Artikler, rapporter og annet (matematikk og statistikk)
https://hdl.handle.net/10037/950
Mon, 11 Nov 2024 04:48:43 GMT2024-11-11T04:48:43ZUnderstanding Coherent Turbulence and the Roll-Cell Transition With Lagrangian Coherent Structures and Frame-Indifferent Fluxes
https://hdl.handle.net/10037/35539
Aksamit, Nikolas Olson; Katurji, Marwan; Zhang, Jiawei<br />
We present the first analysis of frame-indifferent (objective) fluxes and material vortices in Large Eddy Simulations of atmospheric boundary layer turbulence. We extract rotating fluid features that maintain structural coherence over time for near-neutral, transitional, and convective boundary layers. In contrast to traditional analysis of coherent structures in turbulent boundary layers, we provide the first identification of vortex boundaries that are mathematically defined to behave as tracer transport barriers. Furthermore, these vortices are indifferent to the choice of observer reference frame and can be identified without user-dependent velocity field decompositions. We find a strong agreement between the geometric qualities of the coherent structures we extract using our new method and classical descriptions of horizontal rolls and convective cells arising from decades of observational studies. We also quantify trends in individual vortex contributions to turbulent and advective fluxes of heat under varying atmospheric stability. Using recently developed tools from the theory of transport barrier fields, we compare diffusive momentum and heat barrier fields with the presence of rolls and cells, and determine a strong connection between heat and momentum orthogonality with the physical drivers of roll-cell transformation. This newly employed frame-indifferent characterization of coherent turbulent structures can be directly applied to numerical model output, and thus provides a new Lagrangian approach to understand complex scale-dependent processes and their associated dynamics.<br />
Wed, 18 Sep 2024 00:00:00 GMThttps://hdl.handle.net/10037/355392024-09-18T00:00:00ZUnderstanding Coherent Turbulence and the Roll-Cell Transition With Lagrangian Coherent Structures and Frame-Indifferent FluxesAksamit, Nikolas OlsonKaturji, MarwanZhang, JiaweiWe present the first analysis of frame-indifferent (objective) fluxes and material vortices in Large Eddy Simulations of atmospheric boundary layer turbulence. We extract rotating fluid features that maintain structural coherence over time for near-neutral, transitional, and convective boundary layers. In contrast to traditional analysis of coherent structures in turbulent boundary layers, we provide the first identification of vortex boundaries that are mathematically defined to behave as tracer transport barriers. Furthermore, these vortices are indifferent to the choice of observer reference frame and can be identified without user-dependent velocity field decompositions. We find a strong agreement between the geometric qualities of the coherent structures we extract using our new method and classical descriptions of horizontal rolls and convective cells arising from decades of observational studies. We also quantify trends in individual vortex contributions to turbulent and advective fluxes of heat under varying atmospheric stability. Using recently developed tools from the theory of transport barrier fields, we compare diffusive momentum and heat barrier fields with the presence of rolls and cells, and determine a strong connection between heat and momentum orthogonality with the physical drivers of roll-cell transformation. This newly employed frame-indifferent characterization of coherent turbulent structures can be directly applied to numerical model output, and thus provides a new Lagrangian approach to understand complex scale-dependent processes and their associated dynamics.WileyJournal articleTidsskriftartikkelPeer reviewedAksamit, Katurji, Zhang. Understanding Coherent Turbulence and the Roll-Cell Transition With Lagrangian Coherent Structures and Frame-Indifferent Fluxes. Journal of Geophysical Research (JGR): Atmospheres. 2024;129(18)Journal of Geophysical Research (JGR): AtmospheresLie Admissible Triple Algebras: The Connection Algebra of Symmetric Spaces
