Michael Schlosser's Home Page
Dr. Michael J. Schlosser
_{X}
Fakultät für
Mathematik
Universität Wien
OskarMorgensternPlatz 1
A1090 Wien
Austria
telephone:

(+431) 427750455

fax:

(+431) 4277850455

email:


Aktuelle Lehre.
Current Teaching.
Sprechstunden nach Vereinbarung,
OMP1,
Zimmer 11.124; (oder online)
Office hours by appointment,
OMP1,
Room 11.124; (or online)
Scientific publications (in reverse chronological order):
 Papers:
 Residue class biases in unrestricted partitions,
partitions into distinct parts, and overpartitions (jointly with
Nian Hong
Zhou) [18 pp.; preprint]
 A finitebound partition identity
generalizing a problem by Andrews and Deutsch (jointly with
Nicolas
Allen Smoot)
[7 pp.; preprint]
 Diagonal operators, qWhittaker
functions and rook theory (jointly with
Samrith Ram)
[41 pp.; preprint]
 An elliptic extension of the
multinomial theorem [14 pp.; preprint; submitted]
 An esoteric identity with many
parameters and other elliptic extensions of elementary identities
(jointly with Gaurav Bhatnagar
and Archna Kumari)
[New Zealand J.
Math.,
17 pp., to appear]
 Expansions of averaged truncations of
basic hypergeometric series (jointly with
Nian Hong
Zhou) [Proc.
Amer. Math. Soc., 14 pp., to appear]

On an identity by Chaundy and Bullard. III. Basic and elliptic extensions
(jointly with Natsuko Hoshi, Makato Katori, and
Tom H. Koornwinder)
[Amer. Math. Soc.,
Providence, R. I., Contemp.
Math., 22 pp., to appear]
 Combinatorial formulas for Ramanujan's
tau function
[Srinivasa Ramanujan: His Life, Legacy, and Mathematical Influence;
K. Alladi,
G. E. Andrews,
B. C. Berndt,
F. Garvan,
K. Ono,
P. Paule,
S. O. Warnaar,
A. J. Yee (eds.),
Springer, 6 pp.,
to appear]
 Determinant evaluations inspired by Di
Francesco's determinant for twentyvertex configurations
(jointly with Christoph Koutschan
and Christian
Krattenthaler)
[J. Symbolic
Comput. 127 (2025), 102352,
34 pp.]
 Supercongruences involving
Domb numbers and binary quadratic forms (jointly with
GuoShuai
Mao)
[Rev. R.
Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM
118, 11 (2024), 14 pp.]
 Lattice paths and negatively indexed
weightdependent binomial coefficients (jointly with
Josef
Küstner
and Meesue Yoo)
[Ann.
Comb. 27(4) (2023), 917955]

A weighted extension of Fibonacci numbers
(jointly with Gaurav Bhatnagar
and Archna Kumari)
[J.
Difference Equ. Appl. 29(7) (2023), 733747]
 Bilateral identities
of the RogersRamanujan type
[Trans. Amer. Math. Soc. Ser. B 10 (2023), 11191140]
 Enumeration of standard barely
setvalued tableaux of shifted shapes (jointly with
Jang Soo Kim
and Meesue Yoo)
[Europ.
J. Combin. 112 (2023), 103727]
 New qanalogues of Van Hamme's (E.2)
supercongruence and of a supercongruence by Swisher
(jointly with
Victor J. W. Guo)
[Results
Math. 78 (2023), 105]
 Proof of a bisymmetric septuple
equidistribution on ascent sequences (jointly with
Emma Yu Jin)
[Combinatorial
Theory 3 (1) (2023), #5, 44 pp.]
 Three families of
qsupercongruences modulo the square and cube of a
cyclotomic polynomial
(jointly with
Victor J. W. Guo)
[Rev. R.
Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM
(2023) 117:9]
 An extension of a supercongruence
of Long and Ramakrishna (jointly with
JiCai
Liu and
Victor J. W. Guo)
[Proc.
Amer. Math. Soc. 151 (2023), 11571166]
 Asymptotics and
statistics on Fishburn Matrices: dimension distribution
and a conjecture of Stoimenow (jointly with
HsienKuei Hwang
and Emma Yu Jin)
[Random Structures & Algorithms 62 (2) (2023), 479518]
 On the infinite Borwein
product raised to a positive real power (jointly with
Nian
Hong Zhou)
[Ramanujan
J. 61 (2023), 515543]
 Lattice paths and negatively
indexed weightdependent binomial coefficients (jointly with
Josef
Küstner
and Meesue Yoo)
[FPSAC'22 extended abstract,
Sém. Lothar. Combin.
86B (2022), Art. 86B.76, 12 pp.]
 A family of qsupercongruences
modulo the cube of a cyclotomic polynomial
(jointly with
Victor J. W. Guo)
[Bull. Aust. Math. Soc.
105 (2) (2022), 296302]
 Bisymmetric multiple
equidistribution on ascent sequences (jointly with
Emma Yu Jin)
[FPSAC'21 extended abstract,
Sém. Lothar. Combin.
85B (2021), Art. 85B.64, 12 pp.]
 Some qsupercongruences
modulo the square and cube of a cyclotomic polynomial
(jointly with
Victor J. W. Guo)
[Rev. R.
Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM
115, 132 (2021)]
 Logconcavity results for a biparametric
and an elliptic extension of the qbinomial coefficients
(jointly with Koushik Senapati and
Ali K. Uncu)
[Int.
J. Number Th. 17(3) (2021), 787804]
 Elliptic solutions of
dynamical Lucas sequences (jointly with
Meesue Yoo)
[Entropy
2021, 23(2), 183]
 Some qsupercongruences from
transformation formulas for basic hypergeometric series
(jointly with
Victor J. W. Guo)
[Constr.
Approx. 53 (2021), 155200]
 A new family of qsupercongruences
modulo the fourth power of a cyclotomic polynomial
(jointly with
Victor J. W. Guo)
[Results Math.
75, 155 (2020)]
 A family of qhypergeometric
congruences modulo the fourth power of a cyclotomic polynomial
(jointly with
Victor J. W. Guo)
[Israel
J. Math. 240 (2020), 821835]
 Multidimensional matrix inversions and
elliptic hypergeometric series on root systems
(jointly with
Hjalmar Rosengren)
[SIGMA
16 (2020), 088, 21 pp.]

