Department of Mathematical Sciences Articles

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25th Annual Computational Neuroscience Meeting: CNS-2016
(BioMed Central, 2016-08-18) Sharpee, Tatyana O.; Destexhe, Alain; Kawato, Mitsuo; Sekulić, Vladisla; Skinner, Frances K.; Wójcik, Daniel K.; Chintaluri, Chaitanya; Cserpán, Dorottya; Somogyvári, Zoltán; Kim, Jae Kyoung; Kilpatrick, Zachary P.; Bennett, Matthew R.; Josić, Kresimir; Elices, Irene; Arroyo, David; Levi, Rafael; Rodriguez, Francisco B.; Varona, Pablo; Hwang, Eunjin; Kim, Bowon; Han, Hio-Been; Kim, Tae; McKenna, James T.; Brown, Ritchie E.; McCarley, Robert W.; Choi, Jee Hyun; Rankin, James; Popp, Pamela Osborn; Rinzel, John; Tabas, Alejandro; Rupp, André; Balaguer‑Ballester, Emili; Maturana, Matias I.; Grayden, David B.; Cloherty, Shaun L.; Kameneva, Tatiana; Ibbotson, Michael R.; Meffin, Hamish; Koren, Veronika; Lochmann, Timm; Dragoi, Valentin; Obermayer, Klaus; Psarrou, Maria; Schilstra, Maria; Davey, Neil; Ju, Huiwen; Hines, Michael L.; Chen, Liang; Kim, Jimin; Leahy, Will; Shlizerman, Eli; Birgolias, Justas; Gerkin, Richard C.; Crook, Sharon M.; Viriyopase, Atthaphon; Memmeshei, Raol-Martin; Dabaghian, Yuri; DeVuti, Justin; Perotti, Luca; Kim, Ammo J.; Fenk, Lisa M.; Lyu, Cheng; Malmon, Gabby; Zhao, Chang; Widmer, Yves; Sprecher, Simon; Halnes, Geir; Tuomo, Maki-Martun; Keller, Daniel; Petterson, Klas H.; Andreassen, Ole A.; Elnevoll, Gaute T.; Yamada, Yasnori; Steyn-Ross, Moira L.; Steyn-Ross, D. Alistair; Meijas, Jorge F.; Murray, John D.; Kennedy, Henry; Kruscha, Alexandra; Grewe, Jan; Lidner, Benjamin; Badel, Laurent; Kasumi, Ohta; Tsuchimoto, Yoshiko; Kazama, Hokto; Kahng, B.; Tam, Nicoladie D.; Pollonini, Luca; Zouridakis, George; Soh, Jaehyun; Kim, DaeEun; Yoo, Minsu; Palmer, S.E.; Culmone, Viviana; Bojak, Ingo; Ferrario, Andrea; Merriosn-Hort, Robert; Borisyuk, Roman; Kim, Chang Sub; Tezuka, Taro; Joo, Pangyu; Young-Ah, Rho; Burton, Shawn D.; Bard, G.; Marsalek, Petr; Kim, Hoon-Hee; Moon, Seok-hun; Lee, Do-won; Molkov, Yaroslav I.; Hamade, Khaldoun; Teka, Wondimu; Barnett, William H.; Kim, Taegyo; Markin, Sergey; Rybak, Ilya A.; Forrow, Csaba; Demutz, Harald; Demkó, László; Vörös, János; Dabaghian, Yuri; Babichev, Andrey; Huang, Haiping; Metzner, Christoph; Schwikard, Achim; Zurowski, Bartosz; Roach, James P.; Sander, Leonard M.; Zochowski, Michal R.; Skilling, Quinton M.; Ognjanovski, Nicolette; Aton, Sara J.; Zochowski, Michal; Wang, Sheng-Ju; Ouyang, Guang; Zhang, Mingsha; Wong, Michael; Zhou, Changsong; Robinson, Peter A.; Sanz-Leon, Paula; Drysdale, Peter M.; Fung, Felix; Abeysuriya, Romesh G.; Rennle, Chris J.; Zhao, Xuelong; Choe, Yoonsuck; Yang, Huei-Fang; Mi, Yuanyuan; Lin, Xiahoan; Wu, Si; Liedtke, Joscha; Schottdorf, Manual; Wolf, Fred; Yamamura, Yorkio; Wickens, Jeffery R.; Rumbell, Timothy; Ramsey, Julia; Reyes, Amy; Draguljić, Daniel; Hof, Patrick R.; Luebke, Jennifer; Weaver, Christina M.; He, Hu; Yang, Xu; Ma, Hailin; Xu, Zhiheng; Wang, Yuzhe; Baek, Kwangyeol; Morris, Laurel