https://hdl.handle.net/10037/35468
Munthe-Kaas, Hans Zanna; Stava, Jonatan<br />
Associated to a symmetric space there is a canonical connection with zero torsion and parallel curvature. This connection acts as a binary operator on the vector space of smooth sections of the tangent bundle, and it is linear with respect to the real numbers. Thus the smooth section of the tangent bundle together with the connection form an algebra we call the connection algebra. The constraints of zero torsion and constant curvature makes the connection algebra into a Lie admissible triple algebra. This is a type of algebra that generalises pre-Lie algebras, and it can be embedded into a post-Lie algebra in a canonical way that generalises the canonical embedding of Lie triple systems into Lie algebras. The free Lie admissible triple algebra can be described by incorporating triple-brackets into the leaves of rooted (non-planar) trees.<br />
Thu, 25 Jul 2024 00:00:00 GMThttps://hdl.handle.net/10037/354682024-07-25T00:00:00ZLie Admissible Triple Algebras: The Connection Algebra of Symmetric SpacesMunthe-Kaas, Hans ZannaStava, JonatanAssociated to a symmetric space there is a canonical connection with zero torsion and parallel curvature. This connection acts as a binary operator on the vector space of smooth sections of the tangent bundle, and it is linear with respect to the real numbers. Thus the smooth section of the tangent bundle together with the connection form an algebra we call the connection algebra. The constraints of zero torsion and constant curvature makes the connection algebra into a Lie admissible triple algebra. This is a type of algebra that generalises pre-Lie algebras, and it can be embedded into a post-Lie algebra in a canonical way that generalises the canonical embedding of Lie triple systems into Lie algebras. The free Lie admissible triple algebra can be described by incorporating triple-brackets into the leaves of rooted (non-planar) trees.SIGMAJournal articleTidsskriftartikkelPeer reviewedMunthe-Kaas, Stava. Lie Admissible Triple Algebras: The Connection Algebra of Symmetric Spaces. SIGMA. Symmetry, Integrability and Geometry. 2024;20SIGMA. Symmetry, Integrability and GeometryOn nonnegative invariant quartics in type A
https://hdl.handle.net/10037/35458
Debus, Sebastian; Goel, Charu; Kuhlmann, Salma; Riener, Cordian Benedikt<br />
The equivariant nonnegativity versus sums of squares question has been solved for any infinite series of essential reflection groups but type <i>A</i>. As a first step to a classification, we analyse <i>A<sub>n</sub></i>
-invariant quartics. We prove that the cones of invariant sums of squares and nonnegative forms are equal if and only if the number of variables is at most 3 or odd.<br />
Wed, 09 Oct 2024 00:00:00 GMThttps://hdl.handle.net/10037/354582024-10-09T00:00:00ZOn nonnegative invariant quartics in type ADebus, SebastianGoel, CharuKuhlmann, SalmaRiener, Cordian BenediktThe equivariant nonnegativity versus sums of squares question has been solved for any infinite series of essential reflection groups but type <i>A</i>. As a first step to a classification, we analyse <i>A<sub>n</sub></i>
-invariant quartics. We prove that the cones of invariant sums of squares and nonnegative forms are equal if and only if the number of variables is at most 3 or odd.ElsevierJournal articleTidsskriftartikkelPeer reviewedDebus, Goel, Kuhlmann, Riener. On nonnegative invariant quartics in type A. Journal of symbolic computation. 2024Journal of symbolic computationTromsø forskningsstiftelse: 17MatteCRThe Universal Equivariance Properties of Exotic Aromatic B-Series
https://hdl.handle.net/10037/35441
Laurent, Adrien; Munthe-Kaas, Hans Zanna<br />
The exotic aromatic Butcher series were originally introduced for the calculation of
order conditions for the high order numerical integration of ergodic stochastic differential equations in Rd and on manifolds. We prove in this paper that exotic aromatic
B-series satisfy a universal geometric property, namely that they are characterised
by locality and equivariance with respect to orthogonal changes of coordinates. This
characterisation confirms that exotic aromatic B-series are a fundamental geometric
