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59E1T4oBgHgl3EQf6wVZ/content/tmp_files/2301.03526v1.pdf.txt

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+QUANTUM MAGNETISM 
+Thermal Hall conductivity of 𝛼-RuCl3 
+Hae-Young Kee 
+Department of Physics, University of Toronto, Toronto, Ontario, Canada 
+Thermal Hall conductivity originating from topological magnons is observed in the Kitaev candidate 𝜶-RuCl3 in 
+broad intervals of temperature and in-plane magnetic field, raising questions on the role of the Majorana mode 
+in heat conduction.  
+The black-coloured ruthenium trichloride (α-RuCl3) has a layered honeycomb structure composed of Ru3+ with a 
+magnetic moment of an effective spin-1/2. Although RuCl3 compounds were discovered back in the early twentieth 
+century, physicists only began to perceive their connection to the Kitaev spin liquid (KSL) — a special kind of quantum 
+spin liquid (QSL) — in 20141. The elementary excitations of a KSL, Majorana fermions and vortices, offer a platform 
+for quantum memory protected from decoherence, as they cannot be annihilated locally but only through fusion with 
+their antiparticle2. The smoking-gun signature of the KSL is 1/2-integer quantized thermal Hall conductivity under a 
+magnetic field, originating from unpaired Majorana moving around the edge of the sample2. Remarkably, observation 
+of 1/2-integer quantized thermal Hall conductivity in narrow ranges of temperature and magnetic field has been 
+reported3. However, such experiments were repeated by a few groups using an in-plane magnetic field4-6, and 
+conflicting conclusions were drawn. Despite similar-looking data, one group concluded robust 1/2-integer 
+quantization4, while another reported no trace of 1/2-integer quantization5, which has generated considerable debate in 
+the community of quantum magnetism. 
+ 
+The thermal Hall experiment measures the temperature change (∆T) transverse to the thermal current (JQ) 
+applied in the sample under a magnetic field (B) (Fig. 1a) which generally measures magnetic excitations in magnetic 
+materials. When spins are ordered or partially aligned by an external magnetic field, that is, a polarized state, the low-
+energy excitation is a collective motion of the spins, referred to as magnon. A topological magnon is characterized by a 
+finite Chern number associated with the Berry phase in momentum space (Fig. 1b) and may exist in the high magnetic 
+field region of the phase diagram. This means that a magnon propagating transverse to the thermal current leads to a 
+finite thermal Hall conductivity with a temperature dependence following bosonic statistics. To differentiate the source 
+of heat carriers, a detailed measurement of the thermal Hall conductivity is required. Writing in Nature Materials, Peter 
+Czajka and colleagues7 report a comprehensive measurement of the thermal Hall conductivity over broad intervals in 
+temperature and in-plane magnetic field (Ba) and demonstrate that the finite but not quantized thermal Hall signal arises 
+from topological magnons, in contrast to the earlier report of the Majorana mode being the heat carrier. A theoretical 
+study also found topological magnons in the polarized state using a widely accepted set of spin exchange parameters for 
+α-RuCl38, consistent with the conclusion drawn by Czajka and colleagues7. 
+ 
+Czajka and colleagues further suggest that there may be a QSL in the intermediate field region bounded by 
+critical magnetic field strengths Bc1	and Bc2	at very low temperatures (Fig. 1b), where deviation from the expected 
+magnon occurs, and oscillation of the longitudinal thermal conductivity is observed5. As the transition between the 
+polarized state and the QSL at Bc2	is only well defined at T = 0 K in two dimensions, this implies that there may be a 
+crossover between a QSL and the polarized state as the temperature increases, where the topological magnons become 
+responsible for the finite thermal Hall signal. 
+ 
+From microscopic theory, the intermediate field-induced KSL is unexpected, as the so-called vison gap 
+protecting the KSL is about 0.07K, where K is the Kitaev interaction2, implying that the KSL is fragile upon 
+introducing other perturbations. Indeed, α-RuCl3 shows a magnetic ordering with a zigzag pattern in lieu of the KSL at 
+low temperatures9-11, despite the dominant Kitaev interaction1,12-15. The survival of the KSL is even less likely when the 
+interaction is ferromagnetic. For example, a field of about 0.02K destroys the KSL and turns it into the polarized state16, 
+whereas for an antiferromagnetic interaction, the KSL is extended up to a field strength of about 0.3K17-23. However, it 
+is possible that non-Kitaev interactions work together with the Kitaev interaction and promote a QSL under a magnetic 
+field. Such possibilities have been investigated by several theoretical groups using various numerical techniques24-28. 
+Given the huge phase space of exchange parameters, the focus was near the ferromagnetic Kitaev interaction regime 
+relevant for α-RuCl3. A direct transition from the zigzag order to the polarized state was found when the magnetic field 
+is applied in the plane25,26, contradicting the experimental observations. Strikingly, when the field is oriented out of the 
+plane, a magnetically disordered intermediate phase was found26-28. The strong anisotropic field response is due to the 
+non-Kitaev interaction called Γ26,28,29. Whether the intermediate state boosted by the positive Γ interaction is a QSL or 
+not remains to be resolved, as a finite Γ allows for mobile visons, and the free Majorana picture of the pure Kitaev 
+model does not work. However, the effects of the Γ interaction and a magnetic field somehow cancel, and the field-
+induced intermediate phase may map to the effective KSL with a perturbing magnetic field. While this scenario seems 
+unlikely, it has not been ruled out. 
+
+ 
+There are experimental challenges owing to a strong sample dependence30,31. The layers of α-RuCl3 are stacked 
+via a weak van der Waals interaction and different types of stacking are naturally expected10,11,30-34. Depending on the 
+stacking pattern of α-RuCl3, the in-plane spin exchange parameters vary, because of the changes in the Ru–Ru ion bond 
+length and the angle between Ru–Cl–Ru bonds32. As this sensitivity traces back to the spin–orbit entangled 
+wavefunction35, it is difficult to avoid. If the Kitaev interaction is antiferromagnetic in certain samples, a more robust 
+spin liquid and a proposed U(1) spin liquid may occur under the magnetic field21-23. If so, the moment direction in the 
+ordered states may differ from the samples with a dominant ferromagnetic Kitaev interaction36,37. Looking ahead, 
+thorough experimental studies on a given sample that give a full set of information, including the layer stackings, the 
+moment direction of the magnetic order, the anisotropy in the susceptibilities, the dynamic excitations and the thermal 
+Hall measurements in different field directions, will advance our search for a material realization of a KSL. 
+ 
+ 
+References: 
+1. Plumb, K.W. et al. Phys. Rev. B 90, 04112(R) (2014). 
+2. Kitaev, A. Ann. Phys. 321, 2 (2006). 
+3. Kasahara, Y. et al. Nature 559, 227 (2018). 
+4. Bruin, J. A. N. et al. Nat. Phys. 18, 401 (2022). 
+5. Czajka, P. et al.  Nat. Phys. 17, 915 (2021). 
+6. Lefrancois, E. et al. Phys. Rev. X 12, 021025 (2022). 
+7. Czajka, P. et al. Nat. Materials 22, 36 (2023): https://doi.org/10.1038/s41563-022-01397-w 
+8. Zhang, E., Chern, L. E. & Kim, Y. B. Phys. Rev. B 103, 174402 (2021).  
+9. Sears, J. A. et al. Phys. Rev. B 91, 144420 (2015).  
+10. Johnson, R. D. et al. Phys. Rev. B 92, 235119 (2015). 
+11. Cao, H. B. et al. Phys. Rev. B 93, 134423 (2016). 
+12. Kim, H. S. et al. Phys. Rev. B 91, 24110 (2015). 
+13. Sandilands, L. J. et al. Phys. Rev. Lett. 114, 147201 (2015). 
+14. Banerjee, A. et al. Nature Materials 15, 733 (2016). 
+15. Sandilands, L. J. et al. Phys. Rev. B 93, 075144 (2016). 
+16. Jiang, H.-C., Gu, Z.-C., Qi, X.-L. & Trebst, S. Phys. Rev. B 83, 245104 (2011). 
+17. Zhu, Z., Kimchi, I., Sheng, D. N. & Fu, L. Phys. Rev. B 97, 24110 (2018). 
+18. Nasu, J., Kato, Y., Kamiya, Y. & Motome, Y. Phys. Rev. B 98, 060416 (2018). 
+19. Liang, S. et al. Phys. Rev. B 98, 054433 (2018). 
+20. Gohlke, M., Moessner, R. & Pollmann, F. Phys. Rev. B 98, 014418 (2018). 
+21. Jiang, H.-C., Wang, C.-Y., Huang, B. & Lu, Y.-M. arXiv:1809.08247 (2018). 
+22. Hickey, C. & Trebst, S. Nature Commun. 10, 530 (2019). 
+23. Patel, N. D. & Trivedi, N. Proc. Natl. Acad. Sci. 116, 12199 (2019). 
+24. Yadav, R. et al. Sci. Rep. 6, 37925 (2016). 
+25. Winter, S. et al. Phys. Rev. Lett. 120, 077203 (2018). 
+26. Gordon, J. S., Catuneanu, A., Sorensen, E. & Kee, H.-Y. Nature Commun. 10, 2470 (2019). 
+27. Kaib, D. A. S., Winter, S. & Valenti., R. Phys. Rev. B 100, 14445 (2019). 
+28. Lee., H.-Y. et al. Nature Commun. 11, 1639 (2020). 
+29. Rau, J. G., Lee, E. K.-H. & Kee, H.-Y. Phys. Rev. Lett. 112, 077204 (2014). 
+30. Yamashita, M. et al. Phys. Rev. B 102, 220404 (2020). 
+31. Kasahara, Y. et al. Phys. Rev. B 106, L060410 (2022).  
+32. Kim, H.-S. & Kee, H.-Y., Phys. Rev. B 93, 155143 (2016). 
+33. Winter, S., Li, Y., Jeschke, H. O. & Valenti, R. Phys. Rev. B 93, 214431 (2016). 
+34. Balz. C., et al. Phys. Rev. B 103, 174417 (2021). 
+35. Jackeli, G. & Khaliullin, G.  Phys. Rev. Lett. 102, 017205 (2009). 
+36. Chaloupka, J. & Khaliullin, G. Phys. Rev. B 94, 064435 (2016). 
+37. Sears, J. A. et al. Nature Physics 16, 837 (2020). 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+ 
+Figure 1 Thermal Measurements and Phase Diagram of α-RuCl3. a, Thermal Hall experimental set-up. The 
+magnetic field (B) and thermal current (JQ) are applied along the a axis (in-plane direction perpendicular to one of the 
+honeycomb bonds), and the change of temperature (ΔT) is measured across the b axis. The yellow arrow represents a 
+heat carrier leading to a finite thermal Hall conductivity. For the other in-plane direction, that is, the b axis (parallel to 
+one of the honeycomb bonds) set-up, the thermal Hall conductivity vanishes due to the ac-mirror plane (or C2 rotation 
+about the b axis) symmetry. b, Phase diagram with respect to the a-axis field strength (Ba) and temperature (T). At low 
+temperatures without the magnetic field, a magnetic order with a zigzag pattern is found, indicated by red and blue 
+arrows. As the field strength increases, the zigzag pattern along the layers is modified (zz2)34, indicating a weak 
+interlayer spin interaction altered by the in-plane field. A puzzling QSL between Bc1	and Bc2	before the polarized state 
+(PS) was suggested at very low temperatures. As temperature increases, a crossover to the polarized state may occur. 
+The rainbow-coloured hexagon in the polarized state denotes the Berry curvature in the first Brillouin zone, leading to a 
+finite thermal Hall conductivity. 
+
+(a)
+△T
+B
+JQ
+(b)
+T
+topologicalmagnon
+zig-zag
+PS
+ZZ2
+QSL?
+Bc1
+Bc2
+Ba

59E1T4oBgHgl3EQf6wVZ/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf,len=324
+page_content='QUANTUM MAGNETISM Thermal Hall conductivity of 𝛼-RuCl3 Hae-Young Kee Department of Physics, University of Toronto, Toronto, Ontario, Canada Thermal Hall conductivity originating from topological magnons is observed in the Kitaev candidate 𝜶-RuCl3 in broad intervals of temperature and in-plane magnetic field, raising questions on the role of the Majorana mode in heat conduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' The black-coloured ruthenium trichloride (α-RuCl3) has a layered honeycomb structure composed of Ru3+ with a magnetic moment of an effective spin-1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' Although RuCl3 compounds were discovered back in the early twentieth century, physicists only began to perceive their connection to the Kitaev spin liquid (KSL) — a special kind of quantum spin liquid (QSL) — in 20141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' The elementary excitations of a KSL, Majorana fermions and vortices, offer a platform for quantum memory protected from decoherence, as they cannot be annihilated locally but only through fusion with their antiparticle2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' The smoking-gun signature of the KSL is 1/2-integer quantized thermal Hall conductivity under a magnetic field, originating from unpaired Majorana moving around the edge of the sample2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' Remarkably, observation of 1/2-integer quantized thermal Hall conductivity in narrow ranges of temperature and magnetic field has been reported3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' However, such experiments were repeated by a few groups using an in-plane magnetic field4-6, and conflicting conclusions were drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content='  Despite similar-looking data, one group concluded robust 1/2-integer quantization4, while another reported no trace of 1/2-integer quantization5, which has generated considerable debate in the community of quantum magnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content='  The thermal Hall experiment measures the temperature change (∆T) transverse to the thermal current (JQ) applied in the sample under a magnetic field (B) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' 1a) which generally measures magnetic excitations in magnetic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' When spins are ordered or partially aligned by an external magnetic field, that is, a polarized state, the low- energy excitation is a collective motion of the spins, referred to as magnon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' A topological magnon is characterized by a finite Chern number associated with the Berry phase in momentum space (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' 1b) and may exist in the high magnetic field region of the phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' This means that a magnon propagating transverse to the thermal current leads to a finite thermal Hall conductivity with a temperature dependence following bosonic statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' To differentiate the source of heat carriers, a detailed measurement of the thermal Hall conductivity is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' Writing in Nature Materials, Peter Czajka and colleagues7 report a comprehensive measurement of the thermal Hall conductivity over broad intervals in temperature and in-plane magnetic field (Ba) and demonstrate that the finite but not quantized thermal Hall signal arises from topological magnons, in contrast to the earlier report of the Majorana mode being the heat carrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' A theoretical study also found topological magnons in the polarized state using a widely accepted set of spin exchange parameters for α-RuCl38, consistent with the conclusion drawn by Czajka and colleagues7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content='  Czajka and colleagues further suggest that there may be a QSL in the intermediate field region bounded by critical magnetic field strengths Bc1 and Bc2 at very low temperatures (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' 1b), where deviation from the expected magnon occurs, and oscillation of the longitudinal thermal conductivity is observed5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' As the transition between the polarized state and the QSL at Bc2 is only well defined at T = 0 K in two dimensions, this implies that there may be a crossover between a QSL and the polarized state as the temperature increases, where the topological magnons become responsible for the finite thermal Hall signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content='  From microscopic theory, the intermediate field-induced KSL is unexpected, as the so-called vison gap protecting the KSL is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content='07K, where K is the Kitaev interaction2, implying that the KSL is fragile upon introducing other perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' Indeed, α-RuCl3 shows a magnetic ordering with a zigzag pattern in lieu of the KSL at low temperatures9-11, despite the dominant Kitaev interaction1,12-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' The survival of the KSL is even less likely when the interaction is ferromagnetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' For example, a field of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content='02K destroys the KSL and turns it into the polarized state16, whereas for an antiferromagnetic interaction, the KSL is extended up to a field strength of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content='3K17-23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' However, it is possible that non-Kitaev interactions work together with the Kitaev interaction and promote a QSL under a magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' Such possibilities have been investigated by several theoretical groups using various numerical techniques24-28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' Given the huge phase space of exchange parameters, the focus was near the ferromagnetic Kitaev interaction regime relevant for α-RuCl3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' A direct transition from the zigzag order to the polarized state was found when the magnetic field is applied in the plane25,26, contradicting the experimental observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' Strikingly, when the field is oriented out of the plane, a magnetically disordered intermediate phase was found26-28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' The strong anisotropic field response is due to the non-Kitaev interaction called Γ26,28,29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content='  Whether the intermediate state boosted by the positive Γ interaction is a QSL or not remains to be resolved, as a finite Γ allows for mobile visons, and the free Majorana picture of the pure Kitaev model does not work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' However, the effects of the Γ interaction and a magnetic field somehow cancel, and the field- induced intermediate phase may map to the effective KSL with a perturbing magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' While this scenario seems unlikely, it has not been ruled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content='  There are experimental challenges owing to a strong sample dependence30,31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' The layers of α-RuCl3 are stacked via a weak van der Waals interaction and different types of stacking are naturally expected10,11,30-34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' Depending on the stacking pattern of α-RuCl3, the in-plane spin exchange parameters vary, because of the changes in the Ru–Ru ion bond length and the angle between Ru–Cl–Ru bonds32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' As this sensitivity traces back to the spin–orbit entangled wavefunction35, it is difficult to avoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' If the Kitaev interaction is antiferromagnetic in certain samples, a more robust spin liquid and a proposed U(1) spin liquid may occur under the magnetic field21-23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' If so, the moment direction in the ordered states may differ from the samples with a dominant ferromagnetic Kitaev interaction36,37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' Looking ahead, thorough experimental studies on a given sample that give a full set of information, including the layer stackings, the moment direction of the magnetic order, the anisotropy in the susceptibilities, the dynamic excitations and the thermal Hall measurements in different field directions, will advance our search for a material realization of a KSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
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+page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' Hickey, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
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+page_content=' Nature Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
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+page_content=' Patel, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
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+page_content=' 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' Sears, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' Nature Physics 16, 837 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content='  Figure 1 Thermal Measurements and Phase Diagram of α-RuCl3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' a, Thermal Hall experimental set-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' The magnetic field (B) and thermal current (JQ) are applied along the a axis (in-plane direction perpendicular to one of the honeycomb bonds), and the change of temperature (ΔT) is measured across the b axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' The yellow arrow represents a heat carrier leading to a finite thermal Hall conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' For the other in-plane direction, that is, the b axis (parallel to one of the honeycomb bonds) set-up, the thermal Hall conductivity vanishes due to the ac-mirror plane (or C2 rotation about the b axis) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' b, Phase diagram with respect to the a-axis field strength (Ba) and temperature (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' At low temperatures without the magnetic field, a magnetic order with a zigzag pattern is found, indicated by red and blue arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' As the field strength increases, the zigzag pattern along the layers is modified (zz2)34, indicating a weak interlayer spin interaction altered by the in-plane field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' A puzzling QSL between Bc1 and Bc2 before the polarized state (PS) was suggested at very low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' As temperature increases, a crossover to the polarized state may occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content=' The rainbow-coloured hexagon in the polarized state denotes the Berry curvature in the first Brillouin zone, leading to a finite thermal Hall conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content='  (a)  △T  B  JQ  (b)  T  topologicalmagnon  zig zag  PS  ZZ2  QSL?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}
+page_content='  Bc1  Bc2  Ba' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQf6wVZ/content/2301.03526v1.pdf'}

AdAyT4oBgHgl3EQfq_lh/content/tmp_files/2301.00552v1.pdf.txt

@@ -0,0 +1,2011 @@
+1 
+NEURAL SOURCE/SINK PHASE CONNECTIVITY IN 
+DEVELOPMENTAL DYSLEXIA BY MEANS OF INTERCHANNEL CAUSALITY 
+I. RODRÍGUEZ-RODRÍGUEZ, A. ORTIZ, N.J. GALLEGO-MOLINA, M.A. FORMOSO 
+Departamento de Ingeniería de Comunicaciones, Universidad de Málaga, 29004 Málaga, Spain 
+{ignacio.rodriguez, aortiz, njgm, marco.a.formoso}@ic.uma.es 
+W. L. WOO 
+Department of Computer and Information Sciences, Northumbria University, Newcastle upon Tyne NE1 8ST, UK 
+wailok.woo@northumbria.ac.uk 
+While the brain connectivity network can inform the understanding and diagnosis of developmental dyslexia, its 
+cause-effect relationships have not yet enough been examined. Employing electroencephalography signals and band-
+limited white noise stimulus at 4.8 Hz (prosodic-syllabic frequency), we measure the phase Granger causalities among 
+channels to identify differences between dyslexic learners and controls, thereby proposing a method to calculate 
+directional connectivity. As causal relationships run in both directions, we explore three scenarios, namely channels’ 
+activity as sources, as sinks, and in total. Our proposed method can be used for both classification and exploratory 
+analysis. In all scenarios, we find confirmation of the established right-lateralized Theta sampling network anomaly, 
+in line with the temporal sampling framework’s assumption of oscillatory differences in the Theta and Gamma bands. 
+Further, we show that this anomaly primarily occurs in the causal relationships of channels acting as sinks, where it 
+is significantly more pronounced than when only total activity is observed. In the sink scenario, our classifier obtains 
+0.84 and 0.88 accuracy and 0.87 and 0.93 AUC for the Theta and Gamma bands, respectively. 
+Keywords: Developmental Dyslexia, EEG, Granger causality, functional connectivity, anomaly detection; 
+1. Introduction 
+Developmental dyslexia (DD) is a learning difficulty that 
+typically causes various reading difficulties, including 
+letter migration and frequent spelling errors. In any given 
+population, between 5% and 12% of learners are likely to 
+have DD, depending on the test battery used 1. DD is 
+traditionally diagnosed using behavioral tests of reading 
+and writing skills, but these are vulnerable to exogenous 
+factors, such as attitude or disposition, leading to 
+diagnoses that may be fundamentally unsound 2. It is, 
+therefore, imperative to develop more objective metrics 
+that can offer a more accurate diagnosis among young 
+learners. A stimulus system that remains uninfluenced by 
+the learner’s behavior, actions and context (e.g., native 
+language or learning level) would be extremely valuable. 
+If the stimulus further involves the simulation of prosody, 
+i.e. the white noise at the usual frequency of the language 
+envelope, it may also inform our understanding of the 
+brain areas active in auditory processing, indicating the 
+differences between learners with and without dyslexia. 
+While various neuroscience methods for gathering 
+functional brain data exist, including functional magnetic 
+resonance imaging (fMRI) 3, magnetoencephalography 
+(MEG) and functional near-infrared spectroscopy 
+(fNIRS), electroencephalography (EEG) continues to be 
+the most widely used and least costly method to assess 
+cortical brain activity with enhanced temporal resolution.  
+An EEG measures several frequency bands, namely the 
+Delta, Theta, Alpha, Beta, and Gamma bands, which do 
+not experience stimulation equally, and it is thus 
+generally held that the stimulation of one band can 
+transfer to the others. Using EEG to separately 
+investigate the patterns emerging in these bands may 
+offer valuable insights for the research on DD. 
+EEG is well-established in DD studies exploring the 
+functional network connectivity and organization of the 
+brain. Functional connectivity means the level of 
+coordination between the activities in different areas of 
+the brain while the learner is engaging in a task. Prior 
+research has produced various techniques that employ 
+EEG to assess functional connectivity to, for example, 
+determine what patterns are characteristic of neurological 
+conditions, including Parkinson’s disease 4. Studies in 
+cognitive neuroscience have also used brain connectivity 
+to identify brain areas crucial to language and learning 5. 
+
+Rodríguez-Rodríguez et al. 
+ 
+2 
+Connectivity analysis has allowed neuroscience to 
+provide even deeper insights 
+6 by analyzing the 
+parameters linking two signals gathered through two 
+distinct channels, such as their correlation, causality and 
+covariance 7. Employing connectivity analysis to brain 
+signals measured in different regions allows us to explore 
+the neural network, in line with the notion that the brain 
+is hyper-connected 8. 
+Notably, brain connectivity is not solely limited to the 
+interactions between areas, with regions potentially 
+influencing each other through, e.g., phase-phase or 
+phase-amplitude modulations among bands 9. We here 
+consider areas that primarily exert an influence as 
+sources, while those more likely to be subject to this 
+influence are sinks. This novel consideration of 
+connectivity in terms of sources and sinks can not only 
+serve classification but also facilitate exploratory 
+analysis. 
+We propose extracting the frequency components of each 
+band from the EEG signals acquired under prosodic 
+auditory stimuli, subsequently using them to generate a 
+connectivity model based on inter-channel Granger 
+causality 10. By modelling connectivity as sources and 
+sinks, we seek to clarify the existence of abnormalities 
+between learners with and without DD, aiming to offer 
+an 
+enhanced understanding 
+of 
+the 
+mechanisms 
+underlying DD, ultimately allowing early diagnosis. 
+The remainder of the paper is structured as follows. 
+Section 2 describes the most relevant extant work in this 
+field, and Section 3 outlines the data and methodology, 
+including the preprocessing, Granger causality matrices 
+and connectivity matrices construction, and classification 
+algorithms. Section 4 presents our main results, leading 
+to the discussion in Section 5. Finally, Section 6 presents 
+the main conclusions and contributions of this work. 
+2. Related works 
+Previous studies have indicated that the phonological 
+deficit that causes DD may be due to an impairment in 
+the neural encoding of low-frequency speech envelopes, 
+relating to speech prosody 11. There is evidence of 
+significant difficulties among that learners with DD in 
+tasks relying on prosodic awareness, e.g. identifying 
+syllable stress, compared to controls at an earlier reading 
+level 12. This indicates the presence of atypical oscillatory 
+functioning in low-frequency brain rhythms in DD 13. 
+There has been substantial research on the important role 
+played by the ability to perceive prosodic frequency. 
+After directly measuring the neural encoding of 
+children’s speech using EEG, Power et al. 
+11 
+reconstructed the participants’ speech stimulus envelopes 
+using the emergent patterns. The EEG recordings were 
+done while the participants were performing a word 
+report task using noise-vocoded speech, i.e. still with a 
+low-frequency envelope yet with a degraded temporal 
+fine structure (TFS) of speech. Due to this degradation, 
+the participants necessarily derived the spoken words and 
+sentences from the information given by the envelope. If 
+the learners could accurately perceive the words and 
+sentences, it was possible to evaluate the functioning of 
+their neural encoding of the low-frequency envelopes in 
+speech, which is likely impaired in learners with DD 
+according to temporal sampling theory. 
+Brain activity, and thus the connectivity network, occurs 
+across various frequency bands, as demonstrated via the 
+temporal sampling framework (TSF). Temporal coding 
+is thought to be partially attributed to synchronous 
+auditory cortex activity, wherein the network neurons 
+synchronize 
+endogenous 
+oscillations 
+at 
+different 
+preferred rates while matching the temporal information 
+of the acoustic speech signal 14 15 16. The auditory and 
+visual parts of speech unfold across different timescales, 
+and thus, when the neurons in auditory and visual cortices 
+oscillate, they are believed to phase-align their activity to 
+match the input’s modulation rates 17. 
+TSF proposes that atypical oscillatory sampling at 
+various temporal rates may be the cause of the 
+phonological impairment in DD. Furthermore, a potential 
+biological mechanism for DD has recently been 
+suggested, highlighting the presence of atypical 
+dominant neural entrainment 18 for the slow rhythmic 
+prosodic (0.5–1 Hz), syllabic (4–8 Hz) and phoneme (12–
+40Hz) rhythm categories 19. Following this line of 
+thought, we might consider learners with DD to have 
+atypical oscillatory sampling for at least one temporal 
+rate, leading to difficulties in phonologically capturing 
+linguistic units such as syllables or phonemes.  
+However, this phenomenon is not likely to be 
+experienced equally across all frequency bands (i.e. 
+Delta, Theta, Alpha, Beta, and Gamma). Thus, it seems 
+pertinent to examine these bands’ connectivity patterns 
+separately using EEG. Prior research has indeed used 
+EEG or MEG to investigate the fundamental mechanisms 
+underlying DD, implementing speech-based stimuli 
+under the premise that DD is essentially derived from a 
+lesser awareness of individual speech units 20. Using 
+visual and auditory stimulus, Power et al. 21, for example, 
+identified differences between learners with DD and a 
+control group in the preferred entrainment phase of the 
+Delta and Theta bands. Based on changes in the 
+frequency, phase, and power spectrum, it thus becomes 
+feasible to derive measures of spectral connectivity. In 
+line with this, there are techniques showing the statistical 
+
+ 
+Neural source/sink phase connectivity in Developmental Dyslexia 
+ 
+3 
+relationship between electrodes on the same frequency 
+band 22. 
+The prior research has also explored the inference from 
+connectivity patterns during reading tasks. For example, 
+Žarić et al. 23 used visual word and false font processing 
+tasks to investigate disruptions in the connectivity 
+between the visual and language processing networks. 
+They hereby calculated the connectivity patterns based 
+on how statistically significant the differences in the 
+power spectral density (PSD) were for each EEG band. 
+Language-based reading or writing-related tasks have 
+also been used in previous studies identifying 
+discriminant patterns in EEG signals. For instance, using 
+graph theory, González et al. 24 compared the EEG 
+measurements of participants performing audiovisual 
+tasks or at rest to determine differences in the 
+connectivity patterns. Meanwhile, Stam et al. 25 used a 
+phase lag index to compute multiple weighted 
+connectivity matrices for multiple frequency bands.  
+Assessing the connectivity of two channels requires a 
+separate analysis of their respective phases. A signal’s 
+phase, φ(t), changes over time when being captured with 
+an electrode, and thus it must be measured for each 
+channel i, referring to the instantaneous phase, captured 
+using a Hilbert transform and computed via band-pass 
+filtered signals. Consequently, the phase value can be 
+pinpointed at each time point, allowing inter-channel 
+correlation and causality to be determined. Using this 
+method to track changes in the phase synchronization of 
+epileptic patients, Mormann et al. 26 showed that epileptic 
+episodes are often preceded by characteristic changes in 
+synchronization.  
+Following this, we can estimate the inter-channel 
+connectivity based on the cause-effect relationships. The 
+Granger causality test can hereby show whether one of 
+the factors is a time series, allowing the characteristics of 
+additional time series to be predicted. First employed in 
+the 1980s in the economics field, Granger causality is a 
+statistical hypothesis test that has been used to produce 
+good results in a wide range of other fields 10. 
+Neuroscience 
+research 
+has 
+applied 
+it 
+to 
+EEG 
+measurements, producing findings on brain activity in 
+emotion recognition 27, Vagus nerve stimulation 28, and 
+pain perception 29. 
+Connectivity based on causality implies cause-effect 
+relationships between various areas of the brain, but these 
+are not necessarily bidirectional. Thus, some brain areas 
+will be very active because they are influencing others, 
+and other areas may be very active because they are being 
+influenced by remote areas. Likewise, it could be the case 
+that high activity may be due to both situations.  While 
+this concept of sources/sinks is not new, it has been 
+subject to a variety of different approaches. For example, 
+Rimehaug et al. 30 integrated it into their model of the 
+visual cortex’s local field potential, while Sotero et al. 31 
+used it to explain the laminar distribution of phase-
+amplitude coupling of spontaneous current sources and 
+sinks in rat brains. However, neither of those studies 
+based their modeling of sources and sinks on causality 
+relationships, instead using the electrical activity in the 
+cerebral cortex. 
+The concepts of Granger causality and source/sink 
+relationships have been used to address the clinical issue 
+of surgical resection planning by capturing high-
+frequency ictal and preictal oscillations on an intracranial 
+EEG 32, although no connectivity maps were constructed; 
+furthermore, the study did not use machine learning to 
+examine whether this approach could be applied in the 
+differential diagnosis of impairments. 
+Building on the work outlined above, we apply machine 
+learning classification algorithms to assess the potential 
+of diagnosing DD via a learner’s sources, sinks and total 
+activity under stimulus, identified using Granger 
+causality matrices. Due to the impenetrable nature of 
+EEG signal classification and the complexity of the 
+problem being addressed, machine learning is highly 
+suitable 33. Briefly, we seek to demonstrate that different 
+connectivity patterns are induced in certain brain 
+networks by low-level auditory processing. To this end, 
+we delineate this connectivity by establishing the source 
+and sink relationships through the application of Granger 
+causality to the phase synchronization among EEG 
+channels. 
+3. Materials and methods 
+3.1. Data acquisition 
+The dataset comprised EEG data from the University of 
+Málaga’s Leeduca Study Group 34, gathered from 48 age-
+matched child participants (32 skilled readers and 16 
+dyslexic readers) (t(1) = -1.4, p > 0.05, age range: 88-100 
+months). All participants were righthanded native 
+Spanish speakers with normal or corrected-to-normal 
+vision; none had a hearing impairment. All participants 
+in the dyslexic group had been formally diagnosed with 
+dyslexia at school. All participants in the skilled reader 
+group were free from reading and writing difficulties and 
+had not been formally diagnosed with dyslexia. The 
+participants’ 
+legal 
+guardians 
+expressed 
+their 
+understanding of the study, gave their written consent, 
+and were present throughout the experiment. 
+All participants experienced an auditory stimulus in 15-
+minute sessions. The stimulus, which was modulated at 
+
+Rodríguez-Rodríguez et al. 
+ 
+4 
+4.8 Hz (prosodic-syllabic frequency) in 2.5-minute 
+segments, was band-limited white noise. This type of 
+stimulus was chosen to identify what synchronicity 
+patterns the low-level auditory processing would induce 
+and on the basis of the expert knowledge of linguistic 
+psychologists 
+concerning 
+the 
+main 
+frequency 
+components representing words in the human voice. The 
+participants’ EEG signals were recorded with a 
+BrainVision actiCHamp Plus with 32 active electrodes 
+(actiCAP, Brain Products GmbH, Germany) at a 500 Hz 
+sampling rate. The 10–20 standardized system was used 
+to place the 32 electrodes. 
+3.2. Preprocessing 
+The preprocessing involved removing all eye-blinking 
+and movement/impedance variation artifacts from the 
+EEG signals. The former were eliminated via 
+independent component analysis (ICA) 35 based on the 
+eye movements observed in the EOG channel, while for 
+the latter the relevant EEG segments were excluded. The 
+channels were then referenced to the Cz channel. 
+Then, a band-pass filter was applied to the EEG channels 
+to collect information for the five EEG frequency bands 
+(Delta, 1.5–4 Hz; Theta, 4–8 Hz, Alpha, 8–13 Hz; Beta, 
+13–30 Hz; and Gamma, 30–80 Hz). We used finite 
+impulse response (FIR) filters because these ensure a 
+constant phase lag that can later be corrected. To be 
+specific, each signal was sent forward and backward 
+through the two-way zero-phase lag band-pass FIR least-
+squares filter, producing a zero-lag phase in the overall 
+filtering process that addressed the issue of phase lag 36. 
+As low-pass filtering with an 80 Hz threshold was 
+employed, we added a 50 Hz notch filter during 
+preprocessing to eliminate this frequency component. 
+3.3. Hilbert Transform 
+A Hilbert transform (HT) transforms real signals into 
+analytic signals, i.e. complex-valued time series without 
+negative frequency components, allowing the time-
+varying amplitude, phase and frequency, i.e., the 
+instantaneous amplitude, phase and frequency, to be 
+calculated from the analytic signal. 
+ 
+We define HT for a signal x(t) as: 
+ 
+ℋ[𝑥(𝑡)] = 1
+𝜋 ∫
+𝑥(𝑡)
+𝑡 − 𝜏 𝑑𝜏
++∞
+−∞
+ 
+(1) 
+ 
+and we obtain the analytic signal zi(t) for signal x(t) as: 
+ 
+𝑧𝑖(𝑡) = 𝑥𝑖(𝑡) + 𝑗ℋ{𝑥𝑖(𝑡)} = 𝑎(𝑡)𝑒𝑗𝜙(𝑡) 
+(2) 
+ 
+From zi(t), computing the instantaneous amplitude is 
+straightforward: 
+ 
+𝑎(𝑡) = √𝑟𝑒(𝑧𝑖(𝑡))2 + 𝑖𝑚(𝑧𝑖(𝑡))2 
+(3) 
+ 
+with the instantaneous, unwrapped phase as: 
+ 
+𝜙(𝑡) = 𝑡𝑎𝑛−1 𝑖𝑚(𝑧𝑖(𝑡))
+𝑟𝑒(𝑧𝑖(𝑡)) 
+(4) 
+ 
+The above technique gives the phase value for each time 
+point, allowing the inter-channel synchronization to be 
+estimated based on the phase variation. 
+3.4. Granger Causality test 
+Developed for the field of econometrics by Clive 
+Granger, 
+Granger 
+causality 
+37 
+describes 
+causal 
+interactions occurring between continuous-valued time 
+series. As a statistical hypothesis test, it essentially states 
+that “the past and present may cause the future, but the 
+future cannot cause the past”; hence, knowing a cause 
+will be more helpful in predicting future effects than an 
+auto-regression will. Specifically, variable x will 
+Granger-cause y if the auto-regression for y that uses past 
+values of x and y is significantly more accurate than one 
+using only past values of y. We may exemplify this by 
+taking two stationary time-series sequences, xt and yt, 
+whereby xt−k and yt−k are, respectively, the past k values 
+of xt and yt. We then use two regressions to perform 
+Granger causality:  
+ 
+𝑦𝑡̂ 1 = ∑ 𝑎𝑘
+𝑙
+𝑘=1
+𝑦𝑡−𝑘 + 𝜀𝑡 
+(5) 
+ 
+𝑦𝑡̂ 2 = ∑ 𝑎𝑘
+𝑙
+𝑘=1
+𝑦𝑡−𝑘 + ∑ 𝑏𝑘
+𝑤
+𝑘=1
+𝑥𝑡−𝑘 + 𝜂𝑡 
+(6) 
+ 
+where 𝑦𝑡̂ 1 and 𝑦𝑡̂ 2 are, respectively, the fitting values of 
+the first and second regressions; l and w are the maximum 
+numbers of the lagged observations of xt and yt; ak; bk ∈ 
+R are the regression coefficient vectors estimated using 
+least squares; and εt and ηt are white noise (prediction 
+errors). Note that even though w can be infinite, due to 
+the finite nature of our data, we consider w finite and give 
+it a length well below the time series length, estimated 
+using model selection, such as the Akaike information 
+criterion (AIC) 38. Next, an F-test is applied to give a p-
+value indicating whether the regression model produced 
+
+ 
+Neural source/sink phase connectivity in Developmental Dyslexia 
+ 
+5 
+by Eq. (5) is statistically better than that of Eq. (6). If it 
+is, then x Granger-causes y.  
+We perform Granger causality testing for each 
+participant and evaluate the channels’ interactions, 
+producing an n x n square matrix of p-values (n = number 
+of channels). 
+Using Granger causality to analyze the neural network’s 
+directed functional connectivity intuitively demonstrates 
+the directionality with which information is transmitted 
+between neurons or brain regions. Previous studies have 
+already applied this technique to EEG analysis with great 
+success 39 40. 
+3.5. Connectivity vectors 
+The field of neuroscience tends to consider the brain as a 
+network using functional information  
+41 
+42 
+43, 
+culminating in the so-called connectome. This refers to 
+the complete mapping of all connections between brain 
+regions as an adjacency matrix, and often includes the 
+covariance, as well as other metrics, between fMRI 
+signals measured for different regions. Several studies 
+have also examined the temporal covariance between 
+EEG electrodes. 
+Once we had assembled the Granger causality matrices 
+for each participant subject, we established a threshold 
+value that evidenced a causal relationship between the 
+channels. Then, we formulated the three scenarios used 
+to produce each participant’s feature set:  
+• Sources: Array of n x 1 elements; each element 
+relates each channel with the number of channels that 
+it influences. 
+• Sinks: Array of n x 1 elements; each element relates 
+each channel with the number of channels that it is 
+influenced by. 
+• Total activity: Array of n x 1 elements; the sum of 
+the two previous scenarios, acting as a reference for 
+each channel’s global activity. 
+By organizing the information thus, we receive the same 
+number of features as there are channels for each 
+participant, each with a number that indicates its activity 
+as a source, as a sink, or the total. A summary of this 
+process is presented in Figure 1. 
+ 
+ 
+Fig. 1. Assembling the source and sink connectivity arrays for 
+a participant, given the relevant Granger matrix. [P(k)] is an 
+Iverson bracket function. 
+3.6. Ensemble feature selection 
+If the model includes many features, it will be more 
+complex, potentially leading to data overfitting. 
+Moreover, some of the features may be noise and could 
+adversely affect the model. Thus, we removed such 
+features to ensure the better generalization of the model. 
+We hereby selected the variables based on majority 
+voting through the application of several techniques. If a 
+variable was chosen by an algorithm, it received one 
+vote.  The votes were then summed for each variable, and 
+those with the most votes were selected. (Fig. 2). This 
+method has been found to be suitable for datasets that are 
+high-dimensional yet have few instances 44. The voting 
+strategy used a variety of feature selection methods 45, as 
+outlined in the following:  
+Information value (IV) using weight of evidence 
+(WOE): This indicates the predictive power of an 
+independent variable concerning the dependent variable 
+46. It allows a continuous independent variable to be 
+transformed into a set of groups or bins based on the 
+similarity of the dependent variable distribution (i.e. 
+numbers of events and non-events). Using WOE allows 
+outliers and missing values to be addressed and 
+eliminates the need for dummy variables 47: 
+ 
+𝑊𝑂𝐸 = ln (
+𝐸𝑣𝑒𝑛𝑡%
+𝑁𝑜𝑛 𝐸𝑣𝑒𝑛𝑡%) 
+(7) 
+ 
+𝐼𝑉 = Σ[(𝐸𝑣𝑒𝑛𝑡% − 𝑁𝑜𝑛 𝐸𝑣𝑒𝑛𝑡%) ∗ 𝑊𝑂𝐸] 
+(8) 
+ 
+An IV statistic above 0.3 is held to indicate a strong 
+relationship between the predictor and the event/non-
+event odds ratio 48. 
+
+FP1
+FP2
+F7
+PO10
+Sinks =
+FP1
+1
+pvalue<0.01
+GrM[FP1,k],
+[P(R)
+FP2
+1
+, GrM[F7,k].
+F7
+pvalue<0.01
+1
+pvalue<0.01
+P(a)
+...
+PO10
+pvalue<0.01
+1
+ GrM[P010, k]
+P(k)
+, GrM[k, FP2]
+Sources =
+GrM[k, FP1],
+[P(k)
+l0otherwiseRodríguez-Rodríguez et al. 
+ 
+6 
+Variable importance using random forest/extra trees 
+classifier: Calculated using a tree-based estimator, this 
+can be used to eliminate irrelevant features. Variable 
+importance is conventionally computed using the mean 
+decrease in impurity (i.e., gini importance 
+49) 
+mechanism, wherein the improvement in the split 
+criterion for each split of each tree is the importance 
+measure assigned to the splitting variable. For each 
+variable, this is separately accumulated over all the trees 
+in the forest. This measure is similar to the R2 in the 
+training set regression. 
+Recursive Feature Elimination: This can be used to 
+select features by recursively considering feature sets 
+with diminishing size based on an external estimator (a 
+linear regression model) that assigns weights to the 
+features 50. The estimator is trained on the first feature 
+set, noting each feature’s importance based on a given 
+attribute. The least important features are subsequently 
+removed from the current set. The process is performed 
+recursively on the pruned set until the desired number of 
+features is achieved. 
+Chi-square best variables: This uses a chi-square (χ2) 
+test to assess the correlations among a dataset’s features 
+and identify multicollinearity. The aim is revealing any 
+relationships between the dependent variable and any of 
+the independent variables 51. In the chi-square test, H₀ 
+(null hypothesis) assumes that two features are 
+independent, while H₁ (alternative hypothesis) predicts 
+that they are related.  We set a α=0.05 and a p-value of 
+0.05 or greater is considered critical, anything less means 
+the deviations are significant hence the hypothesis must 
+be rejected. 
+L1-based feature selection: Some features can be 
+eliminated using a linear model with an L1 penalty. This 
+method involves regularization, wherein a penalty is 
+added to various parameters of a machine learning model 
+to reduce the model’s freedom and prevent overfitting. 
+When regularizing linear models, the penalty is applied 
+in addition to the coefficients multiplying the predictors 
+52. Unlike other forms of regularization, L1 can reduce 
+some coefficients to zero, meaning the feature is 
+removed. 
+Once the best variables had been chosen by voting, we 
+performed a multicollinearity check on them. 
+ 
+ 
+Fig. 2. The feature selection procedure for the ‘Sources’ 
+scenario using a vote-based approach. 
+3.7. Classification process 
+In an ensemble method, multiple models are first 
+generated and then integrated to produce higher-quality 
+results. The respective predictions are hereby combined 
+using weighted majority voting to make the final 
+prediction. At each boosting iteration, the data are 
+modified by applying w1, w2 , …, wn to each training 
+sample. As the weights are initially wi=1/N, a weak 
+learner is trained in the first step using the raw data. At 
+each successive iteration, the sample weights are 
+modified individually, and the algorithm is then applied 
+to the reweighted data. Training examples that are 
+incorrectly predicted relative to the previous step’s 
+boosted model are given increased weights; correctly 
+predicted examples are given decreased weights. As a 
+result, the examples that were difficult to predict become 
+increasingly influential as the number of iterations 
+increases, and the weak learners that follow are forced to 
+focus on the examples previously missed. 
+Ensemble methods deliver more accurate results than 
+single models, and are particularly suitable for improving 
+binary prediction on small data sets. We use the Gradient 
+Boosting classifier, as well as an Ada Boost for results 
+verification. This latter classifier 53 is a meta-estimator 
+that initially fits to the data, with further copies then being 
+fit to the same data, while incorrectly classified 
+instances’ weights are modified to force subsequent 
+classifiers to focus on them. The Gradient Boosting 
+classifier 54 creates an additive model based on a forward 
+stage-wise construction, allowing the optimization of the 
+arbitrary differentiable loss function. At each stage, n 
+regression trees are fit to the multinomial or deviance 
+binomial loss function’s negative gradient, with a single 
+regression tree being used for the special case of binary 
+classification. To identify the best parameter set, we 
+cross-validate with 20 folds and a parameter grid, as 
+shown in Table 1. 
+
+Method:IVusingWOE-→
+Feature subset: f1, f2, f3,.., fn
+All features for Scenario Sources:
+Voting
+FP1. FP2,F7... PO10
+Method:Var.Imp.using RF→
+Feature subset:f'1, f'2,f3, ...,f'n
+M
+Method:Var.Imp.using Trees →
+Feature subset: f"1, f"2, f"3,., f"n
+Method:RFE→Feature subset
+Reducedranked feature
+subsetbasedonvotes
+→fr1,f2,f3,.,frn
+Method:Chi Squared→Feature subset 
+Neural source/sink phase connectivity in Developmental Dyslexia 
+ 
+7 
+Table 1. Parameter grid of machine learning classifiers. 
+Algorithm 
+Parameter 
+Range 
+Gradient  
+n_estimators 
+1 to 12 
+Boosting 
+Loss 
+deviance, exponential 
+ 
+Learning rate 
+0.05 to 1.5 
+ 
+Criterion 
+friedm_mse, sq_error, mse, mae 
+ 
+Min_samples_split 
+0.01 to 3 
+ 
+Min_samples_leaf 
+0.01 to 3 
+ 
+ 
+Max_depth 
+1 to 4 
+Ada Boost 
+n_estimators 
+1 to 25 
+ 
+Learning rate 
+1 to 3.5 
+ 
+Boosting algorithm 
+SAMME, SAMME.R 
+4. Results 
+Plotting each learner’s array of sources and sinks permits 
+the visual extraction of the respective patterns of the 
+dyslexic and control groups. To this end, we examined 
+the channel distributions for both groups by calculating 
+the means and dispersions and producing a box-and-
+whisker plot. We also constructed a topoplot as this can 
+illustrate the results with greater clarity. For example, 
+Fig. 3 shows the Theta band connectivity of the control 
+and dyslexic groups specifically for total activity. Please 
+note that Fig. 3 and Fig. 4 do not directly represent the 
+electrical activity of the cerebral cortex, but rather show 
+the levels of the cause-effect relationships between the 
+channels, i.e. in one direction or in the other direction or 
+in total. It immediately becomes clear that despite the 
+similarity of the patterns, the dyslexic group has a 
+significantly higher activity level in the Theta band. 
+ 
+ 
+ 
+ 
+ 
+Fig. 3. Boxplot of the total activity in the Alpha band. 
+Fig 4. The equivalent graphical representation of Fig. 3 in a 
+topoplot. 
+ 
+ 
+ 
+Fig. 5. Source/sink activity in the Theta, Beta and Gamma bands in the control and dyslexic groups. Numbers represent how many 
+channels are affected by each channel as a source, or how many channels are affecting each channel as a sink. 
+
+ActivityofsourcesinThetaband
+Control
+22.5
+Dyslexic
+20.0
+nels
+7.5
+5.0
+2.5 -
+Fp1 Fp2 F7 F3 FZ F4 F8 FC5 FC1 FC2FC6 T7 C3 C4 T8 TP9CP5 CP1CP2CP6IP1O P7 P3 PZ P4 P8 PO9 O1 OZ O2PO10
+ChannelActivityofsources inThetaband
+Control group
+Dyslexic group
+19
+19
+Fp1
+Fp2
+F1
+2
+F8
+IFG
+13
+FC6
+FQ1
+FG2
+FG6
+.
+13
+TZ
+T8
+G3
+18
+TP9CR5
+CP1
+CPEFP.i
+CP5
+CBI
+Cp2
+CP6
+P3
+RZ
+P4
+P8
+P7
+PZ
+F4
+P8
+kod
+01
+02
+Pig
+60
+01
+02
+Poip
+QzActivity of sources in Theta band
+19
+Control group
+Dyslexlc group
+19
+13
+13Activity of sources in Beta band
+19
+Control group
+Dyslexlc group
+19
+F8
+13
+13Activityof sources in Gamma band
+19
+Control group
+Dyslexlc group
+19
+13
+13Activity of sinks in Theta band
+19
+Control group
+Dyslexic group
+19
+Fp2
+13
+13Activity of sinks in Beta band
+19
+Control group
+Dyslexic group
+19
+13
+6
+13Activity of sinks in Gamma band
+19
+Controlgroup
+Dyslexicgroup
+19
+Fp.
+F4
+13
+T651
+FE6
+13
+4
+9P5
+EPFTotal activity per channel in Theta band
+Control group
+Dyslexic group
+38
+38
+27
+27
+15
+15Total activity per channel in Beta band
+Control group
+Dyslexic group
+38
+38
+27
+27
+CP
+CPI
+15
+TE
+15Total activity per channel in Gamma band
+Control group
+Dyslexic group
+38
+38
+H
+27
+Fe6
+27
+4
+15
+15Rodríguez-Rodríguez et al. 
+ 
+8 
+Fig. 5 compares the channel activity in the Theta, Beta 
+and Gamma bands, and can be viewed separately as 
+sources, sinks, or total activity for both the control and 
+dyslexic groups. Please note that the range of 
+visualization is the same in all sinks/sources topoplots, 
+while different in the total activity ones, for better 
+representation. Once more, it is immediately clear that 
+while the patterns are broadly similar, the activity level is 
+higher in the dyslexic group, primarily observed in the 
+sink activity (less in the source activity). Thus, although 
+the sources, broadly speaking, behave similarly between 
+the groups, the dyslexic group has significantly more 
+concentrated sinks and more activity. Consequently, the 
+overall activity level is also affected. 
+ 
+ 
+Fig. 6. Feature importance in Theta, Beta and Gamma bands considering sources, sinks and total activity. 
+With as many arrays as subjects, and with each array 
+having as many components as channels, we performed 
+feature selection to identify channels that can help 
+differentiate between the control and dyslexic groups. 
+The feature selection procedure outlined above was thus 
+applied for the cases of sources, sinks and total activity, 
+according to the band. Fig. 6 presents the results for the 
+Theta, Beta and Gamma bands, whereby the importance 
+values are normalized to permit fair and simple 
+comparison. Channels showing a higher significance are 
+those with more dissimilarity between the control and 
+
+Feature importance in Theta band
+1.0
+Activity of sources
+Activity of sinks
+Total activity per channel
+0.8
+0.2 -
+0.0-
+Fp1
+Fp2
+F7
+F3
+Fz
+F4
+F8 
+FC5
+FC1
+FC2
+FC6
+T7
+C3
+C4
+T8
+CP5
+CP1
+CP2
+CP6 TP10
+P3
+Pz
+P4
+P8
+PO9
+Q1
+Qz
+ZO
+PO10
+ChannelFeature importance in Beta band
+1.0
+Activity of sources
+I Activity of sinks
+Total activity per channel
+0.8-
+Fea
+0.2
+0.0-
+Fp1
+Fp2
+F7
+F3
+Fz
+F4
+F8
+FC5
+FC1
+FC2
+FC6
+T7
+C3
+C4
+T8
+CP5
+CP1
+CP2
+CP6TP10
+P7
+P3
+Zd
+P4
+P8
+PO9
+Q1
+Qz
+02
+PO10
+ChannelFeatureimportanceinGammaband
+1.0
+Activity of sources
+Activity of sinks
+Total activity per channel
+0.8
+0.2
+0.0-
+Fp1
+Fp2
+F7
+F3
+Fz
+F4
+F8
+FC5
+FC1
+FC2
+FC6
+C3
+C4
+T8
+CP5
+CP1
+CP2
+CP6TP10
+3
+P4
+P8
+PO9
+Q1
+Qz
+02PO10
+Channel 
+Neural source/sink phase connectivity in Developmental Dyslexia 
+ 
+9 
+dyslexic groups, directing us to where we can find 
+different patterns of functioning. 
+After performing the feature selection for each band, for 
+each case (sources, sinks and total activity), we optimize 
+the Gradient Boosting classifier to obtain the best 
+performance. The results are summarized in Table 2, with 
+performances achieving at least 80% marked bold. 
+According to the results, the greatest differences between 
+the control and dyslexic groups (i.e., the best classifier 
+results) emerge in the Theta and Gamma bands when 
+accounting for the activity sink role of the different 
+channels, achieving accuracies of 84% and 88%, 
+respectively. We also wish to highlight the results for the 
+Beta band for the activity sources regarding the Area 
+Under the Curve (AUC), in addition to accuracy. 
+Table 2. Results of the Gradient Boosting machine 
+learning classifier. 
+Band 
+Features set 
+Accuracy 
+AUC 
+Delta 
+Sources 
+0.77 ± 0.14 
+0.65 ± 0.31 
+ 
+Sinks 
+0.79 ± 0.20 
+0.70 ± 0.29 
+ 
+Total activity 
+0.74 ± 0.19 
+0.76 ± 0.25 
+Theta 
+Sources 
+0.77 ± 0.17 
+0.77 ± 0.30 
+ 
+Sinks 
+0.84 ± 0.15 
+0.87 ± 0.18 
+ 
+Total activity 
+0.74 ± 0.17 
+0.72 ± 0.28 
+Alpha 
+Sources 
+0.79 ± 0.19 
+0.74 ± 0.25 
+ 
+Sinks 
+0.76 ± 0.21 
+0.71 ± 0.29 
+ 
+Total activity 
+0.79 ± 0.17 
+0.77 ± 0.21 
+Beta 
+Sources 
+0.80 ± 0.17 
+0.86 ± 0.18 
+ 
+Sinks 
+0.79 ± 0.24 
+0.81 ± 0.27 
+ 
+Total activity 
+0.76 ± 0.23 
+0.75 ± 0.32 
+Gamma 
+Sources 
+0.81 ± 0.18 
+0.83 ± 0.22 
+ 
+Sinks 
+0.88 ± 0.14 
+0.93 ± 0.16 
+ 
+Total activity 
+0.82 ± 0.12 
+0.87 ± 0.18 
+ 
+The Receiver Operating Curve (ROC) space is a valuable 
+data interpretation tool that can be used to assess the 
+performance of a binary classifier, wherein it indicates 
+the cutoff point at which sensitivity is traded for 
+specificity. Hence, it can be used to evaluate the 
+classifier’s performance in distinguishing positive and 
+negative samples. Related to this, AUC is the probability 
+that the classifier will assign a random positive instance 
+a more extreme value than a random negative instance. 
+Fig. 7 presents the ROC curves for the Theta, Beta and 
+Gamma bands, to identify those with the best 
+performance. Notably, the Gamma band with the 
+channels’ sinks activity as the features presents a 93% 
+under the curve. 
+The obtained results were verified by repeating the 
+classification process using the Ada Boost algorithm. 
+Table 3 presents the results for the Gamma band while 
+Fig. 8 shows the ROC curve. While the performance is 
+slightly diminished, it remains consistent across all bands 
+and cases (sources, sinks and total activity) with the 
+results from the Gradient Boosting. 
+ 
+ 
+ 
+Fig. 7. ROC curves for the Theta, Beta and Gamma bands with 
+the Gradient Boosting classifier. 
+ 
+
+Thetaband
+1.0
+: Rate (Positive label:
+0.8
+0.6
+Positive
+0.4
+True
+0.2
+Chance
+SourcesMeanROC(AUC =0.77±0.31)
+Sinks Mean ROC (AUC = 0.87±0.18)
+0.0
+Total activityMeanROC (AUC = 0.72 ± 0.28)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+False Positive Rate (Positive label: 1)Betaband
+1.0
+: Rate (Positive label:
+0.8
+0.6
+Positive
+0.4
+True
+0.2 
+Chance
+SourcesMeanROC(AUC=0.86±0.18)
+Sinks Mean ROC (AUC = 0.81 ± 0.27)
+0.0
+Total activityMeanROC (AUC = 0.75 ± 0.33)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+False Positive Rate (Positive label:1)Gammaband
+1.0
+ Rate (Positive label: 1)
+0.8
+0.6
+Positive
+0.4
+True
+0.2
+Chance
+SourcesMeanROC(AUC=0.83±0.23)
+SinksMeanROC(AUC=0.93± 0.17)
+0.0
+Total activityMean ROC (AUC = 0.87 ± 0.18)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+False Positive Rate (Positive label: 1)Rodríguez-Rodríguez et al. 
+ 
+10 
+Table 3. Results for the Ada Boost classifier for 
+the Gamma band. 
+Band 
+Feature set 
+Accuracy 
+AUC 
+Gamma 
+Sources 
+0.83 ± 0.17 
+0.82 ± 0.27 
+ 
+Sinks 
+0.88 ± 0.11 
+0.86 ± 0.21 
+ 
+Total activity 
+0.77 ± 0.19 
+0.76 ± 0.31 
+ 
+ 
+Fig. 8. ROC curves for the Gamma band with the Ada Boost 
+classifier. 
+As is often the case in biomedical studies, statistical tests 
+are required to check that the number of samples has not 
+introduced bias in the classification stage (e.g., through 
+overfitting). Moreover, there is a need to check the 
+probability of these results having been obtained by 
+chance. For large datasets, such tests need not be as 
+stringent, but real-world studies demand special attention 
+due to the small sample sizes and unbalanced classes. 
+Specifically, in experimental studies the prevalence of 
+the disorder among the population being treated must be 
+taken into account. For DD, this is around 5-12%, as 
+mentioned above. 
+To this end, a null distribution is generated by estimating 
+the classifier’s accuracy for 1000 permutations of the 
+labels. This indicates the distribution for the null 
+hypothesis that the features are not dependent on the 
+labels, and enables the estimation of the probability that 
+the classification results will be reproduced with shuffled 
+labels. The result is an empirical p-value determined by: 
+ 
+𝑝 − 𝑣𝑎𝑙𝑢𝑒 = #𝑝𝑒𝑟𝑚 𝑤𝑖𝑡ℎ 𝑎𝑐𝑐. ℎ𝑖𝑔ℎ𝑒𝑟 𝑡ℎ𝑎𝑛 𝑏𝑎𝑠𝑒𝑙𝑖𝑛𝑒
+#𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑒𝑟𝑚𝑢𝑡𝑎𝑡𝑖𝑜𝑛𝑠
+ 
+(9) 
+ 
+Fig. 9 gives the permutation test results for the Theta, 
+Beta, and Gamma bands for sources, sinks and total 
+activity. 
+The 
+null 
+distribution 
+from 
+the 
+label 
+permutations, as outlined above, is in blue, while the 
+vertical red line represents the accuracy obtained for the 
+non-permuted case. At each permutation iteration, a 20-
+fold stratified cross-validation is performed, and based on 
+the average of the results obtained at these 20 folds, the 
+corresponding permutation iteration is determined. 
+Hence, Fig. 9 presents the classification’s probability 
+density. According to the permutation tests, the results 
+have low p-values and are significant. 
+5. Discussion 
+The participants were subjected to white noise at 4.8 Hz, 
+i.e. between the syllabic and prosodic frequencies, as the 
+sole stimulus. DD has been shown to link to impairments 
+in syllabic and prosodic perception 55, suggesting general 
+difficulties in identifying the different modulation 
+frequencies. This influences the slower temporal rates of 
+speech processing in particular, as well as the tracking of 
+the amplitude envelope of speech, diminishing learners’ 
+syllabic segmentation efficiency. 
+Multi-time resolution models of speech processing 16 
+have evidenced that phonetic segment identification 
+associates with faster temporal modulations (Gamma 
+rate, 30–80 Hz), syllable identification is linked to slower 
+modulations (Theta rate, 4–10 Hz), and syllable stress 
+and prosodic patterning information correlates with very 
+slow modulations (Delta rate, 1.5–4 Hz). Nonetheless, 
+anomalies can emerge in various frequency ranges due to 
+inter-band entrainment. 
+As it offers adequate time resolution, examining the 
+patterns occurring in EEG channels at different bands can 
+unveil the speech encoding linked to problems with 
+speech prosody and sensorimotor synchronization. 
+Exemplifying this, previous research 18 used speech-
+based stimuli and time-frequency descriptors to reveal 
+the link between speech features and neural dynamics. 
+We find that the classifier performs better in the Theta 
+and Gamma bands. The results for the Theta band are 
+expected as the TSF suggests that the phonological 
+deficit of DD – regardless of language – may be partially 
+attributed to functionally atypical or impaired phonology 
+entrainment mechanisms in the auditory cortex, 
+especially as oscillations at slower temporal rates, i.e. 
+Theta and Delta, relate to syllabic and prosodic 
+processing 56. 
+ 
+ 
+
+Gammaband
+1.0
+: Rate (Positive label:
+0.8
+0.6
+Positive
+0.4
+True
+0.2
+Chance
+SourcesMeanROC(AUC=0.78±0.27)
+Sinks MeanROC (AUC=0.86±0.21)
+0.0
+Total activity Mean ROC(AUC=0.76± 0.31)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+False Positive Rate (Positive label:1) 
+Neural source/sink phase connectivity in Developmental Dyslexia 
+ 
+11 
+ 
+ 
+ 
+Fig. 9. Permutation tests for Gradient Boosting classifier in Theta, Beta and Gamma bands. 
+As per the TSF, group differences are expected in 
+neuronal oscillatory entrainment at slower rates (approx. 
+4 Hz, in line with the stimulus used) 57. Higher causality 
+relationships emerged in the frontal area in all scenarios 
+for the Theta band. In addition, the number of channels 
+that g-causes causality is higher in the dyslexic domain, 
+which was the case for the sources, sinks and total 
+activity. This higher activity in terms of overall causality 
+relations was evident across all bands. However, in the 
+participants with DD there was significantly less 
+entrainment in the auditory networks of the right 
+hemisphere in the Theta band. As Fig. 6 (feature 
+selection) shows, the C4 channel in the upper part, i.e. the 
+Theta band, is predominantly influential for the causality 
+regarding the sources, as well as the sinks and total 
+activity. It has already been established that the right-
+lateralized Theta sampling network tends to involve 
+slower temporal rates and codes the speech signal’s lower 
+modulation frequencies 57, facilitating syllable-scale 
+temporal integration. In other words, spoken sentences 
+are tracked and distinguished by the Theta band phase 
+pattern, allowing the incoming speech signal to be broken 
+into syllable-sized packets and speech dynamics to be 
+tracked through resetting and sliding, such as with 
+varying rates of speech 58. Fig. 5 (topoplots) clearly 
+demonstrates that the C4 channel is the most interesting 
+as it has the most Granger causality (causing and being 
+caused) for all scenarios for the dyslexic group. For the 
+sources, the frontal area contains other noteworthy 
+channels (FP2, F7, F3 and Fz) that show differences 
+between the control and dyslexic groups in terms of 
+activity. The most influential channels in the sinks are F3 
+and F4 (frontal area) and P3.  
+Hence, it seems pertinent to suggest that the main 
+differences in the causality relationships of the Theta 
+band lie in the so-called dorsal and ventral pathways. In 
+particular, the right area seems critical, as evidenced in 
+
+Thetaband-Sources
+8 
+7 
+Score on original data: o.77
+6 
+(p-value:0.002)
+5 
+I
+-
+4 -
+-
+FE
+2
+1 
+0-
+0.40
+0.45
+0.50
+0.55
+0.60
+0.65
+0.70
+0.75
+0.80
+Accuracy scoreTheta band - Sinks
+-
+8 
+score
+on original data: 0.84
+6 -
+(p-value:0.001)
+*.
+-
+-
+-
+0
+0.45
+0.50
+0.55
+0.60
+0.65
+0.70
+0.75
+0.80
+0.85
+Accuracy scoreTheta band -Total activity
+-
+8 -
+-
+Score on original data: o.74
+(p-value:0.001)
+6 -
+-
+robability
+ 4
+-
+2 -
+0
+0.50
+0.55
+0.60
+0.65
+0.70
+0.75
+Accuracy scoreBeta band-Sources
+10 -
+8 
+Score onoriginal data: o.80
+6
+(p-value:0.001)
+4
+2 -
+0
+0.45
+0.50
+0.55
+0.60
+0.65
+0.70
+0.75
+0.80
+Accuracy scoreBeta band -Sinks
+8 -
+！！
+7 -
+Score on original data: o.79
+6 -
+(p-value:0.001)
+.
+4.
+E
+2 
+1 
+0
+0.45
+0.50
+0.55
+0.60
+0.65
+0.70
+0.75
+0.80
+Accuracy scoreBeta band -Total activity
+7
+Score on original data: o.76
+6
+D-
+value:0.002
+5 
+3 -
+2 -
+1
+0
+0.45
+0.50
+0.55
+0.60
+0.65
+0.70
+0.75
+Accuracy scoreGammaband-Sources
+10 
+8
+Score on original data: 0.81
+(p-value: 0.001)
+4
+2 .
+0
+0.50
+0.55
+0.60
+0.65
+0.70
+0.75
+0.80
+Accuracy scoreGammaband-Sinks
+8
+Score 0n original data: 0.88
+6
+(p-value:0.001)
+2 
+0
+0.45
+0.50
+0.55
+0.60
+0.65
+0.70
+00.75
+0.80
+0.85
+Accuracy scoreGammaband-Totalactivity
+-8
+7
+Scoreor
+n original data: 0.82
+6 ·
+(p-value:0.001)
+3 
+2
+1 -
+0
+0.50
+0.55
+0.60
+0.65
+0.70
+0.75
+0.80
+Accuracy scoreRodríguez-Rodríguez et al. 
+ 
+12 
+the prior research and especially demonstrated here with 
+the sinks scenario. 
+Another interesting result worth discussing is that for the 
+Beta band. Here, more activity was observed for all three 
+scenarios in the dyslexic group; this agrees with the 
+results for the Theta band as well as those from previous 
+studies 59. For the sources, differences in the causal 
+relationships were mainly identified in the C3 and C4 
+channels, pointing to areas responsible for motor 
+processing 11. It is becoming increasingly clear that 
+speech perception is at least partially located in the motor 
+areas, especially under less-than-optimal listening 
+conditions. This cruciality of the C4 channel was 
+similarly seen in the Theta band and is in line with prior 
+research evidencing the important role played by the 
+lower frequency bands in general and Beta band coupling 
+in particular 60. Hence, inefficient phase locking in the 
+auditory cortex may affect visual and motor processing 
+development, which may in turn cause some of the 
+visual, motor and attentional difficulties seen in DD 61. 
+It should be noted, however, that the C3-C4 interaction is 
+mostly relevant for the sources and is not important for 
+either the C3 for the sinks or, as a result, for the total 
+activity. Meanwhile, the causal activity in the Beta band 
+is different in the occipital area in the sinks scenario, and 
+it is remarkably different in the frontal area, especially in 
+FP1 for all three scenarios and in the F3 channel for the 
+sinks scenario. 
+In the Gamma band the activity is higher than in the Theta 
+band for maximum values, although the occipital area 
+shows more concentrated activity among the causality 
+relations, as Fig. 5 shows. Nevertheless, the effect is 
+different between the control and dyslexic groups, 
+whereby the participants with DD show higher activity 
+for the sinks, which increases their total activity.  
+For the sources, the channels with the most explicit 
+differences are FC1 and, more generally, TP9 in the left 
+temporal area. In the case of sinks, this is also an 
+important channel, although O1 and, as highlighted 
+above, C3 also play a role.  
+Meanwhile, 
+in 
+the 
+Gamma 
+band, 
+despite 
+the 
+discrepancies between the dorsal and ventral pathways, 
+the latter offers the main difference for the classification 
+of TP9 for both sources and sinks. FC1 is linked to 
+sources and C3 to sinks, suggesting a significant cause-
+effect relationship, albeit with potentially less activity in 
+the dyslexic group, facilitating classification. 
+We can confirm that the classifier performs better in the 
+Theta and Gamma bands, which can evidence atypical 
+oscillatory differences based on both speech and non-
+speech stimuli 56. According to Leong’s models 62, the 
+slower rates (Delta and Theta) temporally constrain 
+entrainment at the faster rates, such as Gamma.  
+Lehongre et al. 65 contended that the oscillatory nesting 
+seen between the Theta/Delta phase and the Gamma 
+power 63 64 offers a way to integrate information at the 
+phonemic (Gamma) rate into the syllabic rate. 
+Meanwhile, the integration of the various acoustic 
+features that contribute to the same phoneme being 
+perceived may be hindered by impairments in the phase 
+locking by Theta generators. Otherwise, flaws in certain 
+Theta mechanisms could influence the development of 
+the phonological system, which thus tends to code 
+information bilaterally with the Gamma oscillations 
+independently and then link them perceptually with the 
+Theta oscillator output. In this case, the impaired phase 
+locking of the right hemisphere Theta oscillatory 
+networks causes difficulties with lower frequency 
+modulations 17 66. 
+In addition, the spontaneous oscillatory neural activity 
+identified in the auditory cortex in both the Theta and 
+Gamma bands is known to associate with spontaneous 
+activity in the visual and premotor areas 66. 
+A bilateral Gamma sampling network codes the signal’s 
+higher frequency modulations, thereby facilitating 
+temporal integration at the phonetic (i.e., phoneme) scale.  
+If we apply this model to DD, it is indicated that impaired 
+processing at the syllable level (i.e., less efficient Theta 
+phase locking) occurs alongside unimpaired Gamma 
+sampling, meaning more weight is assigned to phonetic 
+feature information during phonological development. 
+Hence, as is the case in typical infant development, 
+children with DD may have sensitivity to all phonetic 
+contrasts of human languages 67. 
+Leong and Goswami 62 found that learners with DD show 
+a preference for different phase alignment between 
+amplitude modulations (AMs) when these respectively 
+convey syllable and phoneme information (Theta and 
+Gamma-AMs). A different phase locking angle suggests 
+a discrepancy in the integration of speech information 
+that arrives at a temporal rate different to that of the final 
+perception of the speech 14. Our results concerning the 
+interaction between the Theta and Gamma bands support 
+this. 
+Finally, our results also seem to confirm that the dyslexic 
+brain is less efficient at encoding the amplitude 
+modulation hierarchy’s highest levels, i.e. those bearing 
+information on the prosodic-syllabic structure, leading to 
+cascade effects that impact the encoding of the 
+phonological structure’s levels nested within the Delta 
+band, such as the syllable-level (Theta band) and 
+phoneme-level (Gamma band) AM information. 
+Importantly, our results have been validated using a 
+demanding permutation test, with the aim of ensuring 
+that the results are not coincidental, despite the medium 
+sample size. 
+
+ 
+Neural source/sink phase connectivity in Developmental Dyslexia 
+ 
+13 
+6. Conclusion and future works 
+Our results support the main assumption of the TSF that 
+DD involves a specific deficit in the low-frequency phase 
+locking mechanisms in the auditory cortex, thereby 
+potentially affecting phonological development 56.  
+In confirmation of this, we find an anomaly that emerges 
+primarily in the causal relationships of channels that 
+function as sinks, which is significantly more pronounced 
+than when only the total activity is considered. Hence, it 
+is reasonable to consider a division into Granger-causing 
+or Granger-caused relationships. This, in turn, suggests 
+that the main differences contributing to DD emerge 
+when certain brain areas must function as receptors in the 
+interactions between channels.  
+Furthermore, our results are in line with previous 
+research, which has already detected an anomaly in the 
+right-lateralized Theta band. We have clearly identified 
+this here across all three scenarios (sources, sinks, total 
+activity). 
+We also find confirmation for the higher brain activity in 
+learners with DD, although differences are more 
+significant for the sinks in the Theta and Gamma bands, 
+in turn leading to more total activity. The highest 
+classifier performance (accuracy and AUC) is hereby 
+found in the sink scenario. For the Beta band, the 
+difference in activity is more consistent across all three 
+scenarios. The classifier also performs well for the Beta 
+band in all three scenarios, with few differences 
+observed, thereby confirming the important role played 
+by this band in the sensorimotor coding of speech. 
+The results reflect the causal activity generated in the 
+brain subjected to prosodic-syllabic stimulus at 4.8 Hz. 
+Consequently, future work could consider the Granger 
+causality relationships in the phases across channels and 
+bands using higher frequency stimuli to stimulate 
+syllabic-phonetic and phonetic activity. 
+Acknowledgements 
+This work was supported by projects PGC2018-098813-
+B-C32 (Spanish “Ministerio de Ciencia, Innovación y 
+Universidades”), UMA20-FEDERJA-086 (Consejería de 
+econnomía y conocimiento,Junta de Andalucía) and by 
+European Regional Development Funds (ERDF). 
+ 
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+