S.; Kundu, Prantik; Voon, Valerie; Agnes, Everton J.; Vogels, Tim P.; Giese, Martin; Kuravi, Pradeep; Vogels, Rufin; Seeholzer, Alexander; Podlaski, William; Ranjan, Rajnish; Vogels, Tim; Torres, Joaquin J.; Baroni, Fabiano; Latorre, Roberto; Varona, Pablo; Gips, Bart; Lowet, Eric; Roberts, Mark J.; de Weerd, Peter; Jensen, Ole; van der Eerden, Jan; Goodarzinic, Abdorreza; Niry, Mohammad; Valizadeh, Alireza; Pariz, Aref; Parsi, Shervin S.; Valizadeh, Alireza; Warburton, Julia M.; Marucci, Lucia; Tamagnini, Francesco; Brown, John; Tsaneva‑Atanasova, Krasimira; Kleberg, Florence I.; Triesch, Jochen; Moezzi, Bahar; Iannella, Nicolangelo; Schaworonkow, Natalie; Plogmacher, Lukas; Goldsworthy, Mitchell R.; Hordacre, Brenton; McDonnell, Mark D.; Ridding, Michael C.; Trisch, Jochen; Zaptocky, Martin; Smit, Daniel; Fouquet, Coralie; Trembleau, Alain; Dasgupta, Sakyasingha; Nishikawa, Isao; Aihara, Kazuyuki; Toyoizumi, Taro; Robb, Daniel T.; Mellen, Nick; Toporikova, Natalia; Tang, Rongxiang; Tang, Yi-Yuan; Kiser, Seth A.; Howard Jr., James H.; Tang, Yi-Yuan; Goncharenko, Julia; Davey, Neil; Schilstra, Marla; Steuber, Volker; Voronenko, Sergej O.; Linder, Benjamin; Ahamed, Tosif; Stephens, Greg; Yger, Pierre; Lefebvre, Baptiste; Spampinato, Giulia Lia Beatrice; Esposito, Elric; Stimberg, Marcel; Marre, Olivier; Choi, Hansol; Song, Min-Ho; Chung, SueYeon; Lee, Dan D.; Sompolinsky, Haim; Phillips, Ryan S.; Smith, Jeffrey; Chatzikalymniou, Alexandra Pierri; Ferguson, Katie; Skinner, Frances K.; Gajic, N. Alex Cayco; Clopath, Claudia; Silver, R. Angus; Gleeson, Padraig; Marin, Boris; Sadeh, Sadra; Quintana, Adrian; Cantarelli, Matteo; Dura‑Bernal, Salvador; Lytton, William W.; Davison, Andrew; Silver, Angus; Li, Luozheng; Zhang, Wenhao; Mi, Yuanyuan; Wang, Dahui; Wu, Sl; Song, Youngjo; Park, Sol; Choi, Ilhwan; Jeong, Jaeseung; Shin, Hee‑sup; Choi, Hannah; Pasupathy, Anitha; Shea-Brown, Eric; Huh, Dongsung; Sejnowski, Terrence J.; Vogt, Simon M.; Kumar, Arvind; Schmidt, Robert; Werdt, Stephen Van; Schiff, Steven J.; Veale, Richard; Scheutz, Matthias; Lee, Sang Wan; Gallinaro, Júlia; Rotter, Stefan; Sanz‑Leon, Paula; Robinson, Peter A.; Rubchinsky, Leonid L.; Cheung, Chung Ching; Ratnadurai‑Giridharan, Shivakeshavan; Shomali, Safura Rashid; Ahmadabadi, Majid Nili; Shimazaki, Hideaki; Rasuli, Nader; Zhao, Xiaochen; Rasch, Malte J.; Witting, Jens; Priesemann, Viola; Levina, Anna; Priesemann, Viola; Lizler, Joseph T.; Spinney, Richard E.; Rubinov, Mikail; Wibral, Michael; Bak, Ji Hyun; Pillow, Jonathan; Zaho, Yuan; Park, Memming; Kang, Jiyoung; Park, Hae‑Jeong; Jang, Jaeson; Paik, Se-Bum; Choi, Woochul; Lee, Changju; Jang, Jaeson; Paik, Se‑Bum; Song, Min; Lee, Hyeonsu; Yilmaz, Ergin; Baysal, Velt; Ozer, Mahmut; Koren, Veronika; Obermayer, Klaus; Saska, Daniel; Nowotny, Thomas; Chan, Ho Ka; Diamond, Alan; Hermann, Christoph S.; Murray, Micha M.; Ionta, Silvlo; Hutt, Axel; Lefebvre, Jérémie; Weidel, Philipp; Duarte, Renato; Morrison, Abigail; Iyer, Ramakrishnan; Mihalas, Stefan; Petrovici, Mihai