Given a function (more generally, a measure) on a locally compact Abelian group, one can define t... more Given a function (more generally, a measure) on a locally compact Abelian group, one can define the Toeplitz operators as certain integral transforms of functions on the dual group, where the kernel is the Fourier transform of the original function or measure. In the case of the unit circle, this corresponds to forming a matrix out of the Fourier coefficients in a particular way. We will study the asymptotic eigenvalue distributions of these Toeplitz operators.
Let P 0 be a stochastic matrix with no eigenvalue other than 1 on the unit circle and let P (ε) =... more Let P 0 be a stochastic matrix with no eigenvalue other than 1 on the unit circle and let P (ε) = P 0 + A(ε) be an analytic perturbation which is still stochastic for small enough positive ε. We determine the limit lim ε↓0 1 N(ε) N(ε) l=1 P l (ε), when N(ε) ↑ ∞ at various rates.
In his 1972Periodica Mathematica Hungarica paper, H. Bergström stated a theorem on convergence in... more In his 1972Periodica Mathematica Hungarica paper, H. Bergström stated a theorem on convergence in distribution for triangular arrays of dependent random variables satisfying, a ϕ-mixing condition. A gap in his proof of this theorem is explained and a more general version is proved under weakened hypotheses. The method used consists of comparisons between the given array and associated arrays which are parameterized by a truncation variable. In addition to the main theorem, this method yields a proof of equality of limiting finite-dimensional distributions for processes generated by the given associated arrays as well as the result that if a limit distribution for the centered row sums does exist, it must be infinitely divisible. Several corollaries to the main theorem specialize this result for convergence to distributions within certain subclasses of the infinitely divisible laws.
A stable marriage problem of size 2n is constructed which contains stable matchings. This constru... more A stable marriage problem of size 2n is constructed which contains stable matchings. This construction provides a new lower bound on the maximum number of stable matchings for problems of even size and is comparable to a known lower bound when the size is a power of 2. The method of construction makes use of special properties of the latin marriage problem, which we develop.
Given the depth of I4 arbitrary points in a rectangular region of the ocean. our team created det... more Given the depth of I4 arbitrary points in a rectangular region of the ocean. our team created detailed three-dimensional and two-dimensional grid and contour maps that clearly highlighted dangerous shallows. Because the ocean floor was assumed to be smooth and wythout cliffs. the team chose to use a smooth surface fitting bicubic spline method to interpolate the given data. However. the team needed to first create an irregularly spaced I4 x I4 grid in the x-x plane with one grid line passing through each of the data points. The unknown points on the grid were approximated using an inverse-distance-squared scheme with factors that accounted not only for the depths of given data points, but also for trends in the change of depths between any two original data points. Once all the irregular grid points were known. either by approximation or from the original data. the team ran an I?+lSL bicubic spline subroutine to generate a regularly spaced, fine mesh grid of interpolated depths. which were then plotted with the aid of a Mathlib graphics program.
Let P 0 be a stochastic matrix with no eigenvalue other than 1 on the unit circle and let P (ε) =... more Let P 0 be a stochastic matrix with no eigenvalue other than 1 on the unit circle and let P (ε) = P 0 + A(ε) be an analytic perturbation which is still stochastic for small enough positive ε. We determine the limit lim ε↓0 1 N(ε) N(ε) l=1 P l (ε), when N(ε) ↑ ∞ at various rates.
We developed algorithms to determine the optimal location of emergency service centers in a given... more We developed algorithms to determine the optimal location of emergency service centers in a given city. We give consideration both to the peculiarities of a particular town, Rio Rancho, and to the procedure by which problems of this nature can be solved in general. The streets of Rio Ranch0 are laid out in a uniform grid, except for two obstacles through which there are no streets. We are given data for the demand for emergency service at each block in the grid. We developed algorithms to find the transit times between any two points on the grid for Rio Rancho. To do this. we calculated a distance on a grid with no obstacles, then adjusted for the particular obstructions. We first assumed that demand was concentrated at the center of each block and that stations were located at street intersections. Under these assumptions, the problem is amenable to an exhaustive trial of all possibilities. We then made the more sophisticated assumption that the distribution of demand is continuous, and that the emergency facilities could be located anywhere on the streets. We prove two results which greatly reduce this problem: (I) it is necessary that the location of such emergency statton be at an intersection; and (2) it is an equivalent problem to consider the demand to be located discretely at the center of each block, rather than being continuously distributed over the block. Due to these results, it is possible to run an exhaustive case study in this case as well. Statistical distributions were calculated to provide a means of comparison of the optimal result with the average result. We studied the stability of the solution with respect to the demand distribution and obstacle configuration and found it to be very stable in the first case, and reasonably stable in the latter. Finally, we discuss how similar problems would be approached, and in particular, how an approximation scheme could be implemented if the large number of possibilities prohibits an exhaustive analysis.
