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Articles 1 - 15 of 15
Full-Text Articles in Physical Sciences and Mathematics
Computing With Functions In Spherical And Polar Geometries I. The Sphere, Alex Townsend, Heather Wilber, Grady B. Wright
Computing With Functions In Spherical And Polar Geometries I. The Sphere, Alex Townsend, Heather Wilber, Grady B. Wright
Mathematics Faculty Publications and Presentations
A collection of algorithms is described for numerically computing with smooth functions defined on the unit sphere. Functions are approximated to essentially machine precision by using a structure-preserving iterative variant of Gaussian elimination together with the double Fourier sphere method. We show that this procedure allows for stable differentiation, reduces the oversampling of functions near the poles, and converges for certain analytic functions. Operations such as function evaluation, differentiation, and integration are particularly efficient and can be computed by essentially one-dimensional algorithms. A highlight is an optimal complexity direct solver for Poisson's equation on the sphere using a spectral method. …
Random Multipliers Numerically Stabilize Gaussian And Block Gaussian Elimination: Proofs And An Extension To Low-Rank Approximation, Victor Pan, Xiaodong Yan
Random Multipliers Numerically Stabilize Gaussian And Block Gaussian Elimination: Proofs And An Extension To Low-Rank Approximation, Victor Pan, Xiaodong Yan
Publications and Research
We study two applications of standard Gaussian random multipliers. At first we prove that with a probability close to 1 such a multiplier is expected to numerically stabilize Gaussian elimination with no pivoting as well as block Gaussian elimination. Then, by extending our analysis, we prove that such a multiplier is also expected to support low-rank approximation of a matrix without customary oversampling. Our test results are in good accordance with this formal study. The results remain similar when we replace Gaussian multipliers with random circulant or Toeplitz multipliers, which involve fewer random parameters and enable faster multiplication. We formally …
The Lp Relaxation Orthogonal Array Polytope And Its Permutation Symmetries, Andrew J. Geyer, Dursun A. Bulutoglu, Steven J. Rosenberg
The Lp Relaxation Orthogonal Array Polytope And Its Permutation Symmetries, Andrew J. Geyer, Dursun A. Bulutoglu, Steven J. Rosenberg
Faculty Publications
Symmetry plays a fundamental role in design of experiments. In particular, symmetries of factorial designs that preserve their statistical properties are exploited to find designs with the best statistical properties. By using a result proved by Rosenberg [6], the concept of the LP relaxation orthogonal array polytope is developed and studied. A complete characterization of the permutation symmetry group of this polytope is made. Also, this characterization is verified computationally for many cases. Finally, a proof is provided.
Values Of Minors Of An Infinite Family Of D-Optimal Designs And Their Application To The Growth Problem, C. Koukouvinos, M. Mitrouli, Jennifer Seberry
Values Of Minors Of An Infinite Family Of D-Optimal Designs And Their Application To The Growth Problem, C. Koukouvinos, M. Mitrouli, Jennifer Seberry
Professor Jennifer Seberry
We obtain explicit formulae for the values of the 2v — j minors, j = 0, 1, 2 of D-optimal designs of order 2v = x2 + y2, v odd, where the design is constructed using two circulant or type 1 incidence matrices of either two SBIBD(2s2 + 2s + 1, s2, s2-s/2) or 2 — {2s2 + 2s + 1; s2, s2; s(s–1)} sds. This allows us to obtain information on the growth problem for families of matrices with moderate growth. Some of our theoretical formulae imply growth greater than 2(2s2 + 2s + 1) but experimentation has not …
Values Of Minors Of (1,-1) Incidence Matrices Of Sbibds And Their Application To The Growth Problem, C Koukouvinos, M Mitrouli, Jennifer Seberry
Values Of Minors Of (1,-1) Incidence Matrices Of Sbibds And Their Application To The Growth Problem, C Koukouvinos, M Mitrouli, Jennifer Seberry
Professor Jennifer Seberry
We obtain explicit formulae for the values of the v j minors, j = 0, 1,2 of (1, -1) incidence matrices of SBIBD(v, k, λ). This allows us to obtain explicit information on the growth problem for families of matrices with moderate growth. An open problem remains to establish whether the (1, -1) CP incidence matrices of SBIBD(v, k, λ), can have growth greater than v for families other than Hadamard families.
