On an Orthonormal Basis of Solid Spherical Monogenics Recursively Generated by Anti-Holomorphic z̄-Powers

2009 ◽  
Author(s):  
S. Bock ◽  
K. Gürlebeck ◽  
Theodore E. Simos ◽  
George Psihoyios ◽  
Ch. Tsitouras
Keyword(s):  
Orthonormal Basis ◽  
Spherical Monogenics

scholarly journals Cayley Graphs Without a Bounded Eigenbasis

2020 ◽  
Author(s):  
Ashwin Sah ◽  
Mehtaab Sawhney ◽  
Yufei Zhao
Keyword(s):  
Cayley Graph ◽  
Orthonormal Basis ◽  
Cayley Graphs ◽  
The Other ◽  
Transitive Graph ◽  
Classic Result ◽  
Other Hand ◽  
Small Set ◽  
Vertex Transitive ◽  
Vertex Transitive Graph

Abstract Does every $n$-vertex Cayley graph have an orthonormal eigenbasis all of whose coordinates are $O(1/\sqrt{n})$? While the answer is yes for abelian groups, we show that it is no in general. On the other hand, we show that every $n$-vertex Cayley graph (and more generally, vertex-transitive graph) has an orthonormal basis whose coordinates are all $O(\sqrt{\log n / n})$, and that this bound is nearly best possible. Our investigation is motivated by a question of Assaf Naor, who proved that random abelian Cayley graphs are small-set expanders, extending a classic result of Alon–Roichman. His proof relies on the existence of a bounded eigenbasis for abelian Cayley graphs, which we now know cannot hold for general groups. On the other hand, we navigate around this obstruction and extend Naor’s result to nonabelian groups.

A generalized orthonormal basis for linear dynamical systems

1995 ◽  
Vol 40 (3) ◽  
pp. 451-465 ◽  
Author(s):  
P.S.C. Heuberger ◽  
P.M.J. Van den Hof ◽  
O.H. Bosgra
Keyword(s):  
Dynamical Systems ◽  
Orthonormal Basis ◽  
Linear Dynamical Systems

ON THE LOCATION AND CONTINUATION OF HOPF BIFURCATIONS IN LARGE-SCALE PROBLEMS

2008 ◽  
Vol 18 (05) ◽  
pp. 1589-1597 ◽  
Author(s):  
M. FRIEDMAN ◽  
W. QIU
Keyword(s):  
Dynamical Systems ◽  
Large Scale ◽  
Orthonormal Basis ◽  
Invariant Subspaces ◽  
Hopf Bifurcations ◽  
Augmented System ◽  
Large Scale Problems ◽  
System Technique ◽  
Matlab Package ◽  
Parameter Dependent

CL_MATCONT is a MATLAB package for the study of dynamical systems and their bifurcations. It uses a minimally augmented system for continuation of the Hopf curve. The Continuation of Invariant Subspaces (CIS) algorithm produces a smooth orthonormal basis for an invariant subspace [Formula: see text] of a parameter-dependent matrix A(s). We extend a minimally augmented system technique for location and continuation of Hopf bifurcations to large-scale problems using the CIS algorithm, which has been incorporated into CL_MATCONT. We compare this approach with using a standard augmented system and show that a minimally augmented system technique is more suitable for large-scale problems. We also suggest an improvement of a minimally augmented system technique for the case of the torus continuation.

Dirac Delta Function from Closure Relation of Orthonormal Basis and its Use in Expanding Analytic Functions

2020 ◽  
Vol 6 (2) ◽  
pp. 158-163
Author(s):  
B. B. Dhanuk ◽  
K. Pudasainee ◽  
H. P. Lamichhane ◽  
R. P. Adhikari
Keyword(s):  
Analytic Functions ◽  
Orthonormal Basis ◽  
Special Functions ◽  
Single Point ◽  
Functional Space ◽  
Delta Function ◽  
Hamiltonian Operator ◽  
Dirac Delta Function ◽  
Closure Relation ◽  
Dirac Delta

One of revealing and widely used concepts in Physics and mathematics is the Dirac delta function. The Dirac delta function is a distribution on real lines which is zero everywhere except at a single point, where it is infinite. Dirac delta function has vital role in solving inhomogeneous differential equations. In addition, the Dirac delta functions can be used to explore harmonic information’s imbedded in the physical signals, various forms of Dirac delta function and can be constructed from the closure relation of orthonormal basis functions of functional space. Among many special functions, we have chosen the set of eigen functions of the Hamiltonian operator of harmonic oscillator and angular momentum operators for orthonormal basis. The closure relation of orthonormal functions  used to construct the generator of Dirac delta function which is used to expand analytic functions log(x + 2),exp(-x2) and x within the valid region of arguments.

scholarly journals A sieve algorithm based on overlattices

2014 ◽  
Vol 17 (A) ◽  
pp. 49-70 ◽  
Author(s):  
Anja Becker ◽  
Nicolas Gama ◽  
Antoine Joux
Keyword(s):  
Heuristic Algorithm ◽  
Orthonormal Basis ◽  
Complexity Analysis ◽  
Computer Architectures ◽  
High Dimensions ◽  
Average Case ◽  
Solution Vector ◽  
New Approach ◽  
Bounded Norm

AbstractIn this paper, we present a heuristic algorithm for solving exact, as well as approximate, shortest vector and closest vector problems on lattices. The algorithm can be seen as a modified sieving algorithm for which the vectors of the intermediate sets lie in overlattices or translated cosets of overlattices. The key idea is hence no longer to work with a single lattice but to move the problems around in a tower of related lattices. We initiate the algorithm by sampling very short vectors in an overlattice of the original lattice that admits a quasi-orthonormal basis and hence an efficient enumeration of vectors of bounded norm. Taking sums of vectors in the sample, we construct short vectors in the next lattice. Finally, we obtain solution vector(s) in the initial lattice as a sum of vectors of an overlattice. The complexity analysis relies on the Gaussian heuristic. This heuristic is backed by experiments in low and high dimensions that closely reflect these estimates when solving hard lattice problems in the average case.This new approach allows us to solve not only shortest vector problems, but also closest vector problems, in lattices of dimension$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}n$in time$2^{0.3774\, n}$using memory$2^{0.2925\, n}$. Moreover, the algorithm is straightforward to parallelize on most computer architectures.

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