Given the depth of I4 arbitrary points in a rectangular region of the ocean. our team created det... more Given the depth of I4 arbitrary points in a rectangular region of the ocean. our team created detailed three-dimensional and two-dimensional grid and contour maps that clearly highlighted dangerous shallows. Because the ocean floor was assumed to be smooth and wythout cliffs. the team chose to use a smooth surface fitting bicubic spline method to interpolate the given data. However. the team needed to first create an irregularly spaced I4 x I4 grid in the x-x plane with one grid line passing through each of the data points. The unknown points on the grid were approximated using an inverse-distance-squared scheme with factors that accounted not only for the depths of given data points, but also for trends in the change of depths between any two original data points. Once all the irregular grid points were known. either by approximation or from the original data. the team ran an I?+lSL bicubic spline subroutine to generate a regularly spaced, fine mesh grid of interpolated depths. which were then plotted with the aid of a Mathlib graphics program.
The problem of global optimization of M incoherent phase signals in N complex dimensions is formu... more The problem of global optimization of M incoherent phase signals in N complex dimensions is formulated. Then, by using the geometric approach of Landau and Slepian, conditions for optimality are established for N 2, and the optimal signal sets are determined for M 2, 3, 4, 6 and 12. The method is the following: The signals are assumed to be equally probable and to have equal energy, and thus are represented by points sj, j 1, 2, ..., M, on the unit sphere $1 in CN. If Wjk is the half-space determined by sj and Sk and containing sj, i.e., Wjk {r CN:l(r, sj)l __> I(r, sk)l}, then {j f3 k Wk :j 1, 2, ..., M}, the maximum likelihood decision regions, partition $1. For additive complex Gaussian noise n and a received signal r sj e i + n, where 0 is uniformly distributed over [0, 2hi, the probability of correct decoding for the signal-to-noise ratio A2 is where r2N-e-r2+AZ)U(r) dr,-Pc-U(r)-1 fg Io(2Ar[(s, sj)[) da(s), j=l Rj j {"I S For N 2, it is proved that U(r) Io(2Nrl(s, %)1) d(s). h [M(C) (S)] where C {seS :l(s,s)l }, 2K is the total number of half-spaces that actually determine the decision regions, and h is the strictly increasing, strictly convex function of a(C, W) (where Wis a half-space not containing sj), given by h Io(2arl<s, sj>[) c C C W. Conditions for equality are established and these give rise to the globally optimal signal sets for M 2, 3, 4, 6 and 12.
In his 1972 Periodica Mathematica Hungarica paper, H. Bergstr6m stated a theorem on convergence i... more In his 1972 Periodica Mathematica Hungarica paper, H. Bergstr6m stated a theorem on convergence in distribution for triangular arrays of dependent random variables satisfying a ~b-mixing condition. A gap in his proof of this theorem is explained and a more general version is proved under weakened hypotheses. The method used consists of comparisons between the given array and associated arrays which are parameterized by a truncation variable. In addition to the main theorem, this method yields a proof of equality of limiting finite-dimensional distributions for processes generated by the given associated arrays as well as the result that if a limit distribution for the centered row sums does exist, it must be infinitely divisible. Several corollaries to the main theorem specialize this result for convergence to distributions within certain subclasses of the infinitely divisible laws.
A stable marriage problem of size 2n is constructed which contains 0(2"$) stable matchings. This ... more A stable marriage problem of size 2n is constructed which contains 0(2"$) stable matchings. This construction provides a new lower bound on the maximum number of stable matchings for problems of even size and is comparable to a known lower bound when the size is a power of 2. The method of construction makes use of special properties of the latin marriage problem, which we develop.
