equation in a metal plate. The heat equation is a partial differential equation describing the distribution of heat in a body over time.
- Prior to Fourier's work, there was no known solution to the heat equation in the general case. Fourier's idea was to model a complicated heat source as a superposition of simple sine and cosine waves.
- This superposition or linear combination of sine and cosine waves is called the Fourier series. It allows any periodic function to be decomposed into the sum of simple oscillating functions. Although originally introduced for heat problems, Fourier series have wide applications in mathematics and physics.
The document discusses using the Laplace transform method to solve initial value problems for linear differential equations with constant coefficients. It shows that taking the Laplace transform of the differential equation transforms it into an algebraic equation, avoiding the need to separately solve homogeneous and nonhomogeneous parts. It also explains that determining the inverse Laplace transform to obtain the original function y(t) is the main difficulty, as it requires partial fraction decomposition and knowledge of Laplace transform pairs. Examples are provided to demonstrate solving initial value problems using this method.
Dr. Arpan Bhattacharyya (Indian Institute Of Science, Bangalore)Rene Kotze
1. The document discusses entanglement entropy functionals for higher derivative gravity theories. It proposes new area functionals for computing entanglement entropy in higher derivative theories containing polynomials of curvature tensors.
2. These functionals are derived using the Lewkowycz-Maldacena interpretation of generalized entropy. However, attempting to derive the extremal surface equations from these functionals using bulk equations of motion leads to inconsistencies and ambiguities in some higher derivative theories like Gauss-Bonnet gravity.
3. The document suggests that the source of ambiguity lies in the limiting procedure used to extract the divergences near the conical singularity. Different limiting paths can lead to different extremal surface equations, indicating no unique prescription
Introduction to harmonic analysis on groups, links with spatial correlation.Valentin De Bortoli
This document introduces harmonic analysis on groups and its connections to spatial correlation. It discusses motivations like defining convolution on the sphere S2. Representation theory provides tools to study this, like spherical harmonics which form an orthonormal basis of L2(S2). Spherical CNNs can be understood through the irreducible unitary representations of SO3(R), which are the Wigner D-matrices. The document explores different types of convolutions defined using representations of a group G, like the G-convolution and the (G,π)-convolution. Wavelet transforms provide a link between these convolutions and representations. The goals are to introduce analogues of convolution and Fourier transforms for general groups beyond R2.
Maksim Zhukovskii – Zero-one k-laws for G(n,n−α)Yandex
We study asymptotical behavior of the probabilities of first-order properties for Erdős-Rényi random graphs G(n,p(n)) with p(n)=n-α, α ∈ (0,1). The following zero-one law was proved in 1988 by S. Shelah and J.H. Spencer [1]: if α is irrational then for any first-order property L either the random graph satisfies the property L asymptotically almost surely or it doesn't satisfy (in such cases the random graph is said to obey zero-one law. When α ∈ (0,1) is rational the zero-one law for these graphs doesn't hold.
Let k be a positive integer. Denote by Lk the class of the first-order properties of graphs defined by formulae with quantifier depth bounded by the number k (the sentences are of a finite length). Let us say that the random graph obeys zero-one k-law, if for any first-order property L ∈ Lk either the random graph satisfies the property L almost surely or it doesn't satisfy. Since 2010 we prove several zero-one $k$-laws for rational α from Ik=(0, 1/(k-2)] ∪ [1-1/(2k-1), 1). For some points from Ik we disprove the law. In particular, for α ∈ (0, 1/(k-2)) ∪ (1-1/2k-2, 1) zero-one k-law holds. If α ∈ {1/(k-2), 1-1/(2k-2)}, then zero-one law does not hold (in such cases we call the number α k-critical).
We also disprove the law for some α ∈ [2/(k-1), k/(k+1)]. From our results it follows that zero-one 3-law holds for any α ∈ (0,1). Therefore, there are no 3-critical points in (0,1). Zero-one 4-law holds when α ∈ (0,1/2) ∪ (13/14,1). Numbers 1/2 and 13/14 are 4-critical. Moreover, we know some rational 4-critical and not 4-critical numbers in [7/8,13/14). The number 2/3 is 4-critical. Recently we obtain new results concerning zero-one 4-laws for the neighborhood of the number 2/3.
