Locally repairable codes (LRCs) have received significant recent attention as a method of designi... more Locally repairable codes (LRCs) have received significant recent attention as a method of designing data storage systems robust to server failure. Optimal LRCs offer the ideal trade-off between minimum distance and locality, a measure of the cost of repairing a single codeword symbol. For optimal LRCs with minimum distance greater than or equal to 5, block length is bounded by a polynomial function of alphabet size. In this paper, we give an explicit construction of optimal-length (in terms of alphabet size), optimal LRCs with minimum distance equal to 5. Erratum In an earlier version we presented a construction of explicit optimal-length locally repairable codes of distance 5 using cyclic codes, which however was incorrect and only had distance 4. For that reason the cyclic construction is omitted on this version.
The Gauss circle problem in classical number theory concerns the estimation of N (x) = { (m 1 , m... more The Gauss circle problem in classical number theory concerns the estimation of N (x) = { (m 1 , m 2) ∈ Z 2 : m 2 1 + m 2 2 ≤ x }, the number of integer lattice points inside a circle of radius √ x. Gauss showed that P (x) = N (x) − πx satisfies P (x) = O(√ x). Later Hardy and Landau independently proved that P (x) = Ω − (x 1/4 (log x) 1/4). It is conjectured that inf θ ∈ R : P (x) = O(x θ) = 1 4. I. Kátai [10] showed that X 0 |P (x)| 2 dx = βX 3/2 + O(X(log X) 2). Similar results to those of the circle have been obtained for regions D ⊂ R 2 which contain the origin and whose boundary ∂D satisfies sufficient smoothness conditions. Denote by P D (x) the similar error term to P (x) only for the domain D. W. G. Nowak showed that, under appropriate conditions on ∂D, P D (x) = Ω − (x 1/4 (log x) 1/4) ([12]) and that X 0 |P D (x)| 2 dx = O(X 3/2) ([13]). A result similar to Nowak's mean square estimate is given in the case where only "primitive" lattice points, { (m 1 , m 2) ∈ Z 2 : gcd(m 1 , m 2) = 1 }, are counted in a region D, on assumption of the Riemann Hypothesis.
The presence of a body in an orbit around a close eclipsing binary star manifests itself through ... more The presence of a body in an orbit around a close eclipsing binary star manifests itself through the light time effect influencing the observed times of eclipses as the close binary and the circumbinary companion both move around the common centre of mass. This fact combined with the periodicity with which the eclipses occur can be used to detect the companion. Given a sufficient precision of the times of eclipses, the eclipse timing can be employed to detect substellar or even planetary mass companions. The main goal of the paper is to investigate the potential of the photometry based eclipse timing of binary stars as a method of detecting circumbinary planets. In the models we assume that the companion orbits a binary star in a circular Keplerian orbit. We analyze both the space and ground based photometry cases. In particular, we study the usefulness of the ongoing COROT and Kepler missions in detecting circumbinary planets. We also explore the relations binding the planet discovery space with the physical parameters of the binaries and the geometrical parameters of their light curves. We carry out detailed numerical simulations of the eclipse timing by employing a relatively realistic model of the light curves of eclipsing binary stars. We study the influence of the white and red photometric noises on the timing precision. We determine the sensitivity of the eclipse timing technique to circumbinary planets for the ground and space based photometric observations. We provide suggestions for the best targets, observing strategies and instruments for the eclipse timing method. Finally, we compare the eclipse timing as a planet detection method with the radial velocities and astrometry.
