Publication venue Digital Commons@Georgia Southern
Publication date 12/10/2004
Field of study Get PDF
Publication venue Digital Commons@Georgia Southern
Publication date 29/03/2005
Field of study Get PDF
Publication venue Digital Commons@Georgia Southern
Publication date 12/10/2004
Field of study Get PDF
Publication venue Digital Commons@Georgia Southern
Publication date 22/01/2007
Field of study Get PDF
Publication venue
Publication date 26/11/2017
Field of study Full text link In this paper, we deal with the Orthogonal Procrustes Problem, in which two
point configurations are compared in order to construct a map to optimally
align the two sets. This extends this to Ξ΅ \varepsilon Ξ΅ -diffeomorphisms,
introduced by [1] Damelin and Fefferman. Examples will be given for when
complete maps can not be constructed, for if the distributions do match, and
finally an algorithm for partitioning the configurations into polygons for
convenient construction of the maps.Comment: version: 11.26.1
Publication venue
Publication date 06/05/2018
Field of study Full text link We formulate an Algebraic-Coding Equivalence to the Maximum Distance
Separable Conjecture. Specifically, we present novel proofs of the following
equivalent statements. Let ( q , k ) (q,k) ( q , k ) be a fixed pair of integers satisfying q q q
is a prime power and 2 β€ k β€ q 2\leq k \leq q 2 β€ k β€ q . We denote by P q \mathcal{P}_q P q β the vector
space of functions from a finite field F q \mathbb{F}_q F q β to itself, which can be
represented as the space P q : = F q [ x ] / ( x q β x ) \mathcal{P}_q := \mathbb{F}_q[x]/(x^q-x) P q β := F q β [ x ] / ( x q β x ) of
polynomial functions. We denote by O n β P q \mathcal{O}_n \subset \mathcal{P}_q O n β β P q β the
set of polynomials that are either the zero polynomial, or have at most n n n
distinct roots in F q \mathbb{F}_q F q β . Given two subspaces Y , Z Y,Z Y , Z of P q \mathcal{P}_q P q β ,
we denote by β¨ Y , Z β© \langle Y,Z \rangle β¨ Y , Z β© their span. We prove that the following are
equivalent.
[A] Suppose that either: 1. q q q is odd 2. q q q is even and k βΜΈ { 3 , q β 1 } k \not\in \{3,
q-1\} k ξ β { 3 , q β 1 } .
Then there do not exist distinct subspaces Y Y Y and Z Z Z of P q \mathcal{P}_q P q β
such that:
3. d i m ( β¨ Y , Z β© ) = k dim(\langle Y, Z \rangle) = k d im (β¨ Y , Z β©) = k 4. d i m ( Y ) = d i m ( Z ) = k β 1 dim(Y) = dim(Z) = k-1 d im ( Y ) = d im ( Z ) = k β 1 . 5. β¨ Y , Z β© β O k β 1 \langle Y,
Z \rangle \subset \mathcal{O}_{k-1} β¨ Y , Z β© β O k β 1 β 6. Y , Z β O k β 2 Y, Z \subset \mathcal{O}_{k-2} Y , Z β O k β 2 β 7.
Y β© Z β O k β 3 Y\cap Z \subset \mathcal{O}_{k-3} Y β© Z β O k β 3 β .
[B] Suppose q q q is odd, or, if q q q is even, k βΜΈ { 3 , q β 1 } k \not\in \{3, q-1\} k ξ β { 3 , q β 1 } . There is
no integer s s s with q β₯ s > k q \geq s > k q β₯ s > k such that the Reed-Solomon code
R \mathcal{R} R over F q \mathbb{F}_q F q β of dimension s s s can have s β k + 2 s-k+2 s β k + 2 columns
B = { b 1 , β¦ , b s β k + 2 } \mathcal{B} = \{b_1,\ldots,b_{s-k+2}\} B = { b 1 β , β¦ , b s β k + 2 β } added to it, such that:
8. Any s Γ s s \times s s Γ s submatrix of R βͺ B \mathcal{R} \cup \mathcal{B} R βͺ B containing
the first s β k s-k s β k columns of B \mathcal{B} B is independent. 9. B βͺ { [ 0 , 0 , β¦ , 0 , 1 ] } \mathcal{B} \cup
\{[0,0,\ldots,0,1]\} B βͺ {[ 0 , 0 , β¦ , 0 , 1 ]} is independent.
[C] The MDS conjecture is true for the given ( q , k ) (q,k) ( q , k ) .Comment: This is version: 5.6.18. arXiv admin note: substantial text overlap
with arXiv:1611.0235
Publication venue
Publication date 17/04/2022
Field of study Full text link We study the best approximation problem: min β‘ Ξ± β R m max β‘ 1 β€ i β€ n β£ y i β β j = 1 m Ξ± j Ξ j ( x i ) β£ . \displaystyle \min_{\alpha\in
\mathbb R^m}\max_{1\leq i\leq n}\left|y_i -\sum_{j=1}^m \alpha_j \Gamma_j ({\bf
x}_i) \right|. Ξ± β R m min β 1 β€ i β€ n max β β y i β β j = 1 β m β Ξ± j β Ξ j β ( x i β ) β . Here: Ξ : = { Ξ 1 , . . . , Ξ m } \Gamma:=\left\{\Gamma_1,...,\Gamma_m\right\} Ξ := { Ξ 1 β , ... , Ξ m β } is a
list of functions where for each 1 β€ j β€ m 1\leq j\leq m 1 β€ j β€ m , Ξ j : Ξ β R \Gamma_j:\Delta \rightarrow
\mathbb R Ξ j β : Ξ β R with Ξ \Delta Ξ a set of evaluation points { x 1 , . . . , x n } \left\{{\bf x_1},...,{\bf
x_n}\right\} { x 1 β , ... , x n β } . { y 1 , . . . , y n } \left\{y_1,...,y_n\right\} { y 1 β , ... , y n β } is a set of real values and
R m : = { ( Ξ± 1 , . . . , Ξ± m ) , β Ξ± j β R , β 1 β€ j β€ m } \mathbb R^m:=\left\{(\alpha_1,...,\alpha_m),\, \alpha_j\in \mathbb R,\, 1\leq
j\leq m\right\} R m := { ( Ξ± 1 β , ... , Ξ± m β ) , Ξ± j β β R , 1 β€ j β€ m }