In this paper, we propose a new structured Grassmannian constellation for noncoherent communicati... more In this paper, we propose a new structured Grassmannian constellation for noncoherent communications over single-input multiple-output (SIMO) Rayleigh block-fading channels. The constellation, which we call Grass-Lattice, is based on a measure preserving mapping from the unit hypercube to the Grassmannian of lines. The constellation structure allows for on-the-fly symbol generation,
Construction of Conformally Invariant Differential Operators
ABSTRACT We present a method of computation of the explicit form of conformally invariant differe... more ABSTRACT We present a method of computation of the explicit form of conformally invariant differential operators on ℝn defined using the ambient metric construction. The action of the conformal group on the conformal compactification of ℝn is realised as the action of SO(n+1, 1) on the projectivisation of the null cone in the ambient space ℝ n+1, 1. We first review a class of differential operators on the ambient space, which give rise to the conformally invariant differential operators on ℝn, and then we present a method how to write down the explicit coefficients of the induced operator by means of a suitable adapted frame on the ambient space. The procedure gives an alternative and direct method how to compute the so-called higher symmetry operators for the Laplace equation introduced by M. Eastwood. KeywordsSymmetry operators-Laplace-conformally invariant operators
medRxiv (Cold Spring Harbor Laboratory), Feb 19, 2021
There are many epidemiological models at hand to cope with the present pandemic; it is, however, ... more There are many epidemiological models at hand to cope with the present pandemic; it is, however, difficult to calibrate these models when data are noisy, partial or observed only indirectly. It is also difficult to distinguish relevant data from noise, and to distinguish the impact of individual determinants of the epidemic. In mathematical statistics, the tools to handle all of these phenomena exist; however, they are seldom used for epidemiological models. The goal of this paper is to start filling this gap by proposing a general stochastic epidemiological model, which we call SEIR Filter. Technically our model is a heterogeneous partially observable vector autoregression model, in which we are able to express closed form formulas for the distribution of compartments and observations, so both maximum likelihood and least square estimators are analytically tractable. We give conditions for vanishing, explosion and stationary behaviour of the epidemic and we are able to express a closed form formula for reproduction number. Finally, we present several examples of the model's application. We construct an estimate age-cohort model of the COVID-19 pandemic in the Czech Republic. To demonstrate the strengths of the model, we employ it to analyse and compare three vaccination scenarios.
In the 1980s, Enright, Howe and Wallach [EHW] and independently Jakobsen [J] gave a complete clas... more In the 1980s, Enright, Howe and Wallach [EHW] and independently Jakobsen [J] gave a complete classification of the unitary highest weight modules. In this paper we give a more direct and elementary proof of the same result for the (universal covers of the) Lie groups Sp(2n, R), SO * (2n) and SU (p, q). We also show how to describe the set of unitary highest weight modules with a given infinitesimal character.
We prove that the exceptional complex Lie group F4 has a transitive action on the hyperplane sect... more We prove that the exceptional complex Lie group F4 has a transitive action on the hyperplane section of the complex Cayley plane OP 2. Our proof is direct and constructive. We use an explicit realization of the vector and spin actions of Spin(9, C) ≤ F4. Moreover, we identify the stabilizer of the F4-action as a parabolic subgroup P4 (with Levi factor B3T1) of the complex Lie group F4. In the real case we obtain an analogous realization of F (−20) 4 /P4.
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
IEEE Transactions on Wireless Communications, 2023
In this paper, we propose a new structured Grassmannian constellation for noncoherent communicati... more In this paper, we propose a new structured Grassmannian constellation for noncoherent communications over single-input multiple-output (SIMO) Rayleigh block-fading channels. The constellation, which we call Grass-Lattice, is based on a measure preserving mapping from the unit hypercube to the Grassmannian of lines. The constellation structure allows for on-the-fly symbol generation,
Invariant differential operators for 1-graded geometries
In this thesis we classify singular vectors in scalar parabolic Verma modules for those pairs (sl... more In this thesis we classify singular vectors in scalar parabolic Verma modules for those pairs (sl(n, C), p) of complex Lie algebras where the homogeneous space SL(n, C)/P is the Grassmannian of k-planes in Cn . We calculate cohomology of nilpotent radicals with values in certain unitarizable highest weight modules. According to [BH09] these modules have BGG resolutions with weights determined by this cohomology. Such resolutions induce complexes of invariant differential operators on sections of associated bundles over Hermitian symmetric spaces. We describe formal completions of unitarizable highest weight modules that one can use to modify method from [CD01] that constructs sequences of differential operators over any 1-graded (aka almost Hermitian) geometry. We suggest uniform description of octonionic planes that could serve as a basis for better understanding of the exceptional Hermitian symmetric space for group E6
Let G be a connected simply connected noncompact exceptional simple Lie group of Hermitian type. ... more Let G be a connected simply connected noncompact exceptional simple Lie group of Hermitian type. In this paper, we work with the Dirac inequality which is a very useful tool for the classification of unitary highest weight modules.