References
[1] S. Shelah, J.H. Spencer, Zero-one laws for sparse random graphs, J. Amer. Math. Soc.
1: 97–115, 1988.
The document summarizes several results on metric embeddings. It begins by defining metric embeddings and distortion. It then states three theorems:
1) There is a randomized polynomial-time algorithm that embeds any metric space into a tree metric with expected distortion O(log n).
2) Any n-point 2-metric can be embedded into R^O(log n) with distortion 1+ε.
3) There is an algorithm that embeds any metric space into l_1 with distortion O(log k) such that it preserves distances between k given terminal pairs up to a factor of O(log k).
The document then discusses properties and algorithms for embeddings into l_1, l
Introduction to Fourier transform and signal analysis宗翰 謝
The document discusses Fourier analysis techniques. It introduces continuous and discrete Fourier transforms, and covers properties like orthogonality, completeness of basis functions (e.g. cosines and sines), and Fourier series representations of periodic functions like step functions. It also defines the Fourier transform and its properties like linearity, translation, modulation, scaling, and conjugation. Concepts like Dirac delta functions and convolution theory are explained in relation to Fourier analysis.
Density theorems for Euclidean point configurationsVjekoslavKovac1
1. The document discusses density theorems for point configurations in Euclidean space. Density theorems study when a measurable set A contained in Euclidean space can be considered "large".
2. One classical result is that for any measurable set A contained in R2 with positive upper Banach density, there exist points in A whose distance is any sufficiently large real number. This has been generalized to higher dimensions and other point configurations.
3. Open questions remain about determining all point configurations P for which one can show that a sufficiently large measurable set A contained in high dimensional Euclidean space must contain a scaled copy of P.
On the k-Riemann-Liouville fractional integral and applications Premier Publishers
Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary non-integer order. The subject is as old as differential calculus and goes back to times when G.W. Leibniz and I. Newton invented differential calculus. Fractional integrals and derivatives arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of a complex medium. Very recently, Mubeen and Habibullah have introduced the k-Riemann-Liouville fractional integral defined by using the -Gamma function, which is a generalization of the classical Gamma function. In this paper, we presents a new fractional integration is called k-Riemann-Liouville fractional integral, which generalizes the k-Riemann-Liouville fractional integral. Then, we prove the commutativity and the semi-group properties of the -Riemann-Liouville fractional integral and we give Chebyshev inequalities for k-Riemann-Liouville fractional integral. Later, using k-Riemann-Liouville fractional integral, we establish some new integral inequalities.
Eng. Abd Elrhman has over 10 years of experience in network administration and seeks a challenging position to further develop his skills. He has extensive experience designing, implementing, and troubleshooting networks and infrastructures. Some of his qualifications include Cisco certifications, experience with Active Directory, WAN/LAN administration, real-time systems, security, and team leadership.
Quantitative norm convergence of some ergodic averagesVjekoslavKovac1
The document summarizes quantitative estimates for the convergence of multiple ergodic averages of commuting transformations. Specifically, it presents a theorem that provides an explicit bound on the number of jumps in the Lp norm for double averages over commuting Aω actions on a probability space. The proof transfers the structure of the Cantor group AZ to R+ and establishes norm estimates for bilinear averages of functions on R2+. This allows bounding the variation of the double averages and proving the theorem.
A T(1)-type theorem for entangled multilinear Calderon-Zygmund operatorsVjekoslavKovac1
This document summarizes a talk given by Vjekoslav Kovač at a joint mathematics conference. The talk concerned establishing T(1)-type theorems for entangled multilinear Calderón-Zygmund operators. Specifically, Kovač discussed studying multilinear singular integral forms where the functions partially share variables, known as an "entangled structure." He outlined establishing generalized modulation invariance and Lp estimates for such operators. The talk motivated further studying related problems involving bilinear ergodic averages and forms with more complex graph structures. Kovač specialized his techniques to bipartite graphs, multilinear Calderón-Zygmund kernels, and "perfect" dyadic models.