An Infinite Family Of Hadamard Matrices With Fourth Last Pivot N/2, C. Koukouvinos, M. Mitrouli, Jennifer Seberry
An Infinite Family Of Hadamard Matrices With Fourth Last Pivot N/2, C. Koukouvinos, M. Mitrouli, Jennifer Seberry
Professor Jennifer Seberry
We show that the equivalence class of Sylvester Hadamard matrices give an infinite family of Hadamard matrices in which the fourth last pivot is n/2 . Analytical examples of Hadamard matrices of order n having as fourth last pivot n/2 are given for n = 16 and 32. In each case this distinguished case with the fourth pivot n/2 arose in the equivalence class containing the Sylvester Hadamard matrix.
On The Pivot Structure For The Weighing Matrix W(12,11), C. Kravvaritis, M. Mitrouli, Jennifer Seberry
On The Pivot Structure For The Weighing Matrix W(12,11), C. Kravvaritis, M. Mitrouli, Jennifer Seberry
Faculty of Informatics - Papers (Archive)
C. Koukouvinos, M. Mitrouli and Jennifer Seberry, in "Growth in Gaussian elimination for weighing matrices, W(n, n — 1)", Linear Algebra and its Appl., 306 (2000), 189-202, conjectured that the growth factor for Gaussian elimination of any completely pivoted weighing matrix of order n and weight n — 1 is n — 1 and that the first and last few pivots are (1, 2, 2, 3 or 4, ... , n — 1 or n — 1) for n > 14. In the present paper we concentrate our study on the growth problem for the weighing matrix W(12, 11) and we …
Osprey: A Practical Type System For Validating Dimensional Unit Correctness Of C Programs, Lingxiao Jiang, Zhendong Su
Osprey: A Practical Type System For Validating Dimensional Unit Correctness Of C Programs, Lingxiao Jiang, Zhendong Su
Research Collection School Of Computing and Information Systems
Misuse of measurement units is a common source of errors in scientific applications, but standard type systems do not prevent such errors. Dimensional analysis in physics can be used to manually detect such errors in physical equations. It is, however, not feasible to perform such manual analysis for programs computing physical equations because of code complexity. In this paper, we present a type system to automatically detect potential errors involving measurement units. It is constraint-based: we model units as types and flow of units as constraints. However, standard type checking algorithms are not powerful enough to handle units because of …
On The Growth Problem For Skew And Symmetric Conference Matrices, C. Kravvaritis, M. Mitrouli, Jennifer Seberry
On The Growth Problem For Skew And Symmetric Conference Matrices, C. Kravvaritis, M. Mitrouli, Jennifer Seberry
Faculty of Informatics - Papers (Archive)
C. Koukouvinos, M. Mitrouli and Jennifer Seberry, in "Growth in Gaussian elimination for weighing matrices, W (n, n — 1)", Linear Algebra and its Appl., 306 (2000), 189-202, conjectured that the growth factor for Gaussian elimination of any completely pivoted weighing matrix of order n and weight n— 1 is n— 1 and that the first and last few pivots are (1,2,2,3 or 4, ..., n–1 or (n–1)/2, , (n–1)/2, n–1) for n > 14. In the present paper we study the growth problem for skew and symmetric conference matrices. An algorithm for extending a k × k matrix with elements …
Values Of Minors Of Some Infinite Families Of Matrices Constructed From Supplementary Difference Sets And Their Application To The Growth Problem, C. Koukouvinos, M. Mitrouli, Jennifer Seberry
Values Of Minors Of Some Infinite Families Of Matrices Constructed From Supplementary Difference Sets And Their Application To The Growth Problem, C. Koukouvinos, M. Mitrouli, Jennifer Seberry
Faculty of Informatics - Papers (Archive)
We obtain explicit formulae for the values of the 2v - j minors, j = 0,1, 2 of (1, -1) matrices of order 2v, v odd, where the matrix is constructed using two circulant or type 1 incidence matrices of 2— {v; k1, k2, λ} sds. This allows us to obtain information on the growth problem for families of matrices with moderate growth. Some of our theoretical formulae imply growth close to the order 2v but experimentation has not yet supported this result. An open problem remains to establish whether the (1, -1) CP incidence matrices of certain SBIBDs, can …
An Infinite Family Of Hadamard Matrices With Fourth Last Pivot N/2, C. Koukouvinos, M. Mitrouli, Jennifer Seberry
An Infinite Family Of Hadamard Matrices With Fourth Last Pivot N/2, C. Koukouvinos, M. Mitrouli, Jennifer Seberry
Faculty of Informatics - Papers (Archive)
We show that the equivalence class of Sylvester Hadamard matrices give an infinite family of Hadamard matrices in which the fourth last pivot is n/2 . Analytical examples of Hadamard matrices of order n having as fourth last pivot n/2 are given for n = 16 and 32. In each case this distinguished case with the fourth pivot n/2 arose in the equivalence class containing the Sylvester Hadamard matrix.
