When Γ is a row-finite digraph we classify all finite dimensional modules of the Leavitt path alg... more When Γ is a row-finite digraph we classify all finite dimensional modules of the Leavitt path algebra L(Γ) via an explicit Morita equivalence given by an effective combinatorial (reduction) algorithm on the digraph Γ. The category of (unital) L(Γ)-modules is equivalent to a full subcategory of quiver representations of Γ. However the category of finite dimensional representations of L(Γ) is tame in contrast to the finite dimensional quiver representations of Γ which are almost always wild.
Abstract In this article, we determine the number of yes/no questions required to figure out a bi... more Abstract In this article, we determine the number of yes/no questions required to figure out a birthday (a number between 0 and 31) when up to r incorrect answers to these questions are allowed. In the language of error-correcting codes, we construct optimal (shortest possible length) codes of size 32 and minimal distance d for each positive integer d.
We classify all functions on a locally compact, abelian, compactly generated group giving equalit... more We classify all functions on a locally compact, abelian, compactly generated group giving equality in an entropy inequality generalizing the Heisenberg Uncertainty Principle.
Rocky Mountain Journal of Mathematics, Dec 1, 1989
Research partially supported by N.S.F. Hall acknowledges Weisner's priority for generalizing the ... more Research partially supported by N.S.F. Hall acknowledges Weisner's priority for generalizing the number-theoretic version of /x to a lattice in [29].
Leavitt path algebras are associated to di(rected)graphs and there is a combinatorial procedure (... more Leavitt path algebras are associated to di(rected)graphs and there is a combinatorial procedure (the reduction algorithm) making the digraph smaller while preserving the Morita type. We can recover the vertices and most of the arrows of the completely reduced digraph from the module category of a Leavitt path algebra of polynomial growth. We give an explicit classification of all irreducible representations of when the coefficients are a commutative ring with 1. We define a Morita invariant filtration of the module category by Serre subcategories and as a consequence we obtain a Morita invariant (the weighted Hasse diagram of the digraph) which captures the poset of the sinks and the cycles of Γ, the Gelfand-Kirillov dimension and more. When the Gelfand-Kirillov dimension of the Leavitt path algebra is less than 4, the weighted Hasse diagram (equivalently, the complete reduction of the digraph) is a complete Morita invariant.
We study representations of a Leavitt path algebra L of a finitely separated digraph Γ over a fie... more We study representations of a Leavitt path algebra L of a finitely separated digraph Γ over a field. We show that the category of L-modules is equivalent to a full subcategory of quiver representations. When Γ is a (non-separated) row-finite digraph we determine all possible finite dimensional quotients of L after giving a necessary and sufficient graph theoretic criterion for the existence of a nonzero finite dimensional quotient. This criterion is also equivalent to L having UGN (Unbounded Generating Number) as well as being algebraically amenable. We also realize the category of L-modules as a retract, hence a quotient by an explicit Serre subcategory of the category of quiver representations (that is, FΓ-modules) via a new colimit model for M ⊗ FΓ L.
We prove Csorba's conjecture that the Lovász complex Hom(C 5 , K n) of graph multimorphisms from ... more We prove Csorba's conjecture that the Lovász complex Hom(C 5 , K n) of graph multimorphisms from the 5-cycle C 5 to the complete graph K n is Z/2Zequivariantly homeomorphic to the Stiefel manifold, V n−1,2 , the space of (ordered) orthonormal 2-frames in R n−1. The equivariant piecewise-linear topology that we need is developed along the way.