This document discusses multilinear twisted paraproducts, which are generalizations of classical paraproduct operators to higher dimensions. It begins by reviewing classical paraproducts on the real line and their generalization to higher dimensions using dyadic squares. It then discusses complications that arise, such as twisted paraproducts. The document presents a unified framework for studying such operators using bipartite graphs and selections of vertices. It proves a main boundedness result and discusses special cases like classical dyadic paraproducts and dyadic twisted paraproducts. It introduces tools like Bellman functions and calculus of finite differences to analyze estimates for paraproduct-like operators on finite trees of dyadic squares.
Variants of the Christ-Kiselev lemma and an application to the maximal Fourie...VjekoslavKovac1
1. The document discusses variants of the Christ-Kiselev lemma and its application to maximal Fourier restriction estimates.
2. The Christ-Kiselev lemma allows block-diagonal and block-triangular truncations of operators while controlling their operator norms.
3. These lemmas can be used to prove maximal and variational estimates for the restriction of the Fourier transform to surfaces, which has applications in harmonic analysis.
This document summarizes research on norm-variation estimates for ergodic bilinear and multiple averages. It begins by motivating the study of ergodic averages and their convergence properties. Previous results are discussed that provide pointwise convergence and norm estimates for certain cases. The document then presents new norm-variation estimates obtained by the authors for bilinear and multiple ergodic averages over general measure-preserving systems. These estimates bound the number of jumps in the L2 norm as the averages converge. Finally, analogous results are discussed for bilinear averages on R2 and Z2, linking the estimates to established bounds for singular integrals.
Scattering theory analogues of several classical estimates in Fourier analysisVjekoslavKovac1
This document summarizes some classical estimates in Fourier analysis and their analogues in nonlinear Fourier analysis. It discusses Carleson's theorem on convergence of Fourier series and Fourier transforms, Hausdorff-Young inequalities bounding Lp norms, and results on lacunary trigonometric series and products. Open questions are presented about extending these classical estimates to the nonlinear setting of the SU(1,1) Fourier transform and lacunary SU(1,1) trigonometric products.
The document discusses scaling sets and MRA wavelet sets, which are measurable sets associated with multiresolution analyses and wavelets. It provides definitions and theorems characterizing scaling sets and MRA wavelet sets. Some simple examples of scaling sets and MRA wavelet sets are given as finite unions of intervals. The document then poses questions about the properties of general wavelet sets and provides counterexamples to ideas about possible restrictions on their structure. Finally, more complex examples of scaling sets and MRA wavelet sets are constructed using Rademacher functions.
This document summarizes paraproduct operators with general dilations. It defines paraproducts and provides classical examples. It then introduces a non-classical example of paraproducts with respect to general dilations defined by groups of dilations on Cartesian product spaces. The author and co-author establish Lp estimates for such paraproduct operators by applying martingale estimates to dyadic structures and using square functions. The estimates depend only on the dilation structure and hold for certain exponent ranges.
Fourier Series for Continuous Time & Discrete Time SignalsJayanshu Gundaniya
- Fourier introduced Fourier series in 1807 to solve the heat equation in a metal plate. The heat equation is a partial differential equation describing the distribution of heat in a body over time.
- Prior to Fourier's work, there was no known solution to the heat equation in the general case. Fourier's idea was to model a complicated heat source as a superposition of simple sine and cosine waves.
- This superposition or linear combination of sine and cosine waves is called the Fourier series. It allows any periodic function to be decomposed into the sum of simple oscillating functions. Although originally introduced for heat problems, Fourier series have wide applications in mathematics and physics.
The document discusses using the Laplace transform method to solve initial value problems for linear differential equations with constant coefficients. It shows that taking the Laplace transform of the differential equation transforms it into an algebraic equation, avoiding the need to separately solve homogeneous and nonhomogeneous parts. It also explains that determining the inverse Laplace transform to obtain the original function y(t) is the main difficulty, as it requires partial fraction decomposition and knowledge of Laplace transform pairs. Examples are provided to demonstrate solving initial value problems using this method.
Dr. Arpan Bhattacharyya (Indian Institute Of Science, Bangalore)Rene Kotze
1. The document discusses entanglement entropy functionals for higher derivative gravity theories. It proposes new area functionals for computing entanglement entropy in higher derivative theories containing polynomials of curvature tensors.
2. These functionals are derived using the Lewkowycz-Maldacena interpretation of generalized entropy. However, attempting to derive the extremal surface equations from these functionals using bulk equations of motion leads to inconsistencies and ambiguities in some higher derivative theories like Gauss-Bonnet gravity.