Let G be a connected simply connected noncompact classical simple Lie group of Hermitian type. Th... more Let G be a connected simply connected noncompact classical simple Lie group of Hermitian type. Then G has unitary highest weight representations. The proof of the classification of unitary highest weight representations of G given by Enright, Howe and Wallach is based on the Dirac inequality of Parthasarathy, Jantzen's formula and Howe's theory of dual pairs where one group in the pair is compact. In this paper we focus on the Dirac inequality which can be used to prove the classification in a more direct way.
GLOBECOM 2022 - 2022 IEEE Global Communications Conference, Dec 4, 2022
In this paper, we propose a new structured Grassmannian constellation for noncoherent communicati... more In this paper, we propose a new structured Grassmannian constellation for noncoherent communications over single-input multiple-output (SIMO) Rayleigh block-fading channels. The constellation, which we call Grass-Lattice, is based on a measure preserving mapping from the unit hypercube to the Grassmannian of lines. The constellation structure allows for onthe-fly symbol generation, low-complexity decoding, and simple bit-to-symbol Gray coding. Simulation results show that Grass-Lattice has symbol error rate performance close to that of a numerically optimized unstructured constellation, and is more power efficient than other structured constellations proposed in the literature. Index Terms-Noncoherent communications, Grassmannian constellations, SIMO channels, measure-preserving mapping.
2022 30th European Signal Processing Conference (EUSIPCO)
In this paper, we propose a new algorithm for designing unstructured Grassmannian constellations ... more In this paper, we propose a new algorithm for designing unstructured Grassmannian constellations for noncoherent MIMO communications over Rayleigh block-fading channels. The algorithm minimizes the maximum coherence between subspaces, which is shown to be equivalent to the diversity product previously proposed in the literature. The coherence criterion is optimized by means of a gradient ascent algorithm on the Grassmann manifold. The method is generalized to optimize a weighted cost function that takes into account several neighboring codewords. Simulation results suggest that the constellations designed with the proposed algorithm achieve better SER performance than existing algorithms for unstructured Grassmannian constellation designs.
Background Short-term forecasts of infectious disease burden can contribute to situational awaren... more Background Short-term forecasts of infectious disease burden can contribute to situational awareness and aid capacity planning. Based on best practice in other fields and recent insights in infectious disease epidemiology, one can maximise the predictive performance of such forecasts if multiple models are combined into an ensemble. Here we report on the performance of ensembles in predicting COVID-19 cases and deaths across Europe between 08 March 2021 and 07 March 2022. Methods We used open-source tools to develop a public European COVID-19 Forecast Hub. We invited groups globally to contribute weekly forecasts for COVID-19 cases and deaths reported from a standardised source over the next one to four weeks. Teams submitted forecasts from March 2021 using standardised quantiles of the predictive distribution. Each week we created an ensemble forecast, where each predictive quantile was calculated as the equally-weighted average (initially the mean and then from 26th July the media...
We consider the chordal product determinant, a measure of the distance between two subspaces of t... more We consider the chordal product determinant, a measure of the distance between two subspaces of the same dimension. In information theory, collections of elements in the complex Grassmannian are searched with the property that their pairwise chordal products are as large as possible. We characterize this function from an statistical perspective, which allows us to obtain bounds for the minimal chordal product and related energy of such collections.
Natural convection during directional solidification of an AlZn eutectic
Crystal Research and Technology, 1987
A study was made of the buoyancy convection of melt provoked by temperature gradient of an orient... more A study was made of the buoyancy convection of melt provoked by temperature gradient of an orientation identical to that of the gravity vector. The convection was assessed on the basis of temperature fluctuations in the melt. The critical Rayleigh numbers for periodic and turbulent flow were determined. Both periodic and turbulent flow gives rise to transverse “bands” in the structure of the eutection obtained by directional solidification. The intensity of unstable flow affects the number of eutectic crystallites in the sample.