An Integer Invariant of a Group Action (Topology, Fibrations, Transfer)
Restrictions imposed on the topology of a space X by the action of a group G are investigated via... more Restrictions imposed on the topology of a space X by the action of a group G are investigated via an invariant recently defined by Gottlieb. Gottlieb\u27s trace divides many classical integers associated with the pair (G,X) such as Euler characteristics of invariant subspaces, Lefschetz numbers of equivariant self maps, and characteristic numbers. The necessary definitions and the fundamental results of Gottlieb are given in Section 1. In Section 2 we are interested in the behavior of the trace when the action is restricted to a subgroup. We show that for a compact connected Lie group G the trace does not change when the action is restricted to the normalizer of a maximal torus; for a finite group the trace is the product of the traces of Sylow p-subgroups. As an application we extend a theorem of Browder and Katz to the non-free case. To analyze the trace on an invariant subspace in Section 3 we work in the category of finite G-cw complexes with G a compact Lie group. The obstruction to removing an equivariant cell is, without changing the trace, is shown to lie in the cohomology of the isotropy subgroup of the cell. Thus we establish that the trace of an action depends only on the singular set. Another consequence is that the trace is completely determined by the family of isotropy subgroups in low dimensions. For a finite group G, the greatest common divisor of orbit sizes is divisible by the trace. In general equality holds only in low dimensions (Section 3) or when G is an elementary abelian p-group acting smoothly on a compact manifold (a result of W. Browder). In the appendix we prove this greatest common divisor divides the Lefschetz number of an equivariant self map. This leads to several Borsuk-Ulam type results
The traditional Heisenberg-Weyl measure quantifies the joint localization, uncertainty, or concen... more The traditional Heisenberg-Weyl measure quantifies the joint localization, uncertainty, or concentration of a signal in the phase plane based on a product of energies expressed as signal variances in time and in frequency. In the image processing literature, the term compactness also has been used to refer to this same notion of joint localization, in the sense of a signal representation that is efficient simultaneously in time (or space) and frequency. In this paper, we consider Hirschman uncertainty principles based not on energies and variances directly but rather on entropies computed with respect to normalized energy densities in time and frequency. Unlike the Heisenberg-Weyl measure, this entropic Hirschman notion of joint uncertainty extends naturally from the case of infinitely supported continuous-time signals to the cases of both finitely and infinitely supported discrete-time signals. For the first time, we consider these three cases together and study them relative to one another. In the case of infinitely supported continuous-time signals, we find that, consistent with the energybased Heisenberg principle, the optimal time-frequency concentration with respect to the Hirschman uncertainty principle is realized by translated and modulated Gaussian functions. In the two discrete cases, however, the entropy-based measure yields optimizers that may be generated by applying compositions of operators to the Kronecker delta. Study of the discrete cases yields two interesting results. First, in the finitely supported case, the Hirschmanoptimal functions coincide with the so-called "picket fence" functions that are also optimal with respect to the joint time-frequency counting measure of Donoho and Stark. Second, the Hirschman optimal functions in the infinitely supported case can be reconciled with continuous-time Gaussians through a certain limiting process. While a different limiting process can be used to reconcile the finitely and infinitely supported discrete cases, there does not appear to be a straightforward limiting process that unifies all three cases: The optimizers from the finitely supported discrete case are decidedly non-Gaussian. We perform a very simple experiment that indicates that the Hirschman optimal transform (HOT) is superior to the discrete Fourier transform (DFT) and discrete cosine transform (DCT) in terms of its ability to separate or resolve two limiting cases of localization in frequency, viz. pure tones and additive white noise. We believe that these differences arise from the use of entropy rather than energy as an optimality criterion and are intimately related to the apparent incongruence between Manuscript