We developed algorithms to determine the optimal location of emergency service centers in a given... more We developed algorithms to determine the optimal location of emergency service centers in a given city. We give consideration both to the peculiarities of a particular town, Rio Rancho, and to the procedure by which problems of this nature can be solved in general. The streets of Rio Ranch0 are laid out in a uniform grid, except for two obstacles through which there are no streets. We are given data for the demand for emergency service at each block in the grid. We developed algorithms to find the transit times between any two points on the grid for Rio Rancho. To do this. we calculated a distance on a grid with no obstacles, then adjusted for the particular obstructions. We first assumed that demand was concentrated at the center of each block and that stations were located at street intersections. Under these assumptions, the problem is amenable to an exhaustive trial of all possibilities. We then made the more sophisticated assumption that the distribution of demand is continuous, and that the emergency facilities could be located anywhere on the streets. We prove two results which greatly reduce this problem: (I) it is necessary that the location of such emergency statton be at an intersection; and (2) it is an equivalent problem to consider the demand to be located discretely at the center of each block, rather than being continuously distributed over the block. Due to these results, it is possible to run an exhaustive case study in this case as well. Statistical distributions were calculated to provide a means of comparison of the optimal result with the average result. We studied the stability of the solution with respect to the demand distribution and obstacle configuration and found it to be very stable in the first case, and reasonably stable in the latter. Finally, we discuss how similar problems would be approached, and in particular, how an approximation scheme could be implemented if the large number of possibilities prohibits an exhaustive analysis.
International Journal of Mathematics and Mathematical Sciences, 1991
The non-decreasing functions whicl are star-shaped and supported above at each point of a non-emp... more The non-decreasing functions whicl are star-shaped and supported above at each point of a non-empty closed proper subset of the real line induce an ordering, on the class of distribution functions with finite first moments, that is strictly weaker than first degree stochastic dominance and strictly stronger thau second degree stochastic dominance. Several characterizations of this ordering are developed, both joint distribution criteria and those involving only marginals. Tle latter are deduced from a decomposition theorem, which reduces the problem to consideration of certai functions which are star-shaped on the complement of an open interval.
Given a function (more generally, a measure) on a locally compact Abelian group, one can define t... more Given a function (more generally, a measure) on a locally compact Abelian group, one can define the Toeplitz operators as certain integral transforms of functions on the dual group, where the kernel is the Fourier transform of the original function or measure. In the case of the unit circle, this corresponds to forming a matrix out of the Fourier coefficients in a particular way. We will study the asymptotic eigenvalue distributions of these Toeplitz operators.
Let P 0 be a stochastic matrix with no eigenvalue other than 1 on the unit circle and let P (ε) =... more Let P 0 be a stochastic matrix with no eigenvalue other than 1 on the unit circle and let P (ε) = P 0 + A(ε) be an analytic perturbation which is still stochastic for small enough positive ε. We determine the limit lim ε↓0 1 N(ε) N(ε) l=1 P l (ε), when N(ε) ↑ ∞ at various rates.
In his 1972Periodica Mathematica Hungarica paper, H. Bergström stated a theorem on convergence in... more In his 1972Periodica Mathematica Hungarica paper, H. Bergström stated a theorem on convergence in distribution for triangular arrays of dependent random variables satisfying, a ϕ-mixing condition. A gap in his proof of this theorem is explained and a more general version is proved under weakened hypotheses. The method used consists of comparisons between the given array and associated arrays which are parameterized by a truncation variable. In addition to the main theorem, this method yields a proof of equality of limiting finite-dimensional distributions for processes generated by the given associated arrays as well as the result that if a limit distribution for the centered row sums does exist, it must be infinitely divisible. Several corollaries to the main theorem specialize this result for convergence to distributions within certain subclasses of the infinitely divisible laws.
A stable marriage problem of size 2n is constructed which contains stable matchings. This constru... more A stable marriage problem of size 2n is constructed which contains stable matchings. This construction provides a new lower bound on the maximum number of stable matchings for problems of even size and is comparable to a known lower bound when the size is a power of 2. The method of construction makes use of special properties of the latin marriage problem, which we develop.