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+A neural network potential with self-trained atomic fingerprints:
+a test with the mW water potential
+Francesco Guidarelli Mattioli, Francesco Sciortino, and John Russo∗
+Sapienza University of Rome, Piazzale Aldo Moro 2, 00185 Rome, Italy
+(Dated: January 30, 2023)
+We present a neural network (NN) potential based on a new set of atomic fingerprints built upon
+two- and three-body contributions that probe distances and local orientational order respectively.
+Compared to existing NN potentials, the atomic fingerprints depend on a small set of tuneable
+parameters which are trained together with the neural network weights. To tackle the simultaneous
+training of the atomic fingerprint parameters and neural network weights we adopt an annealing
+protocol that progressively cycles the learning rate, significantly improving the accuracy of the NN
+potential. We test the performance of the network potential against the mW model of water, which
+is a classical three-body potential that well captures the anomalies of the liquid phase. Trained on
+just three state points, the NN potential is able to reproduce the mW model in a very wide range of
+densities and temperatures, from negative pressures to several GPa, capturing the transition from
+an open random tetrahedral network to a dense interpenetrated network. The NN potential also
+reproduces very well properties for which it was not explicitly trained, such as dynamical properties
+and the structure of the stable crystalline phases of mW.
+I.
+INTRODUCTION
+Machine learning (ML) potentials represent one of the
+emerging trends in condensed matter physics and are
+revolutionising the landscape of computational research.
+Nowadays, different methods to derive ML potentials
+have been proposed, providing a powerful methodology
+to model liquids and solid phases in a large variety of
+molecular systems [1–16, 16, 17]. Among these methods,
+probably the most successful representation of a ML po-
+tential so far is given by Neural Network (NN) potentials,
+where the potential energy surface is the output of a feed-
+forward neural network [18–35].
+In short, the idea underlying NN potentials construc-
+tion is to train a neural network to represent the po-
+tential energy surface of a target system.
+The model
+is initially trained on a set of configurations generated
+ad-hoc, for which total energies and forces are known,
+by minimizing a suitable defined loss-function based on
+the error in the energy and force predictions.
+If the
+training set is sufficiently broad and representative, the
+model can then be used to evaluate the total energy and
+forces of any related atomic configuration with an accu-
+racy comparable to the original potential. Typically the
+original potential will include additional degrees of free-
+dom, such as the electron density for DFT calculations,
+or solvent atoms in protein simulations, which make the
+full computation very expensive.
+By training the net-
+work only on a subset of the original degrees of freedom
+one obtaines a coarse-grained representation that can be
+simulated at a much reduced computational cost. NN
+potentials thus combine the best of two worlds, retain-
+ing the accuracy of the underlying potential model, at
+the much lower cost of coarse-grained classical molecu-
+∗Corresponding author: john.russo@uniroma1.it
+lar dynamics simulations. The accuracy of the NN po-
+tential depends crucially on how local atomic positions
+are encoded in the input of the neural network, which
+needs to retain the symmetries of the underlying Hamil-
+tonian, i.e. rotational, translational, and index permu-
+tation invariance. Several methods have been proposed
+in the literature [12, 36], such as the approaches based
+on the Behler-Parrinello (BP) symmetry functions [18],
+the Smooth Overlap of Atomic Positions (SOAP) [37],
+N-body iterative contraction of equivariants (NICE) [38]
+and polynomial symmetry functions [39], or frameworks
+like the DeepMD [23], SchNet [22] and RuNNer [18]. In
+all cases, atomic positions are transformed into atomic
+fingerprints (AFs). The choice of the AFs is particularly
+relevant, as it greatly affects the accuracy and generality
+of the resulting NN potential.
+We develop here a fully learnable NN potential in
+which the AFs, while retaining the simplicity of typi-
+cal local fingerprints, do not need to be fixed beforehand
+but instead are learned during the training procedure.
+The coupled training of the atomic fingerprint param-
+eters and of the network weights makes the NN train-
+ing process more efficient since the NN representation is
+spontaneously built on a variable atomic fingerprint rep-
+resentation. To tackle the combined minimization of the
+AF parameters and of the network weights we adopt an
+efficient annealing procedure, that periodically cycles the
+learning rate, i.e. the step size of the minimization algo-
+rithm, resulting in a fast and accurate training process.
+We validate the NN potential on the mW model of
+water [40], which is a one-site classical potential that
+has found widespread adoption to study water’s anoma-
+lies [41, 42] and crystallization phenomena [43, 44]. Since
+the first pioneering MD simulations [45, 46], water is of-
+ten chosen as a prototypical case study, as the large num-
+ber of distinct local structures that are compatible with
+its tetrahedral coordination make it the molecule with
+the most complex thermodynamic behavior [47], for ex-
+arXiv:2301.11612v1  [cond-mat.soft]  27 Jan 2023
+
+2
+ample displaying a liquid-liquid critical point at super-
+cooled conditions [48–52]. NN potentials for water have
+been developed starting from density functional calcu-
+lations, with different levels of accuracy [53–60].
+NN
+potentials have also been proposed to parametrise accu-
+rate classical models for water with the aim of speeding
+up the calculations when multi-body interactions are in-
+cluded [61], as in the MBpol model [62–64] or for testing
+the relevance of the long range interactions, as for the
+SPC/E model [65]. We choose the mW potential as our
+benchmark system because its explicit three-body poten-
+tial term offers a challenge to the NN representation that
+is not found in molecular models built from pair-wise
+interactions. We stress that we train the NN-potential
+against data which can be generated easily and for which
+structural and dynamic properties are well known (or
+can be evaluated with small numerical errors) in a wide
+range of temperatures and densities. In this way, we can
+perform a quantitative accurate comparison between the
+original mW model and the hereby proposed NN model.
+Our results show that training the NN potential at
+even just one density-temperature state point provides
+an accurate description of the mW model in a surround-
+ing phase space region that is approximately a hundred
+kelvins wide. A training based on three different state
+points extends the convergence window extensively, ac-
+curately reproducing state points at extreme conditions,
+i.e.
+large negative and (crushingly) positive pressures.
+We will show that the NN reproduces thermodynamic,
+structural and dynamical properties of the mW liquid
+state, as well as structural properties of all the stable
+crystalline phases of mW water.
+The paper is organized as follows. In Section II we de-
+scribe the new atomic fingerprints and the details about
+the Neural Network potential implementation, including
+the warm restart procedure used to train the weights
+and the fingerprints at the same time.
+In Section III
+we present the results, which include the accuracy of the
+models built from training sets that include one or three
+state points, and a comparison of the thermodynamic,
+structural and dynamic properties with those of the orig-
+inal mW model. We conclude in Section IV.
+II.
+THE NEURAL NETWORK MODEL
+The most important step in the design of a feed-
+forward neural network potential is the choice on how to
+define the first and the last layers of the network, respec-
+tively named the input and output layers. We start with
+the output layer, as it determines the NN potential ar-
+chitecture to be constructed. Here we follow the Behler
+Parrinello NN potential architecture [18], in which the
+total energy of the system is decomposed as the sum of
+local fields (Ei), each one representing the contribution of
+a local environment centered around atom i. Being this
+a many-body contribution, it is important to note that
+Ei is not the energy of the single atom i, but of all its
+environment (see also the Appendix A). With this choice,
+the total energy of the system is simply the sum over all
+atoms, E = � Ei, and the force ⃗fi acting on atom i is
+the negative gradient of the total energy with respect to
+the coordinates ν of atom i, e.g. fiν = ∂E/∂xiν. We
+have to point out that a NN potential is differentiable
+and hence it is possible to evaluate the gradient of the
+energy analytically. This allows to compute forces of the
+NN potential in the same way of other force fields, e.g.
+by the negative gradient of the total potential energy.
+The input layer is built from two-body (distances) and
+three-body (angles) descriptors of the local environment,
+⃗D(i) and T (i) respectively, ensuring translational and ro-
+tational invariance. The first layer of the neural network
+is the Atomic Fingerprint Constructor (AFC), as shown
+in Fig. 1, which applies an exponential weighting on the
+atomic descriptors, restoring the invariance under per-
+mutations of atomic indexes. The outputs of this first
+layer are the atomic fingerprints (AFs) and in turn these
+are given to the first hidden layer. We will show how
+this organization of the AFC layer allows for the inter-
+nal parameters of the exponential weighting to be trained
+together with the weights in the hidden layers of the net-
+work. In the following we describe in detail the construc-
+tion of the inputs and the calculation flow in the first
+layers.
+A.
+The atomic fingerprints
+The choice of input layer presents considerably more
+freedom, and it is here that we deviate from previous NN
+potentials. The data in this layer should retain all the in-
+formation needed to properly evaluate forces and energies
+of the particles in the system, possibly exploiting the in-
+ternal symmetries of the Hamiltonian (which in isotropic
+fluids are the rotational, translational and permutational
+invariance) to reduce the number of degenerate inputs.
+Given that the output was chosen as Ei, the energy of the
+atomic environment surrounding atom i, the input uses
+an atom-centered representation of the local environment
+of atom i.
+In the input layer, we define an atom-centered repre-
+sentation of the local environment of atom i, consider-
+ing both the distances rij with the nearest neighbours j
+within a spatial cut-off Rc, and the angles θjik between
+atom i and the pair of neighbours jk that are within a
+cut-off Rc′. More precisely, for each atom j within Rc
+from i we calculate the following descriptors
+D(i)
+j (rij; Rc) =
+�
+1
+2
+�
+1 + cos
+�
+π rij
+Rc
+��
+rij ≤ Rc
+0
+rij > Rc
+(1)
+and, for each triplet j − i − k within Rc′ from i,
+T (i)
+jk (rij, rik, θjik) =
+(2)
+1
+2 [1 + cos (θjik)] D(i)
+j (rij; Rc
+′) D(i)
+k (rik; Rc
+′)
+
+3
+C
+Ei
+α
+γ
+δ
+β
+Compression
+⃗
+D(i)
+⃗
+D(i)
+T(i)
+θjik
+rij
+≡ (
+⃗
+D(i), T(i))
+A
+B
+Atomic Fingerprint 
+Constructor 
+Hidden 
+ layers
+FIG. 1: Schematic representation of the Neural Network Potential flow. (A) Starting from the relative distances and the triplets
+angles between neighbouring atoms, the input layer evaluates the atomic descriptors ⃗D(i) = {D(i)
+j } (Eq. 1) and T (i) = {T (i)
+jk }
+(Eq. 2). (B) The first layer is the Atomic Fingerprint Constructor (AFC) and it combines the atomic descriptors into atomic
+fingerprints, weighting them with an exponential function. The red nodes perform the calculation of Eq. 5, where from the two-
+body descriptors a weighting vector ⃗D(i)
+w (α) = {eαD(i)
+j } is calculated (square with α) and then the scalar product ⃗D(i) · ⃗D(i)
+w (α)
+is computed (square with point) and finally a logarithm is applied (circle). The blue nodes perform the calculation of Eq. 7,
+where two weighting vectors are calculated from the two-body descriptors namely ⃗D(i)
+w (γ) and ⃗D(i)
+w (δ) and one weighting
+matrix from the three-body descriptors T (i)
+w (β) = {eβT (i)
+jk /2}. Finally in the compression unit (Eq 6) values are combined as
+0.5[ ⃗D(i) ◦ ⃗D(i)
+w (γ)]T [T (i) ◦T (i)
+w (β)][ ⃗D(i) ◦ ⃗D(i)
+w (δ)] where we use the circle symbol for the element-wise multiplication. The output
+value of the compression unit is given to the logarithm function (circle). The complete network (D) is made of ten AFC units
+and two hidden layers with 25 nodes per layer and here is depicted 2.5 times smaller.
+Here i indicates the label of i-th particle while in-
+dex j and k run over all other particles in the system.
+In Eq. 1, D(i)
+j (rij; Rc) is a function that goes continu-
+ously to zero at the cut-off (including its derivatives).
+The choice of this functional form guarantees that D(i)
+j
+is able to express contributions even from neighbours
+close to the cut-off.
+Other choices, based on polyno-
+mials or other non-linear functions, have been tested
+in the past [31].
+For example, we tested a parabolic
+cutoff function which produced considerably worse re-
+sults than the cutoff function in Eq. 1.
+The function
+T (i)
+jk (rij, rik, θjik) is also continuous at the triplet cutoff
+R′
+c.
+The angular function
+1
+2 [1 + cos (θjik)] guarantees
+that 0 ≤ T (i)
+jk (rij, rik, θjik) ≤ 1. We note that the use of
+relative distances and angles in Eq. 1-2 guarantees trans-
+lational and rotational invariance.
+The pairs and triplets descriptors are then fed to the
+AFC layer to compute the atomic fingerprints, AFs.
+These are computed by projecting the D(i)
+j
+and T (i)
+jk de-
+scriptors on a exponential set of functions defined by
+
+4
+D
+(i)(α) = ln
+�
+��
+j̸=i
+D(i)
+j eαD(i)
+j
++ ϵ
+�
+� − Zα
+(3)
+T
+(i)(β, γ, δ) = ln
+�
+� �
+j̸=k̸=i
+T (i)
+jk eβT (i)
+jk eγD(i)
+j eδD(i)
+k
+2
++ ϵ
+�
+�(4)
+−Zβγδ
+These AFs are built summing over all pairs and all
+triplets involving particle i, making them invariant un-
+der permutations, and multiplying each descriptor by an
+exponential filter whose parameters are called α for dis-
+tance AFs, and β, γ, δ for the triplet AFs. These param-
+eters play the role of feature selectors, i.e. by choosing
+an appropriate list of α, β, γ, δ the AFs can extract the
+necessary information from the atomic descriptors. The
+best choice of α, β, γ, δ will emerge automatically dur-
+ing the training stage. In Eqs. 3-4, the number ϵ is set to
+10−3 and fixes the value of energy in the rare event that
+no neighbors are found inside the cutoff. Parameters Zα
+and Zβγδ are optimized during the training process, shift-
+ing the AFs towards positive or negative values, and act
+as normalization factors that improve the representation
+of the NN.
+The definitions in equations 3-4 can be reformulated in
+terms of product between vectors and matrices in the fol-
+lowing way. The descriptors in equations 1-2 for particle i
+can be represented as a vector ⃗D(i) = {D(i)
+j } and a matrix
+T (i) = {T (i)
+jk } respectively. Given a choice of α, β, γ and
+δ, three weighting vector ⃗D(i)
+w (α) = {eαD(i)
+j }, ⃗D(i)
+w (γ) =
+{eγD(i)
+j } and ⃗D(i)
+w (δ) = {eδD(i)
+j } and one weighting ma-
+trix T (i)
+w (β) = {eβT (i)
+jk /2} are calculated from ⃗D(i) and
+T (i).
+The 2-body atomic fingerprint (Eq. 3) is finally
+computed as
+D
+(i)(α) = ln
+�
+⃗D(i) · ⃗D(i)
+w (α) + ϵ
+�
+− Zα
+(5)
+The 3-body atomic fingerprint (Eq. 4) is computed first
+by what we call compression step in Fig. 1 as
+T c
+(i) = [ ⃗D(i) ◦ ⃗D(i)
+w (γ)]T [T (i) ◦ T (i)
+w (β)][ ⃗D(i) ◦ ⃗D(i)
+w (δ)]
+2
+(6)
+and finally by
+T
+(i)(β, γ, δ) = ln
+�
+T c
+(i)(β, γ, δ) + ϵ
+�
+− Zβγδ
+(7)
+where we use the circle symbol for the element-wise mul-
+tiplication. The NN potential flow is depicted in Figure
+1 following the vectorial representation.
+In summary, our AFs select the local descriptors use-
+ful for the reconstruction of the potential by weight-
+ing them with an exponential factor tuned with expo-
+nents α, β, γ, δ. A similar weighting procedure has been
+showed to be extremely powerful in the selection of com-
+plex patterns and is widely applied in the so-called atten-
+tion layer first introduced by Google Brain [66]. However
+the AFC layer imposes additionally physically motivated
+constraints on the neural network representation.
+We note that the expression for the system energy is a
+sum over the fields Ei, but the local fields Ei are not addi-
+tive energies, involving all the pair distances and triplets
+angles within the cut-off sphere centered on particle i.
+This non-additive feature favours the NN ability to cap-
+ture higher order correlations (multi-body contribution
+to the energy), and has been shown to outperform ad-
+ditive models in complex datasets [67].
+The NN non-
+additivity requires the derivative of the whole energy E
+(as opposed to Ei) to estimate the force on a particle i. In
+this way, contributions to the force on particle i come not
+only from the descriptors of i but also from the descrip-
+tors of all particles who have i as a neighbour, de facto
+enlarging the effective region in space where interaction
+between particles are included. This allows the network
+to include contributions from length-scales larger than
+the cutoffs that define the atomic descriptors. The Ap-
+pendix A provides further information on this point.
+B.
+Hidden layers
+We employ a standard feed-forward fully-connected
+neural network composed of two hidden layers with 25
+nodes per layer and using the hyperbolic tangent (tanh)
+as the activation function. The nodes of the first hidden
+layer are fully connected to the ones in the second layer,
+and these connections have associated weights W which
+are optimized during the training stage.
+The input of the first hidden layer is given by the AFC
+layer where we used five nodes for the two-body AFs
+(Eq. 3) and five nodes for the three-body AFs (Eq. 4)
+for a total of 10 AFs for each atom.
+We explore the
+performance of some combinations for the number of two-
+body and three-body AF in Appendix D and we find that
+the choice of five and five is the more efficient.
+The output is the local field Ei, for each atomic envi-
+ronment i, whose sum E = �N
+i=1 Ei represents the NN
+estimate of the potential energy E of the whole system.
+C.
+Loss function and training strategy
+To train the NN-potential we minimize a loss function
+computed over nf frames, i.e. the number of independent
+configurations extracted from an equilibrium simulation
+of the liquid phase of the target potential (in our case
+the mW potential). The loss function is the sum of two
+contributions.
+The first contribution, H[{∆ϵk, ∆f k
+iν}], expresses the
+difference in each frame k between the NN estimates and
+the target values for both the total potential energy (nor-
+malized by total number of atoms) ϵk and the atomic
+
+5
+forces f k
+iν acting in direction ν on atom i. The nf energy
+ϵk values and 3Nnf force f k
+iν values are combined in the
+following expression
+H[{∆ϵk, ∆f k
+iν}] = pe
+nf
+nf
+�
+k=1
+hHuber(∆ϵk) +
+pf
+3Nnf
+nf
+�
+k=1
+N
+�
+i=1
+3
+�
+ν=1
+hHuber(∆f k
+iν)
+(8)
+where pe = 0.1 and pf = 1 control the relative contri-
+bution of the energy and the forces to the loss function,
+and hHuber(x) is the so-called Huber function
+hHuber(x) =
+�
+0.5x2 if |x| ≤ 1
+0.5 + (|x| − 1) if |x| > 1
+(9)
+pe and pf are hyper-parameters of the model, and we se-
+lected them with some preliminary tests that found those
+values to be near the optimal ones.
+The Huber func-
+tion [68] is an optimal choice whenever the exploration
+of the loss function goes through large errors caused by
+outliers, i.e. data points that differ significantly from pre-
+vious inputs. Indeed when a large deviation between the
+model and data occur, a mean square error minimization
+may gives rise to an anomalous trajectory in parameters
+space, largely affecting the stability of the training pro-
+cedure. This may happen especially in the first part of
+the training procedure when the parameter optimization,
+relaxing both on the energy and forces error surfaces may
+experience some instabilities.
+The second contribution to the loss function is a reg-
+ularization function, R[{αl, βm, γm, δm}], that serves to
+limit the range of positive values of αl and of the triplets
+βm, γm, δm (where the indexes l and m run over the five
+different values of α and five different triplets of values
+for β, γ and δ) in the window −∞ to 5. To this aim we
+select the commonly used relu function
+rrelu(x) =
+�
+x − 5
+if x > 5
+0
+if x ≤ 5
+(10)
+(11)
+and write
+R[{αl, βm, γm, δm}] =
+5
+�
+l=1
+rrelu(αl) +
+5
+�
+m=1
+[rrelu(βm) + rrelu(γm) + rrelu(δm)]
+(12)
+Thus, the R function is activated whenever one param-
+eters of the AFC layer becomes, during the minimization,
+larger than 5.
+To summarize, the global loss function L used in the
+training of the NN is
+L[ϵ, f] = H[{∆ϵk, ∆f k
+iν}] + pbR[{αl, βm, γm, δm}] (13)
+where pb = 1 weights the relative contribution of R com-
+pared to H.
+Compared to a standard NN-potential, we train not
+only the network weights W but also the AFs param-
+eters Σ ≡ {αl, βm, γm, δm} at the same time. The si-
+multaneous optimization of the weights W and AFs Σ
+prevents possible bottleneck in the optimisation of W at
+fixed representation of Σ. Other NN potential approaches
+implement a separate initial procedure to optimise the Σ
+parameters followed by the optimisation of W at fixed
+Σ [69]. The two-step procedure not only requires a spe-
+cific methodological choice for optimising Σ, but also may
+not result in the optimal values, compared to a search in
+the full parameter space (i.e. both Σ and W). Since the
+complexity of the loss function has increased, we have
+investigated in some detail some efficient strategies that
+lead to a fast and accurate training. Firstly, we initial-
+ize the parameters W via the Xavier algorithm, in which
+the weights are extracted from a random uniform distri-
+bution [70].
+To initialize the Σ parameters we used a
+uniform distribution in interval [−5, 5]. We then mini-
+mize the loss function using the warm restart procedure
+proposed in reference [71]. In this procedure, the learn-
+ing rate η is reinitialized at every cycle l and inside each
+cycle it decays as a function of the number of training
+steps t following
+η(l)(t) = Al
+�(1 − ξf)
+2
+�
+1 + cos
+�πt
+Tl
+��
++ ξf
+�
+(14)
+0 ≤ t ≤ Tl
+where ξf = 10−7, Al = η0ξl
+0 is the initial learning rate of
+the l-th cycle with η0 = 0.01 and ξ0 = 0.9, Tl = bτ l is
+the period of the l-th cycle with τ = 1.4 and b = 40. The
+absolute number of training steps n during cycle l can be
+calculated summing over the length of all previous cycles
+as n = τ + �l−1
+m=0 Tm.
+We also select to evaluate the loss function for groups
+of four frames (mini-batch) and we randomly select 200
+frames nf = 200 for a system of 1000 atoms and hence
+we split this dataset in 160 frames (%80) for the training
+set and the 40 frames (%20) for the test set.
+In Fig. 2(A) we represent the typical decay of the learn-
+ing rate of the warm restart procedure, which will be
+compared to the standard exponential decay protocol in
+the Results section.
+D.
+The Target Model
+To test the quality of the proposed novel NN we train
+the NN with data produced with the mW [40] model
+
+6
+100
+101
+102
+103
+104
+105
+n
+0.000
+0.005
+0.010
+A
+0
+1000
+2000
+3000
+ne
+10
+1
+100
+B
+Validation Loss
+Training Loss
+0
+1000
+2000
+3000
+ne
+10
+2
+10
+1
+100
+101
+ (kcal mol
+1)
+C
+0
+1000
+2000
+3000
+ne
+101
+f (kcal mol
+1 nm
+1)
+D
+FIG. 2:
+Model convergence properties:
+(A) Learning rate
+schedule (Eq. 14) as a function of the absolute training step n
+(one step is defined as an update of the network parameters).
+(B) The training and validation loss (see L[ϵ, f] in Eq. 13) evo-
+lution during the training procedure, reported as a function
+of the number of epoch ne (an epoch is defined as a complete
+evaluation of the training dataset). Root mean square (RMS)
+error of the total potential energy per particle (C) and of the
+force cartesian components (D) during the training evaluated
+in the test dataset. Data in panels B-C-D refers to the NN3
+model and the green point shows the best model location.
+of water.
+This potential, a re-parametrization of the
+Stillinger-Weber model for silicon [72], uses a combina-
+tion of pairwise functions complemented with an additive
+three-body potential term
+0
+20
+40
+60
+Seed
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+ (kcal mol
+1)
+A
+Exponential
+Warm restart
+0
+20
+40
+60
+Seed
+1.4
+1.7
+2.0
+2.3
+2.6
+2.9
+3.2
+3.5
+3.8
+f (kcal mol
+1 nm
+1)
+B
+FIG. 3: Comparison of the root mean square error calculated
+on the validation set for 60 replicas differing in the initial
+seed of the training procedure using both an exponential de-
+cay of the learning rate (points) and the warm restart method
+(squares), for the energy (panel A) and for the forces (panel
+B). For the forces, a significant improvement both in the av-
+erage error and in its variance is found for the warm restart
+schedule.
+E =
+�
+i
+�
+j>i
+U2(rij)+λ
+�
+i
+�
+j̸=i
+�
+j>k
+U3 (rij, rik, θjik) (15)
+where the two body contribution between two particles
+i and j at relative distance rij is a generalized Lennard-
+Jones potential
+U2 (rij) = Aϵ
+�
+B
+� σ
+rij
+�p
+−
+� σ
+rij
+�q�
+exp
+�
+σ
+rij − aσ
+�
+(16)
+where the p = 12 and q = 6 powers are substituted
+by q = 0 and p = 4, multiplied by an exponential cut-
+off that brings the potential to zero at aσ, with a =
+1.8 and σ = 2.3925 ˚A. Aϵ (with A = 7.049556277 and
+ϵ = 6.189 kcal mol−1) controls the strength of the two
+body part. B controls the two-body repulsion (with B =
+0.6022245584).
+The three body contribution is computed from all pos-
+sible ordered triplets formed by the central particle with
+the interacting neighbors (with the same cut-off aσ as the
+two-body term) and favours the tetrahedral coordination
+of the atoms via the following functional form
+U3 (rij, rik, θjik) = ϵ [cos (θjik) − cos (θ0)]2 ×
+exp
+�
+γσ
+rij − aσ
+�
+exp
+�
+γσ
+rik − aσ
+�
+(17)
+where θjik is the angle formed in the triplet jik and
+γ = 1.2 controls the smoothness of the cut-off function
+on approaching the cut-off. Finally, θ0 = 109.47◦ and
+λ = 23.15 controls the strength of the angular part of
+the potential.
+
+7
+The mW model, with its three-body terms centered
+around a specific angle and non-monotonic radial interac-
+tions, is based on a functional form which is quite differ-
+ent from the radial and angular descriptors selected in the
+NN model. The NN is thus agnostic with respect to the
+functional form that describes the physical system (the
+mW in this case). But having a reference model with ex-
+plicit three body contributions offers a more challenging
+target for the NN potential compared to potential models
+built entirely from pairwise interactions. The mW model
+is thus an excellent candidate to test the performance of
+the proposed NN potential.
+III.
+RESULTS
+A.
+Training
+We study two different NN models, indicated with the
+labels NN1 and NN3, differing in the number of state
+points included in the training set. These two models are
+built with a cut-off of Rc = 4.545 ˚A
+for the two-body
+atomic descriptors and a cut-off of R′
+c = 4.306 ˚A
+for
+the three-body atomic descriptors. R′
+c is the same as the
+mW cutoff while Rc was made slightly larger to miti-
+gate the suppression of information at the boundaries by
+the cutoff functions.
+The NN1 model uses only train-
+ing information based on mW equilibrium configurations
+from one state point at ρ1 = 1.07 g cm−3, T1 = 270.9 K
+where the stable phase is the liquid.
+The NN3 model
+uses training information based on mW liquid configura-
+tions in three different state points, two state points at
+ρ1 = 0.92 g cm−3, T1 = 221.1 K and ρ2 = 0.92 g cm−3,
+T2 = 270.9 K where the stable solid phase is the clathrate
+Si34/Si136 [73] and one state point at ρ3 = 1.15 g cm−3,
+T2 = 270.9 K.
+This choice of points in the phase diagram is aimed to
+improve agreement with the low temperature-low density
+as well as high density regions of the phase diagram. Im-
+portantly, all configurations come from either stable or
+metastable liquid state configurations. Indeed, the point
+at ρ2 = 0.92 g cm−3, T2 = 270.9 K is quite close to
+the limit of stability (respect to cavitation) of the liquid
+state.
+To generate the training set, we simulate a system of
+N = 1000 mW particles with a standard molecular dy-
+namics code in the NVT ensemble, where we use a time
+step of 4 fs and run 107 steps for each state point. From
+these trajectory, we randomly select 200 configurations
+(frames) to create a dataset of positions, total energies
+and forces. We then split the dataset in the training and
+in the test data sets, the first one containing 80% of the
+data. We then run the training for 4000 epochs with a
+minibatch of 4 frames. At the end of every epoch, we
+check if the validation loss is improved and we save the
+model parameters. In Fig. 2 we plot the loss function for
+the training and test datasets (B), the root mean square
+error of the total energy per particle (C), and of the force
+(D) for the NN3 model. The results show that the learn-
+ing rate schedule of Eq. 14 is very effective in reducing
+both the loss and error functions.
+Interestingly, the neural network seems to avoid over-
+fitting (i.e. the validation loss is decreasing at the same
+rate as the loss on the training data), and the best model
+(deepest local minimum explored), in a given window of
+training steps, is always found at the end of that win-
+dow, which also indicates that the accuracy could be fur-
+ther improved by running more training steps. Indeed we
+found that by increasing the number of training steps by
+one order of magnitude the error in the forces decreases
+by a further 30%. Similar accuracy of the training stage
+is obtained also for the NN1 model (not shown).
+The training procedure always terminates with an
+error
+on
+the
+test
+set
+equal
+or
+less
+than
+∆ϵ
+≃
+0.01 kcal mol−1 (0.43 meV) for the energy, and of ∆f ≃
+1.55 kcal mol−1 nm−1 (6.72 meV ˚A−1) for the forces.
+These values are comparable to the state-of-the-art NN
+potentials [23, 54, 55, 61], and within the typical accuracy
+of DFT calculations [74].
+We can compare the precision of our model with that
+of alternative NN potentials trained on a range of water
+models. An alternative mW neural network potential has
+been trained on a dataset made of 1991 configurations
+of 128 particles system at different pressure and tem-
+perature (including both liquid and ice structures) with
+Behler-Parinello symmetry functions [24].
+The train-
+ing of this model (which uses more atomic fingerprints
+and a larger cutoff radius) converged to an error in en-
+ergy of ∆ϵ ≃ 0.0062 kcal mol−1 (0.27 meV), and ∆f ≃
+3.46 kcal mol−1 nm−1 (15.70 meV ˚A−1) for the forces. In
+a recent work searching for liquid-liquid transition signa-
+tures in an ab-initio water NN model [55], a dataset of
+configurations spanning a temperature range of 0−600 K
+and a pressure range of 0 − 50 GPa was selected. For
+a system of 192 particles, the training converged to an
+error in energy of ∆ϵ ≃ 0.010 kcal mol−1 (0.46 meV),
+and ∆f
+≃
+9.96 kcal mol−1 nm−1 (43.2 meV ˚A−1)
+for the forces.
+In the NN model of MB-POL [61],
+a dataset spanning a temperature range from 198 K
+to 368 K at ambient pressure was selected.
+In this
+case, for a system of 256 water molecule, an accu-
+racy of ∆ϵ ≃ 0.01 kcal mol−1 (0.43 meV) and ∆f ≃
+10 kcal mol−1 nm−1 (43.36 meV ˚A−1) was reached. Fi-
+nally, the NN for water at T = 300 K used in Ref. [54],
+reached precisions of ∆ϵ ≃ 0.046 kcal mol−1 (2 meV) and
+∆f ≃ 25.36 kcal mol−1 nm−1 (110 meV ˚A−1).
+While a direct comparison between NN potentials
+trained on different reference potentials is not a valid test
+to rank the respective accuracies, the comparisons above
+show that our NN potential reaches a similar precision
+in energies, and possibly an improved error in the force
+estimation.
+The accuracy of the NN potential could be further
+improved by extending the size of the dataset and the
+choice of the state points. In fact, while the datasets in
+Ref. [54, 55, 61] have been built with optimized proce-
+
+8
+dures, the dataset used in this study was prepared by
+sampling just one (NN1) or three (NN3) state-points.
+Also the size of the datasets used in the present work is
+smaller or comparable to the ones of Ref. [54, 55, 61].
+In Fig. 3 we compare the error in the energies (A)
+and the forces (B) between sixty independent training
+runs using the standard exponential decay of the learn-
+ing rate (points) and the warm restart protocol (squares).
+The figure shows that while the errors in the energy com-
+putations are comparable between the two methods, the
+warm restart protocol allows the forces to be computed
+with higher accuracy. Moreover we found that the warm
+restart procedure is less dependent on the initial seed and
+that it reaches deeper basins than the standard exponen-
+tial cooling rate.
+B.
+Comparing NN1 with NN3
+The NN potential model was implemented in a custom
+MD code that makes use of the tensorflow C API [75].
+We adopted the same time step (4 fs), the same number
+of particles (N = 1000) and the same number of steps
+(107) as for the simulations in the mW model.
+As described in the Training Section, we compare the
+accuracy of two different training strategies: NN1 which
+was trained on a single state point, and NN3 which is
+instead trained on three different state point. In Fig. 4
+we plot the energy error (∆ϵ) between the NN potential
+and the mW model with both NN1 (panel A) and NN3
+(panel B). Starting from NN1, we see that the model
+already provides an excellent accuracy for a large range
+of temperatures and for densities close to the training
+density. The biggest shortcoming of the NN1 model is
+at densities lower than the trained density, where the
+NN potential model cavitates and does not retain the
+long-lived metastable liquid state displayed by the mW
+model. We speculate that this behaviour is due to the
+absence of low density configurations in the training set,
+which prevents the NN potential model from correctly
+reproducing the attractive tails of the mW potential.
+To overcome this limitation we have included two ad-
+ditional state points at low density in the NN3 model. In
+this case, Fig. 4B shows that NN3 provides a quite ac-
+curate reproduction of the energy in the entire explored
+density and temperature window (despite being trained
+only with data at ρ = 0.92 g cm−3 and ρ = 1.15 g cm−3).
+We can also compare the accuracy obtained during
+production runs against the accuracy reached during
+training, which was ∆ϵ ≃ 0.01 kcal mol−1. Fig. 4B shows
+the error is of the order of 0.032 kcal mol−1 (1.3 meV), for
+density above the training set density. But in the density
+region between 0.92 and 1.15, the error is even smaller,
+around 0.017 kcal mol−1 (0.7 meV) at the lowest density
+boundary.
+We can thus conclude that the NN3 model, which adds
+to the NN1 model information at lower density and tem-
+perature, in the region where tetrahedality in the wa-
+0.9
+1.0
+1.1
+1.2
+ (g cm
+3)
+385
+365
+345
+325
+305
+285
+265
+245
+225
+T (K)
+A
+0.9
+1.0
+1.1
+1.2
+ (g cm
+3)
+B
+0.001
+0.010
+0.020
+0.030
+0.040
+ (kcal mol
+1)
+FIG. 4: Comparison between the mW total energy and the
+NN1 model (A) and NN3 model (B) for different temperatures
+and densities. While the NN3 model is able to reproduce the
+mW total energy with a good agreement in a wide region of
+densities and temperatures, the NN1 provide a good repre-
+sentation only in a limited region of density and temperature
+values. Blue squares represent the state points used for build-
+ing the NN models.
+ter structure is enhanced, is indeed capable to represent,
+with only three state points, a quite large region of the
+phase space, encompassing dense and stretched liquid
+states. This suggests that a training based on few state
+points at the boundary of the density/temperature re-
+gion which needs to be studied is sufficient to produce a
+high quality NN model. In the following we focus entirely
+on the NN3 model.
+C.
+Comparison of thermodynamic, structural and
+dynamical quantities
+In Fig. 5 we present a comparison of thermodynamic
+data between the mW model (squares) and its NN poten-
+tial representation (points) across a wide range of state
+points. Fig. 5A plots the energy as function of density
+for temperatures ranging from melting to deeply super-
+cooled conditions. Perhaps the most interesting result is
+that the NN potential is able to capture the energy min-
+imum, also called the optimal network forming density,
+which is a distinctive anomalous property of water and
+other empty liquids [76].
+Fig. 5(B) shows the pressure as a function of the tem-
+perature for different densities, comparing the mW with
+the NN3 model. Also the pressure shows a good agree-
+ment between the two models in the region of densities
+between ρ = 0.92 g cm−3 and ρ = 1.15 g cm−3, which, as
+for the energy, tends to deteriorate at ρ = 1.22 g cm−3.
+In the large density region explored, the structure of
+the liquid changes considerably. On increasing density, a
+transition from tetrahedral coordinated local structure,
+prevalent at low T and low ρ, towards denser local envi-
+
+9
+0.85
+0.90
+0.95
+1.00
+1.05
+1.10
+1.15
+1.20
+1.25
+ (g cm
+3)
+10.0
+9.5
+9.0
+8.5
+ (kcal mol
+1)
+221.1K
+233.6K
+246.0K
+258.5K
+271.0K
+299.0K
+311.4K
+373.7K
+A
+mW
+NN3
+200
+225
+250
+275
+300
+325
+350
+375
+T (K)
+0
+1
+2
+3
+4
+P (GPa)
+0.92 g cm
+3
+0.99 g cm
+3
+1.07 g cm
+3
+1.15 g cm
+3
+1.19 g cm
+3
+1.22 g cm
+3
+B
+FIG. 5: Comparison between the mW total energy and the
+NN3 total energy as a function of density along different
+isotherm (A) and comparison between the mW pressure and
+the NN3 pressure as a function of temperature along differ-
+ent isochores (B). The relative error of the NN vs the mW
+potential grows with density, but remains within 3% even for
+densities larger than the densities used in the training set.
+ronments with interstitial molecules included in the first
+coordination shell takes place. This structural change is
+well displayed in the radial distribution function, shown
+for different densities at fixed temperature in Fig. 6.
+Fig. 6 also shows the progressive onset of a peak around
+3.5 ˚A
+developing on increasing pressure, which signals
+the growth of interstitial molecules, coexisting with open
+tetrahedral local structures [77, 78]. At the highest den-
+sity, the tetrahedral peak completely merges with the
+interstitial peak. The NN3 model reproduces quite ac-
+curately all features of the radial distribution functions,
+maxima and minima positions and their relative ampli-
+tudes, at all densities, from the tetrahedral-dominated to
+the interstitial-dominated limits. In general, NN3 model
+reproduces quite well the mW potential in energies, pres-
+sure and structures and it appreciably deviates from mW
+pressures and energies quantities only at densities (above
+1.15 g/cm3) which are outside of the training region.
+To assess the ability of NN potential to correctly de-
+scribe also the crystal phases of the mW potential, we
+compare in Fig. 7 the g(r) of mW with the g(r) of the
+NN3 model for four different stable solid phases [73]:
+hexagonal and cubic ice (ρ = 1.00 g cm−3 and T =
+246 K), the dense crystal SC16 (ρ = 1.20 g cm−3 and T =
+234 K) and the clathrate phase Si136 (ρ = 0.80 g cm−3
+and T = 221 K). The results, shown in Fig. 7, show that,
+despite no crystal configurations have been included in
+the training set, a quite accurate representation of the
+crystal structure at finite temperature is provided by the
+NN3 model for all distinct sampled lattices.
+0
+2
+4
+6
+8
+10
+12
+14
+16
+R (Å)
+0
+1
+2
+3
+4
+5
+6
+7
+8
+RDF
+0.92 g cm
+3
+0.99 g cm
+3
+1.15 g cm
+3
+1.22 g cm
+3
+mW
+NN3
+FIG. 6:
+Comparison between the mW radial distribution
+functions g(r) and the NN3 g(r) at T = 270.9 K for four
+different densities.
+The tetrahedral structure (signalled by
+the peak at 4.54 ˚A ) progressively weakens in favour of an in-
+terstitial peak progressively growing at 3.5 − 3.8 ˚A . Different
+g(r) have been progressively shifted by two to improve clarity.
+0
+5
+10
+15
+R (Å)
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+RDF
+Hexagonal diamond
+Cubic diamond
+Si136 Clathrate
+SC16
+mW
+NN3
+FIG. 7:
+Comparison between the mW radial distribution
+functions g(r) and the NN3 g(r) for four different lattices:
+(A) hexagonal diamond (the oxygen positions of the ice Ih);
+(B) cubic diamond (the oxygen positions of the ice Ic; (C)
+the SC16 crystal (the dense crystal form stable at large pres-
+sures in the mW model) and (D) the Si136 clathrate structure,
+which is stable at negative pressures in the mW model. Dif-
+ferent g(r) have been progressively shifted by four to improve
+clarity.
+Finally, we compare in Fig. 8 the diffusion coefficient
+(evaluated from the long time limit of the mean square
+displacement) for the mW and the NN3 model, in a wide
+range of temperatures and densities, where water displays
+a diffusion anomaly.
+Fig. 8 shows again that, also for
+
+10
+0.85
+0.90
+0.95
+1.00
+1.05
+1.10
+1.15
+1.20
+ (g cm
+3)
+0
+100
+200
+300
+400
+500
+600
+700
+800
+D (Å2 ns
+1)
+221.1K
+233.6K
+246.0K
+258.5K
+271.0K
+299.0K
+311.4K
+373.7K
+mW
+NN3
+FIG. 8: Comparison between the mW diffusion coefficient D
+and the NN3 corresponding quantity for different tempera-
+tures and densities, in the interval 221 − 271 K. In this dy-
+namic quantity, the relative error is, for all temperatures,
+around 8%. Note also that in this T window the diffusion
+coefficient shows a clear maximum, reproducing one of the
+well-know diffusion anomaly of water. Diffusion coefficients
+have been calculated in the NVT ensemble using the same
+Andersen thermostat algorithm [79] for mW and NN3 poten-
+tial.
+dynamical quantities, the NN potential offers an excellent
+representation of the mW potential, despite the fact that
+no dynamical quantity was included in the training set. A
+comparison between fluctuations of energy and pressure
+of mW and NN3 potential is reported in Appendix B.
+IV.
+CONCLUSIONS
+In this work we have presented a novel neural net-
+work (NN) potential based on a new set of atomic fin-
+gerprints (AFs) built from two- and three-body local de-
+scriptors that are combined in a permutation-invariant
+way through an exponential filter (see Eq. 3-4). One of
+the distinctive advantages of our scheme is that the AF’s
+parameters are optimized during the training procedure,
+making the present algorithm a self-training network that
+automatically selects the best AFs for the potential of in-
+terest.
+We have shown that the added complexity in the con-
+current training of the AFs and of the NN weights can
+be overcome with an annealing procedure based on the
+warm restart method [71], where the learning rate goes
+through damped oscillatory ramps.
+This strategy not
+only gives better accuracy compared to the commonly
+implemented exponential learning rate decay, but also
+allows the training procedure to converge rapidly inde-
+pendently from the initialisation strategies of the model’s
+parameters.
+Moreover we show in Appendix C that the potential
+hyper-surface of the NN model has the same smoothness
+as the target model, as confirmed by (i) the possibility to
+use the same timestep in the NN and in the target model
+when integrating the equation of motion and (ii) by the
+possibility of simulate the NN model even in the NVE
+ensemble with proper energy conservation.
+We test the novel NN on the mW model [40], a
+one-component model system commonly used to de-
+scribe water in classical simulations. This model, a re-
+parametrization of the Stillinger-Weber model for sili-
+con [72], while treating the water molecule as a simple
+point, is able to reproduce the characteristic tetrahedral
+local structure of water (and its distortion on increasing
+density) via the use of three-body interactions. Indeed
+water changes from a liquid of tetrahedrally coordinated
+molecules to a denser liquid, in which a relevant fraction
+of interstitial molecules are present in the first nearest-
+neighbour shell. The complexity of the mW model, both
+due to its functional form as well as to the variety of dif-
+ferent local structures which characterise water, makes it
+an ideal benchmark system to test our NN potential.
+We find that a training based on configurations ex-
+tracted by three different state points is able to pro-
+vide a quite accurate representation of the mW poten-
+tial hyper-surface, when the densities and temperatures
+of the training state points delimit the region of in which
+the NN potential is expected to work. We also find that
+the error in the NN estimate of the total energy is low,
+always smaller than 0.03 kcal mol−1, with a mean error
+of 0.013 kcal mol−1. The NN model reproduces very well
+not only the thermodynamic properties but also struc-
+tural properties, as quantified by the radial distribution
+function, and the dynamic properties, as expressed by
+the diffusion coefficient, in the extended density interval
+from ρ = 0.92 g cm−3 to ρ = 1.22 g cm−3.
+Interestingly, we find that the NN model, trained only
+on disordered configurations, is also able to properly
+describe the radial distribution of the ordered lattices
+which characterise the mW phase diagram, encompass-
+ing the cubic and hexagonal ices, the SC16 and the Si136
+clathrate structure [73]. In this respect, the ability of the
+NN model to properly represent crystal states suggests
+that, in the case of the mW, and as such probably in the
+case of water, the geometrical information relevant to the
+ordered structures is contained in the sampling of phase
+space typical of the disordered liquid phase. These find-
+ings have been recently discussed in reference [80] where
+it has been demonstrated that liquid water contains all
+the building blocks of diverse ice phases.
+We conclude by noticing that the present approach can
+be generalized to multicomponent systems, following the
+same strategy implemented by previous approaches [18,
+23]. Work in this direction is underway.
+Acknowledgments
+FGM and JR acknowledge support from the European
+Research Council Grant DLV-759187 and CINECA grant
+ISCRAB NNPROT.
+
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+V.
+APPENDIX A
+In this appendix we discuss the effective spacial range
+covered by a NN potential whose fingerprints are defined
+based on pair information confined within a sphere of
+cutoff radius Rc.
+As noted in reference [31], multi-body potentials and
+especially non-additive multibody potentials induce lo-
+cal interactions beyond the cut-off radius, enlarging the
+sphere of interaction.
+Indeed, the force on particle i
+comes from the derivative of the local field of i and of
+all its neighbours with respect to the coordinates of par-
+ticle i.
+2
+1
+3
+2
+1
+3
+4
+5
+A
+B
+FIG. 9: (A) Two-body interactions and (B) three-body inter-
+actions in a non linear local field model Ei. The non linearity
+of the local field enlarges the interaction cut-off where a neigh-
+bour particle (blue) makes a bridge between non-neighboring
+particle (red and blue).
+Fig. 9 graphically explains the effective role of Rc in
+the NN potential.
+In panel A, we describe particle 1
+with only one neighbour (particle 2) within Rc. We also
+represent the sphere centered on particle 3, which also
+includes particle 2 as one of its neighbour. In this case,
+the energy of the system will be represented as a sum
+over the local fields E1, E2 and E3. Due to the intrinsic
+non-linearity of the NN, the field Ei mixes together the
+AFs, and consequently the distances and angles entering
+in the AFs are non-linearly mixed in Ei. The force on
+atom 1 is then written as
+f1ν = −∂E1(r12)
+∂x1ν
+− ∂E2(r21, r23)
+∂x1ν
+= −∂E1(r12)
+∂x1ν
+−∂E2(r21, r23)
+∂r21
+∂r21
+∂x1ν
+− ∂E2(r21, r23)
+∂r23
+∂r23
+∂x1ν
+(A1)
+While the last term vanishes, the next to the last retains
+
+13
+A
+B
+FIG. 10: (A) Standard deviation of total energy (normal-
+ized with the number of particles) and (B) standard deviation
+of virial pressure for both NN3 model (red) and mW model
+(black).
+an intrinsic dependence on the coordinates both of par-
+ticle 2 as well as of particle 3, if the local field E2 is non
+linear. Thus, even if particle 3 is further than Rc, it en-
+ters in the determination of the force acting on particle
+1. A similar effect is also present in the angular part of
+the AFs, as shown graphically in panel B. Indeed, for the
+angular component of the AF the force on particle 1 is
+f1ν = −∂E1(θ512)
+∂x1ν
+− ∂E2(θ123, θ124, θ324)
+∂x1ν
+.
+(A2)
+Also in this case two contributions can be separated: (i)
+the interaction of particle 1 with triplets 123 and 124 is
+an effect of the three-body AF and it is present also in
+additive-models such as the mW model, (ii) the inter-
+action of particle 1 with triplet 324 is an effect of the
+non-additive nature of the NN local field Ei.
+VI.
+APPENDIX B
+In this Appendix we provide further thermodynam-
+ics comparisons between mW and NN3 potential focus-
+ing on the pressure and energy fluctuations. We depict
+in Fig. 10 the standard deviations of the total energy
+(normalized by N) in panel (A) and the standard devia-
+tion of virial pressure in panel (B). Energy fluctuations
+of NN3 follow qualitatively and quantitatively the trend
+of mW potential.
+Pressure fluctuations of NN3 are in
+good agreement with the mW model but, as for the pres-
+sure (Fig. 5.B), the accuracy decreases approaching state
+points outside the density range used for the training.
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+Time (ns)
+10.2
+10.0
+9.8
+9.6
+9.4
+(kcal mol
+1)
+NN3 Total Energy
+NN3 Potential Energy
+mW Total Energy
+mW Potential Energy
+FIG. 11: NVE molecular dynamics at T = 299 K and ρ =
+1.07 g cm−3 for both NN3 and mW model. The time step is
+dt = 4 fs for both models.
+VII.
+APPENDIX C
+In this Appendix we show a comparison between the
+mW and NN3 potentials in terms of the energy conser-
+vation in the NVE ensemble. In Fig. 11 we depict both
+total energy and potential energy for mW and NN3 po-
+tential. The potential energy and total energy of the two
+models are in good agreement.
+VIII.
+APPENDIX D
+In this Appendix we investigate the efficiency of the
+training over different choices for the number and types
+of atomic fingerprints introduced in the Neural Network
+Model section. We start by using only one three-body
+(n3b = 1) and one two-body (n2b = 1) AF and subse-
+quently increasing the number of the AF. For every com-
+bination of n2b and n3b, we run a 4000 epochs training
+and at the end of each training we extract the best model.
+We summarized these results in table I where we com-
+pare the error on forces over the all investigated model.
+From table I it emerges that the choice of n3b = 5 and
+n2b = 5 is the more convenient both for accuracy and
+computational efficiency.
+Doubling the number of the
+three-body AF marginally improves the error on forces
+while increases the computational cost due to the increase
+in the size of the input layer of the first hidden layer and
+due to the additional time to compute the three-body
+AF. Moreover in the RESULTS section we show that the
+choice n3b = 5 and n2b = 5 is sufficient to represent the
+target potential. Finally the accuracy of the training af-
+ter doubling the configurations in the dataset reaches an
+error on forces of ∆f = 5.85 meV ˚A−1 that is 0.87 times
+the error value found with a half of the dataset.
+
+14
+TABLE I:
+Table of errors on forces at the end of the 4000
+epoch-long training procedure for different combination of the
+number and type of the AF.
+n3b
+n2b
+∆f (meV ˚A−1)
+n3b
+n2b
+∆f (meV ˚A−1)
+1
+1
+72.79
+5
+1
+16.53
+1
+2
+67.92
+5
+2
+7.53
+1
+5
+56.25
+5
+5
+6.72
+1
+10
+56.00
+5
+10
+6.87
+1
+15
+56.02
+5
+15
+6.95
+2
+1
+53.76
+10
+1
+7.98
+2
+2
+43.95
+10
+2
+7.17
+2
+5
+32.43
+10
+5
+5.79
+2
+10
+32.39
+10
+10
+6.55
+2
+15
+24.70
+10
+15
+6.19
+