object that naturally generalises aromatic B-series and B-series, as they share similar
equivariance properties. In addition, we provide a classification of the main subsets of
the exotic aromatic B-series, in particular the exotic B-series, using different equivariance properties. Along the analysis, we present a generalised definition of exotic
aromatic trees, dual vector fields, and we explore the impact of degeneracies on the
classification.<br />
Fri, 16 Aug 2024 00:00:00 GMThttps://hdl.handle.net/10037/354412024-08-16T00:00:00ZThe Universal Equivariance Properties of Exotic Aromatic B-SeriesLaurent, AdrienMunthe-Kaas, Hans ZannaThe exotic aromatic Butcher series were originally introduced for the calculation of
order conditions for the high order numerical integration of ergodic stochastic differential equations in Rd and on manifolds. We prove in this paper that exotic aromatic
B-series satisfy a universal geometric property, namely that they are characterised
by locality and equivariance with respect to orthogonal changes of coordinates. This
characterisation confirms that exotic aromatic B-series are a fundamental geometric
object that naturally generalises aromatic B-series and B-series, as they share similar
equivariance properties. In addition, we provide a classification of the main subsets of
the exotic aromatic B-series, in particular the exotic B-series, using different equivariance properties. Along the analysis, we present a generalised definition of exotic
aromatic trees, dual vector fields, and we explore the impact of degeneracies on the
classification.Springer NatureJournal articleTidsskriftartikkelPeer reviewedLaurent, Munthe-Kaas. The Universal Equivariance Properties of Exotic Aromatic B-Series. Foundations of Computational Mathematics. 2024Foundations of Computational MathematicsConvergence rates for sums-of-squares hierarchies with correlative sparsity
https://hdl.handle.net/10037/35278
Rios Zertuche Rios Zertuche, Rodolfo Antonio; Korda, Milan; Magron, Victor<br />
This work derives upper bounds on the convergence rate of the moment-sum-of-squares hierarchy with correlative sparsity for global minimization of polynomials on compact basic semialgebraic sets. The main conclusion is that both sparse hierarchies based on the Schmüdgen and Putinar Positivstellensätze enjoy a polynomial rate of convergence that depends on the size of the largest clique in the sparsity graph but not on the ambient dimension. Interestingly, the sparse bounds outperform the best currently available bounds for the dense hierarchy when the maximum clique size is sufficiently small compared to the ambient dimension and the performance is measured by the running time of an interior point method required to obtain a bound on the global minimum of a given accuracy.<br />
Mon, 25 Mar 2024 00:00:00 GMThttps://hdl.handle.net/10037/352782024-03-25T00:00:00ZConvergence rates for sums-of-squares hierarchies with correlative sparsityRios Zertuche Rios Zertuche, Rodolfo AntonioKorda, MilanMagron, VictorThis work derives upper bounds on the convergence rate of the moment-sum-of-squares hierarchy with correlative sparsity for global minimization of polynomials on compact basic semialgebraic sets. The main conclusion is that both sparse hierarchies based on the Schmüdgen and Putinar Positivstellensätze enjoy a polynomial rate of convergence that depends on the size of the largest clique in the sparsity graph but not on the ambient dimension. Interestingly, the sparse bounds outperform the best currently available bounds for the dense hierarchy when the maximum clique size is sufficiently small compared to the ambient dimension and the performance is measured by the running time of an interior point method required to obtain a bound on the global minimum of a given accuracy.Springer NatureJournal articleTidsskriftartikkelPeer reviewedRios Zertuche Rios Zertuche, Korda, Magron. Convergence rates for sums-of-squares hierarchies with correlative sparsity. Mathematical programming. 2024Mathematical programminginfo:eu-repo/grantAgreement/EC/H2020/813211/Norway/Polynomial Optimization, Efficiency through Moments and Algebra/POEMA/info:eu-repo/grantAgreement/EC/?/?/?/?/ROBOPLOX?/Exploring Pain Reduction through Physical Activity: A Case Study of Seven Fibromyalgia Patients