Given the depth of I4 arbitrary points in a rectangular region of the ocean. our team created det... more Given the depth of I4 arbitrary points in a rectangular region of the ocean. our team created detailed three-dimensional and two-dimensional grid and contour maps that clearly highlighted dangerous shallows. Because the ocean floor was assumed to be smooth and wythout cliffs. the team chose to use a smooth surface fitting bicubic spline method to interpolate the given data. However. the team needed to first create an irregularly spaced I4 x I4 grid in the x-x plane with one grid line passing through each of the data points. The unknown points on the grid were approximated using an inverse-distance-squared scheme with factors that accounted not only for the depths of given data points, but also for trends in the change of depths between any two original data points. Once all the irregular grid points were known. either by approximation or from the original data. the team ran an I?+lSL bicubic spline subroutine to generate a regularly spaced, fine mesh grid of interpolated depths. which were then plotted with the aid of a Mathlib graphics program.
Let P 0 be a stochastic matrix with no eigenvalue other than 1 on the unit circle and let P (ε) =... more Let P 0 be a stochastic matrix with no eigenvalue other than 1 on the unit circle and let P (ε) = P 0 + A(ε) be an analytic perturbation which is still stochastic for small enough positive ε. We determine the limit lim ε↓0 1 N(ε) N(ε) l=1 P l (ε), when N(ε) ↑ ∞ at various rates.
We developed algorithms to determine the optimal location of emergency service centers in a given... more We developed algorithms to determine the optimal location of emergency service centers in a given city. We give consideration both to the peculiarities of a particular town, Rio Rancho, and to the procedure by which problems of this nature can be solved in general. The streets of Rio Ranch0 are laid out in a uniform grid, except for two obstacles through which there are no streets. We are given data for the demand for emergency service at each block in the grid. We developed algorithms to find the transit times between any two points on the grid for Rio Rancho. To do this. we calculated a distance on a grid with no obstacles, then adjusted for the particular obstructions. We first assumed that demand was concentrated at the center of each block and that stations were located at street intersections. Under these assumptions, the problem is amenable to an exhaustive trial of all possibilities. We then made the more sophisticated assumption that the distribution of demand is continuous, and that the emergency facilities could be located anywhere on the streets. We prove two results which greatly reduce this problem: (I) it is necessary that the location of such emergency statton be at an intersection; and (2) it is an equivalent problem to consider the demand to be located discretely at the center of each block, rather than being continuously distributed over the block. Due to these results, it is possible to run an exhaustive case study in this case as well. Statistical distributions were calculated to provide a means of comparison of the optimal result with the average result. We studied the stability of the solution with respect to the demand distribution and obstacle configuration and found it to be very stable in the first case, and reasonably stable in the latter. Finally, we discuss how similar problems would be approached, and in particular, how an approximation scheme could be implemented if the large number of possibilities prohibits an exhaustive analysis.
Given the depth of I4 arbitrary points in a rectangular region of the ocean. our team created det... more Given the depth of I4 arbitrary points in a rectangular region of the ocean. our team created detailed three-dimensional and two-dimensional grid and contour maps that clearly highlighted dangerous shallows. Because the ocean floor was assumed to be smooth and wythout cliffs. the team chose to use a smooth surface fitting bicubic spline method to interpolate the given data. However. the team needed to first create an irregularly spaced I4 x I4 grid in the x-x plane with one grid line passing through each of the data points. The unknown points on the grid were approximated using an inverse-distance-squared scheme with factors that accounted not only for the depths of given data points, but also for trends in the change of depths between any two original data points. Once all the irregular grid points were known. either by approximation or from the original data. the team ran an I?+lSL bicubic spline subroutine to generate a regularly spaced, fine mesh grid of interpolated depths. which were then plotted with the aid of a Mathlib graphics program.