3. The document suggests that the source of ambiguity lies in the limiting procedure used to extract the divergences near the conical singularity. Different limiting paths can lead to different extremal surface equations, indicating no unique prescription
Introduction to harmonic analysis on groups, links with spatial correlation.Valentin De Bortoli
This document introduces harmonic analysis on groups and its connections to spatial correlation. It discusses motivations like defining convolution on the sphere S2. Representation theory provides tools to study this, like spherical harmonics which form an orthonormal basis of L2(S2). Spherical CNNs can be understood through the irreducible unitary representations of SO3(R), which are the Wigner D-matrices. The document explores different types of convolutions defined using representations of a group G, like the G-convolution and the (G,π)-convolution. Wavelet transforms provide a link between these convolutions and representations. The goals are to introduce analogues of convolution and Fourier transforms for general groups beyond R2.
Maksim Zhukovskii – Zero-one k-laws for G(n,n−α)Yandex
We study asymptotical behavior of the probabilities of first-order properties for Erdős-Rényi random graphs G(n,p(n)) with p(n)=n-α, α ∈ (0,1). The following zero-one law was proved in 1988 by S. Shelah and J.H. Spencer [1]: if α is irrational then for any first-order property L either the random graph satisfies the property L asymptotically almost surely or it doesn't satisfy (in such cases the random graph is said to obey zero-one law. When α ∈ (0,1) is rational the zero-one law for these graphs doesn't hold.
Let k be a positive integer. Denote by Lk the class of the first-order properties of graphs defined by formulae with quantifier depth bounded by the number k (the sentences are of a finite length). Let us say that the random graph obeys zero-one k-law, if for any first-order property L ∈ Lk either the random graph satisfies the property L almost surely or it doesn't satisfy. Since 2010 we prove several zero-one $k$-laws for rational α from Ik=(0, 1/(k-2)] ∪ [1-1/(2k-1), 1). For some points from Ik we disprove the law. In particular, for α ∈ (0, 1/(k-2)) ∪ (1-1/2k-2, 1) zero-one k-law holds. If α ∈ {1/(k-2), 1-1/(2k-2)}, then zero-one law does not hold (in such cases we call the number α k-critical).
We also disprove the law for some α ∈ [2/(k-1), k/(k+1)]. From our results it follows that zero-one 3-law holds for any α ∈ (0,1). Therefore, there are no 3-critical points in (0,1). Zero-one 4-law holds when α ∈ (0,1/2) ∪ (13/14,1). Numbers 1/2 and 13/14 are 4-critical. Moreover, we know some rational 4-critical and not 4-critical numbers in [7/8,13/14). The number 2/3 is 4-critical. Recently we obtain new results concerning zero-one 4-laws for the neighborhood of the number 2/3.
References
[1] S. Shelah, J.H. Spencer, Zero-one laws for sparse random graphs, J. Amer. Math. Soc.
1: 97–115, 1988.
The document summarizes several results on metric embeddings. It begins by defining metric embeddings and distortion. It then states three theorems:
1) There is a randomized polynomial-time algorithm that embeds any metric space into a tree metric with expected distortion O(log n).
2) Any n-point 2-metric can be embedded into R^O(log n) with distortion 1+ε.
3) There is an algorithm that embeds any metric space into l_1 with distortion O(log k) such that it preserves distances between k given terminal pairs up to a factor of O(log k).
The document then discusses properties and algorithms for embeddings into l_1, l
Introduction to Fourier transform and signal analysis宗翰 謝
The document discusses Fourier analysis techniques. It introduces continuous and discrete Fourier transforms, and covers properties like orthogonality, completeness of basis functions (e.g. cosines and sines), and Fourier series representations of periodic functions like step functions. It also defines the Fourier transform and its properties like linearity, translation, modulation, scaling, and conjugation. Concepts like Dirac delta functions and convolution theory are explained in relation to Fourier analysis.
Density theorems for Euclidean point configurationsVjekoslavKovac1
1. The document discusses density theorems for point configurations in Euclidean space. Density theorems study when a measurable set A contained in Euclidean space can be considered "large".