The standard SEIR equation-based models represent the state-of-the-art approach in epidemiologica... more The standard SEIR equation-based models represent the state-of-the-art approach in epidemiological modelling. Their drawbacks include unrealistic infectionrelated contact estimates and difficulties in modelling nonpharmaceutical interventions, such as contact reductions or partial closures. In this paper, we present our agent-based model that addresses the above-mentioned issues. It works with a population of individuals (agents) and their contacts are modelled as a multi-graph social network according to real data based on a Czech county. Custom algorithmic procedures simulating testing, quarantine and partial closures of various contact types are implemented. The model can serve as a tool for relative comparison of the efficacy of various policies. It was also used for a study comparing various interventions in Czech primary and secondary schools, using a graph based on real data from a selected Czech school.
Let G be a connected simply connected noncompact exceptional simple Lie group of Hermitian type. ... more Let G be a connected simply connected noncompact exceptional simple Lie group of Hermitian type. In this paper, we work with the Dirac inequality which is a very useful tool for the classification of unitary highest weight modules.
ICASSP 2023 - 2023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)
We consider the design of multiuser constellations for a multiple access channel (MAC) with K use... more We consider the design of multiuser constellations for a multiple access channel (MAC) with K users, with M antennas each, that transmit simultaneously to a receiver equipped with N antennas through a Rayleigh block-fading channel, when no channel state information (CSI) is available to either the transmitter or the receiver. In full-diversity scenarios where the coherence time is at least T ≥ (K + 1)M , the proposed constellation design criterion is based on the asymptotic expression of the multiuser pairwise error probability (PEP) derived by Brehler and Varanasi in [1]. Although this PEP expression was previously considered intractable for optimization, in this work we derive a closed-form formula for its unconstrained gradient and perform Riemannian optimization in the Grassmannian manifold to design multiuser constellations for the MIMO MAC with state-of-the-art performance in terms of symbol error rate (SER).
In this paper, we propose a new structured Grassmannian constellation for noncoherent communicati... more In this paper, we propose a new structured Grassmannian constellation for noncoherent communications over single-input multiple-output (SIMO) Rayleigh block-fading channels. The constellation, which we call Grass-Lattice, is based on a measure preserving mapping from the unit hypercube to the Grassmannian of lines. The constellation structure allows for on-the-fly symbol generation,
Construction of Conformally Invariant Differential Operators
ABSTRACT We present a method of computation of the explicit form of conformally invariant differe... more ABSTRACT We present a method of computation of the explicit form of conformally invariant differential operators on ℝn defined using the ambient metric construction. The action of the conformal group on the conformal compactification of ℝn is realised as the action of SO(n+1, 1) on the projectivisation of the null cone in the ambient space ℝ n+1, 1. We first review a class of differential operators on the ambient space, which give rise to the conformally invariant differential operators on ℝn, and then we present a method how to write down the explicit coefficients of the induced operator by means of a suitable adapted frame on the ambient space. The procedure gives an alternative and direct method how to compute the so-called higher symmetry operators for the Laplace equation introduced by M. Eastwood. KeywordsSymmetry operators-Laplace-conformally invariant operators
medRxiv (Cold Spring Harbor Laboratory), Feb 19, 2021
There are many epidemiological models at hand to cope with the present pandemic; it is, however, ... more There are many epidemiological models at hand to cope with the present pandemic; it is, however, difficult to calibrate these models when data are noisy, partial or observed only indirectly. It is also difficult to distinguish relevant data from noise, and to distinguish the impact of individual determinants of the epidemic. In mathematical statistics, the tools to handle all of these phenomena exist; however, they are seldom used for epidemiological models. The goal of this paper is to start filling this gap by proposing a general stochastic epidemiological model, which we call SEIR Filter. Technically our model is a heterogeneous partially observable vector autoregression model, in which we are able to express closed form formulas for the distribution of compartments and observations, so both maximum likelihood and least square estimators are analytically tractable. We give conditions for vanishing, explosion and stationary behaviour of the epidemic and we are able to express a closed form formula for reproduction number. Finally, we present several examples of the model's application. We construct an estimate age-cohort model of the COVID-19 pandemic in the Czech Republic. To demonstrate the strengths of the model, we employ it to analyse and compare three vaccination scenarios.