The problem of global optimization of M incoherent phase signals in N complex dimensions is formu... more The problem of global optimization of M incoherent phase signals in N complex dimensions is formulated. Then, by using the geometric approach of Landau and Slepian, conditions for optimality are established for N 2, and the optimal signal sets are determined for M 2, 3, 4, 6 and 12. The method is the following: The signals are assumed to be equally probable and to have equal energy, and thus are represented by points sj, j 1, 2, ..., M, on the unit sphere $1 in CN. If Wjk is the half-space determined by sj and Sk and containing sj, i.e., Wjk {r CN:l(r, sj)l __> I(r, sk)l}, then {j f3 k Wk :j 1, 2, ..., M}, the maximum likelihood decision regions, partition $1. For additive complex Gaussian noise n and a received signal r sj e i + n, where 0 is uniformly distributed over [0, 2hi, the probability of correct decoding for the signal-to-noise ratio A2 is where r2N-e-r2+AZ)U(r) dr,-Pc-U(r)-1 fg Io(2Ar[(s, sj)[) da(s), j=l Rj j {"I S For N 2, it is proved that U(r) Io(2Nrl(s, %)1) d(s). h [M(C) (S)] where C {seS :l(s,s)l }, 2K is the total number of half-spaces that actually determine the decision regions, and h is the strictly increasing, strictly convex function of a(C, W) (where Wis a half-space not containing sj), given by h Io(2arl<s, sj>[) c C C W. Conditions for equality are established and these give rise to the globally optimal signal sets for M 2, 3, 4, 6 and 12.
In his 1972 Periodica Mathematica Hungarica paper, H. Bergstr6m stated a theorem on convergence i... more In his 1972 Periodica Mathematica Hungarica paper, H. Bergstr6m stated a theorem on convergence in distribution for triangular arrays of dependent random variables satisfying a ~b-mixing condition. A gap in his proof of this theorem is explained and a more general version is proved under weakened hypotheses. The method used consists of comparisons between the given array and associated arrays which are parameterized by a truncation variable. In addition to the main theorem, this method yields a proof of equality of limiting finite-dimensional distributions for processes generated by the given associated arrays as well as the result that if a limit distribution for the centered row sums does exist, it must be infinitely divisible. Several corollaries to the main theorem specialize this result for convergence to distributions within certain subclasses of the infinitely divisible laws.
A stable marriage problem of size 2n is constructed which contains 0(2"$) stable matchings. This ... more A stable marriage problem of size 2n is constructed which contains 0(2"$) stable matchings. This construction provides a new lower bound on the maximum number of stable matchings for problems of even size and is comparable to a known lower bound when the size is a power of 2. The method of construction makes use of special properties of the latin marriage problem, which we develop.
We developed algorithms to determine the optimal location of emergency service centers in a given... more We developed algorithms to determine the optimal location of emergency service centers in a given city. We give consideration both to the peculiarities of a particular town, Rio Rancho, and to the procedure by which problems of this nature can be solved in general. The streets of Rio Ranch0 are laid out in a uniform grid, except for two obstacles through which there are no streets. We are given data for the demand for emergency service at each block in the grid. We developed algorithms to find the transit times between any two points on the grid for Rio Rancho. To do this. we calculated a distance on a grid with no obstacles, then adjusted for the particular obstructions. We first assumed that demand was concentrated at the center of each block and that stations were located at street intersections. Under these assumptions, the problem is amenable to an exhaustive trial of all possibilities. We then made the more sophisticated assumption that the distribution of demand is continuous, and that the emergency facilities could be located anywhere on the streets. We prove two results which greatly reduce this problem: (I) it is necessary that the location of such emergency statton be at an intersection; and (2) it is an equivalent problem to consider the demand to be located discretely at the center of each block, rather than being continuously distributed over the block. Due to these results, it is possible to run an exhaustive case study in this case as well. Statistical distributions were calculated to provide a means of comparison of the optimal result with the average result. We studied the stability of the solution with respect to the demand distribution and obstacle configuration and found it to be very stable in the first case, and reasonably stable in the latter. Finally, we discuss how similar problems would be approached, and in particular, how an approximation scheme could be implemented if the large number of possibilities prohibits an exhaustive analysis.
International Journal of Mathematics and Mathematical Sciences, 1991
The non-decreasing functions whicl are star-shaped and supported above at each point of a non-emp... more The non-decreasing functions whicl are star-shaped and supported above at each point of a non-empty closed proper subset of the real line induce an ordering, on the class of distribution functions with finite first moments, that is strictly weaker than first degree stochastic dominance and strictly stronger thau second degree stochastic dominance. Several characterizations of this ordering are developed, both joint distribution criteria and those involving only marginals. Tle latter are deduced from a decomposition theorem, which reduces the problem to consideration of certai functions which are star-shaped on the complement of an open interval.
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Papers by Henry Krieger