2. One classical result is that for any measurable set A contained in R2 with positive upper Banach density, there exist points in A whose distance is any sufficiently large real number. This has been generalized to higher dimensions and other point configurations.
3. Open questions remain about determining all point configurations P for which one can show that a sufficiently large measurable set A contained in high dimensional Euclidean space must contain a scaled copy of P.
On the k-Riemann-Liouville fractional integral and applications Premier Publishers
Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary non-integer order. The subject is as old as differential calculus and goes back to times when G.W. Leibniz and I. Newton invented differential calculus. Fractional integrals and derivatives arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of a complex medium. Very recently, Mubeen and Habibullah have introduced the k-Riemann-Liouville fractional integral defined by using the -Gamma function, which is a generalization of the classical Gamma function. In this paper, we presents a new fractional integration is called k-Riemann-Liouville fractional integral, which generalizes the k-Riemann-Liouville fractional integral. Then, we prove the commutativity and the semi-group properties of the -Riemann-Liouville fractional integral and we give Chebyshev inequalities for k-Riemann-Liouville fractional integral. Later, using k-Riemann-Liouville fractional integral, we establish some new integral inequalities.
Eng. Abd Elrhman has over 10 years of experience in network administration and seeks a challenging position to further develop his skills. He has extensive experience designing, implementing, and troubleshooting networks and infrastructures. Some of his qualifications include Cisco certifications, experience with Active Directory, WAN/LAN administration, real-time systems, security, and team leadership.
Math2910 and Math2920 are mathematics courses designed for teachers to take concurrently or separately. While aimed at teachers, they are not education courses but rather focus on mathematics content and thinking. The courses aim to provide a profound understanding of mathematical content and thinking in order to effectively teach those subjects and lead to high student outcomes.
The method of Gauss, also known as the method of simple elimination of Gauss, is a technique used to solve systems of equations. It involves two phases: elimination of the unknowns and back substitution. The method works by multiplying rows by scalars and adding/subtracting rows to eliminate unknowns from the system one by one. Rounding errors can occur and the method fails if a row is divided by zero. Systems that are not well conditioned, where small coefficient changes yield large solution variations, can also be problematic for the Gauss method.
SMS Brandname Marketing hình thức quảng cáo SMS hiển thị tiêu đề tin nhắn SMS (sender) là tên thương hiệu/ công ty mà ai cũng đọc mà không sợ spam.
Công ty FTL rất vui cung cấp cho bạn dịch vụ trên giao diện Guitin.vn
Cutting, drilling, and painting are the primary manufacturing processes. Cutting divides the plastic bottle into three parts for assembly. Drilling makes holes for flies to enter. Painting applies a black coating to attract flies to the internal light.
The document summarizes key takeaways from the PRSA ICON conference in 3 or fewer sentences each. It discusses the importance of PR reporting directly to C-suites, internal corporate social collaboration, overcoming a bad reputation through demonstrated change, the need for regulated industries to participate in social media conversations, integrating loyalty programs via mobile, creating opportunities, the importance of face-to-face communication, recruiting top talent through perks and professional development, using transparency to help consumer brands, gamification as an effective training tool, actions speaking louder than words, and preparing clients to be their own PR representatives.
This document analyzes the similarities and differences between characters in the films "The Shadow" and "Assassins", and between the protagonists in "The Shadow" and "Taken". For the villains in "The Shadow" and "Assassins", it notes that both are males who stalk and hunt their victims in dark clothing without being detected. However, they have different ages, targets, and motives. For the protagonists, it states that both young female leads put themselves in vulnerable situations and were stalked, but "Taken" involves kidnapping while weapons are not present in "The Shadow".
This document defines the thriller genre of film and its key characteristics. Thrillers use suspense, tension, and excitement to stimulate the viewer's emotions. They are fast-paced films that employ literary devices like plot twists and red herrings. Common thriller subgenres include crime, mystery, psychological, and sci-fi films. Thrillers feature tension and suspense, focus on mysteries that must be solved, involve heroes and villains, include climactic sequences, and emphasize fast-paced action over contemplation. Pulp Fiction is provided as an example of a crime thriller, noting its violent and shocking content that contributes to its 18 age rating.