In the 1980s, Enright, Howe and Wallach [EHW] and independently Jakobsen [J] gave a complete clas... more In the 1980s, Enright, Howe and Wallach [EHW] and independently Jakobsen [J] gave a complete classification of the unitary highest weight modules. In this paper we give a more direct and elementary proof of the same result for the (universal covers of the) Lie groups Sp(2n, R), SO * (2n) and SU (p, q). We also show how to describe the set of unitary highest weight modules with a given infinitesimal character.
We prove that the exceptional complex Lie group F4 has a transitive action on the hyperplane sect... more We prove that the exceptional complex Lie group F4 has a transitive action on the hyperplane section of the complex Cayley plane OP 2. Our proof is direct and constructive. We use an explicit realization of the vector and spin actions of Spin(9, C) ≤ F4. Moreover, we identify the stabilizer of the F4-action as a parabolic subgroup P4 (with Levi factor B3T1) of the complex Lie group F4. In the real case we obtain an analogous realization of F (−20) 4 /P4.
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
IEEE Transactions on Wireless Communications, 2023
In this paper, we propose a new structured Grassmannian constellation for noncoherent communicati... more In this paper, we propose a new structured Grassmannian constellation for noncoherent communications over single-input multiple-output (SIMO) Rayleigh block-fading channels. The constellation, which we call Grass-Lattice, is based on a measure preserving mapping from the unit hypercube to the Grassmannian of lines. The constellation structure allows for on-the-fly symbol generation,
Invariant differential operators for 1-graded geometries
In this thesis we classify singular vectors in scalar parabolic Verma modules for those pairs (sl... more In this thesis we classify singular vectors in scalar parabolic Verma modules for those pairs (sl(n, C), p) of complex Lie algebras where the homogeneous space SL(n, C)/P is the Grassmannian of k-planes in Cn . We calculate cohomology of nilpotent radicals with values in certain unitarizable highest weight modules. According to [BH09] these modules have BGG resolutions with weights determined by this cohomology. Such resolutions induce complexes of invariant differential operators on sections of associated bundles over Hermitian symmetric spaces. We describe formal completions of unitarizable highest weight modules that one can use to modify method from [CD01] that constructs sequences of differential operators over any 1-graded (aka almost Hermitian) geometry. We suggest uniform description of octonionic planes that could serve as a basis for better understanding of the exceptional Hermitian symmetric space for group E6
Let G be a connected simply connected noncompact exceptional simple Lie group of Hermitian type. ... more Let G be a connected simply connected noncompact exceptional simple Lie group of Hermitian type. In this paper, we work with the Dirac inequality which is a very useful tool for the classification of unitary highest weight modules.
Let G be a connected simply connected noncompact classical simple Lie group of Hermitian type. Th... more Let G be a connected simply connected noncompact classical simple Lie group of Hermitian type. Then G has unitary highest weight representations. The proof of the classification of unitary highest weight representations of G given by Enright, Howe and Wallach is based on the Dirac inequality of Parthasarathy, Jantzen's formula and Howe's theory of dual pairs where one group in the pair is compact. In this paper we focus on the Dirac inequality which can be used to prove the classification in a more direct way.
GLOBECOM 2022 - 2022 IEEE Global Communications Conference, Dec 4, 2022
In this paper, we propose a new structured Grassmannian constellation for noncoherent communicati... more In this paper, we propose a new structured Grassmannian constellation for noncoherent communications over single-input multiple-output (SIMO) Rayleigh block-fading channels. The constellation, which we call Grass-Lattice, is based on a measure preserving mapping from the unit hypercube to the Grassmannian of lines. The constellation structure allows for onthe-fly symbol generation, low-complexity decoding, and simple bit-to-symbol Gray coding. Simulation results show that Grass-Lattice has symbol error rate performance close to that of a numerically optimized unstructured constellation, and is more power efficient than other structured constellations proposed in the literature. Index Terms-Noncoherent communications, Grassmannian constellations, SIMO channels, measure-preserving mapping.