The traditional Heisenberg-Weyl measure quantifies the joint localization, uncertainty, or concen... more The traditional Heisenberg-Weyl measure quantifies the joint localization, uncertainty, or concentration of a signal in the phase plane based on a product of energies expressed as signal variances in time and in frequency. In the image processing literature, the term compactness also has been used to refer to this same notion of joint localization, in the sense of a signal representation that is efficient simultaneously in time (or space) and frequency. In this paper, we consider Hirschman uncertainty principles based not on energies and variances directly but rather on entropies computed with respect to normalized energy densities in time and frequency. Unlike the Heisenberg-Weyl measure, this entropic Hirschman notion of joint uncertainty extends naturally from the case of infinitely supported continuous-time signals to the cases of both finitely and infinitely supported discrete-time signals. For the first time, we consider these three cases together and study them relative to one another. In the case of infinitely supported continuous-time signals, we find that, consistent with the energybased Heisenberg principle, the optimal time-frequency concentration with respect to the Hirschman uncertainty principle is realized by translated and modulated Gaussian functions. In the two discrete cases, however, the entropy-based measure yields optimizers that may be generated by applying compositions of operators to the Kronecker delta. Study of the discrete cases yields two interesting results. First, in the finitely supported case, the Hirschmanoptimal functions coincide with the so-called "picket fence" functions that are also optimal with respect to the joint time-frequency counting measure of Donoho and Stark. Second, the Hirschman optimal functions in the infinitely supported case can be reconciled with continuous-time Gaussians through a certain limiting process. While a different limiting process can be used to reconcile the finitely and infinitely supported discrete cases, there does not appear to be a straightforward limiting process that unifies all three cases: The optimizers from the finitely supported discrete case are decidedly non-Gaussian. We perform a very simple experiment that indicates that the Hirschman optimal transform (HOT) is superior to the discrete Fourier transform (DFT) and discrete cosine transform (DCT) in terms of its ability to separate or resolve two limiting cases of localization in frequency, viz. pure tones and additive white noise. We believe that these differences arise from the use of entropy rather than energy as an optimality criterion and are intimately related to the apparent incongruence between Manuscript
We continue our previous study of improved Hardy, Rellich and Uncertainty principle inequalities ... more We continue our previous study of improved Hardy, Rellich and Uncertainty principle inequalities on a Riemannian manifold M, started in Kombe-Ozaydin. In the present paper we prove new weighted Hardy-Poincaré, Rellich type inequalities as well as improved version of our Uncertainty principle inequalities on a Riemannian manifold M. In particular, we obtain sharp constants for these inequalities on the hyperbolic space H^n.
1999 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings. ICASSP99 (Cat. No.99CH36258), 1999
We use a new uncertainty measure, H p , that predicts the compactness of digital signal represent... more We use a new uncertainty measure, H p , that predicts the compactness of digital signal representations to determine a good (non-orthogonal) set of basis vectors. The measure uses the entropy of the signal and its Fourier transform in a manner that is similar to the use of the signal and its Fourier transform in the Heisenberg uncertainty principle. The measure explains why the level of discretization of continuous basis signals can be very important to the compactness of representation. Our use of the measure indicates that a mixture of (non-orthogonal) sinusoidal and impulsive or "blocky" basis functions may be best for compactly representing signals.
2011 18th IEEE International Conference on Image Processing, 2011
We introduce new generalized AM and FM functions to perform nonlinear image filtering in the modu... more We introduce new generalized AM and FM functions to perform nonlinear image filtering in the modulation domain with consistent, artifact free phase reconstruction. The new framework enables us to design nonlinear filters in the modulation domain that are capable of producing perceptually motivated signal processing results. As an illustration, we demonstrate that the modulation domain geometric image transformations designed under this framework deliver artifact-free results that are consistent with those of classical intensity-based geometric image transformations.
Research partially supported by N.S.F. Hall acknowledges Weisner's priority for generalizing the ... more Research partially supported by N.S.F. Hall acknowledges Weisner's priority for generalizing the number-theoretic version of /x to a lattice in [29].
In this paper we establish improved Hardy and Rellich type inequalities on Riemannian manifold M.... more In this paper we establish improved Hardy and Rellich type inequalities on Riemannian manifold M. Furthermore, we also obtain sharp constant for the improved Hardy inequality and explicit constant for the Rellich inequality on hyperbolic space H^n.
this paper is to show that the trace of the action, denoted tr(G; X), is precisely equal to the t... more this paper is to show that the trace of the action, denoted tr(G; X), is precisely equal to the trace of the singular set tr(G; X
When Γ is a row-finite digraph we classify all finite dimensional modules of the Leavitt path alg... more When Γ is a row-finite digraph we classify all finite dimensional modules of the Leavitt path algebra L(Γ) via an explicit Morita equivalence given by an effective combinatorial (reduction) algorithm on the digraph Γ. The category of (unital) L(Γ)-modules is equivalent to a full subcategory of quiver representations of Γ. However the category of finite dimensional representations of L(Γ) is tame in contrast to the finite dimensional quiver representations of Γ which are almost always wild.