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+ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A
+METACYCLIC GROUP.
+ARISTIDES KONTOGEORGIS AND ALEXIOS TEREZAKIS
+Abstract. We give a necessary and sufficient condition for a modular rep-
+resentation of a group G = Cph ⋊ Cm in a field of characteristic zero to be
+lifted to a representation over local principal ideal domain of characteristic
+zero containing the ph roots of unity.
+1. Introduction
+The lifting problem for a representation
+ρ : G → GLn(k),
+where k is a field of characteristic p > 0, is about finding a local ring R of char-
+acteristic 0, with maximal ideal mR such that R/mR = k, so that the following
+diagram is commutative:
+GLn(R)
+�
+G
+�
+�
+GLn(k)
+Equivalently one asks if there is a free R-module V , which is also an R[G]-module
+such that V ⊗RR/mR is the k[G]-module corresponding to our initial representation.
+We know that projective k[G]-modules lift in characteristic zero, [16, chap. 15], but
+for a general k[G]-module such a lifting is not always possible, for example, see [10,
+prop. 15]. This article aims to study the lifting problem for the group G = Cq⋊Cm,
+where Cq is a cyclic group of order ph and Cm is a cyclic group of order m, (p, m) = 1
+and give necessary and sufficient condition in order to lift. We assume that the local
+ring R contains the q-roots of unity and k is algebraically closed, and we might need
+to consider a ramified extension of R, in order to ensure that certain q-roots of unit
+are distant in the mR-topology, see remark 35. An example of such a ring R is the
+ring of Witt vectors W(k)[ζq] with the q-roots of unity adjoined to it.
+We notice that a decomposable R[G]-module V gives rise to a decomposable
+R-module modulo mR and also an indecomposable R[G]-module can break in the
+reduction modulo mR into a direct sum of indecomposable k[G]-summands. We
+also give a classification of k[Cq ⋊ Cm]-modules in terms of Jordan decomposition
+and give the relation with the more usual uniserial description in terms of their
+socle [1].
+Date: January 4, 2023.
+Key words and phrases. Lifting of representations, modular representation theory, integral
+representation theory, Generalized Oort conjecture, metacyclic groups.
+1
+arXiv:2301.01032v1  [math.AG]  3 Jan 2023
+
+2
+A. KONTOGEORGIS AND A. TEREZAKIS
+Our interest to this problem comes from the problem of lifting local actions. The
+local lifting problem considers the following question: Does there exist an extension
+Λ/W(k), and a representation
+˜ρ : G �→ Aut(Λ[[T]]),
+such that if t is the reduction of T, then the action of G on Λ[[T]] reduces to the
+action of G on k[[t]]?
+If the answer to the above question is positive, then we say that the G-action
+lifts to characteristic zero. A group G for which every local G-action on k[[t]] lifts to
+characteristic zero is called a local Oort group for k. Notice that cyclic groups are
+always local Oort groups. This result was known as the “Oort conjecture”, which
+was recently proved by F. Pop [15] using the work of A. Obus and S. Wewers [14].
+There are a lot of obstructions that prevent a local action to lift in characteristic
+zero. Probably the most important of these obstructions in the KGB-obstruction
+[4]. It is believed that this is the only obstruction for the local lifting problem, see
+[11], [12]. In [10, Thm. 3] the authors have given a criterion for the local lifting
+which involves the lifting of a linear representation of the same group.
+The case
+G = Cq ⋊Cm and especially the case of dihedral groups Dq = Cq ⋊C2, is a problem
+of current interest in the theory of local liftings, see [12], [6], [18]. For more details
+on the local lifting problem we refer to [3], [4], [5], [11].
+Keep also in mind that the Cq ⋊ Cm groups were important to the study of
+group actions in holomorphic differentials of curves defined over fields of positive
+characteristic p, where the group involved has cyclic p-Sylow subgroup, see [2].
+Let us now describe the method of proof. For understanding the splitting of
+indecomposable R[G]-modules modulo mR, we develop a version of Jordan normal
+form in lemma 16 for endomorphisms T : V → V of order ph, where V is a free
+module of rank d. We give a way to select this basis, by selecting an initial suitable
+element E ∈ V , see lemma 15.
+The normal form (as given in eq.
+(9)) of the
+element T of order q, determines the decomposition of the reduction. We show
+that for every indecomposable summand Vi of V , we can select E as an eigenvalue
+of the generator σ of Cm and then by forcing the relation ΓT = T αΓ to hold, we
+see how the action of σ can be extended recursivelly to an action of σ on Vi, this is
+done in lemma 24. Proving that this gives indeed a well defined action is a technical
+computation and is done in lemmata 26, 27, 28, 32, 33.
+The important thing here is that the definition of the action of σ on E is the
+“initial condition” of a dynamical system that determines the action of Cm on the
+indecomposable summand Vi. The R[Cq⋊Cm] indecomposable module Vi can break
+into a direct sum Vα(ϵν, κν)-modules 1 ≤ ν ≤ s (for a precise definition of them
+see definition 9, notice that κi denotes the dimension). The action of σ on each
+Vα(ϵν, κν) can be uniquely determined by the action of σ on an initial basis element
+as shown in section 3, again by a “dynamical system” approach, where we need s
+initial conditions, one for each Vα(ϵν, κν). The lifting condition essentially means
+that the indecomposable summands Vα(ϵ, κ) of the special fibre, should be able
+to be rearranged in a suitable way, so that they can be obtained as reductions of
+indecomposable R[Cq ⋊ Cq]-modules. The precise expression of our lifting criterion
+is given in the following proposition:
+Proposition 1. Consider a k[G]-module M which is decomposed as a direct sum
+M = Vα(ϵ1, κ1) ⊕ · · · ⊕ Vα(ϵs, κs).
+
+ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP.
+3
+The module lifts to an R[G]-module if and only if the set {1, . . . , s} can be written
+as a disjoint union of sets Iν, 1 ≤ ν ≤ t so that
+a �
+µ∈Iν κµ ≤ q, for all 1 ≤ ν ≤ t.
+b �
+µ∈Iν κµ ≡ a modm for all 1 ≤ ν ≤ t, where a ∈ {0, 1}.
+c For each ν, 1 ≤ ν ≤ t there is an enumeration σ : {1, . . . , #Iν} → Iν ⊂
+{1, .., s}, such that
+ϵσ(2) = ϵσ(1)ακσ(1), ϵσ(3) = ϵσ(3)ακσ(3), . . . ϵσ(s) = ϵσ(s−1)ακσ(s−1)
+In the above proposition, each set Iν corresponds to a collection of modules
+Vα(ϵµ, κµ), µ ∈ Iν which come as the reduction of an indecomposable R[Cq ⋊ Cm]-
+module Vν of V .
+Aknowledgements A. Terezakis is a recipient of financial support in the context
+of a doctoral thesis (grant number MIS-5113934). The implementation of the doc-
+toral thesis was co-financed by Greece and the European Union (European Social
+Fund-ESF) through the Operational Programme—Human Resources Development,
+Education and Lifelong Learning—in the context of the Act—Enhancing Human
+Resources Research Potential by undertaking a Doctoral Research—Sub-action 2:
+IKY Scholarship Programme for Ph.D. candidates in the Greek Universities.
+2. Notation
+Let τ be a generator of the cyclic group Cq and σ be a generator of the cyclic
+group Cm. The group G is given in terms of generators and relations as follows:
+G = ⟨σ, τ|τ q = 1, σm = 1, στσ−1 = τ α for some α ∈ N, 1 ≤ α ≤ ph − 1, (α, p) = 1⟩.
+The integer α satisfies the following congruence:
+(1)
+αm ≡ 1 modq
+as one sees by computing τ = σmτσ−m = τ αm. Also the integer α can be seen as
+an element in the finite field Fp, and it is a (p − 1)-th root of unity, not necessarily
+primitive. In particular the following holds:
+Lemma 2. Let ζm ∈ k be a fixed primitive m-th root of unity. There is a natural
+number a0, 0 ≤ a0 < m − 1 such that α = ζa0
+m .
+Proof. The integer α if we see it as an element in k is an element in the finite field
+Fp ⊂ k, therefore αp−1 = 1 as an element in Fp. Let ordp(α) be the order of α in F∗
+p.
+By eq. (1) we have that ordp(α) | p−1 and ordp(α) | m, that is ordp(α) | (p−1, m).
+The primitive m-th root of unity ζm generates a finite field Fp(ζm) = Fpν for
+some integer ν, which has cyclic multiplicative group Fpν\{0} containing both the
+cyclic groups ⟨ζm⟩ and ⟨α⟩. Since for every divisor δ of the order of a cyclic group
+C there is a unique subgroup C′ < C of order δ we have that α ∈ ⟨ζm⟩, and the
+result follows.
+□
+Definition 3. For each pi | q we define ordpiα to be the smallest natural number
+o such that αo ≡ 1 modpi.
+
+EZNA
+OperationalProgramme
+HumanResourcesDevelopment
+2014-2020
+士
+EducationandLifelongLearning
+avantuen-epyaoia-aaanaeun
+Eupwaikn'Evwon
+Co-financed byGreece and the European Union
+European Social Fund4
+A. KONTOGEORGIS AND A. TEREZAKIS
+It is clear that for ν ∈ N
+αν ≡ 1 modpi ⇒ αν ≡ 1 modpj for all j ≤ i.
+Therefore
+ordpjα | ordpiα for j ≤ i.
+On the other hand α ∈ N and αp−1 ≡ 1 modp so ordpα | p − 1.
+Also since
+σtτσ−t = τ αt we have that αm ≡ 1 modph, therefore ordpα | ordpiα | ordphα | m,
+for 1 ≤ i ≤ h.
+Lemma 4. The center CentG(τ) = ⟨τ, σordphα⟩. Moreover
+|CentG(τ)|
+ph
+=
+m
+ordph(α) =: m′
+Proof. The result follows by observing (τ νσt)τ(τ νσt)−1 = τ αt, for all 1 ≤ ν ≤ q,
+1 ≤ t ≤ m.
+□
+Remark 5. If ordpα = m then ordpiα = m for all 1 ≤ i ≤ h.
+Lemma 6. If the group G = Cq ⋊ Cm is a subgroup of Aut(k[[t]]), then all orders
+ordpiα = m/m′, for all 1 ≤ i ≤ h.
+Proof. We will use the notation of the book of J.P.Serre on local fields [17]. By
+[13, Th.1.1b] we have that the first gap i0 in the lower ramification filtration of the
+cyclic group Cq satisfies (m, i0) = m′.
+The ramification relation [17, prop. 9 p. 69]
+αθi0(τ) = θi0(τ α) = θi0(στσ−1) = θ0(σ)i0θi0(τ),
+implies that θ0(σ)i0 = α ∈ N. From (m, i0) = m′ and the fact that ordθ0(σ) = m
+we obtain
+m
+m′ = ordθ0(σ)i0 = ordp(α).
+Thus
+m
+m′ = ordpα|ordpiα|ordphα = m
+m′ .
+Hence all orders ordpiα = m/m′.
+□
+Remark 7. If the KGB-obstruction vanishes and α ̸= 1, then by [11][prop. 5.9]
+i0 ≡ −1 modm and ordpiα = m for all 1 ≤ i ≤ h.
+3. Indecomposable Cq ⋊ Cm modules, modular representation theory
+In this section we will describe the indecomposable Cq ⋊ Cm-modules. We will
+give two methods in studying them. The first one is needed since it is in accordance
+to the method we will give in order to describe indecomposable R[Cq⋊Cm]-modules.
+The second one, using the structure of the socle, is the standard method of describ-
+ing k[Cq ⋊ Cm]-modules in modular representation theory.
+
+ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP.
+5
+3.1. Linear algebra method. The indecomposable modules of the Cq are deter-
+mined by the Jordan normal forms of the generator τ of the cyclic group Cq. So
+for each 1 ≤ κ ≤ ph there is exactly one Cq indecomposable module denoted by
+Jκ. Therefore we have the following decomposition of an indecomposable Cq ⋊Cm-
+module M considered as a Cq-module.
+(2)
+M = Jκ1 ⊕ · · · ⊕ Jκr.
+Lemma 8. In the indecomposable module Jκ for every element E such that
+(τ − Idκi)κi−1E ̸= 0
+the elements B = {E, (τ − Idκ)E, . . . , (τ − Idκ)κ−1E} form a basis of Jκ such that
+the matrix of τ with respect to this basis is given by
+(3)
+τ = Idκ +
+�
+�
+�
+�
+�
+�
+�
+�
+�
+0
+· · ·
+· · ·
+· · ·
+0
+1
+...
+...
+0
+...
+...
+...
+...
+...
+1
+0
+...
+0
+· · ·
+0
+1
+0
+�
+�
+�
+�
+�
+�
+�
+�
+�
+.
+Proof. Since the set B has k-elements it is enough to prove that it consists of linear
+independent elements. Indeed, consider a linear relation
+λ0E + λ1(τ − Idκ)E + · · · + λκ−1(τ − Idκ)κ−1E = 0.
+By applying (τ −Idκ)κ−1 we obtain λ0(τ −Idκ)κ−1 = 0, which gives us λ0 = 0. We
+then apply (τ − Idκ)κ−2 to the linear relation and by the same argument we obtain
+λ1 = 0 and we continue this way proving that λ0 = · · · = λκ−1 = 0. The matrix
+form of τ in this basis is immediate.
+□
+We will now prove that σ acts on each Jκ of eq.
+(2) proving that r = 1.
+Since the field k is algebraically closed and (m, p) = 1 we know that there is
+a basis of M consisting of eigenvectors of σ.
+There is an eigenvector E of σ,
+which is not in the kernel of (τ − Idκ)κ1−1. Then the elements of the set B =
+{E, (τ − Idκ)E, . . . , (τ − Idκ)κ1−1E} are linearly independent and form a direct Cq
+summand of M isomorphic to Jκ1.
+We will now show that this module is an k[Cq ⋊ Cm]-module. For this, we have
+to show that the generator σ of Cm acts on the basis B. Observe that for every
+0 ≤ i ≤ κ1 − 1 < ph
+σ(τ − 1)i−1 = (τ α − 1)i−1σ.
+Set e = E1 and κ = κ1. This means that the action of σ on e determines the action
+of σ on all other basis elements eν := (τ − 1)ν−1e, 1 ≤ ν ≤ κ1.
+Let us compute:
+σei+1 = σ(τ − 1)ie = (τ α − 1)iζλ
+me
+
+6
+A. KONTOGEORGIS AND A. TEREZAKIS
+On the basis {e1, . . . , eκ1} the matrix τ is given by eq. (3) hence using the binomial
+formula we compute
+(4)
+τ α =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+1
+0
+· · ·
+· · ·
+· · ·
+0
+�α
+1
+�
+1
+...
+...
+�α
+2
+�
+�α
+1
+�
+...
+...
+...
+�α
+3
+�
+�α
+2
+�
+...
+1
+...
+...
+...
+...
+...
+�α
+1
+�
+1
+0
+�α
+k
+�
+� α
+k−1
+�
+· · ·
+�α
+2
+�
+�α
+1
+�
+1
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+.
+Thus τ α − 1 is a nilpotent matrix A = (aij) of the form:
+aij =
+��α
+µ
+�
+if j = i − µ for some µ, 1 ≤ µ ≤ κ
+0
+if j ≥ i
+The ℓ-th power Aℓ = (a(ℓ)
+ij ) of A is then computed by (keep in mind that aij = 0
+for i ≤ j)
+a(ℓ)
+ij =
+�
+i<ν1<···<νℓ−1<j
+ai,ν1aν1,ν2aν2,ν3 · · · aνℓ−1,j
+This means that i − j > ℓ in order to have aij ̸= 0. Moreover for i = j + ℓ (which
+is the the first non zero diagonal below the main diagonal) we have
+ai,i+ℓ = ai,i+1ai+1,i+2 · · · ai+ℓ−1,i+ℓ =
+�α
+1
+�ℓ
+= αℓ.
+Therefore, the matrix of Aℓ is of the following form:
+(5)
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+k − ℓ
+�
+��
+�
+0
+· · ·
+· · ·
+0
+ℓ
+�
+��
+�
+0
+· · ·
+0
+...
+...
+...
+...
+0
+· · ·
+· · ·
+0
+0
+· · ·
+0
+αℓ
+...
+0
+...
+...
+∗
+αℓ
+...
+...
+...
+...
+...
+...
+...
+0
+...
+...
+∗
+· · ·
+∗
+αℓ
+0
+· · ·
+0
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Definition 9. We will denote by Vα(λ, κ) the indecomposable κ-dimensional G-
+module given by the basis elements {(τ − 1)νe, ν = 0, . . . , κ − 1}, where σe = ζλ
+me.
+This definition is close to the notation used in [9].
+Lemma 10. The action of σ on the basis element ei of Vα(λ, κ) is given by:
+(6)
+σei = αi−1ζλ
+mei +
+κ
+�
+ν=i+1
+aνeν,
+for some coefficients ai ∈ k. In particular the matrix of σ with respect to the basis
+e1, . . . , eκ is lower triangular.
+
+ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP.
+7
+Proof. Recall that ei = (τ − 1)i−1e1. Therefore
+σei = σ(τ − 1)i−1e1 = (τ α − 1)i−1σe1 = ζλ
+m(τ α − 1)i−1e1.
+The result follows by eq. (5)
+□
+We have constructed a set of indecomposable modules Vα(λ, κ).
+Apparently
+Vα(λ, κ) can not be isomorphic to Vα(λ′, κ′) if κ ̸= κ′, since they have different
+dimensions.
+Assume now that κ = κ′. Can the modules Vα(λ, κ) and Vα(λ′, κ) be isomorphic
+for λ ̸= λ′?
+The eigenvalues of the prime to p generator σ on Vα(λ, κ)are
+ζλ
+m, αζλ
+m, . . . , ακ−1ζλ
+m.
+Similarly the eigenvalues for σ when acting on Vα(λ′, κ) are
+ζλ′
+m , αζλ′
+m , . . . , ακ−1ζλ′
+m .
+If the two sets of eigenvalues are different then the modules can not be isomorphic.
+But even if λ ̸= λ′ modn the two sets of eigenvalues can still be equal. Even in this
+case the modules can not be isomorphic.
+Lemma 11. The modules Vα(λ1, κ) and Vα(λ2, κ) are isomorphic if and only if
+λ1 ≡ λ2 modm.
+Proof. Indeed, the module Vα(λ1, κ) has an eigenvector for the action of σ which
+generates the Vα(λ1, κ) by powers of (τ − 1), i.e. the vectors
+(7)
+e, (τ − 1)e, (τ − 1)2e, . . . , (τ − 1)κ−1e
+form a basis of Vα(λ1, κ).
+The elements E which can generate Vα(λ1, κ) by powers of (τ − 1) are linear
+combinations
+E =
+κ−1
+�
+ν=0
+λi(τ − 1)νe,
+for λi ∈ k and λ0 ̸= 0.
+On the other hand using eq. (6) we see that σ with respect to the basis given in
+eq. (7) admits the matrix form:
+�
+�
+�
+�
+�
+�
+�
+�
+ζλ
+m
+0
+· · ·
+· · ·
+0
+0
+αζλ
+m
+0
+· · ·
+0
+...
+...
+...
+...
+...
+...
+...
+...
+...
+0
+· · ·
+· · ·
+0
+ακ−1ζλ
+m
+�
+�
+�
+�
+�
+�
+�
+�
+.
+It is now easy to see from the above matrix that every eigenvector of the eigenvalue
+ανλ1, ν > 1 is expressed as a linear combination of the basis given in eq. (7), where
+the coefficient of e is zero.
+Therefore, the eigenvector of the eigenvalue ανζm can not generate the module
+Vα(λ, κ) by powers of (σ − 1)ν.
+□
+
+8
+A. KONTOGEORGIS AND A. TEREZAKIS
+3.2. The uniserial description. We will now give an alternative description of
+the indecomposable Cq ⋊ Cm-modules, which is used in [2].
+It is known that Aut(Cq) ∼= F∗
+p × Q, for some abelian p-group Q. The repre-
+sentation ψ : Cm → Aut(Cq) given by the action of Cm on Cq is known to factor
+through a character χ : Cm → F∗
+p. The order of χ divides p−1 and χp−1 = χ−(p−1)
+is the trivial one dimensional character.
+For all i ∈ Z, χi defines a simple k[Cm]-module of k dimension one, which we
+will denote by Sχi. For 0 ≤ ℓ ≤ m − 1 denote by Sℓ the simple module where
+on which σ acts as ζℓ
+m. Both Sχi, Sℓ can be seen as k[Cq ⋊ Cm]-modules using
+inflation. Finally for 0 ≤ ℓ ≤ m − 1 we define χi(ℓ) ∈ {0, 1, . . . , m − 1} such that
+Sχi(ℓ) ∼= Sℓ ⊗k Sχi.
+There are q · m isomorphism classes of indecomposable k[Cq ⋊ Cm]-modules and
+are all uniserial. An indecomposable k[Cq ⋊ Cm]-module U is unique determined
+by its socle, which is the kernel of the action of τ − 1 on U, and its k-dimension.
+For 0 ≤ ℓ ≤ m − 1 and 1 ≤ µ ≤ q, let Uℓ,µ be the indecomposable k[Cq ⋊ Cm]
+module with socle Sa and k-dimension µ. Then Uℓ,µ is uniserial and its µ ascending
+composition factors are the first µ composition factors of the sequence
+Sℓ, Sχ−1(ℓ), Sχ−2(ℓ), . . . , Sχ−(p−2)(ℓ), Sℓ, Sχ−1(ℓ), Sχ−2(ℓ), . . . , Sχ−(p−2)(ℓ).
+Notice that in our notation Vα(λ, κ) = Uλ+κ,κ.
+Remark 12. The condition ordpi = m for all 1 ≤ i ≤ h, is equivalent to requiring
+that ψi : Cm → Aut(Cpi) is faithful for all i.
+4. Lifting of representations
+Proposition 13. Let G = Cq ⋊ Cm. Assume that for all 1 ≤ i ≤ h, ordpia = m.
+If the G-module V lifts to an R[G]-module ˜V , where K = Quot(R) is a field of
+characterstic zero, then
+m |
+�
+dim( ˜V ⊗R K) − dim( ˜V ⊗R K)Cq�
+.
+Moreover, if ˜V (ζαiκ
+q
+) is the eigenspace of the eigenvalue ζαiκ
+q
+of T acting on ˜V ,
+then
+dim ˜V (ζκ
+q ) = dim ˜V (ζακ
+q ) = dim ˜V (ζα2κ
+q
+) = · · · = dim ˜V (ζαm−1κ
+q
+).
+Proof. Consider a lifting ˜V of V . The generator τ of the cyclic part Cq has eigen-
+values λ1, . . . , λs which are pn-roots of unity. Let ζq be a primitive q-root of unity.
+Consider any eigenvalue λ ̸= 1. It is of the form λ = ζκ
+q for some κ ∈ N, q ∤ κ. If E
+is an eigenvector of T corresponding to λ, that is τE = ζκ
+q E then
+τσ−1E = σ−1τ αE = ζκαm−1
+q
+σ−1E
+and we have a series of eigenvectors E, σ−1E, σ−2E, · · · with corresponding eigen-
+values ζκ
+q , ζκα
+q , ζκa2
+q
+· · · , ζκαo
+q
+, where o = ordq/(q,k). Indeed, the integer o satisfies
+the
+καo ≡ κ modq ⇒ αm ≡ 1 mod
+q
+(q, k).
+Therefore the eigenvalues λ ̸= 1 form orbits of size m, while the eigenspace of the
+eigenvalue 1 is just the invariant space V G and the result follows.
+□
+
+ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP.
+9
+5. Indecomposable Cq ⋊ Cm modules, integral representation theory
+From now on V be a free R-module, where R is an integral local principal ideal
+domain with maximal ideal mR, R has characteristic zero and that R contains all
+q-th roots of unity and has characteristic zero. Let K = Quot(R).
+The indecomposable modules for a cyclic group both in the ordinary and in the
+modular case are described by writing down the Jordan normal form of a generator
+of the cyclic group. Since in integral representation theory there are infinitely many
+non-isomorphic indecomposable Cq-modules for q = ph, h ≥ 3, one is not expecting
+to have a theory of Jordan normal forms even if one works over complete local
+principal ideal domains [7], [8].
+Lemma 14. Let T be an element of order q = ph in End(V ), then the minimal
+polynomial of T has simple eigenvalues and T is diagonalizable when seen as an
+element in End(V ⊗ K).
+Proof. Since T q = IdV , the minimal polynomial of T divides xq − 1, which has
+simple roots over a field of characteristic zero. This ensures that T ∈ End(V ⊗ K)
+is diagonalizable.
+□
+Lemma 15. Let f(x) = (x − λ1)(x − λ2) · · · (x − λd) be the minimal polynomial of
+T on V . There is an element E ∈ V , such that
+E, (T − λ1IdV )E, (T − λ2IdV )(T − λ1IdV )E, . . . , (T − λd−1IdV ) · · · (T − λ1IdV )E
+are linear independent elements in V ⊗ K.
+Proof. Consider the endomorphisms for i = 1, . . . , d
+Πi =
+d
+�
+ν=1
+ν̸=i
+(T − λνIdV ).
+In the above product notice that T − λiIdV , T − λjIdV are commuting endomor-
+phisms. Since the minimal polynomial of T has degree d all R-modules KerΠi are
+strictly less than V . Moreover there is an element E such that E ̸∈ Ker(Πi) for all
+1 ≤ i ≤ d. Consider a relation
+(8)
+d
+�
+µ=0
+γµ
+µ
+�
+ν=0
+(T − λµIdV )E,
+where �0
+ν=0(T − λνIdV )E = E. We fist apply the operator �d
+ν=2(T − λνIdV ) to
+eq. (8) and we obtain
+0 = γ0Π1E,
+and by the selection of E we have that a0 = 0. We now apply �d
+ν=3(T − λνIdV )
+to eq. (8). We obtain that
+0 = γ1
+d
+�
+ν=3
+(T − λνIdV )(T − λ1IdV ) = γ1Π2E,
+and by the selection of E we have that γ1 = 0. We now apply �d
+ν=4(T − λνIdV )
+to eq. (8) and we obtain
+0 = γ2
+d
+�
+ν=4
+(T − λνIdV )(T − λ2IdV )(T − λ1IdV )E = γ2Π3E
+
+10
+A. KONTOGEORGIS AND A. TEREZAKIS
+and by the selection of E we obtain γ3 = 0. Continuing this way we finally arrive
+at γ0 = γ1 = · · · = γd−1 = 0.
+□
+Lemma 16. Let V be a free R-module of rank R acted on by an automorphism
+T : V → V of order ph. Assume that the minimal polynomial of T is of degree d
+and has roots λ1, . . . , λd. Then T can be written as a matrix with respect to the
+basis as follows:
+(9)
+�
+�
+�
+�
+�
+�
+�
+�
+�
+λ1
+0
+· · ·
+· · ·
+0
+a1
+λ2
+...
+...
+0
+a2
+λ3
+...
+...
+...
+...
+...
+...
+0
+0
+· · ·
+0
+ad−1
+λd
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Proof. By lemma 15 the elements
+E, (T − λ1IdV )E, (T − λ2IdV )(T − λ1IdV )E, . . . , (T − λd−1IdV ) · · · (T − λ1IdV )E
+form a free submodule of V of rank d. The theory of submodules of principal ideal
+domains, there is a basis E1, E2, . . . , Ed of the free module V such that
+E1 = E,
+(10)
+a1E2 = (T − λ1IdV )E1,
+a2E3 = (T − λ2IdV )E2,
+. . .
+as−1Ed = (T − λd−1IdV )Ed−1.
+Let us consider the module V1 = ⟨E1, . . . , Ed⟩ ⊂ V . By construction, the map T
+restricts to an automorphism V1 → V1 with respect to the basis E1, . . . , Ed has the
+desired form. We then consider the free module V/V1 and we repeat the procedure
+for the minimal polynomial of T, which again acts on V/V1. The desired result
+follows.
+□
+Remark 17. The element T as defined in eq. (9) has order equal to the higher
+order of the eigenvalues λ1, . . . , λd involved. Indeed, since we have assumed that
+the eigenvalues are different the matrix is diagonalizable in Quot(R) and has order
+equal to the maximal order of the eigenvalues involved. In particular it has order q
+if there is at least one λi that is a primitive q-root of unity. The statement about
+the order of T is not necessarily true if some of the eigenvalues are the same. For
+instance the matrix
+�
+1
+0
+1
+1
+�
+has infinite order over a field of characteristic zero.
+Remark 18. The number of indecomposable R[T]-summands of V is given by
+#{i : ai = 0} + 1.
+A lift of a sum of indecomposable kCq-modules Jκ1 ⊕ · · · ⊕ Jκn can form an
+indecomposable RCq-module. For example the indecomposable module where the
+
+ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP.
+11
+generator T of Cq has the form
+T =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+λ1
+0
+· · ·
+· · ·
+0
+a1
+λ2
+...
+...
+0
+a2
+λ3
+...
+...
+...
+...
+...
+...
+0
+0
+· · ·
+0
+as−1
+λd
+�
+�
+�
+�
+�
+�
+�
+�
+�
+where a1 = · · · = aκ1−1 = 1, aκ1 ∈ mR, aκ1+1, . . . , aκ2+κ1−1 = 1, aκ2+κ1 ∈
+mR , etc reduces to a decomposable direct sum of Jordan normal forms of sizes
+Jκ1, Jκ2−κ1, · · · .
+Remark 19. It is an interesting question to classify these matrices up to conju-
+gation with a matrix in GLd(R). It seems that the valuation of elements ai should
+also play a role.
+Definition 20. Let hi(x1, . . . , xj) be the complete symmetric polynomial of degree
+i in the variables x1, . . . , xj. For instance
+h3(x1, x2, x3) = x3
+1 + x2
+1x2 + x2
+1x3 + x1x2
+2 + x1x2x3 + x1x2
+3 + x3
+2 + x2
+2x3 + x2x2
+3 + x3
+3.
+Set
+L(κ, j, ν) = hκ(λj, λj+1, . . . , λj+ν)
+A(i, j) =
+�
+aiai+1 · · · ai+j
+if j ≥ 0
+0
+if j < 0
+Lemma 21. The matrix T α = (t(α)
+ij ) is given by the following formula:
+t(α)
+ij
+=
+�
+�
+�
+�
+�
+λα
+i
+if i = j
+A(j, i − j − 1) · L(α − (i − j), j, i − j)
+if j < i
+0
+if j > i
+Proof. For j ≥ i the proof is trivial. When j < i and α = 1 it is immediate, since
+L(x, ·, ·) ≡ 0, for every x ≤ 0. Assume this holds for α = n. If α = n + 1,
+t(n+1)
+ij
+= t(n)
+ij tij =
+r
+�
+k=1
+t(α)
+ik tkj = λjt(α)
+ij
++ ajt(α)
+ij+1 = λjA(j, i − j − 1)L(α − (i − j), j, i − j)+
++ ajA(j + 1, i − j − 2)L(α − (i − j − 1), j + 1, i − j − 1) =
+= A(j, i − j − 1)
+�
+λjhα−(i−j)(λj, . . . , λj) + hα−(i−j)+1(λj+1, . . . , λi)
+�
+=
+= A(j, i − j − 1)hα−(i−j)+1(λj, . . . , λi) =
+= A(j, i − j − 1)L(α − (i − j) + 1, i, i − j).
+□
+Remark 22. The space of homogeneous polynomials of degree k in n-variables
+has dimension
+�n−1+c
+n−1
+�
+. Since all q-roots of unity are reduced to 1 modulo mR the
+quantity L(α − (i − j), j, i − j) is reduced to n = (i − j) + 1, c = α − (i − j)
+�n − 1 + c
+n − 1
+�
+=
+� α
+i − j
+�
+.
+This equation is compatible with the computation of τ α given in eq. (4).
+
+12
+A. KONTOGEORGIS AND A. TEREZAKIS
+Lemma 23. There is an eigenvector E of the generator σ of the cyclic group Cm
+which is not an element in
+s�
+i=1
+Ker(Πi ⊗ K).
+Proof. The eigenvectors E1, . . . , Ed of σ form a basis of the space V ⊗ K.
+By
+multiplying by certain elements in R, if necessary, we can assume that all Ei are in
+V and their reductions Ei ⊗ R/mR, 1 ≤ i ≤ d give rise to a basis of eigenvectors of
+a generator of the cyclic group Cm acting on V ⊗ R/mR. If every eigenvector Ei is
+an element of some Ker(Πν) for 1 ≤ i ≤ d, then their reductions will be elements
+in Ker(T − 1)d−1, a contradiction since the later kernel has dimension < d.
+□
+Lemma 24. Let V be a free Cq ⋊ Cm-module, which is indecomposable as a Cq-
+module. Consider the basis given in lemma 16. Then the value of σ(E1) determines
+σ(Ei) for 2 ≤ i ≤ d.
+Proof. Let σ be a generator of the cyclic group Cm. We will use the notation of
+lemma 15. We use lemma 23 in order to select a suitable eigenvector of E1 of σ
+and then form the basis E1, E2, . . . , Ed as given in eq. (10). We can compute the
+action of σ on all basis elements Ei by
+(11)
+σ(ai−1Ei) = σ(T − λi−1IdV )Ei−1 = (T a − λi−1IdV )σ(Ei−1).
+This means that one can define recursively the action of σ on all elements Ei.
+Indeed, assume that
+σ(Ei−1) =
+d
+�
+ν=1
+γν,i−1Eν.
+We now have
+(T a − λi−1IdV )Eν =
+d
+�
+µ=1
+t(α)
+µ,νEµ − λi−1Eν
+= (λα
+ν − λi−1)Eν +
+d
+�
+µ=ν+1
+t(α)
+µ,νEµ
+We combine all the above to
+ai−1σ(Ei) =
+d
+�
+ν=1
+γν,i−1(λα
+ν − λi−1)Eν +
+d
+�
+ν=1
+γν,i−1
+d
+�
+µ=ν+1
+t(α)
+µ,νEµ
+=
+d
+�
+ν=1
+˜γν,iEν,
+(12)
+for a selection of elements γν,i ∈ R, which can be explicitly computed by collecting
+the coefficients of the basis elements E1, . . . , Ed.
+Observe that the quantity on the right hand side of eq. (12) must be divisible
+by ai−1. Indeed, let v be the valuation of the local principal ideal domain R. Set
+e0 = min
+1≤ν≤d{v(˜γν,i)}.
+If e0 < v(ai−1) then we divide eq. (12) by πe0 where π is the local uniformizer of
+R, that is mR = πR. We then consider the divided equation modulo mR to obtain
+
+ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP.
+13
+a linear dependence relation among the elements Ei ⊗ k, which is a contradiction.
+Therefore e0 ≥ v(ai−1) and we obtain an equation
+σ(Ei) =
+d
+�
+ν=1
+˜γν,i
+ai−1
+Eν =
+d
+�
+ν=1
+γν,iEν.
+□
+For example σ(E1) = ζϵ
+mE1. We compute that
+a1σ(E2) = (T α − λ1Id)σ(E1)
+and
+σ(E2) = (λα
+1 − λ1)
+a1
+ζϵ
+µE1 + ζϵ
+m
+d
+�
+µ=2
+t(α)
+µ,1
+a1
+Eµ
+= (λα
+1 − λ1)
+a1
+ζϵ
+µE1 + ζϵ
+m
+d
+�
+µ=2
+A(1, µ − 2)L(α − (µ − 1), 1, µ − 1)
+a1
+Eµ
+= (λα
+1 − λ1)
+a1
+ζϵ
+µE1 + ζϵ
+m
+d
+�
+µ=2
+a1a2 · · · aµ−1hα−(µ−1)(λ1, λ2, . . . , λµ)
+a1
+Eµ.
+Proposition 25. Assume that no element a1, . . . , ad−1 given in eq. (9) is zero.
+Given α ∈ N, α ≥ 1 and an element E1, which is not an element in �d
+i=1 Ker(Πi ⊗
+K), if there is a matrix Γ = (γij), such that ΓTΓ−1 = T α and ΓE1 = ζϵ
+mE1, then
+this matrix Γ is unique.
+Proof. We will use the idea leading to equation (11) replacing σ with Γ. We will
+compute recursively and uniquely the entries γµ,i, arriving at the explicit formula
+of eq. (18).
+Observe that trivially γν,1 = 0 for all ν < 1 since we only allow 1 ≤ ν ≤ d. We
+compute
+˜γµ,i = γµ,i−1(λα
+µ − λi−1) +
+µ−1
+�
+ν=1
+γν,i−1t(α)
+µ,ν
+(13)
+= γµ,i−1(λα
+µ − λi−1) +
+µ−1
+�
+ν=1
+γν,i−1A(ν, µ − ν − 1)L
+�
+α − (µ − ν), ν, µ − ν)
+= γµ,i−1(λα
+µ − λi−1) +
+µ−1
+�
+ν=1
+γν,i−1aνaν+1 · · · aµ−1hα−µ+ν(λν, λν+1, . . . , λµ)
+Define
+[λα
+m − λx]j
+i =
+j�
+x=i
+(λα
+µ − λx)
+[a]j
+i =
+j�
+x=i
+ax
+for i ≤ j. If i > j then both of the above quantities are defined to be equal to 1.
+
+14
+A. KONTOGEORGIS AND A. TEREZAKIS
+Observe that for µ = 1 eq. (13) becomes
+(14)
+γ1,i =
+1
+ai−1
+γ1,i−1(λα
+1 − λi−1)
+and we arrive at (assuming that Γ(E1) = ζϵ
+mE1)
+(15)
+γ1,i =
+ζϵ
+m
+a1a2 · · · ai−1
+i−1
+�
+x=1
+(λα
+1 − λx) =
+ζϵ
+m
+a1a2 · · · ai−1
+[λα
+1 − λx]i−1
+1
+.
+For µ ≥ 2 we have γµ,1 = 0, since by assumption TE1 = ζϵ
+mE1. Therefore eq. (13)
+gives us
+γµ,i =
+i−2
+�
+κ1=0
+[λα
+µ − λx]i−1
+i−κ1
+[a]i−1
+i−1−κ1
+µ−1
+�
+µ2=1
+γµ2,i−1−κ1[a]µ−1
+µ2 hα−µ+µ2(λµ2, . . . , λµ)
+=
+µ−1
+�
+µ2=1
+[a]µ−1
+µ2 hα−µ+µ2(λµ2, . . . , λµ)
+i−2
+�
+κ1=0
+[λα
+µ − λx]i−1
+i−κ1
+[a]i−1
+i−1−κ1
+γµ2,i−1−κ1.
+(16)
+We will now prove eq. (16) by induction on i. For i = 2, µ ≥ 2 we have
+γµ,2 = 1
+a1
+γµ,1(λα
+µ − λ1) + 1
+a1
+µ−1
+�
+µ2=1
+γµ2,1[a]µ−1
+µ2 hα−µ+µ2(λµ2, . . . , λµ)
+= 1
+a1
+[a]µ−1
+1
+hα−µ+1(λ1, . . . , λµ)γ1,1.
+
+ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP.
+15
+Assume now that eq. (16) holds for computing γµ,i−1. We will treat the γµ,i case.
+We have
+γµ,i = (λα
+µ − λi−1)
+ai−1
+γµ,i−1 +
+1
+ai−1
+µ−1
+�
+µ2=1
+γµ2,i−1[a]µ−1
+µ2 hα−µ+µ2(λµ2, . . . , λµ)
+= (λα
+µ − λi−1)
+ai−1
+µ−1
+�
+µ2=1
+[a]µ−1
+µ2 hα−µ+µ2(λµ2, . . . , λµ)
+i−3
+�
+κ1=0
+[λα
+µ − λx]i−2
+i−1−κ1
+[a]i−2
+i−2−κ1
+γµ2,i−2−κ1
++
+1
+ai−1
+µ−1
+�
+µ2=1
+γµ2,i−1[a]µ−1
+µ2 hα−µ+µ2(λµ2, . . . , λµ)
+=
+µ−1
+�
+µ2=1
+[a]µ−1
+µ2 hα−µ+µ2(λµ2, . . . , λµ)
+i−3
+�
+κ1=0
+[λα
+µ − λx]i−1
+i−1−κ1
+[a]i−1
+i−2−κ1
+γµ2,i−2−κ1
++
+1
+ai−1
+µ−1
+�
+µ2=1
+γµ2,i−1[a]µ−1
+µ2 hα−µ+µ2(λµ2, . . . , λµ)
+=
+µ−1
+�
+µ2=1
+[a]µ−1
+µ2 hα−µ+µ2(λµ2, . . . , λµ)
+i−2
+�
+κ1=1
+[λα
+µ − λx]i−1
+i−κ1
+[a]i−1
+i−1−κ1
+γµ2,i−1−κ1
++
+µ−1
+�
+µ2=1
+[a]µ−1
+µ2 hα−µ+µ2(λµ2, . . . , λµ)
+1
+ai−1
+γµ2,i−1
+=
+µ−1
+�
+µ2=1
+[a]µ−1
+µ2 hα−µ+µ2(λµ2, . . . , λµ)
+i−2
+�
+κ1=0
+[λα
+µ − λx]i−1
+i−κ1
+[a]i−1
+i−1−κ1
+γµ2,i−1−κ1
+and equation (16) is now proved.
+We proceed recursively applying eq. (16) to each of the summands γµ2,i−1−κ1
+if µ2 > 1 and i − 1 − κ1 > 1. If µ2 = 1, then γµ2,i−1−κ1 is computed by eq. (14)
+and if µ2 > 1 and i − 1 − κ1 ≤ 1 then γµ2,i−1−κ1 = 0. We can classify all iterations
+needed by the set Σµ of sequences (µs, µs−1, . . . , µ3, µ2) such that
+(17)
+1 = µs < µs−1 < · · · < µ3 < µ2 < µ = µ1.
+For example for µ = 5 the set of such sequences is given by
+Σµ = {(1), (1, 2), (1, 3), (1, 2, 3), (1, 4), (1, 2, 4), (1, 3, 4), (1, 2, 3, 4)}
+corresponding to the tree of iterations given in figure 1. The length of the sequence
+(µs, µs−1, . . . , µ2) is given in eq. (17) is s − 1. In each iteration the i changes to
+i − 1 − k thus we have the following sequence of indices
+i1 = i → i2 = i−1−κ1 → i3 = i−2−(κ1+κ2) → · · · → is = i−(s−1)−(κ1+· · ·+κs−1)
+For the sequence i1, i2, . . . , we might have it = 1 for t < s − 1. But in this case,
+we will arrive at the element γµt+1,it = γµt,1 = 0 since µt > 1. This means that we
+will have to consider only selections κ1, . . . , κs−1 such that is−1 ≥ 1. Therefore we
+
+16
+A. KONTOGEORGIS AND A. TEREZAKIS
+µ = 5
+µ2 = 1
+µ2 = 2
+µ3 = 1
+µ2 = 3
+µ3 = 1
+µ3 = 2
+µ4 = 1
+µ2 = 4
+µ3 = 1
+µ3 = 2
+µ4 = 1
+µ3 = 3
+µ4 = 1
+µ4 = 2
+µ5 = 1
+Figure 1.
+Iteration tree for µ = 5
+arrive at the following expression for µ ≥ 2
+γµ,i =
+�
+(µs,...,µ2)∈Σµ
+[a]µ−1
+µ2 [a]µ2−1
+µ3
+· · · [a]µs−1−1
+µs
+s
+�
+ν=2
+hα−µν−1+µν(λµν, . . . , λµν−1)
+·
+�
+i=i1>i2>···>is≥1
+s−1
+�
+ν=1
+[λα
+µν − λx]iν−1
+iν+1+1
+[a]iν−1
+iν+1
+· γ1,is.
+=
+�
+(µs,...,µ2)∈Σµ
+s
+�
+ν=2
+hα−µν−1+µν(λµν, . . . , λµν−1)
+·
+�
+i=i1>i2>···>is≥1
+[a]µ−1
+1
+[a]i−1
+is
+s−1
+�
+ν=1
+[λα
+µν − λx]iν−1
+iν+1+1
+ζϵ
+m[λα
+1 − λx]is−1
+1
+[a]is−1
+1
+=
+�
+(µs,...,µ2)∈Σµ
+s
+�
+ν=2
+hα−µν−1+µν(λµν, . . . , λµν−1)[a]µ−1
+1
+[a]i−1
+1
+ζϵ
+m
+�
+i=i1>i2>···>is≥1
+s
+�
+ν=1
+[λα
+µν − λx]iν−1
+iν+1+1
+(18)
+where is+1 + 1 = 1 that is is+1 = 0.
+□
+We will now prove that the matrix Γ of lemma 25 exists by cheking that ΓT =
+T αΓ. Set (aµ,i) = ΓT, (bµ,i) = T αΓ. For i < d we have
+aµ,i =
+d
+�
+ν=1
+γµ,νtν,i = γµ,itii + γµ,i+1ti+1,i
+= γµ,iλi + γµ,i(λα
+µ − λi) +
+µ−1
+�
+ν=1
+γν,it(α)
+µ,ν
+= γµ,iλα
+µ +
+µ−1
+�
+ν=1
+γν,it(α)
+µ,ν =
+µ
+�
+ν=1
+t(α)
+µ,νγν,i = bµ,i.
+For i = d we have:
+aµ,d =
+d
+�
+ν=1
+γµ,νtν,d = γµ,dtd,d = γµ,dλd
+
+ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP.
+17
+while
+bµ,d =
+d
+�
+ν=1
+t(α)
+µ,νγν,d =
+µ−1
+�
+ν=1
+t(α)
+µ,νγν,d + λα
+µγµ,d
+This gives us the relation
+(19)
+(λd − λa
+µ)γµ,d =
+µ−1
+�
+ν=1
+t(α)
+µ,νγν,d
+For µ = 1 using eq. (15) we have
+γ1,dλd = γ1,dλα
+1 ⇒ [λα
+1 − λx]d
+1 = 0.
+This relation is satisfied if λα
+1 is one of {λ1, . . . , λd}. Without loss of generality we
+assume that
+(20)
+λ(a)
+i
+=
+�
+λi+1
+if m ∤ i
+λi−m+1
+if m | i
+We have the following conditions:
+µ = 2
+(λd − λα
+2 )γ2,d = t(α)
+2,1 γ1,d
+µ = 3
+(λd − λα
+3 )γ3,d = t(α)
+3,1 γ1,d + t(α)
+3,2 γ2,d
+µ = 4
+(λd − λα
+4 )γ4,d = t(α)
+4,1 γ1,d + t(α)
+4,2 γ2,d + t(α)
+4,3 γ3,d
+...
+...
+µ = d − 1
+(λd − λα
+d−1)γd−1,d = t(α)
+d−1,1γ1,d + t(α)
+d−1,2γ2,d + · · · + t(α)
+d−1,d−2γd−1,d
+All these equations are true provided that γ1,d, . . . , γd−2,d = 0. Finally, for µ = d,
+we have
+(21)
+(λd − λα
+d )γd,d =
+d−1
+�
+ν=1
+t(α)
+d,νγν,d
+which is true provided that (λd − λα
+d )γd,d = t(a)
+d,d−1γd−1,d.
+Lemma 26. For n ≥ 2 the vertical sum Sn of the products of every line of the
+following array
+y
+1
+1
+(x1 − x2)
+(x1 − x3)
+· · ·
+· · ·
+(x1 − xn)
+2
+(z − x1)
+1
+(x1 − x3)
+· · ·
+· · ·
+(x1 − xn)
+3
+(z − x1)
+(z − x2)
+1
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+n − 1
+(z − x1)
+(z − x2)
+· · ·
+(z − xn−2)
+1
+(x1 − xn)
+n
+(z − x1)
+(z − x2)
+· · ·
+(z − xn−2)
+(z − xn−1)
+1
+is given by
+Sn =
+n
+�
+y=1
+n
+�
+ν=y+1
+(x1 − xν)
+y−1
+�
+µ=1
+(z − xµ) = (z − x2) · · · (z − xn).
+
+18
+A. KONTOGEORGIS AND A. TEREZAKIS
+In particular when z = xn the sum is zero.
+Proof. We will prove the lemma by induction. For n = 2 we have S2 = (x1 − x2) +
+(z−x1) = z−x2. Assume that the equality holds for n. The sum Sn+1 corresponds
+to the array:
+y
+1
+1
+(x1 − x2)
+(x1 − x3)
+· · ·
+(x1 − xn)
+(x1 − xn+1)
+2
+(z − x1)
+1
+(x1 − x3)
+· · ·
+(x1 − xn)
+(x1 − xn+1)
+3
+(z − x1)
+(z − x2)
+1
+...
+...
+...
+...
+...
+...
+...
+...
+...
+n − 1
+(z − x1)
+· · ·
+(z − xn−2)
+1
+(x1 − xn)
+(x1 − xn+1)
+n
+(z − x1)
+(z − x2)
+· · ·
+(z − xn−1)
+1
+(x1 − xn+1)
+n + 1
+(z − x1)
+(z − x2)
+· · ·
+(z − xn−1)
+(z − xn)
+1
+We have by definition Sn+1 = Sn(x1 − xn+1) + (z − x1)(z − x2) · · · (z − xn), which
+by induction gives
+Sn+1 = (z − x2) · · · (z − xn)(x1 − xn+1) + (z − x1)(z − x2) · · · (z − xn)
+= (z − x2) · · · (z − xn)(x1 − xn+1 + z − x1)
+and gives the desired result.
+□
+Lemma 27. Consider A < l < L < B. The quantity
+�
+l≤y≤L
+[λa − λx]y−1
+A
+· [λb − λx]B
+y+1
+equals to
+[λa − λx]l−1
+A
+· [λb − λx]B
+L+1 · [λa − λx]L
+l − [λb − λx]L
+l
+(λa − λb)
+Proof. We write
+�
+l≤y≤L
+[λa − λx]y−1
+A
+· [λb − λx]B
+y+1
+= [λa − λx]l−1
+A
+· [λb − λx]B
+L+1 ·
+�
+l≤y≤L
+[λa − λx]y−1
+l
+· [λb − λx]L
+y+1
+The last sum can be read as the vertical sum S of the products of every line in the
+following array:
+y
+l
+1
+(λb − λl+1)(λb − λl+2)
+· · ·
+(λb − λL−1)(λb − λL)
+l + 1 (λa − λl)
+1
+(λb − λl+2)
+· · ·
+(λb − λL−1)(λb − λL)
+l + 2 (λa − λl)(λa − λl+1)
+1
+...
+...
+...
+...
+...
+...
+...
+...
+...
+L − 2(λa − λl)(λa − λl+1)
+· · ·
+1
+(λb − λL−1)(λb − λL)
+L − 1(λa − λl)(λa − λl+1)
+· · ·
+(λa − λL−2)
+1
+(λb − λL)
+L
+(λa − λl)(λa − λl+1)
+· · ·
+(λa − λL−2)(λa − λL−1)
+1
+If l = b, then lemma 26 implies that S = [λa − λx]L
+b+1. Furthermore, if L = a then
+S = 0.
+
+ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP.
+19
+The quantity S cannot be directly computed using lemma 26, if l ̸= b.
+We
+proceed by forming the array:
+y
+b
+1
+(λb − λb+1)
+· · ·
+(λb − λl)
+· · ·
+· · ·
+· · ·
+· · ·
+(λb − λL)
+...
+...
+...
+l − 1 (λa − λb)
+· · ·
+1
+(λb − λl)
+· · ·
+· · ·
+· · ·
+· · ·
+(λb − λL)
+l
+(λa − λb)
+· · ·
+(λa − λl−1)
+1
+(λb − λl+1)(λb − λl+2)
+· · ·
+(λb − λL−1)(λb − λL)
+l + 1 (λa − λb)
+· · ·
+(λa − λl−1)(λa − λl)
+1
+(λb − λl+2)
+· · ·
+(λb − λL−1)(λb − λL)
+l + 2 (λa − λb)
+· · ·
+(λa − λl−1)(λa − λl)(λa − λl+1)
+1
+...
+...
+...
+...
+...
+...
+...
+...
+...
+L − 2(λa − λb)
+· · ·
+(λa − λl−1)(λa − λl)(λa − λl+1)
+· · ·
+1
+(λb − λL−1)(λb − λL)
+L − 1(λa − λb)
+· · ·
+(λa − λl−1)(λa − λl)(λa − λl+1)
+· · ·
+(λa − λL−2)
+1
+(λb − λL)
+L
+(λa − λb)
+· · ·
+(λa − λl−1)(λa − λl)(λa − λl+1)
+· · ·
+(λa − λL−2)(λa − λL−1)
+1
+The value of this array is computed using lemma 26 to be equal to [λa −λx]L
+b+1. We
+observe that the sum of the products of the top left array can be computed using
+lemma 26, while the sum of the products of the lower right array is S.
+[λa − λx]l−1
+b
+· S + [λa − λx]l−1
+b+1 · [λb − λx]L
+l = [λa − λx]L
+b+1
+we arrive at
+[λa − λx]l−1
+b
+S = [λa − λx]l−1
+b+1
+�
+[λa − λx]L
+l − [λb − λx]L
+l
+�
+or equivalently
+(λa − λb) · S = [λa − λx]L
+l − [λb − λx]L
+l
+□
+Lemma 28. For all 1 ≤ µ ≤ d − 2 we have γµ,d = 0.
+Proof. Let µ1 = µ > µ2 > · · · > µs = 1 ∈ Σµ be a selection of iterations and
+d = i1 > i2 > · · · · · · is ≥ 1 > is+1 = 0 be the sequence of i’s. Using eq. (20) we
+see that the quantity [λα
+µν − λx]iν−1
+iν+1+1 ̸= 0 if and only if one of the following two
+inequalities hold:
+either
+iν+1 >µν − mf(µν)
+(22)
+or
+iν <µν + 2 − mf(µν),
+(23)
+where
+f(x) =
+�
+1
+if m | x
+0
+if m ∤ x
+We will denote the above two inequalities by (22)ν,(23)ν when applied for the
+integer ν. Assume, that for all 1 ≤ ν ≤ s one of the two inequalities (22)ν,(23)ν
+hold, that is [λα
+µν − λx]iν−1
+iν+1+1 ̸= 0. Inequality (22)s can not hold for ν = s since it
+gives us 0 = is+1 > 1 = µs, we have m ∤ 1 = µs.
+We will keep the sequence ¯µ : µ1 > µ2 > · · · > µs fixed and we will sum over all
+possible selections of sequences of i1 > · · · is > is+1 = 0, that is we will show that
+the sum
+(24)
+Γ¯µ,i :=
+�
+i=i1>i2>···>is≥1
+s
+�
+ν=1
+[λα
+µν − λx]iν−1
+iν+1+1
+is zero, which will show that γµ,d = 0 using eq. (18).
+
+20
+A. KONTOGEORGIS AND A. TEREZAKIS
+Observe now that if (23)ν holds and m ∤ ν, ν −1, then (23)ν−1 also holds. Indeed
+the combination of (23)ν and (22)ν−1 gives the impossible inequality
+µν + 2
+(23)ν
+>
+iν
+(22)ν−1
+>
+µν−1.
+Assume now that m | ν and (23)ν holds, then (23)ν−1 also holds.
+Indeed the
+combination of (23)ν and (22)ν−1 gives us
+µν + 2 − m
+(23)ν
+>
+iν
+(22)ν−1
+>
+µν−1 − mf(µν−1).
+If m ∤ µν−1, then the above inequality is impossible since it implies that
+µν + 2 − m > µν−1 > µν.
+If m | µν−1, then the inequality is also impossible since it implies that µν + 2 >
+µν−1 so if we write µν−1 = k′m and µν = km, k, k′ ∈ N, k′ > k, we arrive at
+2 > (k′ − k)m ≥ m. This proves the following
+Lemma 29. The inequality (22)ν−1 might be correct only in cases where m | µν−1,
+m ∤ µν.
+Assume that for all ν inequality (23) holds. Then for ν = 1 it gives us (recall
+that µ ≤ d − 2)
+(25)
+µ + 2 ≤ d = i1 < µ1 + 2 − mf(µ1) = µ + 2 − mf(µ),
+which is impossible. Therefore either there are ν such that none of the two inequal-
+ities (22)ν, (23)ν hold (in this case the contribution to the sum is zero) or there are
+cases where (22) holds.
+The sumands appearing in eq. (24) can be zero, for example the sequence µ1 =
+m > µ2 = 1 with i2 = 2 < i1 = d, s = 2 give the contribution
+[λα
+µ2 − λx]i2−1
+1
+[λα
+µ1 − λx]d−1
+i2
+= [λα
+1 − λx]1
+1[λα
+m − λx]d−1
+i2+1 = (λ2 − λ1)[λ1 − λx]d−1
+3
+while for i2 = 1 < i1 = d it gives the contribution
+[λα
+µ2 − λx]i2−1
+1
+[λα
+µ1 − λx]d−1
+i2+1 = [λα
+1 − λx]0
+1[λα
+m − λx]d−1
+2
+= [λ1 − λx]d−1
+2
+It is clear that these non-zero contributions cancel out when added.
+Lemma 30. Assume that m | µν0−1 and m ∤ µν0, where (23)ν0 and (22)ν0−1 hold.
+Then, we can eliminate µν0−1 and iν0 from both selections of the sequence of µ’s
+and i’s, i.e. we can form the sequence of length s − 1
+¯µs−1 = µs < ¯µs−2 = µs−1 < · · · < ¯µν0−1 = µν0 < ¯µν0���2 = µν0−2 < · · · < ¯µ1 = µ1.
+and the corresponding sequence of equal length
+¯is−1 = is < ¯is−2 = is−1 < · · · < ¯iν0−1 = iν0−1 < ¯iν0 = iν0+1 < · · · < ¯i1 = i1 = d,
+so that
+Γ¯µ,i =
+�
+i1>···>is
+s
+�
+ν=1
+[λα
+µν − λx]iν−1
+iν+1+1 = (⋆)
+�
+¯i1>···>¯is−1
+s
+�
+ν=1
+ν̸=ν0−1
+[λα
+µν − λx]iν−1
+iν+1+1,
+where (⋆) is a non zero element.
+
+ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP.
+21
+Proof. (of lemma 30) We are in the case m | µν0−1 and m ∤ µν0, where (23)ν0 and
+(22)ν0−1 hold,
+(26)
+µν0−1 − m
+(22)ν0−1
+<
+iν0
+(23)ν0
+<
+µν0 + 2,
+or equivalently
+µ0 := µν0−1 − m + 1 ≤ iν0 ≤ µν0 + 1
+For iν0+1 the inequality (22)ν0 iν0+1 > µν0 − mf(µν0) can not hold, since it implies
+iν0+1 < iν0
+(23)ν0
+<
+µν0 + 2 < iν0+1 + 2.
+Observe that also
+iν0+1 + 1 ≤ iν0 ≤ iν0−1 − 1.
+Set l = max{µ0, iν0+1 + 1} and L = min{µν0 + 1, iν0−1 − 1}. Then y = iν0 satisfies
+l ≤ y ≤ L.
+By lemma 27 the quantity
+�
+l≤y≤L
+[λµν0+1 − λx]y−1
+iν0+1+1 · [λµ0 − λx]
+iν0−1−1
+y+1
+equals to
+[λµν0+1 − λx]l−1
+iν0+1+1 · [λµ0 − λx]
+iν0−1−1
+L+1
+· [λµν0+1 − λx]L
+l − [λµ0 − λx]L
+l
+(λµν0+1 − λµ0)
+(27)
+[λµν0+1 − λx]L
+iν0+1+1 · [λµ0 − λx]
+iν0−1−1
+L+1
+− [λµν0+1 − λx]l−1
+iν0+1+1 · [λµ0 − λx]
+iν0−1−1
+l
+(λµν0+1 − λµ0)
+Case A1 l = µ0 ≥ iν0+1 + 1. Then [λµ0 − λx]L
+l = 0.
+Case A2 l = iν0+1 + 1 > µ0. We set z := iν0+1, which is bounded by eq. (23)ν0+1
+that is
+µ0
+Case A2
+≤
+z
+(23)ν0+1
+≤
+µν0+1 + 1.
+Notice that in this case m ∤ µν0+1. Indeed, we have assumed that inequality (23)ν0+1
+holds wich gives us
+µν0−1 − m = µ0 − 1
+(Case A2)
+<
+iν0+1
+(23)ν0+1
+<
+µν0+1 + 2 − m,
+which implies that µν0−1 < µν0+1 + 2, a contradiction. Thus for l = z + 1 we
+compute
+�
+µ0≤z≤µν0+1+1
+[λα
+µν0+1 − λx]
+iν0+1−1
+iν0+2+1 · [λµ0 − λx]L
+l =
+=
+�
+µ0≤z≤µν0+1+1
+[λµν0+1+1 − λx]z−1
+iν0+2+1 · [λµ0 − λx]L
+z+1 =
+= (⋆) · [λµν0+1+1 − λx]
+µν0+1+1
+µ0
+− [λµ0 − λx]
+µν0+1+1
+µ0
+λµν0+1+1 − λµ0+1
+= 0.
+Case B1 L = µν0 + 1 ≤ iν0−1 − 1. In this case [λµν0+1 − λx]L
+l = 0.
+
+22
+A. KONTOGEORGIS AND A. TEREZAKIS
+Case B2 L = iν0−1 − 1 < µν0 + 1. In this case eq. (27) is reduced to
+[λµν0+1 − λx]
+iν0−1−1
+iν0+1+1
+(λµν0+1 − λµ0)
+This means that we have erased the µν0−1 from the product and we have
+�
+i1>···>is
+s
+�
+ν=1
+[λα
+µν − λx]iν−1
+iν+1+1 = (⋆)
+�
+i1>···>is
+s
+�
+ν=1
+ν̸=ν0−1
+[λα
+µν − λx]iν−1
+iν+1+1,
+where (⋆) is a non zero element. This procedure gives us that the original quantity
+[λα
+µν0 − λx]
+iν0−1
+iν0+1+1 · [λα
+µν0−1 − λx]
+iν0−1−1
+iν0+1
+after summing over iν0 becomes the quantity
+[λα
+µν0 − λx]
+iν0−1−1
+iν0+1+1 = [λα
+¯µν0−1 − λx]
+¯iν0−1−1
+¯iν0+1
+,
+that is we have eliminated the µν0−1 and iν0 from both selections of the sequence
+of µ’s and i’s, i.e. we have the sequence of length s − 1
+¯µs−1 = µs < ¯µs−2 = µs−1 < · · · < ¯µν0−1 = µν0 < ¯µν0−2 = µν0−2 < · · · < ¯µ1 = µ1.
+and the corresponding sequence of equal length
+¯is−1 = is < ¯is−2 = is−1 < · · · < ¯iν0−1 = iν0−1 < ¯iν0 = iν0+1 < · · · < ¯i1 = i1 = d.
+□
+Remark 31. One should be careful here since ¯iν0−1 = iν0−1 > iν0 > ¯iν0 = iν0+1,
+so ¯iν0−1 > ¯iν0 + 1. This means that the new sequence of ¯is−1 > · · · > ¯i1 satisfies a
+stronger inequality in the ν0 position, unless ν0 − 1 = d in the computation of γd,d.
+Consider the set s, s − 1, . . . , ν0 such that m ∤ µν for s ≥ ν ≥ ν0 and assume
+that m | µν0−1 and (23)ν0 and (22)ν0−1 hold. We apply lemma 30 and we obtain
+a new sequence of µ’s with µν0−1 removed, provided that ν0 − 1 > 1. We continue
+this way and in the sequence of µ’s we eliminate all possible inequalities like (26)
+obtaining a series of µ which involves only inequalities of type (23). But this is not
+possible if µ ≤ d − 2, according to equation (25). This proves that all γµ,d = 0 for
+1 ≤ µ ≤ d − 2, this completes the proof of lemma 28.
+□
+Lemma 32. If µ2 ̸= d − 1, then the contribution of the corresponding summand
+Γ¯µ,i to γd,d is zero.
+Proof. We are in the case µ = d = i. We begin the procedure of eliminating all
+sequences of inequalities of the form (23)ν0, (22)ν0−1, where m | ν0−1, m ∤ ν0, using
+lemma 30. For ν = 1 inequality (23)1 can not hold since it implies the impossible
+inequality d = i1 < d + 2 − m. Therefore, (22)1 holds, that is i2 > d − m. On the
+other hand we can assume that (23)2 holds by the elimination process, so we have
+d − m
+(22)1
+< i2
+(23)2
+< µ2 + 2.
+Following the analysis of the proof of lemma 28 we see that the contribution to γd,d
+is non zero if case B2 holds, that is (ν0 = 2 in this case) d − 1 = iν0−1 − 1 < µ2 + 1,
+obtaining that µ2 = d − 1.
+□
+
+ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP.
+23
+Lemma 33. Equation (21) holds, that is
+(λd − λα
+d )γd,d =
+d−1
+�
+ν=1
+t(α)
+d,νγν,d = t(α)
+d,d−1γd−1,d.
+Proof. We will use the procedure of the proof of lemma 30. We recall that for
+each fixed sequence of µs > · · · > µ1 we summed over all possible sequences i1 >
+· · · > is+1 = 0.
+In the final step the inequality (26) appears, for ν0 = 2, and
+µν0 = µ2 = d − 1 and ν0 − 1 = 1 and µν0−1 = µ = d, that is:
+0 = µν0−1 − m
+(22)2
+< iν0
+(23)1
+< µν0 + 2 = d + 1.
+As in the proof of lemma 30 we sum over y = iν and the result is either zero in case
+B1 or in the B2 case, where µν0 = µ2 = d − 1 and µ0 = µν0−1 − m + 1 = d − m + 1,
+the contribution is computed to be equal to
+[λα
+µν0+1 − λx]
+iν0−1−1
+iν0+1+1
+(λµν0+1 − λm0)
+= [λα
+d − λx]d−1
+i3+1
+λd − λα
+d
+.
+The last µν0−1 = µ1 = d is eliminated in the above expression. This means that
+for a fixed sequence µ1 > . . . > µs the contribution of the inner sum in eq. (24) is
+given by
+1
+λd − λα
+d
+·
+�
+d−1=i2>i3>···>is≥1
+s
+�
+ν=2
+[λα
+µν − λx]iν−1
+iν+1+1.
+Observe that µ1 = d does not appear in this expression and this expression corre-
+sponds to the sequence ¯µ1 = µ2 = d − 1 > ¯µ2 = µ3 > · · · > ¯µs−1 = ¯µs = 1. Notice,
+also that the problem described in remark 31 does not appear here, sence we erased
+i1 which is not between some i’s but the first one. Therefore, we can relate it to
+a similar expression that contributes to γd−1,d. Conversely every contribution of
+γd−1,d gives rise to a contribution in γd,d, by multiplying by λd − λα
+d . The desired
+result follows by the expression of γµ,d given in eq. (18).
+□
+We have shown so far how to construct matrices Γ, T so that
+(28)
+T q = 1, ΓTΓ−1 = T α.
+We will now prove that Γ has order m. By equation (28) Γk should satisfy equation
+ΓkTΓ−k = T αk.
+Using proposition 25 asserting the uniqueness of such Γk with α replaced by αk we
+have that the matrix multiplication of the entries of Γ giving rise to (γ(k)
+µ,i ) = Γk
+coincide to the values by the the recursive method of proposition (28) applied
+for Γ′ = Γk, α′ = αk and Γ′E1 = ζϵk
+m E1.
+In particular for k = m, we have
+αm ≡ 1 modpν for all 1 ≤ ν ≤ h, that is the matrix Γk should be recursively
+constructed using proposition (28) for the relation ΓmTΓm = T, ΓmE1 = E1,
+leading to the conclusion Γm = Id.
+Notice that the first eigenvalue of Γ is a
+primitive root of unity, therefore Γ has order exactly m.
+By lemma 10 the action of σ in the special fibre is given by a lower triangular
+matrix. Therefore, we must have
+(29)
+γν,i ∈ mr for ν < i.
+
+24
+A. KONTOGEORGIS AND A. TEREZAKIS
+Proposition 34. If
+(30)
+v(λi − λj) > v(aν) for all 1 ≤ i, j ≤ d and 1 ≤ ν ≤ d − 1
+then the matrix (γµ,i) has entries in the ring R and is lower triangular modulo mR.
+Proof. Assume that the condition of eq. (30) holds. In equation (18) we compute
+the fraction
+(31)
+[a]µ−1
+1
+[a]i−1
+1
+=
+�
+�
+�
+�
+�
+1
+[a]i−1
+µ
+if i > µ
+1
+if i = µ
+[a]µ−1
+i
+if i < µ
+The number of (λα
+µ − λx) factors in the numerator is equal to (recall that is+1 = 0)
+s
+�
+ν=1
+(iν − 1 − iν+1 − 1 + 1) = i − s,
+and i > µ ≥ s, so i − s > 0. Therefore, for the upper part of the matrix i > µ we
+have i − s factors of the form (λα
+i − λj) in the numerator and i − µ factors ax in
+the denominator. Their difference is equal to (i − s) − (i − µ) = µ − s ≥ 0. By
+assumption the matrix reduces to an upper triangular matrix modulo mR.
+□
+Remark 35. The condition given in equation (30) can be satisfied in the following
+way: It is clear that λi − λj ∈ mR. Even in the case vmR(λi − λj) = 1 we can
+consider a ramified extension R′ of the ring R with ramification index e, in order to
+make the valuation vmR′ (λi − λj) = e and then there is space to select vmR′ (ai) <
+vmR′ (λi − λj).
+Proposition 36. We have that
+(32)
+γi,i ≡ ζϵ
+mαi−1 modmR
+Let A = {a1, . . . , ad−1} ∈ R be the set of elements below the diagonal in eq. (9). If
+ai ∈ mR, then
+γµ,i ∈ mR for µ ̸= i,
+that is Ei is an eigenvector for the reduced action of Γ modulo mR. If aκ1, . . . , aκr
+the elements of the set A which are in mR, then the reduced matrix of Γ has the
+form:
+�
+�
+�
+�
+�
+�
+Γ1
+0
+· · ·
+0
+0
+Γ2
+...
+...
+...
+...
+...
+0
+0
+· · ·
+0
+Γr
+�
+�
+�
+�
+�
+�
+where Γ1, Γ2, . . . , Γr+1 for 1 ≤ ν ≤ r + 1 are (κν − κν−1) × (κν − κν−1) lower
+triangular matrices (we set κ0 = 0, κr+1 = d).
+
+ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP.
+25
+Proof. Consider the matrix Γ:
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+γ11
+...
+...
+0
+γκ1,1
+· · ·
+γκ1,κ1
+γ11
+γκ1+1,κ1+1
+...
+...
+γµ,i
+γκ2,κ1+1
+· · ·
+γκ2,κ2
+γκ1+1,κ1+1
+γµ,i
+...
+γκr+1,κr+1
+· · ·
+...
+...
+γκ1,κ1
+γκ2,κ2
+γd,κr+1
+· · · γd,d
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+1 ≤ i ≤ κ1 < m ≤ d
+κ1 < i ≤ κ2 < µ ≤ d
+We have that µ = i and the only element in Σµ which does not have any factor of
+the form (λα
+y − λx) is the sequence
+1 = µs = µs−1 − 1 < µs−1 < · · · < µ2 = µ1 − 1 < µ1 = µ
+For this sequence eq. (18) becomes
+γi,i =
+s
+�
+ν=2
+hα−1(λµν, λµν−1)ζϵ
+m modmR,
+which gives the desired result since hα−1(λµν, λµν−1) ≡
+�α
+1
+�
+= α modmR.
+For proving that all entries γµ,i ∈ mR for κν < i ≤ κν+1 < µ ≤ d, that is for
+all entries bellow the central blocks, we observe that from equation (18) combined
+with eq. (31) that γµ,i is divisible by [a]µ−1
+i
+= aiai+1 · · · aκν+1 · · · aµ−1 ∈ mR.
+□
+Recall that by lemma 2 there is an 1 ≤ a0 ≤ m such that α = ζa0
+m .
+Proposition 37. The indecomposable module V modulo mR breaks into a direct
+sum of r + 1 indecomposable k[Cq ⋊ Cm] modules Vν, 1 ≤ ν ≤ r + 1. Each Vν is
+isomorphic to Vα(ϵ + a0κν−1, κν − κν−1).
+Proof. By eq. (32) the first eigenvalue of the reduced matrix block Γν is
+ζϵ
+mακν−1 = ζϵ+(κν−1)a0
+m
+.
+Since that first eigenvalue together with the size of the block determine the last
+eigenvalue, that is the action of Cm on the socle the reduced block is uniquely
+determined up to isomorphism.
+□
+This way we arrive at a new obstruction.
+Assume that the indecomposable
+representation given by the matrix T as in lemma 16 reduces modulo mR to a sum
+of Jordan blocks. Then the σ action on the leading elements of each Jordan block
+in the special fibre should be described by the corresponding action of σ on the
+leading eigenvector E of V . The corresponding actions on the special fibre should
+be compatible.
+This observation is formally given in proposition 1, which we now prove: Each set
+Iν, 1 ≤ ν ≤ t corresponds to an indecomposable R[G]-module, which decomposes
+to the indecomposables Vα(ϵµ, κµ), ν ∈ Iν of the special fiber. Indecomposable
+
+26
+A. KONTOGEORGIS AND A. TEREZAKIS
+summands have different roots of unity in R, therefore �
+µ∈Iν kν ≤ q, this is con-
+dition (1.a). The second condition (1.b) comes from proposition 13. If 1 is one of
+the possible eigenvalues of the lift T, then �
+µ∈Iν κµ ≡ 1 modm. If all eigenvalues
+of the lift T are different than one, then �
+µ∈Iν κµ ≡ 0 modm. If #Iν = q, then
+there is one zero eigenvalue and the sum equals 1 modm.
+It is clear by eq. (32) that condition (1.c) is a necessary condition. On the other
+hand if (1.c) is satisfied we can write (after a permutation if necessary) the set
+{1, . . . , S}, S = �t
+ν=1 #Iν as
+J1 = {1, 2, . . . , κ(1)
+1 , κ(1)
+1
++ 1, . . . , κ(1)
+1
++ κ(1)
+2 , . . . ,
+r1
+�
+j=1
+κ(1)
+j
+= b1}, I1 = {κ(1)
+1 , . . . , κ(1)
+r1 }
+J2 = {b1 + 1, b1 + 2, . . . , b2 = b1 +
+r2
+�
+j=1
+κ(2)
+j }, I2 = {κ(2)
+1 , . . . , κ(2)
+r2 }
+· · · · · ·
+Js = {bs−1 + 1, bt−1 + 2, . . . , bt = S}, Is = {κ(s)
+1 , . . . , κ(s)
+rs }
+The matrix given in eq. (9), where
+ai =
+�
+�
+�
+�
+�
+0
+if i ∈ {b1, . . . , bs−1}
+π
+if i ∈ {κ(ν)
+1 , κ(ν)
+1
++ κ(ν)
+2 , κ(ν)
+1
++ κ(ν)
+2
++ κ(ν)
+3 , . . . , κ(ν)
+1
++ κ(ν)
+2
++ · · · + κ(ν)
+rν−1}
+1
+otherwise
+lifts the τ generator, and by (12) there is a well defined extended action of the σ
+as well.
+Example: Consider the group q = 52, m = 4, α = 7,
+G = C52 ⋊ C4 = ⟨σ, τ|σ4 = τ 25 = 1, στσ−1 = τ 7⟩.
+Observe that ord57 = ord527 = 4.
+• The module V (ϵ, 25) is projective and is known to lift in characteristic zero.
+This fits well with proposition 1, since 4 | 25 − 1 = 4 · 6.
+• The modules V (ϵ, κ) do not lift in characteristic zero if 4 ∤ κ or 4 ∤ κ − 1.
+Therefore only V (ϵ, 1), V (ϵ, 4), V (ϵ, 5), V (ϵ, 8), V (ϵ, 9), V (ϵ, 12), V (ϵ, 13),
+V (ϵ, 16), V (ϵ, 17), V (ϵ, 20), V (ϵ, 21), V (ϵ, 24), V (ϵ, 25) lift.
+• The module V (1, 2) ⊕ V (3, 2) lift to characteristic zero, where the matrix
+of T with respect to a basis E1, E2, E3, E4 is given by
+T =
+�
+�
+�
+�
+ζq
+0
+0
+0
+1
+ζ2
+q
+0
+0
+0
+π
+ζ3
+q
+0
+0
+0
+1
+ζ4
+q
+�
+�
+�
+�
+and σ(E1) = ζqE1.
+• The module V (1, 2) ⊕ V (1, 2) does not lift in characteristic zero. There is
+no way to permute the direct summands so that the eigenvalues of σ are
+given by ζϵ
+m, αζϵ
+m, α2ζϵ
+m, α3ζϵ
+m. Notice that α = 2 = ζm.
+• The module V (ϵ1, 21)⊕V (221·ϵ1, 23) does not lift in characteristic zero. The
+sum 21+24 is divisible by 4, ϵ2 = 221ϵ1 is compatible, but 21+23 = 44 > 25
+so the representation of T in the supposed indecomposable module formed
+
+ON THE LIFTING PROBLEM OF REPRESENTATIONS OF A METACYCLIC GROUP.
+27
+by their sum can not have different eigenvalues which should be 25-th roots
+of unity.
+References
+[1] J. L. Alperin. Local representation theory, volume 11 of Cambridge Studies in Advanced
+Mathematics. Cambridge University Press, Cambridge, 1986. Modular representations as an
+introduction to the local representation theory of finite groups.
+[2] Frauke M. Bleher, Ted Chinburg, and Aristides Kontogeorgis. Galois structure of the holo-
+morphic differentials of curves. J. Number Theory, 216:1–68, 2020.
+[3] T. Chinburg, R. Guralnick, and D. Harbater. Oort groups and lifting problems. Compos.
+Math., 144(4):849–866, 2008.
+[4] Ted Chinburg, Robert Guralnick, and David Harbater. The local lifting problem for actions
+of finite groups on curves. Ann. Sci. ´Ec. Norm. Sup´er. (4), 44(4):537–605, 2011.
+[5] Ted Chinburg, Robert Guralnick, and David Harbater. Global Oort groups. J. Algebra,
+473:374–396, 2017.
+[6] Huy Dang, Soumyadip Das, Kostas Karagiannis, Andrew Obus, and Vaidehee Thatte. Local
+oort groups and the isolated differential data criterion, 2019.
+[7] A. Heller and I. Reiner. Representations of cyclic groups in rings of integers. I. Ann. of Math.
+(2), 76:73–92, 1962.
+[8] A. Heller and I. Reiner. Representations of cyclic groups in rings of integers. II. Ann. of Math.
+(2), 77:318–328, 1963.
+[9] Sotiris Karanikolopoulos and Aristides Kontogeorgis. Representation of cyclic groups in pos-
+itive characteristic and Weierstrass semigroups. J. Number Theory, 133(1):158–175, 2013.
+[10] Aristides Kontogeorgis and Alexios Terezakis. The canonical ideal and the deformation theory
+of curves with automorphisms, 2021.
+[11] Andrew Obus. The (local) lifting problem for curves. In Galois-Teichm¨uller theory and arith-
+metic geometry, volume 63 of Adv. Stud. Pure Math., pages 359–412. Math. Soc. Japan,
+Tokyo, 2012.
+[12] Andrew Obus. A generalization of the Oort conjecture. Comment. Math. Helv., 92(3):551–
+620, 2017.
+[13] Andrew Obus and Rachel Pries. Wild tame-by-cyclic extensions. J. Pure Appl. Algebra,
+214(5):565–573, 2010.
+[14] Andrew Obus and Stefan Wewers. Cyclic extensions and the local lifting problem. Ann. of
+Math. (2), 180(1):233–284, 2014.
+[15] Florian Pop. The Oort conjecture on lifting covers of curves. Ann. of Math. (2), 180(1):285–
+322, 2014.
+[16] Jean-Pierre Serre. Linear representations of finite groups. Springer-Verlag, New York, 1977.
+Translated from the second French edition by Leonard L. Scott, Graduate Texts in Mathe-
+matics, Vol. 42.
+[17] Jean-Pierre Serre. Local fields. Springer-Verlag, New York, 1979. Translated from the French
+by Marvin Jay Greenberg.
+[18] Bradley Weaver. The local lifting problem for D4. Israel J. Math., 228(2):587–626, 2018.
+Department of Mathematics, National and Kapodistrian University of Athens Pane-
+pistimioupolis, 15784 Athens, Greece
+Email address: kontogar@math.uoa.gr
+Department of Mathematics, National and Kapodistrian University of Athens, Panepis-
+timioupolis, 15784 Athens, Greece
+Email address: aleksistere@math.uoa.gr
+