A.; Leng, Luziwei; Breitwieser, Oliver; Stöckel, David; Bytschok, Ilja; Martel, Roman; Bill, Johannes; Schemmel, Johannes; Meier, Karlheinz; Esler, Timothy B.; Burkitt, Anthony N.; Grayden, David B.; Kerr, Robert R.; Tahayori, Bahman; Meffin, Hamish; Moezzi, Bahar; Iannella, Nicolangelo; McDonnell, Mark D.; Nolte, Max; Reimann, Michael W.; Muler, Eilif; Markram, Henry; Parziale, Antonio; Senatore, Rosa; Marcelli, Angelo; Maouene, M.; Skiker, K.; Neymotin, Samuel A.; Dura‑Bernal, Salvador; Seidenstein, Alexandra; Lakatos, Peter; Sanger, Terence D.; Lytton, William W.; Dura‑Bernal, Salvador; Menzies, Rosemary J.; McLauchlan, Campbell; van Albada, Sacha J.; Kedziora, David J.; Neymotin, Samuel; Kerr, Cliff C.; Ryu, Juhyoung; Lee, Sang-Hun; Lee, Joonwon; Lee, Hyang Jung; Lim, Daeseob; Lee, Jung H.; Wang, Jisung; Lee, Heonsoo; Jung, Nam; Quang, Le Anh; Maeng, Seung Eu; Lee, Tae Ho; Lee, Jae Woo; Park, Chang-hyun; Ahn, Sora; Moon, Jangsup; Choi, Yun Seo; Kim, Juhee; Jun, Sang Beom; Lee, Seungjun; Lee, Hyang Woon; Jo, Sumin; Jun, Eunji; Yu, Suin; Goetze, Felix; Lai, Pik‑Yin; Kwag, Jeehyun; Liang, Guangsheng; Jang, Hyun Jae; Filipovi, Marko; Reig, Ramon; Aertsen, Ad; Silberberg, Gilad; Kumar, Arvind; Bachmann, Claudia; Buttler, Simone; Jacobs, Heidi; Dillen, Kim; Fink, Gereon R.; Kukolja, Juraj; Kepple, Daniel; Giaffar, Hamza; Rinberg, Dima; Shea, Steven; Koulakov, Alex; Bahuguna, Jyotika; Tetzlaff, Tom; Kotaleski, Jeanette Hellgren; Kunze, Tim; Peterson, Andre; Knösche, Thomas; Kim, Minjung; Kim, Hojeong; Park, Ji Sung; Yeon, Ji Won; Kim, Sung-Phil; Lee, Chungho; Kim, Sung-Phil; Spiegler, Andreas; Petkoski, Spase; Palva, Matias J.; Jirsa, Viktor K.; Saggio, Maria L.; Siep, Silvan F.; Stacey, William C.; Bernard, Christophe; Choung, Oh‑hyeon; Jeong, Yong; Lee, Yong‑il; Jeong, Jaesung; Kim, Su Hyun; Lee, Jeungmin; Kwon, Jaehyung; Kralik, Jerald D.; Hwang, Dong‑Uk; Park, Sang-Min; Kim, Seongkyun; Kim, Hyoungkyu; Lim, Sewoong; Yoon, Sangsup; Park, Choongseok; Miller, Thomas; Clements, Katie; Hye Jr., Eoon; Issa, Fadi A.; Baek, JeongHun; Oba, Shigeyuki; Yoshimoto, Junichiro; Doya, Kenji; Ishii, Shin; Mosqueiro, Thiago S; Strube‑Bloss, Martin F.; Smith, Brian; Huerta, Ramon; Hadrava, Michal; Hlinka, Jaroslav; Bos, Hannah; Helias, Moritz; Welzig, Charles M.; Harper, Zachary J.; Kim, Won Sup; Shin, In-Seob; Baek, Hyeon-Man; Han, Seung Kee; Richter, René; Vitay, Julien; Beuth, Frederick; Hamker, Fred H.; Kameneva, Tatiana; Graham, Bruce P.; Kale, Penelope J.; Gollo, Leonardo L.; Stern, Merav; Abbott, L.F.; Fedorov, Leonid A.; Giese, Martin A.; Ardestani, Mohammad Hovaidi; Giese, Martin; Chakravarthy, V.Srinivasa; Chhabria, Karishma; Philips, Ryan T.; Ardestani, Mohammad Hovaidi; Faraji, Mohammad Java; Preuschoff, Kerstin; Gerstner, Wulfram; Briaire`, Jeroen J.; Kalkman, Randy K.; Frijns, Johan H. M.; Lee, Won Hee; Frangou, Sophia; Fulcher, Ben D.; Tran, Patricia H. P.; Fornito, Alex; Gliske, Stephen V.; Stacey, William C.; Holman, Katherine