Abstract In this article, we determine the number of yes/no questions required to figure out a bi... more Abstract In this article, we determine the number of yes/no questions required to figure out a birthday (a number between 0 and 31) when up to r incorrect answers to these questions are allowed. In the language of error-correcting codes, we construct optimal (shortest possible length) codes of size 32 and minimal distance d for each positive integer d.
We classify all functions on a locally compact, abelian, compactly generated group giving equalit... more We classify all functions on a locally compact, abelian, compactly generated group giving equality in an entropy inequality generalizing the Heisenberg Uncertainty Principle.
Rocky Mountain Journal of Mathematics, Dec 1, 1989
Research partially supported by N.S.F. Hall acknowledges Weisner's priority for generalizing the ... more Research partially supported by N.S.F. Hall acknowledges Weisner's priority for generalizing the number-theoretic version of /x to a lattice in [29].
Leavitt path algebras are associated to di(rected)graphs and there is a combinatorial procedure (... more Leavitt path algebras are associated to di(rected)graphs and there is a combinatorial procedure (the reduction algorithm) making the digraph smaller while preserving the Morita type. We can recover the vertices and most of the arrows of the completely reduced digraph from the module category of a Leavitt path algebra of polynomial growth. We give an explicit classification of all irreducible representations of when the coefficients are a commutative ring with 1. We define a Morita invariant filtration of the module category by Serre subcategories and as a consequence we obtain a Morita invariant (the weighted Hasse diagram of the digraph) which captures the poset of the sinks and the cycles of Γ, the Gelfand-Kirillov dimension and more. When the Gelfand-Kirillov dimension of the Leavitt path algebra is less than 4, the weighted Hasse diagram (equivalently, the complete reduction of the digraph) is a complete Morita invariant.
We study representations of a Leavitt path algebra L of a finitely separated digraph Γ over a fie... more We study representations of a Leavitt path algebra L of a finitely separated digraph Γ over a field. We show that the category of L-modules is equivalent to a full subcategory of quiver representations. When Γ is a (non-separated) row-finite digraph we determine all possible finite dimensional quotients of L after giving a necessary and sufficient graph theoretic criterion for the existence of a nonzero finite dimensional quotient. This criterion is also equivalent to L having UGN (Unbounded Generating Number) as well as being algebraically amenable. We also realize the category of L-modules as a retract, hence a quotient by an explicit Serre subcategory of the category of quiver representations (that is, FΓ-modules) via a new colimit model for M ⊗ FΓ L.
We prove Csorba's conjecture that the Lovász complex Hom(C 5 , K n) of graph multimorphisms from ... more We prove Csorba's conjecture that the Lovász complex Hom(C 5 , K n) of graph multimorphisms from the 5-cycle C 5 to the complete graph K n is Z/2Zequivariantly homeomorphic to the Stiefel manifold, V n−1,2 , the space of (ordered) orthonormal 2-frames in R n−1. The equivariant piecewise-linear topology that we need is developed along the way.