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+JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, FEB. 2019
+1
+Real-time Feedback Based Online Aggregate EV
+Power Flexibility Characterization
+Dongxiang Yan, Shihan Huang, and Yue Chen, Member, IEEE
+Abstract—As an essential measure to combat global warming,
+electric vehicles (EVs) have witnessed rapid growth. Meanwhile,
+thanks to the flexibility of EV charging, vehicle-to-grid (V2G)
+interaction has captured great attention. However, the direct con-
+trol of individual EVs is challenging due to their small capacity,
+large number, and private information. Hence, it is the aggregator
+that interacts with the grid on behalf of EVs by characterizing
+their aggregate flexibility. In this paper, we focus on the aggregate
+EV power flexibility characterization problem. First, an offline
+model is built to obtain the lower and upper bounds of the
+aggregate power flexibility region. It ensures that any trajectory
+within the region is feasible. Then, considering that parameters
+such as real-time electricity prices and EV arrival/departure
+times are not known in advance, an online algorithm is developed
+based on Lyapunov optimization techniques. We prove that the
+charging time delays for EVs always meet the requirement even
+if they are not considered explicitly. Furthermore, real-time
+feedback is designed and integrated into the proposed online
+algorithm to better unlock EV power flexibility. Comprehensive
+performance comparisons are carried out to demonstrate the
+advantages of the proposed method.
+Index Terms—Aggregate flexibility, charging station, electric
+vehicle, Lyapunov optimization, online algorithm.
+I. INTRODUCTION
+T
+HANKS to the low carbon emissions, electric vehicles
+(EVs) have been considered a promising solution to
+climate change and proliferate in recent years [1]. However,
+the uncontrolled charging of a large number of EVs can cause
+voltage deviation, line overload, and huge transmission loss
+[2], threatening the reliability of the power system. Unlike
+inelastic loads, the charging power and charging period of
+EVs are more flexible [3]. Therefore, unlocking the power
+flexibility hidden in EVs is a promising way to lessen the
+adverse impact of EVs on the power grid.
+There are extensive literature aiming to design coordinated
+charging strategies to optimally schedule EV charging. For
+example, to promote local renewable generation consumption,
+a dynamic charging strategy was proposed to allow the EV
+charging power to dynamically track the PV generation [4]
+and wind generation [5]. To save the electricity cost, a
+deterministic optimal charging strategy was proposed for a
+home energy management system based on the time-of-use
+tariffs [6]. A model predictive control (MPC) algorithm was
+proposed to minimize the operational cost of EV charging
+stations [7] relying on short-term forecasts. To address the
+uncertainties related to EV charging, reference [8] proposed a
+D. Yan, S. Huang, and Y. Chen are with the Department of Me-
+chanical and Automation Engineering, the Chinese University of Hong
+Kong,
+Hong
+Kong
+SAR,
+China
+(e-mail:
+dongxiangyan@cuhk.edu.hk,
+shhuang@link.cuhk.edu.hk, yuechen@mae.cuhk.edu.hk).
+stochastic charging strategy based on the probabilistic model
+related to EV daily travels. A combined robust and stochastic
+MPC method was developed in [9] to handle the uncertain EV
+charging behaviors and renewable generations. A multi-stage
+energy management strategy including day-ahead and real-
+time stages was developed for a charging station integrated
+with PV generation and energy storage [10]. In addition,
+a pricing mechanism was suggested in [11] to guide EVs
+for economical charging. A double-layer optimization model
+was built to reduce the voltage violations caused by EV
+charging [12]. Despite the efforts mentioned above that intend
+to determine the EV charging power, it is challenging to
+directly control a large number of individual EVs due to the
+high computational complexity.
+To get around this problem, some other literature en-
+deavored to characterize the EV charging power flexibility.
+Reference [13] proposed to model the aggregate EV charging
+flexibility region by the lower and upper bounds of power and
+cumulative energy. This aggregate EV model was adopted by
+[14] to evaluate the achievable vehicle-to-grid capacity of an
+EV fleet and by [15] to quantify the value of EV flexibility in
+terms of maintaining distribution system reliability. Reference
+[16] further considered the spatio-temporal distribution of the
+probability that an EV is available for charging during the
+aggregation and clustering processes. An EV dispatchable
+region was proposed to allow charging stations to participate in
+market bidding [17]. In addition, the aggregate flexibility issue
+was also studied in the fields of thermostatically controllable
+loads (TCLs) [18], distributed energy resources [19], and
+virtual power plant [20]. For example, a geometric approach
+was utilized to model the aggregate flexibility of TCLs [21].
+An inner box approximation method was proposed to charac-
+terize the power flexibility region of various distributed energy
+resources [22].
+The above works provide sound techniques for evaluating
+EV flexibility in an offline manner. It means that the aggregator
+is assumed to have complete information of future uncertainty
+realizations, e.g., EV arrival/departure time, and electricity
+prices. In practice, those data are usually unavailable or
+inaccurate, making the obtained region fail to reflect the actual
+EV aggregate flexibility in real-time. Thus, an online algorithm
+is desired. A straightforward approach is the greedy algorithm
+that decomposes the offline problem into subproblems in each
+time slot by neglecting the time-coupling constraints [23].
+Obviously, the result could be far from optimum. Hence, we
+resort to another approach, Lyapunov optimization, that can
+run in an online manner but with an outcome near to the
+offline optimum [24]. Lyapunov optimization has been used in
+arXiv:2301.03342v1  [math.OC]  9 Jan 2023
+
+JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, FEB. 2019
+2
+microgrid control [25], energy storage sharing [26], and data
+center energy management [27]. For EV charging, a charging
+strategy based on Lyapunov optimization was proposed to
+minimize the total electricity cost [28]. However, it cannot
+guarantee that EVs will depart with desired amount of energy.
+To meet the EV charging requirement, virtual delay queues
+were introduced to minimize the charging cost under uncertain
+renewable generations and electricity prices [29], [30].
+Though the optimal online EV charging strategy has been
+widely studied as above, the online aggregate EV power
+flexibility characterization problem has not been well explored
+yet. The latter problem is more complicated than the former
+one, requiring a new model and algorithm design. This paper
+proposes a real-time feedback based online aggregate EV
+power flexibility characterization method. Our main contribu-
+tions are two-fold:
+1) Model. We first propose an offline optimization model
+to characterize the aggregate EV power flexibility region.
+It decomposes the time-coupled flexibility region into each
+time slot and gives their lower and upper bounds. We prove
+that any trajectory within the region is achievable. Then,
+by categorizing the EVs according to their allowable time
+delays, we develop a counterpart of the offline model that
+enables the further utilization of the Lyapunov optimization
+framework. The proposed model has not been reported in
+previous literature.
+2) Algorithm. A real-time feedback based online algorithm
+is developed to derive the aggregate EV power flexibility
+region sequentially. First, to fit into the Lyapunov optimization
+framework, charging task queues and delay-aware virtual
+queues are introduced to reformulate the model. Then, a
+drift-plus-penalty term is constructed and by minimizing its
+upper bound, an online algorithm is developed. We prove
+that the charging time delays for EVs will not exceed their
+maximum allowable values even if they are not explicitly
+considered. The bound of optimality gap between the offline
+and online outcomes is provided theoretically. Furthermore,
+real-time dispatch strategy based feedback is designed and
+integrated into the online algorithm. The proposed real-time
+feedback based online algorithm is prediction free and can
+adapt to uncertainties such as random electricity prices and EV
+charging behaviors. Furthermore, it can make use of the most
+recent information, allowing it to even outperform the offline
+model with full knowledge of future uncertainty realizations
+but without the updated dispatch information.
+The rest of this paper is organized as follows. Section
+II formulates the offline model for deriving aggregate EV
+charging power flexibility region. Section III and IV introduce
+the Lyapunov optimization method and real-time feedback
+design, respectively, to generate flexibility region in an online
+manner. Simulation results are presented in Section V. Finally,
+Section VI concludes this paper.
+II. PROBLEM FORMULATION
+In this section, we first introduce the concept of aggregate
+EV power flexibility and then formulate an offline optimization
+problem to approximate it.
+A. Aggregate EV Charging Power Flexibility
+As shown in Fig. 1, when an EV v ∈ V arrives at the
+charging station, it submits its charging task to the aggregator.
+The task is described by (ta
+v, td
+v, ea
+v, ed
+v), where ta
+v is its arrival
+time, td
+v is its departure time, ea
+v is the initial battery energy
+level at ta
+v, and ed
+v is the desired energy level when it leaves.
+For the EV v, the maximum allowable charging time delay is
+Rv = td
+v−ta
+v. The EV charging task needs to be finished within
+this declared time duration. With the submitted information,
+the aggregator can flexibly schedule the EV charging to meet
+the charging requirement. Two possible trajectories to meet the
+EV charging need are depicted in Fig. 1. Let {pc
+v,t, ��t} be the
+charging power of EV v over time. The range that the charging
+power can vary within is called the power flexibility of EV
+v. If we sum the power flexibility of all EVs in a charging
+station up, we can get the aggregate EV power flexibility of
+the charging station.
+(𝑡1
+𝑎, 𝑡1
+𝑑,
+𝑒1
+𝑎, 𝑒1
+𝑑)
+(𝑡2
+𝑎, 𝑡2
+𝑑,
+𝑒2
+𝑎, 𝑒2
+𝑑)
+(𝑡𝑣𝑎, 𝑡𝑣𝑑, 
+𝑒𝑣𝑎, 𝑒𝑣𝑑)
+Aggregator
+Distribution System Operator
+Aggregate 
+dispatch power
+Aggregate power 
+flexibility region
+Ƽ𝑝𝑑,𝑡, Ƹ𝑝𝑑,𝑡
+𝑝𝑑,𝑡
+𝑑𝑖𝑠𝑝
+𝑝1,𝑡
+𝑑𝑖𝑠𝑝
+𝑝2,𝑡
+𝑑𝑖𝑠𝑝
+𝑝𝑣,𝑡
+𝑑𝑖𝑠𝑝
+EV 1
+EV 2
+EV v
+1. Generate EV aggregate power flexibility region
+2. Disaggregation
+Fig. 1. System diagram and illustration of EV power flexibility.
+However, it is difficult to characterize the EV power flexibil-
+ity for each time slot due to the temporal-coupled EV charging
+constraints. The EV power flexibility in the current time slot is
+affected by those in the past time slots and further affects those
+in the future time slots. This is different from the traditional
+controllable generators whose flexibility can be described by
+the minimum and maximum power outputs in each time slot.
+In the following, we aim to derive an aggregate EV power
+flexibility region that: 1) is time-decoupled so that it can be
+used in real-time power system operation; and 2) any trajectory
+within it can meet the EV charging requirement.
+B. Offline Problem Formulation
+Suppose there are T time slots, indexed by t ∈ T
+=
+{1, ..., T}. The desired time-decoupled aggregate EV power
+flexibility region can be represented by a series of intervals
+[ˇpd,t, ˆpd,t], ∀t ∈ T . The intervals can be specified by a lower
+power trajectory {ˇpd,t, ∀t} and an upper power trajectory
+{ˆpd,t, ∀t}. To obtain the lower and upper power trajectories,
+we formulate the following offline optimization problem:
+P1 :
+max
+ˆpd,t,ˇpd,t,∀t lim
+T →∞
+1
+T
+T
+�
+t=1
+E
+�
+Ft
+�
+,
+(1a)
+
+Power
+Aggregate power
+flexibility region
+flexibilitJOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, FEB. 2019
+3
+where
+Ft = πt(ˆpd,t − ˇpd,t), ∀t,
+(1b)
+subject to
+ˆpd,t =
+�
+v∈V
+ˆpc
+v,t, ∀t,
+(1c)
+0 ≤ ˆpc
+v,t ≤ pmax
+v
+, ∀v, ∀t,
+(1d)
+ˆev,t+1 = ˆev,t + ˆpc
+v,t∆t, ∀v, ∀t ̸= T,
+(1e)
+ˆev,tav = eini
+v , ˆev,tdv ≥ ed
+v, ∀v,
+(1f)
+emin
+v
+≤ ˆev,t ≤ emax
+v
+, ∀v, ∀t,
+(1g)
+ˇpd,t =
+�
+n∈V
+ˇpc
+v,t, ∀t,
+(1h)
+0 ≤ ˇpc
+v,t ≤ pmax
+v
+, ∀v, ∀t,
+(1i)
+ˇev,t+1 = ˇev,t + ˇpc
+v,t∆t, ∀v, ∀t ̸= T,
+(1j)
+ˇev,tav = eini
+v , ˇev,tdv ≥ ed
+v, ∀v,
+(1k)
+emin
+v
+≤ ˇev,t ≤ emax
+v
+, ∀v, ∀t,
+(1l)
+ˇpd,t ≤ ˆpd,t, ∀t,
+(1m)
+ˆpc
+v,t/ˇpc
+v,t =
+� ˆpc
+v,t/ˇpc
+v,t,
+if t ∈ [ta
+v, td
+v]
+0,
+if t < ta
+v ∪ t > td
+v
+.
+(1n)
+In the objective function (1a)-(1b), πt, ∀t are the real-time
+electricity prices, showing the unit value of power flexibility
+in different time slots. Hence, the objective function aims to
+maximize the value of total aggregate EV power flexibility.
+Constraint (1c) defines the upper bound of aggregate EV
+power flexibility region. The charging power of an EV v is
+limited by (1d), where pmax
+v
+is the maximum charging power.
+Constraint (1f) defines the EV’s initial energy level and the
+charging requirement. (1e) and (1g) describe the EV’s energy
+dynamics and battery capacity. Similarly, (1h)-(1l) are the
+constraints related to the lower bound of the aggregate EV
+power flexibility. (1m) is the joint constraint to ensure that
+{ˆpd,t, ∀t} and {ˇpd,t, ∀t} provide the upper and lower bounds,
+respectively. (1n) limits that charging only happens during the
+EV’s declared parking time.
+Proposition 1: Any aggregate EV charging power trajectory
+within [ˇpd,1, ˆpd,1] × · · · × [ˇpd,T , ˆpd,T ] is achievable.
+The proof of Proposition 1 can be found in Appendix A.
+Despite this nice property, the offline optimization problem
+above cannot be solved directly since it requires complete
+knowledge of the future EV charging tasks and future elec-
+tricity prices, which are usually not available in practice.
+Therefore, an online algorithm is necessary. To this end, in
+the next section, we will first propose a closely related but
+more flexible form of the problem studied. Then, we adopt the
+Lyapunov optimization framework to reformulate the offline
+problem into an online one. We construct charging task queues
+and delay-aware virtual queues to ensure the satisfaction of
+charging requirements. Furthermore, considering the impact
+of real-time dispatch decisions on the future aggregate EV
+power flexibility, a real-time feedback based online flexibility
+characterization method is developed in Section IV to avoid
+the potential underestimate of EV power flexibility.
+III. ONLINE ALGORITHM
+In this section, we adopt the Lyapunov optimization frame-
+work to solve the offline problem P1 in an online manner.
+The proposed algorithm can output an aggregate EV power
+flexibility value with an economic value close to P1.
+A. Problem Modification
+As mentioned above, the charging station serves dozens of
+EVs every day, and each EV arrives along with a charging task,
+i.e., (ta
+v, td
+v, ea
+v, ed
+v). Those EV charging tasks can be first stored
+in a queue and be served later according to a first-in-first-
+out basis. Since different EVs may have different allowable
+charging time delays, we use multiple queues to classify and
+collect the EV charging tasks. Suppose there are G types
+of charging time delays Rgs, each of which is indexed by
+g ∈ {1, 2, ..., G}. Correspondingly, we construct G queues to
+collect the respective charging tasks, and each queue is denoted
+by Qg. For queue Qg, Qg,t refers to its charging task backlog
+in time slot t. The queue backlog growth is described by
+Qg,t+1 = max[Qg,t − xg,t, 0] + ag,t,
+(2)
+where xg,t is the charging power for EVs in group g at time
+t, and ag,t is the arrival rate of EV charging tasks of group g
+at time t. In particular, ag,t sums up the energy demand of all
+EVs that arrive at the beginning of time t, i.e.,
+ag,t =
+�
+v∈Vg
+ag,v,t,
+(3)
+where ag,v,t is the charging demand of EV v of group g in
+time slot t. Vg is the set of EVs in group g.
+Recalling that our target in P1 is to derive an upper bound
+and a lower bound for the aggregate EV power flexibility
+region, we correspondingly define the upper bound queue ˆQg,t
+and the lower bound queue ˇQg,t. Similar to (2), we have
+ˆQg,t+1 = max[ ˆQg,t − ˆxg,t, 0] + ˆag,t,
+(4)
+ˇQg,t+1 = max[ ˇQg,t − ˇxg,t, 0] + ˇag,t,
+(5)
+where ˆxg,t and ˇxg,t are the charging power for upper and
+lower bound queues, respectively, i.e., ˆxg,t = �
+v∈Vg ˆpc
+v,t and
+ˇxg,t = �
+v∈Vg ˇpc
+v,t.
+The upper and lower bounds of arriving charging demand,
+i.e., ˆag,t and ˇag,t, are determined by
+ˆag,t =
+�
+v∈Vg
+ˆag,v,t, ˇag,t =
+�
+v∈Vg
+ˇag,v,t,
+(6)
+Particularly, the lower bound of arriving charging demand
+ˇag,v,t can be determined in the following charging as soon as
+possible way,
+ˇag,v,t =
+�
+�
+�
+pmax
+v
+,
+ta
+v ≤ t < ⌊ˇtmin
+v
+⌋ + ta
+v
+ˇecha
+v
+/ηc − ⌊ˇtmin
+v
+⌋pmax
+v
+,
+t = ⌊ˇtmin
+v
+⌋ + ta
+v
+0,
+otherwise
+,
+(7)
+where ˇecha
+v
+= ed
+v −ea
+v, ˇtmin
+v
+is the minimum required charging
+time determined by ˇtmin
+v
+=
+ˇecha
+v
+pmax
+v
+ηc , and ⌊.⌋ means rounding
+down to the nearest integer.
+
+JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, FEB. 2019
+4
+Different from ˇag,v,t, the upper bound of arrival charging
+demand ˆag,v,t is determined using the maximum charging
+demand emax
+v
+instead of ed
+v. Denote ˆecha
+v
+= emax
+v
+− ea
+v,
+ˆtmin
+v
+=
+ˆecha
+v
+pmax
+v
+ηc , and then ˆag,v,t can be determined by
+ˆag,v,t =
+�
+�
+�
+pmax
+v
+,
+ta
+v ≤ t < ⌊ˆtmin
+v
+⌋ + ta
+v
+ˆecha
+v
+/ηc − ⌊ˆtmin
+v
+⌋pmax
+v
+,
+t = ⌊ˆtmin
+v
+⌋ + ta
+v
+0,
+otherwise.
+(8)
+We then formulate the aggregate EV power flexibility char-
+acterization problem as follows:
+P2 :
+min
+ˆxg,t,ˇxg,t lim
+T →∞
+1
+T
+T
+�
+t=1
+E
+�
+− Ft
+�
+,
+(9a)
+subject to
+lim
+T →∞
+1
+T
+T
+�
+t=1
+E[ˆag,t − ˆxg,t] ≤ 0, ∀g
+(9b)
+lim
+T →∞
+1
+T
+T
+�
+t=1
+E[ˇag,t − ˇxg,t] ≤ 0, ∀g
+(9c)
+0 ≤ ˆxg,t ≤ min{xg,max, ˆQg,t}, ∀g
+(9d)
+0 ≤ ˇxg,t ≤ min{xg,max, ˇQg,t}, ∀g
+(9e)
+ˆxg,t ≥ ˇxg,t, ∀g
+(9f)
+where xg,max = �
+v∈Vg pmax
+v
+. Constraint (9b) ensures that if
+using the upper bound trajectory {ˆxg,t, ∀t}, the total charg-
+ing requirement can be satisfied in the long run. Constraint
+(9c) poses a similar requirement for the lower bound trajec-
+tory {ˇxg,t, ∀t}. Based on (9b), (9c), and the definitions of
+ˆQg,t+1, ˇQg,t+1 in (4)-(5), we can prove that the queues ˆQg,t
+and ˇQg,t are mean rate stable. To be specific,
+ˆQg,t+1 − ˆag,t ≥ ˆQg,t − ˆxg,t, ∀t
+(10)
+Summing (10) up over all t and divide both sides by T yields
+0 ≤ E[ ˆQg,T ]
+T
+≤
+�T
+t=1 E[ˆag,t − ˆxg,t]
+T
+(11)
+Hence, lim
+T →∞
+E[ ˆ
+Qg,T ]
+T
+= 0. Similarly, lim
+T →∞
+E[ ˇ
+Qg,T ]
+T
+= 0.
+Constraints (9d) and (9e) give the upper and lower bounds
+of the aggregate EV charging power for group g, respectively.
+The upper bound is no less than the lower bound, as shown in
+(9f). The P2 provides a counterpart problem for P1. Similar
+to the proof of Proposition 1, we can prove that any trajectory
+between [ˇxg,t, ˆxg,t] is achievable. However, the allowable
+charging delay is not considered in P2, which may result in
+unfulfilled EV charging tasks upon departure.
+B. Construct Virtual Queues
+To overcome the aforementioned charging delay issue, we
+introduce delay-aware virtual queues,
+ˆZg,t+1 = max{ ˆZg,t + ηg
+Rg
+I ˆ
+Qg,t>0 − ˆxg,t, 0}, ∀g, ∀t
+(12)
+ˇZg,t+1 = max{ ˇZg,t + ηg
+Rg
+I ˇ
+Qg,t>0 − ˇxg,t, 0}, ∀g, ∀t
+(13)
+where I ˆ
+Qg,t>0 and I ˇ
+Qg,t>0 are indicator functions of ˆQg,t and
+ˇQg,t, respectively. They are equal to 1 if there exists unserved
+charging tasks in the queues, i.e., ˆQg,t > 0 and ˇQg,t > 0.
+Using ηg
+Rg to times it, this whole term constitutes a penalty to
+the virtual queue backlog. ηg is a user-defined parameter that
+can adjust the growth rate of the virtual queues. For instance,
+increasing the value of ηg leads to a fast queue growth and a
+larger backlog value, calling for more attention to accelerate
+the charging process. We prove that, when Qg,t and Zg,t have
+finite upper bounds, with a proper ηg, the charging time delay
+for EVs in group g is bounded.
+Proposition 2: Suppose ˆQg,t, ˇQg,t, ˆZg,t, and ˇZg,t have finite
+upper bounds, e.g., ˆQg,t ≤ ˆQg,max, ˇQg,t ≤ ˇQg,max ˆZg,t ≤
+ˆZg,max, and ˇZg,t ≤ ˇZg,max. The charging time delay of all
+EVs in group g is upper bounded by ˆδg,max and ˇδg,max time
+slots, where
+ˆδg,max := ( ˆQg,max + ˆZg,max)Rg
+ηg
+,
+(14)
+ˇδg,max := ( ˇQg,max + ˇZg,max)Rg
+ηg
+.
+(15)
+The proof of Proposition 2 can be found in Appendix B.
+It ensures that the charging tasks can always be fulfilled
+within the available charging periods by properly setting the
+parameters ηg, ∀g.
+C. Lyapunov Optimization
+Based on the charging task queues and delay-aware virtual
+queues, the Lyapunov optimization framework is applied as
+follows.
+1) Lyapunov
+Function:
+First,
+we
+define
+Θt
+=
+( ˆ
+Qt, ˆ
+Zt, ˇ
+Qt, ˇ
+Zt) as the concatenated vector of queues,
+where
+ˆ
+Qt = ( ˆQ1,t, ..., ˆQG,t),
+(16a)
+ˆ
+Zt = ( ˆZ1,t, ..., ˆZG,t),
+(16b)
+ˇ
+Qt = ( ˇQ1,t, ..., ˇQG,t),
+(16c)
+ˇ
+Zt = ( ˇZ1,t, ..., ˇZG,t).
+(16d)
+The Lyapunov function is then defined as
+L(Θt) = 1
+2
+�
+g∈G
+ˆQ2
+g,t + 1
+2
+�
+g∈G
+ˆZ2
+g,t + 1
+2
+�
+g∈G
+ˇQ2
+g,t + 1
+2
+�
+g∈G
+ˇZ2
+g,t,
+(17)
+where L(Θt) can be considered as a measure of the queue
+size. A smaller L(Θt) is preferred to push (virtual) queues
+ˆQg,t, ˆZg,t, ˇQg,t, and ˇZg,t to be less congested.
+2) Lyapunov Drift: The conditional one-time slot Lyapunov
+drift is defined as follows:
+∆(Θt) = E[L(Θt+1) − L(Θt)|Θt],
+(18)
+where the expectation is taken with respect to the random Θt.
+The Lyapunov drift is a measure of the expectation of the
+queue size growth given the current state Θt. Intuitively, by
+minimizing the Lyapunov drift, virtual queues are expected
+to be stabilized. However, only minimizing the Lyapunov
+drift may lead to a low aggregate EV power flexibility value.
+
+JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, FEB. 2019
+5
+Therefore, we include the expected aggregate flexibility value
+(1b) for the time slot t to (18). The drift-plus-penalty term is
+obtained, i.e.,
+∆(Θt) + V E[−Ft|Θt],
+(19)
+where V is a weight parameter that controls the trade-off
+between (virtual) queues stability and aggregate EV power
+flexibility maximization.
+3) Minimizing the Upper Bound: (19) is still time-coupled
+due to the definition of ∆(Θt). To adapt to online imple-
+mentation, instead of directly minimizing the drift-plus-penalty
+term, we minimize the upper bound to obtain the upper and
+lower bounds of aggregate EV power flexibility region. We
+first calculate the one-time slot Lyapunov drift:
+L(Θt+1) − L(Θt)
+= 1
+2
+�
+g∈G
+� �
+ˆQ2
+g,t+1 − ˆQ2
+g,t
+�
++
+�
+ˆZ2
+g,t+1 − ˆZ2
+g,t
+�
++
+� ˇQ2
+g,t+1 − ˇQ2
+g,t
+�
++
+� ˇZ2
+g,t+1 − ˇZ2
+g,t
+� �
+.
+(20)
+Use queue ˆQg,t as an example, based on the queue update
+equations (4), we have
+ˆQ2
+g,t+1 = {max[ ˆQg,t − ˆxg,t, 0] + ˆag,t}2
+≤ ˆQ2
+g,t + ˆa2
+g,max + ˆx2
+g,max + 2 ˆQg,t(ˆag,t − ˆxg,t).
+(21)
+Thus,
+1
+2
+�
+ˆQ2
+g,t+1 − ˆQ2
+g,t
+�
+≤ 1
+2
+�
+ˆx2
+g,max + ˆa2
+g,max
+�
++ ˆQg,t (ˆag,t − ˆxg,t) .
+(22)
+Similarly, for queue ˇQg,t, ˆZg,t, and ˇZg,t, we have
+1
+2
+� ˇQ2
+g,t+1 − ˇQ2
+g,t
+�
+≤ 1
+2
+�
+ˇx2
+g,max + ˇa2
+g,max
+�
++ ˇQg,t (ˇag,t − ˇxg,t) .
+(23)
+1
+2[ ˆZ2
+g,t+1 − ˆZ2
+g,t] ≤ 1
+2 max[( ηg
+Rg
+)2, ˆx2
+g,max]
++ ˆZg,t[ ηg
+Rg
+− ˆxg,t].
+(24)
+1
+2[ ˇZ2
+g,t+1 − ˇZ2
+g,t] ≤ 1
+2 max[( ηg
+Rg
+)2, ˇx2
+g,max]
++ ˇZg,t[ ηg
+Rg
+− ˇxg,t].
+(25)
+We then substitute inequalities (22),(23), (24) and (25) into
+drift-plus-penalty term and yield
+∆(Θt) + V E[−Ft|Θt]
+≤ A + V E[−Ft|Θt] +
+�
+g∈G
+ˆQg,tE [ˆag,t − ˆxg,t|Θt]
++
+�
+g∈G
+ˇQg,tE [ˇag,t − ˇxg,t|Θt] +
+�
+g∈G
+ˆZg,tE [−ˆxg,t|Θt]
++
+�
+g∈G
+ˇZg,tE [−ˇxg,t|Θt] ,
+(26)
+where A is a constant, i.e.,
+A = 1
+2
+�
+g∈G
+(ˆx2
+g,max + ˆa2
+g,max) + 1
+2
+�
+g∈G
+max[( ηg
+Rg
+)2, ˆx2
+g,max]
++ 1
+2
+�
+g∈G
+(ˇx2
+g,max + ˇa2
+g,max) + 1
+2
+�
+g∈G
+max[( ηg
+Rg
+)2, ˇx2
+g,max]
++
+�
+g∈G
+[ ˆZg,max
+ηg
+Rg
+] +
+�
+g∈G
+[ ˇZg,max
+ηg
+Rg
+].
+By reorganizing the expression in (26) and ignoring the con-
+stant terms, we can obtain the following online optimization
+problem,
+P3 :
+min
+ˆxg,t,ˇxg,t,∀g,∀t
+�
+g��G
+(−V πt − ˆQg,t − ˆZg,t)ˆxg,t
++
+�
+g∈G
+(V πt − ˇQg,t − ˇZg,t)ˇxg,t,
+(27)
+s.t. (9d) − (9f),
+where ˆQg,t, ˇQg,t, ˇZg,t, and ˇZg,t are first updated based on
+(4),(5),(12), and (13) before solving P3 in each time slot.
+In each time slot t, given the current system queue state
+Θt, the proposed method determines the current upper and
+lower aggregate EV power flexibility bounds ˆxg,t and ˇxg,t by
+solving problem P3. Hence, the original offline optimization
+problem P1 has been decoupled into simple online (real-time)
+problems, which can be executed in each time slot without
+requiring prior knowledge of future uncertain states. Since
+the modified problem P3 is slightly different from the offline
+one P1, an important issue we care about is: what’s the gap
+between the optimal solutions of the online problem P3 the
+and offline problem P1?
+Proposition 3: Denote the obtained long-term time-average
+aggregate EV power flexibility value of P1 and P3 by F ∗
+and F pro, respectively. We have
+0 ≤ −F pro + F ∗ ≤ 1
+V A,
+(28)
+where A is a constant defined in (26).
+The proof of Proposition 3 can be found in Appendix C.
+The optimality gap can be controlled by the parameter V . A
+bigger V value leads to a smaller optimality gap but increased
+queue sizes. In contrast, a smaller V value makes the queues
+more stable but results in a larger optimality gap.
+IV. DISAGGREGATION
+AND REAL-TIME FEEDBACK DESIGN
+In each time slot t, given the aggregate EV power flexibility
+region [�
+g ˇx∗
+g,t, �
+g ˆx∗
+g,t], the distribution system operator
+(DSO) can determine the optimal aggregate dispatch strategy
+for EVs. This aggregate dispatch strategy should be further
+disaggregated to obtain the control strategy for each EV,
+which is studied in this section. Moreover, considering that
+the current dispatch strategy will influence the aggregate EV
+power flexibility in future time slots, real-time feedback is
+designed and integrated with the proposed online flexibility
+characterization method in Section III.
+
+JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, FEB. 2019
+6
+A. Disaggregation
+Suppose the dispatch strategy in time slot t is pdisp
+agg,t ∈
+[�
+g ˇx∗
+g,t, �
+g ˆx∗
+g,t]. This can be determined by the DSO by
+solving an economic dispatch problem based on the up-to-
+date information (e.g., the electricity price πt, the grid-side
+renewable generation). Since this paper focuses on the online
+characterization of aggregate EV power flexibility, the eco-
+nomic dispatch problem by the DSO is omitted for simplicity.
+Interested readers can refer to [31].
+Let the dispatch ratio αt be
+αt =
+pdisp
+agg,t − �
+g ˇx∗
+g,t
+�
+g ˆx∗
+g,t − �
+g ˇx∗
+g,t
+.
+(29)
+Then, the dispatched power pdisp
+g,t
+for each group g can be
+determined according to the ratio, i.e.,
+pdisp
+g,t
+= (1 − αt)ˇx∗
+g,t + αtˆx∗
+g,t,
+(30)
+which satisfies
+pdisp
+agg,t =
+�
+g
+pdisp
+g,t .
+The next step is to allocate pdisp
+g,t
+to the EVs in the group
+g. All EVs in the group g are sorted according to their arrival
+time. Then, we follow the first-in-first-service principle to
+allocate the energy; namely, the EV that comes earlier will
+be charged with the maximum charging power. We have
+pdisp
+v,t
+= min
+�
+pdisp
+g,t , pmax
+v
+, emax
+v
+− ev,t
+∆t
+�
+, ∀v ∈ Vg,
+(31)
+where Vg refers to the set of EVs in group g. The third term
+on the right side is used to ensure that the EV will not exceed
+its allowable maximum energy level.
+After this charging assignment for an earlier EV is com-
+pleted, the following update procedures will execute
+pdisp
+g,t
+← (pdisp
+g,t
+− pdisp
+v,t ),
+(32)
+which means deducting pdisp
+v,t
+from the total remaining dis-
+patched power pdisp
+g,t .
+Then, the pdisp
+g,t
+is allocated to the next earlier arrival EVs
+until the aggregate EV charging power is completely assigned.
+At this time, the disaggregation is finished.
+The dispatched power disaggregation algorithm is presented
+in Algorithm 1.
+B. State Update to Improve Power Flexibility Region
+Following the disaggregation procedures in Algorithm 1, we
+can get the actual EV dispatched charging power pdisp
+v,t , ∀v.
+By now, we can move on to the next time slot t + 1 and
+evaluate the EV power flexibility by solving problem P3,
+determine the dispatch strategy, disaggregate the dispatched
+power, and so on. But considering that the current actual
+dispatched EV charging power can affect the future aggregate
+EV power flexibility, which is ignored in the aforementioned
+processes. Therefore, we propose a real-time feedback method
+to integrate the current actual EV dispatched charging power
+into the future aggregate EV power flexibility characterization.
+Algorithm 1 EV Dispatched Power Disaggregation
+1: Initialization: aggregate EV dispatched power pdisp
+agg,t.
+2: Calculate the dispatched aggregate charging power pdisp
+g,t
+for each group g using (30).
+3: for Each group g ∈ G do
+4:
+for Each EV v in group g do
+5:
+if the EV is not available for charging then
+6:
+Let EV v’s charging power pdisp
+v,t
+= 0, ∀v.
+7:
+else
+8:
+Calculate pdisp
+v,t
+according to (31).
+9:
+Update the remaining aggregate power via (32).
+10:
+if If the updated pdisp
+g,t
+= 0 then
+11:
+Break and return to Step 3.
+12:
+end if
+13:
+end if
+14:
+end for
+15: end for
+Disaggregation
+State update
+Dispatch
+Aggregate power flexibility 
+region
+State 
+feedback
+t=t+1
+Fig. 2. Overall procedure of the proposed method.
+The overall procedure is shown in Fig. 2 with the right-hand
+side blue box showing the real-time feedback.
+To be specific, we change the constraints (9d)-(9e) for time
+slot t into
+ˆxg,t = pdisp
+g,t , ˇxg,t = pdisp
+g,t .
+(33)
+Since pdisp
+g,t
+∈ [ˇx∗
+g,t, ˆx∗
+g,t], after replacing (9d)-(9e) with (33),
+the problem P3 is still solvable and the optimal solution is
+ˆxupdate∗
+g,t
+= pdisp
+g,t , ˇxupdate∗
+g,t
+= pdisp
+g,t , ∀g. With these updated
+lower and upper bounds, we update the queues ˆQg,t+1, ˇQg,t+1,
+ˆZg,t+1, ˇZg,t+1 according to (4), (5), (12) and (13), respectively.
+Then, we move on the solve problem P3 for time slot t + 1
+using the updated ˆQg,t+1, ˇQg,t+1, ˆZg,t+1, ˇZg,t+1.
+So far, we have developed a real-time feedback based online
+aggregate EV power flexibility characterization method as well
+as the EV dispatched charging power disaggregation approach.
+A completed description of the proposed method is shown in
+Algorithm 2.
+V. SIMULATION RESULTS AND DISCUSSIONS
+In this section, we evaluate the performance of the proposed
+online algorithm and compare it with other approaches.
+A. System Setup
+The time resolution is set as 10 minutes. The entire sim-
+ulation duration considered is 24 hours, i.e., 144 time slots
+
+JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, FEB. 2019
+7
+Algorithm 2 Real-time Feedback Based Online Aggregate EV
+Power Flexibility Characterization and Disaggregation
+I. Aggregation
+1: Aggregator classifies the arriving EVs and pushes them
+into different queues ˆQg, ˇQg, ˆZg and ˇZg according to
+their declared charging delay time Rg.
+2: Solve problem P3 and obtain the aggregate EV power
+flexibility region [ˇx∗
+g,t, ˆx∗
+g,t].
+3: Update queues ˆQg,t+1, ˇQg,t+1, ˆZg,t+1, and ˇZg,t+1 ac-
+cording to (4), (5), (12) and (13), respectively.
+II. Dispatch and Disaggregation
+4: Receive the dispatch decision from the DSO, and decom-
+pose it to each group according to (29) and (30).
+5: Perform EV dispatched power disaggregation according to
+Algorithm 1.
+III. Real-Time Feedback and Update
+6: Update the lower and upper bounds ˆxupdate∗
+g,t
+, ˇxupdate∗
+g,t
+, ∀g
+of EV aggregate power flexibility region.
+7: Update queues ˆQg,t+1, ˇQg,t+1, ˆZg,t+1, and ˇZg,t+1 ac-
+cording to (4), (5), (12) and (13), respectively.
+8: Move to the next time slot t = t+1, and repeat the above
+steps I-III.
+each with a time interval of 10 minutes. To reflect the actual
+fluctuations in the electricity prices, we use the real-time
+electricity price data obtained from the PJM market [32]. The
+dynamic electricity price data profile is shown in Fig. 3. For
+the setting of EVs, we consider 30 EVs that are divided into
+three groups with different allowable charging time delays, i.e.,
+G = 3. Each group has 10 EVs. The EVs in the same group
+have identical allowable time delay, i.e., Rg. Particularlly,
+R1 = 8 hours, R2 = 6 hours, and R3 = 7 hours. In addition,
+for EV battery parameters, we refer to the Nissan Leaf EV
+model with a battery pack of 40 kWh and a maximum charging
+power of 6.6 kW [33]. Considering that the EV charging
+behavior is uncertain, the EVs’ arriving times are randomly
+generated. The initial battery energy level of each EV is
+selected from a uniform distribution in [0.3, 0.5] × 40 kWh
+randomly [29]. We set the required state-of-charge (SOC) 1
+upon departure as 0.5 and the maximum SOC upon departure
+as 0.9. The weight parameter value of V is chosen as 1000,
+and the value of ηg is set as 648, 540, and 756 for the three
+groups, respectively.
+Fig. 3. Real-time electricity price profile.
+1The SOC of an EV is the ratio between the battery energy level and the
+battery capacity.
+B. Effectiveness of the proposed method
+We first show how the obtained aggregate EV power
+flexibility region looks like. Since the power grid dispatch
+determined by the DSO is beyond the scope of this paper,
+here the dispatch ratio αt in (29) is assumed to be randomly
+generated within the range of [0, 1] in each time slot, as shown
+in Fig. 4. Based on the generated dispatch ratio, we apply
+the proposed online flexibility characterization method and
+real-time feedback in turns (as in Algorithm 2) to obtain the
+aggregate EV power flexibility region (grey area) over time for
+each group and the charging station as a whole. The results are
+shown in Fig. 5. As seen, the power flexibility region varies
+over time. This is because EVs dynamically arrive and leave.
+Fig. 4. Randomly generated dispatch ratio αt.
+Fig. 5. The obtained aggregate EV power flexibility region.
+To validate the effectiveness of the proposed algorithm, dis-
+aggregation of the dispatched EV charging power is performed
+and we check if the SOC curves of EVs satisfy the charging
+requirements. Here, if the final EV SOC value can reach
+or exceed the EV owner’s requirement (SOC ≥ 0.5) upon
+leaving, then it means that the proposed method is effective.
+The left of Fig. 6 shows the actual EV charging SOC curves
+under the randomly generated dispatch ratio αt in Fig. 4. Each
+curve represents an EV. As we can see from the figure, all
+EVs’ final SOC is between 0.58 and 0.7, greater than the
+required value 0.5 and less than the maximum value 0.9. The
+right-hand side of Fig. 6 shows the number of delayed time
+slots (NDTS) to reach the requirement SOC=0.5. We can find
+that the maximum NDST is 10 for group 1, 7 for group 2,
+and 12 for group 3. All of them are within their respective
+declared allowable charging delay, i.e. Rg. This validates the
+proposed algorithm in providing maximum power flexibility
+while meeting the charging requirement.
+Furthermore, Fig. 7 shows the queue backlog evolution of
+the three groups over time. Taking group 1 for example, the
+lower and upper bound queues ˇQ1 and ˆQ1 first increase be-
+cause EVs continue to arrive with their charging tasks pushing
+
+JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, FEB. 2019
+8
+Rg=8h=48 
+time slots
+Rg=6h=36 
+time slots
+Rg=7h=42 
+time slots
+Fig. 6.
+EV charging SOC of each group and the number of delayed time
+slots needed to reach SOC=0.5.
+into the queues. Then as time moves on, through the EV
+charging dispatch pdisp
+v,t , ∀v, ∀t determined by disaggregation,
+the EV SOC gradually increases and reaches the minimum
+charging requirement 0.5. Hence, the lower bound queue ˇQ1
+becomes zero because the ˇag,t, ∀g, ∀t is set using the charging
+as soon as possible method to meet the minimum charging
+requirement as in (7). The upper bound queue ˆQ1 is still larger
+than zero since the EV SOC has not reach the maximum value
+0.9 (see Fig. 6), so there is charging flexibility. At the same
+time, since ˆQ1 is nonnegative, the delay aware upper bound
+queue ˆZ1 keeps growing, aiming to increase the charging
+power. The queue evolution in groups 2 and 3 can be analyzed
+similarly.
+Fig. 7. Queue backlog of each group.
+C. Performance Evaluation
+To show the advantage of the proposed online algorithm,
+two widely used benchmarks in the literature are performed.
+• Benchmark 1 (B1): This is a greedy algorithm that EVs
+start charging at the maximum charging power upon
+arrival. Let us denote the arrival time as t0. When the EV
+SOC reaches the minimum charging requirement 0.5, the
+lower bound of charging power ˇpd,t turns to be zero (time:
+t1), and the upper bound of charging power ˆpd,t remains
+the maximum charging power until the EV SOC reaches
+the maximum value 0.9 (time: t2). The aggregate power
+flexibility region for [t0, t1] is empty and for t ∈ [t1, t2] is
+the region between 0 and the maximum charging power.
+• Benchmark 2 (B2): This is the offline method. It directly
+solves P1 to obtain the aggregate EV power flexibility
+regions over the whole time horizon by assuming known
+future information. Though not realistic, it provides a
+theoretical benchmark to verify the performance of other
+methods. But it is worth noting that since it does not
+take into account the real-time actual dispatch strategy
+when calculating the aggregate flexibility, its performance
+may be worse than the proposed real-time feedback based
+method even though it is an offline method.
+Fig. 8 shows the accumulated flexibility values (�t
+τ=1 Fτ)
+under the three different methods, and TABLE I summarizes
+the total flexibility value (�T
+t=1 Ft) under different methods.
+The B1, i.e., greedy algorithm, has the worst performance and
+the lowest total flexibility value due to the myopic strategy. For
+B2, since it has complete future knowledge of EV behaviors
+and real-time electricity prices, it outperforms B1. However,
+this method is usually impossible in practice since the accurate
+future information is hardly available. Though predictions on
+future uncertainty realizations may be obtained, the potential
+prediction errors limit B2’s performance. In contrast, the
+proposed online algorithm achieves the best performance with
+the highest total power flexibility value. This is owing to the
+fact that it runs in a online manner with real-time feedback
+that allows it to utilize the most recent dispatch information
+to update its state. In addition, compared to the offline method
+B2, it does not require prior knowledge of future information
+or forecasts, which is more practical.
+Fig. 8. Accumulated flexibility value under different methods.
+TABLE I
+TOTAL FLEXIBILITY VALUE COMPARISON BETWEEN B1, B2, AND THE
+PROPOSED ALGORITHM (UNIT: USD).
+Methods
+B1
+B2
+Proposed
+Value
+517.69
+586
+647.21
+Improvement
+-
+13.2%
+25%
+The above result is obtained under the random dispatch ratio
+αt (see Fig. 4). In fact, the dispatch ratio can affect the actual
+charging power of each EV and further affect their aggregate
+power flexibility. Therefore, it is interesting to investigate the
+
+JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, FEB. 2019
+9
+impact of αt on the aggregate EV power flexibility (or the total
+power flexibility value (9)). Here, we use a uniform α over
+time, i.e., αt = α, ∀t. We change α from 0 to 1 and record the
+total power flexibility value in Fig. 9. As seen, the total power
+flexibility value depends on the dispatch ratio α. Generally,
+a larger α leads to a larger power flexibility value. However,
+this increase is nonlinear. When the dispatch ratio α exceeds
+0.5, the total power flexibility value no longer increases. In the
+extreme case when α = 0, the total power flexibility value is
+520 USD, which is still larger than the greedy algorithm B1.
+In addition, it can be concluded that if the average dispatch
+ratio α is greater than 0.2, then the proposed online algorithm
+more likely outperforms the offline method. This demonstrates
+the advantage of the proposed algorithm. We also present the
+aggregate EV power flexibility region under different α in Fig.
+10. As α decreases, the aggregate EV power flexibility region
+gradually narrows. This is because under a low dispatch ratio,
+the EVs are charged at a low charging rate and more likely to
+fail to meet the charging requirement; hence, the lower bound
+of aggregate EV power flexibility region is raised to ensure
+the EVs can meet the charging requirement in the remaining
+time.
+Fig. 9. The impact of α on total power flexibility value.
+0
+50
+Power
+[kW]
+ub
+lb
+=0
+0
+50
+Power
+[kW]
+ub
+lb
+=0.1
+0
+50
+Power
+[kW]
+ub
+lb
+=0.2
+0
+50
+Power
+[kW]
+ub
+lb
+=0.3
+0
+50
+Power
+[kW]
+ub
+lb
+=0.4
+0
+20
+40
+60
+80
+100
+120
+140
+Time [10 min]
+0
+50
+Power
+[kW]
+ub
+lb
+=0.5
+0
+50
+Power
+[kW]
+ub
+lb
+=0
+0
+50
+Power
+[kW]
+ub
+lb
+=0.1
+0
+50
+Power
+[kW]
+ub
+lb
+=0.2
+0
+50
+Power
+[kW]
+ub
+lb
+=0.3
+0
+50
+Power
+[kW]
+ub
+lb
+=0.4
+0
+20
+40
+60
+80
+100
+120
+140
+Time [10 min]
+0
+50
+Power
+[kW]
+ub
+lb
+=0.5
+Fig. 10. The aggregate EV power flexibility under different α.
+D. Impact of Parameters
+According to (19), the parameter V controls the trade-
+off between stabilizing the queues and maximizing the total
+power flexibility value in the objective function (27). Here, we
+change the value of V to investigate its impact on the total
+power flexibility value. As shown in Fig. 11, the total power
+flexibility value becomes larger with an increasing V .
+Fig. 11. The impact of V on the total power flexibility value.
+Fig. 12. The impact of V on the average time delay.
+Fig. 12 depicts the impact of V on the number of time slots
+needed for EVs to meet their required battery energy level ed
+v.
+We calculate the maximum/minimum/average number of time
+slots for the EVs in each group. As seen, with the growth
+of V , the number of delayed time slots slightly increases.
+This is because a larger V means putting more emphasis on
+maximizing the total power flexibility, which may result in a
+reduced lower bound ˇxg,tof the aggregate EV power flexibility
+region. Consequently, the charging time needed to reach the
+required energy level becomes longer. Comparing the three
+groups, we can find that the time delays of groups 1 and 3 are
+generally longer than that of group 2, which is owing to the
+shorter allowable time delay Rg.
+Fig. 13 shows the impact of ηg on the total flexibility value
+and the number of time slots needed for EVs to meet their
+required battery energy level ed
+v. We can find that a larger ηg
+results in a lower total power flexibility value and less number
+of delayed time slots. This is because a larger ηg forces the
+virtual delay-aware queue ˇZg to grow rapidly, allowing EVs
+to get charged quickly. Meanwhile, the power flexibility is
+sacrificed.
+VI. CONCLUSION
+With the proliferation of EVs, it is necessary to better
+utilize their charging power flexibility, making them valuable
+resources rather than burdens on the power grid. This paper
+proposes a real-time feedback based online aggregate EV
+power flexibility characterization method. It can output the
+aggregate flexibility region for each time slot in an online
+manner, with a total flexibility value over time similar to the
+offline counterpart. We prove that by choosing an aggregate
+
+JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, FEB. 2019
+10
+Fig. 13.
+The impact of ηg on the total flexibility value and average time
+delay.
+dispatch strategy within the obtained flexibility region for
+each time slot, the corresponding disaggregated EV control
+strategies allow all EVs to satisfy their charging requirements.
+Simulations demonstrate the effectiveness and benefits of the
+proposed method. It is worth noting that the proposed method
+can even outperform the offline method in some cases since
+it can utilize up-to-date dispatch information via real-time
+feedback. Future research may take into account the conflicting
+interests between the operator, aggregator, and EVs when
+deriving the flexibility region.
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+APPENDIX A
+PROOF OF PROPOSITION 1
+Let {pd,t, ∀t} be the aggregate power trajectory. For each
+time slot t ∈ T , since pd,t ∈ [ˇp∗
+d,t, ˆp∗
+d,t], we can define an
+
+JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, FEB. 2019
+11
+auxiliary coefficient:
+βt :=
+ˆp∗
+d,t − pd,t
+ˆp∗
+d,t − ˇp∗
+d,t
+∈ [0, 1]
+(A.1)
+so that pd,t = βtˇp∗
+d,t + (1 − βt)ˆp∗
+d,t. Then, we can construct a
+feasible EV dispatch strategy by letting
+pc
+v,t = βtˇpc∗
+v,t + (1 − βt)ˆpc∗
+v,t,
+(A.2a)
+ev,t = βtˇec∗
+v,t + (1 − βt)ˆec∗
+v,t.
+(A.2b)
+for all time slots t ∈ T .
+We prove that it is a feasible EV dispatch strategy as
+follows,
+pd,t = βtˇp∗
+d,t + (1 − βt)ˆp∗
+d,t
+= βt
+�
+v∈V
+ˇpc∗
+v,t + (1 − βt)
+�
+v∈V
+ˆpc∗
+v,t
+=
+�
+v∈V
+�
+βtˇpc∗
+v,t + (1 − βt)ˆpc∗
+v,t
+�
+=
+�
+v∈V
+pc
+v,t
+(A.3)
+Hence, constraint (1c) holds for pd,t and pc
+v,t, ∀v. Similarly,
+we can prove that constraints (1d)-(1g) are met. Therefore,
+we have constructed a feasible EV dispatch strategy, which
+completes the proof.
+■
+APPENDIX B
+PROOF OF PROPOSITION 2
+Here, we use the contradiction. If a charging request ˆag,t
+arrives in time slot t cannot be fulfilled on or before time
+slot t + ˆδg,max. Then, queue ˆQg,τ > 0always holds for τ ∈
+[t + 1, ..., t + ˆδg,max]. Thus, we have I ˆ
+Qg,t>0 = 1. According
+to delay virtual queue dynamics (12), for all τ ∈ [t+1, ..., t+
+ˆδg,max], we have
+ˆZg,τ+1 ≥ ˆZg,τ + ηg
+Rg
+− ˆxg,τ, ∀g, ∀t.
+(B.1)
+By summing the above inequalities over τ ∈ [t + 1, ..., t +
+ˆδg,max], we have
+ˆZg,t+ˆδg,max+1 − ˆZg,t+1 ≥ ηg
+Rg
+ˆδg,max +
+t+ˆδg,max
+�
+τ=t+1
+(−ˆxg,τ).
+(B.2)
+Since ˆZg,t+ˆδg,max+1 ≤ ˆZg,max and ˆZg,t+1 ≥ 0, we have
+ˆZg,max ≥ ηg
+Rg
+ˆδg,max +
+t+ˆδg,max
+�
+τ=t+1
+(−ˆxg,τ).
+(B.3)
+Since the charging tasks are processed in a first-in-first-out
+manner, and the charging request is not fulfilled by t+ˆδg,max,
+we have
+t+ˆδg,max
+�
+τ=t+1
+(ˆxg,τ) < ˆQg,max
+(B.4)
+Combining the above two inequalities, we obtain
+ˆZg,max > ηg
+Rg
+ˆδg,max − ˆQg,max,
+(B.5)
+which implies
+ˆδg,max < ( ˆQg,max + ˆZg,max)Rg
+ηg
+.
+(B.6)
+However, this result contradicts the definition of ˆδg,max in
+(14). Therefore, the worst case delay should be less than or
+equal to ˆδg,max as defined in (14).
+The proof of (15) follows a similar procedure, and we omit
+it here for brevity.
+■
+APPENDIX C
+PROOF OF PROPOSITION 3
+Denote the solution of P3 by the proposed algorithm by
+ˆxpro
+g,t and ˇxpro
+g,t , and the optimal solution of P1 by ˆx∗
+g,t and
+ˇx∗
+g,t. According to (26), we have
+∆(Θt) + V E[−F pro
+t
+|Θt]
+≤ A + V E[−F pro
+t
+|Θt] +
+�
+g∈G
+ˆQg,tE
+�
+ˆag,t − ˆxpro
+g,t |Θt
+�
++
+�
+g∈G
+ˇQg,tE
+�
+ˇag,t − ˇxpro
+g,t |Θt
+�
++
+�
+g∈G
+ˆZg,tE
+�
+−ˆxpro
+g,t |Θt
+�
++
+�
+g∈G
+ˇZg,tE
+�
+−ˇxpro
+g,t |Θt
+�
+,
+≤ A + V E[−F ∗
+t |Θt] +
+�
+g∈G
+ˆQg,tE
+�
+ˆag,t − ˆx∗
+g,t|Θt
+�
++
+�
+g∈G
+ˇQg,tE
+�
+ˇag,t − ˇx∗
+g,t|Θt
+�
++
+�
+g∈G
+ˆZg,tE
+�
+−ˆx∗
+g,t|Θt
+�
++
+�
+g∈G
+ˇZg,tE
+�
+−ˇx∗
+g,t|Θt
+�
+,
+≤ A + V E[−F ∗
+t |Θt]
+(C.1)
+The result is based on the fact that
+lim
+T →∞
+1
+T
+T
+�
+t=1
+E [ˆag,t − ˆxg,t|Θt] ≤ 0
+(C.2)
+lim
+T →∞
+1
+T
+T
+�
+t=1
+E [ˇag,t − ˇxg,t|Θt] ≤ 0
+(C.3)
+lim
+T →∞
+1
+T
+T
+�
+t=1
+E [−ˆxg,t|Θt] ≤ 0
+(C.4)
+lim
+T →∞
+1
+T
+T
+�
+t=1
+E [−ˇxg,t|Θt] ≤ 0
+(C.5)
+which is due to constraints (9b)-(9e).
+By summing the above inequality (C.1) over time slots t ∈
+{1, 2, . . . , T}, we have
+T
+�
+t=1
+V E[−F pro
+t
+]
+≤ AT + V
+T
+�
+t=1
+E[−F ∗
+t ] − E[L(ΘT +1)] + E[L(Θ1)].
+
+JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, FEB. 2019
+12
+Based on the fact that L(ΘT +1) and L(Θ1) are finite, we
+divide both sides of the above inequalities by V T and let
+T → ∞, then we have
+lim
+T →∞
+1
+T
+T
+�
+t=1
+E(−F pro
+t
+) ≤ A
+V + lim
+T →∞
+1
+T
+T
+�
+t=1
+E(−F ∗
+t ),
+which completes the proof.
+■
+

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+arXiv:2301.04477v1  [gr-qc]  11 Jan 2023
+Reconstruction Methods and the Amplification of the Inflationary Spectrum
+Leonardo Chataignier,∗ Alexander Yu. Kamenshchik,† Alessandro Tronconi,‡ and Giovanni Venturi§
+Dipartimento di Fisica e Astronomia, Università di Bologna, via Irnerio 46, 40126 Bologna, Italy
+I.N.F.N., Sezione di Bologna, I.S. FLAG, viale B. Pichat 6/2, 40127 Bologna, Italy
+We analyze the consequences of different evolutions of the Hubble parameter on the spectrum of
+scalar inflationary perturbations. The analysis is restricted to inflationary phases described by a
+transient evolution, when uncommon features arise in the inflationary spectra which may lead to
+an amplitude enhancement. We then discuss how the spectrum is, respectively, amplified or blue-
+tilted in the presence or absence of a growing solution of the Mukhanov-Sasaki equation. The cases
+of general relativity with a minimally coupled inflaton and that of induced gravity are considered
+explicitly. Finally, some remarks on constant roll inflation are discussed.
+I.
+INTRODUCTION
+The possibility that a large amount of the Dark Matter (DM) content in our Universe is made of (primordial)
+black holes (PBHs) has been seriously considered in the last few years [1]. This idea seems compelling because it
+could improve our understanding of cosmological evolution and, in particular, of inflation [2]. Moreover, the PBH
+hypothesis is also intriguing due to the increasing amount of direct and indirect observations of black holes (BHs)
+out of the astrophysical range, as well as the current lack of evidence for particle models of DM that go beyond the
+Standard Model of particle physics.
+According to the present observational bounds [3], it is possible that even the whole DM content of the Universe
+today is comprised of PBHs originated from the collapse of matter overdensities in a certain wavelength interval
+of inflationary perturbations. In this scenario, the abundance of PBHs is related to the amplitude of the inflaton
+fluctuations, the enhancement of which must be by several orders of magnitude with respect to (w.r.t.) the amplitude
+probed by Cosmic Microwave Background (CMB) radiation. Nonetheless, the microscopic physics that originate such
+a mechanism of amplification is still debated. For example, the amplification needed can be generated by a phase of
+ultra slow roll (USR) inflation in the presence of an inflection point of the inflaton potential [4]. This USR phase
+is the consequence of a transient period of inflatonary evolution, when slow-roll conditions are violated, and the
+inflaton then relaxes towards a de Sitter attractor. In contrast to the case of the fluctuations imprinted in the CMB,
+the perturbations [5] do not freeze at horizon exit in this case, as a growing solution of the Mukhanov-Sasaki (MS)
+equation is present, and it is responsible for the amplification of the modes. Other possibilities have been considered
+in the literature, such as an inflationary model able to generate a blue-tilted spectrum without the presence of the
+growing solution [6].
+In this article, these two mechanisms of amplification are considered. Instead of analyzing the possible consequences
+of different inflationary models obtained by varying the form of the inflaton-gravity action, we shall here consider
+different evolutions of the Hubble parameter and correspondingly obtain the inflaton action. Within this approach,
+even if the inflaton potential cannot be exactly reconstructed, the features of the resulting spectra can still be
+calculated, and one may verify whether their amplitude is amplified. For simplicity, our starting point is the case
+of a minimally coupled inflaton, then some non-minimally coupled models are also investigated. Moreover, different
+techniques for the reconstruction are adopted.
+The article is organised as follows. In Section II, we review the general formalism of the dynamics of the inflationary
+perturbations adopting a slightly unconventional formalism, and we derive the conditions for the existence of a growing
+solution in the MS equation or simply a blue-tilted spectrum in the absence of this solution. Furthermore, a useful
+relation between the odd and even slow-roll (SR) parameters in a certain hierarchy is obtained. This relation is valid
+for transient periods described by a certain time evolution, and it will be employed across the entire article. In Section
+III, different models are analysed, and the procedure for reconstruction is illustrated. In Section IV, the application
+of the formalism to constant roll inflation is studied. Finally, the conclusions are drawn in Section V.
+∗ leonardo.chataignier@unibo.it
+† kamenshchik@bo.infn.it
+‡ tronconi@bo.infn.it
+§ giovanni.venturi@bo.infn.it
+
+2
+II.
+INFLATIONARY PERTURBATIONS
+Let us first review the formalism of the inflationary perturbations. On adopting a slightly unconventional approach,
+we find the conditions which must hold in order to have an amplification of the inflationary spectrum either as the
+wavenumber k grows or as time evolves. In a realistic inflationary scenario, wherein amplification starts at some
+given time, both mechanisms essentially lead to an enhancement of the shortest wavelength part of the spectrum
+(k > kCMB).
+The conditions are then expressed in a model-independent form, which is valid provided the SR
+parameters are “constant”, and we use it in what follows to discuss different scenarios.
+In general, after some manipulations, the Mukhanov-Sasaki (MS) equation takes the following form
+v′′
+k +
+�
+k2 − z′′
+z
+�
+vk = 0 ,
+(1)
+where the prime denotes the derivative w.r.t. conformal time η and z is a time-dependent function that depends on
+the specific model of inflation. For example, in the case of general relativity (GR) with a minimally coupled inflaton,
+one has z = a√ǫ1, which leads to (see e.g. [7])
+z′′
+z = a2H2
+�
+2 − ǫ1 + ǫ2
+�3
+2 + ǫ2
+4 − ǫ1
+2 + ǫ3
+2
+��
+≡ a2H2fMS(ǫi) ,
+(2)
+with ǫ1 = − ˙H/H2, ǫi+1 = ǫ−1
+i dǫi/dN for i > 0 and N = ln a. The infinite set of ǫi’s form the so-called hierarchy of
+“Hubble flow functions” of SR parameters. It is important to note that, depending on the model of inflation, other
+hierarchies are commonly used, and they are associated with the evolution of different (homogeneous) degrees of
+freedom.
+In general, one has
+z′′
+z ≡ a2H2fMS ,
+(3)
+where fMS is a dimensionless quantity that takes a different form depending on the inflationary model. It can then
+be expressed as a function of the SR parameters ǫi’s.
+It is now convenient to define the new independent variable ξ = k/(aH), where dξ/dη = −aH(1 − ǫ1)ξ < 0 during
+inflation. Due to
+d
+dη = −aH(1 − ǫ1)ξ d
+dξ
+(4)
+and
+d2
+dη2 = a2H2(1 − ǫ1)2
+�
+ξ2 d2
+dξ2 +
+ǫ1ǫ2
+(1 − ǫ1)2 ξ d
+dξ
+�
+,
+(5)
+we are led to
+�
+ξ2 d2vk
+dξ2 +
+ǫ1ǫ2
+(1 − ǫ1)2 ξ dvk
+dξ
+�
++ ξ2 − fMS(ǫi)
+(1 − ǫ1)2
+vk = 0 .
+(6)
+On rewriting the MS equation in terms of ξ one eliminates its explicit dependence on aH.
+In the regime where the SR parameters are constant, and in the long wavelength limit (ξ → 0) Eq. (6) can be
+algebraically solved and the features of the primordial spectra can be derived in a straightforward manner. Indeed,
+in this limit, the two independent solutions of Eq. (6) have the form vk = ξα, where α satisfies the algebraic equation
+α2 +
+�
+ǫ1ǫ2
+(1 − ǫ1)2 − 1
+�
+α − fMS(ǫi)
+(1 − ǫ1)2 = 0 ,
+(7)
+with
+α1,2 =
+−
+�
+ǫ1ǫ2
+(1−ǫ1)2 − 1
+�
+±
+��
+ǫ1ǫ2
+(1−ǫ1)2 − 1
+�2
++ 4 fMS(ǫi)
+(1−ǫ1)2
+2
+.
+(8)
+
+3
+For instance, when fMS is defined by Eq. (2), and in the pure de Sitter case (ǫi = 0), we obtain
+α1,2 = 1 ± 3
+2
+.
+(9)
+For this case, the positive solution, α1 = 2, decreases in time, while the negative solution, α2 = −1, increases, and it
+remains nontrivial in the limit ξ → 0, which leads to
+vk,dS ∼ k−1/2
+� k
+aH
+�−1
+, Rk,dS ∼ k−3/2 aH
+z
+= k−3/2H ,
+(10)
+where Rk ≡ vk/z is the curvature perturbation (z = a in the de Sitter case), and the prefactor k−1/2 is essentially
+fixed by the initial (Bunch-Davies) conditions. The quantity Rk is independent of time, and the spectral index can
+be straightforwardly computed to be
+ns − 1 = d ln ∆2
+s
+d ln k
+,
+(11)
+with ∆2
+s ≡ |Rk,dS|2k3/(2π2), which leads to the well known de Sitter result (ns − 1)dS = 0.
+In the SR case (|ǫi| ≪ 1), the SR parameters can be approximated by constants and the expressions (8) are still
+valid but must be expanded to first order for consistency. One then obtains
+α1,2 = 1 ± √9 + 12ǫ1 + 6ǫ2
+2
+≃ 1 ± (3 + 2ǫ1 + ǫ2)
+2
+,
+(12)
+which implies (ns − 1)SR = −2ǫ1 − ǫ2.
+We note that there is a caveat one must take into account for USR. In this case, one finds the same solutions for
+the α’s as the de Sitter case, but the definition of the curvature perturbations is different, since zUSR ∝ a√ǫ1 → 0.
+Then, the amplitude of primordial curvature perturbations depends on time and is amplified. In the USR case, the
+spectral index cannot be calculated analytically with the same procedure as illustrated for de Sitter and SR.
+One can better illustrate the differences among the three cases just mentioned by solving the equation for Rk,
+R′′
+k + 2z′
+z Rk + k2Rk = 0 .
+(13)
+In terms of ξ, Eq. (13) can be conveniently rewritten as
+ξ2 d2Rk
+dξ2
++
+�
+ǫ1ǫ2 − 2 (1 − ǫ1) d ln z
+dN
+(1 − ǫ1)2
+�
+ξ dRk
+dξ
++
+ξ2
+(1 − ǫ1)2 Rk = 0 .
+(14)
+In GR with a minimally coupled inflaton, we have d ln z/dN = 1 + ǫ2/2. Then, for constant SR parameters and in
+the long wavelength limit, the last term is negligible, and the equation admits a constant solution and a solution
+proportional to ξβ, where
+β = 3 − 4ǫ1 + ǫ2 + ǫ1 (ǫ1 − 2ǫ2)
+(1 − ǫ1)2
+.
+(15)
+If ξβ decreases in time, the constant solution dominates in the ξ → 0 limit. This is what happens for de Sitter and
+SR. In contrast, if ξβ increases in time, it dominates in the ξ → 0 limit. This is what occurs for USR leading to
+results that are very different from de Sitter and SR, namely, an amplitude of the spectrum that increases in time.
+The non-constant solution is
+Rk ∝
+� k
+aH
+�β
+∼ e−β(1−ǫ1) N ,
+(16)
+and it increases or decreases depending on the sign of
+Φ ≡ β (1 − ǫ1) = 3 − 4ǫ1 + ǫ2 + ǫ1 (ǫ1 − 2ǫ2)
+(1 − ǫ1)
+,
+(17)
+
+4
+increasing if Φ < 0 and decreasing if Φ > 0. Only in the latter case the spectral index of primordial spectrum can be
+analytically calculated by using the definition (11). For a general inflationary model, one finds
+∆2
+s ∝ k2+2α2 = k
+2−
+�
+ǫ1ǫ2
+(1−ǫ1)2 −1
+�
+−
+��
+ǫ1ǫ2
+(1−ǫ1)2 −1
+�2+4 fMS(ǫi)
+(1−ǫ1)2 ,
+(18)
+and
+ns − 1 = 2 −
+�
+ǫ1ǫ2
+(1 − ǫ1)2 − 1
+�
+−
+��
+ǫ1ǫ2
+(1 − ǫ1)2 − 1
+�2
++ 4 fMS(ǫi)
+(1 − ǫ1)2 .
+(19)
+A.
+Evolutions with “Constant” SR Parameters
+Let us now illustrate an important point.
+The results obtained above are exact when the SR parameters are
+constant. However, given the recursive definition of the SR parameters (ǫi+1 = ǫ−1
+i dǫi/dN), a constant set of ǫi’s
+corresponds either to H = const and ǫi = 0 (de Sitter case), or ǫ1 = const and ǫi = 0 for i > 1 (power law inflation).
+It may thus seem redundant to present the general formalism for such a restricted range of applications. Nevertheless,
+we note that the above results can be applied to a wider set of problems. First, as we already mentioned, the general
+results for Φ and ns − 1 can be applied to the SR case, in which the expressions must be expanded to the first order
+for consistency, since the SR parameters are approximately constant when they are small. Furthermore, the large a
+limit of some transient phase (such as the USR phase) leads to non-trivial sequences of “constant” SR parameters. In
+these cases, one obtains a hierarchy of, for example, ǫi’s with constant, non-zero SR parameters for either even or odd
+values of i, while the remaining SR parameters are zero. For instance, let ǫi
+N→∞
+=
+li +Li(N) with limN→∞ Li(N) = 0.
+Then, due to their recursive definition, one obtains
+ǫiǫi+1 ≡ dǫi
+dN
+a→∞
+=
+Li,N(N) ,
+(20)
+which leads to limN→∞ ǫi+1 = 0, provided limN→∞ Li,N(N) = 0, and, in particular,
+ǫi+1
+N→∞
+=
+Li,N(N)
+li + Li(N) .
+(21)
+Moreover,
+ǫi+2 ≡ dǫi+1/dN
+ǫi+1
+N→∞
+=
+Li,NN(N)
+Li,N(N) + ǫi+1 .
+(22)
+Let us now suppose Li(N) ∝ e−γN ∼ a−γ, with γ > 0. In this case:
+ǫi+2
+N→∞
+=
+−γ + ǫi+1 ,
+(23)
+and the subsequent terms of the hierarchy take values equal to zero and −γ:
+lim
+N→∞ ǫi = li,
+lim
+N→∞ ǫi+1+2n = 0,
+lim
+N→∞ ǫi+2n = −γ .
+(24)
+Therefore, due to their definition, an infinite sequence of SR parameters may take alternate “constant” values in the
+large a limit. This property is crucial in the analysis that follows, and it depends on the form of Li(N). Indeed,
+exponential forms lead to the result (24) but, in contrast, if Li ∝ N −γ, then the sequence obtained is limN→∞ ǫj = 0
+for j > i.
+It is also worthwhile to mention that similar results can be generalized to other hierarchies of SR parameters because
+they only depend on the recursive definition of the SR parameters [analogously to Eq. (20)] and on the form of Li.
+For example, the same results can be extended to the hierarchy of “scalar field flow functions” that is defined by
+δ0 = φ/φ0 and δiδi+1 = dδi/dN. In general, the ǫi’s and the δi’s are related through the homogeneous Friedmann
+and Klein-Gordon equations, and, in some scenarios, it is useful to use one or both hierarchies.
+
+5
+III.
+MODEL RECONSTRUCTION
+We are interested in reconstructing scalar field potentials that describe transient inflationary solutions, which are
+associated with varying SR parameters with a “constant” behaviour in the future (and necessarily ǫ1 < 1). Therefore,
+the results illustrated in the previous section can be adopted to study such models and to verify whether they can
+generate an amplification of the primordial spectrum. Finding the entire evolution of the scalar field is not necessary
+for this purpose, and we will only calculate the potential and the asymptotic behaviour of the homogeneous quantities
+in terms of the corresponding SR parameters. The potentials that lead to an amplification can then be used to build
+an inflationary model that fits the CMB observations and which produces a large amount of DM in the form of PBHs
+at the end of inflation.
+A.
+GR with a Minimally Coupled Inflaton
+In order to proceed with the reconstruction, let us first briefly review the homogeneous Einstein equation,
+H2 =
+1
+3MP
+2
+�1
+2
+˙φ2 + V (φ)
+�
+,
+(25)
+˙H = −
+˙φ2
+2MP
+2 ,
+(26)
+which leads to
+MP
+2H2 (3 − ǫ1) = V .
+(27)
+This last equation can be used to reconstruct the potential. The Eqs. (26) and (27) can be conveniently used for the
+reconstructions starting from some ansatz for H = H(a). In this case, Eq. (26) becomes
+ǫ1 =
+1
+2MP
+2
+� dφ
+d ln a
+�2
+,
+(28)
+which can be integrated to obtain, when possible, a = a(φ).
+Let us first consider the following evolution of the Hubble constant:
+H = H0
+�
+α + A
+an
+�m
+,
+(29)
+where A, α, n > 0. Similarly to USR, the evolution described by Eq. (29) has a de Sitter attractor in the future,
+and, indeed, H(a) is that of USR when n = 6 and m = 1/2. (It is interesting to note that this evolution represents
+a general solution in the model with a minimally coupled scalar field and a constant potential, or, in other words, in
+a universe driven by a mixture of two fluids: a cosmological constant and stiff matter. It is curious that n = 6 and
+m = 1/4 yield the general solution for the universe driven by the Chaplygin gas [8].) We also note that the transient
+is described by A/an ∼ e−nN and that a result similar to Eq. (24) is then expected. This is easily verified if we
+explicitly calculate the hierarchy of SR parameters:
+ǫ1 = m · n
+A
+αan + A = m ǫ3 = m ǫ5 = . . .
+a→+∞
+−→ 0 ,
+(30)
+and
+ǫ2 = −n
+αan
+αan + A = ǫ4 = ǫ6 = . . .
+a→+∞
+−→ −n ,
+(31)
+where a > [(m · n − 1)A/α]1/n is necessary for inflation to occur. We can integrate and invert Eq. (28) to obtain
+exp
+�φ − φ0
+MP
+� n
+2m
+�
+= x + 1
+x − 1
+x0 − 1
+x0 + 1 ,
+(32)
+
+6
+with x ≡ A−1/2√
+αan + A and x, x0 > 1. Notice that φ = φ0 when x = x0. Conversely, φ = φ∞, with
+φ∞ ≡ φ0 + MP
+�
+2m
+n ln B0 ,
+(33)
+for x → ∞. Eq. (32) can be solved for x, which yields
+x = e
+φ−φ0
+MP
+√ n
+2m + B0
+e
+φ−φ0
+MP
+√ n
+2m − B0
+,
+(34)
+where B0 = (x0 − 1)/(x0 + 1), and the reconstructed potential is finally
+V = H2
+0
+� αx2
+x2 − 1
+�2m �
+3 − n · m
+x2
+�
+.
+(35)
+For n = 6 and m = 1/2, one recovers a constant potential and the USR evolution, as expected. For other choices of the
+parameters n and m, the expression for the potential in terms of φ is a complicated function with exponentials that
+need not be written here explicitly. However, this cumbersome expression is exact. Since the asymptotic behaviour
+of the potential at φ ∼ φ∞ determines the limiting values of the SR parameters, we simply give the form of V around
+φ∞, which is
+V ≃ 3H2
+0α2m
+�
+1 + n
+4
+�
+1 − n
+2
+� �φ − φ∞
+MP
+�2�
+.
+(36)
+Finally, let us calculate the consequences of the background evolution given by Eq; (29) on the inflationary spectrum.
+The value of Φ is
+Φ = 3 − 4ǫ1 + ǫ2 + ǫ1 (ǫ1 − 2ǫ2)
+(1 − ǫ1)
+a→+∞
+−→ 3 − n ,
+(37)
+and for n > 3 the curvature perturbations Rk are amplified, after their horizon exit, as time passes. In contrast, if
+0 < n < 3, from the constant solution for Rk, one finds
+ns − 1 = n > 0 ,
+(38)
+which implies the amplitude is that of a blue-tilted spectrum, which grows as the wavenumber k increases.
+We
+conclude that for GR with a minimally coupled inflaton, the inflationary evolution described by Eq. (29), with a
+transient phase and a de Sitter attractor in the future, leads to an inflationary enhancement. The corresponding
+inflaton dynamics is driven by the potential (35) and similar behaviours can be obtained from potentials of the form
+(36) with the field close to φ∞.
+B.
+Power Law solutions
+In this section, we generalise the results obtained from Eq. (29), and study the transient phase with a Power Law
+inflation attractor. For this case, in contrast to de Sitter, it is only possible to reconstruct the inflaton potential
+exactly for particular choices of the parameters. Close to the attractor, an approximate reconstruction can always
+be obtained, and that is enough for the purposes of model building. The amplification of the primordial spectrum
+can still be studied in full generality, as it depends on the asymptotic values of the SR parameters, which can be
+calculated exactly. In this case, and in the large a limit, one obtains
+ǫ1 → const + L(a) ,
+(39)
+with L(a) → 0. In analogy to the previous case, we consider
+ǫ1 =
+�
+β + B
+an
+�m
+→ βm ,
+(40)
+with β, B, n > 0 and βm < 1 (so as to have acceleration close to the attractor). Notice that, when β = 0, one finds
+a transient phase with a de Sitter attractor, but ǫ1 in Eq. (40) is different from that in the set (30). This case is
+
+7
+expected to generate a hierarchy of the form (24) in the large a limit. Indeed, the ansatz (40) leads to the following
+hierarchy of SR parameters:
+ǫ2 = −mǫ4 = −mǫ6 = · · · = − n m B
+B + β an → 0 ,
+(41)
+and
+ǫ3 = ǫ5 = · · · = −
+n βan
+B + β an → −n ,
+(42)
+where, in contrast to the de Sitter case examined in the previous section, now the even SR parameters tend to zero.
+By proceeding with reconstruction and integrating Eqs. (40) and (28), one finds, respectively,
+H = H0 exp
+�
+−
+�
+β + B
+an
+�1+m
+(1 + m)nβ
+2F1
+�
+1, 1 + m, 2 + m, 1 + B
+βan
+��
+,
+(43)
+and
+φ − φ0 = f(a) − f(a0) ,
+(44)
+where
+f(a) =
+2
+√
+2MP
+�
+β + B
+an
+� 2+m
+2
+2F1
+�
+1, 1 + m
+2 , 2 + m
+2 , 1 +
+B
+βan
+�
+(2 + m)nβ
+.
+(45)
+In this case, the exact reconstruction of the potential is rather complicated unless one adopts simplifying assumptions.
+For example, let m = −1 and 0 < β < 1. Then,
+H =
+H0
+[n (B + βan)]
+1
+nβ ,
+(46)
+and
+φ − φ0 = MP
+ln
+�
+1+
+�
+ǫ1(a)
+β
+1−
+�
+ǫ1(a)
+β
+1−
+�
+ǫ1(a0)
+β
+1+
+�
+ǫ1(a0)
+β
+�
+n√β
+.
+(47)
+In the a → ∞ limit, one obtains
+φ∞ = φ0 + MP
+ln
+�
+β+1
+β−1A0
+�
+n√β
+,
+(48)
+with A0 ≡
+�
+1 −
+�
+ǫ1(a0)/β
+�
+/
+�
+1 +
+�
+ǫ1(a0)/β
+�
+. The relation (47) can be inverted to obtain an = an(φ):
+an = an
+0
+��
+1 −
+�
+ǫ1(a0)
+β
+�
++
+�
+1 +
+�
+ǫ1(a0)
+β
+�
+en√β(φ−φ0)/MP
+�2
+4en√β(φ−φ0)/MP
+,
+(49)
+and finally the potential can be reconstructed, provided we substitute Eq. (49) into Eq. (27). In terms of an, it then
+takes the following form:
+V =
+H2
+0
+[n (B + βan)]
+2
+nβ
+�
+3 −
+�
+β + B
+an
+�m�
+.
+(50)
+The expression in terms of φ is cumbersome and it will not be needed. It is also worthwhile to note that such a
+potential depends on the homogeneous inflaton through the exponential function exp
+�
+n√βφ/MP
+�
+. This functional
+dependence is expected as it is the generalization of the standard Power-Law inflation potential, which only contains
+one exponential function of the inflaton. Moreover, various approximate reconstruction methods can be used to obtain
+
+8
+the shape of the potential close to the attractor but we omit this discussion here. Whereas the exact reconstruction can
+be obtained for certain values of the parameters, the behaviour of the resulting inflationary spectra can be calculated
+exactly from Eqs. (41) and (42). For generic values of m, one may compute
+Φ = β2m − 4βm + 3
+(1 − βm)
+= 3 − βm > 0 .
+(51)
+This shows the absence of the growing solution for Eq. (13). The spectral index is then simply given by
+ns − 1 = − 2βm
+1 − βm < 0 .
+(52)
+This is the same result as the one obtained for the Power Law attractor solution. In contrast to the de Sitter case,
+the resulting primordial spectrum, if evaluated on the trajectory which approaches the attractor (and close to it),
+coincides with the spectrum calculated on the attractor itself, and no amplification or peculiar features emerge. It is
+also noteworthy that this result is not restricted to the evolution given by Eq. (40), as it only depends on the limits
+(41) and (42), which are not partcular to Eq. (40). For instance, starting from the ansatz
+H(a) = H0
+�a0
+a
+�βm �
+1 + A0
+an
+�m
+,
+(53)
+where n, A0 > 0, the resulting spectra are the same.
+The observed absence of amplification in the cases of Power-Law inflation considered here is relevant because Power-
+Law is the exactly solvable inflationary model that is most akin to SR. One may then conjecture that similar results
+(and, in particular, the lack of amplification) hold for SR inflation when the inflaton approaches the attractor solution,
+and the enhancement is a peculiarity of de Sitter.
+C.
+Non-Minimally Coupled Inflaton
+Let us now consider the different scenario of a non-minimally coupled inflaton. In order to perform the reconstruc-
+tion, we first review the basic homogeneous equations for this model:
+H2 =
+1
+3F(φ)
+� ˙φ2
+2 + V − 3HF,φ ˙φ
+�
+,
+(54)
+and
+˙H = −
+1
+2F(φ)
+�
+(1 + F,φφ) ˙φ2 + F,φ
+�
+¨φ − H ˙φ
+��
+,
+(55)
+where F(φ) represents a general non-minimal coupling and F = MP reproduces the minimally coupled case. In contrast
+to the previous cases, the homogeneous equations and the reconstruction procedure now become more involved. Then,
+for simplicity, we shall henceforth limit our study to the induced gravity (IG) case, where F(φ) = ξφ2 [9, 10]. This
+simplifying choice is also justified by the fact that both Higgs inflation and Starobinsky inflation (in the Einstein
+Frame) occur in a regime that is very close to pure IG.
+Reconstructing the inflaton potential for a given H(a) is not as straightforward as for GR with a minimally coupled
+inflaton, and we found exact potentials only for certain values of the parameters and for the de Sitter attractor case
+[cf. Eq. (29)]. Nevertheless, we can still predict the shape of the inflationary spectra or, at least, the possibility of an
+amplification in the large a limit.
+D.
+De Sitter limit
+Let us consider H(a) given by Eq. (29). In IG, the following exact relations hold between some SR parameters:
+ǫ1 =
+δ1
+1 + δ1
+� δ1
+2ξ + 2δ1 + δ2 − 1
+�
+,
+(56)
+
+9
+ǫ1 =
+1
+2ξ(1 + 6ξ)
+�
+(1 + 2ξ)δ2
+1 − 8ξδ1 − 6ξ2
+�
+1 + 2δ1 − δ2
+1
+6ξ
+� �d ln V
+d ln φ − 4
+��
+.
+(57)
+Before discussing the reconstruction of the inflaton potential, we must first calculate the asymptotic values of the SR
+parameters, which are pivotal in the analysis of the amplification of the spectrum. From Eq. (54), the potential can
+then be obtained as
+V = 3ξφ2H2
+�
+1 + 2δ1 − 1
+6ξ δ2
+1
+�
+,
+(58)
+provided H = H(φ) and δ1 = δ1(φ) are known [indeed, Eq. (58) is the IG counterpart of Eq. (27) in GR].
+Let us first calculate the SR parameters in the large a limit. Since lima→∞ ǫ1 = 0, one either has lima→∞ δ1 = 0
+and lima→∞ δ2 ̸= 0, or lima→∞ δ2 = 0 and lima→∞ δ1 ̸= 0 but satisfying the relation
+δ1,∞ =
+2ξ
+1 + 4ξ .
+(59)
+These results follow from the functional dependence of H(a) on a inherited by ǫ1 and δi’s and by the general result
+given in Eq. (24), which is applied here to the SR hierarchy δi. Notice that, in contrast to GR, two different de Sitter
+trajectories are present in IG, and they are associated with two different evolutions of the inflaton field. Using Eq.
+(57) in the same limit for a, one obtains that the potential, on the attractor, must satisfy
+d ln V∞
+d ln φ − 4 = 0 ⇒ V∞ ∝ φ4
+(60)
+in the former case and
+d ln V∞
+d ln φ − 4 = 0 ⇒ V∞ ∝ φ2
+(61)
+in the latter case.
+We can now proceed to evaluate the full hierarchy of δi’s. Starting from Eq. (56) and differentiating, we find
+ǫ2 = δ2
+�
+(1 + 4ξ)δ2
+1 + 2ξ (δ2 + δ3 − 1) + 2δ1 (1 + 4ξ + ξδ3)
+�
+(1 + δ1) [(1 + 4ξ)δ1 + 2ξ(δ2 − 1)]
+,
+(62)
+and, by further differentiating, the ǫi’s with arbitrary large i can be obtained. In the large-a limit, we have already
+calculated ǫ2i = −n and ǫ2i+1 = 0 [cf. Eqs. (30) and (31)], and one then obtains two possible hierarchies for the δi’s:
+δ2i+1,∞ = 0, δ2i,∞ = ǫ2,∞ = −n ,
+(63)
+and
+δ1,∞ =
+2ξ
+1 + 4ξ , δ2i+1,∞ = −n, δ2i,∞ = 0 .
+(64)
+This latter statement cannot be simply verified by substitution because the limits involved do not commute. For
+example, on substituting first δ2 = 0 in Eq. (62), one obtains ǫ2 = 0, which is not correct. One must solve (at least
+perturbatively in the large-a limit) Eq. (56) and then evaluate the limits with the help of the solution found. The
+above results are correctly reproduced only if we proceed in this manner.
+The exact reconstruction of the inflaton potential is not possible in general. Nonetheless, in specific cases, the
+potential may be derived exactly as follows. Consider the following ansatz for δ1:
+δ1 = n0 + n1a−n
+d0 + d1a−n ,
+(65)
+which is suggested by the expression for ǫ1 and Eq. (56). If n0 = 0 and n1 ̸= 0, then Eq. (65) can be integrated, and
+the resulting φ(a) is inverted as follows
+φ(a) = φ0
+�
+d0 + d1a−n�− n1
+n d1 ⇒ a−n =
+�
+φ(a)
+φ0
+�− n d1
+n1 − d0
+d1
+.
+(66)
+
+10
+The coefficients n1, d0, d1 and ξ can finally be fixed by the requirement that Eq. (65) be a solution of Eq. (56). Two
+nontrivial solutions can be found:
+d0 = −(1 + n) n1α
+A m n
+, d1 = −(1 + n)n1
+m n
+, ξ =
+m
+2(1 − 3m + n + m n) ,
+(67)
+or
+d0 = −(1 + n) n1α
+A m n
+, d1 = −(1 + n + m n)n1
+m n
+, ξ =
+m
+2(1 − 2m + n + m n) .
+(68)
+Notice that more exact solutions for δ1 can be found if we start from the ansatz (65) and n0 ̸= 0. However, by further
+integrating these solutions to obtain φ(a), one is led to non-invertible functions, and the reconstruction cannot be
+completed. For both Eqs. (67) and (68) one has
+δ1,∞ = 0 ,
+(69)
+and one can explicitly verify that the hierarchies belong to the set (63). Notice that n1 in Eqs. (67) and (68) can be
+arbitrarily chosen, as should be due to the form of the ansatz (65). Let us, for simplicity, complete the reconstruction
+choosing n and m to reproduce USR in the IG context (n = 6, m = 1/2). In this case, Eqs. (67) and (68) take the
+following form:
+n0 = 0, d0 = − 7α
+3An1, d1 = −7
+3n1, ξ = 1/10 ⇒ δ1 = −
+3A
+7 (A + αa6) ,
+(70)
+n0 = 0, d0 = − 7α
+3An1, d1 = −10
+3 n1, ξ = 1/36 ⇒ δ1 = −
+3A
+10A + 7αa6 .
+(71)
+From Eq. (29), a(φ) in (66) and Eq. (70), one finds
+δ1 = 3
+7
+�
+α
+�φ0
+φ
+�14
+− 1
+�
+and
+H2 = H2
+0
+� φ
+φ0
+�14
+,
+(72)
+with φ/φ0
+a→∞
+−→ α1/14 and φ > φ0, while for Eq. (71) one finds
+δ1 = 3
+10
+�
+α
+�φ0
+φ
+�20
+− 1
+�
+and
+H2 = H2
+0
+�� 7
+10
+φ
+φ0
+�20
++ 3α
+10
+�
+,
+(73)
+with φ/φ0
+a→∞
+−→ α1/20 and φ > φ0. Finally, by using Eq. (58), one obtains
+V = −
+3H2
+0
+490φ12φ14
+0
+�
+8φ28 − 72αφ14
+0 φ14 + 15α2φ28
+0
+�
+(74)
+for the first exact solution, and
+V = −
+H2
+0
+6000φ38φ20
+0
+�
+7φ20 + 3αφ20
+0
+� �
+7φ40 − 84αφ20
+0 φ20 + 27α2φ40
+0
+�
+(75)
+for the second. In the a → ∞ limit, the potentials (74) and (75) satisfy the condition d ln V/d ln φ = 4. The potential
+can have negative values but, in the vicinity of φ ≃ φ∞, the potential is positive and V∞ > 0.
+We discuss at last the behaviour of the primordial scalar curvature spectrum. The general formulae illustrated in
+the Sec. II can be easily generalised to the IG case wherein
+zIG = aφδ1
+�
+1 + 6ξ
+1 + δ1
+,
+(76)
+and Φ is given by
+Φ =
+�
+1 − ǫ1 −
+ǫ1ǫ2
+(1 − ǫ1) +
+�
+2 + 2δ1 + 2δ2 − δ1δ2
+1 + δ1
+��
+.
+(77)
+
+11
+If we evaluate Φ w.r.t. to the hierarchies (63) and (64), one observes that only constants and terms linear in the SR
+parameters remain. Moreover ǫ1,∞ = 0 and Φ then simplifies to
+Φ = 3 + 2δ1 + 2δ2 ,
+(78)
+which can be negative only for the hierarchy (63) but is strictly positive for the hierarchy (64), provided we restrict
+ourselves to positive values of the non-minimal coupling ξ. In the former case, Φ = 3 − 2n, which implies that the
+growing solution exists for n > 3/2.
+If no growing solution exists [as is the case for (64) or (63) with n < 3/2], an amplification of the spectrum is only
+possible if the spectrum is blue-tilted. Let us then evaluate ns − 1. In the IG case, fMS(ǫi) in the MS equation is
+given by
+fMS = δ2
+1 + δ2
+2 + (3 − ǫ1) (1 + δ1 + δ2) + δ2δ3 +
+δ1δ2
+�
+ǫ1 + δ1 − 3δ2 − δ3 + 2δ1δ2
+1+δ1 − 2
+�
+1 + δ1
+− 1 ,
+(79)
+and, as usual, it can be simplified to obtain the following expression for the scalar spectral index:
+ns − 1 = 3 −
+�
+1 + 4 (δ2
+1 + δ2
+2 + 3 (1 + δ1 + δ2) − 1) .
+(80)
+Then, for the hierarchy (63) and n < 3/2, we obtain
+ns − 1 = 3 − |3 − 2n| = 2n ,
+(81)
+which is indeed blue-tilted, while for the hierarchy (64), we find
+ns − 1 = −
+4ξ
+1 + 4ξ ,
+(82)
+which is red-tilted.
+We therefore conclude that solutions having H of the form given in Eq. (29), in the IG context, may lead to a
+spectrum enhancement for evolutions asymptotically described by the hierarchy (63) and either for n > 3/2 (due to
+the presence of the growing solution) or 0 < n < 3/2 (in absence of the growing solution but with the blue-tilted
+spectrum).
+IV.
+APPLICATIONS
+We have so far studied the consequences of cosmological evolutions with a transient phase, which is crucial to
+potentially obtain the amplification required by the formation of PBHs. Indeed, the presence of the transient generates,
+in the large-a limit, a sequence of values for the SR parameters that is otherwise not obtained. We then reconstructed,
+when possible, the potentials that led to the desired evolution. In this section, our approach will be slightly different,
+as we shall study the presence of the transient solutions in the particular dynamical regime of constant roll (CR)
+inflation [11], which is the natural generalisation of USR.
+CR solutions satisfy the equation
+¨φ + BH ˙φ = 0 ,
+(83)
+where B > 0, and one recovers the USR solution for B = 3, while the case of |B| ≪ 1 reproduces standard SR. We
+observe that the CR condition (83) can be rewritten, in terms of the SR parameters, as
+δ2 + δ1 − ǫ1 + B = 0 .
+(84)
+Eq. (84) is model independent, since it only depends on the definitions of ǫi’s and δi’s, and can be easily integrated
+to obtain
+dφ
+d ln aH
+� a
+a0
+�B
+= C3 ⇒ ˙φ = C3
+�a0
+a
+�B
+,
+(85)
+where C3 is an integration constant.
+In the minimally coupled case, CR can generate an amplification of the primordial scalar spectrum. In what follows,
+after a revision of this result (which was analysed in [7]) we shall consider CR in IG and study its consequences.
+
+12
+A.
+Constant Roll in GR with a Minimally Coupled Inflaton
+In GR with a minimally coupled inflaton, on imposing CR conditions and adopting the Hamilton-Jacobi (HJ)
+formalism, it is possible to reconstruct the evolution of the Hubble parameter and the corresponding potential [7]. In
+particular, one finds that H(φ) is the following superposition of two exponential functions:
+H(φ) = C1 exp
+��
+B
+2
+φ
+MP
+�
++ C2 exp
+�
+−
+�
+B
+2
+φ
+MP
+�
+.
+(86)
+In [7], the solution (86) with one exponential (C1 = 0 or C2 = 0), as well as the cosh and sinh cases, are analysed
+with the aim of finding the exact solutions compatible with CMB observations [12] (and thus not amplified).
+Here, in a slightly different approach, we consider the general case, and we study the power enhancement of the
+spectrum. Eq. (26) can be rewritten in terms of the SR parameters
+ǫ1 =
+φ2
+2MP
+2 δ2
+1 ,
+(87)
+from which, using the chain rule, we obtain
+ǫ1 = −δ1
+d ln H
+d ln φ .
+(88)
+Eq. (87) then becomes
+ǫ1 = 2MP
+2
+φ2
+�d ln H
+d ln φ
+�
+.
+(89)
+The potential can subsequently be reconstructed by substituting Eqs. (86) and (89) into Eq. (27):
+V (φ) = MP
+2H(φ)2
+�
+3 − 2MP
+2
+φ2
+�d ln H
+d ln φ
+�2�
+.
+(90)
+In order to obtain the corresponding evolution, one must integrate and invert the equation
+δ1 = −2MP
+2
+φ2
+d ln H
+d ln φ ,
+(91)
+which can be easily derived from Eq. (87) by using (88). One finds
+� a
+a0
+�B
+=
+x
+B (C2 − C1x2) ,
+(92)
+where x = exp
+��
+B
+2
+φ
+MP
+�
+. It is straightforward to invert Eq. (92) so as to obtain x = x(a). Correspondingly, one has
+H(a) = ±
+4C1C2 +
+� a0
+a
+�2B ∓
+� a0
+a
+�B �
+4C1C2 +
+� a0
+a
+�2B
+∓
+� a0
+a
+�B +
+�
+4C1C2 +
+� a0
+a
+�2B
+a→∞
+−→ ±8C1C2 +
+� a0
+a
+�2B
+4√C1C2
+.
+(93)
+Notice that the same result can be obtained if one uses the CR definition (84) instead of Eq. (26).
+The last, approximate, equality in Eq. (93) is the large-a limit of H(a), and this shows that the CR evolution is
+asymptotically equivalent to the evolution given in Eq. (29) with m = 1 and n = 2B. The results obtained in the
+Sec. II for large a are therefore inherited by CR. Thus, one obtains ǫ2i+1,∞ = 0 and ǫ2i,∞ = 2B. Correspondingly,
+Φ = 3 − 2B ,
+(94)
+which shows that the curvature perturbations are amplified for B > 3/2 due to the presence of a growing solution. In
+contrast, if 0 < B < 3/2, one finds a blue-tilted spectrum
+ns − 1 = 3 −
+�
+(3 − 2B)2 = 2B > 0 ,
+(95)
+i.e., a spectrum enhancement in the absence of growing solutions. Therefore, CR inflation admits transient solutions
+that always lead to an amplification. Finally, it is worthwhile to mention that the solutions with C1 = 0 or C2 = 0
+simply correspond to the attractor solutions for power-law inflation, and thus they are not associated with any
+amplification effect.
+
+13
+B.
+Constant Roll with a Non-Minimally Coupled Inflaton
+Let us now consider CR in the IG context. For this case, the HJ formalism leads to [13]
+H(φ) = C1φ(B+p)/2 +
+C2
+φ(p−B)/2 ,
+(96)
+where p =
+�
+(B + 2)2 + 2B(2 + ξ−1), and (B + p)/2 and (p − B)/2 are both positive with (p − B)/2 < (B + p)/2.
+For simplicity, we shall take C1,2 > 0 and we restrict the analysis to the φ > 0 interval. Studying the spectrum
+enhancement for CR in the IG case is more complicated than for GR. This is essentially a consequence of the
+complicated form of Eq. (56) in comparison to Eq. (26) in the GR case. However, the simple relation (84) holds, and
+it can be used to simplify the equations. First, with Eq. (84), one may eliminate δ2 from Eq. (56) and obtain
+ǫ1 = 1 + 2ξ
+2ξ
+δ2
+1 − (B + 1)δ1 .
+(97)
+Subsequently, by using Eq. (88), one finds
+δ1 =
+2ξ
+1 + 2ξ
+�
+B + 1 − d ln H
+d ln φ
+�
+,
+(98)
+and the potential can be reconstructed by substituting Eqs. (96) and (98) into Eq. (58).
+The evolution could be obtained by integrating Eq. (98) and inverting the result. However, analytically inverting
+the resulting equation for arbitrary values of the parameters is impossible. As we are only interested in the asymptotic
+form of H(a), one can employ a perturbative approach. Integration of Eq. (98) yields
+�a0
+a
+�B
+= φ
+2+B
+2
+�
+(B + p + 2) C1φ
+p
+2 + (B − p + 2) C2
+φ
+p
+2
+�
+,
+(99)
+where B + p + 2 > 0 and B − p + 2 < 0. Therefore, in the large-a limit, the inversion of Eq. (99) leads to
+φ(a) = φ∞ +
+�
+i>1
+φi
+�a0
+a
+�i B
+∼ φ∞ + φ1
+�a0
+a
+�B
+,
+(100)
+where φ∞ is positive. By substituting Eq. (100) into Eq. (96) and expanding for large a (properly accounting for the
+next-to-leading-order contributions), one finally obtains the asymptotic form of H(a), which reads
+H ∼ H∞ + H1
+�a0
+a
+�B
+.
+(101)
+Comparison to Eq. (29) shows that m = 1 and n = B, and the corresponding hierarchy of δi’s is given by Eq.(63)
+since
+δ1,∞ = lima→∞ ˙φ
+H∞φ∞
+= 0 ,
+(102)
+where ˙φ is given by (85). One then obtains
+Φ = 3 − 2B, ns − 1 = 2B ,
+(103)
+and, when 0 < B < 3/2,
+ns − 1 = 2B ,
+(104)
+which are the same results as GR with a minimally coupled inflaton. Indeed, in the a → ∞ limit, the homogeneous
+inflaton is frozen at a certain value and one essentially recovers the evolution of the minimally coupled case, where
+“Newton’s constant” is now reproduced by the (constant) asymptotic value of the inflaton. Furthermore the a depen-
+dence of the solution is a consequence of the fact that the CR condition (84) is independent of the specific inflationary
+model, provided H∞ and φ∞ are found to be (finite) constants.
+
+14
+C.
+Jordan and Einstein frame mapping
+In the previous section, we found the same asymptotic behaviour for the spectra in the minimally coupled case and
+in the IG case. This result was obtained in spite of the fact that CR condition is not frame invariant; i.e., the CR
+condition in the Einstein Frame (EF) is not mapped, in general, into a CR condition in the Jordan Frame (JF). In
+this section, we briefly review this statement and discuss its consequences.
+It is well known that, by a suitable conformal transformation and a redefinition of the scalar field (inflaton), one
+can map a minimally coupled theory (defined in the so-called EF) into a non-minimally coupled one (in the JF). In
+partcular, for IG, the mapping is given by the following transformation rules (see e.g. [14]):
+a(t) = MP
+√ξσ ˜a(t), N(t) = MP
+√ξσ
+˜
+N(t) ,
+(105)
+and
+φ = MP
+�
+1 + 6ξ
+ξ
+ln σ
+σ0
+, ˜V (φ(σ)) = MP
+2
+ξ2σ4 V (σ) ,
+(106)
+where the tilde refers to the Einstein frame, φ is the scalar field in the EF, σ is that in the JF. The mapping induces
+the following transformations of the Hubble parameter:
+˜H = d˜a/dt
+˜N˜a
+= (1 + δ1) MP
+√ξσ H ,
+(107)
+where
+H(t) = da(t)/dt
+a(t)N(t), ǫi+1 =
+dǫi/dt
+ǫiN(t)H(t), δi+1 =
+dδi/dt
+δiN(t)H(t)
+(108)
+are the Hubble and SR parameters in the JF. From the relation (105), one also finds that
+d
+d ln ˜a = (1 + δ1)−1
+d
+d ln a .
+(109)
+It is now straightforward to obtain the relations between SR parameters in the two frames:
+˜ǫ1 ≡ −d ln ˜H
+d ln ˜a = − (1 + δ1)−1
+d
+d ln a ln
+�
+(1 + δ1) MP
+√ξσ H
+�
+=
+δ1 + ǫ1 − δ1δ2
+1+δ1
+1 + δ1
+.
+(110)
+Given the relation (56), one then finds
+˜ǫ1 = (1 + 6ξ)δ2
+1
+2ξ(1 + δ1)2 .
+(111)
+From Eqs.
+(56) and (111), given that CR for a minimally coupled inflaton has ˜ǫ1,∞ = 0, one concludes that,
+correspondingly, in the JF one has δ1,∞ = 0 and ǫ1,∞ = 0. If we differentiate Eq. (111), we obtain the following
+relations among other SR parameters in the two frames
+˜ǫ2 =
+2δ2
+(1 + δ1)2 ,
+(112)
+˜ǫ3 = δ3 − 2δ1δ2 + δ1δ3
+(1 + δ1)2
+,
+(113)
+˜ǫ4 =
+δ1δ2
+�
+2δ2 − 2δ1δ2 + 3δ3 + 3δ1δ3 − (1 + δ1)2 δ3δ4
+�
+(1 + δ1)2 (2δ1δ2 − δ1δ3 − δ3)
+,
+(114)
+and further ˜ǫi’s can be found by iterating the procedure but are useless for what follows.
+
+15
+Similarly, one can directly calculate the relations of the ˜δi’s with the dynamical variables in the JF:
+˜δ1 ≡
+˙φ
+˜N ˜Hφ
+=
+�
+1 + 6ξ
+ξ
+MP
+φ
+δ1
+1 + δ1
+,
+(115)
+and
+˜δ2 ≡ d˜δ1/dt
+˜N ˜H˜δ1
+= −˜δ1 +
+δ2
+(1 + δ1)2 .
+(116)
+From the last relation and Eq. (56), one has that the CR condition in the EF,
+˜δ2 + ˜δ1 − ˜ǫ1 + B = 0 ,
+(117)
+is mapped into the following condition in the JF
+δ2 + (B − 1) δ1 − ǫ1 + B = 0 .
+(118)
+Notice that only for B = 2 the CR condition is frame invariant. Nonetheless, both equations reduce to δ2,∞ = ˜δ2,∞ =
+−B at late times, and the evolution is indistinguishable, at least as far as the homogeneous degrees of freedom and
+the inflationary spectra are concerned.
+We conclude that, whereas the scalar spectral index ns − 1 is frame invariant, Φ is generally not frame invariant.
+This can be checked directly by substitution. However, assuming CR holds in the EF, one verifies that Φ and ns − 1
+are both frame invariant in the asymptotic regime.
+TABLE I. Results summary
+Inflation Asymptotic Growing
+Blue-Tilted
+Model
+Svolution
+Solution Spectral Index
+GR
+dS
+n > 3
+0 < n < 3
+GR
+PL
+−
+−
+IG
+dS, δ1,∞ = 0
+n > 3/2
+0 < n < 3/2
+IG
+dS, δ1,∞ ̸= 0
+-
+-
+CR+GR
+dS
+B > 3/2
+0 < B < 3/2
+CR+IG
+dS, δ1,∞ = 0
+B > 3/2
+0 < B < 3/2
+V.
+CONCLUSIONS
+In this article, we have analyzed the effects of different transient phases, which may occur during inflation due to a
+particularity of the inflaton potential, on the primordial inflationary spectrum of scalar perturbations. These transients
+have been studied in the last few years as sources of amplification of the amplitude of the curvature spectrum. It
+is important to notice that if the amplitude of scalar perturbations grows large enough, it may induce gravitational
+collapse and consequently seed the formation of primordial black holes after inflation ends. In the literature, several
+mechanisms for such an amplification during inflation have been proposed. In particular, the presence of an ultra
+slow-roll or, more generally, a constant-roll phase has been studied. Whereas in the former case the amplification is
+due to the existence of a growing solution to the equation of motion of the curvature perturbations, in the latter case
+the amplification can also be generated by a blue-tilted spectrum in absence of the growing solution.
+The purpose of this paper was precisely to examine general features of the aforementioned models starting from
+a rather generic ansatz for the Hubble parameter as a function of the scale factor.
+This general description of
+the transient phase is model independent, and many results obtained can be readily applied to several modified
+gravity models. The matter-gravity dynamics is described in terms of the hierarchies of SR parameters, both at the
+homogeneous level and at the level of perturbations. These hierarchies, when the transient phase that describes the
+approach to some inflationary attractor is considered, have been shown to take a peculiar form wherein either odd or
+even terms of the hierarchy are null and the remaining ones are different for zero. This general feature is a peculiarity
+of the asymptotic form of the SR parameters close to the attractor, and it is then used as a simplifying assumption
+
+16
+throughout the entire article. The resulting hierarchies, in the large-a limit and for the cases considered, were used
+to calculate the behavior of the primordial curvature spectrum as the parametrisation of H(a) was varied. Then,
+when possible, the corresponding inflaton potential was fully reconstructed. An overview of the spectra enhancement
+results was presented in Table (I).
+For simplicity, only the induced gravity case has been considered here as a generalisation of general relativity with
+a minimally coupled inflaton. Induced gravity is particularly relevant since both Higgs and Starobinsky inflationary
+models (which are in good agreement with observations) take place in the ‘induced gravity phase’. We note that while
+transient evolutions that have the de Sitter universe as a limit (such as USR) can lead to an amplification, the results
+differ when power-law inflation is considered as the limit of a transitory dynamics and, for the cases we were able to
+solve explicitly, no modification of the scalar spectrum was obtained. Finally, the constant-roll case was discussed in
+more detail as an application of the preceding results, and the issue of the transition from the Einstein frame to the
+Jordan frame was also scrutinized.
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+