https://hdl.handle.net/10037/35272
Jenssen, Marit Dagny Kristine; Salvi, Elisa; Fors, Egil Andreas; Nilsen, Ole Andreas; Ngo, Phuong Dinh; Tejedor Hernandez, Miguel Angel; Bellika, Johan Gustav; Godtliebsen, Fred<br />
Fibromyalgia is a chronic disease that affects a considerable fraction of the global population,
primarily women. Physical activity is often recommended as a tool to manage the symptoms. In this
study, we tried to replicate a positive result of pain reduction through physical activity. After collecting
pain and physical activity data from seven women with fibromyalgia, one patient experienced a
considerable reduction in pain intensity. According to the patient, the improvement was related
to physical activity. Our study was conducted to investigate the replicability of this result through
personalized activity recommendations. Out of the other six patients, three experienced a reduction
in pain. The remaining three patients did not experience any pain relief. Our results show that two
of these were not able to follow the activity recommendations. These results indicate that physical
activity may have a positive effect on chronic pain patients. To estimate how effective physical activity
can be for this patient group, an intervention with longer follow-ups and larger sample sizes needs to
be performed in the future.<br />
Mon, 29 Jul 2024 00:00:00 GMThttps://hdl.handle.net/10037/352722024-07-29T00:00:00ZExploring Pain Reduction through Physical Activity: A Case Study of Seven Fibromyalgia PatientsJenssen, Marit Dagny KristineSalvi, ElisaFors, Egil AndreasNilsen, Ole AndreasNgo, Phuong DinhTejedor Hernandez, Miguel AngelBellika, Johan GustavGodtliebsen, FredFibromyalgia is a chronic disease that affects a considerable fraction of the global population,
primarily women. Physical activity is often recommended as a tool to manage the symptoms. In this
study, we tried to replicate a positive result of pain reduction through physical activity. After collecting
pain and physical activity data from seven women with fibromyalgia, one patient experienced a
considerable reduction in pain intensity. According to the patient, the improvement was related
to physical activity. Our study was conducted to investigate the replicability of this result through
personalized activity recommendations. Out of the other six patients, three experienced a reduction
in pain. The remaining three patients did not experience any pain relief. Our results show that two
of these were not able to follow the activity recommendations. These results indicate that physical
activity may have a positive effect on chronic pain patients. To estimate how effective physical activity
can be for this patient group, an intervention with longer follow-ups and larger sample sizes needs to
be performed in the future.MDPIJournal articleTidsskriftartikkelPeer reviewedJenssen MDK, Salvi E, Fors EA, Nilsen O, Ngo P, Tejedor Hernandez MA, Bellika JG, Godtliebsen F. Exploring Pain Reduction through Physical Activity: A Case Study of Seven Fibromyalgia Patients. Bioengineering. 2024BioengineeringEffects of plasma resistivity in FELTOR simulations of three-dimensional full-F gyro-fluid turbulence
https://hdl.handle.net/10037/35115
Wiesenberger, M.; Held, Markus<br />
A full-F, isothermal, electromagnetic, gyro-fluid model is used to simulate plasma turbulence in
a COMPASS-sized, diverted tokamak. A parameter scan covering three orders of magnitude of
plasma resistivity and two values for the ion to electron temperature ratio with otherwise fixed
parameters is setup and analysed. Two transport regimes for high and low plasma resistivities
are revealed. Beyond a critical resistivity the mass and energy confinement reduces with
increasing resistivity. Further, for high plasma resistivity the direction of parallel acceleration is
swapped compared to low resistivity.
Three-dimensional visualisations using ray tracing techniques are displayed and discussed.
The field-alignment of turbulent fluctuations in density and parallel current becomes evident.
Relative density fluctuation amplitudes increase from below 1% in the core to 15% in the edge
and up to 40% in the scrape-off layer.