Through collecting audience feedback on his music video "Lucky Strike", Symon Aguilar learned that his target audience had changed from young male teenagers to young female teenagers. He discovered this by analyzing comments and social media posts about the band Maroon 5, whose song was featured in the video. Symon conducted focus groups to get direct feedback on the rough cut. Participants said the narrative was unclear and needed strengthening. Symon incorporated this feedback to improve the music video's understandability.
O documento resume a implantação da República em Portugal após o regicídio de 1908, incluindo a sucessão de D. Manuel II, as razões para a queda da monarquia, a nova bandeira e hino nacional da república.
Let G be a simple graph with n vertices, and λ1, · · · , λn be the eigenvalues of its adjacent matrix. The Estrada index of G is a graph invariant, defined as EE = i e n i 1 ,, is proposed as a measure of branching in alkanes. In this paper, we obtain two candidates which have the fourth largest EE among trees with n vertices
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Abstract: We extend some existing results on the zeros of polar derivatives of polynomials by considering more general coefficient conditions. As special cases the extended results yield much simpler expressions for the upper bounds of zeros than those of the existing results.
Mathematics Subject Classification: 30C10, 30C15.Keywords: Zeros of polynomial, Eneström - Kakeya theorem, Polar derivatives.
Title: On the Zeros of Polar Derivatives
Author: P. Ramulu, G.L. Reddy
ISSN 2350-1022
International Journal of Recent Research in Mathematics Computer Science and Information Technology
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This document presents a new version of the quasi-boundary value method for solving the 1-D nonlinear ill-posed heat equation. The method approximates the solution of the nonlinear backward problem by solving a regularized problem. Error estimates between the exact and approximate solutions are provided. Under additional smoothness assumptions on the exact solution, an improved error estimate of O(ε(t+m)/(T+m)) is proved, where ε is a regularization parameter and t, T, m are time variables. This error estimate converges to zero faster than previous estimates as t approaches zero. A numerical experiment is also presented to demonstrate the effectiveness of the new method.
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The document describes a method for canonizing graphs of bounded treewidth in AC1 complexity. It presents the following:
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11.fixed point theorem of discontinuity and weak compatibility in non complet...Alexander Decker
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The document presents an orthogonal decomposition of the Hilbert space L2(I) over the unit interval I = [0,1]. It establishes that L2(I) can be written as the orthogonal direct sum of two closed subspaces: A2(I), the space of square integrable functions whose first derivative is zero, and the image of the derivative operator applied to a traceless Sobolev space W1,2_0(I). It defines the corresponding projection operators and proves several properties, including that the projections are idempotent and their images are orthogonal. It also provides examples that illustrate the decomposition for some elementary functions.
Geometric and viscosity solutions for the Cauchy problem of first orderJuliho Castillo
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IJCER (www.ijceronline.com) International Journal of computational Engineering research
1. International Journal Of Computational Engineering Research (ijceronline.com) Vol. 2 Issue. 5
Convexity of Minimal Total Dominating Functions Of Quadratic
Residue Cayley Graphs
1
S.Jeelani Begum 2 B.Maheswari
1
Dept. of Mathematics, Madanapalle Institute of Technology & Science, Madanapalle, Andhra Pradesh, India
2
Dept. of Applied Mathematics, S.P.Women’s University, Tirupati, Andhra Pradesh, India
Abstract
Nathanson [17] paved the way for the emergence of a new class of graphs, namely,Arithmetic Graphs by introducing
the concepts of Number Theory, particularly, the Theory of congruences in Graph Theory. Cayley graphs are another
class of graphs associated with the elements of a group. If this group is associated with some arithmetic function then the
Cayley graph becomes an arithmetic graph. Domination theory is an important branch of Graph Theory and has many
applications in Engineering, Communication Networks and many others. In this paper we study the minimal total
dominating functions of Quadratic Residue Cayley graphs and discuss the convexity of these functions in different cases.
Keywords: Arithmetic graph, Cayley graph, Total dominating set, Neighbourhood set, Quadratic Residue Cayley Graph,
Total Dominating Functions, Minimal Total Dominating Functions, Convexity.
1. Introduction
There is a class of graphs, namely, Cayley graphs, whose vertex set V is the set of elements of a group (G, .) and
two vertices x and y of G are adjacent if and only if xy-1 is in some symmetric subset S of G. A subset S of a group (G, .) is
called a symmetric subset of G if s-1 is in S for all s in S. If the group (G, .) is the additive group (Zn , ) of integers
0,1,2,……..,n-1 modulo n, and the symmetric set S is associated with some arithmetic function, then the Cayley Graph may
be treated as an arithmetic graph. In this paper we consider Quadratic Residue Cayley graphs. A detailed study of
convexity and minimality of dominating functions and total dominating functions are given in Cockayne et al. [2,3-12]
Chesten et al. [1], Yu [18] and Domke et al. [13,14]. Algorithmic complexity results for these parameters are given in
Laskar et al. [15] and Cockayne et al.[3].We start with the definition of a Quadratic Residue Cayley graph.
Quadratic Residue Cayley Graph
Let p be an odd prime and n, a positive integer such that n ≡ 0 (mod p). If the quadratic congruence,
x 2 n (mod p ) has a solution then, n is called a quadratic residue mod p.
The Quadratic Residue Cayley graph G(Zp , Q), is the Cayley graph associated with the set of quadratic residues
modulo an odd prime p, which is defined as follows. Let p be an odd prime, S, the set of quadratic residues modulo p and
let S* = {s, n - s / s S, s ≠ n }. The quadratic residue Cayley graph G(Zp , Q) is defined as the graph whose vertex set
is Zp = { 0 , 1 , 2 , …… p – 1} and the edge set is E = { ( x , y ) / x – y or y - x S*}.
For example the graph of G(Z19, Q) is given below.
Issn 2250-3005(online) September| 2012 Page 1249
2. International Journal Of Computational Engineering Research (ijceronline.com) Vol. 2 Issue. 5
2. Total Dominating Functions
Total Dominating Set : Let G(V, E) be a graph without isolated vertices. A subset T of V is called a total dominating set
(TDS) if every vertex in V is adjacent to at least one vertex in T.
Minimal Total Dominating Set : If no proper subset of T is a total dominating set, then T is called a minimal total
dominating set (MTDS) of G.
Neighbourhood Set : The open neighbourhood of a vertex u is the set of vertices adjacent to u and is denoted by N(u).
Total Dominating Function : Let G(V, E) be a graph without isolated vertices. A function f : V [0,1] is called
a total dominating function (TDF) if f ( N (v))
uN ( v )
f (u ) 1 for all v V.
Minimal Total Dominating Function : Let f and g be functions from V→ [0,1]. We define f < g if f(u) g(u), u V,
with strict inequality for at least one vertex u. A TDF f is called a minimal total dominating function (MTDF) if for all
g < f, g is not a TDF.
We require the following results whose proofs are presented in [16].
Lemma 1: The Quadratic Residue Cayley graph G(Zp , Q) is S * - regular, and the number of edges in G(Zp, Q)
Z p S*
is .
2
Theorem 1: The Quadratic Residue Cayley graph G(Zp , Q) is complete if p is of the form 4m+3.
Suppose p = 4m + 3. Then G(Zp, Q) is complete. Then each vertex is of degree p – 1. That is the graph G(Zp, Q)
is (p – 1) – regular.
S* = p – 1.
Hence each N(v) consists of p-1 vertices , vV.
We consider the case p = 4m+3 of G(Zp , Q ) and prove the following results.
3. MAIN RESULTS
Theorem 3.1: Let T be a MTDS of G(Zp, Q). Let f : V [0,1] be a function defined by
1, if v T ,
f (v )
0, if v V T .
Then f becomes a MTDF of G(Zp, Q).
Proof: Consider G(Zp, Q). Let T be a MTDS of G(Zp, Q).
Since G(Zp, Q) is complete, T 2.
Also every neighbourhood N(v) of v V consists of (p-1) –vertices.
Let f be a function defined on V as in the hypothesis.
Then the summation values taken over the neighbourhood N(v) of v V is
2, if u V T ,
f (u )
uN ( v ) 1, if u T .
Therefore
uN ( v )
f (u ) 1 , v V.
This implies that f is a TDF.
We now check for the minimality of f.
Define a function g : V [0,1] by
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3. International Journal Of Computational Engineering Research (ijceronline.com) Vol. 2 Issue. 5
r , if v T , v vk ,
g (v) 1, if v T {vk },
0, if v V T .
where 0 < r < 1 and vk V.