Mean Square Estimate for Primitive Lattice Points in Convex Planar Domains Ryan Coatney Departmen... more Mean Square Estimate for Primitive Lattice Points in Convex Planar Domains Ryan Coatney Department of Mathematics Master of Science The Gauss circle problem in classical number theory concerns the estimation of N(x) = { (m1,m2) ∈ Z : m1 +m2 ≤ x }, the number of integer lattice points inside a circle of radius √ x. Gauss showed that P (x) = N(x) − πx satisfies P (x) = O( √ x). Later Hardy and Landau independently proved that P (x) = Ω−(x (log x)). It is conjectured that inf { θ ∈ R : P (x) = O(x) } = 1 4 . I. Kátai [10] showed that ∫ X 0 |P (x)| dx = βX + O(X(logX)). Similar results to those of the circle have been obtained for regions D ⊂ R which contain the origin and whose boundary ∂D satisfies sufficient smoothness conditions. Denote by PD(x) the similar error term to P (x) only for the domain D. W. G. Nowak showed that, under appropriate conditions on ∂D, PD(x) = Ω−(x(log x)) ([12]) and that ∫ X 0 |PD(x)| dx = O(X) ([13]). A result similar to Nowak’s mean square estimate is give...
Let S = {a1,a2, . . .} be a real sequence, ak+1−ak ≥ σ > 0 (k = 1,2, . . .). Weyl proved in 19... more Let S = {a1,a2, . . .} be a real sequence, ak+1−ak ≥ σ > 0 (k = 1,2, . . .). Weyl proved in 1916 that the sequence Sx : a1x,a2x,a3x, . . . is uniformly distributed(mod 1) for almost allx ∈ R. Interpreted in the obvious way, this remains valid if x varies overRd. The following results are proved about the null set E(S) = {x∈ R : Sx is not uniformly distributed(mod 1)}. (i) If ak = O(kp), thenE(d)(S) has dimension≤ d−1/p. Givenp, this bound is attained for suitableS . (ii) The intersection ofE(d)(S) with a curveC satisfying natural conditions is a null subset ofC , and has dimension ≤ 1−1/pd if ak = O(kp). (iii) Let d = 1. Supposeak ∈ N (k ≥ 1), ak ≤ Ck for infinitely manyk. The subsetBδ(S) of E (d)(S)∩ [0,1) consisting ofx for which Sx has biasb(x) ≥ δ > 0 (defined below) isfinite. A cardinality bound is given, and is strengthened for the set HI (S) = {x : Sx omits the interval I (mod 1)}. Roger C. Baker Department of Mathematics, Brigham Young University, Provo, UT 84602, USA,...
Locally repairable codes (LRCs) have received significant recent attention as a method of designi... more Locally repairable codes (LRCs) have received significant recent attention as a method of designing data storage systems robust to server failure. Optimal LRCs offer the ideal trade-off between minimum distance and locality, a measure of the cost of repairing a single codeword symbol. For optimal LRCs with minimum distance greater than or equal to 5, block length is bounded by a polynomial function of alphabet size. In this paper, we give an explicit construction of optimal-length (in terms of alphabet size), optimal LRCs with minimum distance equal to 5. Erratum In an earlier version we presented a construction of explicit optimal-length locally repairable codes of distance 5 using cyclic codes, which however was incorrect and only had distance 4. For that reason the cyclic construction is omitted on this version.
The Gauss circle problem in classical number theory concerns the estimation of N (x) = { (m 1 , m... more The Gauss circle problem in classical number theory concerns the estimation of N (x) = { (m 1 , m 2) ∈ Z 2 : m 2 1 + m 2 2 ≤ x }, the number of integer lattice points inside a circle of radius √ x. Gauss showed that P (x) = N (x) − πx satisfies P (x) = O(√ x). Later Hardy and Landau independently proved that P (x) = Ω − (x 1/4 (log x) 1/4). It is conjectured that inf θ ∈ R : P (x) = O(x θ) = 1 4. I. Kátai [10] showed that X 0 |P (x)| 2 dx = βX 3/2 + O(X(log X) 2). Similar results to those of the circle have been obtained for regions D ⊂ R 2 which contain the origin and whose boundary ∂D satisfies sufficient smoothness conditions. Denote by P D (x) the similar error term to P (x) only for the domain D. W. G. Nowak showed that, under appropriate conditions on ∂D, P D (x) = Ω − (x 1/4 (log x) 1/4) ([12]) and that X 0 |P D (x)| 2 dx = O(X 3/2) ([13]). A result similar to Nowak's mean square estimate is given in the case where only "primitive" lattice points, { (m 1 , m 2) ∈ Z 2 : gcd(m 1 , m 2) = 1 }, are counted in a region D, on assumption of the Riemann Hypothesis.