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@@ -0,0 +1,1414 @@
+Astronomy & Astrophysics manuscript no. bt_spidi2
+©ESO 2023
+January 30, 2023
+Spectroscopic and interferometric signatures of magnetospheric
+accretion in young stars
+B. Tessore1, A. Soulain1, G. Pantolmos1, J. Bouvier1, C. Pinte1, 2, and K. Perraut1
+1 Université Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France
+2 School of Physics and Astronomy, Monash University, VIC 3800, Australia
+xx/xx/xx; yy/yy/yy
+ABSTRACT
+Aims. We aim to assess the complementarity between spectroscopic and interferometric observations in the characterisation of the
+inner star-disc interaction region of young stars.
+Methods. We use the code MCFOST to solve the non-LTE problem of line formation in non-axisymmetric accreting magnetospheres.
+We compute the Brγ line profile originating from accretion columns for models with different magnetic obliquities. We also derive
+monochromatic synthetic images of the Brγ line emitting region across the line profile. This spectral line is a prime diagnostics of
+magnetospheric accretion in young stars and is accessible with the long baseline near-infrared interferometer GRAVITY installed at
+the ESO Very Large Telescope Interferometer.
+Results. We derive Brγ line profiles as a function of rotational phase and compute interferometric observables, visibilities and phases,
+from synthetic images. The line profile shape is modulated along the rotational cycle, exhibiting inverse P Cygni profiles at the time
+the accretion shock faces the observer. The size of the line’s emission region decreases as the magnetic obliquity increases, which is
+reflected in a lower line flux. We apply interferometric models to the synthetic visibilities in order to derive the size of the line-emitting
+region. We find the derived interferometric size to be more compact than the actual size of the magnetosphere, ranging from 50 to
+90% of the truncation radius. Additionally, we show that the rotation of the non-axisymmetric magnetosphere is recovered from the
+rotational modulation of the Brγ-to-continuum photo-centre shifts, as measured by the differential phase of interferometric visibilities.
+Conclusions. Based on the radiative transfer modelling of non-axisymmetric accreting magnetospheres, we show that simultaneous
+spectroscopic and interferometric measurements provide a unique diagnostics to determine the origin of the Brγ line emitted by young
+stellar objects and are ideal tools to probe the structure and dynamics of the star-disc interaction region.
+Key words. Radiative transfer – Line: profiles – Stars: variables: T Tauri, Herbig Ae/Be – Accretion, accretion disks
+1. Introduction
+The early evolution of low mass stars (M∗ < 2 M⊙) during the
+classical T Tauri (CTT) phase depends on the interaction be-
+tween the star and its accretion disc, on a distance of a few stel-
+lar radii. At the truncation radius, matter from the disc surface is
+channelled onto the stellar surface following the magnetic field
+lines and forming an accretion funnel or column (Ghosh et al.
+1977; Zanni & Ferreira 2009; Romanova & Owocki 2016; Pan-
+tolmos et al. 2020). The star-disc interaction is responsible for
+accretion and ejection phenomena that have a strong impact on
+spectral lines formed in the close vicinity of the star’s surface.
+Ghosh et al. (1977) developed an analytical model of mag-
+netospheric accretion around a rotating neutron star with a dipo-
+lar magnetic field. Hartmann et al. (1994) applied this magne-
+tospheric accretion model to the formation of emission lines in
+the spectrum of T Tauri stars. This fundamental paper sets the
+general theoretical framework for the density and temperature
+distributions in aligned axisymmetric magnetospheres. The cou-
+pling between this representation of magnetospheric accretion in
+T Tauri systems with radiative transfer calculations has provided
+a crucial tool to interpret spectroscopic, photometric, and inter-
+ferometric observations. The sensitivity of hydrogen lines to the
+parameters of the magnetospheric models was studied in detail
+by Muzerolle et al. (2001), improving the earlier calculations by
+Hartmann et al. (1994).
+Near-infrared observations of the Brackett γ (Brγ) line with
+the Very Large Telescope Interferometer (VLTI) GRAVITY in-
+strument (Gravity Collaboration et al. 2017) also probe the inner
+part of the star-disc interaction region (Gravity Collaboration
+et al. 2020; Bouvier et al. 2020a). However, it is still difficult
+to associate the characteristic sizes derived from interferometry
+with the actual size of the magnetospheric accretion region, a
+key parameter in our understanding of the star-disc interaction.
+In this paper, we aim at studying the formation of the Brγ
+line and compute its spectroscopic and interferometric signa-
+tures for non-axisymmetric models of the inner star-disc inter-
+action region, akin to state-of-art MHD simulations (Romanova
+& Owocki 2016). In particular, we want to clarify the meaning
+of the sizes inferred through near-infrared interferometric obser-
+vations and how they compare with the overall size of the mag-
+netospheric accretion region.
+In sections §2 and §3, we describe the model used to com-
+pute the line formation in accreting magnetospheres. We discuss
+spectroscopic and interferometric signatures in sections §4 and
+§5, respectively.
+2. Radiative transfer framework
+We use the code MCFOST 1(Pinte et al. 2006, 2009; Tessore
+et al. 2021) to compute emergent line fluxes from multidimen-
+1 https://github.com/cpinte/mcfost
+Article number, page 1 of 13
+arXiv:2301.11628v1  [astro-ph.SR]  27 Jan 2023
+
+A&A proofs: manuscript no. bt_spidi2
+sional models of magnetospheres for a 20-level hydrogen atom.
+The atomic model, with 19 bound levels and the ground state of
+HII, consists of 171 bound-bound transitions (atomic lines) and
+19 bound-free transitions (continua). We focus here on the Brγ
+line at 2.1661 µm although, the Balmer lines Hα and Hβ and
+the Paschen β line (Paβ) are modelled as well. These specific
+hydrogen lines are commonly used to characterise accretion and
+ejection phenomena in young systems (Folha & Emerson 2001;
+Alencar et al. 2012; Bouvier et al. 2020a; Pouilly et al. 2020;
+Sousa et al. 2021). The method to solve for the non-LTE pop-
+ulations of hydrogen and the microphysics are the same as in
+Tessore et al. (2021). The updated version of the code we use
+now simultaneously solves the charge equation and the statis-
+tical equilibrium equations, which has been proven to increase
+the convergence in chromospheric conditions (Leenaarts et al.
+2007). We tested our code for different magnetospheric models
+taken as benchmarks in Muzerolle et al. (2001) and Kurosawa
+et al. (2006). The results of this comparison are presented and
+discussed in Appendix A.
+3. Magnetospheric accretion model
+Matter from the circumstellar disc is channelled onto the stel-
+lar surface along the dipolar magnetic field lines. The stellar
+magnetic field truncates the disc at a distance Rt from the star,
+the truncation radius. In practice, the interaction between the
+stellar magnetic field and the disc takes place over a small re-
+gion between Rt and Rt + δr. Both Rt and δr are used to de-
+fine the size of the disc region magnetically connected to the
+star. As the gas approaches the stellar surface, it decelerates in
+a shock and is heated at coronal temperatures. Theoretical mod-
+els of accretion shocks by Calvet & Gullbring (1998) show that
+the optically thin emission of the pre/post-shock dominates be-
+low the Balmer jump and that the optically thick emission of
+the heated photosphere contributes to the total continuum emis-
+sion at larger wavelengths. In the following, we only consider
+the contribution of the heated photosphere to the shock radia-
+tion. The shock2 temperature is computed from the energy of the
+gas infalling onto the stellar surface following the prescription
+of Romanova et al. (2004) unless specified. This approach as-
+sumes energy conservation and that the shock radiates as a black
+body, meaning that its temperature is determined by the specific
+kinetic energy and enthalpy of the gas deposited at the stellar
+surface. The shock temperature hence derived is of the order of
+4500 K - 6000 K.
+3.1. The stellar surface
+The stellar surface is considered as the inner boundary of the
+model and emits as a blackbody whose temperature is deter-
+mined by the stellar parameters. Throughout the paper, the stellar
+parameters are T∗ = 4, 000 K, M∗ = 0.5 M⊙, and R∗ = 2 R⊙. We
+set the distance to the star at 140 pc, which is typical of the near-
+est star forming regions such as Upper Scorpius (≈ 146 pc Galli
+et al. 2018a) or Taurus (≈ 130 pc Galli et al. 2018b).
+3.2. Geometry of the accretion funnels
+We consider 3D non-axisymmetric models of the magneto-
+spheric accretion region. These models are parametrised by the
+same set of parameters as the axisymmetric magnetospheric
+2 We assume that the shock region is unresolved and is part of the
+stellar surface.
+Fig. 1: Density distribution of a non-axisymmetric model with an
+obliquity of 10◦. The rotation axis of the star Ω is shown with a
+white arrow and the dipole axis, µ, with a red arrow. The density
+is computed from Eqs. (1) and (2). The colour map scales with
+the density.
+model of Hartmann et al. (1994) (see also Muzerolle et al. 1998,
+2001; Kurosawa et al. 2006; Lima et al. 2010; Kurosawa et al.
+2011; Dmitriev et al. 2019).The density and the velocity fields of
+the accretion columns are fully described with a set of indepen-
+dent parameters: the mass accretion rate ˙M, the rotation period
+Prot, Rt, and δr.
+For our study, the value of ˙M, Rt and δr, and of the temper-
+ature of the magnetosphere are fixed. The impact of these pa-
+rameters on the line formation has been discussed thoroughly in
+Muzerolle et al. (1998, 2001, see also App. A). The line’s re-
+sponse to the mass accretion rate and to the temperature is an es-
+sential proxy for understanding the physics of the star-disc inter-
+action region. We use a mass accretion rate ˙M = 10−8 M⊙ yr−1,
+a truncation radius Rt = 4 R∗, and δr = 1 R∗. The value of the
+rotation period is deduced from the maximum truncation ra-
+dius (Rt + δr), imposing that stable accretion occurs at 90% of
+the corotation radius, consistent with the work of Blinova et al.
+(2016). The rotation period is therefore fixed at Prot = 6 days,
+corresponding to slowly rotating T Tauri stars (see Herbst et al.
+2007; Bouvier et al. 2014, for a review). The rotational velocity
+for that period is thus of the order of 80 km s−1 at the outer edge
+of the magnetosphere.
+When the magnetic field axis (µ) is misaligned with respect
+to the rotational axis (Ω) of the star, the geometry of the accre-
+tion flow changes dramatically. The equations for the magnetic
+field components of a non-axisymmetric dipole, i.e. with a non-
+zero obliquity, are provided in Mahdavi & Kenyon (1998). The
+parameter βma describes the angle between the dipole moment
+and the star’s rotational axis, the magnetic obliquity.
+We approximate the density, ρ, along the non-axisymmetric
+magnetic field lines with,
+ρ = α B
+� = αB ρaxi
+Baxi
+,
+(1)
+where α is a constant along a given field line and B the ana-
+lytic misaligned dipolar field. �, ρaxi, and Baxi denote the veloc-
+ity field, density, and dipolar magnetic field, respectively, and
+they are taken from the axisymmetric model of Hartmann et al.
+(1994). In other words, the 3D density structure is computed
+from Eq. (1) under the assumption that the infalling gas has a
+velocity field on the poloidal plane. The value of α is computed
+Article number, page 2 of 13
+
+1.2e-08
+5e-9
+[s-'b] 
+2e-9
+1e-9
+5e-10
+2.3e-10
+RtB. Tessore & A. Soulain et al.: Spectroscopic and interferometric signatures of magnetospheric accretion
+from the numerical integration 3 of the mass flux over the shock
+area,
+˙M =
+�
+ρv · dS,
+(2)
+where dS is the surface element and v the velocity field. In our
+model, the value of ˙M is an input parameter and is held constant.
+Therefore, α is obtained to ensure consistency between Eqs. (1)
+and (2). We compute five models with an obliquity βma ranging
+from five to forty degrees in step of ten degrees, representative
+of what has been measured for T Tauri stars with spectroscopy
+(McGinnis et al. 2020) and spectropolarimetry (Donati et al.
+2008, 2010, 2013; Johnstone et al. 2014; Pouilly et al. 2020). For
+these non-axisymmetric models, the shortest field lines – defin-
+ing the main accretion columns4 – carry most of the gas density.
+We remove the longest field lines – the secondary columns –
+in our modelling as in Esau et al. (2014). This yields models
+with one crescent-shaped accretion spot per stellar hemisphere
+reminiscent of numerical simulations of misaligned dipoles (Ro-
+manova et al. 2003).
+Figure 1 shows the density of a non-axisymmetric magneto-
+sphere with an obliquity of 10◦.
+3.3. Temperature of the funnels
+The temperature of the magnetospheric accretion region is not
+well constrained. The determination of the temperature by Mar-
+tin (1996) from first principles was not able to reproduce the
+observations. A self-consistent calculation of the temperature of
+the magnetosphere is beyond the scope of this paper. Instead,
+we adopt here the temperature profile of Hartmann et al. (1994),
+which has been extensively used in the past to model line fluxes
+from accreting T Tauri stars. The temperature is computed using
+a volumetric heating rate (∝ r−3) and balancing the energy input
+with the radiative cooling rates of Hartmann et al. (1982). The
+exact balance between the heating and cooling mechanisms is
+unknown. Instead, the temperature profile is normalised to a free
+parameter, Tmax, that sets the value of the maximum temperature
+in the funnel flow. In the following, we have set the temperature
+maximum to Tmax = 8, 000 K.
+4. Spectroscopic signatures
+Thanks to the Doppler shift of the funnel flow, it is possible
+to reconstruct the origin of the emission line by looking at the
+brightness maps in various velocity channels. Figure 2 shows
+the contribution of the different parts of the magnetosphere to
+the total integrated Brγ line flux at a given velocity for an in-
+clination of 30◦, matching the model illustrated in Fig. 1. At
+those density and temperature, the continuum emission comes
+from the stellar surface (Isurf/Imag > 100). Locally, the contin-
+uum emission from the shock is three times larger than the emis-
+sion from the star. Overall, given the small covering area of the
+accretion shock (around 1%), the total continuum emission at the
+frequency of the Brγ line is dominated by the star’s radiation,
+Fshock/F∗ = 3%. The low-velocity components (< 50 km s−1) of
+the line form in the regions where the projected velocity along
+the line-of-sight is close to zero and near the disc. The geometry
+3 For this 3D magnetospheric accretion model, an explicit formula for
+the shock area does not exist (see also Mahdavi & Kenyon 1998)
+4 Geometrically, the shortest field lines obey the following criterion
+cos φ′ × z > 0 where φ′ is the azimuth in the frame aligned with the
+dipole axis and z the coordinate parallel to the rotation axis.
+of the non-axisymmetric model, defined in §3, is responsible for
+a rotational modulation of the integrated line flux. Many classical
+T Tauri stars shows modulated photometric variability (e.g Cody
+et al. 2014) and, more directly related to the magnetospheric re-
+gion, many also show rotational modulation of the longitudinal
+component of the stellar magnetic field (e.g Donati et al. 2020).
+Indeed, the periodic variability of optical and emission line pro-
+files has been reported in various systems (for instance Sousa
+et al. 2016; Alencar et al. 2018; Bouvier et al. 2020a), which in-
+dicates that the emission region is stable on a timescale of several
+rotation periods.
+Figure 3 shows the variability of the Brγ line at different
+phases of rotation at an inclination of 60◦ for different obliq-
+uities. The origin of the rotational phase is defined such that at
+phase 0.5, the accretion shock is facing the observer. The red-
+shifted absorption seen for the Brγ line at phases 0.250, 0.47 and
+0.69, results from a lower source function of the gas above the
+shock (see App. A). From observations, red-shifted absorption
+in the Paβ and Brγ lines are seen in less than 34% and 20% of
+the line profiles, respectively (Folha & Emerson 2001). The in-
+verse P Cygni profile disappears when the shock, or a significant
+fraction of it, is hidden on the opposite side of the star. The line,
+with either a double-peaked profile or a moderate red-shifted ab-
+sorption, is reminiscent of Reipurth et al. (1996) cases II and IV.
+While the profiles with redshifted absorption agree with observa-
+tions, those that display an M-shape are usually not observed in
+young stellar objects. This suggests that magnetospheric accre-
+tion is not the only contribution to the profile, which can also be
+impacted by various types of outflows (e.g., stellar, interface, and
+disk winds Lima et al. 2010; Kurosawa et al. 2011). The optically
+thick accretion disc is not included in our models. The effect of
+the disc emission and absorption on the spectroscopic and inter-
+ferometric observables will be discussed in a subsequent paper.
+We also observe a decrease of the line flux as the obliquity
+increases. Figure 4 shows the radius encompassing 90% of the
+total line flux, R90, at an inclination of 60◦ for non-axisymmetric
+models with different obliquities for the Hα, Hβ, Paβ and Brγ
+lines.
+5
+10
+15
+20
+25
+30
+35
+40
+ma [ ]
+2.8
+3.0
+3.2
+3.4
+3.6
+3.8
+R90 [R ]
+H
+H
+Pa
+Br
+Fig. 4: Radius encompassing 90% of the total flux (R90) for each
+line as a function of the obliquity, βma. Hydrogen lines are la-
+belled with different colours.
+As βma increases, the volume of the magnetospheric accre-
+tion region decreases because the arc length of the accreting field
+lines shortens. Therefore, the total flux, for all lines, decreases
+accordingly, independently of the viewing angle of the system.
+However, we also note a dependence of R90 with the line. The
+Article number, page 3 of 13
+
+A&A proofs: manuscript no. bt_spidi2
+II
+III
+IV
+I
+200
+0
+200
+v [km. s
+1]
+1.0
+1.2
+1.4
+F/Fc
+I
+II
+III
+IV
+V
+V
+0%
+5%
+30%
+50%
+80%
+Fig. 2: Origin of the emission seen across the Brackett γ line. The contribution of individual images to the total line flux is indicated
+on the central image showing the line profile. The brightness maps are in units of the maximum emission. The emission of the stellar
+surface is saturated. Orange to red colours indicates the regions of maximum emission. The system is seen at an inclination of 30◦
+and an rotational phase of ∼0.25, similar to Fig. 1.
+value of R90 represents the size of the emitting region in a given
+line, which is a function of density and temperature, and of the
+viewing angle.
+5. Interferometric signatures
+In this section, we compute the size of the Brγ line-emitting re-
+gion inferred from interferometric observations, and we compare
+it to model flux radii (see §4).
+5.1. Interferometric observables
+The interferometric observables are derived from the radiative
+transfer (RT) model using the ASPRO25 software developed by
+the Jean-Marie Mariotti Center (JMMC). These observables rep-
+resent what would be observed with GRAVITY in the near-
+infrared. We consider the configuration obtained with the Very
+Large Telescope (i.e. 4x8m telescopes), encompassing a range
+of baselines from 35 to 135 m. With a typical night of 8 hours,
+we compute one observing point per hour for the six baselines
+5 Available at https://www.jmmc.fr
+of the VLTI to increase the Fourier sampling, namely u-v cov-
+erage, which is crucial for the fitting part of our approach. As
+described in Bourgès & Duvert (2016), we derive the observ-
+ables from the RT images (see Fig. 2) by computing the com-
+plex visibility in each spectral channel around the Brγ line and
+interpolating them to match GRAVITY’s spectral resolution (R
+= 4000). Specifically, we simulate a total of 37 spectral chan-
+nels (from 2.161 to 2.171 µm with a step of 2.8 10−4 µm) for the
+six projected baselines repeated eight times. Within this range,
+31 spectral channels are used to measure the K-band continuum
+and six channels sample the Brγ line emitting region.
+Figure B.1 illustrates the resulting u-v plane projected on-
+sky for a typical object observed at the VLTI with a declination
+of -34◦ (e.g. TW Hydrae).
+Figure 5 shows the interferometric observables along the ro-
+tational cycle for a model with an inclination of 60◦ and an obliq-
+uity of 10◦. The two main observables are: the modulus of the
+complex visibility – the visibility amplitude – and the differen-
+tial phase – its argument – dispersed in wavelength. The phase
+is normalised to zero in the continuum. The visibility amplitude
+can then be used to estimate the object’s size, while the phase
+measures the photo-centre shifts between the line-emitting re-
+Article number, page 4 of 13
+
+B. Tessore & A. Soulain et al.: Spectroscopic and interferometric signatures of magnetospheric accretion
+250
+125
+0
+125
+250
+0.8
+0.9
+1.0
+1.1
+1.2
+1.3
+1.4
+phase = 0.03
+250
+125
+0
+125
+250
+phase = 0.25
+250
+125
+0
+125
+250
+phase = 0.47
+250
+125
+0
+125
+250
+phase = 0.69
+250
+125
+0
+125
+250
+phase = 0.92
+ma = 5
+ma = 10
+ma = 20
+ma = 30
+ma = 40
+v [km s
+1]
+F/Fc
+Fig. 3: Brackett γ line variability along the rotational cycle. Each column corresponds to a specific rotational phase. At phase 0,
+the shock area is unseen on the stellar surface, while phase of 0.5, the shock is fully seen on the visible hemisphere. The colours
+correspond to different values of the obliquity. All fluxes are computed with an inclination of 60◦.
+gion and the continuum. The phase can only be used as a rela-
+tive measurement (e.g. between the line and the continuum), the
+absolute phase being lost due to a combination of atmospheric
+and instrumental effects. We repeat the simulated observations
+and compute nine datasets over a rotational cycle sampled every
+40 degrees ( 0.11 in phase). In this study, we are interested in the
+line’s emitting region only. Therefore, we use pure line quanti-
+ties, instead of total visibilities and phases, to remove the contri-
+bution from the stellar surface (see appendix B for the derivation
+of the pure line interferometric quantities).
+5.2. Physical characteristic and sizes
+Once the interferometric observables are computed, we apply
+standard modelling methods to interpret the data (Berger 2003).
+Firstly, we average the visibility amplitude of the six spectral
+channels available within the Brγ line6. We use the average vis-
+ibilities to recover the global size of the Brγ emitting region,
+where the different velocities probe specific parts of the mov-
+ing material within the magnetosphere. Then, we fit the aver-
+aged visibility amplitude using elongated Gaussian or uniform
+disc models. Such models are typically used in interferometry to
+estimate the system’s characteristic size and on-sky orientation.
+The source’s brightness distribution is defined by its half-flux ra-
+dius in the case of a Gaussian disc or its radius for the uniform
+disc model, and an elongation factor and a position angle. In the
+following, we adopt the definition of "radius" for both models,
+which corresponds to the half-flux semi-major axis for the Gaus-
+sian model and the semi-major axis for the uniform disc model.
+The recovered sizes and orientations are represented in the top
+panel of Fig. 5. While neither model can fully account for the
+size of the magnetosphere, the uniform disc probes a larger area
+of the magnetosphere, while the Gaussian disc seems limited to
+the most luminous parts. We note that the fit of the visibility is
+6 Five and four spectral channels only were used at phase 0.25 and
+0.47, respectively, due to a limited line-to-continuum ratio (see Ap-
+pendix B for details).
+equally good for both models and, thus, does not allow us to dis-
+criminate between the models from the synthetic visibilities only
+(middle-top, Fig. 5).
+In order to quantify the physical meaning of the interfero-
+metric measurements, we compare the interferometric sizes with
+reference flux radii of the RT models. We set these radii to repre-
+sent 50, 80, 90, and 99% of the total flux emitted by the magne-
+tospheric accretion region. Figure 6 compares the sizes derived
+with interferometry to the characteristic radii of the RT models.
+We find that the size derived from the uniform disc model is
+modulated around an average value of 3.5 R∗ corresponding to
+90% of the Brγ emitting region. The size obtained by interferom-
+etry appears to be modulated by the position of the funnel flows
+close to the star, with a minimum located around phase 0.8. The
+Gaussian model exhibits the same modulation but with a lower
+amplitude (2.1 ± 0.4 R∗) and appears sensitive to the magneto-
+sphere’s innermost region, close to the 50% flux radius. The size
+derived from the uniform disc model emerges as being the most
+appropriate to recover the reference model size, accounting for
+at least 80% of the total flux emitted by the magnetosphere.
+Article number, page 5 of 13
+
+A&A proofs: manuscript no. bt_spidi2
+0.0
+0.2
+0.4
+0.6
+0.8
+Phase
+2.0
+2.5
+3.0
+3.5
+4.0
+4.5
+5.0
+Radius [R
+]
+Rt +
+r
+50%
+80%
+90%
+99%
+Uniform disc
+Gaussian disc
+Fig. 6: Interferometric radii as a function of the rotational phase.
+Uniform and Gaussian disc models are shown with green and
+yellow markers, respectively. Blue lines correspond to the radii
+encompassing 50, 80, 90 and 99% of the total RT model’s flux.
+The blue shaded areas represent the standard deviation of these
+radii across the rotational phase.
+The red shaded area indicates the inner (Rt) and outer radius (Rt+
+δr) of the RT model.
+The derived orientations obtained from interferometry seem
+to be particularly representative of the position of the accretion
+funnel flow and the on-sky orientation of the Brγ emitting re-
+gion (Fig. 5). The measured position angle agrees with the mag-
+netosphere’s orientation, particularly when the shock faces the
+observer (phase = 0.5). Nevertheless, it appears somewhat haz-
+ardous to decipher the shape and orientation of the emitting re-
+gion across the rotational cycle from this observable only, as
+different magnetospheric configurations can be described by a
+very similar interferometric model (e.g. phases 0.03 and 0.25).
+A stronger constraint on the orientation of the funnel flows arises
+from differential phase measurements.
+5.3. Differential phases and photo-centre shifts
+From the differential phases, we can derive the photo-centre shift
+between the continuum and the Brγ line emitting region. In the
+regime of marginally resolved sources, there is a direct relation-
+ship between the projected photo-centre displacement vector (P)
+and the phase along each baseline (Lachaume 2003):
+φi = −2π Bi
+λ P,
+(3)
+where φi is the differential phase measured for the ith baseline,
+Bi is the length of the corresponding baseline, and λ is the effec-
+tive wavelength of the spectral channel. A four telescope beam-
+combiner like GRAVITY gives access to six projected baselines
+that enable us to accurately retrieve the value and orientation
+of the photo-centre shifts in each spectral channel (Le Bouquin
+et al. 2009; Waisberg et al. 2017). Such a measurement results in
+a position-velocity plot of the displacement of the photo-centre
+across the Brγ line relative to the continuum. This is illustrated
+in the bottom panels of Figure 5.
+The photo-centre shifts trace the accretion funnel flow’s di-
+rection and follow the stellar rotation. For instance, when the
+northern accretion shock (N-shock) is located behind the star
+(phase = 0), the accreting material falls onto the stellar surface in
+the direction of the observer. Accordingly, the photo-centre mea-
+sured in the blue-shifted part of the line profile (≃ -75 km s−1)
+lies on the blue-shifted part of the velocity map, corresponding
+to the approaching funnel flow. Equivalently, the photo-centre
+measured in positive velocity channels of the line profile (≃
++75 km s−1) is shifted towards the receding funnel flow. In con-
+trast, when the shock faces the observer (phase = 0.5), the veloc-
+ity map goes from blue to red in the east-west direction, and the
+photo-centre shifts recover this trend as demonstrated at phase
+0.47.
+We can thus identify three privileged directions and shapes
+of the photo-centre shifts: – linear north-south at phase ≃ 0 (N-
+shock behind), – S-shape at phase 0.25 and 0.69 and – linear
+east-west at phase ≃ 0.5 (N-shock in front). The differential
+phase is, therefore, a key ingredient to recover the geometry and
+orientation of the line-emitting region, tracing the moving mate-
+rial along a rotational cycle.
+5.4. Signal-to-noise considerations
+As a proof-of-concept, the results presented above assume infi-
+nite signal-to-noise ratio. The goal is to predict the spectroscopic
+and interferometric signatures of the magnetospheric accretion
+process. Thus, the models predict typical visibility amplitudes
+ranging from 1 down to 0.97 (see Fig. 5). Such a modest inter-
+ferometric signal requires a measurement accuracy of about 1%
+to be securely detected. Similarly, the models predict a deviation
+of the differential phases by 1 to 2 degrees (Fig. 5), which re-
+quires an accuracy of order of a fraction of a degree to yield a
+robust detection. Recent interferometric studies performed with
+VLTI/GRAVITY in the K-band demonstrate that these levels of
+accuracy can be routinely obtained indeed with reasonable ex-
+posure times on young stellar objects (e.g. Bouvier et al. 2020b;
+Gravity Collaboration et al. 2020, 2022), or active galactic nuclei
+(Gravity Collaboration et al. 2018).
+6. Summary and conclusion
+We presented non-LTE radiative transfer modelling of the Brack-
+ett γ line emission for non-axisymmetric models of accreting
+magnetospheres. We used the equations of a misaligned dipo-
+lar magnetic field to derive the geometry of the magnetospheric
+accretion region for different obliquities of the magnetic dipole.
+We used MCFOST to compute radiative signatures of the Brγ
+line along a full stellar rotational cycle. Further, we derived near-
+infrared interferometric observables for the line, comparable to
+what the GRAVITY instrument has already measured for T Tauri
+stars.
+The main conclusions of this study are the following:
+1) The total flux in the line, and the line-to-continuum ratio,
+depends on the obliquity of the dipole. As the obliquity in-
+creases, the size of the emitting region decreases, leading
+to a lower integrated flux. Also, projection effects make the
+emission region of lines forming close to the stellar surface
+appearing narrower.
+2) The Brγ line total flux varies with the rotational phase due to
+the non-axisymmetry of the models induced by the magnetic
+obliquity. The line profiles exhibit a red-shifted absorption,
+that is an inverse P Cygni profile, when a significant fraction
+of the accretion shock is aligned with the observer’s line of
+sight. When the shock is hidden on the opposite side of the
+star, the line profiles exhibit a double-peaked shape, reminis-
+cent of the lines formed in rotating envelope. The latter is
+due to the relatively large rotational velocity of the magneto-
+spheric model (∼80 km s−1).
+Article number, page 6 of 13
+
+B. Tessore & A. Soulain et al.: Spectroscopic and interferometric signatures of magnetospheric accretion
+3) Near-infrared interferometric observations in the Brγ line di-
+rectly probe the size of the magnetospheric accretion region.
+The Gaussian disc model is sensitive to the brightest parts of
+the magnetosphere, up to 50% of the truncation radius, while
+a uniform disc model grasps 90% of the magnetosphere. It is
+of prime importance to consider this aspect when estimating
+magnetospheric radius from interferometric measurements.
+In both cases, the measured radius varies with the rotational
+phase (due to the non-axisymmetry of the dipole). A robust
+interferometric estimate of the magnetospheric radius there-
+fore requires monitoring the system over a full rotational cy-
+cle.
+4) The combined knowledge of the differential phase and the
+associated photo-centre shifts gives hints on the object ori-
+entation and geometry. More specifically, the relative direc-
+tion of the photo-centre shifts indicates the changing orien-
+tation of the accreting material along the rotational cycle in
+the non-axisymmetric case.
+Near-infrared interferometry of the Brackett γ line is used
+to characterise the inner star-disc interaction region, and offers a
+good estima te of the size of the line’s forming region, at sub-au
+precision. Comparing this size with reference model radii, such
+as the truncation radius, allows us to distinguish between mul-
+tiple origins of the Brγ line, within or beyond these radii (e.g.
+magnetosphere, stellar and disc winds, jets). Further, simultane-
+ous spectroscopic and interferometric observations along a rota-
+tional cycle, have the potential to unveil the geometry and ori-
+entation of the line’s emitting region. The variability of the line
+associated with the photo-centre shifts, provides a unique and
+unambiguous proxy of the physical processes occurring in the
+magnetosphere of young accreting systems, within a few hun-
+dredths of an astronomical unit around the central star.
+Article number, page 7 of 13
+
+A&A proofs: manuscript no. bt_spidi2
+Fig. 5: Synthetic interferometric measurements and modelling across the rotational phase of the system. Top: integrated images over the Brγ line. Green (uniform disc) and
+yellow (Gaussian disc) ellipses are the characteristic sizes measured with a GRAVITY-like instrument. Middle-top: Pure line visibility amplitude observables associated with
+the corresponding models. The visibility variation (for a given baseline) as the u-v plane rotates is the specific signature of an elongated object. Middle-bottom: Pure phase
+visibility across the line profile for the six baselines of the VLTI. Colours encode the observing time. Bottom: Velocity map of the radiative transfer model. The coloured dots
+represent the measurement of the photo-centre derived from the phase visibility in each available spectral channel (see appendix B). In each figure, the magnetosphere and the
+stellar surface have been normalised independently, for display purpose.
+Article number, page 8 of 13
+
+phase = 0.03
+phase = 0.47
+phase = 0.92
+rud = 3.60 R*
+phase = 0.25
+rud = 3.88 R*
+phase = 0.69
+rud = 3.06 R*
+rud = 3.30 R*
+rgd = 2.12 R
+rgd = 2.43 R,
+rgd = 2.29 R*
+rgd = 1.80 R*
+1.95 R
+0.8
+0.8
+0.8
+0.8
+0.8
+0.6
+0.6
+0.6
+0.6
+0.6
+0.
+0.01 AU
+0.01 AU
+0.01 AU
+0.01 AU
+0.01 AU
+1.00-
+1.00-
+1.00
+1.00
+1.00
+0.99
+m0.991
+≥0.99 -
+visibil
+visibil
+visibil
+visibil
+visibil
+.-GD fit
+..GD fit
+.. GD fit
+.. GD fit
+.-GD fit
+UD fit
+UD fit
+UD fit
+UD fit
+UD fit
+UT1-UT2
+In-TIn
+UT1-UT2
+UT1-UT2
+UT1-UT2
+UT1-UT3
+UT1-UT4
+UT1-UT4
+UT1-UT4
+UT1-UT4
+UT1-UT4
+UT2-UT3
+UT2-UT3
+UT2-UT3
+UT2-UT3
+UT2-UT3
+-L60
+F260
+L60
+L60
+-L60
+.
+UT2-UT4
+UT2-UT4
+UT2-UT4
+UT2-UT4
+UT3-UT4
+40
+120
+2 0
+50
+50
+50
+Diff. Φ [deg]
+Diff.  [deg]
+Diff.  [deg]
+Diff.  [deg]
+Diff.  [deg
+UT2
+Φ [deg]
+p[deg]
+[deg]
+[deg]
+[deg]
+Diff. 
+[deg]
+[deg]
+[deg]
+[deg]
+[deg]
+Diff. 
+Diff.
+Diff.
+Diff.
+Diff.
+100
+100
+100
+100
+100
+100
+100
+100
+100
+100
+100
+.00
+100
+100
+elocity [km/s]
+Velocity [km/s]
+elocity [km/s]
+F00m
+F00m
+F00m
+-00m
+-00m
+75
+75
+75
+75
+200-
+200-
+Photocenter shift[μas]
+-002
+200
+Photocenter shift [μas]
+50
+[μas]
+[μas]
+[μas]
+100-
+100-
+100-
+100-
+hotocenter shift ["
+100-
+hotocenter shift [
+hotocenter shift [
+25
+25
+25
+25
+-0
+-100
+-100-
+-100-
+-100-
+-100-
+-50
+-50
+-50
+-50
+-50
+-200-
+-200-
+-200-
+200-
+-75 
+-75 
+-75 
+-75 
+-75
+-300-
+-300-
+-300-
+-300 -200 -
+-300 -200 -
+-300 -200 -
+-300 -200 -
+-300 -200 -
+-100
+300
+-100
+300
+-100
+300
+-100
+300
+-100
+300
+Photocenter shift [μasB. Tessore & A. Soulain et al.: Spectroscopic and interferometric signatures of magnetospheric accretion
+Appendix A: benchmark
+We present here the comparison between line profiles obtained
+with MCFOST and previous studies. The magnetospheric model
+corresponds to the axisymmetric and compact configuration of
+Muzerolle et al. (2001) with a fixed shock temperature at 8 000
+K, a rotation period of 5 days and the following canonical T Tauri
+parameters: M∗ = 0.8 M⊙, R∗ = 2 R⊙ and T∗ = 4 000 K. The in-
+clination of the system is 60 degrees. The continuum emission
+of the stellar surface (shock and photosphere) is constant for all
+models. Figures A.1, A.2, A.3, and A.4 show the Hα, Hβ, Paβ
+and Brγ lines profiles for different values of Tmax and ˙M. An in-
+verse P Cygni profile, with a red-shifted absorption, is seen for
+all lines although it is dependent on the value of the mass ac-
+cretion rate and of the maximum temperature. For a given mass
+accretion rate, an increase of the maximum temperature results
+in a higher line emission peak and a shallower red-shifted ab-
+sorption. As the temperature increases, the line source function
+increases, which is the cause of a higher emission above the
+continuum emission. The appearance of the red-shifted absorp-
+tion component is caused by absorption from the gas above the
+stellar surface. It is controlled by the ratio between the source
+function of the line in the accretion funnel and that of the un-
+derlying continuum from the stellar surface, especially at low
+mass accretion rates and temperatures. Eventually, for the high-
+est mass accretion rate and temperature, the lines become so op-
+tically thick that the red-shifted absorption is washed out by the
+large wings of the line. The red-shifted absorption is more pro-
+nounced for lines forming closer to the accretion shock like the
+Hβ line. At a temperature larger than 8,000 K and a mass accre-
+tion rate above 10−8 M⊙ yr−1, the continuum emission from the
+magnetosphere becomes important and the line-to-continuum ra-
+tio decreases. This effect is seen for instance in the Hα line (see
+Fig. A.1). When the mass accretion rate increases for a given
+temperature, the density of the magnetosphere increases. As a
+consequence, the line source function increases. At high temper-
+ature and high density, the background continuum emission of
+the magnetosphere dominates for certain wavelengths, and ab-
+sorption occurs. The latter effect is seen in the Paβ (Fig. A.3) and
+Brγ (Fig. A.4) lines where the strong continuum contribution at
+the disc surface leads to absorption at low velocities, where the
+lines source function is small. These results are consistent with
+the previous studies and demonstrate the robustness of our code
+for modelling the close environment of T Tauri stars (Tessore
+et al. 2021).
+Appendix B: Derivation of the interferometric
+pure-line phase and visibility
+We focus on the magnetospheric emission probed by the Brγ line
+and, therefore, aim to remove any additional contributions (stel-
+lar photosphere, the accretion shocks, dusty disc, etc...). Follow-
+ing Kraus et al. (2008); Bouvier et al. (2020b), we compute the
+continuum-subtracted observables, the so-called pure line visi-
+bility and phase, by using the emission line profiles computed in
+Sect. 3. This is of prime importance in the case of Brγ line as
+the magnetospheric emission is quite faint in the infrared (≈ 1.3
+excess flux compared to the continuum, Fig. 3). The deriva-
+tion of the pure line quantities is only possible if the source is
+marginally resolved (i.e, size < λ/2B).
+In this case, the pure line visibility Vline and phase Φline are
+given by:
+VLine = FL/CVTot − VCont
+FL/C − 1
+,
+(B.1)
+ΦLine = arcsin
+�
+FL/C
+FL/C − 1
+VTot
+VLine
+sin ΦTot
+�
+.
+(B.2)
+Where FL/C denotes the line-to-continuum flux ratio as taken
+from the normalised spectrum (Fig. 3), VCont is the visibility
+computed in the continuum (star+shock only), and VTot, ΦTot
+are the total complex quantities measured by GRAVITY. In
+Eq. (B.1), we note that in the case when FL/C is close to one,
+the derived VLine cannot exist (converges to infinity). Such non-
+ideal profiles appear if the red absorption becomes too important.
+Therefore, we assume to discard the affected spectral channels
+for phases 0.25 and 0.47, where FL/C is too close to one: – one
+point (v = 53 km s−1) at phases 0.25 and – two points (v = 15,
+53 km s−1) at phase 0.47.
+Acknowledgements. The authors thank Claudio Zanni, Lucas Labadie, Cather-
+ine Dougados, and Alexander Wojtczak for fruitful discussions. This project
+has received funding from the European Research Council (ERC) under the
+European Union’s Horizon 2020 research and innovation programme (grant
+agreement No 742095; SPIDI: Star-Planets-Inner Disk-Interactions, http://
+www.spidi-eu.org). B. Tessore thanks the french minister of Europe and
+foreign affairs and the minister of superior education, research and innova-
+tion (MEAE and MESRI) for research funding through FASIC partnership.
+C. Pinte acknowledges funding from the Australian Research Council via
+FT170100040 and DP180104235. The numerical simulations presented in this
+paper were performed with the Dahu supercomputer of the GRICAD infrastruc-
+ture (https://gricad.univ-grenoble-alpes.fr), which is supported by Grenoble re-
+search communities.
+Article number, page 9 of 13
+
+A&A proofs: manuscript no. bt_spidi2
+0.8
+1.0
+1.2
+M = 10
+9.0 M /yr
+1.0
+1.5
+M = 10
+8.0 M /yr
+1
+2
+6000 K
+M = 10
+7.0 M /yr
+0.75
+1.00
+1.25
+1
+2
+2
+4
+7000 K
+1.0
+1.5
+2.0
+2
+4
+6
+5
+10
+8000 K
+1000
+500
+0
+500
+1000
+2
+4
+1000
+500
+0
+500
+1000
+5
+10
+15
+1000
+500
+0
+500
+1000
+1
+2
+10000 K
+H
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+Fig. A.1: Dependence of Hα line flux with mass accretion rates ˙M and maximum temperatures Tmax. The inclination of the system
+is 60◦.
+0.9
+1.0
+M = 10
+9.0 M /yr
+0.8
+1.0
+M = 10
+8.0 M /yr
+0.75
+1.00
+1.25
+6000 K
+M = 10
+7.0 M /yr
+0.8
+1.0
+1.0
+1.5
+1
+2
+3
+7000 K
+0.75
+1.00
+1.25
+1
+2
+3
+2
+4
+6
+8000 K
+1000
+500
+0
+500
+1000
+1.0
+1.5
+2.0
+1000
+500
+0
+500
+1000
+2.5
+5.0
+7.5
+1000
+500
+0
+500
+1000
+1
+2
+3
+10000 K
+H
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+Fig. A.2: Same as Fig. A.1 for Hβ
+Article number, page 10 of 13
+
+B. Tessore & A. Soulain et al.: Spectroscopic and interferometric signatures of magnetospheric accretion
+1.00
+1.02
+M = 10
+9.0 M /yr
+0.98
+1.00
+1.02
+M = 10
+8.0 M /yr
+0.9
+1.0
+1.1
+6000 K
+M = 10
+7.0 M /yr
+0.98
+1.00
+1.02
+1.0
+1.2
+1.0
+1.5
+7000 K
+1.0
+1.1
+1.0
+1.5
+1
+2
+8000 K
+1000
+500
+0
+500
+1000
+1.0
+1.2
+1.4
+1000
+500
+0
+500
+1000
+1
+2
+1000
+500
+0
+500
+1000
+1.0
+1.1
+1.2
+10000 K
+Pa
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+Fig. A.3: Same as Fig. A.1 for Paβ.
+0.99
+1.00
+1.01
+M = 10
+9.0 M /yr
+0.99
+1.00
+1.01
+M = 10
+8.0 M /yr
+1.00
+1.05
+6000 K
+M = 10
+7.0 M /yr
+0.99
+1.00
+1.01
+0.95
+1.00
+1.05
+1.0
+1.2
+1.4
+7000 K
+1.000
+1.025
+1.0
+1.2
+1.0
+1.5
+8000 K
+1000
+500
+0
+500
+1000
+1.0
+1.1
+1.2
+1000
+500
+0
+500
+1000
+1.0
+1.5
+1000
+500
+0
+500
+1000
+1.00
+1.05
+1.10
+10000 K
+Br
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+v [km/s]
+F/Fc
+Fig. A.4: Same as Fig. A.1 for Brγ.
+Article number, page 11 of 13
+
+A&A proofs: manuscript no. bt_spidi2
+50
+25
+0
+25
+50
+U [M ]
+60
+40
+20
+0
+20
+40
+60
+V [M ]
+UT1-UT2
+UT1-UT3
+UT1-UT4
+UT2-UT3
+UT2-UT4
+UT3-UT4
+150
+100
+50
+0
+50
+100
+150
+V [m] - East
+150
+100
+50
+0
+50
+100
+150
+U [m] (2.17 µm) - North
+Fig. B.1: Fourier sampling (u-v coverage) of the simulated data.
+The different colours correspond to the six different baselines
+of the VLTI. The eight points per baseline represent a typical
+observational sequence with one data point per hour.
+Article number, page 12 of 13
+
+B. Tessore & A. Soulain et al.: Spectroscopic and interferometric signatures of magnetospheric accretion
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+