Finally, the integration of exact conservation laws over the closed field line region allows for
an identification of numerical errors within the simulations. The electron force balance and
energy conservation show relative errors on the order of 10<sup>−3</sup> while the particle conservation
and ion momentum balance show errors on the order of 10<sup>−2</sup>
.
All simulations are performed with a new version of the FELTOR code, which is fully
parallelized on GPUs. Each simulation covers a couple of milliseconds of turbulence.<br />
Thu, 18 Apr 2024 00:00:00 GMThttps://hdl.handle.net/10037/351152024-04-18T00:00:00ZEffects of plasma resistivity in FELTOR simulations of three-dimensional full-F gyro-fluid turbulenceWiesenberger, M.Held, MarkusA full-F, isothermal, electromagnetic, gyro-fluid model is used to simulate plasma turbulence in
a COMPASS-sized, diverted tokamak. A parameter scan covering three orders of magnitude of
plasma resistivity and two values for the ion to electron temperature ratio with otherwise fixed
parameters is setup and analysed. Two transport regimes for high and low plasma resistivities
are revealed. Beyond a critical resistivity the mass and energy confinement reduces with
increasing resistivity. Further, for high plasma resistivity the direction of parallel acceleration is
swapped compared to low resistivity.
Three-dimensional visualisations using ray tracing techniques are displayed and discussed.
The field-alignment of turbulent fluctuations in density and parallel current becomes evident.
Relative density fluctuation amplitudes increase from below 1% in the core to 15% in the edge
and up to 40% in the scrape-off layer.
Finally, the integration of exact conservation laws over the closed field line region allows for
an identification of numerical errors within the simulations. The electron force balance and
energy conservation show relative errors on the order of 10<sup>−3</sup> while the particle conservation
and ion momentum balance show errors on the order of 10<sup>−2</sup>
.
All simulations are performed with a new version of the FELTOR code, which is fully
parallelized on GPUs. Each simulation covers a couple of milliseconds of turbulence.IOP PublishingJournal articleTidsskriftartikkelPeer reviewedWiesenberger, Held. Effects of plasma resistivity in FELTOR simulations of three-dimensional full-F gyro-fluid turbulence. Plasma Physics and Controlled Fusion. 2024;66(6)Plasma Physics and Controlled Fusioninfo:eu-repo/grantAgreement/Funder/H2020/101052200/EU/Implementation of activities described in the Roadmap to Fusion during Horizon Europe through a joint programme of the members of the EUROfusion consortium/EUROfusion/Combinatorial mutations of Gelfand–Tsetlin polytopes, Feigin–Fourier–Littelmann–Vinberg polytopes, and block diagonal matching field polytopes
https://hdl.handle.net/10037/35111
Clarke, Oliver; Higashitani, Akihiro; Mohammadi, Fatemeh<br />
The Gelfand-Tsetlin and the Feigin–Fourier–Littelmann–Vinberg polytopes for the
Grassmannians are defined, from the perspective of representation theory, to
parametrize certain bases for highest weight irreducible modules. These polytopes
are Newton-Okounkov bodies for the Grassmannian and, in particular, the GT
polytope is an example of a string polytope. The polytopes admit a combinatorial
description as the Stanley’s order and chain polytopes of a certain poset, as
shown by Ardila, Bliem and Salazar. We prove that these polytopes occur among
matching field polytopes. Moreover, we show that they are related by a sequence
of combinatorial mutations that passes only through matching field polytopes. As
a result, we obtain a family of matching fields that give rise to toric degenerations
for the Grassmannians. Moreover, all polytopes in the family are Newton-Okounkov