The presence of a body in an orbit around a close eclipsing binary star manifests itself through ... more The presence of a body in an orbit around a close eclipsing binary star manifests itself through the light time effect influencing the observed times of eclipses as the close binary and the circumbinary companion both move around the common centre of mass. This fact combined with the periodicity with which the eclipses occur can be used to detect the companion. Given a sufficient precision of the times of eclipses, the eclipse timing can be employed to detect substellar or even planetary mass companions. The main goal of the paper is to investigate the potential of the photometry based eclipse timing of binary stars as a method of detecting circumbinary planets. In the models we assume that the companion orbits a binary star in a circular Keplerian orbit. We analyze both the space and ground based photometry cases. In particular, we study the usefulness of the ongoing COROT and Kepler missions in detecting circumbinary planets. We also explore the relations binding the planet discovery space with the physical parameters of the binaries and the geometrical parameters of their light curves. We carry out detailed numerical simulations of the eclipse timing by employing a relatively realistic model of the light curves of eclipsing binary stars. We study the influence of the white and red photometric noises on the timing precision. We determine the sensitivity of the eclipse timing technique to circumbinary planets for the ground and space based photometric observations. We provide suggestions for the best targets, observing strategies and instruments for the eclipse timing method. Finally, we compare the eclipse timing as a planet detection method with the radial velocities and astrometry.
Mean Square Estimate for Primitive Lattice Points in Convex Planar Domains Ryan Coatney Departmen... more Mean Square Estimate for Primitive Lattice Points in Convex Planar Domains Ryan Coatney Department of Mathematics Master of Science The Gauss circle problem in classical number theory concerns the estimation of N(x) = { (m1,m2) ∈ Z : m1 +m2 ≤ x }, the number of integer lattice points inside a circle of radius √ x. Gauss showed that P (x) = N(x) − πx satisfies P (x) = O( √ x). Later Hardy and Landau independently proved that P (x) = Ω−(x (log x)). It is conjectured that inf { θ ∈ R : P (x) = O(x) } = 1 4 . I. Kátai [10] showed that ∫ X 0 |P (x)| dx = βX + O(X(logX)). Similar results to those of the circle have been obtained for regions D ⊂ R which contain the origin and whose boundary ∂D satisfies sufficient smoothness conditions. Denote by PD(x) the similar error term to P (x) only for the domain D. W. G. Nowak showed that, under appropriate conditions on ∂D, PD(x) = Ω−(x(log x)) ([12]) and that ∫ X 0 |PD(x)| dx = O(X) ([13]). A result similar to Nowak’s mean square estimate is give...
Let S = {a1,a2, . . .} be a real sequence, ak+1−ak ≥ σ > 0 (k = 1,2, . . .). Weyl proved in 19... more Let S = {a1,a2, . . .} be a real sequence, ak+1−ak ≥ σ > 0 (k = 1,2, . . .). Weyl proved in 1916 that the sequence Sx : a1x,a2x,a3x, . . . is uniformly distributed(mod 1) for almost allx ∈ R. Interpreted in the obvious way, this remains valid if x varies overRd. The following results are proved about the null set E(S) = {x∈ R : Sx is not uniformly distributed(mod 1)}. (i) If ak = O(kp), thenE(d)(S) has dimension≤ d−1/p. Givenp, this bound is attained for suitableS . (ii) The intersection ofE(d)(S) with a curveC satisfying natural conditions is a null subset ofC , and has dimension ≤ 1−1/pd if ak = O(kp). (iii) Let d = 1. Supposeak ∈ N (k ≥ 1), ak ≤ Ck for infinitely manyk. The subsetBδ(S) of E (d)(S)∩ [0,1) consisting ofx for which Sx has biasb(x) ≥ δ > 0 (defined below) isfinite. A cardinality bound is given, and is strengthened for the set HI (S) = {x : Sx omits the interval I (mod 1)}. Roger C. Baker Department of Mathematics, Brigham Young University, Provo, UT 84602, USA,...
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