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+arXiv:2301.05338v1  [cs.DS]  13 Jan 2023
+Computing matching statistics on Wheeler DFAs
+Alessio Conte1, Nicola Cotumaccio2,3, Travis Gagie3, Giovanni Manzini1,
+Nicola Prezza4 and Marinella Sciortino5
+1 University of Pisa, Italy, alessio.conte@unipi.it, giovanni.manzini@unipi.it
+2 GSSI, L’Aquila, Italy, nicola.cotumaccio@gssi.it
+3 Dalhousie University, Halifax, Canada, nicola.cotumaccio@dal.ca, travis.gagie@dal.ca
+4 Ca’ Foscari Unversity, Venice, Italy, nicola.prezza@unive.it
+5 University of Palermo, Italy, marinella.sciortino@unipa.it
+Abstract
+Matching statistics were introduced to solve the approximate string matching problem, which is
+a recurrent subroutine in bioinformatics applications.
+In 2010, Ohlebusch et al. [SPIRE 2010]
+proposed a time and space efficient algorithm for computing matching statistics which relies on
+some components of a compressed suffix tree - notably, the longest common prefix (LCP) array. In
+this paper, we show how their algorithm can be generalized from strings to Wheeler deterministic
+finite automata. Most importantly, we introduce a notion of LCP array for Wheeler automata, thus
+establishing a first clear step towards extending (compressed) suffix tree functionalities to labeled
+graphs.
+Introduction
+Given a string T and a pattern π, the classical formulation of the pattern matching problem
+requires to decide whether the pattern π occurs in the string T and, possibly, count the
+number of such occurrences and report the positions where they occur. The invention of
+the FM-index [1], which is based on the Burrows-Wheeler transform [2], opened a new
+line of research in the pattern matching field. The indexing and compression techniques
+behind the FM-index deeply rely on the idea of suffix sorting, and over the years have
+been generalized from strings to trees [3], De Brujin graphs [4,5], Wheeler graphs [6,7] and
+arbitrary graphs [8, 9]. In particular, the class of Wheeler graphs is probably the one that
+captures the intuition behind the FM-index in the simplest way, and indeed the notion of
+Wheeler order has relevant consequences in automata theory [7,10].
+However, in bioinformatics we are not only interested in exact pattern matching, but
+also in a myriad of variations of the pattern matching problem [11]. In particular, matching
+statistics were introduced to solve the approximate pattern matching problem [12]. A pow-
+erful data structure that is able to address the variations of the pattern matching problem
+at once is the suffix tree [13]. The main drawback of the suffix tree is its space consumption,
+which is non-negligible both in theory and in practice. As a consequence, the suffix tree has
+been replaced by the suffix array [14]. While suffix arrays do not have all the functionalities
+of suffix trees, it has been shown that they can be augmented with some additional data
+structures — notably, the longest common prefix (LCP) array — so that it is possible to
+retrieve the full functionalities of a suffix trees [15]. All these components can be successfully
+compressed, leading to the so-called compressed suffix trees [16].
+
+The natural question is whether it is possible to provide suffix tree functionalities not
+only to strings, but also to graphs, and in particular Wheeler graphs. In this paper, we
+provide a first partial affirmative answer by considering the problem of computing matching
+statistics. In 2010, Ohlebusch et al. [17] proposed a time and space efficient algorithm for
+computing matching statistics which relies on some components of a compressed suffix tree.
+In this paper, we show how their algorithm can be generalized from strings to Wheeler deter-
+ministic finite automata. Most importantly, we introduce a notion of longest common prefix
+(LCP) array for Wheeler automata, thus establishing an important step towards extending
+(compressed) suffix tree functionalities to labeled graphs.
+Notation and first definitions
+Throughout the paper, we consider an alphabet Σ and a fixed total order ⪯ on Σ. We
+denote by Σ∗ the set of all finite strings on Σ and by Σω the set of all (countably) infinite
+strings on Σ. The empty word is ǫ. If α ∈ Σ∗, then αR is the reverse string of α. We extend
+the total order ⪯ from Σ to Σ∗ ∪ Σω lexicographically. If i and j are integers, with i ≤ j,
+define [i, j] = {i, i + 1, . . . , j − 1, j}. If T is a string, the i-th character of T is T[i], and
+T[i..j] = T[i]..T[j].
+We will consider deterministic automata A = (Q, E, s0, F), where Q is the set of states,
+E ⊆ Q × Q × Σ is the set of labeled edges, s0 ∈ Q is the initial state and F ⊆ Q is the set
+of final states. The definition implies that for every u ∈ Q and for every a ∈ Σ there exists
+at most one edge labeled a leaving u. Following [7,10], we assume that s0 has no incoming
+edges, and every state is reachable from the initial state; moreover, all edges entering the
+same state have the same label (input-consistency), so that for every u ∈ Q \ {s0} we can
+let ��(u) be the label of all edges entering u. We define λ(s0) = #, where # ̸∈ Σ is a special
+character for which we assume # ≺ a for every a ∈ Σ (the character # is an analogous of
+the termination character $ used for suffix trees and suffix arrays). As a consequence, an
+edge (u′, u, a) can be simply written as (u′, u), because it must be a = λ(u).
+We assume familiarity with the notions of suffix array (SA), Burrows Wheeler transform
+(BWT), FM-index and backward search [1].
+The matching statistics of a pattern π = π[1..m] with respect to a string T = T[1..n] are
+defined as follows. Assume that T[n] = $ ̸∈ Σ, where $ ≺ a for every a ∈ Σ. Determining
+the matching statistics of π with respect to T means determining, for 1 ≤ i ≤ m, (i) the
+longest prefix π′ of π[i..m] which occurs in T, and (ii) the interval corresponding to the
+set of all strings starting with π′ in the list of all lexicographically sorted suffixes. We can
+describe (i) and (ii) by means of three values: the length ℓi of π′, and the endpoints li and
+ri of the interval considered in (ii). For example, let T = mississippi$ (see Figure 1), and
+π = stpissi. For i = 1, we have π′ = s, so ℓ1 = 1 and [l1, r1] = [9, 12] (suffixes starting with
+s). For i = 2, we have π′ = ǫ, so ℓ2 = 0 and [l2, r2] = [1, n] = [1, 12] (all suffixes start with
+the empty string). For i = 3, we have π′ = pi, so ℓ3 = 2, and [l3, r3] = [7, 7] (suffixes starting
+with pi). For i = 4, we have π′ = issi, so ℓ4 = 4, and [l4, r4] = [4, 5] (suffixes starting with
+issi). One can proceed analogously for i = 5, 6, 7.
+
+i
+Sorted suffixes
+LCP
+SA
+BWT
+1
+$
+12
+i
+2
+i$
+0
+11
+p
+3
+ippi$
+1
+8
+s
+4
+issippi$
+1
+5
+s
+5
+ississippi$
+4
+2
+m
+6
+mississippi$
+0
+1
+$
+7
+pi$
+0
+10
+p
+8
+ppi$
+1
+9
+i
+9
+sippi$
+0
+7
+s
+10
+sissippi$
+2
+4
+s
+11
+ssippi$
+1
+6
+i
+12
+ssissippi$
+3
+3
+i
+Figure 1: The sorted suffixes of “mississippi$” and the LCP, SA, and BWT arrays.
+Computing matching statistics for strings
+We will first describe the algorithm by Ohlebusch et al. [17], emphasizing the ideas that we
+will generalize when switching to Wheeler DFAs. The algorithms computes the matching
+statistics using a number of iterations linear in m by exploiting the backward search. We
+start from the end of π, and we use the backward search (starting from the interval [1, n]
+which corresponds to the set of suffixes prefixed by the empty string) to find the interval of
+all occurrences of the last character of π in T (if any). Then, starting from the new interval,
+we use the backward search to find all the occurrences of the suffix of length 2 of π in T (if
+any), and so on. At some point, it may happen that for some i ≤ m+1 we have that π[i..m]
+occurs in T, but the next application of the backward search returns the empty interval, so
+that π[i − 1..m] does not occur in T (the case i = m + 1 corresponds to the initial setting
+when π[i..m] is the empty string). We distinguish two cases:
+• (Case 1) If li = 1 and ri = n, this means that all suffixes of T are prefixed by π[i..m].
+This may happen in particular if i = m + 1: this means that the first backward search
+has been unsuccessful. We immediately conclude that character π[i−1] does not occur
+in T, so ℓi−1 = 0 and [li−1, ri−1] = [1, n] (because all suffixes start with the empty
+string). In this case, in the following iterations of the algorithm, we can simply discard
+π[i − 1, m]: when for i′ ≤ i − 2 we will be searching for the longest prefix of π[i′, m]
+occurring in T, it will suffice to search for the longest prefix of π[i′, i − 2] occurring in
+T.
+• (Case 2) If li > 1 or ri < n, this means that the number of suffixes of T starting with
+π[i..m] is less than n. Now, every suffix starting with π[i..m] also starts with π[i..m−1].
+If the number of suffixes starting with π[i..m − 1] is equal to the number of suffixes
+starting with π[i..m], then also π[i−1..m−1] does not occur in T. More in general, for
+j ≤ m−1 we can have that π[i−1..j] occurs in T only if the number of suffixes starting
+with π[i..j] is larger than the number of suffixes starting with π[i..m]. Since we are
+interested in maximal matches, we want j to be as large as possible: we will show later
+
+how to compute the largest integer j such that the number of suffixes starting with
+π[i..j] is larger than the number of suffixes starting with π[i..m]. Notice that j always
+exists, because all n suffixes start with the empty string, but less than n suffixes start
+with π[i..m]. After determining j we discard π[j + 1..m] (so in the following iterations
+of the algorithm we will simply consider π[1..j]), and we recursively apply the backward
+search starting from the interval associated with the occurrences of π[i..j] — we will
+also see how to compute this interval.
+Let us apply the above algorithm to T = mississippi$ and π = stpissi. We start with
+the interval [1, n] = [1, 12], corresponding to the empty pattern, and character π[7] = i. A
+backward step yields the interval [l7, r7] = [2, 5] (suffixes starting with i), so ℓ7 = 1. Now, we
+apply a backward step from [2, 5] and π[6] = s, obtaining [l6, r6] = [9, 10] (suffixes starting
+with si), so ℓ6 = 2. Again, we apply a backward step from [9, 10] and π[5] = s, obtaining
+[l5, r5] = [11, 12] (suffixes starting with ssi), so ℓ5 = 3. Again, we apply a backward step
+from [11, 12] and π[4] = i, obtaining [l4, r4] = [4, 5] (suffixes starting with issi), so ℓ4 = 4.
+We now apply a backward step from [4, 5] and π[3] = p, and we obtain the empty interval.
+This means that no suffix starts with pissi. Notice in Figure 1 that the number of suffixes
+starting with issi is equal to the number of suffixes starting with iss or is, but the number
+of suffixes starting with i is bigger. As a consequence, we consider the interval of all suffixes
+starting with i — which is [2, 5] — and we apply a backward step with π[3] = p. This time
+the backward step is successful, and we obtain [l3, r3] = [7, 7] (suffixes starting with pi), and
+ℓ3 = 2. We now apply a backward step from [7, 7] and π[2] = t, obtaining the empty interval.
+This means that no suffix starts with tpi. Notice in Figure 1 that the number of suffixes
+starting with p is bigger than the number of suffixes starting with pi. The corresponding
+interval is [7, 8], but a backward step with π[2] = t is still unsuccessful (so no suffix starts
+with tp).
+The number of suffixes starting with p is smaller than the number of suffixes
+starting with the empty string (which is equal to n = 12), so we apply a backward step with
+[1, 12] and π[2] = t. Since the backward step is still unsuccessful, we conclude that π[2] = t
+does not occur in S, so [l2, r2] = [1, n] = [1, 12] and ℓ2 = 0. Finally, we start again from the
+whole interval [1, 12], and a backward step with π[1] = s returns [l1, r1] = [9, 12] (suffixes
+starting with s), so ℓ1 = 1.
+It is easy to see that the number of iterations is linear in m. Indeed, every time we apply
+a backward step, either we move to the left across π to compute a new matching statistic,
+or we increase by at least 1 the length of the suffix of π which is forever discarded. This
+implies that the number of iterations is bounded by 2|π| = 2m.
+We are only left with showing (i) how to compute j and (ii) the interval of all suffixes
+starting with π[i..j] in Case 2 of the algorithm.
+To this end, we introduce the longest
+common prefix (LCP) array LCP = LCP[2, n] of T. We define LCP[i] to be the length of the
+longest common prefix of the (i − 1)-st lexicographically smallest suffix of T and the i-th
+lexicographically smallest suffix of T. In Figure 1 we have LCP[5] = 4 because the fourth
+lexicographically smallest suffix of T is issippi$, the fifth lexicographically smallest suffix of
+T is ississippi$, and the longest common prefix of issippi$ and ississippi$ is issi, which has
+length 4. Remember that in the example the backward search starting from [4, 5] (suffixes
+starting with issi) and p was unsuccessful, so computing j means determining the longest
+prefix of issi such that the the number of suffixes starting with such a prefix is bigger than 2.
+
+2
+5
+6
+7
+8
+9
+3
+4
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+1
+start
+a
+a
+a
+a
+a
+b
+b
+b
+c
+c
+d
+d
+e
+e
+e
+f
+g
+h
+i
+l
+a
+Figure 2: A Wheeler DFA. States are numbered according to their positions in the Wheeler
+order.
+This is easy to compute by using the LCP array: the longest such prefix is the one of length
+max{LCP[4], LCP[6]} = max{1, 0} = 1, so that the desired prefix is i. As a consequence,
+we are only left with showing how to compute the interval of all suffixes starting with the
+prefix i — which is [2, 5]. Notice that in order to compute this interval, it is enough to
+expand the interval [4, 6] in both directions as long as the LCP value does not go below 1.
+Since LCP[4] = 1, LCP[3] = 1, and LCP[2] = 0, and we already know that LCP[6] = 0, we
+conclude that the desired interval is [2, 5]. In other words, given a position t, we must be
+able to compute the biggest integer k less than t such that LCP[k] < LCP[t], and the smallest
+integer k bigger than t such that LCP[k] < LCP[t] (in our case, t = 4). These queries are
+called PSV (“previous smaller value”) and NSV (“next smaller value”) queries. The LCP
+array can be augmented in such a way that PSV and NSV queries can be solved efficiently:
+different space-time trade-offs are possible, we refer the reader to [17] for details.
+Matching statistics for Wheeler DFAs
+Let us define Wheeler DFAs [7].
+Definition 1. Let A = (Q, E, s0, F) be a DFA. A Wheeler order on A is a total order ≤ on
+Q such that s0 ≤ u for every u ∈ Q and:
+(Axiom 1) If u, v ∈ Q and u < v, then λ(u) ⪯ λ(v).
+(Axiom 2) If (u′, u), (v′, v) ∈ E, λ(u) = λ(v) and u < v, then u′ < v′.
+A DFA A is Wheeler if it admits a Wheeler order.
+It is immediate to check that this definition is equivalent to the one in [7], where it was
+shown that if a DFA A admits a Wheeler order ≤, then ≤ is uniquely determined (that is,
+
+≤ is the Wheeler order on A). In the following, we fix a Wheeler DFA A = (Q, E, s0, F),
+where we assume Q = {u1, . . . , un}, with u1 < u2 < · · · < un in the Wheeler order, and u1
+coincides with the initial state s0. See Figure 2 for an example.
+We now show that a Wheeler order can be seen of as a permutation of the set of all states
+playing the same role as the suffix array of a string. In the following, it will be expedient
+to (conceptually) assume that s0 has a self-loop labeled # (this is consistent with Axiom 1,
+because # ≺ a for every a ∈ Σ). This implies that every state has at least one incoming
+edge, so for every state ui there exists at least one infinite string α ∈ Σω that can be read
+starting from ui and following edges in a backward fashion (for example, in Figure 2 for u9
+such a string is cel### . . . ). We denote by Iui the set of all such strings. Formally:
+Definition 2. Let i ∈ [1, n]. For every state ui ∈ Q define:
+Iui = {α ∈ Σω | there exist integers f1, f2, . . . in [1, n] such that (i) f1 = i,
+(ii) (ufk+1, ufk) ∈ E for every k ≥ 1 and (iii) α = λ(uf1)λ(uf2) . . . }.
+For example, in Figure 2 we have Iu3 = {abdg### . . . , abeh### . . . , acei### . . . }.
+The following lemma shows that the permutation of the states defined by the Wheeler
+order is the one lexicographically sorting the strings entering each state, just like the permu-
+tation defined by the suffix array lexicographically sorts the suffixes of the strings (a suffix
+is seen as a string “leaving” a text position).
+Lemma 3. Let i, j ∈ [1, n], with i < j. Let α ∈ Iui and β ∈ Iuj. Then, α ⪯ β.
+Proof. Let f1, f2, . . . in [1, n] be such that (i) f1 = i, (ii) (ufk+1, ufk) ∈ E for every k ≥ 1 and
+(iii) α = λ(uf1)λ(uf2) . . . . Analogously, let g1, g2, . . . in [1, n] be such that (i) g1 = j, (ii)
+(ugk+1, ugk) ∈ E for every k ≥ 1 and (iii) β = λ(ug1)λ(ug2) . . . . Let α ̸= β. We must prove
+that α ≺ β. Let p ≥ 1 be the smallest integer such that the p-th character of α is different
+than the p-th character of β. In other words, we know that λ(uf1) = λ(ug1), λ(uf2) = λ(ug2),
+. . . , λ(ufp−1) = λ(ugp−1), but λ(ufp) ̸= λ(ugp). We must prove that λ(ufp) ≺ λ(ugp). Since
+λ(uf1) = λ(ug1) f1 = i < j = g1, and (uf2, uf1), (ug2, ug1) ∈ E, from Axiom 2 we obtain
+f2 < g2. Since λ(uf2) = λ(ug2), f2 < g2, and (uf3, uf2), (ug3, ug2) ∈ E, from Axiom 2 we
+obtain f3 < g3. By iterating this argument, we conclude fp < gp. By Axiom 1, we obtain
+λ(ufp) ⪯ λ(ugp). Since λ(ufp) ̸= λ(ugp), we conclude λ(ufp) ≺ λ(ugp).
+If we think of a string as a labeled path, then the suffix array sorts the strings that can
+be read from each position by moving forward (that is, the suffixes of the string), while the
+Wheeler order sorts the strings that can be read from each position by moving backward
+towards the initial state. The underlying idea is the same: the forward vs backward difference
+is only due to historical reasons [6]. To compute the matching statistics on Wheeler DFA
+we reason as in the previous section replacing backward search with the forward search [6]
+defined as follows: given an interval [i, j] in [1, n] and a ∈ Σ, find the (possibly empty)
+interval [i′, j′] in [1, n] such that a state vk′ is reachable from some state vk, with i ≤ k ≤ j,
+through an edge labeled a, if and only if i′ ≤ k′ ≤ j′ (this easily follows by using the axioms of
+Definition 1). For a constant size alphabet, given [i, j] and a then [i′, j′] can be determined in
+constant time. Given a string π ∈ Σ∗, if we start from the whole set of states and repeatedly
+apply the forward search we reach the set of all states ui for which there exists α ∈ Iui
+
+prefixed by πR; this is an interval with respect to the Wheeler order: in the following we call
+this interval T(π).
+Because of the forward vs backward difference the problem of matching statistics will be
+defined in a symmetrical way on Wheeler DFAs. Given a pattern π = π[1..m], for every
+1 ≤ i ≤ m we want to determine (i) the longest suffix π′ of π[1..i] which occurs in the
+Wheeler DFA A (that is, that can be read somewhere on A by concatenating edges), and
+(ii) the endpoints of the interval T(π′).
+Broadly speaking, we can apply the same idea of the algorithm for strings, but in a
+symmetrical way. We start from the beginning of π (not from the end of π), and initially we
+consider the whole set of states. We repeatedly apply the forward search (not the backward
+search), until the forward search returns the empty interval for some i ≥ 0. This means that
+π[1..i+1] does not occur in A. Then, if T(π[1..i]) is the whole set of states, we conclude that
+the character π[i + 1] labels no edge in the graph. Otherwise, we must find the smallest j
+such that T(π[1..i]) is strictly contained in T(π[j..i]) (that is, we must determine the longest
+suffix π[j..i] of π[1..i] which reaches more states than π[1..i]). Then we must determine the
+endpoints of the interval T(π[j..i]) so that we can go on with the forward search.
+The challenge now is to find a way to solve the same subproblems that we identified in
+Case 2 of the algorithm for strings. In other words, we must find a way to determine j and
+find the endpoints of the interval T(π[j..i]). We will show that the solution is not as simple
+as the one for the algorithm on strings.
+The LCP array and matching statistics for Wheeler DFAs
+We start observing that Iui may be an infinite set. For example, in Figure 2, we have
+Iu2 = {aaaaa . . . , abdf### . . . , aabdf### . . . , aaabdf### . . . , . . . }.
+In general, an infinite set of (lexicographically sorted) strings in Σω need not admit a
+minimum or a maximum. For example, the set {baaaa . . . , abaaa . . . , aabaa . . . , aaaba . . . }
+does not admit a minimum (but only the infimum string aaaaa . . . ). Nonetheless, Lemma 3
+implies that each Iui admits both a minimum and a maximum. For example, the minimum
+is obtained as follows. Let f1 = i, and for every k ≥ 1, recursively let fk+1 be the smallest
+integer in [1, n] such that (ufk+1, ufk) ∈ E. Then, the minimum of Iui is λ(uf1)λ(uf2) . . . ,
+and analogously one can determine the maximum.
+In the following, we will denote the minimum and the maximum of Iui by mini and maxi,
+respectively (for example, in Figure 2 we have min2 = aaaaa . . . , and max2 = abdf### . . . ).
+Lemma 3 implies that:
+min1 ⪯ max1 ⪯ min2 ⪯ max2 ⪯ · · · ⪯ maxn−1 ⪯ minn ⪯ maxn.
+This suggests to generalize the LCP array as follows. Given α, β ∈ Σ∗ ∪ Σω, let lcp(α, β) be
+the length of the longest common prefix of α and β (if α = β ∈ Σω, define lcp(α, β) = ∞).
+Definition 4. The LCP-array of a Wheeler automaton A is the array LCPA = LCPA[2, 2n]
+which contains the following 2n − 1 values in this order: lcp(min1, max1), lcp(max1, min2),
+lcp(min2, max2), . . . , lcp(maxn−1, minn), lcp(minn, maxn).
+
+From the above characterization of mini and maxi, one can prove that for every entry
+either LCPA[i] = ∞ or LCPA[i] < 3n (it follows from Fine and Wilf Theorem [18,19]), and
+one can design a polynomial time algorithm to compute LCPA.
+Unfortunately, the array LCPA alone is not sufficient for computing matching statistics.
+Assume that T(π) = {ur, ur+1, . . . , us−1, us}, and that when we apply the forward search by
+adding a character c, we obtain T(πc) = ∅. We must then determine the largest suffix π′
+of T(π) such that T(π) is strictly contained in T(π′). Suppose that every string in Iur is
+prefixed by πR, and every string in Ius is prefixed by πR. In particular, both minr and maxs
+are prefixed by πR. In this case, we can proceed like in the algorithm for strings: the desired
+suffix π′ is the one having length max{lcp(maxr−1, minr), lcp(maxs, mins+1)}, which can be
+determined using LCPA. However, in general, even if some string in Iur must be prefixed
+by πR, the string minr need not be prefixed by πR, and similarly maxs need not be prefixed
+by πR. The worst-case scenario occurs when r = s. Consider Figure 2, and assume that
+π = heba. Then, we have r = s = 3 (note that abeh### . . . is a string in Iu3 prefixed by
+πR). However, both min3 = abdg### . . . , and max3 = acei### . . . , are not prefixed by
+πR. Notice that lcp(max2, min3) = 3 and lcp(max3, min4) = 3, but π′ is not the suffix of
+length 3 of π. Indeed, since min3 is only prefixed by the prefix of πR of length 2, and max3
+is only prefixed by the prefix of πR of length 1, we conclude that it must be |π′| = 2. In
+general, the desired suffix π′ is the one having length |π′| given by:
+max
+�
+min{lcp(maxr−1, minr),lcp(minr, πR)}, min{lcp(πR, maxs),lcp(maxs, mins+1)}
+�
+. (1)
+The above formula shows that, in order to compute π′, in addition to LCPA it suffices to
+know the values lcp(minr, πR) and lcp(πR, maxs) (π′ is a suffix of π, so it is determined by
+its length). We now show how our algorithm can efficiently maintain the current pattern π,
+the set T(π) = {ur, ur+1, . . . , us−1, us} and the values lcp(minr, πR) and lcp(πR, maxs) during
+the computation of the matching statistics. We assume that the input automaton is encoded
+with the rank/select data structures supporting the execution of a step of forward search in
+O(log |Σ|) time, see [6] for details. In addition, we will use the following result.
+Lemma 5. Let A[1, n] be a sequence of values over an ordered alphabet Σ. Consider the
+following queries: (i) given i, j ∈ [1..n], compute the minimum value in S[i..j], and (ii)
+given t ∈ [1..n] and c ∈ Σ, determine the biggest k < t (or the smallest k > t) such that
+A[k] < c. Then, A can be augumented with a data structure of 2n+o(n) bits such that query
+(i) can be answered in constant time and query (ii) can be answered in O(log n) time.
+Proof. There exists a data structure of 2n + o(n) bits that allows to solve range minimum
+queries in constant time [20], so using A we can solve queries (i) in constant time. Now, let
+us show how to solve queries (ii). Let f1 be the answer of query (i) on input i = ⌈t/2⌉ and
+j = t − 1. If f1 < c, then we must keep searching in the interval [⌈t/2⌉, t − 1], otherwise, we
+must keep searching in the interval [1, ⌈t/2⌉ − 1]. In other words, we can answer a query (ii)
+by means of a binary search on [1, t − 1], which takes O(log t) (and so O(log n)) time.
+Notice that query (ii) can be seen as a variant of PSV and NSV queries. In the following,
+we assume that the array LCPA has been augmented with the data structure of Lemma 5.
+At the beginning we have π = ǫ, so T(ǫ) = {1, 2, . . . , n} and trivially lcp(minr, πR) =
+lcp(πR, maxs) = 0. At each iteration we perform a step of forward search computing T(πc)
+given T(π); then we distinguish two cases according to whether T(πc) is empty or not.
+
+Case 1. T(πc) = {ur′, ur′+1, . . . , us′−1, us′} is not empty. In that case πc will become the
+pattern at the next iteration. Since we already have T(πc) we are left with the task of com-
+puting lcp(minr′, cπR) and lcp(cπR, maxs′). We only show how to compute lcp(minr′, cπR),
+the latter computation being analogous. Let k be the smallest integer in [1, n] such that
+(uk, ur′) ∈ E. Notice that we can easily compute k by means of standard rank/select opera-
+tions on the compact data structure used to encode A. Since ur′ ∈ T(πc), it must be k ≤ s.
+Moreover, the characterization of minr′ that we described above implies that minr′ = c mink,
+hence lcp(minr′, cπR) = lcp(c mink, cπR) = 1 + lcp(mink, πR). To compute lcp(mink, πR) we
+distinguish two subcases:
+a) k > r, hence r < k ≤ s. Since ur, us ∈ T(π), there exist α ∈ Iur and β ∈ Ius both
+prefixed by πR. But α ⪯ maxr ⪯ mink ⪯ mins ⪯ β, so mink is also prefixed by πR,
+and we conclude lcp(mink, πR) = |π|.
+b) k ≤ r. In this case, we have mink ⪯ maxk ⪯ mink+1 ≺ maxk+1 ⪯ · · · ⪯ minr ≺ πR,
+and therefore lcp(mink, πR) is equal to
+min{lcp(mink, maxk), lcp(maxk, mink+1), lcp(mink+1, maxk+1), . . . , lcp(minr, πR)}.
+With the above formula we can compute lcp(mink, πR) using query (i) of Lemma 5 over
+the range LCPA[2k, 2r − 1] and the value lcp(minr, πR).
+Case 2. T(πc) is empty. In this case at the next iteration the pattern will be largest suffix
+π′ of π such that T(π) is strictly contained in T(π′) = {ur′′, . . . , us′′}. We compute |π′|
+using (1); if |π′| > lcp(minr, πR) we set r′′ = r, otherwise we apply query (ii) of Lemma 5 to
+find the rightmost entry r′′ in LCPA[2, 2r − 1] smaller than |π′|. Computing s′′ is analogous.
+Given T(π′) = {ur′′, ur′′+1, . . . , us′′−1, us′′}, where r′′ ≤ r, s ≤ s′′, and at least one inequal-
+ity is strict, we want to compute lcp(minr′′, (π′)R) and lcp((π′)R, maxs′′). We only consider
+lcp(minr′′, (π′)R), the latter computation being analogous. We distinguish two subcases:
+a) r′′ = r. Then lcp(minr′′, (π′)R) = lcp(minr, (π′)R) = min{lcp(minr, πR), |π′|}.
+b) r′′ < r. In particular, since ur′′ is the left endpoint of T(π′) and |T(π′)| ≥ 2, one can
+prove like in Case 1a) that maxr′′ is prefixed by (π′)R. We immediately conclude that
+lcp(minr′′, (π′)R) = min{lcp(minr′′, maxr′′), |π′|}, which can be immediately computed
+since lcp(minr′′, maxr′′) is a value stored in LCPA.
+We can summarize the above discussion as follows.
+Theorem 6. Given a Wheeler DFA A, there exists a data structure occupying O(|A|) words
+which can compute the pattern matching statistics of a pattern P in time O(|P| log |A|).
+Funding
+TG funded by National Institutes of Health (NIH) NIAID (grant no. HG011392),
+the National Science Foundation NSF IIBR (grant no. 2029552) and a Natural Science and
+Engineering Research Council (NSERC) Discovery Grant (grant no. RGPIN-07185-2020).
+GM funded by the Italian Ministry of University and Research (PRIN 2017WR7SHH). MS
+funded by the INdAM-GNCS Project (CUP E55F22000270001). NP funded by the European
+Union (ERC, REGINDEX, 101039208). Views and opinions expressed are however those
+of the author(s) only and do not necessarily reflect those of the European Union or the
+European Research Council. Neither the European Union nor the granting authority can be
+held responsible for them.
+
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+Springer Berlin Heidelberg.
+

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+1 
+ 
+Exploring Euroleague History using Basic Statistics 
+ 
+ 
+ 
+Christos Katris1,2 
+ 
+1Adjunct Lecturer, Department of Mathematics, University of Patras 
+2Customs Officer (Statistician), Independent Authority for Public Revenue, Greece 
+ 
+1chriskatris@upatras.gr, 2c.katris1@aade.gr 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+2 
+ 
+Abstract 
+  
+In this paper are used historical statistical data to track the evolution of the game in the European-wide 
+top-tier level professional basketball club competition (until 2017-2018 season) and also are answered 
+questions by analyzing them. The term basic is referred because of the nature of the data (not available 
+detailed statistics) and of the level of aggregation (not disaggregation to individual level). We are 
+examining themes such as the dominance per geographic area, the level of the competition in the game, 
+the evolution of scoring pluralism and possessions in the finals, the effect of a top scorer in the 
+performance of a team and the existence of unexpected outcomes in final fours. For each theme under 
+consideration, available statistical data is specified and suitable statistical analysis is applied. The analysis 
+allows us to handle and answer the above themes and interesting conclusions are drawn. This paper can 
+be an example of statistical thinking in basketball problems by the means of using efficiently available 
+statistical data.  
+ 
+Keywords:  Statistical analysis, basketball statistics, Euroleague evolution.    
+ 
+1. Introduction 
+The field of basketball is ideal for the application of statistical methods in order to extract useful 
+conclusions which can help in analyzing many aspects of the game. The origin of many ideas is from 
+persons outside academia. The book of Oliver (2004) was a worthy attempt to develop and apply 
+statistical concepts in the area of basketball. Much information is included on this book and can offer to a 
+reader a statistical way of thinking for the game of basketball. There are also many academic papers 
+which use advanced statistical methods for basketball analysis in themes such as performance evaluation 
+of players and teams, home advantage effect etc. The field of basketball analytics is not yet entirely 
+unified and new ideas which are based on quantitative analysis are appearing continuously from diverse 
+academic fields. In many cases, there are used advanced statistics for the analysis of many situations. The 
+majority of studies – not only with USA origin - are related to NBA and this is not just a coincidence. The 
+
+3 
+ 
+tracking system of statistics is superior to other leagues in terms of quality (calculation of more advanced 
+statistics) and quantity (calculation of more statistical categories), and the discrepancy was larger 
+especially in the past.  
+ 
+ The paper of Kubatko et al (2007) presents the general accepted basics of the analysis of basketball. 
+Furthermore, most of the statistics are based on the concept of possessions. However, this is not the case 
+for other leagues, including Euroleague. Given the available statistical data is difficult or even impossible 
+(for older years) to calculate neither possessions nor advanced statistics. Only after 2001 in the modern 
+era of Euroleague, plenty of statistics are available.  
+ 
+This paper is an attempt to utilize available statistical information through statistical analysis in order to 
+explore the evolution of the game in Euroleague. It is demonstrated that even simple available statistics 
+can offer insights about the game and can be extracted useful conclusions. Graphical analysis, statistical 
+hypothesis testing and correlation measures are our weapons in this chase of insights related to the 
+evolution of Euroleague. The next section is a brief description of Euroleague and are referred the sources 
+of statistical data. Section 3 is the main part of the paper and contains statistical analysis and methods to 
+deal with questions related to the historical evolution of the tournament. Finally, in Section 4 are 
+presented the conclusions of the analysis.         
+  
+2. A Brief History of Euroleague and Statistical Data 
+In this paper is examined the evolution of the European-wide top-tier level professional basketball club 
+competition. Briefly the history of the competition is following. The FIBA European Champions Cup 
+competition has established in 1958 and FIBA was organizing its operation until 2000. Then Euroleague 
+Basketball was created. The next year, the two competitions were unified again under the umbrella of 
+Euroleague Basketball (for more details: https://en.wikipedia.org/wiki/EuroLeague). Also the competition 
+has changed names across time. From 1958 to 1991 was the FIBA European Champions Cup, from 1991 
+to 1996 the name of the competition was FIBA European League, from 1996 to 2000 the name was FIBA 
+
+4 
+ 
+Euroleague. In season 2000-2001 there were 2 competitions: FIBA Suproleague which was organized by 
+FIBA and Euroleague which was organized by Euroleague Basketball. From the next year there was a 
+unique competition for the top-tier level under the name Euroleague which was organized by Euroleague 
+Basketball. In 2016 the name changed to EuroLeague. For the rest of the article the name Euroleague is 
+used for the whole competition. The concept of final four applied for 1965-1966 and 1966-1967 seasons 
+and was included permanently in the competition from the season of 1987-1988. In this paper, we 
+consider as final four teams before 1986-1987, the teams which have reached the semi-finals in order to 
+generate a consistent system for studying the evolution of the tournament.   
+ 
+There is not a unique data source which contains all information from the beginning of the tournament in 
+1958. Statistical data sources which were used are: Wikipedia, http://pearlbasket.altervista.org, 
+http://www.linguasport.com and http://www.fibaeurope.com/ and http://www.euroleague.net/ for stats 
+after 2001.    
+ 
+3. Statistical Analysis of Euroleague Historical Data 
+In this section is made an attempt to shed light to questions related to the historical evolution of the game 
+with the use of suitable statistical methods. The graphs are created in excel, whilst for the implementation 
+of the methods is used statistical software R.  
+ 
+ 
+3.1 Dominance on the Game per Geographic Area  
+Firstly, we can derive some quick conclusions about the dominance in the game in terms of geographic 
+location. Table 1 displays per country the winners, the runners-up and the number of teams which had 
+appeared to final fours. We consider only the teams which participated to final fours since 1958. From 
+this limited statistical information we will explore briefly the game over time.   
+ 
+
+5 
+ 
+Based on Table 1, we consider the following Geographic areas: Spain and Italy which are leading the 
+table in all categories are considered separately, Ex USSR and ex Yugoslavian countries form the next 
+area and every other country is assigned to a fourth group (other).  
+Fig.1 displays the titles per time period of teams from each geographic area and Fig.2 displays the 
+appearances in final fours of teams from each geographic area.   
+ 
+Table 1. Titles and appearances per country 
+Country 
+Winner 
+Runner-Up 
+Final Four Appearances 
+Number of Teams 
+Spain 
+13 
+16 
+57 
+6 
+Italy  
+13 
+13 
+44 
+9 
+Greece 
+9 
+7 
+13 
+5 
+Russia1 
+7 
+6 
+17 
+2 
+Israel 
+6 
+9 
+20 
+1 
+Croatia2 
+5 
+1 
+3 
+3 
+Latvia1 
+3 
+1 
+4 
+1 
+Turkey 
+1 
+2 
+6 
+2 
+Lithuania1 
+1 
+1 
+3 
+1 
+Georgia1 
+1 
+1 
+3 
+1 
+Bosnia2 
+1 
+0 
+4 
+1 
+Serbia2 
+1 
+0 
+10 
+4 
+France 
+1 
+0 
+9 
+4 
+Czech Republic3 
+0 
+3 
+9 
+2 
+Bulgaria 
+0 
+2 
+2 
+1 
+Slovenia2 
+0 
+0 
+3 
+1 
+Poland 
+0 
+0 
+2 
+2 
+Romania 
+0 
+0 
+1 
+1 
+Netherlands 
+0 
+0 
+1 
+1 
+Hungary 
+0 
+0 
+1 
+1 
+Belgium 
+0 
+0 
+1 
+1 
+1 Trophies won before 1991 were under the umbrella of Soviet Union 
+2 Trophies won before 1995 were under the umbrella of Yugoslavia 
+3 Trophies won before 1991 were under the umbrella of Czechoslovakia 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+6 
+ 
+Fig.1. Titles evolution per geographic area 
+ 
+ 
+ 
+Fig.2. Appearances to Final Four per geographic area 
+ 
+ 
+ 
+ 
+ 
+ 
+1958-1970
+1971-1980
+1981-1990
+1991-2000
+2001-2010
+2011-2018
+Spain
+4
+3
+0
+2
+2
+2
+Italy
+2
+4
+5
+1
+1
+0
+Ex USSR and Yugoslavian
+7
+2
+4
+3
+2
+1
+Other
+0
+1
+1
+4
+6
+5
+0
+1
+2
+3
+4
+5
+6
+7
+8
+Titles
+Titles Evolution per Geographic Area
+1958-1970
+1971-1980
+1981-1990
+1991-2000
+2001-2010
+2011-2018
+Spain
+10
+9
+7
+11
+10
+10
+Italy
+6
+12
+9
+7
+9
+1
+Ex USSR and Yugoslavian
+20
+9
+12
+6
+10
+9
+Other
+16
+10
+12
+16
+15
+12
+0
+5
+10
+15
+20
+25
+Appearances
+Appearances to Final Four per geographic area
+
+7 
+ 
+To test formally if there are significant differences to the appearances and to the titles per geographic 
+area, we perform Friedman tests with titles (or appearances) per geographic area as treatments and time 
+periods as blocks (a blocking factor is a source of variability which is not of primary interest). We want to 
+check for significant differences to the titles and appearances per geographic area. Note that we want to 
+overall check the titles and appearances and not the trend, and we consider the time periods as blocks in 
+order to reduce their effect to the variability of titles and appearances respectively. The non-parametric 
+Friedman test is used in order not to have distributional assumptions, because normality assumption (data 
+to follow normal distribution) does not seem very likely. Details about the test can be found in every book 
+of non-parametric statistics such as that of (Hollander and Wolfe, 1999).  
+ 
+The null hypothesis (𝐻0) is that apart of the effect of time period (blocks) there is no difference in titles 
+(or appearances) are even between the considered regions. The level of significance is considered at 5% 
+(0.05). To reject the null hypothesis, the p-value should be less than 0.05.  
+Friedman Test 
+ 
+Statistic 
+df 
+p-value 
+Appearances 
+6.5789 
+3 
+0.0866 
+Titles 
+0.7627 
+3 
+0.8584 
+ 
+From the application of the test we do not have enough evidence to suppose significant differences 
+between the performance of geographic areas in appearances and titles. However there are trends which 
+have been described graphically and discussed previously.   
+ 
+ 
+3.2 Dominance of the Champion 
+Next, is examined the dominance of the champion to its opponents and is measured in terms of scoring 
+points. The considered data are the points per game for and against the champions after the quarterfinals 
+because the potential existence of weak teams in earlier rounds may lead to instability of point 
+performance. In Fig.3 there are displayed the Points per Game (PPG) of the champion team and of their 
+opponents, while Fig.4 displays the Point difference as % of the points of the opponents of the champion 
+
+8 
+ 
+team. This is considered as a metric of the dominance of the champion team against its opponents in 
+terms of scoring.  
+ 
+ 
+Fig.3. PPG for and against the champion team 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig.4. Point difference as % of the opponents of the champion team 
+ 
+ 
+ 
+ 
+30.00%
+40.00%
+50.00%
+60.00%
+70.00%
+80.00%
+90.00%
+1958-1970 1971-1980 1981-1990 1991-2000 2001-2010 2011-2018
+CR4
+CR5
+CR6
+CR7
+CR8
+0
+20
+40
+60
+80
+100
+120
+1958 1962 1966 1970 1974 1978 1982 1986 1990 1994 1998 2001 2005 2009 2013 2017
+Points per Game
+Points  per game for and against the Champions
+champion
+opponent
+
+9 
+ 
+The above graph displays the points scored by the champion minus the points scored by opponents on 
+average, as percentage of opponent points. This could show in a sense how dominant was a champion. 
+Only in six seasons the champions scored more than 20% of their opponent points with Real Madrid to be 
+the only team which scored more than 30% of their opponents’ points in 1978. Extreme cases like this 
+should be examined in more detail. For example this team scored only 75 points in the final.   
+To better track the change of the game over time, we calculate the average points of the champions and 
+opponents in every decade and the average points per team and we draw their evolution across time in 
+Table 2 and Fig.5.  
+Table 2. Euroleague For and Against Points for Champion after Quarterfinals on average 
+Time Period 
+Average Points per Time Period 
+champion 
+opponent 
+Points per Team 
+1958-1970 
+82.61 
+72.90 
+77.75 
+1971- 1980 
+91.20 
+77.88 
+84.54 
+1981-1990 
+88.27 
+81.92 
+85.09 
+1991-2000 
+72.92 
+65.61 
+69.27 
+2001-2010 
+82.94 
+74.38 
+78.66 
+2011-2018 
+81.86 
+75.18 
+78.52 
+ 
+ 
+ 
+Fig.5. Evolution of Average Points per team  
+ 
+ 
+ 
+ 
+ 
+60
+65
+70
+75
+80
+85
+90
+Points per Team
+Points per Team
+
+10 
+ 
+ 
+From the above graph we can see the changes of the mentality of the game across time. From 1958 to 
+1990 there was an upward trend in scoring, with a sudden drop in 90s, something which indicates a 
+significant change in game mentality, and a return to the levels of 1958-1970 period.  
+Finally, the Fig.6 displays the decade average of the difference of points scored by champions minus the 
+points scored by opponents as percentage of the opponent points. It’s an indication of the dominance of 
+the champions of every decade. On average, the champions were more dominant in 60s and 70s, but in 
+the 80s they scored only 8% more than their opponents, a clear sign that the competition was more intense 
+in this decade. We also notice that the competitiveness of the last years (2010-2018) tends to similar 
+levels.  
+ 
+Fig.6. Evolution of Point difference as % of the opponents of the champion team  
+ 
+ 
+ 
+ 
+ 
+ 
+3.3 Analyze Scoring Pluralism in the Finals: Evolution of the style of the game 
+In this subsection we want to follow the evolution of the game as pictured in finals. We use raw data which 
+are the first scorers and the team points of the finals since 1958. It is commonly assumed that in the runner-up 
+team there is a more dominant scorer, in terms of first scorer points as % of team points. To test this 
+hypothesis we perform a Wilcoxon rank sum test for pairs of observations for data from all finals since 
+1958. The null hypothesis (𝐻0) of the test is that the differences between the pairs follows a symmetric 
+0.00%
+2.00%
+4.00%
+6.00%
+8.00%
+10.00%
+12.00%
+14.00%
+16.00%
+18.00%
+1958-1970 1971- 1980 1981-1990 1991-2000 2001-2010 2011-2018
+Dominance of the Champion
+Point Difference pct.
+
+11 
+ 
+distribution around zero. This is a non parametric test and through its application we avoid the 
+distributional assumption of normality of the data. A detailed description of the test can be found in 
+(Hollander and Wolfe, 1999). The test suggests that there is no reason to assume that in a specific year is 
+more probable the runner-up team to have a more dominant player in the scoring in final.  
+ 
+Wilcoxon Signed Rank Test for paired Samples 
+Statistic 
+p-value 
+824 
+0.1494 
+ 
+Additionally, we explore whether first scorer in terms of % of team points appears randomly or is more 
+probable to appear in sequences either from the champion or from the runner-up team. This can be 
+achieved through the application of a runs test in the difference of first scorer points as % of team points 
+between the two teams and we generate from this variable a sequence of + signs (if the variable is larger 
+than a threshold) and – signs (if the variable is smaller than a threshold). A run of a sequence is defined as 
+a series consisting of adjacent equal elements. We are testing the null hypothesis (H0) that each element 
+in the sequence is independently drawn from the same distribution. The threshold in our case is set to 
+zero. A description of the test can be found in (Gibbons and Chakraborti, 2003) and its implementation 
+performed via the randtests package of R (Caeiro and Mateus, 2014).  
+Through the application of the test, we can decide if over and under zero values are random. There is no 
+sign of non-randomness for this variable, so we can assume that the first scorer appears randomly from 
+the champion or from the runner-up team.   
+Runs Test 
+Statistic 
+Observations>0 
+Observations<0 
+Runs 
+p-value 
+~0 
+24 
+40 
+31 
+~1 
+ 
+The above tests are for the whole time period and they don’t reveal anything about the evolution of the game. The 
+rest of this section examines the evolution of the game and the statistical tests are adjusted accordingly.  
+ 
+At first, we present graphically the 10 year moving average of the points scored by the first scorer in final as % of 
+team points in Fig.7. It is displayed the 10 year moving average for decreasing the effect of extreme cases 
+and is easier to follow the trend of the game.  
+
+12 
+ 
+Fig.7. Moving average (10-year) of the points scored by the first scorer in final as % of team points 
+ 
+ 
+ 
+ 
+ 
+ 
+We observe that the game has been transformed over time from finals with offences which are based on 
+top scorers to finals with more pluralism. From the beginning until the 80s the trend was the one player 
+star in scoring, but since then, there was a slow but continuing turn to games based on pluralism.  
+ 
+At the next step, in Fig. 8 we present graphically the 10 year moving average of the difference between 
+the points of first scorer as % of team points for champion team minus the same metric of runner-up team.  
+ 
+Fig.8. Moving average (10-year) of the difference of points of first scorers as % of their team points  
+ 
+ 
+ 
+ 
+0%
+5%
+10%
+15%
+20%
+25%
+30%
+35%
+40%
+45%
+1966-1967
+1981-1982
+1996-1997
+2008-2009
+Points scored by 1st scorer as % of team points
+10 year Moving Avarage
+-8.00%
+-6.00%
+-4.00%
+-2.00%
+0.00%
+2.00%
+4.00%
+1966-1967
+1978-1979
+1990-1991
+2002-2003
+2013-2014
+10-year Moving Average of difference
+ 10-year Moving Average
+of difference
+
+13 
+ 
+From 1998 there is a downward trend until 2009 and a new cycle begins after 2009 and evolves but in a 
+lower level than the past. After 2001 there is no single year where the champion team has a more 
+dominant scorer than the runner-up team in terms of 10 year moving averages.  
+ 
+We make an assumption that there is a structural break in this variable and is very important to specify the 
+time when it happened because is a clue that the game has changed at this moment. To achieve this, is 
+performed a Zivot-Andrew test (Zivot and Andrews, 1992) to test for the existence of a structural break 
+(null hypothesis 𝐻0) against the hypothesis of nonstationarity.  
+ 
+ 
+ 
+Andrew-Zivot Test* 
+Statistic 
+p-value 
+Potential Break 
+-4.3499 
+>0.1 
+1997-1998 final 
+     *We assume both level and linear trend and 5 lags 
+ 
+ 
+The existence of a structural break is in favour compared to non-stationarity. Moreover, it is important to 
+specify when the structural break occurs. The potential structural break occurs in 1997-1998 final. The 
+history of Euroleague can be break into 2 periods: before and after 1998, let’s say after 1998 is the 
+modern period of Euroleague. For this reason, we perform again the Wilcoxon sign rank test and the Runs 
+test for the modern period of Euroleague. In the modern period of Euroleague, we can assume that there is 
+a more dominant scorer in the runner-up team, but we cannot predict this fact for a specific year.  
+ 
+Wilcoxon Signed Rank Test for paired Samples 
+Statistic 
+p-value 
+42.5 
+0.02056 
+ 
+ 
+ 
+Runs Test 
+Statistic 
+Observations>0 
+Observations<0 
+Runs 
+p-value 
+1.5607 
+5 
+12 
+11 
+0.1186 
+ 
+ 
+ 
+ 
+ 
+ 
+
+14 
+ 
+3.4 Pace in the Finals: The concept of possessions 
+In this subsection we include to our analysis the central concept of possessions (Kubatko et al, 2007). 
+Larger number of possessions displays a quicker pace of a game and the intension is to track the evolution 
+of the game.    
+ 
+We assume that both teams have the same number of possessions, but there is no unique formula for the 
+calculation of exact possessions in a game. For this reason, there are considered two formulas and we 
+average them in order to approximate more accurately the actual possessions. The used formulas are the 
+possessions lost (1) and the possessions gained (2) respectively: 
+ 𝑃𝑂𝑆𝑆𝑡 = 𝐹𝐺𝐴𝑡 + 𝜆 × 𝐹𝑇𝐴𝑡 − 𝑂𝑅𝐸𝐵𝑡 + 𝑇𝑂𝑡                                                                     (1) 
+𝑃𝑂𝑆𝑆𝑡 = 𝐹𝐺𝑀𝑡 + 𝜆 × 𝐹𝑇𝑀𝑡 + 𝐷𝑅𝐸𝐵𝑜 + 𝑇𝑂𝑡                                                                    (2) 
+ 
+After the calculation of the positions, we perform a line graph for the 5 year moving average of the 
+possessions in order to track their evolution (Fig.9). There is a downward trend and stability in low game 
+pace in the 90s, but from the beginning of the millennium there is a growing trend in game pace and from 
+2002 only four times there were fewer than 70 positions.  
+ 
+Fig.9. Evolution of possessions as 5-year moving average  
+ 
+ 
+ 
+60
+62
+64
+66
+68
+70
+72
+74
+76
+1987-1988
+1997-1998
+2006-2007
+2016-2017
+Possessions
+5 year moving average
+
+15 
+ 
+Considering a break at 1997-1998 we perform a Mann-Whitney test for the equality of possessions before 
+and after 1998. This test is a non parametric equivalent of a t-test for comparing the means of 2 groups 
+when the data do not follow Normal distribution. Details can be found on (Hollander and Wolfe, 1999).   
+ 
+Possessions 
+Before 1998 
+After 1998 
+66.25 
+71.33 
+Mann-Whitney Test for possessions 
+Statistic 
+p-value 
+84 
+0.01028 
+ 
+ 
+We detect a significant difference in possessions before and after 1998 finals at the 5% significance level. 
+This result is in accordance with our assumption that the game has changed after 1998 final and supports 
+the assumption that the triumph of Zalgiris in 1999 was the start of the change.   
+ 
+ 
+ 
+3.5 Correlation of First Scorer with Team Performance 
+Another interesting question is whether the existence of a first scorer of a tournament is correlated with 
+the performance of the team. The most popular opinion is that first scorers belong to weak teams which 
+do not have offensive many good offensive players. The other opinion is that a very gifted scorer can 
+affect the performance of the team positively and relies to the coach to keep the balance of the team. We 
+have the first scorers of the tournament after 1992 and we measure the strength of the link between their 
+scoring performances (PPG) with the success of their team in the season using the Pearson (r) and 
+Spearman (ρ) correlation coefficients (Hollander and Wolfe, 1999; Best and Roberts, 1975). The 
+Spearman coefficient is non parametric and correlates the ranks of the variables and assesses monotonic 
+relationships between them (instead of linear relationships which are assessed from Pearson correlation). 
+Except from the coefficient, we perform a statistical test for testing the null hypothesis that the coefficient 
+(either r or ρ) is zero, thus there is not significant correlation between the variables. We assign values for 
+the performance of the teams: 1 for regular season, 2 for Top 16, 3 for quarterfinals, 4 for final four, 4.5 if 
+
+16 
+ 
+the team was runner-up and 5 if the team won the trophy. Table 3 displays the first scorer, the team 
+position and the assigned values of the position.  
+ 
+ 
+𝑯𝟎: 𝝆 = 𝟎  𝒐𝒓   𝒓 = 𝟎 
+ 
+Coefficient 
+Statistic 
+p-value 
+Pearson Correlation 
+-0.4002 
+-2.2271 
+0.03481 
+Spearman Correlation 
+-0.3925 
+5088.032 
+0.03886 
+ 
+Both Pearson and Spearman correlation coefficients indicate that there is a significant negative 
+relationship between the first scorer and the performance of the team.  This finding rather favors the first 
+opinion where the first scorers are rarely parts of top teams (exception of Nando De Colo in 2015-2016 
+confirms the general rule).  
+Table 3. First scorer of the tournament, team position and assigned values of the position 
+Season 
+Player 
+PPG 
+Team 
+Performance 
+Assigned Score 
+1991-1992 
+Nikos Galis 
+32.3 
+Aris 
+Regular season 
+1 
+1992-1993 
+Zdravko Radulović 
+23.9 
+Cibona 
+Regular season 
+1 
+1993-1994 
+Nikos Galis 
+23.8 
+Panathinaikos 
+3rd place 
+4 
+1994-1995 
+Sašha Danilović 
+22.1 
+Buckler Bologna 
+Quarterfinals 
+3 
+1995-1996 
+Joe Arlauckas 
+26.4 
+Real Madrid 
+4th place 
+4 
+1996-1997 
+Carlton Myers 
+22.9 
+Teamsystem Bologna 
+Quarterfinals 
+3 
+1997-1998 
+Peja Stojaković 
+20.9 
+PAOK 
+Top 16 
+2 
+1998-1999 
+İbrahim Kutluay 
+21.4 
+Fenerbahçe 
+Top 16 
+2 
+1999-2000 
+Miljan Goljović 
+20.2 
+Pivovarna Laško 
+Regular season 
+1 
+2000-2001 
+(FIBA) 
+Miroslav Berić 
+23.3 
+Partizan 
+Top 16 
+2 
+2000-2001 
+(Euroleague) 
+Alphonso Ford 
+26 
+Peristeri 
+Top 16 
+2 
+2001-2002 
+Alphonso Ford 
+24.8 
+Olympiacos 
+Top 16 
+2 
+2002-2003 
+Miloš Vujanić 
+25.8 
+Partizan 
+Regular season 
+1 
+2003-2004 
+Lynn Greer 
+25.1 
+Śląsk Wrocław 
+Regular season 
+1 
+2004-2005 
+Charles Smith 
+20.7 
+Scavolini Pesaro 
+Quarterfinals 
+3 
+2005-2006 
+Drew Nicholas 
+18.5 
+Benetton Treviso 
+Top 16 
+2 
+2006-2007 
+Igor Rakočević 
+16.2 
+Tau Cerámica 
+4th place 
+4 
+2007-2008 
+Marc Salyers 
+21.8 
+Roanne 
+Regular season 
+1 
+2008-2009 
+Igor Rakočević 
+18 
+Tau Cerámica 
+Quarterfinals 
+3 
+2009-2010 
+Linas Kleiza 
+17.1 
+Olympiacos 
+2nd place 
+4.5 
+2010-2011 
+Igor Rakočević 
+17.2 
+Efes Pilsen 
+Top 16 
+2 
+2011-2012 
+Bo McCalebb 
+16.9 
+Montepaschi Siena 
+Quarterfinals 
+3 
+2012-2013 
+Bobby Brown 
+18.8 
+Montepaschi Siena 
+Top 16 
+2 
+
+17 
+ 
+2013-2014 
+Keith Langford 
+17.6 
+EA7 Milano 
+Quarterfinals 
+3 
+2014-2015 
+Taylor Rochestie 
+18.9 
+Nizhny Novgorod 
+Top 16 
+2 
+2015-2016 
+Nando de Colo 
+18.9 
+CSKA Moscow 
+Winner 
+5 
+2016-2017 
+Keith Langford 
+21.8 
+UNICS 
+Regular season 
+1 
+2017-2018 
+Alexey Shved 
+21.8 
+Khimki 
+Quarterfinals 
+3 
+ 
+3.6 Unexpected Outcomes in the Final Fours 
+In this section it is examined the unexpected of the Final-Fours in terms of outcomes based on previous 
+attempts with the use of Binomial Distribution. Can we make the assumption that each final four is an 
+experiment with each team to have the same probabilities of winning the tournament (25%)?   
+To answer this question, we consider each final four as an experiment and teams are considered as 
+independent random variables. Each experiment can be described by the binomial distribution and the 
+whole situation with multinomial distribution (Forbes et al, 2011) which is a generalization of binomial 
+distribution and describes n trials. There is performed a multinomial goodness of fit test and to strengthen 
+the results a binomial test for each team, in order to decide if there is any significant difference from 
+binomial distribution.  
+Multinomial Testing 
+p-value 
+ 
+0.54499±0.001575 
+Binomial Testing* 
+ 
+ 
+p-value 
+Cibona 
+2 attempts - 2 trophies 
+0.0625 
+Jugoplastica 
+4 attempts - 3 trophies 
+0.05078 
+ASK Riga 
+4 attempts - 3 trophies 
+0.05078 
+Panathinaikos 
+12 attempts - 6 trophies 
+0.08608 
+*Only cases with p-value<0.1 
+ 
+Table on the appendix displays the final four teams, the expected titles according to Binomial distribution, 
+the observed values and their difference. Indeed there is no evidence that there are significant 
+discrepancies from the binomial distribution at the 5% level of significance.  
+In the modern period of Euroleague (1999-2018), again there is no evidence of significant discrepancy 
+from the multinomial distribution, but the case of Panathinaikos could be seen as an exception, with 
+significant larger success rate than the expected. 
+ 
+
+18 
+ 
+Multinomial Testing 
+p-value 
+ 
+0.68173±0.001473 
+Binomial Testing* 
+ 
+ 
+p-value 
+Panathinaikos 
+8 attempts - 5 trophies 
+0.02730 
+*Only cases with p-value<0.1 
+ 
+ 
+4 Summary and Conclusions 
+To sum up, in this paper is made an attempt to address questions related to historical evolution of 
+Euroleague using statistical analysis to draw conclusions. One main problem is the lack of plenty 
+available statistical data from the beginning of the competition. This paper demonstrates that by applying 
+suitable statistical designs we can draw interesting conclusions even with limited data. Firstly, is made a 
+brief exploration of the historical evolution of the Euroleague and the tracking statistics.  
+Then, some questions are answered and some conclusions are drawn which are briefly the following:  
+Although overall there is no difference in success between more traditional powers such as Italy, Spain 
+and ex USSR and Yugoslavian countries and other countries, there is a clear trend of other countries to 
+expand their presence (in terms of titles and final four appearances) in the tournament after the 90s. In 
+terms of scoring, there was an upward trend from the beginning of the competition, with a sudden drop in 
+90s, something which indicates a significant change in game mentality in terms of defence and/or game 
+pace. The champions were more dominant in 60s and 70s, but in the 80s they scored only 8% more than 
+their opponent, which indicates that the competition was more intense in this decade. The last years 
+(2010-2018), the competitiveness of the tournament tends to similar levels.  
+There is a popular belief that the first scorer in the majority of cases come from the runner-up team. 
+However, there is no reason to assume that in a specific year is more probable the runner-up team to have 
+a more dominant player in the scoring in the final. In the modern period of Euroleague (after 1998), we 
+can assume that there is a more dominant scorer in the runner-up team, but we cannot predict this fact for 
+a specific year. According to the game evolution in finals, we observe that the game has been transformed 
+from finals with offences which are based on top scorers to finals with more pluralism. From the 
+beginning until the 80s the trend was the one player star in scoring, but since then, there was a slow but 
+
+19 
+ 
+continuing turn to games based on pluralism. Moreover, we detect a significant difference in possessions 
+before and after 1998 finals, which is in accordance with our assumption that the game has changed after 
+1998 final and supports the assumption that the triumph of Zalgiris in 1999 was the start of the change. 
+Furthermore, it is found a significant negative relationship between the first scorer and the performance of 
+his team. This finding favors the opinion that the first scorers are rarely parts of top teams. Finally, there 
+is no evidence to reject the hypothesis that in a final four there are equal chances of winning overall. In 
+modern era, again the hypothesis of the final four as a random experiment is not rejected, however in the 
+case of Panathinaikos we observe significantly higher success rate than the expected.   
+ 
+ 
+References 
+Best, D. J., & Roberts, D. E. 1975. “Algorithm AS 89: the upper tail probabilities of Spearman's rho.” Journal of the 
+Royal Statistical Society. Series C (Applied Statistics), 24(3), 377-379. 
+Caeiro, F., & Mateus, A. 2014. randtests: Testing randomness in R. R package version, 1. 
+Forbes, C., Evans, M., Hastings, N., & Peacock, B. 2011.Statistical distributions. John Wiley & Sons. 
+Gibbons, J. D., & Chakraborti, S. 2003. Nonparametric Statistical Inference.  Marcel Dekker. Inc. New York. 
+Hollander, M., & Wolfe, D. A. 1999. Nonparametric statistical methods. 2nd Edition, John Wiley & Sons.   
+Kubatko, J., Oliver, D., Pelton, K., & Rosenbaum, D. T. 2007. “A starting point for analyzing basketball statistics.”  
+Journal of Quantitative Analysis in Sports, 3(3). 
+Oliver, D. (2004). Basketball on paper: rules and tools for performance analysis. Potomac Books, Inc.  
+Zivot, E., & Andrews, D. W. K. 2002. “Further evidence on the great crash, the oil-price shock, and the unit-root 
+hypothesis.” Journal of business & economic statistics, 20(1), 25-44. 
+ 
+ 
+ 
+ 
+
+20 
+ 
+Appendix 
+Table. Euroleague Final Four Teams 
+Year 
+Winner 
+Runner-Up 
+3rd Place 
+4th Place 
+1958 
+ASK Riga 
+Academic 
+Honved 
+Real Madrid 
+1958- 1959 
+ASK Riga 
+Academic 
+Lech Poznan 
+OKK Beograd 
+1959-1960 
+ASK Riga 
+Dinamo Tbilisi 
+Pologna Warzawa 
+Slovan Orbis Praha 
+1960-1961 
+CSKA Moscow 
+ASK Riga 
+Real Madrid 
+Steaua Bucarest 
+1961-1962 
+Dinamo Tbilisi 
+Real Madrid 
+ASK Olimpija 
+CSKA Moscow 
+1962-1963 
+CSKA Moscow 
+Real Madrid 
+Dinamo Tbilisi 
+Spartak Brno 
+1963-1964 
+Real Madrid 
+Spartak Brno 
+OKK Beograd 
+Simmenthal Milano 
+1964-1965 
+Real Madrid 
+CSKA Moscow 
+Ignis Varese 
+OKK Beograd 
+1965-1966 
+Simmenthal Milano 
+Slavia Praha 
+CSKA Moscow 
+AEK 
+1966-1967 
+Real Madrid 
+Simmenthal Milano 
+ASK Olimpija 
+Slavia Praha 
+1967-1968 
+Real Madrid 
+Spartak Brno 
+Simmenthal Milano 
+Zadar 
+1968-1969 
+CSKA Moscow 
+Real Madrid 
+Spartak Brno 
+Standard Liege 
+1969-1970 
+Ignis Varese 
+CSKA Moscow 
+Real Madrid 
+Slavia Praha 
+1970-1971 
+CSKA Moscow 
+Ignis Varese 
+Real Madrid 
+Slavia Praha 
+1971-1972 
+Ignis Varese 
+Jugoplastica1 
+Real Madrid 
+Panathinaikos 
+1972-1973 
+Ignis Varese 
+CSKA Moscow 
+Crvena Zvezda 
+Simmenthal Milano 
+1973-1974 
+Real Madrid 
+Ignis Varese 
+Berck 
+Radniski Belgrade 
+1974-1975 
+Ignis Varese 
+Real Madrid 
+Berck 
+Zadar 
+1975-1976 
+Mobilgirgi Varese 
+Real Madrid 
+ASVEL 
+Forst Cantù 
+1976-1977 
+Maccabi Tel Aviv 
+Mobilgirgi Varese 
+CSKA Moscow 
+Real Madrid 
+1977-1978 
+Real Madrid 
+Mobilgirgi Varese 
+ASVEL 
+Maccabi Tel Aviv 
+1978-1979 
+Bosna 
+Emerson Varese 
+Maccabi Tel Aviv 
+Real Madrid 
+1979-1980 
+Real Madrid 
+Maccabi Tel Aviv 
+Bosna 
+Sinudyne Bologna3 
+1980-1981 
+Maccabi Tel Aviv 
+Sinudyne Bologna3 
+Nashua EBBC 
+Bosna 
+1981-1982 
+Squibb Cantù 
+Maccabi Tel Aviv 
+Partizan 
+FC Barcelona 
+1982-1983 
+Ford Cantù 
+Billy Milano 
+Real Madrid 
+CSKA Moscow  
+1983-1984 
+Virtus Roma 
+FC Barcelona 
+Jollycolombani Cantù 
+Bosna 
+1984-1985 
+Cibona 
+Real Madrid 
+Maccabi Tel Aviv 
+CSKA Moscow 
+1985-1986 
+Cibona 
+Zalgiris 
+Simac Milano 
+Real Madrid 
+1986-1987 
+Tracer Milano 
+Maccabi Tel Aviv 
+Orthez 
+Zadar 
+1987-1988 
+Tracer Milano 
+Maccabi Tel Aviv 
+Partizan 
+Aris 
+1988-1989 
+Jugoplastica1 
+Maccabi Tel Aviv 
+Aris 
+FC Barcelona 
+1989-1990 
+Jugoplastica1 
+FC Barcelona 
+Limoges 
+Aris 
+1990-1991 
+POP 84 1 
+FC Barcelona 
+Maccabi Tel Aviv 
+Scavolini Pezaro 
+1991-1992 
+Partizan 
+Joventut 
+Phillips Milano 
+Estudiantes 
+1992-1993 
+Limoges 
+Benneton Treviso 
+PAOK 
+Real Madrid 
+1993-1994 
+Joventut 
+Olympiacos 
+Panathinaikos 
+FC Barcelona 
+1994-1995 
+Real Madrid 
+Olympiacos 
+Panathinaikos 
+Limoges 
+1995-1996 
+Panathinaikos 
+FC Barcelona 
+CSKA Moscow 
+Real Madrid 
+1996-1997 
+Olympiacos 
+FC Barcelona 
+Smelt Olimpija 
+ASVEL 
+1997-1998 
+Kinder Bologna3 
+AEK 
+Benneton Treviso 
+Partizan 
+1998-1999 
+Zalgiris 
+Kinder Bologna3 
+Olympiacos 
+Teamsystem Bologna4 
+1999-2000 
+Panathinaikos 
+Maccabi Tel Aviv 
+Efes Pilsen 
+FC Barcelona 
+2000-2001 (FIBA) 
+Kinder Bologna3 
+TAU  Ceramica2 
+AEK 
+Paf Wennington Bologna4 
+2000-2001 (Euroleague) 
+Maccabi Tel Aviv 
+Panathinaikos 
+Efes Pilsen 
+CSKA Moscow 
+2001-2002 
+Panathinaikos 
+Kinder Bologna3 
+Benneton Treviso 
+Maccabi Tel Aviv 
+2002-2003 
+FC Barcelona 
+Benneton Treviso 
+Montepaschi Siena 
+CSKA Moscow 
+2003-2004 
+Maccabi Tel Aviv 
+Skipper Bologna4 
+CSKA Moscow 
+Montepaschi Siena 
+2004-2005 
+Maccabi Tel Aviv 
+TAU  Ceramica2 
+Panathinaikos 
+CSKA Moscow 
+2005-2006 
+CSKA Moscow 
+Maccabi Tel Aviv 
+TAU  Ceramica2 
+FC Barcelona 
+2006-2007 
+Panathinaikos 
+CSKA Moscow 
+Unicaja Malaga 
+TAU  Ceramica2 
+2007-2008 
+CSKA Moscow 
+Maccabi Tel Aviv 
+Montepaschi Siena 
+Real Madrid 
+2008-2009 
+Panathinaikos 
+CSKA Moscow 
+FC Barcelona 
+Olympiacos 
+2009-2010 
+FC Barcelona 
+Olympiacos 
+CSKA Moscow 
+Partizan 
+2010-2011 
+Panathinaikos 
+Maccabi Tel Aviv 
+Montepaschi Siena 
+Real Madrid 
+2011-2012 
+Olympiacos 
+CSKA Moscow 
+FC Barcelona 
+Panathinaikos 
+2012-2013 
+Olympiacos 
+Real Madrid 
+CSKA Moscow 
+FC Barcelona 
+2013-2014 
+Maccabi Tel Aviv 
+Real Madrid 
+FC Barcelona 
+CSKA Moscow 
+2014-2015 
+Real Madrid 
+Olympiacos 
+CSKA Moscow 
+Fenerbahce 
+2015-2016 
+CSKA Moscow 
+Fenerbahce 
+Lokomotiv Kuban 
+Laboral Kutxa2 
+2016-2017 
+Fenerbahce 
+Olympiacos 
+CSKA Moscow 
+Real Madrid 
+2017-2018 
+Real Madrid 
+Fenerbahce 
+Zalgiris 
+CSKA Moscow 
+1Croatia Split 
+2Club Deportivo Saski Baskonia, S.A.D.  
+3Virtus Pallacanestro Bologna 
+4 Fortitudo Pallacanestro Bologna 103 
+
+21 
+ 
+Table . Teams and appearances to final four 
+Team 
+Winner 
+Appearances 
+Expected Titles 
+Observed Titles 
+Difference 
+Lokomotiv Kuban 
+0 
+1 
+0.25 
+0 
+-0.25 
+Unicaja Malaga 
+0 
+1 
+0.25 
+0 
+-0.25 
+PAOK 
+0 
+1 
+0.25 
+0 
+-0.25 
+Estudiantes 
+0 
+1 
+0.25 
+0 
+-0.25 
+Scavolini Pezaro 
+0 
+1 
+0.25 
+0 
+-0.25 
+Orthez 
+0 
+1 
+0.25 
+0 
+-0.25 
+Nashua EBBC 
+0 
+1 
+0.25 
+0 
+-0.25 
+Radniski Belgrade 
+0 
+1 
+0.25 
+0 
+-0.25 
+Crvena Zvezda 
+0 
+1 
+0.25 
+0 
+-0.25 
+Standard Liege 
+0 
+1 
+0.25 
+0 
+-0.25 
+Steaua Bucarest 
+0 
+1 
+0.25 
+0 
+-0.25 
+Lech Poznan 
+0 
+1 
+0.25 
+0 
+-0.25 
+Honved 
+0 
+1 
+0.25 
+0 
+-0.25 
+Pologna Warzawa 
+0 
+1 
+0.25 
+0 
+-0.25 
+Efes Pilsen 
+0 
+2 
+0.5 
+0 
+-0.5 
+Berck 
+0 
+2 
+0.5 
+0 
+-0.5 
+Zadar 
+0 
+3 
+0.75 
+0 
+-0.75 
+Olimpija Ljubliana 
+0 
+3 
+0.75 
+0 
+-0.75 
+OKK Beograd 
+0 
+3 
+0.75 
+0 
+-0.75 
+ASVEL 
+0 
+3 
+0.75 
+0 
+-0.75 
+Aris 
+0 
+3 
+0.75 
+0 
+-0.75 
+Montepaschi Siena 
+0 
+4 
+1 
+0 
+-1 
+Fortitudo Bologna 
+0 
+3 
+0.75 
+0 
+-0.75 
+AEK 
+0 
+3 
+0.75 
+0 
+-0.75 
+Praha 
+0 
+5 
+1.25 
+0 
+-1.25 
+Baskonia 
+0 
+5 
+1.25 
+0 
+-1.25 
+Treviso 
+0 
+4 
+1 
+0 
+-1 
+Brno 
+0 
+4 
+1 
+0 
+-1 
+Academic 
+0 
+2 
+0.5 
+0 
+-0.5 
+Limoges 
+1 
+3 
+0.75 
+1 
+0.25 
+Partizan 
+1 
+5 
+1.25 
+1 
+-0.25 
+Virtus Roma 
+1 
+1 
+0.25 
+1 
+0.75 
+Bosna 
+1 
+4 
+1 
+1 
+0 
+Zalgiris 
+1 
+3 
+0.75 
+1 
+0.25 
+Joventut Badalona 
+1 
+2 
+0.5 
+1 
+0.5 
+Dinamo Tbilisi 
+1 
+3 
+0.75 
+1 
+0.25 
+Fenerbahce 
+1 
+4 
+1 
+1 
+0 
+Cibona 
+2 
+2 
+0.5 
+2 
+1.5 
+Cantu 
+2 
+4 
+1 
+2 
+1 
+Virtus Bologna 
+2 
+6 
+1.5 
+2 
+0.5 
+FC Barcelona 
+2 
+16 
+4 
+2 
+-2 
+Split 
+3 
+4 
+1 
+3 
+2 
+ASK Riga (Latvia) 
+3 
+4 
+1 
+3 
+2 
+Olympia Milano 
+3 
+10 
+2.5 
+3 
+0.5 
+Olympiacos 
+3 
+10 
+2.5 
+3 
+0.5 
+Varese 
+5 
+11 
+2.75 
+5 
+2.25 
+Panathinaikos 
+6 
+12 
+3 
+6 
+3 
+Maccabi Tel Aviv 
+6 
+20 
+5 
+6 
+1 
+CSKA Moscow 
+7 
+29 
+7.25 
+7 
+-0.25 
+Real Madrid 
+10 
+32 
+8 
+10 
+2 
+ 
+ 
+ 
+ 
+