2022 30th European Signal Processing Conference (EUSIPCO)
In this paper, we propose a new algorithm for designing unstructured Grassmannian constellations ... more In this paper, we propose a new algorithm for designing unstructured Grassmannian constellations for noncoherent MIMO communications over Rayleigh block-fading channels. The algorithm minimizes the maximum coherence between subspaces, which is shown to be equivalent to the diversity product previously proposed in the literature. The coherence criterion is optimized by means of a gradient ascent algorithm on the Grassmann manifold. The method is generalized to optimize a weighted cost function that takes into account several neighboring codewords. Simulation results suggest that the constellations designed with the proposed algorithm achieve better SER performance than existing algorithms for unstructured Grassmannian constellation designs.
Background Short-term forecasts of infectious disease burden can contribute to situational awaren... more Background Short-term forecasts of infectious disease burden can contribute to situational awareness and aid capacity planning. Based on best practice in other fields and recent insights in infectious disease epidemiology, one can maximise the predictive performance of such forecasts if multiple models are combined into an ensemble. Here we report on the performance of ensembles in predicting COVID-19 cases and deaths across Europe between 08 March 2021 and 07 March 2022. Methods We used open-source tools to develop a public European COVID-19 Forecast Hub. We invited groups globally to contribute weekly forecasts for COVID-19 cases and deaths reported from a standardised source over the next one to four weeks. Teams submitted forecasts from March 2021 using standardised quantiles of the predictive distribution. Each week we created an ensemble forecast, where each predictive quantile was calculated as the equally-weighted average (initially the mean and then from 26th July the media...
We consider the chordal product determinant, a measure of the distance between two subspaces of t... more We consider the chordal product determinant, a measure of the distance between two subspaces of the same dimension. In information theory, collections of elements in the complex Grassmannian are searched with the property that their pairwise chordal products are as large as possible. We characterize this function from an statistical perspective, which allows us to obtain bounds for the minimal chordal product and related energy of such collections.
Natural convection during directional solidification of an AlZn eutectic
Crystal Research and Technology, 1987
A study was made of the buoyancy convection of melt provoked by temperature gradient of an orient... more A study was made of the buoyancy convection of melt provoked by temperature gradient of an orientation identical to that of the gravity vector. The convection was assessed on the basis of temperature fluctuations in the melt. The critical Rayleigh numbers for periodic and turbulent flow were determined. Both periodic and turbulent flow gives rise to transverse “bands” in the structure of the eutection obtained by directional solidification. The intensity of unstable flow affects the number of eutectic crystallites in the sample.
The standard SEIR equation-based models represent the state-of-the-art approach in epidemiologica... more The standard SEIR equation-based models represent the state-of-the-art approach in epidemiological modelling. Their drawbacks include unrealistic infectionrelated contact estimates and difficulties in modelling nonpharmaceutical interventions, such as contact reductions or partial closures. In this paper, we present our agent-based model that addresses the above-mentioned issues. It works with a population of individuals (agents) and their contacts are modelled as a multi-graph social network according to real data based on a Czech county. Custom algorithmic procedures simulating testing, quarantine and partial closures of various contact types are implemented. The model can serve as a tool for relative comparison of the efficacy of various policies. It was also used for a study comparing various interventions in Czech primary and secondary schools, using a graph based on real data from a selected Czech school.
Let G be a connected simply connected noncompact exceptional simple Lie group of Hermitian type. ... more Let G be a connected simply connected noncompact exceptional simple Lie group of Hermitian type. In this paper, we work with the Dirac inequality which is a very useful tool for the classification of unitary highest weight modules.
ICASSP 2023 - 2023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)
We consider the design of multiuser constellations for a multiple access channel (MAC) with K use... more We consider the design of multiuser constellations for a multiple access channel (MAC) with K users, with M antennas each, that transmit simultaneously to a receiver equipped with N antennas through a Rayleigh block-fading channel, when no channel state information (CSI) is available to either the transmitter or the receiver. In full-diversity scenarios where the coherence time is at least T ≥ (K + 1)M , the proposed constellation design criterion is based on the asymptotic expression of the multiuser pairwise error probability (PEP) derived by Brehler and Varanasi in [1]. Although this PEP expression was previously considered intractable for optimization, in this work we derive a closed-form formula for its unconstrained gradient and perform Riemannian optimization in the Grassmannian manifold to design multiuser constellations for the MIMO MAC with state-of-the-art performance in terms of symbol error rate (SER).
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