Hypergeometric and basic hypergeometric series and integrals
associated with root systems
[Encyclopedia of Special Functions: The
AskeyBateman Project, ser. eds. M. E. H. Ismail
and W. van Assche.
Volume 2. Multivariable
Special Functions,
vol. eds. T. H. Koornwinder and
J. Stokman,
Cambridge Univ. Press, Cambridge, 2020, pp. 122158]

A noncommutative weightdependent generalization of the binomial theorem
[Sém. Lothar. Combin.
81 (2020), Art. B81j, 24 pp.]
 Some new qcongruences for
truncated basic hypergeometric series: even powers
(jointly with
Victor J. W. Guo)
[Results Math.
75, 1 (2020)]
 A tribute to Dick Askey. A small
contribution to the September 2019 Richard Askey Liber Amicorum
[Celebratio Mathematica,
Richard
Allen Askey Liber Amicorum (2022), Contribution #64, pp. 104106]

A partial theta function Borwein conjecture
(jointly with Gaurav Bhatnagar)
[Ann.
Comb. 23(3) (2019), 561572]
 Some new qcongruences for
truncated basic hypergeometric series
(jointly with
Victor J. W. Guo)
[Symmetry
2019, 11(2), 268]
 Proof of a basic hypergeometric
supercongruence modulo the fifth power of a cyclotomic polynomial
(jointly with
Victor J. W. Guo)
[J.
Difference Equ. Appl. 25(7) (2019), 921929]

Wronskians of theta functions and series for 1/π
(jointly with Alex Berkovich
and Heng Huat Chan)
[Adv.
Math. 338 (2018), 266304]

Combinatorial interpretations of Ramanujan's tau function
(jointly with Frank Garvan)
[Discrete
Math. 341 (2018), 28312840]

Weightdependent commutation relations and combinatorial
identities (jointly with
Meesue Yoo)
[Discrete
Math. 341 (2018), 23082325]

Elliptic wellpoised Bailey transforms and lemmas on root systems
(jointly with Gaurav Bhatnagar)
[SIGMA
14 (2018), 025, 44 pp.]

Summation formulae for noncommutative hypergeometric series
[in
Frontiers
in Orthogonal Polynomials and qSeries
(Z. Nashed and
X. Li, eds.),
World Scientific, 2018, pp. 451496]

qAnalogues of two product formulas of hypergeometric
functions by Bailey [in
Frontiers
in Orthogonal Polynomials and qSeries
(Z. Nashed and
X. Li, eds.),
World Scientific, 2018, pp. 445449]

Basic hypergeometric summations from rook theory
(jointly with Meesue Yoo)
[in Analytic Number
Theory, Modular Forms and qHypergeometric Series in Honor of
Krishna Alladi's
60th birthday, University of Florida,
Gainesville, March 2016
(G.E. Andrews
and F. Garvan, eds.),
Springer
Proc. Math. Stat. 221 (2017), 677692]

Elliptic rook and file numbers (jointly with
Meesue Yoo)
[Electron. J. Combin.
24(1) (2017), #P1.31, 47 pp.]

Elliptic extensions of the alphaparameter model
and the rook model for matchings (jointly with
Meesue Yoo)
[Adv. Appl. Math. 84 (2017), 833]

qStirling numbers of the second kind and
qBell numbers for graphs (jointly with Zsófia R.
Kereskényiné Balogh)
[Electronic
Notes Discr. Math. 54 (2016), 361366]

Elliptic hypergeometric summations by Taylor series
expansion and interpolation (jointly with
Meesue Yoo)
[SIGMA
12 (2016), 039, 21 pp.]

An elliptic extension of the general product formula for augmented
rook boards (jointly with
Meesue Yoo)
[Europ.
J. Combin. 58 (2016), 247266]

Elliptic rook and file numbers
(jointly with Meesue Yoo),
DMTCS
proc. FPSAC'16, 2016, 10871098.

Some combinatorial identities involving noncommuting variables
(jointly with Meesue Yoo),
DMTCS
proc. FPSAC'15, 2015, 961972.

The major index generating function of standard Young tableaux of shapes
of the form "staircase minus rectangle" (jointly with
Christian
Krattenthaler)
[in Ramanujan 125
(K. Alladi
and F. Garvan, eds.),
Amer. Math. