Guarding Terrains though the Lens of Parameterized ComplexityAkankshaAgrawal55
The Terrain Guarding problem is a well-studied visibility problem in Discrete and Computational Geometry. So far, the understanding of the parameterized complexity of Terrain Guarding has been very limited, and, more generally, exact (exponential-time) algorithms for visibility problem are extremely scarce. In this talk we will look at two results regarding Terrain Guarding, from the viewpoint of parameterized complexity. Both of these results will utilize new and known structural properties of terrains. The first result that we will see is a polynomial kernel for Terrain Guarding, when parameterized by the number of reflex vertices. (A reflex vertex is a vertex of the terrain where the angle is at least 180 degrees.) The next result will be regarding a special version of Terrain Guarding, called Orthogonal Terrain Guarding. We will consider the above problem when parameterized by the number of minima in the input terrain, and obtain a dynamic programming based XP algorithm for it.
This presentation is the one that I gave at the Parameterized Complexity Seminar (https://sites.google.com/view/pcseminar).
This is the talk that I gave at the Dagstuhl seminar ''New Horizons in Parameterized Complexity'', 2019. This is about a polynomial kernel for Interval Vertex Deletion. The main focus of this talk is to obtain a polynomial kernel for a slightly larger parameter, which is the vertex cover number of the input graph.
Guarding Terrains though the Lens of Parameterized ComplexityAkankshaAgrawal55
The Terrain Guarding problem is a well-studied visibility problem in Discrete and Computational Geometry. So far, the understanding of the parameterized complexity of Terrain Guarding has been very limited, and, more generally, exact (exponential-time) algorithms for visibility problem are extremely scarce. In this talk we will look at two results regarding Terrain Guarding, from the viewpoint of parameterized complexity. Both of these results will utilize new and known structural properties of terrains. The first result that we will see is a polynomial kernel for Terrain Guarding, when parameterized by the number of reflex vertices. (A reflex vertex is a vertex of the terrain where the angle is at least 180 degrees.) The next result will be regarding a special version of Terrain Guarding, called Orthogonal Terrain Guarding. We will consider the above problem when parameterized by the number of minima in the input terrain, and obtain a dynamic programming based XP algorithm for it.
This presentation is the one that I gave at the Parameterized Complexity Seminar (https://sites.google.com/view/pcseminar).
This is the talk that I gave at the Dagstuhl seminar ''New Horizons in Parameterized Complexity'', 2019. This is about a polynomial kernel for Interval Vertex Deletion. The main focus of this talk is to obtain a polynomial kernel for a slightly larger parameter, which is the vertex cover number of the input graph.
Ilya Shkredov – Subsets of Z/pZ with small Wiener norm and arithmetic progres...Yandex
It is proved that any subset of Z/pZ, p is a prime number, having small Wiener norm (l_1-norm of its Fourier transform) contains a subset which is close to be an arithmetic progression. We apply the obtained results to get some progress in so-called Littlewood conjecture in Z/pZ as well as in a quantitative version of Beurling-Helson theorem.
The Art Gallery problem is a fundamental visibility problem in Computational Geometry, introduced by Klee in 1973. The input consists of a simple polygon P, (possibly infinite) sets X and Y of points within P, and an integer k, and the objective is to decide whether at most k guards can be placed on points in X so that every point in Y is visible to at least one guard. In the classic formulation of Art Gallery, X and Y consist of all the points within P. Other well-known variants restrict X and Y to consist either of all the points on the boundary of P or of all the vertices of P. The above mentioned variants of Art Gallery are all W[1]-hard with respect to k [Bonnet and Miltzow, ESA'16]. Given the above result, the following question was posed by Giannopoulos [Lorentz Center Workshop, 2016].
``Is Art Gallery FPT with respect to the number of reflex vertices?''
In this talk, we will obtain a positive answer to the above question, for some variants of the Art Gallery problem. By utilising the structural properties of ``almost convex polygons'', we design a two-stage reduction from (Vertex,Vertex)-Art Gallery to a new CSP problem where constraints have arity two and involve monotone functions. For the above special version of CSP, we obtain a polynomial time algorithm. Sieving these results, we obtain an FPT algorithm for (Vertex,Vertex)-Art Gallery, when parameterized by the number of reflex vertices. We note that our approach also extends to (Vertex,Boundary)-Art Gallery and (Boundary,Vertex)-Art Gallery.
This slides are from a talk that I gave at the Algorithms Seminar at Tel-Aviv University.
Solving connectivity problems via basic Linear Algebracseiitgn
Directed reachability and undirected connectivity are well studied problems in Complexity Theory. Reachability/Connectivity between distinct pairs of vertices through disjoint paths are well known but hard variations. We talk about recent algorithms to solve variants and restrictions of these problems in the static and dynamic settings by reductions to the determinant.
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.
Slide set presented for the Wireless Communication module at Jacobs University Bremen, Fall 2015.
Teacher: Dr. Stefano Severi, assistant: Andrei Stoica
Slides from my introductory talk on Lossy Kernelization at the Parameterized Complexity Summer School 2017, co-located with ALGO 2017 and held at TU Wien, Vienna.
Approximation Algorithms for the Directed k-Tour and k-Stroll ProblemsSunny Kr
In the Asymmetric Traveling Salesman Problem (ATSP), the input is a directed n-vertex graph G = (V; E) with nonnegative edge lengths, and the goal is to nd a minimum-length tour, visiting
each vertex at least once. ATSP, along with its undirected counterpart, the Traveling Salesman
problem, is a classical combinatorial optimization problem
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.
Quantum bialgebras derivable from Uq(sl2) which contain idempotents and von Neumann regular Cartan-like generators are introduced and investigated. Various types of antipodes (invertible and von Neumann regular) on these bialgebras are constructed, which leads to a Hopf algebra structure and a von Neumann-Hopf algebra structure, respectively. For them, explicit forms of some particular R-matrices (also, invertible and von Neumann regular) are presented, and the latter respects the Pierce decomposition.
Building Compatible Bases on Graphs, Images, and ManifoldsDavide Eynard
Spectral methods are used in computer graphics, machine learning, and computer vision, where many important problems boil down to constructing a Laplacian operator and finding its eigenvalues and eigenfunctions. We show how to generalize spectral geometry to multiple data spaces. Our construction is based on the idea of simultaneous diagonalization of Laplacian operators. We describe this problem and discuss numerical methods for its solution. We provide several synthetic and real examples of manifold learning, object classification, and clustering, showing that the joint spectral geometry better captures the inherent structure of multi-modal data.
Talk at SIAM-IS 2014 (http://www.math.hkbu.edu.hk/SIAM-IS14/). A big thanks to Michael Bronstein for providing a great set of slides this presentation is a mere extension of.
Ilya Shkredov – Subsets of Z/pZ with small Wiener norm and arithmetic progres...Yandex
It is proved that any subset of Z/pZ, p is a prime number, having small Wiener norm (l_1-norm of its Fourier transform) contains a subset which is close to be an arithmetic progression. We apply the obtained results to get some progress in so-called Littlewood conjecture in Z/pZ as well as in a quantitative version of Beurling-Helson theorem.
The Art Gallery problem is a fundamental visibility problem in Computational Geometry, introduced by Klee in 1973. The input consists of a simple polygon P, (possibly infinite) sets X and Y of points within P, and an integer k, and the objective is to decide whether at most k guards can be placed on points in X so that every point in Y is visible to at least one guard. In the classic formulation of Art Gallery, X and Y consist of all the points within P. Other well-known variants restrict X and Y to consist either of all the points on the boundary of P or of all the vertices of P. The above mentioned variants of Art Gallery are all W[1]-hard with respect to k [Bonnet and Miltzow, ESA'16]. Given the above result, the following question was posed by Giannopoulos [Lorentz Center Workshop, 2016].
``Is Art Gallery FPT with respect to the number of reflex vertices?''
In this talk, we will obtain a positive answer to the above question, for some variants of the Art Gallery problem. By utilising the structural properties of ``almost convex polygons'', we design a two-stage reduction from (Vertex,Vertex)-Art Gallery to a new CSP problem where constraints have arity two and involve monotone functions. For the above special version of CSP, we obtain a polynomial time algorithm. Sieving these results, we obtain an FPT algorithm for (Vertex,Vertex)-Art Gallery, when parameterized by the number of reflex vertices. We note that our approach also extends to (Vertex,Boundary)-Art Gallery and (Boundary,Vertex)-Art Gallery.
This slides are from a talk that I gave at the Algorithms Seminar at Tel-Aviv University.
Solving connectivity problems via basic Linear Algebracseiitgn
Directed reachability and undirected connectivity are well studied problems in Complexity Theory. Reachability/Connectivity between distinct pairs of vertices through disjoint paths are well known but hard variations. We talk about recent algorithms to solve variants and restrictions of these problems in the static and dynamic settings by reductions to the determinant.
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.
Slide set presented for the Wireless Communication module at Jacobs University Bremen, Fall 2015.
Teacher: Dr. Stefano Severi, assistant: Andrei Stoica
Slides from my introductory talk on Lossy Kernelization at the Parameterized Complexity Summer School 2017, co-located with ALGO 2017 and held at TU Wien, Vienna.
Approximation Algorithms for the Directed k-Tour and k-Stroll ProblemsSunny Kr
In the Asymmetric Traveling Salesman Problem (ATSP), the input is a directed n-vertex graph G = (V; E) with nonnegative edge lengths, and the goal is to nd a minimum-length tour, visiting
each vertex at least once. ATSP, along with its undirected counterpart, the Traveling Salesman
problem, is a classical combinatorial optimization problem
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.
Quantum bialgebras derivable from Uq(sl2) which contain idempotents and von Neumann regular Cartan-like generators are introduced and investigated. Various types of antipodes (invertible and von Neumann regular) on these bialgebras are constructed, which leads to a Hopf algebra structure and a von Neumann-Hopf algebra structure, respectively. For them, explicit forms of some particular R-matrices (also, invertible and von Neumann regular) are presented, and the latter respects the Pierce decomposition.
Building Compatible Bases on Graphs, Images, and ManifoldsDavide Eynard
Spectral methods are used in computer graphics, machine learning, and computer vision, where many important problems boil down to constructing a Laplacian operator and finding its eigenvalues and eigenfunctions. We show how to generalize spectral geometry to multiple data spaces. Our construction is based on the idea of simultaneous diagonalization of Laplacian operators. We describe this problem and discuss numerical methods for its solution. We provide several synthetic and real examples of manifold learning, object classification, and clustering, showing that the joint spectral geometry better captures the inherent structure of multi-modal data.
Talk at SIAM-IS 2014 (http://www.math.hkbu.edu.hk/SIAM-IS14/). A big thanks to Michael Bronstein for providing a great set of slides this presentation is a mere extension of.
Nodal Domain Theorem for the p-Laplacian on Graphs and the Related Multiway C...Francesco Tudisco
We consider the p-Laplacian on discrete graphs, a nonlinear operator that generalizes the standard graph Laplacian (obtained for p=2). We consider a set of variational eigenvalues of this operator and analyze the nodal domain count of the corresponding eigenfunctions. In particular, we show that the famous Courant’s nodal domain theorem for the linear Laplacian carries over almost unchanged to the nonlinear case. Moreover, we use the nodal domains to prove a higher-order Cheeger inequality that relates the k-way graph cut to the k-th variational eigenvalue of the p-Laplacian
On the equality of the grundy numbers of a graphijngnjournal
Our work becomes integrated into the general problem of the stability of the network ad hoc. Some, works attacked(affected) this problem. Among these works, we find the modelling of the network ad hoc in the form of a graph. Thus the problem of stability of the network ad hoc which corresponds to a problem of allocation of frequency amounts to a problem of allocation of colors in the vertex of graph. we present use a parameter of coloring " the number of Grundy”. The Grundy number of a graph G, denoted by Γ(G), is the largest k such that G has a greedy k-coloring, that is a coloring with colours obtained by applying the greedy algorithm according to some ordering of the vertices of G. In this paper, we study the Grundy number of the lexicographic, Cartesian and direct products of two graphs in terms of the Grundy numbers of these graphs.
On the Equality of the Grundy Numbers of a Graphjosephjonse
Our work becomes integrated into the general problem of the stability of the network ad hoc. Some, works attacked(affected) this problem. Among these works, we find the modelling of the network ad hoc in the form of a graph. Thus the problem of stability of the network ad hoc which corresponds to a problem of allocation of frequency amounts to a problem of allocation of colors in the vertex of graph. we present use a parameter of coloring " the number of Grundy”. The Grundy number of a graph G, denoted by Γ(G), is the largest k such that G has a greedy k-coloring, that is a coloring with colours obtained by applying the greedy algorithm according to some ordering of the vertices of G. In this paper, we study the Grundy number of the lexicographic, Cartesian and direct products of two graphs in terms of the Grundy numbers of these graphs.
A crystallographic group is a group acting on R^n that contains a translation subgroup Z^n as a finite index subgroup. Here we consider which Coxeter groups are crystallographic groups. We also expose the enumeration in dimension 2 and 3. Then we shortly give the principle under which the enumeration of N dimensional crystallographic groups is done.
Complexity Classes and the Graph Isomorphism Problemcseiitgn
The Graph Isomorphism problem is one of the few problems in NP, but not expected to be NP complete and not known to be in P.In this talk I will review some of the attempts that have been made in order to provide a better classification of the problem in terms of complexity classes reviewing upper and lower bounds and illustrating in this way the utility of several complexity classes.
Safalta Digital marketing institute in Noida, provide complete applications that encompass a huge range of virtual advertising and marketing additives, which includes search engine optimization, virtual communication advertising, pay-per-click on marketing, content material advertising, internet analytics, and greater. These university courses are designed for students who possess a comprehensive understanding of virtual marketing strategies and attributes.Safalta Digital Marketing Institute in Noida is a first choice for young individuals or students who are looking to start their careers in the field of digital advertising. The institute gives specialized courses designed and certification.
for beginners, providing thorough training in areas such as SEO, digital communication marketing, and PPC training in Noida. After finishing the program, students receive the certifications recognised by top different universitie, setting a strong foundation for a successful career in digital marketing.
Normal Labour/ Stages of Labour/ Mechanism of LabourWasim Ak
Normal labor is also termed spontaneous labor, defined as the natural physiological process through which the fetus, placenta, and membranes are expelled from the uterus through the birth canal at term (37 to 42 weeks
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
1. Preliminaries
Spectral synthesis
Spectral synthesis for A(G)
Spectral synthesis for A(G/K)
Spectral synthesis for A(KG/K)
References
SPECTRAL SYNTHESIS PROBLEM
FOR
FOURIER ALGEBRAS
Kunda Chowdaiah
NISER
November 25, 2011
Kunda Chowdaiah SPECTRAL SYNTHESIS PROBLEM FOR FOURIER AL
2. Preliminaries
Spectral synthesis
Spectral synthesis for A(G)
Spectral synthesis for A(G/K)
Spectral synthesis for A(KG/K)
References
Preliminaries
Banach algebra
A Banach algebra A is a Banach space that is also an algebra and
satisfying the following condition
xy ≤ x y , ∀ x , y ∈ A
.
An algebra A is called commutative if xy = yx ∀x , y ∈ A.
For any Banach algebra A, We Dene
∆(A) = {ϕ : A → C | ϕ is non-zero homomorphism}.
∆(A) ⊆ A∗ with ϕ ≤ 1 ∀ ϕ ∈ ∆(A)
Let A be a commutative Banach algebra,∆(A) with the weak *
topology forms a locally compact space called Gelfand space.
Kunda Chowdaiah SPECTRAL SYNTHESIS PROBLEM FOR FOURIER AL
3. Preliminaries
Spectral synthesis
Spectral synthesis for A(G)
Spectral synthesis for A(G/K)
Spectral synthesis for A(KG/K)
References
For every x ∈ A we dene x : ∆(A) → C by x (ϕ) = ϕ(x ) then x is
a continuous function which vanish at innity.
x → x is called Gelfand transform.The Gelfand transform is an
algebra homomorphism from A into C0(∆(A)).
Semisimple Banach algebra
Let A be a commutative Banach algebra, A is called semisimple if
{ker ϕ : ϕ ∈ ∆(A)} = {0}
Regular Banach algebra
A commutative Banach algebra A is called regular if given any
closed sub set E of ∆(A) , ϕ0 ∈ ∆(A)E then there exist x ∈ A
such that x (ϕ0) = 1 and x (ϕ) = 0 for all ϕ ∈ E
Kunda Chowdaiah SPECTRAL SYNTHESIS PROBLEM FOR FOURIER AL
4. Preliminaries
Spectral synthesis
Spectral synthesis for A(G)
Spectral synthesis for A(G/K)
Spectral synthesis for A(KG/K)
References
Denition
Union problem
Results and examples
Spectral synthesis problem
Let A be a regular semisimple commutative Banach algebra.
For any closed ideal I ofA the hull h(I ) of I is dened by
h(I ) = {ϕ ∈ ∆(A) : ϕ(I ) = 0}
Associated to each closed subset E of ∆(A) two distinguished
ideals with hull equal to E namely
I (E ) = {x ∈ A : x (ϕ) = 0 for all ϕ ∈ E }
j (E ) = {x ∈ A : supp(x ) is compact and supp(x ) ∩ E = φ}
J(E ) = j (E )
Then I (E ) is the largest ideal with hull E and J(E ) is the smallest
ideal with hull E .
Kunda Chowdaiah SPECTRAL SYNTHESIS PROBLEM FOR FOURIER AL
5. Preliminaries
Spectral synthesis
Spectral synthesis for A(G)
Spectral synthesis for A(G/K)
Spectral synthesis for A(KG/K)
References
Denition
Union problem
Results and examples
Denition
Let A be semisimple regular commutative Banach algebra , E be a
closed sub set of ∆(A)
(i) E is called set of spectral synthesis if I (E ) = j (E )
We say that spectral synthesis holds for A if every closed subset of
∆(A) is a set of spectral synthesis.
(ii) E is called Ditkin set if given x ∈ I (E ) there exist a sequence
(yk )k in j (E ) such that xyk → x as k → ∞. i.e., x ∈ xj (E ).
Remark
Every Ditkin set is a set of spectral synthesis.
Kunda Chowdaiah SPECTRAL SYNTHESIS PROBLEM FOR FOURIER AL
6. Preliminaries
Spectral synthesis
Spectral synthesis for A(G)
Spectral synthesis for A(G/K)
Spectral synthesis for A(KG/K)
References
Denition
Union problem
Results and examples
Tauberian
Let A be semisimple regular commutative Banach algebra ,A is
called Tauberian if the set of all x ∈ A , such that x has compact
support is dense in A.
Remark
A is Tauberian if and only if φ is a set of spectral synthesis A.
Theorem
Let A be a Tauberian, suppose that ∆(A) is discrete. Then
spectral synthesis holds for A if and only if x ∈ x A for each x ∈ A.
Kunda Chowdaiah SPECTRAL SYNTHESIS PROBLEM FOR FOURIER AL
7. Preliminaries
Spectral synthesis
Spectral synthesis for A(G)
Spectral synthesis for A(G/K)
Spectral synthesis for A(KG/K)
References
Denition
Union problem
Results and examples
Union problem
The union of two Ditkin sets is Ditkin set.
Question
Is the union of two sets of spectral synthesis is a set of spectral
synthesis ?
Suppose that E1 and E2 are closed subsets of ∆(A) such that
E1 ∩ E2 is a Ditkin set. Then E1 ∪ E2 is a set of spectral
synthesis if and only if both E1 and E2 are sets of spectral
synthesis.
This does not remain true if the hypothesis that E1 ∩ E2 be a
Ditkin set is dropped. In fact, the so-called Mirkil algebra M
M = {f ∈ L
2(T) : f |[−π
2 ,π
2 ] is continuous}.
f =
√
2π f 2 + f |[−π
2 , π
2 ] ∞
Kunda Chowdaiah SPECTRAL SYNTHESIS PROBLEM FOR FOURIER AL
8. Preliminaries
Spectral synthesis
Spectral synthesis for A(G)
Spectral synthesis for A(G/K)
Spectral synthesis for A(KG/K)
References
Denition
Union problem
Results and examples
Results and examples
Let X be a locally compact Hausdor space then Spectral
synthesis holds for C0(X )
Let G be a compact abelian group. Then spectral synthesis
holds for L
1(G ).
[L. Schwartz 1948] The sphere S
n−1 ⊆ R
n
fails to be a set of
spectral synthesis for L
1(R
n
) if n ≥ 3.
A famous theorem due to Malliavin[1959] states that for any
noncompact locally compact abelian group G , spectral
synthesis fails for L
1(G ).
Spectral synthesis fails for the algebra C
1[0, 1] of continuously
dierentiable functions on [0, 1].
Kunda Chowdaiah SPECTRAL SYNTHESIS PROBLEM FOR FOURIER AL
9. Preliminaries
Spectral synthesis
Spectral synthesis for A(G)
Spectral synthesis for A(G/K)
Spectral synthesis for A(KG/K)
References
Fourier algebra
Results of Eymand(1964)
Spectral synthesis for A(G)
Fourier algebra
Let G be a locally compact group, G denote the equivalence class
of unitary representations G If π ∈ G and ξ, η ∈ Hπ where Hπ is a
Hilbert space associated to π.
Then the continuous function πξ,η(x ) = π(x )ξ, η is called
coecient function of π
B (G ) = {πξ,η : π ∈ G ; ξ, η ∈ Hπ}
B (G ) is a commutative Banach algebra with respect to point wise
multiplication called the Fourier-Stieltjes algebra of G .
B (G ) ∩ Cc (G ) in B (G ) is called Fourier algebra and is denoted by
A(G ).
Kunda Chowdaiah SPECTRAL SYNTHESIS PROBLEM FOR FOURIER AL
10. Preliminaries
Spectral synthesis
Spectral synthesis for A(G)
Spectral synthesis for A(G/K)
Spectral synthesis for A(KG/K)
References
Fourier algebra
Results of Eymand(1964)
Spectral synthesis for A(G)
Results of Eymand(1964)
A(G ) is a semisimple regular Tauberian commutative Banach
algebra.
∆(A(G )) = G Here Gelfand transform is an identity map.
A(G )∗ = VN(G )
Here VN(G ) is the closure of linear span of {λ(x ) : x ∈ G } in
B (L
2(G )) with respect to weak operator topology. Here λ is a
left regular representation of G dened by
λ(x )f (y ) = f (x
−1y ) for any x ∈ G , f ∈ L
2(G )
VN(G ) is called von Neumann algebra of group G
Kunda Chowdaiah SPECTRAL SYNTHESIS PROBLEM FOR FOURIER AL
11. Preliminaries
Spectral synthesis
Spectral synthesis for A(G)
Spectral synthesis for A(G/K)
Spectral synthesis for A(KG/K)
References
Fourier algebra
Results of Eymand(1964)
Spectral synthesis for A(G)
We write T , u for the value of T at u. There is a natural action
of A(G ) on VN(G ) given by v .T , u = T , uv .
Let T ∈ VN(G ) the support of T is dened as
suppT = {x ∈ G : λ(x ) is a w-* limit of some uα.T , uα ∈ A(G }
Kunda Chowdaiah SPECTRAL SYNTHESIS PROBLEM FOR FOURIER AL
12. Preliminaries
Spectral synthesis
Spectral synthesis for A(G)
Spectral synthesis for A(G/K)
Spectral synthesis for A(KG/K)
References
Fourier algebra
Results of Eymand(1964)
Spectral synthesis for A(G)
Spectral synthesis for A(G), of abelian group
If G be a locally compact abelian group G is the dual group G then
A(G ) = L
1(G ).
Result
If G is discrete abelian group then spectral synthesis holds for A(G ).
Kunda Chowdaiah SPECTRAL SYNTHESIS PROBLEM FOR FOURIER AL
13. Preliminaries
Spectral synthesis
Spectral synthesis for A(G)
Spectral synthesis for A(G/K)
Spectral synthesis for A(KG/K)
References
Fourier algebra
Results of Eymand(1964)
Spectral synthesis for A(G)
Spectral synthesis for A(G), of non-abelian group
Denition(Kaniuth and Lau, 2001)
A closed subset E of G is called a set of spectral synthesis for
VN(G ) if T ∈ VN(G ) and suppT ⊆ E implies that T ∈ I (E )⊥.
We say that A(G ) admits VN(G )-spectral synthesis if every closed
subset of G is a set of spectral synthesis for VN(G )
Theorem(Kaniuth and Lau, 2001)
Let E be a closed subset of G. Then E is a set of synthesis
synthesis if and only if E a set of spectral synthesis for VN(G ).
Theorem
Let H be a any closed subgroup of a locally compact group G , then
H is a set of spectral synthesis.
Kunda Chowdaiah SPECTRAL SYNTHESIS PROBLEM FOR FOURIER AL
14. Preliminaries
Spectral synthesis
Spectral synthesis for A(G)
Spectral synthesis for A(G/K)
Spectral synthesis for A(KG/K)
References
Results of Forrest (1998)
Spectral synthesis for A(G/K)
Let K be a compact subgroup of G , G /K denote the the
homogeneous space of left cosets of K . we will denote ˜x for the
left coset xK as a element of G /K
Let ϕ : G → G /K be the quotient map then , given any continuous
map ˜u on G /K we can identify ˜u with continuous function on G
denoted by u = ˜u ◦ ϕ
B (G : K ) = {u ∈ B (G ) : u(x ) = u(xk ), ∀x ∈ G , ∀k ∈ K }
A(G : K ) = {u ∈ B (G : K ) : ϕ(supp(u)) be compact in G /K }− . B(G)
Kunda Chowdaiah SPECTRAL SYNTHESIS PROBLEM FOR FOURIER AL
15. Preliminaries
Spectral synthesis
Spectral synthesis for A(G)
Spectral synthesis for A(G/K)
Spectral synthesis for A(KG/K)
References
Results of Forrest (1998)
Results of Forrest (1998)
Properties of A(G /K )
Let K1 and K2 be compact subgroups of G , then
A(G : K1) = A(G : K2) i K1 = K2
A(G /K ) ≡ A(G : K )
∆(A(G /K )) = G /K
A(G /K ) is a semisimple regular commutative Banach algebra .
Kunda Chowdaiah SPECTRAL SYNTHESIS PROBLEM FOR FOURIER AL
16. Preliminaries
Spectral synthesis
Spectral synthesis for A(G)
Spectral synthesis for A(G/K)
Spectral synthesis for A(KG/K)
References
Results of Forrest (1998)
Results of Forrest (1998)
Theorem
Let K be a compact subgroup of G , E ⊆ G /K be a closed subset
for which spectral synthesis fails in A(G /K ) then spectral synthesis
fails for ϕ−1(E ) in A(G ). In particular if spectral synthesis fails for
A(G /K ) then spectral synthesis fails for A(G ).
Corollary
Let K be a compact subgroup of G , then each singleton set
{x } ⊆ G /K is a set of spectral synthesis for A(G /K ).
Kunda Chowdaiah SPECTRAL SYNTHESIS PROBLEM FOR FOURIER AL
17. Preliminaries
Spectral synthesis
Spectral synthesis for A(G)
Spectral synthesis for A(G/K)
Spectral synthesis for A(KG/K)
References
Spectral synthesis for A(KG/K)
Suppose that K is a compact subgroup of G . Let
K G /K = {KxK : x ∈ G } denote the space of all double cosets of
K in G .
Example
M(2)= Euclidian Motion group . G = R2 T and K = T
G = SL(2, R), K =
cos θ sin θ
− sin θ cos θ
: θ ∈ [−π, π)
One wants to study the Fourier algebras on K G /K .
Kunda Chowdaiah SPECTRAL SYNTHESIS PROBLEM FOR FOURIER AL
18. Preliminaries
Spectral synthesis
Spectral synthesis for A(G)
Spectral synthesis for A(G/K)
Spectral synthesis for A(KG/K)
References
If K is a compact subgroup of G then we dene
L
1
(K G /K ) = {f ∈ L
1
(G ) : f (k1xk2) = f (x ) ∀x ∈ G ∀k1, k2 ∈ K }
Gelfand pair
We say that (G , K ) is Gelfand pair if L
1(K G /K ) is abelian under
convolution.
Above two examples are Gelfand pairs.
Remark
Let (G , K ) is a Gelfand pair then Plancherel measure d π and
Inverse Fourier transform I, are available for L
1(K G /K ).
Dene A(K G /K ) = {I(f ) : f ∈ (L
1(K G /K ), d π)}
Kunda Chowdaiah SPECTRAL SYNTHESIS PROBLEM FOR FOURIER AL
19. Preliminaries
Spectral synthesis
Spectral synthesis for A(G)
Spectral synthesis for A(G/K)
Spectral synthesis for A(KG/K)
References
Theorem (VM, 2008)
A(K G /K ) is a Tauberian regular semisimple commutative Banach
algebra.
Remark
A(K G /K ) is yet to be understood completely.
Problem for my thesis
We want to study spectral synthesis problem for this Banach
algebra A(K G /K ).
Kunda Chowdaiah SPECTRAL SYNTHESIS PROBLEM FOR FOURIER AL
20. Preliminaries
Spectral synthesis
Spectral synthesis for A(G)
Spectral synthesis for A(G/K)
Spectral synthesis for A(KG/K)
References
P. Eymard, L'algèbre de Fourier d'un groupe localement
compact, Bull. Soc. Math. France. 92 (1964), 181236.
B. Forrest, Fourier analysis on coset spaces, Rocky Mountain J.
Math. 28 (1998), 173-190.
E. Hewitt and K.A. Ross, Abstract harmonic analysis, II,
Springer-Verlag, Berlin-Heidelberg -New York, 1969.
C. Herz, Harmonic synthesis for subgroups, Ann. Inst. Fourier
23 (1973), 91-123.
E. Kaniuth A course in commutative Banach algebras
,Graduate texts in mathematics, Springer, 2009.
E. Kaniuth and A. T. Lau Spectral synthesis for A(G ) and
subspaces of VN(G ), Proc. Amer. Math. Soc. 129(2001),
3253-3263.
Kunda Chowdaiah SPECTRAL SYNTHESIS PROBLEM FOR FOURIER AL
21. Preliminaries
Spectral synthesis
Spectral synthesis for A(G)
Spectral synthesis for A(G/K)
Spectral synthesis for A(KG/K)
References
P. Malliavin, Impossibilitsè de la synthsèse spectrale sur les
groupes abèliens non compacts, Inst. Hautes Èt. Sci. Publ.
Math. 2 (1959), 61-68.
V. Muruganandam, The Fourier algebra of a Hypergroup- I,
Journal of Australian Mathematical Society, 82 (2007), 59-83.
V. Muruganandam, The Fourier algebra of a Hypergroup- II
Spherical hypergroups, Mathematische Nachrichten, 281,(11),
(2008), 1590-1603.
W. Rudin, Fourier analysis on groups, Interscience, New York -
London, 1962.
M. Takesaki and N. Tatsuuma, Duality and subgroups. II, J.
Funct. Anal. 11 (1972), 184-190.
Kunda Chowdaiah SPECTRAL SYNTHESIS PROBLEM FOR FOURIER AL