An Integer Invariant of a Group Action (Topology, Fibrations, Transfer)
Restrictions imposed on the topology of a space X by the action of a group G are investigated via... more Restrictions imposed on the topology of a space X by the action of a group G are investigated via an invariant recently defined by Gottlieb. Gottlieb\u27s trace divides many classical integers associated with the pair (G,X) such as Euler characteristics of invariant subspaces, Lefschetz numbers of equivariant self maps, and characteristic numbers. The necessary definitions and the fundamental results of Gottlieb are given in Section 1. In Section 2 we are interested in the behavior of the trace when the action is restricted to a subgroup. We show that for a compact connected Lie group G the trace does not change when the action is restricted to the normalizer of a maximal torus; for a finite group the trace is the product of the traces of Sylow p-subgroups. As an application we extend a theorem of Browder and Katz to the non-free case. To analyze the trace on an invariant subspace in Section 3 we work in the category of finite G-cw complexes with G a compact Lie group. The obstruction to removing an equivariant cell is, without changing the trace, is shown to lie in the cohomology of the isotropy subgroup of the cell. Thus we establish that the trace of an action depends only on the singular set. Another consequence is that the trace is completely determined by the family of isotropy subgroups in low dimensions. For a finite group G, the greatest common divisor of orbit sizes is divisible by the trace. In general equality holds only in low dimensions (Section 3) or when G is an elementary abelian p-group acting smoothly on a compact manifold (a result of W. Browder). In the appendix we prove this greatest common divisor divides the Lefschetz number of an equivariant self map. This leads to several Borsuk-Ulam type results
The traditional Heisenberg-Weyl measure quantifies the joint localization, uncertainty, or concen... more The traditional Heisenberg-Weyl measure quantifies the joint localization, uncertainty, or concentration of a signal in the phase plane based on a product of energies expressed as signal variances in time and in frequency. In the image processing literature, the term compactness also has been used to refer to this same notion of joint localization, in the sense of a signal representation that is efficient simultaneously in time (or space) and frequency. In this paper, we consider Hirschman uncertainty principles based not on energies and variances directly but rather on entropies computed with respect to normalized energy densities in time and frequency. Unlike the Heisenberg-Weyl measure, this entropic Hirschman notion of joint uncertainty extends naturally from the case of infinitely supported continuous-time signals to the cases of both finitely and infinitely supported discrete-time signals. For the first time, we consider these three cases together and study them relative to one another. In the case of infinitely supported continuous-time signals, we find that, consistent with the energybased Heisenberg principle, the optimal time-frequency concentration with respect to the Hirschman uncertainty principle is realized by translated and modulated Gaussian functions. In the two discrete cases, however, the entropy-based measure yields optimizers that may be generated by applying compositions of operators to the Kronecker delta. Study of the discrete cases yields two interesting results. First, in the finitely supported case, the Hirschmanoptimal functions coincide with the so-called "picket fence" functions that are also optimal with respect to the joint time-frequency counting measure of Donoho and Stark. Second, the Hirschman optimal functions in the infinitely supported case can be reconciled with continuous-time Gaussians through a certain limiting process. While a different limiting process can be used to reconcile the finitely and infinitely supported discrete cases, there does not appear to be a straightforward limiting process that unifies all three cases: The optimizers from the finitely supported discrete case are decidedly non-Gaussian. We perform a very simple experiment that indicates that the Hirschman optimal transform (HOT) is superior to the discrete Fourier transform (DFT) and discrete cosine transform (DCT) in terms of its ability to separate or resolve two limiting cases of localization in frequency, viz. pure tones and additive white noise. We believe that these differences arise from the use of entropy rather than energy as an optimality criterion and are intimately related to the apparent incongruence between Manuscript