A.; Fink, Christian G.; Kim, Jinseop; Mu, Shang; Briggman, Kevin L; Seung, H. Sebastian; Wegener, Detlef; Bohnenkamp, Lisa; Ernst, Udo A.; Mäki‑Marttunen, Tuomo; Halnes, Geir; Devor, Anna; Dale, Anders M.; Andreassen, Ole A.; Einevoll, Gaute T.; Hagen, Espen; Lines, Glenn T.; Edwards, Andy; Tveito, Aslak; Senk, Johanna; van Albada, Sacha J; Diesmann, Markus; Schmidt, Maximilian; Bakker, Rembrandt; Shen, Kelly; Bezgin`, Gleb; Hilgetag`, Claus‑Christian; Sun, Haoqi; Sourina, Olga; Huang, Guang-Bin; Klanner, Felix; Denk, Cornelia; Glomb, Katharina; Ponce‑Alvarez, Adrián; Gilson, Matthieu; Ritter, Petra; Deco, Gustavo; Witek, Maria A. G.; Clarke, Eric F.; Hansen, Mads; Wallentin, Mikkel; Kringelbach, Morten L.; Vuust, Peter; Klingbeil, Guido; Schutter, Erik De; Chen, Weiliang; Hong, Sungho; Takashima, Akira; Zamora, Criseida; Gallimore, Andrew R.; Karoly, Philippa J.; Freestone, Dean R.; Soundry, Daniel; Kuhlmann, Levin; Paninski, Liam; Cook, Mark; Lee, Jaejin; Fishman, Yonatan I.; Cohen, Yale E.; Cocchi, Luca; Sweeney, Yann; Lee, Soohyun; Jung, Woo-Sung; Kim, Bowon; Kim, Youngsoo; Jung, Younginha; Rankin, James; Chavane, Frédéric; Soman, Karthik; Muralidharan, Vignesh; Shivkumar, Sabyasach; Mandall, Alekhya; Priyadharsini, B. Praga; Mehta, Hima; Brinkman, Braden A.; Kekona, Tyler; Rieke, Fred; Shea‑Brown, Eric; Buice, Michael; Pittà, Maurizio De; Berry, Hugues; Brunel, Nicolas; Breakspear, Michael; Marsat, Gary; Drew, Jordan; Chapman, Phillip D.; Daly, Kevin C.; Bradley, Samual P.; Seo, Sat Byul; Su, Jianzhong; Kavalali, Enge T.; Blackwell, Justin; Shiau, LieJune; Buhry, Laure; Basnayake, Kanishka; Lee, Sue-Hyun; Levy, Brandon A.; Baker, Chris I.; Leleu, Timothée; Aihara, Kazuyuki; Department of Mathematical Sciences, School of Science
3D simulation of a viscous flow past a compliant model of arteriovenous-graft annastomosis
(Elsevier, 2019-03) Bai, Zengding; Zhu, Luoding; Mathematical Sciences, School of Science
Hemodialysis is a common treatment for end-stage renal-disease patients to manage their renal failure while awaiting kidney transplant. Arteriovenous graft (AVG) is a major vascular access for hemodialysis but often fails due to the thrombosis near the vein-graft anastomosis. Almost all of the existing computational studies involving AVG assume that the vein and graft are rigid. As a first step to include vein/graft flexibility, we consider an ideal vein-AVG anastomosis model and apply the lattice Boltzmann-immersed boundary (LB-IB) framework for fluid-structure-interaction. The framework is extended to the case of non-uniform Lagrangian mesh for complex structure. After verification and validation of the numerical method and its implementation, many simulations are performed to simulate a viscous incompressible flow past the anastomosis model under pulsatile flow condition using various levels of vein elasticity. Our simulation results indicate that vein compliance may lessen flow disturbance and a more compliant vein experiences less wall shear stress (WSS).