bodies for the Grassmannians.<br />
Mon, 19 Feb 2024 00:00:00 GMThttps://hdl.handle.net/10037/351112024-02-19T00:00:00ZCombinatorial mutations of Gelfand–Tsetlin polytopes, Feigin–Fourier–Littelmann–Vinberg polytopes, and block diagonal matching field polytopesClarke, OliverHigashitani, AkihiroMohammadi, FatemehThe Gelfand-Tsetlin and the Feigin–Fourier–Littelmann–Vinberg polytopes for the
Grassmannians are defined, from the perspective of representation theory, to
parametrize certain bases for highest weight irreducible modules. These polytopes
are Newton-Okounkov bodies for the Grassmannian and, in particular, the GT
polytope is an example of a string polytope. The polytopes admit a combinatorial
description as the Stanley’s order and chain polytopes of a certain poset, as
shown by Ardila, Bliem and Salazar. We prove that these polytopes occur among
matching field polytopes. Moreover, we show that they are related by a sequence
of combinatorial mutations that passes only through matching field polytopes. As
a result, we obtain a family of matching fields that give rise to toric degenerations
for the Grassmannians. Moreover, all polytopes in the family are Newton-Okounkov
bodies for the Grassmannians.ElsevierJournal articleTidsskriftartikkelPeer reviewedClarke, Higashitani, Mohammadi. Combinatorial mutations of Gelfand–Tsetlin polytopes, Feigin–Fourier–Littelmann–Vinberg polytopes, and block diagonal matching field polytopes. Journal of Pure and Applied Algebra. 2024;228(7)Journal of Pure and Applied AlgebraOn the degree of varieties of sum of squares
https://hdl.handle.net/10037/34924
Ferguson, Andrew; Ottaviani, Giorgio; Safey el Din, Mohab; Teixeira Turatti, Ettore<br />
We study the problem of how many different sum of squares decompositions a general polynomial <i>f</i> with SOS-rank <i>k</i> admits. We show that there is a link between the variety SOS<sub><i>k</sub></i>(<i>f</i>) of all SOS-decompositions of <i>f</i> and the orthogonal group O(<i>k</i>). We exploit this connection to obtain the dimension of SOS<sub><i>k</sub></i>(<i>f</i>) and show that its degree is bounded from below by the degree of O (<i>k</i>). In particular, for <i>k</i> = 2 we show that SOS<sub>2</sub>(<i>f</i>) is isomorphic to O(2)
and hence the degree bound becomes an equality. Moreover, we compute the dimension of the space of polynomials of SOS-rank <i>k</i> and obtain the degree in the special case <i>k</i> = 2.<br />
Sat, 10 Feb 2024 00:00:00 GMThttps://hdl.handle.net/10037/349242024-02-10T00:00:00ZOn the degree of varieties of sum of squaresFerguson, AndrewOttaviani, GiorgioSafey el Din, MohabTeixeira Turatti, EttoreWe study the problem of how many different sum of squares decompositions a general polynomial <i>f</i> with SOS-rank <i>k</i> admits. We show that there is a link between the variety SOS<sub><i>k</sub></i>(<i>f</i>) of all SOS-decompositions of <i>f</i> and the orthogonal group O(<i>k</i>). We exploit this connection to obtain the dimension of SOS<sub><i>k</sub></i>(<i>f</i>) and show that its degree is bounded from below by the degree of O (<i>k</i>). In particular, for <i>k</i> = 2 we show that SOS<sub>2</sub>(<i>f</i>) is isomorphic to O(2)
and hence the degree bound becomes an equality. Moreover, we compute the dimension of the space of polynomials of SOS-rank <i>k</i> and obtain the degree in the special case <i>k</i> = 2.ElsevierJournal articleTidsskriftartikkelPeer reviewedFerguson, Ottaviani, Safey el Din, Teixeira Turatti. On the degree of varieties of sum of squares. Journal of Pure and Applied Algebra. 2024;228(7)Journal of Pure and Applied AlgebraThe span of singular tuples of a tensor beyond the boundary format
https://hdl.handle.net/10037/34918
Sodomaco, Luca; Teixeira Turatti, Ettore<br />