The traditional Heisenberg-Weyl measure quantifies the joint localization, uncertainty, or concen... more The traditional Heisenberg-Weyl measure quantifies the joint localization, uncertainty, or concentration of a signal in the phase plane based on a product of energies expressed as signal variances in time and in frequency. In the image processing literature, the term compactness also has been used to refer to this same notion of joint localization, in the sense of a signal representation that is efficient simultaneously in time (or space) and frequency. In this paper, we consider Hirschman uncertainty principles based not on energies and variances directly but rather on entropies computed with respect to normalized energy densities in time and frequency. Unlike the Heisenberg-Weyl measure, this entropic Hirschman notion of joint uncertainty extends naturally from the case of infinitely supported continuous-time signals to the cases of both finitely and infinitely supported discrete-time signals. For the first time, we consider these three cases together and study them relative to one another. In the case of infinitely supported continuous-time signals, we find that, consistent with the energybased Heisenberg principle, the optimal time-frequency concentration with respect to the Hirschman uncertainty principle is realized by translated and modulated Gaussian functions. In the two discrete cases, however, the entropy-based measure yields optimizers that may be generated by applying compositions of operators to the Kronecker delta. Study of the discrete cases yields two interesting results. First, in the finitely supported case, the Hirschmanoptimal functions coincide with the so-called "picket fence" functions that are also optimal with respect to the joint time-frequency counting measure of Donoho and Stark. Second, the Hirschman optimal functions in the infinitely supported case can be reconciled with continuous-time Gaussians through a certain limiting process. While a different limiting process can be used to reconcile the finitely and infinitely supported discrete cases, there does not appear to be a straightforward limiting process that unifies all three cases: The optimizers from the finitely supported discrete case are decidedly non-Gaussian. We perform a very simple experiment that indicates that the Hirschman optimal transform (HOT) is superior to the discrete Fourier transform (DFT) and discrete cosine transform (DCT) in terms of its ability to separate or resolve two limiting cases of localization in frequency, viz. pure tones and additive white noise. We believe that these differences arise from the use of entropy rather than energy as an optimality criterion and are intimately related to the apparent incongruence between Manuscript
We continue our previous study of improved Hardy, Rellich and Uncertainty principle inequalities ... more We continue our previous study of improved Hardy, Rellich and Uncertainty principle inequalities on a Riemannian manifold M, started in Kombe-Ozaydin. In the present paper we prove new weighted Hardy-Poincaré, Rellich type inequalities as well as improved version of our Uncertainty principle inequalities on a Riemannian manifold M. In particular, we obtain sharp constants for these inequalities on the hyperbolic space H^n.
1999 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings. ICASSP99 (Cat. No.99CH36258), 1999
We use a new uncertainty measure, H p , that predicts the compactness of digital signal represent... more We use a new uncertainty measure, H p , that predicts the compactness of digital signal representations to determine a good (non-orthogonal) set of basis vectors. The measure uses the entropy of the signal and its Fourier transform in a manner that is similar to the use of the signal and its Fourier transform in the Heisenberg uncertainty principle. The measure explains why the level of discretization of continuous basis signals can be very important to the compactness of representation. Our use of the measure indicates that a mixture of (non-orthogonal) sinusoidal and impulsive or "blocky" basis functions may be best for compactly representing signals.
2011 18th IEEE International Conference on Image Processing, 2011
We introduce new generalized AM and FM functions to perform nonlinear image filtering in the modu... more We introduce new generalized AM and FM functions to perform nonlinear image filtering in the modulation domain with consistent, artifact free phase reconstruction. The new framework enables us to design nonlinear filters in the modulation domain that are capable of producing perceptually motivated signal processing results. As an illustration, we demonstrate that the modulation domain geometric image transformations designed under this framework deliver artifact-free results that are consistent with those of classical intensity-based geometric image transformations.
Research partially supported by N.S.F. Hall acknowledges Weisner's priority for generalizing the ... more Research partially supported by N.S.F. Hall acknowledges Weisner's priority for generalizing the number-theoretic version of /x to a lattice in [29].
In this paper we establish improved Hardy and Rellich type inequalities on Riemannian manifold M.... more In this paper we establish improved Hardy and Rellich type inequalities on Riemannian manifold M. Furthermore, we also obtain sharp constant for the improved Hardy inequality and explicit constant for the Rellich inequality on hyperbolic space H^n.
this paper is to show that the trace of the action, denoted tr(G; X), is precisely equal to the t... more this paper is to show that the trace of the action, denoted tr(G; X), is precisely equal to the trace of the singular set tr(G; X
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Papers by Murad Ozaydin