A conclusive theorem on Finsler metrics of sectional flag curvature
(arXiv, 2018-12-22) Huang, Libing; Shen, Zhongmin; Mathematical Sciences, School of Science
If the flag curvature of a Finsler manifold reduces to sectional curvature, then locally either the Finsler metric is Riemannian, or the flag curvature is isotropic.
A Literature Review of Similarities Between and Among Patients With Autism Spectrum Disorder and Epilepsy
(Springer Nature, 2023-01-18) Assuah, Freda B.; Emanuel, Bryce; Lacasse, Brianna M.; Beggs, John; Lou, Jennie; Motta, Francis C.; Nemzer, Louis R.; Worth, Robert; Cravens, Gary D.; Mathematical Sciences, School of Science
Autism spectrum disorder (ASD) has been shown to be associated with various other conditions, and most commonly, ASD has been demonstrated to be linked to epilepsy. ASD and epilepsy have been observed to exhibit high rates of comorbidity, even when compared to the co-occurrence of other disorders with similar pathologies. At present, nearly one-half of the individuals diagnosed with ASD also have been diagnosed with comorbid epilepsy. Research suggests that both conditions likely share similarities in their underlying disease pathophysiology, possibly associated with disturbances in the central nervous system (CNS), and may be linked to an imbalance between excitation and inhibition in the brain. Meanwhile, it remains unclear whether one condition is the consequence of the other, as the pathologies of both disorders are commonly linked to many different underlying signal transduction mechanisms. In this review, we aim to investigate the co-occurrence of ASD and epilepsy, with the intent of gaining insights into the similarities in pathophysiology that both conditions present with. Elucidating the underlying disease pathophysiology as a result of both disorders could lead to a better understanding of the underlying mechanism of disease activity that drives co-occurrence, as well as provide insight into the underlying mechanisms of each condition individually.
A System with Two Spare Units, Two Repair Facilities, and Two Types of Repairers
(MDPI, 2022-03-08) Andalib, Vahid; Sarkar, Jyotirmoy; Mathematical Sciences, School of Science
Assuming exponential lifetime and repair time distributions, we study the limiting availability 𝐴∞ as well as the per unit time-limiting profit 𝜔 of a one-unit system having two identical, cold standby spare units using semi-Markov processes. The failed unit is repaired either by an in-house repairer within an exponential patience time T or by an external expert who works faster but charges more. When there are two repair facilities, we allow the regular repairer to begin repair or to continue repair beyond T if the expert is busy. Two models arise accordingly as the expert repairs one or all failed units during each visit. We show that (1) adding a second spare to a one-unit system already backed by a spare raises 𝐴∞ as well as 𝜔; (2) thereafter, adding a second repair facility improves both criteria further. Finally, we determine whether the expert must repair one or all failed units to maximize these criteria and fulfill the maintenance management objectives better than previously studied models.
Accessibility of the Boundary of the Thurston Set
(Taylor & Francis, 2023) Silvestri, Stefano; Pérez, Rodrigo A.; Mathematical Sciences, School of Science
Consider two objects associated to the Iterated Function System (IFS) {1+𝜆⁢𝑧,−1+𝜆⁢𝑧}: the locus ℳ of parameters 𝜆∈𝔻∖{0} for which the corresponding attractor is connected; and the locus ℳ0 of parameters for which the related attractor contains 0. The set ℳ can also be characterized as the locus of parameters for which the attractor of the IFS {1+𝜆⁢𝑧,𝜆⁢𝑧,−1+𝜆⁢𝑧} contains 𝜆−1. Exploiting the asymptotic similarity of ℳ and ℳ0 with the respective associated attractors, we give sufficient conditions on 𝜆∈∂ℳ or ∂ℳ0 to guarantee it is path accessible from the complement 𝔻∖ℳ.