LNE2T4oBgHgl3EQfpwjv/content/tmp_files/2301.04033v1.pdf.txt

@@ -0,0 +1,1202 @@
+1 
+ 
+Numerical investigation of progressive damage and associated seismicity on a 
+laboratory fault 
+Qi Zhao1,2*, Nicola Tisato3, Aly Abdelaziz2, Johnson Ha2, and Giovanni Grasselli2 
+1Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University, 
+Hung Hom, Hong Kong SAR, China. 
+2Department of Civil and Mineral Engineering, The University of Toronto, 35 St. George Street, 
+Toronto, Ontario M5S 1A4, Canada. 
+3Department of Geological Sciences, Jackson School of Geosciences, The University of Texas at 
+Austin, 2305 Speedway Stop C1160, Austin, TX 78712-1692, USA. 
+*Corresponding author: Qi Zhao (qi.qz.zhao@polyu.edu.hk) 
+ 
+Abstract 
+Understanding rock shear failure behavior is crucial to gain insights into slip-related geohazards 
+such as rock avalanches, landslides, and earthquakes. However, descriptions of the progressive 
+damage on the shear surface are still incomplete or ambiguous. In this study, we use the hybrid 
+finite-discrete element method (FDEM) to simulate a shear experiment and obtain a detailed 
+comprehension of shear induced progressive damage and the associated seismic activity. We built 
+a laboratory fault model from high resolution surface scans and micro-CT imaging. Our results 
+show that under quasi-static shear loading, the fault surface experiences local dynamic seismic 
+activities. We found that the seismic activity is related to the stress concentration on interlocking 
+asperities. This interlocking behavior (i) causes stress concentration at the region of contact that 
+could reach the compressive strength, and (ii) produces tensile stress up to the tensile strength in 
+the region adjacent to the contact area. Thus, different failure mechanisms and damage patterns 
+including crushing and sub-vertical fracturing are observed on the rough surface. Asperity failure 
+creates rapid local slips resulting in significant stress perturbations that alter the overall stress 
+condition and may trigger the slip of adjacent critically stressed asperities. We found that the 
+
+2 
+ 
+spatial distribution of the damaged asperities and the seismic activity is highly heterogeneous; 
+regions with intense asperity interactions formed gouge material, while others exhibit minimal to 
+no damage. These results emphasize the important role of surface roughness in controlling the 
+overall shear behavior and the local dynamic seismic activities on faults. 
+Keywords 
+Shear behavior; surface roughness; asperity; shear induced damage; seismicity  
+ 
+
+3 
+ 
+1 Introduction 
+Understanding shear behavior along rock discontinuities at various scales, such as joints 
+and faults, is essential to rock engineering projects and geohazard mitigation. Rock discontinuities 
+are planes of weakness and are responsible for many geohazards, for example, rock avalanches, 
+landslides, and earthquakes. Numerous laboratory shear experiments have been conducted on a 
+large variety of rock types under different conditions (e.g., Bandis et al., 1983; Beeler, 1996; 
+Marone, 1998; Di Toro et al., 2004; Grasselli, 2001; Reches and Lockner, 2010; Tisato et al., 2012; 
+Kim and Jeon, 2019; Zhao et al., 2020; Morad et al., 2022). Among these studies, many suggested 
+the importance of surface roughness and contact condition in controlling the shear behavior. 
+However, the progressive damaging process on faults is still not well understood because fault 
+surfaces cannot be observed directly during shear, with the exception of a few studies utilizing 
+transparent halite samples (e.g., Renard et al., 2012) and in situ and operando testes conducted 
+under X-ray micro-computed tomography (micro-CT) (e.g., Zhao et al, 2018; Zhao et al., 2020). 
+Shear processes control coseismic damage and friction on fault, but the constitutive friction 
+theories are not yet fully understood. 
+To observe and gain insights into damage processes on rough rock surfaces undergoing 
+shear deformation, Tatone and Grasselli (2013) used micro-CT to image the joint surfaces after 
+the shear test; and Crandall et al. (2017) used micro-CT to obtain geometrical information from 
+fractured shale core that is incrementally sheared. Recently, direct and detailed observations of the 
+evolution of laboratory fault were achieved by using an in situ rotary shear experimental apparatus 
+under X-ray micro-CT (Zhao et al., 2017). Such experimental results help draw the connections 
+between microscopic damage and macroscopic shear behavior, the formation and accumulation of 
+gouge material, and shear-induced secondary fractures (Zhao, 2017; Zhao et al., 2018; Zhao et al., 
+
+4 
+ 
+2020). However, due to technological limitations, time-continuous observations of the shear 
+surface evolution and the in situ stress condition remain shortfalls. 
+Numerical simulation methods have been extensively used to study the shear behavior of 
+rock discontinuities. To simulate the interaction and breakage of asperities and the frictional 
+sliding behavior, numerical methods that can capture solid fracturing and interaction are typically 
+used. For example, the particle-based lattice solid model (Mora and Place 1993) uses a numerical 
+concept similar to the discrete element method (DEM) to simulate frictional behavior and 
+fracturing in solids. Mora and Place (1998) and Place and Mora (2000) used their model to study 
+the role fault gouge on the frictional behavior of faults, offering a possible explanation for the heat 
+flow paradox (Henyey and Wasserburg, 1971; Lachenbruch and Sass, 1992). Bonded particle-
+based methods, such as the particle flow code (PFC) (Cundall and Strack, 1979), are commonly 
+used for simulating rock shear behavior. Park and Song (2009) used PFC3D to simulate direct 
+shear tests and demonstrated that the method can simulate typical rock joint shear behavior, and 
+they found that the peak shear strength and peak dilation angle was strongly influenced by the 
+friction coefficient, roughness, and bond strength, while the residual shear strength and residual 
+friction angle was influenced by the particle size, friction coefficient, and bond strength. Asadi et 
+al. (2012) used a similar approach in two dimensions (PFC2D) to simulate direct shear experiments 
+on synthetic joint profiles of varying roughness and boundary conditions to assess asperity 
+degradation and intact material damage. They showed that as the joint is sheared, highly localized 
+asperity interaction on the joint surface and the geometry of the asperities has a significant 
+influence on how joints fail. Types of failure include asperity sliding, cut-off, separation, and 
+crushing, typically associated with tensile failure into the intact material in conditions with steep 
+asperities and/or high normal stress. However, Bahaaddini et al. (2013) identified a significant 
+
+5 
+ 
+shortcoming of the particle-based methods due to the unrealistic shear and dilation behavior of 
+joints as a result of particle interlocking due to the inherent micro-scale roughness of the joint. To 
+overcome this limitation, they implemented the smooth-joint model (Pierce et al., 2007), where 
+the blocks associated with either side of the joint are generated separately to appropriately define 
+and apply the smooth joint model to the interface. Lambert and Coll (2014) created a synthetic 
+rock joint by importing the real morphology of the joint surface into a bonded particle assembly 
+and studied the shear behavior using the smooth-joint model. Their results reproduced the 
+progressive degradation of the asperities upon shearing. 
+The hybrid finite-discrete element method (FDEM) is increasingly used to investigate rock 
+shear behavior. FDEM is a micromechanical numerical method first introduced by Munjiza et al., 
+(1995) combining the finite element method and discrete element method. In doing so, the 
+numerical method can model the transition of a continuous material to a discontinuous material as 
+it deforms, yields, and breaks. Karami and Stead (2008) and Tatone (2014) used FDEM to model 
+direct shear tests and relate progressive asperity degradation mechanisms with the measured shear 
+stress and dilation during shearing. In addition, Tatone (2014) verified the numerical modelling 
+observations by coupling their study with X-ray micro-CT imaging on post-mortem specimens. It 
+was found that tensile fractures develop in asperities at and near the peak shear stress, followed by 
+a reduction in shear resistance as asperities continue to fail in both tension and shear, and finally, 
+a residual shear resistance is reached once asperities are completely broken and gouge is formed. 
+In this study, we used the two-dimensional (2D) FDEM to simulate an experiment on the 
+gradual evolution of deformation of a laboratory fault, and we improve the understanding of shear 
+behavior of rough faults through combined interpretation of the simulation and experimental 
+results. First, we provide a brief review of the FDEM, emphasizing the modeling of damage and 
+
+6 
+ 
+seismic activity. Second, we develop a clustering algorithm to improve the comprehension of the 
+simulated fractures and seismic events. Next, we build a model based on the laboratory experiment 
+and analyze the simulated results focusing on three aspects that are hardly accessible by 
+experiments: (1) the time-continuous variation of stress conditions on the shear surface, (2) the 
+progressive failure of the asperities and accumulation of gouge, and (3) the seismic activity related 
+to shear-induced damage.  
+The carefully built and calibrated numerical model is able to simulate the emergent rock 
+mechanical and frictional behaviors. We observe that shear-induced damage and seismic activities 
+are heterogeneously distributed along the fault surface due to the surface roughness. Seismic 
+events occur at the locations of asperity failure due to the interlocking-induced stress concentration. 
+Such events radiate seismic waves and significantly change the overall stress conditions. Some 
+areas on the fault were covered by gouge material and free from damage. These results agree with 
+the laboratory observations and further elaborate on the importance of surface roughness in 
+controlling shear behavior, which is critical to rock engineering practices and earthquake studies. 
+2 Material and methods 
+2.1 In situ shear test under X-ray micro-CT 
+The numerical simulation in this study is based on the experimental work using in situ 
+shear tests under micro-CT reported by Zhao et al. (2018) and Zhao et al. (2020), and a brief review 
+is provided here for completeness. The tested specimen was a cylindrical Flowstone (microfine 
+calcium sulfate cement mortar) 32 mm in length and 12 mm in diameter. The specimen was divided 
+into top and bottom parts by a three-point bending test that created two semi-samples divided by 
+a discontinuity (i.e., laboratory fault) with two matching rough surfaces. An unconfined rotary 
+shear test was conducted on the two semi-samples by shearing the fault under the initial normal 
+
+7 
+ 
+stress of 2.5 MPa. The top semi-sample was forced to slip incrementally against the fixed bottom 
+semi-sample. Normal force and torque were recorded during rotation and used to calculate the 
+friction coefficient. After each incremental slip of 6°, a three-dimensional (3D) micro-CT scan was 
+conducted, which allows for imaging of the gradual morphological evolution of the specimen (Fig. 
+1a). This experimental work provided detailed information of the shear-induced secondary 
+fractures (Fig. 1b) and the progressive damage on the slipping surface (Fig. 1c) in the sample 
+volume; however, the observation of the shear surface damage evolution was only available at 
+discrete time points coincident with each shear step, while an actual time-continuous observation 
+of the shear surface evolution and the local stress condition on the rough surface was not available. 
+ 
+Fig. 1 Summary of the laboratory set-up and results. (a) Schematic of the in situ shear test and a 
+zoom-in view of the shear surface (i.e., the zone of interest). (b) 3D visualization of the 
+development of shear induced fractures with increasing shear displacement. (c) 2D unwrapped 
+micro-CT image slice showing the progressive damage on the slipping surface with increasing 
+shear displacement, viewed at the radius (R = 5.6 mm) corresponding to the highest asperity. Red 
+
+8 
+ 
+dashed boxes in (c) indicate (from left to right) shear-induced aperture opening, fracturing, and 
+progressive damage and gouge formation (modified from Zhao et al. (2018) and Zhao et al. (2020)). 
+2.2 The hybrid finite-discrete element method 
+The hybrid finite-discrete element method (FDEM) combines continuum mechanics 
+principles with discrete element principles to simulate interaction, deformation, and fracturing of 
+materials (Munjiza et al., 1995; Munjiza, 2004). FDEM has been used to investigate a wide range 
+of rock mechanics and geophysics problems including, but not limited to, tunneling and excavation, 
+micromechanics, rock joint shear behavior, hydraulic fracturing, thermal-mechanical/hydro-
+thermal-mechanical coupling, and fault dynamics (e.g., Karami and Stead 2008; Mahabadi et al., 
+2012; Lisjak et al., 2014; Zhao et al., 2014; Yan et al., 2016; Huang et al., 2017; Lei et al., 2017; 
+Ma et al., 2017; Fukuda et al., 2019; Okubo et al., 2019; Knight et al., 2020). Simulating the entire 
+shear behavior and evolution of a rough surface is a challenging task that requires advanced 
+computational resources encompassing, for example, the 3D FDEM method. However, 3D models 
+explicitly capturing the surface roughness at sub-millimeter resolution and the entire shear process 
+is not practical due to the demanded computation power. On the other hand, 2D FDEM simulations 
+has the merit of reducing the computational demand, and it has been shown to provide insights 
+into the mechanical behavior of rock joints and faults (e.g., Karami and Stead, 2008; Tatone, 2014; 
+Okubo et al., 2019). 
+FDEM models synthesize the macroscopic behavior of materials from the interaction of 
+the micromechanical constituents. In a 2D FDEM model, the simulated material is first discretized 
+based on a finite element mesh consisting of nodes and triangular elements. Then, the finite 
+element mesh is enriched by inserting a four-node cohesive crack element (CCE) between each 
+adjacent triangular element pair. Motion for the discretized system is calculated by an explicit time 
+
+9 
+ 
+integration scheme, and the nodal coordinates of the elements are updated at each simulation step 
+(Munjiza, 2004). FDEM models the progressive damage and failure of brittle material according 
+to the principles of non-linear elastic fracture mechanics (Dugdale, 1960; Barenblatt, 1962), and it 
+captures the fracturing behavior of solids by modeling the entire failure path, including elastic 
+deformation, yielding, and fracturing (Fig. 2). 
+ 
+Fig. 2 Schematic diagram showing the FDEM approach of simulating fracturing. (a) Propagation 
+of a fracture and the creation of fracture process zone (FPZ). (b) Realization of the fracturing 
+process in FDEM involves the yielded cohesive crack elements and broken cohesive crack 
+elements (BCCE). 
+Depending on the local stress and deformation field, the CCE undergoes elastic 
+deformation, yielding, and breakage, simulating the damage development of the fracture process 
+zone (FPZ) (Fig. 3) (Labuz et al., 1985). During elastic loading, the relationships between bonding 
+stresses (normal bonding stress, σ and shear bonding stress, τ) and the corresponding crack 
+displacement (opening, o and slip, s) are as follows (Munjiza et al., 1999):  
+
+Cracktip10 
+ 
+𝜎 = {
+2𝑜
+𝑜𝑝 𝑓𝑡                         (𝑜 < 0, compression)
+[
+2𝑜
+𝑜𝑝 − (
+𝑜
+𝑜𝑝)
+2
+] 𝑓𝑡     (0 < 𝑜 < 𝑜𝑝, tension)
+     
+ 
+ 
+     (1) 
+𝜏 = [
+2𝑠
+𝑠𝑝 − (
+𝑠
+𝑠𝑝)
+2
+] 𝑓𝑠    (|𝑠| ≤ |𝑠𝑝|, shear)    
+ 
+ 
+ 
+     (2) 
+where ft and fs are the peak tensile and shear bonding strength of a CCE, respectively. The peak 
+shear bonding strength is calculated based on the Mohr-Coulomb failure criterion using the 
+cohesion (c) and internal friction angle (ϕ): 𝑓𝑠 = 𝑐 + 𝜎 tan 𝜙. op and sp are the peak opening and 
+slip values at the peak bonding stresses calculated as 𝑜𝑝 = 2ℎ𝑓𝑡 𝑝𝑓
+⁄
+ and 𝑠𝑝 = 2ℎ𝑓𝑠 𝑝𝑓
+⁄
+, where h is 
+the nominal element edge length, and pf is the fracture penalty value. A CCE yields once the stress 
+reaches the peak, then it experiences a post-peak softening behavior with the bonding stresses 
+gradually decreased (Munjiza et al., 1999): 
+𝜎 = 𝐹(𝐷)𝑓𝑡 
+ 
+ 
+ 
+ 
+ 
+  (3) 
+𝜏 = 𝐹(𝐷)𝑓𝑠   
+ 
+ 
+ 
+ 
+  (4) 
+F(D) is an empirical function that approximates the shape of the experimental stress-displacement 
+failure curve according to Evans and Marathe (1968): 
+F(𝐷) = [1 −
+𝑎+𝑏−1
+𝑎+𝑏 exp (𝐷
+𝑎+𝑐𝑏
+(𝑎+𝑏)(1−𝑎−𝑏))] ∙ [𝑎(1 − 𝐷) + 𝑏(1 − 𝐷)𝑐]    
+ 
+(5) 
+where a, b, c are empirical curve fitting parameters equal to 0.63, 1.8, and 6.0, respectively. The 
+damage coefficient (D) is calculated for Mode I, II, and I-II as  
+𝐷I =
+𝑜−𝑜𝑝
+𝑜𝑟−𝑜𝑝   
+ 
+ 
+ 
+ 
+ 
+  (6) 
+𝐷II =
+𝑠−𝑠𝑝
+𝑠𝑟−𝑠𝑝  
+ 
+ 
+ 
+ 
+ 
+  (7) 
+
+11 
+ 
+𝐷I−II = √𝐷I
+2 + 𝐷II
+2   
+ 
+ 
+ 
+ 
+  (8) 
+with the subscripts indicating the mode of failure. The CCE breaks when D = 1, which corresponds 
+to a residual opening (or) or a residual slip (sr), for pure Mode I or II failure, respectively. For 
+Mode I-II failure, DI-II = 1 corresponds to a mixed failure opening and slip (of and sf). The values 
+of or and sr are calculated using the predefined numerical fracture energy GfI and GfII, for opening 
+failure and shear failure, respectively. The failure mode of the CCE (κ) is computed as  
+𝜅 = {
+1                    (pure tensile, Mode I)
+1 + 𝐷II         (mixed mode, Mode I − II) 
+2                    (pure shear, Mode II)
+  
+ 
+ 
+  (9) 
+ 
+
+a)
+b
+Mode I
+Mode II
+Gf
+0
+Sp
+Opening
+Slip
+O, t
+Normal/tangential bonding stress
+0, s
+Opening/slip
+c)
+f,fs
+Tensile/shear strength
+Internal friction angle
+Pf
+Fracture penalty
+h
+Nominal elementedgelength
+Mode I-HI
+G/G Energy consumed by Mode I/II fracture
+f×f(D)
+0
+Failure path
+Broken
+Mode I-II
+012 
+ 
+Fig. 3 Deformation and failure criteria of the cohesive crack element (CCE). (a) Mode I, tensile 
+mode, (b) Mode II, shear mode, and (c) Mode I-II, mixed-mode. Shaded areas highlight the total 
+fracture energy consumed during the failure process of a CCE. The blue curve (failure path) 
+indicates the stress condition during the yielding and failure processes of the CCE. 
+When both DI and DII are satisfied at the same time, the failure is also considered as Mode 
+I-II, and a value of 1.5 is assigned to these events during post-processing. The broken cohesive 
+crack element (BCCE) is then considered as a new crack with no cohesion, and its behavior is 
+handled by the interaction algorithms, which are discussed in detail in the literature (Munjiza 2004; 
+Mahabadi et al., 2012). 
+2.3 Simulation of fracture propagation and seismicity in FDEM 
+Modeling seismic activity in rocks can provide quantitative information of the rock failure 
+process, and a validated model can improve the understanding of laboratory and field seismic 
+observations. In FDEM, upon breakage of the CCE, the accumulated strain energy is released, 
+resembling seismic activity. The coordinates, failure time, kinetic energy at failure, and failure 
+mode of the related BCCE can be recorded (Lisjak et al., 2013). However, a limitation of this 
+approach is that it considers each BCCE as one single seismic event. Consequently, the properties 
+of the fracture and the associated seismic events are highly dependent on the mesh size and mesh 
+orientation (Munjiza and John, 2002). In nature, the breakage of CCEs can be regarded as acoustic 
+emissions associated with the breakage of several mineral grains and grain boundaries (Zhao et al., 
+2015; Abdelaziz et al., 2018). In most cases, such a mesh dependency needs to be addressed to 
+obtain a better physical meaning of the failure process of CCEs. Zhao et al. (2014) attempted to 
+mitigate the problem with a clustering algorithm considering the temporal and spatial distribution 
+of BCCEs. In their method, each BCCE is viewed as an advancing crack tip, and BCCEs 
+
+13 
+ 
+connecting to the crack tip are clustered together as a continuous fracture. However, this 
+implementation did not consider the physical meaning of fracture propagation. The propagating 
+fracture can arrest and then continue to propagate according to the stress conditions and material 
+heterogeneities (Van der Pluijm and Marshak, 2004), and from an energy dissipation point of view, 
+choosing the yielding point of a BCCE as the fracture tip is more consistent with the cohesive 
+crack model (Shet and Chandra, 2002). 
+Stemming from Zhao et al. (2014) and Zhao (2017), we implemented a new clustering 
+algorithm to mimic fracture propagation process during a seismic event. Note that we consider 
+only seismic activities related to the formation of new fractures, and seismic events created by 
+slipping on existing fracture surfaces are not considered. The algorithm proceeds as follows: 
+(1) The first BCCE that yields at time ty and fails at time tf is considered the initial crack of a 
+cluster. The search algorithm is then executed to include BCCEs connecting to either side of this 
+BCCE (i.e., fracture tips). 
+(2) BCCEs that are connected to the fracture tips and yield within the time window between ty and 
+tf are included in the same cluster and then treated as new fracture tips. At each output frame, the 
+same searching criterion is applied to such new fracture tips until no new BCCEs are found. Then, 
+this cluster of BCCEs is considered to be one continuous fracture, whose growth has produced one 
+seismic event. 
+(3) Repeat steps 1-2, until all recorded BCCEs are processed. 
+(4) Calculate the source parameters of the clustered seismic events as follows (for a cluster of n 
+BCCEs):  
+(a) event time is the breakage time tf of the initial BCCE in this cluster; 
+
+14 
+ 
+(b) the hypocentre location is the centre coordinates of the initial BCCE in this cluster; 
+(c) the kinetic energy, Ee, is calculated as the sum of the kinetic energy of all BCCEs in 
+this cluster, 𝐸e = ∑
+𝐸k
+𝑖
+𝑛
+𝑖=1
+, where 𝐸k
+𝑖 =
+1
+2 ∑
+𝑚𝑗𝑣𝑗
+2
+4
+𝑗=1
+ is the kinetic energy of a BCCE, and 
+mj and vj are the nodal mass and velocity of the BCCE at the time of breakage. We adopt 
+the empirical relation between radiated energy and magnitude to calculate the magnitude 
+of the seismic events: 𝑀𝑒 =
+2
+3 (log𝐸𝑒 − 4.8) (Gutenberg, 1956; Lisjak et al., 2013). 
+(d) the dominant source mechanism (ζ) of each cluster is calculated as a weighted average 
+of the failure modes of all BCCEs in this cluster: 
+𝜁 =
+∑
+𝐸k
+𝑖
+𝑛
+𝑖=1
+𝜅𝑖
+∑
+𝐸k
+𝑖
+𝑛
+𝑖=1
+ 
+ 
+ 
+ 
+ 
+ 
+(10) 
+Where κi is the failure mode of the ith BCCE, and its associated kinetic energy, Eki is taken 
+as its weight. ζ = 1 and 2 represent pure tensile (Mode I) and shear events (Mode II), 
+respectively, while events having 1 < ζ < 2 have tensile and shear failure components 
+(Mode I-II). 
+This algorithm considers multiple BCCEs created by a single fracturing event, resulting in a more 
+realistic representation of the source mechanism and event energy than previous studies. Note that 
+if a series of connected CCEs break simultaneously due to mechanisms such as crushing or 
+pulverization, they will also be clustered as one event under this algorithm. 
+2.4 Numerical model setup 
+
+15 
+ 
+ 
+Fig. 4 Preparation of 2D surface profiles for the FDEM model. (a) The top (left) and bottom (right) 
+parts of the sample used in the rotary shear experiment. (b)–(c) 3D surface scan of the shear 
+surfaces. Red dashed lines indicate the extracted profiles. (d) The initial condition by micro-CT 
+imaging. (e) Comparison of the profiles (red dashed curves) with the micro-CT image showing the 
+initial condition of the shear simulation. Note that profiles are vertically offset for clearer 
+illustration. 
+3D shear simulations would mimic at best the deformation processes, but this is currently 
+impossible due to computational limitations. Instead, we built the 2D FDEM model that considers 
+not only the geometry of the experimental specimen but also the initial contact condition on the 
+rough surface. A 2D circular profile at the radius of 5.6 mm, which corresponds to the roughest 
+region (i.e., highest asperities) on the surface, was extracted (Fig. 4a-c). We chose such a profile 
+because the work by Zhao et al. (2018) suggested that this region with the largest roughness plays 
+an important role in controlling the shear strength and fracture development during the experiment. 
+To capture the geometry of the slipping surface, we digitized the top and bottom surfaces before 
+the experiment using a 3D surface scanner (ATOS II by GOM) at a horizontal grid interval of 
+44 μm. The relative location of the two profiles were adjusted to recreate the initial contact 
+conditions according to the micro-CT image (Fig. 4d-e). We subsampled the profiles to a 0.1 mm 
+
+a)
+b)
+C)
+1 mm
+0.30.20.10.0-0.1-0.2-0.3
+d)
+Elevation(mm)
+e)
+Sheardirection
+0
+5
+10
+15
+20
+25
+30
+(mm)16 
+ 
+nominal grid interval, which was chosen as an acceptable compromise between computation time 
+and accuracy in representing the surface geometry. In addition, to mimic the rotary shear behavior, 
+the two ends of the profiles were extended by 3 mm (i.e., the desired total shear displacement) 
+using the same geometry as their opposite ends to create an effective periodic boundary. These 
+profiles formed the initial shear surfaces of the numerical model (Fig. 4e). 
+ 
+Fig. 5 (a) Mesh topology and boundary conditions of the shear test simulation. The blue dotted 
+line indicates the location of the virtual measurement line. (b) Zoom in view of the refined mesh 
+at the shear surfaces, and the arrows indicate the smallest gap between top and bottom surfaces. 
+The bodies of the top and bottom model were 15 mm in thickness, resulting in a total 
+vertical height of 30 mm, similar to the sample used in the laboratory experiment (Fig. 5a). The 
+corners at the ends of the shear surfaces were filleted with a radius of 0.2 mm to avoid stress 
+concentrations that may result in unrealistic damage. To reduce computational time in applying 
+the normal stress during the simulation, the initial vertical distance between the top and bottom 
+semi-sample was adjusted to 2×10−6 mm (Fig. 5b). Moreover, two rigid boxes were added to 
+simulate the sample holders encasing the two semi-samples. The region of interest (i.e., within 
+1 mm distance from the shear surface) was discretized with a constant nominal element size of 
+0.1 mm. The remaining parts of the model were meshed with linearly increasing mesh size as a 
+
+b)
+5 mm
+V
+0.2 mm
+x
+V17 
+ 
+function of the distance from the shear surface, with the coarsest element size being 3 mm. As a 
+result, the model was meshed into 20,240 triangular elements. These elements were assigned with 
+the calibrated numerical properties (Table 1&2), while the shear boxes had properties of stainless 
+steel (Young’s modulus at 200 GPa, density at 8100 kg/m3, and Poisson’s ratio at 0.25). 
+Table 1 Laboratory measured macromechanical properties (i.e., calibration targets) and emergent 
+properties of the calibrated FDEM model (after Tatone and Grasselli, 2015; Zhao, 2017). 
+Properties (unit) 
+Laboratory 
+measurement 
+Calibrated 
+FDEM model 
+     Density (kg·m−3) 
+1704 
+1704 
+     Young’s modulus (GPa) 
+15.0 
+15.0 
+     Poisson’s ratio (-) 
+0.24 
+0.24 
+     Internal friction angle (Degrees) 
+23 
+23 
+     Internal cohesion (MPa) 
+16.4 
+16.4 
+     Tensile strength (MPa)  
+2.6 
+2.7 
+     Uniaxial compressive strength (MPa) 
+50.3 
+49.9 
+ 
+FDEM models synthesize the macroscopic behavior of materials from the interaction of 
+the micromechanical constituents. The overall deformation and failure behavior of the simulated 
+material are controlled by the combined effect of the input parameters defining the elastic 
+triangular elements and CCEs. As a result, the macroscopic mechanical properties (as listed in 
+Table 1, except for the density that needs no calibration) measured by standard laboratory tests 
+cannot be used directly. Rather, an iterative calibration approach is carried out to obtain input 
+parameters representative of the material, and the laboratory measured properties were used as the 
+calibration targets. In this approach, numerical compressive and tensile strength test models are 
+created and simulated using an initial set of input parameters. The macroscopic mechanical 
+properties and failure patterns are obtained from the simuation and compared against laboratory 
+
+18 
+ 
+measurements. In a successful calibration, the numerical model will replicate both the macroscopic 
+mechanical properties measured from the experiments and the overall failure mode of the material. 
+If the simulation result is inadequate, the input parameters are iteratively fine-tuned until the 
+calibration targets are met (Tatone and Grasselli, 2015). The laboratory-measured properties and 
+the emergent macromechanical properties of the calibrated FDEM model are listed in Table 1, and 
+the calibrated FDEM model parameters are listed in Table 2.   
+Table 2 Calibrated FDEM model input parameters (after Zhao, 2017). 
+Parameter (unit) 
+Value 
+Continuum triangular elements 
+ 
+     Density, ρ (kg·m−3) 
+1704 
+     Young’s modulus, E (GPa) 
+15.6 
+     Poisson’s ratio, υ (-) 
+0.22 
+     Viscous damping factor, α 
+1 
+Cohesive crack elements 
+ 
+     Internal cohesion, c (MPa) 
+17.5 
+     Tensile strength, σt (MPa)  
+2.55 
+     Friction angle, ϕ (Degree) 
+24.5 
+     Mode I fracture energy, GIc (J·m−2) 
+3.8 
+     Mode II fracture energy, GIIc (J·m−2) 
+90 
+     Fracture penalty, Pf (GPa) 
+156 
+     Normal contact penalty, Pn (GPa) 
+156 
+     Tangential contact penalty, Pt (GPa) 
+156 
+ 
+2.5 Simulation procedure and boundary conditions 
+The simulation was computed using the Irazu FDEM software (Geomechanica Inc., 2021) 
+with GPU (graphics processing unit) parallelization. The shear test simulation was conducted in 
+three phases (Table 3). In phase 1, the initial normal stress was applied by compressing the sample 
+at a constant vertical velocity of 0.2 m/s until the vertical stress reaches 2.5 MPa, which 
+
+19 
+ 
+corresponds to the initial normal stress condition of the laboratory experiment. In phase 2, the top 
+and bottom boxes were constrained to their vertical position, and the horizontal shear velocity was 
+increased gradually to 0.3 m/s. This transition phase allows the oscillation induced by the 
+instantaneous stop of normal loading to dampen oscillations due to the shear acceleration. In 
+phase 3, the top and bottom boxes were fixed in their vertical positions (i.e., this is a constant 
+normal stiffness shear test) and moved in the horizontal direction at a constant velocity of 0.3 m/s 
+until the desired shear displacement of 3 mm was reached. Note that the loading velocities used in 
+the study are significantly higher (1000 times) than those used in laboratory experiments; however, 
+such a speed has been verified to provide a quasi-static loading condition while allowing a 
+reasonable computation time (Mahabadi, 2012). The model has 26 million simulation time steps, 
+and each step represents a simulation time of 4×10−10 s. 
+Table 3 Simulation phases and boundary conditions applied to the model. The applied velocities 
+in the x (vx) and y (vy) directions, and the resultant shear displacement (u) are listed. 
+Phase Simulation steps  
+vx (m/s)[1] vy (m/s)[2] u (mm) 
+1 
+1–66,400 
+0 
+0.1 
+0 
+2 
+66,401–964,000 
+0–0.15[3] 
+0 
+0–0.02 
+3 
+964,000–26,000,000 0.15 
+0 
+0.02–3.02 
+[1] Positive (→) on the top box and negative (←) on the bottom box. 
+[2] Negative (↓) on the top box and positive (↑) on the bottom box. 
+[3] Linearly interpolated every time step to ramp up the shear velocity gradually. 
+ 
+Normal and shear stresses were measured along a line parallel to the fault and placed 5 mm 
+above the rigid box in the bottom sample (Fig. 5a). This measurement line monitored the stress 
+conditions every 13,000 simulation steps, equivalent to a 200 kHz monitoring rate. The recorded 
+stress values in all elements along the measurement line were averaged to obtain the overall normal 
+
+20 
+ 
+stress (σn) and shear stress (τ), which were used to calculate the friction coefficient μ = τ/σn, similar 
+to the laboratory-measured apparent friction coefficient. 
+3 Results and data analysis 
+3.1 Shear behavior 
+ 
+Fig. 6 Calculated friction coefficient of (a) the laboratory test results plotted as a function of the 
+equivalent slip distance at a radius of 5.6 mm and (b) the numerical simulation. The first ~0.3 mm 
+are detailed in Fig. 7. Red arrows indicate significant drops of frictional resistance associated to 
+seismic events 1, 2, and 3, which are investigated in Section 3.2. 
+The simulated μ showed a similar trend with the experimental data. It reached the peak 
+value of 0.22 at a shear displacement of 0.32 mm, followed by a significant drop (Fig. 6b). The 
+simulated μ experienced many abrupt drops during the slipping process and then stabilized at 
+approximately 0.04 after approximately 1.7 mm of shear displacement. The simulated μ was 
+significantly lower than the value reported in the laboratory experiment, with many more 
+oscillations.  
+
+a)
+Experiment
+1.5
+1.0
+0.5
+0
+0
+0.5
+1.2
+1.8
+2.4
+3.0
+Equivalent slip distance (mm)
+b)
+Simulation
+0.2
+0.1
+0
+Event 2 & 3
+-0.1
+-Event 1
+Fig. 7b
+-0.2
+0
+0.5
+1
+1.5
+2
+2.5
+3
+Shear displacement (mm)21 
+ 
+The simulated stress conditions of the first 0.3 mm showed intriguing similarities to the 
+laboratory experimental data (Fig. 7a&b). In this interval, the shear behavior observed in the 
+experiment can be divided into four stages (Fig. 7a): (I) τ, σn, and the resultant μ ramped up 
+gradually; (II) τ experienced a relatively stable stage with minor change, and σn decreased 
+continuously, causing minor change of μ; (III) τ and σn gradually increased to a peak shear stress, 
+and μ increased to the peak value; and (IV) τ, σn, and μ dropped rapidly.  
+ 
+Fig. 7 Comparison of the overall normal and shear stresses and the friction coefficient of stages I-
+IV between (a) the experimental data and (b) the simulated data. (c) and (d) are the zoom in views 
+of the local shear and normal stresses, respectively, at the asperity responsible for the stress drop 
+at stage IV. Orange circles numbered 1-6 indicate the horizontal shear displacements (u). (e) and 
+(f) are the micro-CT image of the laboratory specimen corresponding to frame 1 and 6 in (c) and 
+(d). The initial surface profiles from the surface scan data (red curves) are placed next to the 
+laboratory fault for comparison. 
+
+a)Experiment
+b) Simulation
+II
+III
+IV
+II
+III
+IV
+70.6
+T(MPa)
+2
+0.4
+(MPa)
+0.2
+0
+0
+3
+On (MPa)
+2.2
+6
+tttttt!
+1.8
+1
+0.2
+0.1
+≥0.5
+、.
+0
+e)
+0
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0.05
+0.15
+0.2
+0.25
+0.3
+Shear displacement (mm)
+Shear displacement, u (mm)
+1
+u=0mm
+②u=0.092mm
+③ u= 0.136 mm
+4u=0.183mm
+5 u=0.320mm
+6u=0.321mm
+5mm
+Localshear/normalstress(MPa)
+0
+5
+10
+2022 
+ 
+The numerical simulation qualitatively captured the general trend of these stages (Fig. 7b); 
+however, simulated σn in stage I decreased gradually, and more oscillations are observed in the 
+curves in the simulated data. To further investigate the mechanisms behind the shear behavior 
+during these stages, we examined the simulated local stress conditions around the asperity whose 
+breakage was responsible for the large and sudden drop of frictional resistance at stage IV (Fig. 
+7c&d). During stage I, the simulation shows that the shear surface is at the initial contact condition. 
+As the shear displacement increases, the shear stress increases gradually due to frictional resistance 
+of the initial contact area. Note that the numerical model did not capture the minor normal stress 
+increase measured in the experiment at this stage. Such an increase may be related to the interaction 
+of the asperities in the direction perpendicular to shear (i.e., out-of-plane motion) that does not 
+exist in the 2D simulation. During stage II, the top and bottom surfaces adjusted to a more 
+conforming contact, which resulted in the decrease of the shear and normal stress. During stage III, 
+new contact points were established, and asperities engage and interlock, causing the shear stress 
+to increase rapidly reaching the peak shear stress at the end of this stage. Asperities survived and 
+climbed onto each other, causing dilation that increased the normal stress. At stage IV, the highly 
+stressed asperity underwent high-stress concentration and failure, releasing the accumulated strain 
+energy that resulted in the sudden and significant drop of stresses and frictional resistance. The 
+simulated failure pattern, in terms of location and mechanism, resembled the laboratory 
+observation (Fig. 7e&f). More importantly, the numerical model can provide the evolution of 
+surface contacts and stress conditions throughout the shear process. 
+3.2 Progressive damage, gouge formation, and seismic activity 
+Progressive damage on the shear surface and fault gouge formation was simulated by 
+BCCEs (Fig. 8). The first several BCCEs occurred when the top and bottom semi-sample were 
+
+23 
+ 
+loaded with the initial normal stress. Before ⁓1 mm of shear displacement, the damage was 
+concentrated in the vicinity of the shear surface. After ⁓1 mm of shear displacement, a number of 
+sub-vertical fractures penetrated the sample body, resembling the fracturing observed in the 
+laboratory (Fig. 9). The distribution of the shear-induced damage was mostly concentrated close 
+to the fault surface and heterogeneously distributed along the fault. Broken asperities formed the 
+gouge layer that accumulated between the semi-samples. As a result, some portions of the bare 
+fracture surface were protected from wearing (Fig. 9a), and this phenomenon is also observed in 
+the laboratory micro-CT image (Fig. 9b). 
+
+24 
+ 
+ 
+Fig. 8 Damage of the shear surface and the accumulation of gouge material with increasing shear 
+displacement. Damage is represented by broken cohesive crack elements. 
+
+25 
+ 
+ 
+Fig. 9 (a) Zoom-in view of a portion of the simulated fault surface at 3 mm of slip. The dashed red 
+lines highlight intact fault walls that were not damaged. (b) Zoom-in view of the micro-CT image 
+of a portion of the laboratory fault at a similar location to (a) (adopted from Zhao et al., (2018)). 
+  
+Fig. 10 Simulated seismic activities. (a) Magnitude, location, and failure mode of the clustered 
+seismic activity. (b) Event count in each bin. 
+A total of 7,557 CCEs were broken throughout the simulation, and they were clustered into 
+1,561 seismic events. Most of the BCCEs near the shear surface failed in shear mode (Mode II), 
+and almost all sub-vertical fractures propagated in tensile mode (Mode I). The magnitude of these 
+seismic events ranged between −11.1 and −4.4, with an average magnitude of −7.3. In general, 
+
+a)
+1mm
+-Intact surface
+Intact surface
+-Gouge
+Gouge
+Shear
+ Shear induced fracture
+induced
+fractures
+mmMode I-IH
+20
+30
+35
+40
+2026 
+ 
+large magnitude events were mostly produced by shear-mode failures, while small magnitude 
+events mostly arose from tensile-mode failures (Fig. 10a). Along the vertical direction (y direction), 
+the spatial distribution of seismic activity coincides with the damage pattern: events were 
+concentrated within ±4 mm of the fault. We divided the horizontal length of the fault, i.e., the x 
+direction, into 100 bins and examined the spatial distribution of the seismic events along such a 
+path (Fig. 10b). Seismic events were distributed heterogeneously along the fault: bins at x = 
+29.8 mm and 34.9 mm had the largest number of events at 44; bins at x ranging 10.5 to 12.2 mm 
+and 20.5 to 21.0 mm had no seismic events. 
+Each asperity failure resulted in the sudden and significant drop of frictional resistance and 
+the release of accumulated strain energy. These large magnitude events caused stick-slip-like 
+responses and released high amplitude stress waves propagating across the model. Prior to these 
+dynamic seismic events, their corresponding locations experienced low shear velocity due to the 
+interlocking of asperities and are referred to as interlocking zones (ILZs) in the following 
+discussion. Stress concentrated at ILZs and eventually broke the asperities, releasing the 
+accumulated strain energy (see animated figures Fig. S1 in Supplementary Material for the velocity 
+fields). Three seismic events (Events 1-3, as indicated in Fig. 6) with distinct wave radiation 
+patterns are chosen as examples for further examination. Event 1 at u⁓0.3 mm was related to the 
+most significant friction drop. Events 2 and 3 were two consecutive events that occurred on the 
+slipping surface 12.5 mm apart from each other with a 0.005 ms time delay. We examined the 
+stress field (Fig. 11) and observed that the magnitude of seismic events was directly correlated to 
+the magnitude of the stress concentration at the asperities that failed. We observed that stress 
+concentration at the ILZs reached values as high as the compressive strength of the material, 
+causing compressive failure. Due to interlocking, the non-interlocking regions slightly ahead (with 
+
+27 
+ 
+respect to the shear direction) of the ILZs were subjected to significant tensile stress that reached 
+the tensile strength of the material, thus, causing tensile fracturing. By examining the particle 
+velocity field (Fig. 12), we found that prior to the seismic events, the locations of the ILZ were 
+experiencing particle velocities lower than the loading velocity (i.e., < 0.1 m/s). As the seismic 
+events occurred, the source region had particle velocities that were two orders of magnitude higher 
+than that of the ILZs (i.e., > 10 m/s). Interestingly, considering that P- and S-wave velocities are 
+2967 m/s and 1884 m/s, respectively, Event 3 occurred right after the arrival of the P-wave induced 
+by Event 2, but prior to the arrival of the S-wave. Therefore, Event 3 may have been triggered by 
+the stress perturbation from Event 2. 
+Fig. 11 Output frames of the numerical model showing the horizontal stress σxx at (a-c) Event 1 
+and (d-g) Events 2 and 3. Interlocking zones (ILZs) that are related to the selected seismic events 
+are highlighted by the yellow arrows. Note that the simulation time interval between two frames 
+is 0.052 ms.  
+ 
+
+evenl
+erem28 
+ 
+ 
+Fig. 12 Output frames of the numerical model showing the particle velocity at (a-c) Event 1 and 
+(d-g) Events 2 and 3. ILZs are highlighted by the yellow arrows and the P-wave wavefront of 
+Event 2 is labelled. Note the color map is in log scale. 
+4 Discussion 
+The FDEM numerical model qualitatively captured the mechanical behavior observed in 
+the laboratory experiments, highlighting the dominant role of surface roughness on the shear 
+behavior of rocks at low-stress conditions. Both the laboratory experiment and the numerical 
+simulation show a slip weakening behavior where the friction coefficient ramps up to the peak 
+value and then decreases to a residual value. In the numerical simulation, the shear stress and 
+friction reached a steady-state and residual value around ~1.7 mm of total displacement, in 
+agreement with the laboratory value (Zhao et al., 2018). 
+During the first ~0.3 mm of shear displacement, the discrepancies between the 
+experimental and simulation results in stage I and the additional stress oscillations in the simulation 
+results may be because the 2D model was not able to capture 3D asperity interactions. However, 
+the overall variation trend of stresses and the damage pattern on the shear surface showed close 
+
+Event!
+front
+ILZ
+Eyent2
+Event329 
+ 
+similarities, suggesting that the 2D profile that we used is a proxy for the laboratory specimen. 
+This also supports our previous interpretation that the highest asperity was responsible for the 
+formation of the large secondary sub-vertical fractures and the associated sudden drop in shear 
+resistance (Zhao et al., 2018). These results suggest that our numerical technique, which uses a 
+combination of surface scanning, X-ray micro-CT imaging, and FDEM modelling, represents a 
+promising approach to simulate realistic fault behavior. Our simulation provides the continuous 
+evolution of contacts on the shear surface and the stress conditions that complement the laboratory 
+observations in achieving a better comprehension of how the interaction between asperities 
+controls the stress conditions and damage patterns in faults.  
+During the shear process, asperities interact in various modes including climbing onto each 
+other, interlocking, and breaking (Scholz, 1990). Our experimental and numerical results show 
+that such interactions directly influenced the stress conditions and damage patterns. When the slip 
+displacement is small (u < 1.5 mm), weak asperities (i.e., millimetric scale unevenness) controlled 
+the frictional behavior, creating gouge material. These observations agree with the laboratory 
+observations on the post-mortem sample and suggest the importance of surface roughness in 
+controlling the formation of the gouge layer. As the slip displacement increases (u > 1.5 mm), the 
+large-scale roughness of the shear surface (i.e., centimetric scale waviness) becomes important to 
+the shear behavior and damage pattern. Large scale waviness causes high stress concentration 
+through interlocking and climbing and may cause sub-vertical secondary fractures. 
+The damage and seismic event distributions are closely related to the stress heterogeneity 
+on the shear surface caused by the surface roughness. Depending on the geometry of the asperity, 
+the stress concentration at the ILZs could reach the compressive strength of the material, causing 
+compressive failure. This mechanism creates gouge material in the vicinity of the shear surface. 
+
+30 
+ 
+On the other hand, the areas ahead of the ILZs experience tensile stress up to the tensile strength 
+of the material, thus, creating tensile fractures. This mechanism creates large sub-vertical 
+secondary fractures. Breakage of strong asperities release the accumulated strain energy in the 
+whole model, causing an overall shear stress drop, giving a stick-slip-like shear behavior. Such a 
+lock-and-fail mechanism is recently found to be the key process of stick-slip behavior of bare 
+surfaces (Chen et al., 2020; Morad et al., 2022). Note that the overall shear loading in our model 
+is considered quasi-static, but the local seismic events are dynamic activities with particle velocity 
+more than 100 times the quasi-static loading velocity. This suggests that on a rough shear surface, 
+quasi-static shear consists of numerous heterogeneously distributed local dynamic seismic 
+activities, and this process may complicate the slip process on rough faults and the estimation of 
+the energy budget (Tinti et al., 2005). 
+Observations on Events 2 and 3 suggest that the stress perturbation from asperities 
+breakage may trigger events on adjacent interlocking zones. From an earthquake perspective, there 
+are two possible mechanisms that may trigger seismic events in the near field: (1) static stress 
+redistribution (e.g., King et al., 1994; Toda et al., 1998) and (2) dynamic stress wave perturbation 
+(e.g., Kilb et al., 2000; Gomberg et al., 2001). In our simulation, the modeled body did not slip as 
+a rigid body, rather, the slipping consisted of pulses of local movements, accompanied by 
+numerous continuously changing of contacts and asperities breakages. When the asperity 
+associated to Event 1 breaks, the dynamic stress perturbation was damped out, and the static stress 
+concentration is transferred to nearby asperities, which eventually caused failure of other asperities. 
+On the other hand, Events 2 and 3 showed a more interesting correlation. Event 3 occurred 
+between the arrival times of P- and S-waves from Event 2. Within this time window, stress 
+redistribution had not reached a steady state, suggesting that the perturbation of the dynamic stress 
+
+31 
+ 
+wave radiated from Event 2 may have triggered Event 3. These results imply that static stress 
+transfer and dynamic stress perturbation triggering may occur on the same fault and contribute to 
+the movement of fault slip. However, due to the limitation of the model output frequency and post-
+processing method, the triggering is not conclusive, Event 2 and 3 may have been independent 
+seismic events occurred in a narrow time window, and more investigation is needed in future 
+research. 
+The numerical simulation has the advantage of continuously modeling the fault shear 
+process, fault surface damage, and associated stress conditions. However, the simulated sample 
+experienced more damage than the laboratory sample, which is probably related to the limitation 
+of 2D simulations not accounting for the motion in the third dimension. For the same reason, the 
+simulated stresses suffered significant fluctuations, and the friction coefficient was much lower 
+than the experimental measurement, which is a common limitation of 2D simulations. The 
+laboratory experiment by Frye and Marone (2002) and the numerical simulation by Hazzard and 
+Mair (2003) demonstrated that 2D numerical models exhibit friction values notably lower than 3D 
+models and suffer from greater stress fluctuations due to the lack of particle motion in the third 
+dimension. In addition, we meshed the shear surface at a relatively high resolution (0.1 mm), 
+resulting in a large number of asperities at various sizes. Hence, the interlocking and breakage of 
+these asperities caused stress oscillations (i.e., microseismic events). Even though we qualitatively 
+captured the shear behavior that matches the laboratory measurements, to fully capture the shear 
+behavior of the rotary shear experiment, a 3D model capturing the surface geometry and asperity 
+interaction on the entire shear surface will be required. 
+5 Conclusion 
+
+32 
+ 
+In this study, we used a carefully built and calibrated FDEM numerical model to simulate a 
+laboratory shear experiment. We introduced a new clustering algorithm to improve the 
+understanding of the simulated fracturing and associated seismic events. The model was able to 
+qualitatively capture the frictional behavior observed in the laboratory experiment, providing the 
+missing information in the experimental observation regarding the continuous variation of stresses 
+and the progressive evolution on the shear surfaces.  
+Our numerical model matches the experimental results particularly well at the beginning 
+of the shear deformation (~0.3 mm). We were able to identify similar stress variation trends and 
+damage patterns. The simulation results provided detailed evolution processes of the contacts on 
+the shear surface and the local stress conditions, which are not available in experimental 
+observations. Combining the numerical and experimental results, we conclude that interlocking of 
+asperities can cause compressive stress concentration on the front side (i.e., facing the shear 
+direction) of the asperity, which could induce compressive failure (e.g., crushing) near the shear 
+surface; on the other hand, tensile stress concentration is generated on the leeward side of the 
+asperity, which could cause sub-vertical tensile fractures that could propagate into the host rock. 
+Progressive surface damage and the associated microseismic events occur at the locations of 
+asperity interactions and is highly heterogeneous. Several locations experienced no damage even 
+after large shear displacement, these locations are either not in contact or were protected by gouge 
+materials. 
+As a result of the interlocking and breakdown of asperities, local dynamic failure events 
+occur, even though the overall loading is quasistatic. These events are considered microseismic 
+events, and their magnitudes range between −11.1 and −4.4. Strain energy stored in the medium 
+was released during these events, causing dynamic perturbation to the overall stress condition, and 
+
+33 
+ 
+the particle velocity in the source reached > 10 m/s, two orders of magnitude larger than the 
+surrounding regions. This high amplitude stress perturbation could even trigger the failure of 
+adjacent critically stressed asperities. 
+Both the numerical model and the experiment suggested the importance of shear surface 
+roughness in controlling slip behavior, and we were able to explain the laboratory observations 
+with the help of numerical results. Shear surface evolution is a complicated process that involves 
+frictional sliding, fracturing, gouge comminution, and seismicity. The high degree of agreement 
+between simulation and experiment data leads to a promising future of predicting fault behavior 
+through, laboratory testing, surface characterization, and numerical simulations. These results 
+improved the understanding of shear behavior and demonstrated that micromechanical based 
+numerical simulation is a capable approach to study fault mechanics. 
+Declaration of competing interest 
+The authors declare that they have no known competing financial interests or personal 
+relationships that could have appeared to influence the work reported in this paper. 
+Acknowledgements  
+Q. Zhao is supported by the FCE Start-up Fund for New Recruits at the Hong Kong Polytechnic 
+University (Project ID P0034042) and the Early Career Scheme of the Research Grants Council of 
+the Hong Kong Special Administrative Region, China (Project No. PolyU 25220021). This work 
+has also been supported through the NSERC Discovery Grants 341275, CFILOF Grant 18285, 
+Carbon Management Canada (CMC), and NSERC/Energi Simulation Industrial Research Chair 
+Program. The authors would like to thank Geomechanica Inc. for providing the Irazu FDEM 
+simulation software. Q. Zhao would like to thank Dr. Andrea Lisjak and Dr. Bin Chen for 
+
+34 
+ 
+discussions and suggestions. The authors appreciate the constructive suggestions and comments 
+from the editor and the reviewers. 
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+

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+Resonant inelastic X-ray scattering in topological semimetal FeSi
+Yao Shen,1 Anirudh Chandrasekaran,2, 3 Jennifer Sears,1 Tiantian Zhang,4, 5 Xin Han,6 Youguo Shi,6
+Jiemin Li,7 Jonathan Pelliciari,7 Valentina Bisogni,7 Mark P. M. Dean,1, ∗ and Stefanos Kourtis8, 2, †
+1Condensed Matter Physics and Materials Science Department,
+Brookhaven National Laboratory, Upton, New York 11973, USA
+2Department of Physics, Boston University, Boston, MA, 02215, USA
+3Department of Physics and Centre for the Science of Materials,
+Loughborough University, Loughborough LE11 3TU, UK
+4Department of Physics, Tokyo Institute of Technology, Okayama, Meguro-ku, Tokyo, Japan
+5Tokodai Institute for Element Strategy, Tokyo Institute of Technology, Nagatsuta, Midori-ku, Yokohama, Kanagawa, Japan
+6Beijing National Laboratory for Condensed Matter Physics,
+Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
+7National Synchrotron Light Source II, Brookhaven National Laboratory, Upton, New York 11973, USA
+8Institut quantique & D´epartement de physique, Universit´e de Sherbrooke, J1K 2R1, Qu´ebec, Canada
+(Dated: January 10, 2023)
+The energy spectrum of topological semimetals contains protected degeneracies in reciprocal space that corre-
+spond to Weyl, Dirac, or multifold fermionic states. To exploit the unconventional properties of these states, one
+has to access the electronic structure of the three-dimensional bulk. In this work, we resolve the bulk electronic
+states of candidate topological semimetal FeSi using momentum-dependent resonant inelastic X-ray scattering
+(RIXS) at the Fe L3 edge. We observe a broad excitation continuum devoid of sharp features, consistent with
+particle-hole scattering in an underlying electronic band structure. Using density functional theory, we calculate
+the electronic structure of FeSi and derive a band theory formulation of RIXS in the fast collision approximation
+to model the scattering process. We find that band theory qualitatively captures the number and position of the
+main spectral features, as well as the overall momentum dependence of the RIXS intensity. Our work paves the
+way for targeted studies of band touchings in topological semimetals with RIXS.
+I.
+INTRODUCTION
+In the last decade and a half, topological matter has become
+a cornerstone of quantum materials science [1]. The discovery
+of three-dimensional topological insulators [2], in particular,
+sparked a flurry of activity in the then nascent field. Elec-
+trons in these crystalline materials are effectively noninteract-
+ing, giving rise to electronic bands in the bulk that are indis-
+tinguishable from those of a trivial band insulator. The elec-
+tronic wavefunction, however, is characterized by topologi-
+cal indices that dictate the presence of symmetry-protected
+Dirac states at the surface of the material, as well as nontrivial
+(magneto-)transport responses.
+More recently, topological semimetals have been added
+to the catalogue of three-dimensional topological materi-
+als [3, 4]. These systems also feature topologically protected
+boundary states and nontrivial (magneto-)transport, but addi-
+tionally have distinct geometric characteristics in their bulk
+band structure. In the simplest case of Weyl semimetals, these
+geometric characteristics are singly degenerate energy sur-
+faces in reciprocal space that contain a band touching point—a
+Weyl node—around which electronic bands disperse linearly
+in all three directions in reciprocal space [5–7]. Such band
+touchings are Berry curvature singularities characterized by
+topological indices. The value of the topological index of a
+nodal point determines the geometry of the band dispersion in
+∗ mdean@bnl.gov
+† Stefanos.Kourtis@usherbrooke.ca
+the vicinity of nodal points [8–10]. Conversely, the geometry
+of the bulk bands becomes a proxy of topology in these ma-
+terials. Measuring the electronic density of states in the bulk
+can therefore reveal the topological nature of a semimetal.
+Resonant inelastic X-ray scattering (RIXS) is a spectro-
+scopic technique that yields momentum- and energy-resolved
+spectra of charge-neutral electronic excitations. While RIXS
+has been extensively used in studying magnetic excitations in
+gapped materials like insulators and superconductors [11, 12],
+it is increasingly applied in studies of compounds that host
+itinerant carriers with small or no charge gaps [13–15]. That
+RIXS can be used to map electronic bands of materials, in-
+cluding semimetals, has long been established [16–23]. Im-
+provements in resolution in recent years have renewed in-
+terest in using RIXS to detect band structure effects at meV
+energy scales in materials of technological interest, such as
+unconventional superconductors [24]. There are even theo-
+retical proposals to use RIXS to measure topological indices
+of nodal points in topological semimetals [25, 26].
+These
+prospects are particularly appealing for probing the bulk of
+three-dimensional materials, since alternative techniques such
+as angle-resolved photoemission (ARPES) and scanning tun-
+neling spectroscopy, probe predominantly the surface rather
+than the bulk. Furthermore, topological nodal points may only
+appear above the Fermi level or as a result of an applied mag-
+netic field, settings in which the resolving power of ARPES is
+limited. As these settings may be relevant to the technologi-
+cal exploitation of topological materials, alternative methods
+to visualize the bulk band structure and to identify topologi-
+cal features are sought after. Before honing in on properties
+of topological origin, however, one has to determine whether
+arXiv:2301.02677v1  [cond-mat.str-el]  6 Jan 2023
+
+2
+bulk band structure effects at large are detectable in RIXS of
+topological semimetals.
+In this work, we use RIXS to probe the bulk of FeSi aim-
+ing to quantitatively test how bulk band structure manifests
+in RIXS spectra of a putative topological semimetal [27, 28].
+We observe broad continua in the RIXS spectra, consistent
+with particle-hole scattering in an underlying band structure.
+We model the RIXS process in the fast-collision approxima-
+tion using the band structure of FeSi as determined by den-
+sity functional theory (DFT) calculations. We find reasonable
+agreement between experiment and theory in the number and
+hierarchy of dominant spectral features. We interpret this find-
+ing as evidence of a bulk band structure in FeSi, with many-
+body effects playing only a secondary role in determining the
+RIXS spectrum. Our results indicate that higher resolution
+experiments — feasible with existing instruments — could vi-
+sualize topological nodal points and thus identify and classify
+topological semimetals.
+II.
+THEORY
+A.
+RIXS cross section and fast collision approximation
+We briefly introduce the theoretical description of RIXS at
+zero temperature. More comprehensive presentations of RIXS
+can be found in Refs. 29 and 30.
+In a RIXS experiment, core electrons of an ion are pro-
+moted to a state above the Fermi level εF by an intense x-
+ray beam, thereby locally exciting the irradiated material into
+a highly energetic and short-lived intermediate state. Subse-
+quently, the core hole recombines with a valence electron. The
+process imparts both energy and momentum to particle-hole
+excitations in the material. In what follows, we will consider
+excitation of core electrons directly into orbital(s) close to εF ,
+which give rise to the low-energy physics in the material. This
+process, which is often referred to as direct RIXS, is illus-
+trated in Fig. 1.
+The double differential cross section is a measure of the
+total RIXS intensity. Up to a constant prefactor, it is given by
+I(kin, kout, ωin, ωout, ϵin, ϵout)
+=
+�
+fg
+|Ffg(kin, kout, ωin, ϵin, ϵout)|2δ(Eg − Ef + ℏ∆ω) ,
+(1)
+where ℏ∆ω = ℏ(ωin − ωout) is the energy transferred to the
+material, kin and kout (ϵin and ϵout) the incoming and outgoing
+photon wavevectors (polarizations) and Eg and Ef the ener-
+gies corresponding to initial and final many-body states |g⟩
+and |f⟩ of the valence electrons. The scattering amplitude
+Ffg in the dipole approximation is
+Ffg(kin, kout, ωin, ϵin, ϵout)
+= ⟨f| �D†(ϵout, kout) �G(ωin) �D(ϵin, kin)|g⟩ ,
+(2)
+where �G is the intermediate-state propagator
+�G(ωin) = (Eg + ℏωin + iΓ − �
+H)−1 ,
+(3)
+with �
+H the Hamiltonian describing the system in the interme-
+diate excited state and Γ the intermediate-state inverse life-
+time. The dipole operators �D and �D † represent the x-ray ab-
+sorption and emission, respectively. For a crystalline material,
+they can be written as
+�D(ϵ, k) = ϵ · �Dk ,
+(4)
+�Dk =
+�
+µ,ν
+⟨µ|�r − rµ|ν⟩
+�
+R
+eik·R �d †
+Rµ �pRν ,
+(5)
+where R is the lattice position. States |µ⟩ and |ν⟩ express
+single-electron valence and core states respectively. The com-
+bined valence (core) index µ (ν) encodes spin, orbital, and
+sublattice degrees of freedom.
+Core states |ν⟩ are conve-
+niently expressed as atomic orbitals, whereas valence states
+|µ⟩ can be appropriately chosen Wannier functions, both lo-
+calized in space around the same position rµ of each atomic
+site within the unit cell. The position operator �r, defined with
+respect to each lattice position R, is the same for all ions.
+For the L2/3 and M2/3 resonant edges, the operators �d †
+Rµ and
+�pRν create a d-orbital electron and a p-orbital core hole, re-
+spectively.
+Physical arguments allow us to simplify the RIXS scatter-
+ing amplitude. First, core holes do not hop appreciably; they
+are created and annihilated at the same site. Taking this into
+account, Ffg becomes [30]
+Ffg(q, ωin, ϵin, ϵout)
+=
+�
+µ,ν,µ′,ν′
+Tµνµ′ν′(ϵin, ϵout)Fµνµ′ν′(q, ωin) ,
+(6)
+where q = kin − kout. The scattering amplitude has been
+factored in the atomic scattering tensor
+Tµνµ′ν′(ϵin, ϵout) = ⟨µ|ϵout · �r|ν⟩∗⟨µ′|ϵin · �r|ν′⟩
+(7)
+and the fundamental scattering amplitude
+Fµνµ′ν′(q, ωin)
+= ⟨f|
+�
+R
+e−iq·R �p †
+Rν �dRµ �G(ωin) �d †
+Rµ′ �pRν′|g⟩ .
+(8)
+The intrinsic spectral characteristics of a material are carried
+by the tensor F, which is typically the main quantity of inter-
+est in theoretical studies. The tensor T modulates the scatter-
+ing amplitude according to the geometry of the localized core
+and valence states. The entries of T can be calculated given
+knowledge of the valency of the targeted ion and the symme-
+try group of the crystal [31–34].
+Then, within the fast-collision approximation, one assumes
+that �G(ωin) ≈ 1/Γ, where Γ is the inverse core-hole lifetime.
+In this approximation, the RIXS process reduces to the intro-
+duction of a particle-hole excitation with fixed momentum and
+energy in the material — see Fig 1 for an example.
+Before proceeding to derive the theory of RIXS in band
+structures, we evaluate the geometric modulation of the RIXS
+spectrum owing purely to the orbital content of the quantum
+
+3
+εk
+k
+εF
+energy
+core level
+εF
+εk
+k
+INITIAL
+energy
+core level
+εF
+εk
+k
+FINAL
+fast
+collision
+approximation
+FIG. 1. Illustration of the direct RIXS process and reduction to effective particle-hole scattering via the fast-collision approximation.
+states involved in the RIXS process. This is obtained by set-
+ting the fundamental scattering amplitude Fµνµ′ν′ to unity:
+T (ϵin, ϵout) =
+������
+�
+µ,ν,µ′,ν′
+Tµνµ′ν′(ϵin, ϵout)
+������
+2
+.
+(9)
+We shall use T as a diagnostic to disentangle contributions to
+the modulation of the RIXS intensity as a function of scatter-
+ing angles. The first contribution comes through the polariza-
+tion vectors, which are angle-dependent — see Fig. 2. The
+second contribution is the intrinsic momentum dependence
+coming from electronic dispersion in the material. In Sec. V
+we calculate the geometric modulation in Eq. (9) for FeSi and
+compare it to the scattering angle dependence of RIXS inten-
+sity.
+B.
+RIXS process in a band structure
+We wish to describe the RIXS response of crystalline mate-
+rials in which electrons are, to a good approximation, non-
+interacting.
+Valence electrons in these materials are well-
+described by band theory. The states |g⟩ and |f⟩ in Eq. (2)
+are then collections of Bloch modes.
+For a given RIXS edge, one then sums over core states
+|ν⟩ and valence Wannier states |µ⟩ and
+��µ′�
+connected by the
+dipole operators �D, �D†. Here we study the Fe L3 edge, hence
+we consider 2p3/2 orbitals for core electrons and the 3d shell
+for valence electrons.
+In k-space, the band eigenbasis is given by a unitary rota-
+tion of a basis of Wannier states |µ⟩ per lattice position R to a
+basis of Bloch states |kµ⟩. The Wannier states have wavefunc-
+tions ϕµ(x) = ⟨x|µ⟩ that are centered about different points
+in the unit cell, possibly atomic sites. Let ϕkµ(x) = ⟨x|kµ⟩
+be the spatial wavefunction of |kµ⟩, which could be a spinor.
+We then have
+ϕkµ(x) =
+1
+√
+N
+�
+R
+eik·Rϕµ(x − R) .
+(10)
+The raising and lowering operators of the Bloch wavefunc-
+tions are �d †
+kµ and �dkµ. They are defined by �d †
+kµ|Ω⟩ = |kµ⟩,
+{�dkµ, �dk′µ′} = 0 and {�dkµ, �d †
+k′µ′} = δk,k′δµ,µ′, where |Ω⟩ is
+the vacuum of valence excitations and core holes. A general
+Hamiltonian describing noninteracting valence electrons is
+�
+Hband =
+�
+k∈BZ
+�
+µ,µ′
+�d †
+kµ Hµµ′(k) �dkµ′ ,
+(11)
+where Hµµ′(k) are the elements of the matrix H(k).
+Let U(k) be a matrix that diagonalizes H(k), such that
+U †(k)H(k)U(k) is a diagonal matrix containing the eigen-
+values εl(k), which constitute the dispersing bands. We can
+then write
+�dkµ =
+�
+l
+Uµl(k) �ψkl,
+(12)
+where l denotes an energy band index and �ψkl annihilates the
+corresponding eigenstate. The ground state at zero tempera-
+ture is obtained by populating all states below the Fermi level:
+|g⟩ =
+�
+� �
+l,k∈BZ
+Θ(εF − εl(k)) �ψ†
+kl
+�
+� |Ω⟩ ,
+(13)
+with Θ the Heaviside step function. The Wannier lowering
+operators at any lattice site R can be expressed in terms of the
+band operators as
+�dRµ = 1
+√
+N
+�
+k∈BZ
+e−ik·R �dkµ
+(14a)
+= 1
+√
+N
+�
+l,k∈BZ
+e−ik·R Uµl(k) �ψkl.
+(14b)
+We now assume, as per the fast collision approximation,
+that the intermediate-state Hamiltonian is well approximated
+by the band Hamiltonian, along with a core hole inverse life-
+time Γ in the intermediate-state propagator. Due to this as-
+sumption, core electron operators cancel out and the interme-
+diate state propagator becomes simply
+�G(ωin) =(Eg + ℏωin + iΓ − �
+Hband)−1 ,
+(15a)
+=
+�
+k
+�
+l
+|k, l⟩⟨k, l|
+Eg + ℏωin + iΓ − εl(k) .
+(15b)
+
+4
+where |k, l⟩ are band eigenstates, and we treat Eg and Γ as
+free parameters to be determined by fitting the x-ray absorp-
+tion spectrum (see App. B).
+Substituting the expression for �dRµ in Eq. (14b) in Eq. (8),
+we obtain the fundamental RIXS scattering amplitude in a
+band structure
+Fµµ′(q, ωin) = ⟨f|
+�
+k,k′∈BZ
+�
+l,l′
+1
+N
+�
+R
+e−i(k+q−k′)·R
+× Uµl(k) U ∗
+µ′l′(k′) �ψkl �G(ωin) �ψ †
+k′l′|g⟩ .
+(16)
+Notice that F is independent of the core orbitals at this level
+of description. The sum over R evaluates to Nδk′,k+q, which
+enforces k′ = k + q. When εl′(k + q) > εF, we have that
+�ψ †
+k+ql′|g⟩ is an eigenstate of the band Hamiltonian with en-
+ergy Eg + εl′(k + q) (otherwise the single particle state is
+already occupied and this term evaluates to zero). The action
+of �G(ωin) on �ψ †
+k+ql′|g⟩ is
+�G(ωin) �ψ †
+k+ql′|g⟩ =
+Θ(εl′(k + q) − εF)
+ℏωin − εl′(k + q) + iΓ
+�ψ †
+k+ql′|g⟩.
+(17)
+Furthermore, the action of �ψkl on �ψ †
+k+ql′|g⟩ is non-zero only
+if εl(k) < εF (we need this single particle level to be occupied
+for the term to be non-zero). Using this we obtain
+Fµµ′(q, ωin) =
+�
+l,l′
+�
+k∈BZ
+�
+⟨f| �ψkl �ψ †
+k+ql′|g⟩
+× Θ(εl′(k + q) − εF)
+× Θ(εF − εl(k))
+×
+Uµl(k) U ∗
+µ′l′(k + q)
+ℏωin − εl′(k + q) + iΓ
+�
+.
+(18)
+The sum over final states |f⟩ can be taken over the eigenstates
+of �
+Hband. The pair of step functions in the fundamental scat-
+tering amplitude given above in Eq. (18) ensures that there
+is a unique |f⟩ that makes the inner product ⟨f| �ψkl �ψ †
+k+ql′|g⟩
+non-zero, since the role of the operator pair is to simply cre-
+ate particle-hole excitations across the Fermi level. Thus, the
+final sum over |f⟩ can be replaced as
+�
+f
+→
+�
+l,l′
+�
+k∈BZ
+Θ(εl′(k + q) − εF) Θ(εF − εl(k)).
+(19)
+This corresponds to summing over final states with one
+particle-hole excitation in the valence bands. The inner prod-
+uct is then redundant and can be removed.
+The final form of the RIXS intensity for systems well-
+described by band theory is
+I(q, ωin, ∆ω, ϵin, ϵout) =
+�
+l,l′
+�
+k∈BZ
+Θ(εl′(k + q) − εF) Θ(εF − εl(k))
+×
+������
+�
+µ,ν,µ′
+⟨µ|ϵout · �r|ν⟩∗⟨µ′|ϵin · �r|ν⟩
+Uµl(k) U ∗
+µ′l′(k + q)
+ℏωin − εl′(k + q) + iΓ
+������
+2
+×
+η
+[εl(k) − εl′(k + q) + ℏ∆ω]2 + η2 ,
+(20)
+where we have replaced the Dirac δ-function with a
+Lorentzian of peak broadening η to represent finite experi-
+mental resolution.
+With respect to a local set of coordinate axes, the incoming
+X-ray polarization ϵ has components (ϵx, ϵy, ϵz). The outgo-
+ing polarization is usually not measured, and hence one sums
+over either polarizations parallel to and perpendicular to the
+scattering plane or over left, linear, and right polarizations.
+We list the polarization matrix elements for the specific case
+of the L3 edge of a 3d transition-metal in Table I.
+III.
+EXPERIMENTAL METHODS
+A.
+Sample preparation
+Single crystals of FeSi were prepared using a Ga flux
+method.
+We mixed the starting materials in a molar ratio
+of 1:1:15 in a glove box filled with argon.
+This mixture
+was placed in an alumina crucible and sealed in an evacuated
+quartz tube. The crucible was heated to 1150◦C and held for
+10 h, before cooling to 950◦C at 2 K/h, after which the flux
+was centrifuged. The crystals were washed with diluted hy-
+drochloric acid in order to remove Ga flux from the surface of
+the samples.
+
+5
+B.
+RIXS and experimental geometry
+FIG. 2. Schematic of the RIXS setup. kin and kout respectively
+denote the ingoing and outgoing scattering vectors. The components
+of the ingoing and outgoing photon polarization within the scattering
+plane are denoted by πin and πout while the σ polarization direction
+is the same for both. The incident angle θi is measured with respect
+to the sample surface, that is the a direction in the sample coordinates
+while the c direction is the normal to the sample surface.
+RIXS measurements were performed at the Soft Inelas-
+tic X-Ray (SIX) beamline at the National Syncrotron Light
+Source-II (NSLS-II). The energy resolution was 23 meV. The
+experimental setup is depicted in Fig. 2. The lab coordinates
+are denoted by x, y, z while the crystallographic axes are la-
+belled a, b, c. We define the incident and outgoing beam an-
+gles with respect to the sample coordinate system, wherein
+the c direction is the normal to the sample surface and the ac
+plane is the scattering plane. Experimental data are corrected
+to account for self-absorption effects.
+With θi denoting the incident angle measured with respect
+to the sample surface and 2θ denoting the angle between the
+incident and outgoing beam, we can easily verify the follow-
+ing in the sample frame:
+kin = kin(− cos θi, 0, − sin θi) ,
+(21a)
+kout = kout (− cos(2θ − θi), 0, sin(2θ − θi)) ,
+(21b)
+ϵπ,in = (− sin θi, 0, cos θi) ,
+(21c)
+ϵπ,out = (sin(2θ − θi), 0, cos(2θ − θi)) ,
+(21d)
+ϵσ,in = ϵσ,out = (0, 1, 0) .
+(21e)
+Although in reality the ingoing and outgoing photon mo-
+mentum magnitudes kin and kout is different owing to non-
+zero energy transfer ∆ω, the difference is negligible since
+∆ω ≪ ωin, and hence kin ≈ kout = k. The momentum
+transfer to the material q is then
+q = k (cos(2θ − θi) − cos θi, 0, − sin(2θ − θi) − sin θi) .
+(22)
+0
+1
+2
+3
+4
+5
+705 706 707 708 709 710 705 706 707 708 709 710
+Energy Loss (eV)
+(a) Experiment
+(b) Theory
+Incident energy (eV)
+705 706 707 708 709 710
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+RIXS (arb. units)
+FIG. 3. Color maps of RIXS intensity with 2θ = 150◦ and θi = 68◦
+for π-polarized incident beam as a function of incident photon energy
+ℏωin and energy loss ∆ω at the L3 edge of Fe in FeSi as obtained (a)
+in experiment and (b) in the band theory formalism of Sec. II B.
+In practice the ingoing and outgoing beam directions are set
+to specific values, which defines a specific 2θ. By rotating
+the sample about the y axis (or equivalently b axis), we can
+change q by changing θi .
+IV.
+DENSITY FUNCTIONAL THEORY AND
+TIGHT-BINDING MODEL
+The band structure of FeSi was simulated in a similar way
+to prior studies of FeSi [35]. We performed first-principles
+calculations based on DFT [36] within the Perdew-Burke-
+Ernzerhof exchange-correlation [37] implemented in the Vi-
+enna ab initio simulation package (VASP) [38]. The plane-
+wave cutoff energy was 450 eV with a 9×9×9 k-mesh in the
+BZ for self-consistent calculation without considering spin-
+orbit coupling. Maximally localized Wannier functions [39]
+were used to obtain the tight-binding model of bulk FeSi with
+the lattice constants a = b = c = 4.48 ˚A.
+V.
+RIXS SPECTRUM OF FESI
+The RIXS intensity at the Fe L3 edge with π-polarized in-
+cident beam is shown in Fig. 3 in the incident energy-energy
+loss plane. The absence of prominent sharp inelastic features
+suggests a particle-hole continuum, consistent with particle-
+hole excitations in a partially filled band structure.
+Momentum-resolved RIXS spectra are shown in Fig. 4 for
+two values of 2θ. Spectral weight from inelastic processes
+lies predominantly in a window of width ∼ 5 eV. Within that
+window, all spectra have similar lineshape, featuring a peak
+around 2 eV that disperses to higher energies with increasing
+θi and a dispersionless shoulder above 3 eV. Overall inelastic
+intensity increases with increasing θi for both values of 2θ.
+We use the band theory formulation of Sec. II B and
+Eq. (20) to theoretically model the RIXS process in FeSi. A
+fit of the absorption spectrum (see App. B) yields Γ = 0.8
+
+6
+0
+1
+2
+3
+4
+5
+0
+1
+2
+3
+4
+5
+Exp
+(a) 2� D 150ı
+(b) 2� D 70ı
+RIXS (arb. units)
+�i
+�i
+Thy
+Energy loss (eV)
+FIG. 4. Momentum-resolved RIXS spectra at ℏωin = 708.7 eV
+with π x-ray polarization for (a) 2θ = 150◦ and (b) 2θ = 70◦
+and comparison to simulations within band theory (bottom pan-
+els) using Eq. (20) with Γ = 0.8 eV and ε0
+= 707.67 eV.
+For scattering angle 2θ = 150◦, the incident angle values are
+θi = 10◦, 30◦, 45◦, 68◦, 120◦, while for 2θ = 70◦ we have θi =
+10◦, 30◦, 60◦.
+eV and Eg = 707.67 eV, and we choose a peak broadening
+η = 100 meV. We use a 48 × 48 × 48 grid of k points in the
+Brillouin zone for the 32-band tight binding model detailed
+above.
+The simulated spectra show a structure similar to that ob-
+served experimentally, with inelastic weight in a ∼ 5 eV win-
+dow containing a peak at ∆ω ∼ 2.5 eV and shoulder at
+∆ω ∼ 3.5 eV. As in experiment, overall inelastic intensity
+increases with increasing θi for both values of 2θ, though to
+a lesser extent. Compared to experiment, features are shifted
+to higher energies in simulated spectra, while for 2θ = 70◦
+and θi = 60◦ the main peak subsides, leading to theory and
+experimental spectra that look qualitatively different. Finally,
+experimental spectra also contain subdominant features close
+to the elastic line (∆ω < 1 eV) that, as discussed later, deviate
+from the band theory predictions.
+To investigate what causes the overall increase in RIXS in-
+tensity with increasing θi, we calculate the atomic scattering
+tensor (7). From this we obtain the modulation of the RIXS
+spectrum (9) coming purely from the orbital content of core
+and valence states. After summing over outgoing polariza-
+tions, we obtain the behavior shown in Fig. 5. Comparing to
+the θi dependence of RIXS spectra in Fig. 4, we see that ge-
+ometric considerations are insufficient to explain the momen-
+tum dependence of RIXS intensity in experiment: the mo-
+mentum dependence of the atomic scattering tensor is differ-
+ent from that observed, even showing a reverse trend in the
+0
+0.5
+1
+1.5
+2
+2.5
+0ı
+30ı
+60ı
+90ı
+120ı 150ı 0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+0ı
+30ı
+60ı
+90ı
+120ı 150ı
+Orbital RIXS
+Incident angle �i
+(a) 2� D 150ı
+(b) 2� D 70ı
+T .�; �/ C T .�; �/
+FIG. 5. The contribution of the dipole matrix elements to the RIXS
+spectrum of FeSi for π ingoing polarisation as given by Eq. (9) after
+summing over σ and π polarisations of the outgoing beam.
+case of 2θ = 70◦. Reinstating the band structure fundamen-
+tal scattering amplitudes in Eq. (20) yields the experimentally
+observed trend of overall momentum dependence, albeit only
+qualitatively, as shown in Fig. 4.
+VI.
+DISCUSSION & CONCLUSION
+We have seen that the theoretical formulation of RIXS
+based on band theory captures the overall momentum depen-
+dence of the experimental Fe L3-edge RIXS spectra of FeSi
+better than a calculation based on just the atomic multiplet.
+Band theory also reproduces the right bandwidth for the in-
+elastic part of the spectrum, as well as the right number of
+features therein, at roughly the right energy.
+Discrepancies between theory and experiment exist. While
+the overall momentum dependence of the RIXS spectrum is
+reproduced by band theory, experimental spectra depend more
+sensitively on θi.
+Spectral features also do not align per-
+fectly between experiment and theory. This includes a fea-
+ture around 0.3 eV in the experiment, which is only present
+as a weak shoulder in the theory. We discuss potential rea-
+sons for these discrepancies. First, the band theory of RIXS
+ignores electronic correlations. The extent to which correla-
+tions play a role in the 3d shell of FeSi is not clear [40, 41]. A
+more detailed theoretical study of the RIXS spectrum would
+require identifying the precise type, range, and magnitude of
+interactions present in FeSi. Fully incorporating the effects
+of interactions in theoretical studies of RIXS is nevertheless a
+challenge, since we are dealing with a three-dimensional ma-
+terial with 32 relevant orbitals per unit cell. Numerical sim-
+ulation of the RIXS spectrum based on dynamical mean field
+theory [42] may eventually be up to this task. Second, in the
+fast collision approximation we ignore the effects of a finite
+core-hole lifetime, which may be appreciable in 3d transition
+metal compounds [43]. Future simulations could be improved
+by incorporating dynamics and interactions with the core hole
+in the intermediate state.
+In conclusion, we have reported RIXS spectra of FeSi at
+
+7
+the Fe L3 edge. We observe an excitation continuum without
+sharp features. Through a band theory formulation of RIXS
+in the fast collision approximation, we model the RIXS pro-
+cess using the ab initio band structure of FeSi. We obtain
+reasonable agreement for the spectrum bandwidth, as well as
+the number and position of main features. Theory also repro-
+duces the dispersion trend of the RIXS spectrum, albeit only
+qualitatively. This work paves the path to ever finer resolution
+of distinctive band structure features in topological materials
+with RIXS.
+ACKNOWLEDGMENTS
+A.C. was supported by DOE Grant No.
+DE-FG02-
+06ER46316 and EPSRC grant EP/T034351/1.
+S.K. ac-
+knowledges support from the Minist`ere de l’´Economie et de
+l’Innovation du Qu´ebec and the Canada First Research Excel-
+lence Fund. Work at Brookhaven National Laboratory (x-ray
+scattering and analysis) was supported by the U.S. Depart-
+ment of Energy, Office of Science, Office of Basic Energy
+Sciences. This research used resources at the SIX beamline
+of the National Synchrotron Light Source II, a U.S. DOE Of-
+fice of Science User Facility operated for the DOE Office of
+Science by Brookhaven National Laboratory under Contract
+No. DE-SC0012704. We acknowledge National Natural Sci-
+ence Foundation of China (U2032204), the Strategic Prior-
+ity Research Program of the Chinese Academy of Sciences
+(XDB33010000) for funding sample synthesis. We thank Yue
+Cao, Siddhant Das, Claudio Chamon, Michael El-Batanouny,
+Jungho Kim, and Karl Ludwig for useful discussions.
+Appendix A: Polarization matrix elements and the atomic
+scattering tensor
+The DFT derived tight-binding model used for the calcula-
+tions presented in the paper involves thirty two basis orbitals
+per unit cell of the crystal lattice. Due to the assumption of
+zero spin orbit coupling for the valence bands, this gives rise
+to thirty-two, two-fold spin-degenerate bands. The orbitals
+used are the five 3d orbitals of each of the four Fe atoms and
+the three 3p orbitals of each of the four silicon atoms within a
+unit cell, giving a total of 32 orbitals per unit cell.
+Since the tight binding model is expressed in terms of 3d or-
+bitals whose local axes are perfectly aligned with crystal axis
+for each of the four Fe atoms in the unit cell, we need to com-
+pute the matrix elements of the 2p3/2 → 3d transitions for
+just one of the atoms. The 2p orbitals all have the same radial
+part of the wavefunction, φ2p(r) and, likewise, the 3d orbitals
+have same radial wavefunction φ3d(r). The radial integral of
+the various matrix elements in the atomic scattering tensor is
+simply the radial integral of the product φ2p(r) · φ3d(r) and
+the radial part of the dipole transition operator. Since this is a
+common term that just provides an overall multiplicative fac-
+tor for the RIXS cross section, we ignore it and compute only
+the azimuthal and polar integrals of the matrix elements. We
+document the relevant matrix elements of the dipole operator
+in Table I, which were verified by comparing to open source
+RIXS code EDRIXS [44].
+Appendix B: X-ray absorption spectrum and theoretical fit
+To align the experimental RIXS spectra with theoretical re-
+sults obtained through ab initio calculations, we need to de-
+termine the absolute energy scale Eg of the initial state. We
+determine Eg through a fit of the experimental X-ray absorp-
+tion intensity with the calculated absorption spectrum
+Iabs(q, ωin, ϵin) =
+�
+ϵout
+�
+l,l′
+�
+k∈BZ
+Θ(εl′(k + q) − εF) Θ(εF − εl(k))
+������
+�
+µ,ν,µ′
+⟨µ|ϵout · �r|ν⟩∗⟨µ′|ϵin · �r|ν⟩ Uµl(k) U ∗
+µ′l′(k + q)
+Eg + ℏωin − εl′(k + q) + iΓ
+������
+2
+,
+(B1)
+which is obtained by integrating over ∆ω the RIXS spectrum
+in Eq. (20). In addition to Eg, we consider the core hole in-
+verse lifetime Γ as a tunable parameter in the fit. We sum over
+outgoing polarizations since the measured spectrum is not po-
+larization resolved. Fig. 6 shows the fit that minimizes the
+average absolute deviation and yields the values Γ = 0.8 eV
+and Eg = 707.67 eV.
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+3045 (2010).
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+and A. Vishwanath, Rev. Mod.
+Phys. 90, 015001 (2018).
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+
+8
+J = 3
+2, Jz = − 3
+2 J = 3
+2, Jz = − 1
+2
+J = 3
+2, Jz = 1
+2
+J = 3
+2, Jz = 3
+2
+d3z2−r2↑
+(0, 0, 0)
+�
+− 1
+6, i
+6, 0
+�
+�
+0, 0, 2
+3
+�
+�
+1
+2
+√
+3,
+i
+2
+√
+3, 0
+�
+d3z2−r2↓
+�
+−
+1
+2
+√
+3,
+i
+2
+√
+3, 0
+�
+�
+0, 0, 2
+3
+�
+� 1
+6, i
+6, 0
+�
+(0, 0, 0)
+dxz↑
+(0, 0, 0)
+�
+0, 0,
+1
+2
+√
+3
+�
+�
+1
+√
+3, 0, 0
+�
+�
+0, 0, − 1
+2
+�
+dxz↓
+�
+0, 0, 1
+2
+�
+�
+1
+√
+3, 0, 0
+�
+�
+0, 0, −
+1
+2
+√
+3
+�
+(0, 0, 0)
+dyz↑
+(0, 0, 0)
+�
+0, 0, −
+i
+2
+√
+3
+�
+�
+0,
+1
+√
+3, 0
+�
+�
+0, 0, − i
+2
+�
+dyz↓
+�
+0, 0, − i
+2
+�
+�
+0,
+1
+√
+3, 0
+�
+�
+0, 0, −
+i
+2
+√
+3
+�
+(0, 0, 0)
+dx2−y2↑
+(0, 0, 0)
+�
+1
+2
+√
+3,
+i
+2
+√
+3, 0
+�
+(0, 0, 0)
+�
+− 1
+2, i
+2, 0
+�
+dx2−y2↓
+� 1
+2, i
+2, 0
+�
+(0, 0, 0)
+�
+−
+1
+2
+√
+3,
+i
+2
+√
+3, 0
+�
+(0, 0, 0)
+dxy↑
+(0, 0, 0)
+�
+−
+i
+2
+√
+3,
+1
+2
+√
+3, 0
+�
+(0, 0, 0)
+�
+− i
+2, − 1
+2, 0
+�
+dxy↓
+�
+− i
+2, 1
+2, 0
+�
+(0, 0, 0)
+�
+−
+i
+2
+√
+3, −
+1
+2
+√
+3, 0
+�
+(0, 0, 0)
+TABLE I. Dipole matrix elements relevant for the L3 edge of FeSi. Only the polar and azimuthal integrals are evaluated since the radial
+integral is the same for all the core-valence pairs, and provides only an overall prefactor to the theoretical RIXS spectrum.
+704
+705
+706
+707
+708
+709
+710
+0
+500
+1000
+1500
+2000
+Incident Energy (eV)
+XAS (arb. units)
+FIG. 6. Optimal average absolute deviation fit to the L3-edge x-
+ray absorption spectrum (XAS) of FeSi. The black dots represent
+the experimental absorption spectrum while the continuous blue line
+represents the theoretical spectrum calculated using the tight-binding
+model described in Sec IV. The fit yields Γ = 0.8 eV and Eg =
+707.67 eV.
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+

MNAyT4oBgHgl3EQfgfgN/content/tmp_files/2301.00358v1.pdf.txt

@@ -0,0 +1,1317 @@
+arXiv:2301.00358v1  [gr-qc]  1 Jan 2023
+Noname manuscript No.
+(will be inserted by the editor)
+Nonsingular Black Holes in Higher dimensions
+Bikash Chandra Paul
+Received: date / Accepted: date
+Abstract We present a class of new nonsingular black holes in higher dimensional
+theories of gravity. Assuming a specific form of the stress energy tensor exact an-
+alytic solutions of the field equation are generated in general theory of relativity
+(GR) and Rastall theory. The non-singular black hole solutions are obtained with
+a finite pressure at the centre in D = 4 dimensions. For D > 4 the transverse pres-
+sure is found finite at the centre for a set of model parameters. In the later case the
+transverse pressure is more than that in the usual four dimensions. The exact ana-
+lytic solution of the field equations in higher dimensions for large r coincides with
+the Schwarzschild black hole solution in the usual four and in higher dimensions
+which is singularity free. The different features of the generalized non-singular
+black hole in GR and modified GR are explored. A new vacuum nonsingular black
+hole is found in Rastall gravity. We also study the motion of massive and massless
+particles around the black holes.
+1 Introduction
+The idea that spacetime dimensions should be extended from four to higher dimen-
+sions came from the seminal work of Kaluza and Klein [1,2] who first tried to unify
+gravity with electromagnetism. The Kaluza-Klein approach has been revived and
+considerably generalized after realizing that many interesting theories of particle
+interactions need spacetime dimensions more than four for their formulation. Dur-
+ing the last few decades considerable research activities in progress to understand
+the quantum properties of gravity. The investigation seems to lead some people to
+believe that a consistent theory of quantum gravity cannot be obtained within the
+framework of point-field theories. For example, superstring theory is considered
+B. C. Paul
+E-mail: bcpaul@nbu.ac.in
+Department of Physics, University of North Bengal, Siliguri, Dist. : Darjeeling 734 013, West
+Bengal, India
+and
+IUCAA Centre for Astronomy Research and Development, North Bengal
+
+2
+Bikash Chandra Paul
+to be the promising candidate which may unify gravity with the other fundamen-
+tal forces in nature which requires ten dimensions for consistent formulation. The
+advent of string theory has opened up new and interesting possibilities in this con-
+text. The discovery that a supergravity theory coupled to Yang-Mills fields with a
+gauge group SO(32) or E8 ×E8 is anomaly-free in ten dimensions had inspired con-
+siderable activities in this area. Although the expected breakthrough has not yet
+come, worldwide hectic activities have served to focus on a number of issues which
+need further investigations in higher dimensions. The present ideas in dimensional
+reduction suggest that our cosmos may be a 3-brane evolving in a D-dimensional
+spacetime. Cadeau and Woolgar [3] addressed this issue in the context of black
+holes which led to homogeneous but non- FRW-braneworld cosmologies. Classical
+General Relativity (GR) in more than the usual four dimensions is thus a sub-
+ject of increasing attention in recent years. A successful development of counting
+of the five dimensional black hole entropy [4] and the AdS/CFT correspondence
+relates the properties of a D-dimensional black hole with those of a quantum field
+theory in (D − 1) dimensions [5]. There has been a growing interest to investigate
+the physics of higher-dimensional black holes [6] which is markedly different, and
+much richer in structure compared to four dimensions.
+In connection with localized sources, higher dimensional generalization of the
+spherically symmetric Schwarzschild, Reisner-Nordstr¨om black holes, Kerr black
+holes can be found in the literature [7,8,9,10,11,12]. The generalization of the
+rotating Kerr black hole [8,9,13,14] and black holes in compactified spacetime [7,
+12] are also found in the literature. The linearized stability of the black holes [15],
+no hair theorems [16], black hole thermodynamics and Hawking radiation have
+also been investigated. Mandelbrot [17] investigted the problem on the variability
+of dimensions and describe how a ball of thin thread is seen as an observer changes
+scale. An object which looks like a point object from a very large distance becomes
+a three-dimensional ball visible at a closer distance. Therefore at various scales the
+ball appears to change shape as an observer moves down. While the embedding
+dimensions for the ball has not changed, the effective dimensions of the contents
+however also remains same. It is possible that there are compact [18] or non-
+compact [19] dimensions present at a certain point. In this case the (3 + 1) metric
+is simply not true, although one obtains a valid description with general relativity.
+We also probe the black hole solution in Rastall theory [20] which is prescribed
+by a modification of GR to accommodate the present accelerating universe [21,22,
+23].
+Regular (i.e. non-singular) black holes have been initiated by Bardeen [24] and
+thereafter a number of black hole solutions in four dimensions have been obtained
+[25,26,27,28,29,30,31,32,33]. In this case one can find metrics which are spher-
+ically symmetric, static, asymptotically flat, with regular centres, and for which
+the resulting Einstein tensor is physically reasonable, satisfying the weak energy
+condition and having components which are bounded and fall off appropriately at
+large distance. Dymnikova [25] obtained nonsingular Schwarzsclid black hole solu-
+tion in vacuum and thereafter extended to obtain nonsingular cosmological black
+hole [26] solutions to include the de Sitter solution in the usual four dimensions.
+Formation and evaporation of non-singular black hole is also discussed [34] from
+an initial vacuum region accommodating Bardeen-like static region supported by
+finite density and pressures, subsequently its pressure vanishes rapidly at large ra-
+dius which however behaves as a cosmological constant at a small radius. In 2019,
+
+Nonsingular Black Holes in Higher dimensions
+3
+Event Horizon Telescope group captured the first ever image of a supermassive
+black hole at the centre of the M87 galaxy which triggers the various possibili-
+ties for the state of the compact object and opened up new horizon in theoretical
+physics. The motivation of the paper is to obtain a nonsingular black hole solution
+in higher dimensions and investigate the different features of such black holes in
+GR and beyond GR. For this we consider Higher dimensional Einstein gravity
+(GR) and Rastall gravity for a comparative study.
+The paper is organised as follows: In sec. 2, the Einstein field equation in a
+static higher dimensional metric is obtained. In sec. 3, non-singular black holes are
+obtained in Rastall theory with extension of spacetime dimensions. In sec. 4, we
+present analytical set up of the non-singular black hole solution to investigate the
+shadow of the black hole. The effective potential and the shadow behaviour of the
+black holes are analyzed in sec. 5. Finally we summarize in sec 6.
+2 Einstein Field Equation in Higher Dimensions
+We consider a higher dimensional gravitational action which is given by
+I = −
+1
+16πGD
+� √−g dDx R + Im
+(1)
+where R is the Ricci scalar, GD is the D-dimensional gravitational constant and
+Im represents the matter action. The Einstein field equation is given by
+RAB − 1
+2gABR = κ2TAB
+(2)
+where A, B = 0, 1, ...D −1 and T A
+B = (−ρ, Pr, P⊥, ...) the energy-momentum tensor,
+ρ the energy density, Pr the radial pressure, P⊥ transverse pressure, κ2 = 8πGD
+c2
+.
+We consider D dimensional spacetime metric given by
+ds2 = −eνdt2 + eλ dr2 + r2dΩ2
+D−2
+(3)
+where ν and λ are functions of radial coordinate r and ΩD−2 is for unit sphere in
+SD−2 dimensions. The components of the Einstein equations and the metric given
+by eq. (26) are given by
+T t
+t = (D − 2)
+2
+�
+e−λ
+�
+D − 3
+r2
+− λ′
+r
+�
+− D − 3
+r2
+�
+,
+(4)
+T r
+r = (D − 2)
+2
+�
+e−λ
+�
+D − 3
+r2
++ ν′
+r
+�
+− D − 3
+r2
+�
+,
+(5)
+T θ1
+θ1 = −(D − 3)(D − 4)
+2r2
++ e−λ
+2 ×
+�
+ν′′ + ν′2
+2 − λ′ν′
+2
++ (D − 3)(ν′ − λ′)
+r
++ (D − 3)(D − 4)
+r2
+�
+,
+(6)
+T θ2
+θ2 = ...... = T θD−2
+θD−2 = T θ1
+θ1
+(7)
+for simplicity we have taken κ2 = 8πGD
+c2
+= 1. The radial null vector lA can be
+selected to have the components lt = eλ/2, lr = ±eν/2 and li = 0. The two
+
+4
+Bikash Chandra Paul
+radial null-null components of the Ricci tensor are equal, and given by RABlAlB =
+eλRtt + eνRrr = (D − 2) (eλ+ν)′
+2eλ+ν , which vanishes if and only if (λ + ν) is a constant.
+A rescaling of the time coordinate can be set to make the sum of the terms equal
+to zero for black hole solution and we write
+λ + ν = 0
+(8)
+Now substituting
+λ = − ln f(r)
+(9)
+we obtain the following components of energy momentum tensors in terms of f(r),
+which are given by
+T t
+t = T r
+r = D − 2
+2
+�
+f(r)
+�
+D − 3
+r2
++
+f′
+rf(r)
+�
+− D − 3
+r2
+�
+(10)
+T θ1
+θ1 = f(r)
+2
+�
+f′′
+f + 2(D − 3)f′
+rf(r)
++ (D − 3)(D − 4)
+r2
+�
+− (D − 3)(D − 4)
+2r2
+(11)
+T θ2
+θ2 = ...... = T θD−2
+θD−2 = T θ1
+θ1
+(12)
+where the prime denotes the derivative with respect to r. The source term satis-
+fying
+T t
+t = T r
+r ,
+and
+T θ2
+θ2 = ...... = T θD−2
+θD−2 = T θ1
+θ1
+(13)
+and the equation of state, T A
+B ;A = 0,.
+Assume the density profile in higher dimensions T t
+t = −ρ as
+ρ = −T t
+t = ρ0 e
+− rD−1
+rD−1
+∗
+(14)
+where r∗ is a dimensional constant connected with a constant density ρ0. The
+density ρ0 also permits a D dimensional de Sitter solution with its size given by
+r2
+0 = (D − 1)(D − 2)
+2Λ
+,
+(15)
+where Λ = ρ0. Using the density profile given eq. (14) in eq. (10) we integrate and
+obtain the metric potential which yields
+f(r) = 1 − rD−3
+g
+rD−3 +
+2ρ0rD−1
+∗
+(D − 1)(D − 2)
+1
+rD−3 e
+− rD−1
+rD−1
+∗
+(16)
+where rD−3
+g
+=
+�
+2ρ0
+(D−1)(D−2)
+�
+rD−1
+∗
+. The higher dimensional metric is now can be
+written as
+ds2 = −
+�
+1 − Rs(r)
+rD−3
+�
+dt2 +
+dr2
+�
+1 − Rs(r)
+rD−3
+� + r2dΩ2
+D−2
+(17)
+where we denote
+Rs(r) = rD−3
+g
+�
+1 − exp
+�
+−rD−1
+rD−1
+∗
+��
+(18)
+
+Nonsingular Black Holes in Higher dimensions
+5
+and
+rD−1
+∗
+= r2
+0 rD−3
+g
+,
+(19)
+where r2
+0 = (D−1)(D−2)
+2ρ0
+. This is an exact spherically symmetric solution of the
+Einstein field equations in D-dimensions. For D = 4 the solution given by eq.
+(17) reduces to the solution obtained by Dymnikova [25]. The other components
+of energy momentum tensor can be obtained using the Einstein’s field equations
+which are given by
+T θ2
+θ2 = ... = T θD−2
+θD−2 =
+�
+D − 1
+D − 2
+� r
+r∗
+�D−1
+− 1
+�
+ρ0e
+− rD−1
+rD−1
+∗
+.
+(20)
+It is evident that in the usual 4 dimensions Dymnikova [25] black hole solutions
+recovered with anisotropic fluid distribution when r = r∗, which is true also in
+higher dimensions. The nonsingular black hole (NSBH) solutions are permitted
+with anisotropic fluid distributions in higher dimensions. The energy density and
+radial pressure follow the vacuum configuration but the tangential pressures do
+not. The tangential pressure is non-zero which remains positive definite for r > r∗.
+At the center the tangential pressure is negative indicating existence of exotic mat-
+ter (P⊥ < 0) at the center of the black hole. The nonsingular black hole solution
+obtained by Dymnikova can not be described in lower dimension D = 2 + 1, how-
+ever, we can extend the concept of NSBH in more than the usual four dimensions.
+The generalization of the black hole solution in higher dimensions accommodates
+a new class of NSBH solutions where the tangential pressure increases to a large
+extent inside the non-singular black hole with a different feature but away from
+the centre of the black hole it decreases exponentially.
+The mass of a massive object in higher dimension is given by
+m(r) = AD−2
+� r
+0
+r′D−2ρ(r′)dr′
+(21)
+where AD−2 =
+2π
+D−1
+2
+Γ ( D−1
+2
+) which at r → ∞ is connected to the whole mass M con-
+nected with rD−3
+g
+by the Schwarzschild relation. The modulus difference between
+Rs(rg) and rD−3
+g
+is rD−3
+g
+e
+− rD−1
+rD−1
+∗
+. The measure of the difference between the higher
+dimensional Schwarzschild mass m(r) ∼ rD−3
+g
+in a singular black hole and Rs of a
+non-singular black hole is given by
+M − m(r)
+M
+= exp
+�
+−rD−1
+rD−1
+∗
+�
+.
+(22)
+Here m(r) becomes M at infinite distance. It is found that the mass difference
+decreases as the dimension in which black hole embedded increases. The metric
+has two event horizons located at
+r+ = rg
+�
+1 − O
+�
+exp
+�
+−r2g
+r2
+0
+���
+,
+r− = r0
+�
+1 − O
+�
+exp
+�
+−r0
+rg
+���
+.
+(23)
+
+6
+Bikash Chandra Paul
+Here r+ is the external event horizon. The metric evaluated at gtt(r+) = 0 de-
+scribes an object with the similar properties properties of a black hole by a dis-
+tant observer, it does not send light signals outside and could not interact with
+its surroundings by the gravitational field. In four dimensions it is found that
+both r+ and r− are removable singularities of the metric. The singularities can be
+eliminated by an appropriate transformation.
+3 Higher Dimensional Rastall gravity
+In this section we explore NSBH solution in the Rastall theory of gravity for D ≥ 4
+dimensions. The Rastall theory [20] is based on the modification of the Einstein
+field equation for a spacetime with Ricci scalar filled by an energy momentum
+source as follows:
+T AB; A = λRB
+(24)
+where λ is the Rastall parameter which is a measure for deviation from the stan-
+dard GR conservation law. Consequently the Rastall field equation can be written
+as
+GAB + κ2λgABR = κ2TAB
+(25)
+where κ2 is the Rastall gravitational constant. The above field equation reduces
+to that of GR in the limit λ → 0 and κ2 = 8πG. However, for a vanishing trace
+of the energy-momentum tensor, for example the electrovacuum solution can be
+obtained when λ = 1
+4 or R = 0. It is important to note that the former possibility
+is not physically acceptable as the trace of the energy momentum tensor vanishes
+T = 0 for any scalar field. Consequently the matter configuration where the energy-
+momentum tensor has null trace, the relativistic solution obtained in Rastall theory
+is same as that one obtains in the general theory of relativity (GR). This feature of
+Rastall theory which is a modified GR led us to look for black holes solutions in a
+background of matter/energy with non-vanishing trace. It may be pointed out here
+that the Rastall gravity is widely used to accommodate acceptable explanation for
+the current acceleration of the universe which has no solution in GR and for this
+it is interesting to explore NSBH in Rastall theory.
+We consider the metric for black hole solution in higher dimensions D ≥ 4:
+ds2 = −f(r)dt2 + dr2
+f(r) + r2dΩ2
+D−2.
+(26)
+Using the metric, we obtain the non-vanishing components of the Rastall tensor
+HAB = GAB + λgABR and κ2 = 1,
+Ht
+t = D − 2
+2r2
+�
+rf′ − (D − 3) + (D − 3)f� + λR,
+(27)
+Hr
+r = D − 2
+2r2
+�
+rf′ − (D − 3) + (D − 3)f� + λR,
+(28)
+Hθi
+θi = r2f′′ + (D − 3)(2rf′ + (D − 4)(f − 1)
+2r2
++ λR
+(29)
+where i = 1, 2, ..., (D − 2), and the Ricci scalar in D dimensions is given by
+R = − 1
+r2
+�
+r2f′′ + 2(D − 2)rf′ + (D − 2)(D − 3)(f − 1)
+�
+(30)
+
+Nonsingular Black Holes in Higher dimensions
+7
+in the above we denote ()′ to represent derivative with respect to the radial co-
+ordinate r. We solve the field equation to obtain higher dimensional non-singular
+black holes in Rastall theory and for this Ht
+t = T t
+t and Hrr = T rr yield
+Pr = D − 2
+2r2
+�
+rf′ − (D − 3) + (D − 3)f�
+− λ
+r2
+�
+r2f′′ + 2(D − 2)rf′ + (D − 2)(D − 3)(f − 1)
+�
+,
+(31)
+and also we consider Hθ1
+θ1 = T θ1
+θ1 , ... and HθD−2
+θD−2 = T θD−2
+θD−2 which yield
+P⊥ =
+1
+2r2
+�
+r2f′′ + 2(D − 3)rf′ + (D − 3)(D − 4)(f − 1)
+�
+− λ
+r2
+�
+r2f′′ + 2(D − 2)rf′ + (D − 2)(D − 3)(f − 1)
+�
+.
+(32)
+In this case we explore the non-singular Black hole obtained in higher dimensional
+Rastall gravity, the general solution of the metric is
+f(r) = 1 − rD−3
+g
+rD−3 +
+2ρ0rD−1
+∗
+(D − 1)(D − 2)
+1
+rD−3 e
+− rD−1
+rD−1
+∗
+(33)
+The energy density and radial pressure are
+ρ =
+
+
+D − 2 − 2λD + 2(D − 1)λ rD−1
+rD−1
+∗
+D − 2
+
+ ρ0e
+− rD−1
+rD−1
+∗
+,
+(34)
+Pr = −
+
+
+D − 2 − 2λD + 2(D − 1)λ rD−1
+rD−1
+∗
+D − 2
+
+ ρ0e
+− rD−1
+rD−1
+∗
+,
+(35)
+the tangential pressure is given by
+P⊥ =
+�
+(1 − 2λ)D − 1
+D − 2
+rD−1
+rD−1
+∗
+− D − 2 − 2λD
+D − 2
+�
+ρ0e
+− rD−1
+rD−1
+∗
+.
+(36)
+The energy density and the transverse pressure in Rastall gravity framework ob-
+tained in eqs. (34) and (36) reduces to the eqs. (14) and (20) in GR for λ → 0.
+The modification introduced in GR by Rastall admits nonsingular Dymnikova
+[25] black hole (NSBH) with normal matter while the radial pressure corresponds
+to vacuum equation of state. At the center of the NSBH the energy density is
+ρ = (D − 2 − 2λD)ρ0, which increases as the. number of spacetime dimension in-
+creases for a given range of Rastall parameter λ < D−2
+2D . It is evident that for a
+given dimension, NSBH admits greater mass for lower values of λ and the lower
+limiting value for λ < D−2
+2D
+and |λ| > D−2
+2D
+(for negative λ). The corresponding
+tangential pressure at the center P⊥ = (2Dλ + 2 − D)ρ0 is finite but negative. In
+D = 4 dimensions, at the center of the black hole, ρ(r = 0) = 2(1 − 4λ)ρ0 and
+tangential pressure P⊥ = −(1−4λ)ρ0 which indicates existence of exotic matter at
+the centre in GR (as λ = 0) as well as in Rastall theory for λ > − 1
+4. Thus NSBH
+can be realized with both central radial pressure and tangential pressure negative
+and equal but an anisotropy in pressure develops away from the center in Rastall
+
+8
+Bikash Chandra Paul
+gravity, normal matter exists when r >
+�
+D−2−2λD
+(D−1)(1−2λ
+�1/(D−1)
+r∗. The tangential
+pressure indicates black hole surrounded by exotic matter in Rastall gravity [35]
+for the range 1
+4 < λ < 1
+2.. For r → ∞, the energy density and pressure vanishes
+asymptotically.
+When λ = D−2
+2D , we get the following :
+ρ = ρ0
+�
+D − 1
+D
+rD−1
+rD−1
+∗
+�
+e
+− rD−1
+rD−1
+∗
+,
+(37)
+Pr = −ρ = ρ0
+�
+D − 1
+D
+rD−1
+rD−1
+∗
+�
+e
+− rD−1
+rD−1
+∗
+,
+(38)
+the tangential pressure is given by
+P⊥ = 2ρ0
+�
+D − 1
+D(D − 2)
+rD−1
+rD−1
+∗
+�
+e
+− rD−1
+rD−1
+∗
+.
+(39)
+one obtains NSBH with ρ > 0, ρ + Pr = 0 and P⊥ > 0. For D = 4 dimensions,
+λ = 1
+4 and the NSBH can be realized in Rastall gravity with normal matter which
+however is not permitted in GR. Also we note that at the centre of the NSBH the
+tangential pressure vanishes, admitting a perfect vacuum NSBH in the usual four
+dimensions. The result obtained in this case is also applicable in higher dimensions.
+This is a new result.
+4 Analytical set up
+The modified Schwarzschild metric for a non-singular black hole is given by
+ds2 = −f(r) dt2 + f(r)−1 dr2 + r2dΩ2
+D−2
+(40)
+where f(r) = 1 − � rg
+r
+�D−3 + � rg
+r
+�D−3 exp
+�
+− � r
+r∗
+�D−1�
+, making use of the as-
+sumption κ2 = 8π made earlier, we write rg =
+�
+16πM
+(D−2)AD−2
+�
+1
+D−3 and the area
+of D dimensional sphere AD−2 =
+2π
+D−1
+2
+Γ( D−1
+2 ), where M represents the mass of the
+non-singular Black hole. The metric function gtt = f(r), whose sign determines
+gravitational trapping [34], we plot to draw a sketch to study the existence of
+black hole solutions. The metric potential f(r) is plotted with r in Fig. (1) for
+D = 4 and Fig. (2) for D = 10. It is evident that both the extreme black hole
+and non-extreme black holes can be obtained for a given set of values of rg and
+r0. We note that extreme black hole exists for rg = 1.0 and r0 = 0.57 in D = 4
+and rg = 2.0 and r0 = 1.57 in D = 10. In the first case no black hole exist for
+rg < 1.0 and the later case for r0 > 1.57. The photon radii are tabulated in Table-I
+for D = 4 and Table-II for D = 10. It is found that for D = 4, it increases with
+decrease of ρ0 ∼ 1/r2
+0 for a given mass but for a given ρ0, photon radius is found
+to increase with mass. In D = 10 dimensions as ρ0 is decreases the photon radius
+decreases then increases and decreases once again. In Fig (3) dimensional varia-
+tion of the photon radius for M = 1 is plotted for non-singular black holes with
+
+Nonsingular Black Holes in Higher dimensions
+9
+1
+2
+3
+4
+5
+6
+r
+�1.0
+�0.5
+0.5
+1.0
+f�r�
+Fig. 1 Radial variation of f(r) for rg = 0.8 (Red), 1.0 (Black), 1.5 (Blue) for r0 = 0.57 in
+D = 4.
+1
+2
+3
+4
+5
+6
+r
+�1.0
+�0.5
+0.5
+1.0
+f�r�
+Fig. 2 Radial variation of the metric function f(r) in D = 10 for r0 = 1.1 (Blue), 1.57 (Black)
+and 2.0 (Red) with rg = 2.
+dimensions. The photon radius is maximum at D = 4 and then decreases sharply
+as the dimensions is increases and remains constant.
+The Lagrangian is given by
+L = 1
+2gAB ˙xA ˙xB.
+(41)
+2
+4
+6
+8
+10
+�1
+0
+1
+2
+3
+r
+V
+Fig. 3 Radial variation of the potential for J= 5 (Blue), 6 (Green), 8 (Dashed), 10 (Red) in
+D = 4 for non-singular BH.
+
+10
+Bikash Chandra Paul
+where ˙() =
+d
+dτ and τ is the affine parameter. Expanding eq. (41) we get
+2L = −f(r)˙t2 +
+1
+f(r) ˙r2 + (r2 ˙θ1
+2 + sin2θ1 ˙θ2
+2 + ......)
+(42)
+To obtain trajectory of light path, we set θi = π
+2 where i = 1, ..., D − 3 and θD−2
+is a free parameter. The momenta are given by
+Pt = ∂L
+∂ ˙t = −f(r)˙t,
+Pr = ∂L
+∂ ˙r =
+1
+f(r) ˙r,
+Pθ1 = ∂L
+∂ ˙θ1
+= r2 ˙θ1, Pθ2 = ∂L
+∂ ˙θ2
+= r2sin2θ1 ˙θ2, ....
+(43)
+Now as defined above, θi = π
+2 , and at the equatorial plane θ1 = π
+2 ,
+∂L
+∂ ˙t = constant
+(44)
+and we determine the energy (E) and angular momentum (J) at r → ∞ as
+f(r)˙t = E, PθD−2 = r2
+˙
+θD−2 = J.
+(45)
+The Hamilton Jacobi equation is the most general method to find the geodesic
+equation of motion around black hole or a compact object, we adopt the technique
+to obtain the photon orbits. In higher dimensions we get
+∂S
+∂τ = H = −1
+2gAB ∂S
+∂xA
+∂S
+∂xB
+(46)
+where gAB is the inverse of the metric and S is the Jacobian. The Jacobian is
+given by
+S = 1
+2m2τ − E + JθD−2 + Sr(r) +
+D−3
+�
+i=1
+Sθi(θi)
+(47)
+where Sr(r) and Sθi(θi) are functions of r and θi respectively and m is the mass
+of the test particle, it is zero for photon. The Hamilton-Jacobi eq. (46) can be
+written as
+r4
+�
+1 −
+Rs
+rD−3
+�2 �
+∂S
+∂τ
+�
+= E2r4 − r2
+�
+1 −
+Rs
+rD−3
+�
+(K + J2)
+(48)
+D−3
+�
+i=1
+1
+Πi−1
+n=1sin2θn
+�∂Sθi
+∂θi
+�2
+= K − ΠD−3
+i=1 J2cot2θi
+(49)
+where K is the Carter constant [36]. Using the above eq. (43) in eq. (46) we get
+the following
+˙t =
+E
+f(r),
+˙θD−2 =
+J
+r2ΠD−3
+i=1 sin2θi
+;
+r2 ˙r = ±
+√
+R,
+r2
+D−3
+�
+i=1
+Πi−1
+n=1sin2θn ˙θi = ±
+�
+Θi
+(50)
+
+Nonsingular Black Holes in Higher dimensions
+11
+in the above ”+” and ”-” sign corresponds to motion of photon in outgoing and
+incoming radial direction and over dot represents derivative w.r.t to the affine
+parameterτ. For the null curves the eqs.(49) can be expressed as
+R(r) = E2r4 − r2f(r)(K2 + J2),
+(51)
+Θi(θi) = K − ΠD−3
+i=1 J2cot2θi.
+(52)
+The characteristics of photon near the black hole can be defined by two impact
+parameters, which are functions of the constants E, J and K. For general orbit we
+define the impact parameters ξ = J
+E and η =
+K
+E2 . The boundary of the shadow of
+a black hole can be estimated from the effective potential. The radial null geodesic
+from eqs. (48) and (50) is given by
+�
+dr
+dτ
+�2
++ Veff = 0,
+(53)
+where Veff is the effective potential, for radial motion we obtain
+Veff = f(r)
+r2 (K + J2) − E2
+= 1
+r2
+�
+1 −
+�rg
+r
+�D−3 �
+1 − e−( r
+r∗ )
+D−1��
+(K + J2) − E2.
+(54)
+The effective potential is identical to the classical equation describing the motion
+of a massless particle in a 1-dimensional potential V (r) provided its energy is
+1
+2E2 (of course the true energy should be E), but we use this form to obtain an
+expression for potential in our study. We plot radial variation of V (r) in Fig. (4) in
+a four dimensional universe for singular as well as non-singular black hole. As the
+angular velocity increases the photons heading towards the black hole are unstable.
+In Fig. 2, it is found that there is no difference of the behaviour of the potential.
+In Fig. (4) we plot radial variation of V (r) for different angular momentum, for a
+non-singular BH, it is evident that as the angular momentum increases the photon
+can approach near to the BH unbounded
+The photon orbits are circular and unstable for a maximum value of the effec-
+tive potential. The unstable circular orbit determines the boundary of the apparent
+shape and can be maximized. The maximal value of the effective potential corre-
+sponds to the circular orbits and the unstable photons satisfies
+Veff
+���
+r=rp
+= dVeff
+dr
+���
+r=rp
+= 0,
+R(r) = dR(r)
+dr
+���
+r=rc
+= 0
+(55)
+The impact parameters are now related as Using eqs. (54) and (55), we get
+f(rp)
+r2p
+(K + J2) − E2 = 0
+rpf′(rp) − 2f(rp)
+r3p
+(K + J2) = 0.
+(56)
+In four dimensions the potential V (r) is plotted in Fig. (4) with different angular
+momentum (J) for rg = 2 and E = 1. The particles are bounded for a radius
+r < rmin and unbounded for the range rmin < r < rmax. The range of values
+
+12
+Bikash Chandra Paul
+rp in
+rp in
+rp in
+r0
+M = 2
+M = 5
+M = 10
+0.5
+0.8757
+1.0192
+1.1532
+0.6
+1.0221
+1.1851
+1.3389
+0.8
+1.3079
+1.5053
+1.6958
+1.0
+1.5880
+1.8142
+2.0383
+1.2
+6.0000
+2.1150
+2.3702
+1.5
+6.0000
+-
+-
+1.8
+5.990
+-
+-
+2
+2.9921
+-
+-
+Table 1 The variation of the photon radius (rp) in D = 4 with r0 =
+�
+(D−1)(D−2)
+4ρ0
+and the
+mass of the BH.
+rp in
+rp in
+r0
+M = 5
+M = 10
+0.5
+0.9369
+1.0746
+0.6
+1.0035
+1.0002
+0.7
+1.0641
+0.9863
+0.8
+1.1203
+1.0613
+0.84
+1.1417
+24.6008
+0.85
+1.1470
+0.9472
+0.89
+1.178
+1.5069
+0.9
+1.1730
+25.2259
+0.91
+1.1781
+0.9598
+0.95
+1.1983
+1.1687
+1.0
+1.2229
+0.9772
+1.2
+1.8426
+1.0105
+2
+3.5942
+6.9106
+Table 2 The variation of the photon radius (rp) in D = 10 with r0 =
+�
+(D−1)(D−2)
+4ρ0
+and the
+mass of the BH.
+3
+4
+5
+6
+7
+8
+9
+10
+D
+2
+4
+6
+8
+10
+Rp
+[t]
+Fig. 4 Dimensional variation of the photon radius for M = 1 for a non-singular BH
+can be determined from the sketch. It is found that rmin decreases as angular
+momentum (J) increases. We note that the potentials for Schwarzschild black hole
+(singular) and that for non-singular black holes overlaps for a set of similar values
+of D, J and E.
+We draw the shadow contour of non-singular black hole in Fig. (8) for a given
+value of ρ0 (say, 0.16 unit) in all dimensions. It is shown that as the spacetime
+dimensions increases the radius of the shadow decreases.
+
+Nonsingular Black Holes in Higher dimensions
+13
+�10
+�5
+0
+5
+10
+�10
+�5
+0
+5
+10
+Fig. 5 Contour plot for an object having M = 2M⊙ with ρ0 = 0.16 unit for D = 4 (Red),
+D = 5 (Dashed) and D = 6 (Green), D = 8 (Thick)
+�40
+�20
+0
+20
+40
+�40
+�20
+0
+20
+40
+Fig. 6 Contour plot for an object for ρ0 = 0.04 unit in D = 4 with M = 2M⊙ (Black),
+M = 4M⊙ (Green), M = 6M⊙ (Red), M = 10M⊙ (Blue)
+5 Effective potential and shadow behaviour
+The effective potential of the Schwarzschild-Tangherlini black holes exhibits a max-
+imum for the photon sphere radius rp corresponding to the real and the positive
+solution of the constraint obtained from eq. (56),
+rpf′(rp) − 2f(rp) = 0.
+(57)
+Defining impact parameters η and ξ that are functions of the energy E, angular
+momentum J and the Carter constant K as
+ξ = J
+E ,
+η = K
+E2
+(58)
+we get from eq. (55) corresponding to Veff
+E2
+= 0 and
+R
+E2 = 0, the following
+η + ξ2 =
+r2p
+f(rp), η + ξ2 =
+4r2p
+rf′(rp) + 2f(rp).
+(59)
+
+14
+Bikash Chandra Paul
+Now we obtain
+η + ξ2 =
+5r2p
+rpf′(rp) + 3f(rp),
+(60)
+where the right hand side corresponds to
+r2
+p
+f(rp), the observer’s frame the shadow
+can be described properly making use of the celestial coordinates α and β as
+introduced earlier [37]. Following the definition introduced by Subrahmanyan as
+follows
+α =
+lim
+rp→∞
+�
+rpP θD−2
+P t
+�
+,
+βi =
+lim
+rp→∞
+�
+rpP θi
+P t
+�
+,
+(61)
+where
+i = 1, ...(D − 3).
+For an observer on the equatorial plane, these equations reduced to
+η + ξ2 = α2 + β2 =
+r2p
+f(rp)
+(62)
+the radius of the shadow is Rbhs =
+rp
+√
+f(rp). The form of f(r) is complex and
+therefore we study numerically. The photon radius depends on the dimensions.
+The photon radius is plotted in Fig. 4, it is evident that as the mass of the black
+hole increases the radius decreases. It is maximum in D = 4 but decreases sharply
+as the dimension increases but almost constant with the increase in dimension. The
+fig (9) shows that as the mass increases the radius of the shadow also increases.
+6 Discussion
+We obtain non-singular black hole (NSBH) solutions in the higher dimensional
+Einstein’s general theory of gravity (GR) and found that the methods in GR can
+be adopted also in Rastall gravity. Considering a specific exponential form of the
+energy density we obtain NSBH which reduces to the Dymnikova NSBH solution
+[25] obtained in the usual four dimensions (D = 4). In 2+1 dimensions no black hole
+solution exists. However, a non-rotating NSBH solutions obtained by Dymnikova
+in four dimensional GR can be accommodated in higher dimensions. We obtained
+NSBH in a vacuum described by T t
+t + T rr = 0 with p⊥ < 0 (where i = 1, ..., D − 2)
+near the center indicating requirement of exotic matter which however extends up
+to certain height thereafter p⊥ > 0 for r > � D−2
+D−1
+�
+1
+D−1 r∗ in GR. But in the Rastall
+gravity p⊥ > 0 for r >
+�
+2−D+2λD
+(D−1)(2λ−1)
+�
+1
+D−1 r∗ when λ ̸= D−2
+2D
+and thereafter at a
+large distance it vanishes because the pressure decreases rapidly i.e., exponentially.
+Both in GR and modified gravity it indicates existence of exotic matter near the
+center of the NSBH but in the later case the Rastall parameter plays an important
+role in determining the distance from the centre where the normal matter exists in
+the tangential direction. In the usual four dimensions at the center of the NSBH
+in the modified theory we get the following estimations ρ(r = 0) = 2(1 − 4λ)ρ0
+and tangential pressure P⊥ = −2(1 − 4λ) which are determined by the Rastall
+parameter λ. It is evident that exotic matter at the center of the black hole requires
+both in GR (as λ = 0) as well as in Rastall theory with the lower limiting value
+of the Rastall parameter λ < 1
+4. The tangential pressure is negative it indicates
+
+Nonsingular Black Holes in Higher dimensions
+15
+NSBH surrounded by exotic matter in Rastall gravity, existence of BH with exotic
+matter also reported in [35]. Thus NSBH is realized with both the central radial
+pressure and tangential pressure negative and equal initially but an anisotropy
+in pressure develops away from the center in Rastall gravity with normal matter
+thereafter when r >
+�
+D−2+2λD
+(1−2λ)(D−2)
+�
+1
+D−1 r∗.
+However, we note a new and interesting result in Rastall theory that permits
+a NSBH with normal matter in the usual four and in higher dimensions when
+λ = D−2
+2D , which however is not permitted in GR. In It is also noted that away from
+the center at a large distance, the tangential pressure remains positive definite at a
+maximum radial distance which is P⊥ = (1−2λ) D−1
+D−2
+rD−1
+rD−1
+∗
+ρ0e
+− rD−1
+rD−1
+∗
+. Thus one gets
+a physically realistic NSBH for 1
+4 < λ < 1
+2 in D = 4 dimensions. For r → ∞, both
+the energy density and pressure vanishes asymptotically. Thus the Rastall gravity
+has rich structure which unearth the structure of non-singular black hole even with
+normal matter for a restricted domain of the Rastall parameter depending on the
+embedding spacetime dimensions. Thus, we see that for λ ̸= 0 the Rastall theory
+plays an important role leading to distinct solutions relative to GR.
+The sketch of the potentials permissible in the theory are plotted in Figs. (1)
+and (2), which show that both extreme and non-extreme black holes exist.
+The contour plots in Fig. (5) for NSBH shows that the circular shadow radius
+decreases as the spacetime dimension is increased for a given mass. The circular
+shadow radius in Fig. (6) show that the radii increases with the mass of the
+compact objects for a given dimensions. The rotating NSBH will be taken up
+elsewhere.
+Acknowledgment The author would like to thank IUCAA , Pune and IUCAA
+Centre for Astronomy Research and Development (ICARD), NBU for extending
+research facilities and North Bengal University for a research grant. BCP acknowl-
+edge the suggestions and constructive criticism of the anonymous Referee.
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+

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@@ -0,0 +1,1133 @@
+CLIP2Scene: Towards Label-efficient 3D Scene Understanding by CLIP
+Runnan Chen1, Youquan Liu2, Lingdong Kong3, Xinge Zhu6, Yuexin Ma5,
+Yikang Li4, Yuenan Hou4, Yu Qiao4, Wenping Wang7
+1The University of Hong Kong
+2Hochschule Bremerhaven
+3National University of Singapore
+4Shanghai AI Lab
+5ShanghaiTech University
+6The Chinese University of Hong Kong
+7Texas A&M University
+Abstract
+Contrastive
+language-image
+pre-training
+(CLIP)
+achieves promising results in 2D zero-shot and few-shot
+learning.
+Despite the impressive performance in 2D
+tasks, applying CLIP to help the learning in 3D scene
+understanding has yet to be explored. In this paper, we
+make the first attempt to investigate how CLIP knowledge
+benefits 3D scene understanding. To this end, we propose
+CLIP2Scene, a simple yet effective framework that transfers
+CLIP knowledge from 2D image-text pre-trained models to
+a 3D point cloud network. We show that the pre-trained
+3D network yields impressive performance on various
+downstream tasks, i.e., annotation-free and fine-tuning
+with labelled data for semantic segmentation. Specifically,
+built upon CLIP, we design a Semantic-driven Cross-modal
+Contrastive Learning framework that pre-trains a 3D
+network via semantic and spatial-temporal consistency
+regularization. For semantic consistency regularization, we
+first leverage CLIP’s text semantics to select the positive
+and negative point samples and then employ the contrastive
+loss to train the 3D network. In terms of spatial-temporal
+consistency regularization, we force the consistency be-
+tween the temporally coherent point cloud features and
+their corresponding image features.
+We conduct experi-
+ments on the nuScenes and SemanticKITTI datasets. For
+the first time, our pre-trained network achieves annotation-
+free 3D semantic segmentation with 20.8% mIoU. When
+fine-tuned with 1% or 100% labelled data, our method
+significantly outperforms other self-supervised methods,
+with improvements of 8% and 1% mIoU, respectively.
+Furthermore, we demonstrate its generalization capability
+Semantic and 
+Spatial-Temporal
+Consistency
+Regularization
+Image
+Encoder
+Annotation-free
+1% annotation
+CLIP2Scene
+Text
+Encoder
+CLIP
+How CLIP benefits 
+3D scene 
+understanding?
+100% annotation
+Semantic-driven Cross-modal  Contrastive Learning
+car, bus
+pedestrian
+car
+Figure 1. We explore whether and how CLIP knowledge benefits
+3D scene understanding. To this end, we propose CLIP2Scene, a
+semantic-driven cross-modal contrastive learning framework that
+leverages CLIP knowledge to pre-train a 3D point cloud seg-
+mentation network via semantic and spatial-temporal consistency
+regularization.
+CLIP2Scene yields impressive performance on
+annotation-free 3D semantic segmentation and significantly out-
+performs other self-supervised methods when fine-tuning on an-
+notated data.
+for handling cross-domain datasets.
+1. Introduction
+3D scene understanding is fundamental in autonomous
+driving, robot navigation, etc [24, 26].
+Current deep
+arXiv:2301.04926v1  [cs.CV]  12 Jan 2023
+
+learning-based methods have shown inspirational perfor-
+mance on 3D point cloud data [37, 50, 29, 44, 15, 45].
+However, some drawbacks hinder their real-world applica-
+tions. The first one comes from their heavy reliance on the
+large collection of the annotated point clouds, especially
+when high-quality 3D annotations are expensive to acquire
+[34, 40]. Besides, they typically fail to recognize novel ob-
+jects that are never seen in the training data [11, 35]. As
+a result, it may need extra annotation efforts to train the
+model on recognizing these novel objects, which is both te-
+dious and time-consuming.
+Contrastive Vision-Language Pre-training (CLIP) [38]
+provides a new perspective that mitigates the above issues
+in 2D vision. It was trained on large-scale free-available
+image-text pairs from websites and built vision-language
+correlation to achieve promising open-vocabulary recogni-
+tion. MaskCLIP [49] further explores semantic segmen-
+tation based on CLIP. With minimal modifications to the
+CLIP pre-trained network, MaskCLIP can be directly used
+for the semantic segmentation of novel objects without ad-
+ditional training efforts. PointCLIP [48] reveals that the
+zero-shot classification ability of CLIP can be generalized
+from the 2D image to the 3D point cloud. It perspectively
+projects a point cloud frame into different views of 2D depth
+maps that bridge the modal gap between the image and
+the point cloud. The above studies indicate the potential
+of CLIP on enhancing the 2D segmentation and 3D clas-
+sification performance. However, whether and how CLIP
+knowledge benefits 3D scene understanding is still under-
+explored.
+In this paper, we explore how to leverage CLIP’s 2D
+image-text pre-learned knowledge for 3D scene understand-
+ing. Previous cross-modal knowledge distillation methods
+[40, 34] suffer from the optimization-conflict issue, i.e.,
+some of the positive pairs are regarded as negative sam-
+ples for contrastive learning, leading to unsatisfactory rep-
+resentation learning and hammering the performance of
+downstream tasks. Besides, they also ignore the tempo-
+ral coherence of the multi-sweep point cloud, failing to
+utilize the rich inter-sweep correspondence.
+To handle
+the mentioned problems, we propose a novel Semantic-
+driven Cross-modal Contrastive Learning framework that
+fully leverages CLIP’s semantic and visual information to
+regularize a 3D network. Specifically, we propose Seman-
+tic Consistency Regularization and Spatial-Temporal Con-
+sistency Regularization. In semantic consistency regular-
+ization, we utilize CLIP’s text semantics to select the posi-
+tive and negative point samples for less-conflict contrastive
+learning. For spatial-temporal consistency regularization,
+we take CLIP’s image pixel feature to impose a soft con-
+straint on points within local space and time. Such oper-
+ation also prevents the network from degenerating due to
+image-to-point calibration errors.
+We conduct several downstream tasks on nuScenes to
+verify how the pre-trained network benefits the 3D scene
+understanding.
+The first one is annotation-free semantic
+segmentation. Following MaskCLIP, we place class names
+into multiple hand-crafted templates as prompts and av-
+erage the text embeddings generated by CLIP to conduct
+the annotation-free segmentation. For the first time, our
+method achieves 20.8% mIoU annotation-free 3D semantic
+segmentation without any labelled data for training. Sec-
+ondly, we compare with other self-supervised methods to
+verify the superiority of our method in label-efficient learn-
+ing. When fine-tuning the 3D network with 1% or 100% la-
+belled data, our method significantly outperforms state-of-
+the-art self-supervised methods, with improvements of 8%
+and 1% mIoU, respectively. Besides, to verify the general-
+ization capability, we pre-train the network on the nuScenes
+dataset and evaluate it on the SemanticKITTI dataset. Our
+method still significantly outperforms state-of-the-art meth-
+ods.
+The contributions of our work are summarized as fol-
+lows.
+• The first work that distils CLIP knowledge to a 3D net-
+work for 3D scene understanding.
+• We propose a novel Semantic-driven Cross-modal
+Contrastive Learning framework that pre-trains a 3D
+network via spatial-temporal and semantic consistency
+regularization.
+• We
+propose
+a
+novel
+Semantic-guided
+Spatial-
+Temporal Consistency Regularization that forces the
+consistency between the temporally coherent point
+cloud features and their corresponding image features.
+• For the first time, our method achieves promising
+performance on annotation-free 3D scene segmenta-
+tion and significantly outperforms state-of-the-art self-
+supervised methods when fine-tuning with labelled
+data.
+2. Related Work
+Zero-shot Learning in 3D. The objective of zero-shot
+learning (ZSL) is to recognize objects that are unseen in
+the training set. Many efforts have been devoted to the 2D
+recognition tasks [8, 30, 47, 36, 31, 1, 43, 32, 4, 2, 19, 33,
+23], and few works concentrate on performing ZSL in the
+3D domain [18, 11, 35, 16, 17]. [18] makes the first at-
+tempt to apply ZSL to 3D tasks, where they train PointNet
+[37] on ”seen” samples and test on ”unseen” samples. Sub-
+sequent work [16] addresses the hubness problem caused
+by the low-quality point cloud features. [17] proposes the
+triplet loss to boost the performance under the transductive
+setting, where the ”unseen” class is observed and unlabeled
+
+Spatial-Temporal Consistency Regularization
+Image
+Encoder
+Text
+Encoder
+car, bus
+Pedestrian
+…
+A 
+photo 
+of a { }; 
+This is 
+the { } 
+in the 
+scene; 
+…
+Semantic Consistency Regularization
+Point
+Encoder
+CLIP
+pixel-to-text mapping
+pixel-point-text pairs
+pixel-to-point mapping
+3D Network
+point-text pairs
+… … …
+…
+…
+…
+…
+Multi-sweeps 
+calibration
+…
+…
+grid 1
+grid 2
+grid 3
+pulling force
+Semantic-guided fusion features 
+text embedding
+point feature
+text embedding
+point feature
+pixel feature
+point feature
+prompts
+𝑃1
+𝑃2
+𝑃3
+image feature
+… …
+…
+…
+grid 1
+grid 2
+grid 3
+Figure 2. Illustration of the Semantic-driven Cross-modal Contrastive Learning. Firstly, we obtain the text embeddings ti, image pixel
+feature xi, and point feature pi by text encoder, image encoder, and point encoder, respectively. Secondly, we leverage CLIP knowledge to
+construct positive and negative samples for contrastive learning. Thus we obtain point-text pairs {xi, ti}M
+i=1 and all pixel-point-text pairs
+in a short temporal {ˆxk
+i , ˆpk
+i , tk
+i }
+ˆ
+M,K
+i=1,k=1. Here, {xi, ti}M
+i=1 and {ˆxk
+i , ˆpk
+i , tk
+i }
+ˆ
+M,K
+i=1,k=1 are used for Semantic Consistency Regularization and
+Spatial-Temporal Consistency Regularization, respectively. Lastly, we perform Semantic Consistency Regularization by pulling the point
+features to their corresponding text embedding and Spatial-Temporal Consistency Regularization by mimicking the temporally coherent
+point features to their corresponding pixel features.
+in the training phase. Recently, some studies introduced
+CLIP into zero-shot learning. MaskCLIP [49] investigates
+the problem of utilizing CLIP to help the 2D dense pre-
+diction tasks and exhibits encouraging zero-shot semantic
+segmentation performance. PointCLIP [48] is the pioneer-
+ing work that applies CLIP to 3D recognition. As opposed
+to previous approaches that require training on the labelled
+point cloud, PointCLIP is free from any 3D training and
+shows impressive performance on zero-shot and few-shot
+classification tasks. Our work takes a step further to inves-
+tigate whether the rich semantic and visual knowledge in
+CLIP can benefit the 3D semantic segmentation tasks.
+Self-supervised Representation Learning. The purpose
+of self-supervised learning is to learn a good representa-
+tion that benefits the downstream tasks. The dominant ap-
+proaches resort to contrastive learning to pre-train the net-
+work [27, 25, 21, 20, 14, 13, 7, 10, 12, 9]. Recently, inspired
+by the success of CLIP, leveraging the pre-trained model
+of CLIP to the downstream tasks has raised the commu-
+nity’s attention. DenseCLIP [39] utilizes the CLIP’s pre-
+trained knowledge for dense image pixel prediction. Det-
+CLIP [46] proposes a pre-training method equipped with
+CLIP for open-world detection. In this paper, we make the
+first attempt to pre-train a 3D network with CLIP’s knowl-
+edge for 3D scene understanding.
+Cross-modal Knowledge Distillation.
+Recently, an in-
+creasing number of researchers have focused on transferring
+the knowledge in 2D images to 3D point cloud [34, 40].
+PPKT [34] proposes the contrastive pixel-to-point knowl-
+edge transfer to utilize the rich information in image back-
+bones. SLidR [40] resorts to the InfoNCE loss to help the
+3D network distil rich knowledge from the 2D image back-
+bone. Our work explores leveraging the image-text pre-
+trained CLIP knowledge to help 3D scene understanding.
+3. Methodology
+Considering the impressive open-vocabulary perfor-
+mance achieved by CLIP in image classification and seg-
+mentation, natural curiosities have been raised. Can CLIP
+endow the ability to a 3D network for annotation-free
+scene understanding?
+And further, will it promote the
+network performance when fine-tuned on labelled data?
+To answer the above questions, we study the cross-modal
+knowledge transfer of CLIP for 3D scene understanding,
+termed CLIP2Scene. Our work is a pioneer in exploiting
+CLIP knowledge for 3D scene understanding. In what fol-
+lows, we revisit the CLIP applied in 2D open-vocabulary
+classification and semantic segmentation, then present our
+CLIP2Scene in detail.
+Our approach consists of three
+major components: Semantic Consistency Regularization,
+Semantic-Guided Spatial-Temporal Consistency Regular-
+ization, and Switchable Self-Training Strategy.
+
+car
+road
+bicycle
+...
+building
+Text embeddings
+Figure 3. Illustration of the image pixel-to-text mapping.
+The
+dense pixel-text correspondence {xi, ti}M
+i=1 is extracted by the
+off-the-shelf method MaskCLIP [49].
+3.1. Revisiting CLIP
+Contrastive Vision-Language Pre-training (CLIP) miti-
+gates the following drawbacks that dominate the computer
+vision field: 1. Deep models need a large amount of for-
+matted and labelled training data, which is expensive to ac-
+quire; 2. The model’s generalization ability is weak, mak-
+ing it difficult to migrate to a new scenario with unseen
+objects. CLIP consists of an image encoder (ResNet [28]
+or ViT [6]) and a text encoder (Transformer [42]), both
+respectively project the image and text representation to a
+joint embedding space. During training, CLIP constructs
+positive and negative samples from 400 million image-text
+pairs to train both encoders with a contrastive loss, where
+the large-scale image-text pairs are free-available from the
+Internet and assumed to contain every class of images and
+most concepts of text. Therefore, CLIP can achieve promis-
+ing open-vocabulary recognition.
+For 2D zero-shot classification, CLIP first places the
+class name into a pre-defined template to generate the text
+embeddings and then encodes images to obtain image em-
+beddings.
+Next, it calculates the similarity matrices be-
+tween images and text embeddings to determine the class.
+MaskCLIP further extends CLIP into 2D semantic segmen-
+tation. Specifically, MaskCLIP modifies the attention pool-
+ing layer of the CLIP’s image encoder, thus performing
+pixel-level mask prediction instead of the global image-
+level prediction.
+3.2. CLIP2Scene
+As shown in Fig. 2, we first leverage CLIP and 3D net-
+work to respectively extract the text embeddings, image
+pixel feature and point feature.
+Secondly, we construct
+positive and negative samples based on CLIP’s knowledge.
+Lastly, we impose Semantic Consistency Regularization by
+pulling the point features to their corresponding text embed-
+ding. At the same time, we apply Spatial-Temporal Con-
+sistency Regularization by forcing the consistency between
+temporally coherent point features and their corresponding
+pixel features. In what follows, we present the details and
+insights.
+3.2.1
+Semantic Consistency Regularization
+As CLIP is pre-trained on 2D images and text, our first con-
+cern is the domain gap between 2D images and the 3D point
+cloud. To this end, we build dense pixel-point correspon-
+dence and transfer image knowledge to the 3D point cloud
+via the pixel-point pairs. Specifically, we calibrate the Li-
+DAR point cloud with corresponding images captured by
+six cameras. Therefore, the dense pixel-point correspon-
+dence {xi, pi}M
+i=1 can be obtained accordingly, where xi
+and pi indicates i-th paired image feature and point feature,
+which are respectively extracted by the CLIP’s image en-
+coder and the 3D network. M is the number of pairs. Note
+that it is an online operation and is irreverent to the image
+and point data augmentation.
+Previous methods [40, 34] provide a promising solution
+to cross-modal knowledge transfer.
+They first construct
+positive pixel-point pairs {xi, pi}M
+i=1 and negative pairs
+{xi, pj}(i ̸= j), and then pull in the positive pairs while
+pushing away the negative pairs in the embedding space via
+the InfoNCE loss. Despite the encourageable performance
+of previous methods in transferring cross-modal knowl-
+edge, they are both confronted with the same optimization-
+conflict issue. For example, suppose i-th pixel xi and j-th
+point pj are in the different positions of the same instance
+with the same semantics. However, the InfoNCE loss will
+try to push them away, which is unreasonable and ham-
+mer the performance of the downstream tasks [40]. In light
+of this, we propose a Semantic Consistency Regularization
+that leverages the CLIP’s semantic information to allevi-
+ate this issue. Specifically, we generate the dense pixel-
+text pairs {xi, ti}M
+i=1 by following the off-the-shelf method
+MaskCLIP [49] (Fig. 3), where ti is the text embedding gen-
+erated from the CLIP’s text encoder. Note that the pixel-text
+mappings are free-available from CLIP without any addi-
+tional training. We then transfer pixel-text pairs to point-
+text pairs {pi, ti}M
+i=1 and utilize the text semantics to se-
+lect the positive and negative point samples for contrastive
+
+Image 𝐼
+Pixel-to-point mapping
+𝑃1
+𝑃2
+𝑃3
+Multi-sweeps calibration
+… … …
+…
+…
+grid 1
+grid 2
+grid 3
+…
+…
+grid 1
+grid 2
+grid 3
+{𝑓𝑛}𝑛=1
+𝑁
+ො𝑥𝑖
+𝑘, ො𝑝𝑖
+𝑘
+𝑛=1,𝑘=1
+𝑁,𝐾
+Text embedding
+Figure 4. Illustration of the image pixel-to-point mapping (left)
+and semantic-guided fusion feature generation (right). We build
+the grid-wise correspondence between an image I and the tem-
+porally coherent LiDAR point cloud {Pk}K
+k=1 within S seconds
+and generate semantic-guided fusion features for individual grids.
+Both {ˆxk
+i , ˆpk
+i }
+ˆ
+M,K
+i=1,k=1 and {fn}N
+n=1 are used to perform Spatial-
+Temporal Consistency Regularization.
+learning. The objective function is as follows:
+LS info = −
+C
+�
+c=1
+log
+�
+ti∈c,pi exp(D(ti, pi)/τ)
+�
+ti∈c,tj /∈c,pj exp(D(ti, pj)/τ),
+(1)
+where ti ∈ c indicates that ti is generated by c-th classes
+name, and C is the number of classes. D denotes the scalar
+product operation and τ is a temperature term (τ > 0).
+Since the text is composed of class names placed into
+pre-defined templates, the text embedding represents the se-
+mantic information of the corresponding class. Therefore,
+those points with the same semantics will be restricted near
+the same text embedding, and those with different semantics
+will be pushed away. To this end, our Semantic Consistency
+Regularization causes less conflict in contrastive learning.
+3.2.2
+Semantic-guided Spatial-temporal Consistency
+Regularization
+Besides semantic consistency regularization, we consider
+how image pixel features help to regularize a 3D network.
+The natural alternative directly pulls in the point feature
+with its corresponding pixel in the embedding space. How-
+ever, after trial and error, we observe that the network easily
+degenerates and achieves poor performance in the down-
+stream tasks when following the aforementioned strategy.
+The main reason lies in the noise-assigned semantics of the
+image pixel and the imperfect pixel-point mapping caused
+by the calibration errors. To this end, we propose a novel
+semantic-guided Spatial-Temporal Consistency Regulariza-
+tion to alleviate the problem by imposing a soft constraint
+on points within local space and time.
+Specifically, given an image I and temporally coherent
+LiDAR point cloud {Pk}K
+k=1, where K is the number of
+sweeps within S seconds. Note that the image is matched
+to the first frame of the point cloud P1 with pixel-point pairs
+{ˆx1
+i , ˆp1
+i } ˆ
+M
+i=1. We register the rest of the point cloud to the
+first frame via the calibration matrices and map them to the
+image (Fig. 4). Thus we obtain all pixel-point-text pairs
+in a short temporal {ˆxk
+i , ˆpk
+i , tk
+i }
+ˆ
+M,K
+i=1,k=1. Next, we divide
+the entire stitched point cloud into regular grids {gn}N
+n=1,
+where the temporally coherent points are located in the
+same grid. We impose the spatial-temporal consistency con-
+straint within individual grids by the following objective
+function:
+LSSR =
+�
+gn
+�
+(ˆi,ˆk)∈gn
+(1 − sigmoid(D(ˆp
+ˆk
+ˆi , fn)))/N, (2)
+where (ˆi, ˆk) ∈ gn indicates the pixel-point pair {ˆxk
+i , ˆpk
+i }
+is located in the n-th grid. {fn}N
+n=1 is a semantic-guided
+cross-modal fusion feature formulated by:
+fn =
+�
+(ˆi,ˆk)∈gn
+a
+ˆk
+ˆi ∗ ˆx
+ˆk
+ˆi + b
+ˆk
+ˆi ∗ ˆp
+ˆk
+ˆi ,
+(3)
+where aˆk
+ˆi and bˆk
+ˆi are attention weight calculated by:
+a
+ˆk
+ˆi =
+exp(D(ˆxˆk
+ˆi , t1
+ˆi )/λ)
+�
+(ˆi,ˆk)∈gn exp(D(ˆxˆk
+ˆi , t1
+ˆi )/λ) + exp(D(ˆpˆk
+ˆi , t1
+ˆi )/λ)
+,
+b
+ˆk
+ˆi =
+exp(D(ˆpˆk
+ˆi , t1
+ˆi )/λ)
+�
+(ˆi,ˆk)∈gn exp(D(ˆxˆk
+ˆi , t1
+ˆi )/λ) + exp(D(ˆpˆk
+ˆi , t1
+ˆi )/λ)
+,
+(4)
+where λ is the temperature term.
+Actually, those pixel and point features within the local
+grid gn are restricted near a dynamic centre fn. Thus, such a
+soft constraint alleviates the noisy prediction and calibration
+error issues. At the same time, it imposes Spatio-Temporal
+Regularization on the temporally coherent point features.
+3.2.3
+Switchable Self-training Strategy
+We combine the loss function LS info and LSSR to end-
+to-end train the whole network, where the CLIP’s image
+and text encoder backbone are frozen during training. We
+find that method worked only when the pixel-point feature
+{xi, pi}M
+i=1 and {ˆxk
+i , ˆpk
+i }
+ˆ
+M,K
+i=1,k=1, which are used in LS info
+and LSSR, are generated from different learnable linear
+layer. On top of that, we further put forward an effective
+strategy to promote performance. Specifically, after con-
+trastive learning of the 3D network for a few epochs, we
+randomly switch the point labels between the paired im-
+age pixel’s labels and their own predictions for self-training.
+Merely training the 3D network with their own predictions
+yields satisfactory performance. Essentially, such a Switch-
+able Self-Training Strategy (S3) increases the number of
+
+Table 1. Ablation study experiments on the nuScenes validation
+dataset for annotation-free semantic segmentation.
+Ablation target
+Settings
+mIoU(%)
+-
+baseline
+15.1
+Prompts
+nuScenes
+15.1
+semanticKITTI
+13.9
+Cityscapes
+11.3
+Regularization
+w/o SCR
+19.8
+KL
+0
+Training Strategies
+w/o S3
+18.8
+ST
+10.1
+Sweeps
+1 sweep
+18.7
+3 sweeps
+20.8
+5 sweeps
+20.6
+merged
+18.6
+-
+CLIP2Scene
+20.8
+positive and negative samples by switching the point pseudo
+labels, which benefits cross-modal knowledge distillation.
+4. Experiments
+Datasets.
+We conduct experiments on two large-scale
+outdoor
+LiDAR
+segmentation
+benchmarks,
+i.e.,
+Se-
+manticKITTI [3] and nuScenes [5, 22].
+The nuScenes
+dataset contains 700 scenes for training, 150 scenes for
+validation and 150 scenes for testing, where 16 classes
+are utilized for LiDAR semantic segmentation. As to Se-
+manticKITTI, it contains 19 classes for training and evalu-
+ation. It has 22 sequences, where sequences 00 to 10, 08
+and 11 to 21 are used for training, validation and testing,
+respectively.
+Implementation Details.
+We use the nuScenes [5, 22]
+dataset to pre-train the network.
+Following SLidR, we
+pre-train the network on all key frames from 600 scenes.
+Besides, we fine-tune the pre-trained network on Se-
+manticKITTI [3] to verify the generalization ability. We
+leverage CLIP’s image encoder and text encoder to gener-
+ate image features and text embedding, respectively. Fol-
+lowing MaskCLIP, we modify the attention pooling layer of
+the CLIP’s image encoder, thus extracting the dense pixel-
+text correspondences. We take SPVCNN [41] as the 3D
+network to produce the point-wise feature. The whole net-
+work is trained on the PyTorch platform. The training time
+is about 40 hours for 20 epochs on two NVIDIA Tesla A100
+GPUs. For the switchable self-training strategy, we ran-
+domly switch the point supervision signal after 10 epochs.
+The optimizer is SGD with a cosine scheduler. We set the
+temperature λ and τ to be 1 and 0.5, respectively.
+The
+sweep number is set to be 3 empirically. We apply sev-
+eral data augmentations in contrastive learning, including
+random rotation around the z-axis and random flip on the
+Table 2. Comparison of different self-supervised methods for se-
+mantic segmentation on the nuScenes and SemanticKITTI valida-
+tion datasets.
+Initialization
+nuScenes
+semanticKITTI
+1%
+100%
+1%
+Random
+42.2
+69.1
+32.5
+PPKT [34]
+48.0
+70.1
+39.1
+SLidR [40]
+48.2
+70.4
+39.6
+CLIP2Scene
+56.3
+71.5
+42.6
+point cloud, random horizontal flip and random crop-resize
+on the image.
+4.1. Annotation-free Semantic Segmentation
+After pre-training the network, we show the performance
+of the 3D network when it is not fine-tuned on any annota-
+tions. As no previous method reports the 3D annotation-free
+segmentation performance, we compare our method with
+different setups (Table 1). In what follows, we describe the
+experimental settings and give insights into our method and
+the different settings.
+Settings. We conduct experiments on the nuScenes dataset
+to evaluate the annotation-free semantic segmentation per-
+formance. Following MaskCLIP [49], we place the class
+name into 85 hand-craft prompts and feed it into the CLIP’s
+text encoder to produce multiple text features. We then av-
+erage the text features and feed the averaged features to the
+classifier for point-wise prediction. Besides, to explore how
+to effectively transfer CLIP’s knowledge to the 3D network
+for annotation-free segmentation, We conduct the following
+experiments to highlight the effectiveness of different mod-
+ules in our framework.
+Baseline. The input of the 3D network is only one sweep,
+and we pre-train the framework via semantic consistency
+regularization.
+Prompts (nuScenes, semanticKITTI, Cityscapes). Based
+on the baseline, we respectively replace the nuScenes, se-
+manticKITTI, and Cityscapes class names into the prompts
+to produce the text embedding.
+Regularization (w/o STR, KL). Based on the full method,
+we remove the Spatial-temporal Consistency Regulariza-
+tion (w/o SCR). Besides, we abuse both SR and SCR
+and distill the image feature to the point cloud by Kull-
+back–Leibler (KL) divergence loss.
+Training Strategies (w/o S3, ST). We abuse the Switchable
+Self-Training Strategy (w/o S3) in the full method. Besides,
+we show the performance of only training the 3D network
+by their own predictions after ten epochs (ST).
+Sweeps Number (1 sweep, 3 sweeps, 5 sweeps, and
+merged). We set the sweep number K to be 1, 3, and 5, re-
+spectively. Besides, we also take three sweeps of the point
+cloud as the input to pre-train the network.
+Effect of Different Prompts.
+To verify how text em-
+
+Ground truth
+Ours*
+Ours
+Bus
+Motorcycle
+Car
+Truck
+Figure 5. Qualitative results of annotation-free semantic segmentation on nuScenes dataset. Note that we show the results by individual
+class. From the left to the right column are the bus, motorcycle, car and truck, respectively. The first row is the ground truth; The second
+row (ours*) is our prediction of the highlighted target; the third row is our prediction of full classes (ours).
+bedding affects the performance, we generate various text
+embeddings by the class name from different datasets
+(nuScenes, SemanticKITT, and Cityscapes) for pre-training
+the framework.
+As shown in Table 1, we find that
+even learning with other datasets’ text embedding (se-
+manticKITT and Cityscapes), the 3D network could still
+recognize the nuScenes’s objects with decent performance
+(13.9 and 11.3 mIoU, respectively). The result shows that
+the 3D network is capable of open-vocabulary recognition.
+Effect of Semantic and Spatial-temporal Consistency
+Regularization. We remove Spatial-temporal Consistency
+Regularization (w/o SCR) from our method. Experiments
+show that the performance is dramatically decreased, indi-
+cating the effectiveness of our design. Besides, we also dis-
+till the image feature to the point cloud by KL divergence
+loss, where the text embeddings calculate the logits. How-
+ever, such a method fails to transfer the semantic informa-
+tion from the image. The main reason is the noise-assigned
+semantics of the image pixel and the imperfect pixel-point
+correspondence due to the calibration error.
+Effect of Switchable Self-training Strategy. To examine
+the effect of the Switchable Self-Training Strategy, we ei-
+ther train the network with image supervision (w/o S3) or
+train the 3D network by their own predictions. Both tri-
+als witness the performance drop, indicating our Switch-
+able Self-Training Strategy is efficient in cross-modal self-
+supervised learning. The main reason is that the number of
+positive and negative samples is enlarged by switching the
+supervision signal.
+Effect of Sweep Numbers. Intuitively, the performance of
+our method benefits from more sweeps information. There-
+fore, we also show the performance when restricting sweep
+size to 1, 3, and 5, respectively. However, we observe that
+the performance of 5 sweeps is similar to 3 sweeps but is
+more computationally expensive. Thus, we empirically set
+the sweep number to be 3.
+Qualitative Evaluation. We show the qualitative evalua-
+tion in Fig. 5. Note that we show the results by individ-
+ual class (construction vehicle, truck, and car). The results
+show that our method is able to perceive the objects without
+any annotation training data. However, we also observe the
+false positive predictions around the ground truth objects.
+We will resolve this issue in future work.
+4.2. Annotation-efficient Semantic Segmentation
+Besides annotation-free semantic segmentation, the pre-
+trained 3D network also boosts the performance when it
+is fine-tuned on labelled data. To the best of our knowl-
+edge, only one published method SLidR studies image-to-
+Lidar self-supervised representation distillation. We also
+compared our method with another self-supervised method
+PPKT [34] for 3D network pre-training.
+In the follow-
+ings, we first introduce SLidR [40] and PPKT, then compare
+them in detail.
+PPKT. PPKT is a cross-modal self-supervised method for
+the RGB-D dataset. It performs 2D-to-3D knowledge dis-
+tillation via pixel-to-point contrastive loss. Since there is
+no public code, we re-implement it for a fair comparison.
+
+Input
+Ground Truth
+SLidR
+Ours
+Figure 6. Qualitative results of fine-tuning on 1% nuScenes dataset. From the first row to the last row are the input Lidar scan, ground truth,
+prediction of SLidR, and our prediction, respectively. Note that we show the results by error map, where the red point indicates the wrong
+prediction. Apparently, our method achieves decent performance.
+Specifically, we use the same 3D network and training pro-
+tocol but replace our semantic and Spatio-Temporal Reg-
+ularization with InfoNCE loss. The framework is trained
+on 4, 096 randomly selected image-to-point pairs for 50
+epochs.
+SLidR. SLidR is an image-to-Lidar self-supervised method
+for autonomous driving data.
+Compared with PPKT,
+it introduces image super-pixel into cross-modal self-
+supervised learning. For a fair comparison, we replace our
+loss function with their superpixel-driven contrastive loss.
+Performance. As shown in Table 2, our method signifi-
+cantly outperforms the state-of-the-art methods when fine-
+tuned on 1% and 100% data, with the improvement of
+8.1% and 1.1%, respectively.
+Compared with the ran-
+dom initialization, the improvement is 14.1% and 2.4%, re-
+spectively, indicating the efficiency of our semantic-driven
+cross-modal contrastive learning framework. The qualita-
+tive results are shown in Fig. 6. Besides, we also verify the
+cross-domain generalization ability of our method. When
+pre-training the 3D network on the nuScenes dataset and
+fine-tuning on 1% SemanticKITTI dataset, our method sig-
+nificantly outperforms other state-of-the-art self-supervised
+methods.
+Discussions. PPKT and SLidR reveal that contrastive loss
+is promising for transferring knowledge from image to point
+cloud. Like self-supervised learning, constructing the pos-
+itive and negative samples is vital to unsupervised cross-
+modal knowledge distillation.
+However, previous meth-
+ods suffer from the optimization-conflict issue, i.e., some
+of the negative paired samples are actually positive pairs.
+For example, the road occupies a large proportion of the
+point cloud in a scene and is supposed to have the same
+semantics in the semantic segmentation task. When ran-
+domly selecting training samples, most negatively defined
+road-road points are actually positive. When feedforward-
+ing such training samples into contrastive learning, the con-
+trastive loss will push them away in the embedding space,
+leading to unsatisfactory representation learning and ham-
+mering the downstream tasks’ performance.
+SLidR in-
+troduces superpixel-driven contrastive learning to alleviate
+such issues. The motivation is that the visual representation
+of the image pixel and the projected points are consistent
+intra-superpixel. Although avoiding selecting the negative
+image-point pairs from the same superpixel, the conflict is-
+sue still exists inter-superpixel. In our CLIP2Scene, we in-
+troduce the free-available dense pixel-text correspondence
+to alleviate the optimization conflicts. The text embedding
+represents the semantic information and can be used to se-
+lect more reasonable training samples for contrastive learn-
+ing.
+Besides training sample selection, the previous method
+also ignores the temporal coherence of the multi-sweep
+point cloud. Similar to multi-view consistency, multi-sweep
+consistency emphasizes inter-sweep consistency along time
+series. That is, for those LiDAR points mapping to the same
+image pixel, their feature should be the same. Besides, con-
+sidering the sparsity of the LiDAR scan and the calibration
+error between the LiDAR scan and the camera image. We
+
+专
+. - relax the pixel-to-point mapping to image grid-to-point grid
+mapping and calculate the dynamic centre within the indi-
+vidual grid for consistency regularization. To this end, our
+Spatial-temporal consistency regularization leads to a more
+comprehensive point representation.
+Last but not least, the previous method typically enlarges
+the number of training samples by data augmentation. In
+our CLIP2Scene, we find that randomly switching the su-
+pervision signal benefits self-supervised learning. Essen-
+tially, our Switchable Self-Training Strategy enlarges the
+training samples and prevents the network from deteriorat-
+ing.
+5. Conclusion
+We explored how CLIP knowledge benefits 3D scene
+understanding in this paper, termed CLIP2Scene. To ef-
+ficiently transfer CLIP’s image feature and text feature
+to a 3D network, we propose a novel Semantic-driven
+Cross-modal Contrastive Learning framework including Se-
+mantic Regularization and Spatial-Temporal Regulariza-
+tion. For the first time, our pre-trained 3D network achieves
+annotation-free 3D semantic segmentation with decent per-
+formance. Besides, our method significantly outperforms
+state-of-the-art self-supervised methods when fine-tuning
+the 3D network with labelled data.
+Potential Negative Impacts. Although our approach im-
+proves the 3D semantic segmentation performance in gen-
+eral, its effectiveness under adversarial attack is not con-
+sidered, which could be safety-critical in practical applica-
+tions, such as autonomous driving and robot navigation.
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