Values Of Minors Of An Infinite Family Of D-Optimal Designs And Their Application To The Growth Problem, C. Koukouvinos, M. Mitrouli, Jennifer Seberry
Values Of Minors Of An Infinite Family Of D-Optimal Designs And Their Application To The Growth Problem, C. Koukouvinos, M. Mitrouli, Jennifer Seberry
Faculty of Informatics - Papers (Archive)
We obtain explicit formulae for the values of the 2v — j minors, j = 0, 1, 2 of D-optimal designs of order 2v = x2 + y2, v odd, where the design is constructed using two circulant or type 1 incidence matrices of either two SBIBD(2s2 + 2s + 1, s2, s2-s/2) or 2 — {2s2 + 2s + 1; s2, s2; s(s–1)} sds. This allows us to obtain information on the growth problem for families of matrices with moderate growth. Some of our theoretical formulae imply growth greater than 2(2s2 + 2s + 1) but experimentation has not …
On The Complete Pivoting Conjecture For Hadamard Matrices Of Small Orders, C. Koukouvinos, M. Mitrouli, Jennifer Seberry
On The Complete Pivoting Conjecture For Hadamard Matrices Of Small Orders, C. Koukouvinos, M. Mitrouli, Jennifer Seberry
Faculty of Informatics - Papers (Archive)
In this paper we study explicitly the pivot structure of Hadamard matrices of small orders 16, 20 and 32. An algorithm computing the (n — j) x (n — j) minors of Hadamard matrices is presented and its implementation for n = 12 is described. Analytical tables summarizing the pivot patterns attained are given.
Values Of Minors Of (1,-1) Incidence Matrices Of Sbibds And Their Application To The Growth Problem, C Koukouvinos, M Mitrouli, Jennifer Seberry
Values Of Minors Of (1,-1) Incidence Matrices Of Sbibds And Their Application To The Growth Problem, C Koukouvinos, M Mitrouli, Jennifer Seberry
Faculty of Informatics - Papers (Archive)
We obtain explicit formulae for the values of the v j minors, j = 0, 1,2 of (1, -1) incidence matrices of SBIBD(v, k, λ). This allows us to obtain explicit information on the growth problem for families of matrices with moderate growth. An open problem remains to establish whether the (1, -1) CP incidence matrices of SBIBD(v, k, λ), can have growth greater than v for families other than Hadamard families.
Growth In Gaussian Elimination For Weighing Matrices, W (N, N — 1), C. Koukouvinos, M. Mitrouli, Jennifer Seberry
Growth In Gaussian Elimination For Weighing Matrices, W (N, N — 1), C. Koukouvinos, M. Mitrouli, Jennifer Seberry
Faculty of Informatics - Papers (Archive)
We consider the values for large minors of a skew-Hadamard matrix or conference matrix W of order n and find maximum n x n minor equals to (n — 1)n/2, maximum (n — 1) x (n — 1) minor equals to (n–1)n/2-1 maximum (n — 2) x (n — 2) minor equals to 2(n — 1) n/2–2, and maximum (n — 3) x (n — 3) minor equals to 4(n — 1)n/2-3. This leads us to conjecture that the growth factor for Gaussian elimination of completely pivoted skew-Hadamard or conference matrices and indeed any completely pivoted weighing matrix of order …