Accuracy improvement of the immersed boundary–lattice Boltzmann coupling scheme by iterative force correction
(Elsevier, 2016-01) Zhang, Chunze; Cheng, Yongguang; Zhu, Luoding; Wu, Jiayang; Department of Mathematical Sciences, School of Science
The non-slip boundary condition at solid walls cannot be accurately achieved by the conventional immersed boundary–lattice Boltzmann (IB–LB) coupling schemes due to insufficient interpolation accuracy. To solve this problem, an iterative force correction procedure for the IB–LB coupling scheme is proposed. Cheng’s external forcing term in the LB equation is selected to properly incorporate the present and the next time step effects. The unknown IB force and the corresponding force on fluid at the next time step are calculated by iterative correction, based on the known immersed boundary speed, flow velocity, and the relationship between the IB speed and the IB force. Instead of the Dirac delta function, the Lagrange interpolation polynomial is used to obtain the IB speed from nearby fluid velocity. Typical cases, including the flow around a circular cylinder, shearing flow near a non-slip wall, and circular Couette flow between two inversely rotating cylinders, are simulated to verify and validate the method. It is shown that the present method guarantees the non-slip boundary condition and maintain the overall first-order spatial convergence rate of the conventional immersed boundary method (IBM). The accuracy improvement is obvious for both stationary and moving solid boundaries in both viscous flows and strong shearing flows. To demonstrate application possibility, a mechanical heart valve flow is also simulated, and better agreements with experimental data are achieved compared to those by commercial software.
Affinization of category 𝒪 for quantum groups
(2014-05) Mukhin, Eugene; Young, C. A. S.; Department of Mathematical Sciences, School of Science
Let be a simple Lie algebra. We consider the category of those modules over the affine quantum group whose -weights have finite multiplicity and lie in a finite union of cones generated by negative roots. We show that many properties of the category of the finite-dimensional representations naturally extend to the category . In particular, we define the minimal affinizations of parabolic Verma modules. In types ABCFG we classify these minimal affinizations and conjecture a Weyl denominator type formula for their characters.
Amphetamine enhances endurance by increasing heat dissipation
(APS, 2016-09-01) Morozova, Ekaterina; Yoo, Yeonjoo; Behrouzvaziri, Abolhassan; Zaretskaia, Maria; Rusyniak, Daniel; Zaretsky, Dmitry; Molkov, Yaroslav; Department of Mathematical Sciences, School of Science
Athletes use amphetamines to improve their performance through largely unknown mechanisms. Considering that body temperature is one of the major determinants of exhaustion during exercise, we investigated the influence of amphetamine on the thermoregulation. To explore this, we measured core body temperature and oxygen consumption of control and amphetamine‐trea ted rats running on a treadmill with an incrementally increasing load (both speed and incline). Experimental results showed that rats treated with amphetamine (2 mg/kg) were able to run significantly longer than control rats. Due to a progressively increasing workload, which was matched by oxygen consumption, the control group exhibited a steady increase in the body temperature. The administration of amphetamine slowed down the temperature rise (thus decreasing core body temperature) in the beginning of the run without affecting oxygen consumption. In contrast, a lower dose of amphetamine (1 mg/kg) had no effect on measured parameters. Using a mathematical model describing temperature dynamics in two compartments (the core and the muscles), we were able to infer what physiological parameters were affected by amphetamine. Modeling revealed that amphetamine administration increases heat dissipation in the core. Furthermore, the model predicted that the muscle temperature at the end of the run in the amphetamine‐treated group was significantly higher than in the control group. Therefore, we conclude that amphetamine may mask or delay fatigue by slowing down exercise‐induced core body temperature growth by increasing heat dissipation. However, this affects the integrity of thermoregulatory system and may result in potentially dangerous overheating of the muscles.
An asymptotic expansion for the expected number of real zeros of Kac-Geronimus polynomials
(Rocky Mountain Mathematics Consortium, 2021) Aljubran, Hanan; Yattselev, Maxim L.; Mathematical Sciences, School of Science
Let {φi(z;α)}i=0∞, corresponding to α∈(−1,1), be orthonormal Geronimus polynomials. We study asymptotic behavior of the expected number of real zeros, say 𝔼n(α), of random polynomials Pn(z):= ∑i=0nηiφi(z;α), where η0,…,ηn are i.i.d. standard Gaussian random variables. When α=0, φi(z;0)=zi and Pn(z) are called Kac polynomials. In this case it was shown by Wilkins that 𝔼n(0) admits an asymptotic expansion of the form 𝔼n(0)∼2πlog(n+1)+ ∑p=0∞Ap(n+1)−p (Kac himself obtained the leading term of this expansion). In this work we obtain a similar expansion of 𝔼(α) for α≠0. As it turns out, the leading term of the asymptotics in this case is (1∕π)log(n+1).

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