Since strict inequality holds at the vertex v = vk T of V, it follows that g < f.
Then
1 r , if v V T ,
v) g (u) 1, if v T , v vk ,
uN ( r , if v T , v vk .
Thus g (u ) 1, v V .
uN ( v )
This implies that g is not a TDF. Since r < 1 is arbitrary it follows that there exists no g < f such that g is a TDF.
Thus f is a MTDF.
Theorem 3.2: Let T1 and T2 be two MTDSs of G(Zp, Q) and f1 , f 2 be two functions of G(Zp, Q) defined by
1, if v T1 ,
f1 v
0, otherwise.
1, if v T2 ,
and f2 v
0, otherwise.
Then the convex combination of f1 and f 2 becomes a TDF of G(Zp, Q) but not minimal.
Proof: Let T1 and T2 be two MTDSs of G(Zp, Q) and f1 , f2 be the functions defined as in the hypothesis. Then by
Theorem 3.1, the above functions are MTDFs of G(Zp, Q).
Leth(v) f1 (v) f 2 (v) , where 1 and 0 1, 0 1.
Case 1: Suppose T1 and T2 are such that T1 T2 .
Then the possible values of h(v) are
, if v T1 and v T2 ,
, if v T2 and v T1 ,
h (v )
, if v {T1 T2 },
0,
otherwise.
Since each neighbourhood N(v) of v in V consists of (p-1) vertices of G(Zp, Q), the summation value of h(v) taken over
N(v) is
, if v T1 and v T2 ,
, if v T2 and v T1 ,
v) h(u) , if v {T1 T2 },
uN (
2( ),
otherwise.
This implies that h(u ) 1 , v V.
uN ( v )
Therefore h is a TDF. We now check for the minimality of h.
Define a function g : V [0,1] by
Issn 2250-3005(online) September| 2012 Page 1251
4. International Journal Of Computational Engineering Research (ijceronline.com) Vol. 2 Issue. 5
, if v T1 and v T2 ,
, if v T2 and v T1 ,
g (v )
r , if v {T1 T2 },
0,
otherwise.
where 0 < r < 1.
Since strict inequality holds at v {T1 T2 } , it follows that g < h.
r , if v T1 and v T2 ,
r , if v T2 and v T1 ,
Now v ) g (u) , if v {T1 T2 },
uN (
r ,
otherwise.
where r 1 and r 1 .
Thus g (u ) 1, v V .
uN ( v )
Therefore g is a TDF. Hence it follows that h is a TDF but not minimal.
Case 2: Suppose T1 and T2 are disjoint.
Then the possible values of h(v) are
, if v T1 ,
h(v ) , if v T2 ,
0,
otherwise.
Since each neighbourhood N(v) of v in V consists of (p-1) vertices of G(Zp, Q), the summation value of h(v) taken over
N(v) is
2 , if v T1 ,
v) h(u) 2 , if v T2 ,
uN ( 2( ),
otherwise.
This implies
uN ( v )
h(u ) 1 , v V, since 1 .
Therefore h is a TDF. We now check for the minimality of h.
Define a function g : V [0,1] by
r , if v T1 , v vi ,
, if v T1 , v vi ,
g (v )
, if v T2 ,
0,
otherwise.
where 0 < r < .
Since strict inequality holds at v vi T1 , it follows that g < h.
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5. International Journal Of Computational Engineering Research (ijceronline.com) Vol. 2 Issue. 5
2 , if v T1 , v vi ,
r 2 , if v T1 , v vi ,
Then g (u )
uN ( v ) r , if v T2 ,
r 2 , otherwise.
where r 2 2(1 ) 2 1 .
i.e., r 2 1 .
Thus g (u ) 1, v V .
uN ( v )
Therefore g is a TDF. Hence it follows that h is not a MTDF.
References
[1]. G.A. Chesten , G. Fricke , S.T. Hedetniemi and D.P. Jacobs. On the Computational complexity of upper fractional
domination, Discrete. Appl. Math., 27: 195-207, 1990.
[2]. E.J. Cockayne, R.M.. Dawes and S.T. Hedetniemi. Total domination in graphs, Networks, 10: 211- 219, 1980.
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