A singular <i>k</i>-tuple of a tensor T of format (<i>n<sub>1</sub></i>, ..., <i>n</i><sub><i><k></i></sub>) is essentially a complex critical point of the distance function from <i>T</i> constrained to the cone of tensors of format (<i>n</i><sub>1</sub>, ..., <i>n<sub>k</sub></i>) of rank at most one. A generic tensor has finitely many complex singular <i>k</i>-tuples, and their number depends only on the tensor format. Furthermore, if we fix the first <i>k</i> - 1 dimensions <i>n</i><sub>i</sub></i>, then the number of singular <i>k</i>-tuples of a generic tensor becomes a monotone non-decreasing function in one integer variable <i>n>sub>k</sub></i>, that stabilizes when (<i>n<sub>1</sub>, ..., n<sub>k</sub></i>) reaches a boundary format.<p> <p>In this paper, we study the linear span of singular k-tuples of a generic tensor. Its dimension also depends only on the tensor format. In particular, we concentrate on special order three tensors and order-<i>k</i> tensors of format (2, ..., 2, <i>n</i>). . As a consequence, if again we fix the first <i>k</i> - 1 dimensions <i>n</i> > 3 and let <i>n<sub>k</sub></i> increase, we show that in these special formats, the dimension of the linear span stabilizes as well, but at some concise non-sub-boundary format. We conjecture that this phenomenon holds for an arbitrary format with <i>k</i>>3. . Finally, we provide equations for the linear span of singular triples of a generic order three tensor <i>T</i> of some special non-sub-boundary format. From these equations, we conclude that <i>T</i> belongs to the linear span of its singular triples, and we conjecture that this is the case for every tensor format.<br />
Wed, 10 May 2023 00:00:00 GMThttps://hdl.handle.net/10037/349182023-05-10T00:00:00ZThe span of singular tuples of a tensor beyond the boundary formatSodomaco, LucaTeixeira Turatti, EttoreA singular <i>k</i>-tuple of a tensor T of format (<i>n<sub>1</sub></i>, ..., <i>n</i><sub><i><k></i></sub>) is essentially a complex critical point of the distance function from <i>T</i> constrained to the cone of tensors of format (<i>n</i><sub>1</sub>, ..., <i>n<sub>k</sub></i>) of rank at most one. A generic tensor has finitely many complex singular <i>k</i>-tuples, and their number depends only on the tensor format. Furthermore, if we fix the first <i>k</i> - 1 dimensions <i>n</i><sub>i</sub></i>, then the number of singular <i>k</i>-tuples of a generic tensor becomes a monotone non-decreasing function in one integer variable <i>n>sub>k</sub></i>, that stabilizes when (<i>n<sub>1</sub>, ..., n<sub>k</sub></i>) reaches a boundary format.<p> <p>In this paper, we study the linear span of singular k-tuples of a generic tensor. Its dimension also depends only on the tensor format. In particular, we concentrate on special order three tensors and order-<i>k</i> tensors of format (2, ..., 2, <i>n</i>). . As a consequence, if again we fix the first <i>k</i> - 1 dimensions <i>n</i> > 3 and let <i>n<sub>k</sub></i> increase, we show that in these special formats, the dimension of the linear span stabilizes as well, but at some concise non-sub-boundary format. We conjecture that this phenomenon holds for an arbitrary format with <i>k</i>>3. . Finally, we provide equations for the linear span of singular triples of a generic order three tensor <i>T</i> of some special non-sub-boundary format. From these equations, we conclude that <i>T</i> belongs to the linear span of its singular triples, and we conjecture that this is the case for every tensor format.ElsevierJournal articleTidsskriftartikkelPeer reviewedSodomaco, Teixeira Turatti. The span of singular tuples of a tensor beyond the boundary format. Journal of symbolic computation. 2024;120Journal of symbolic computation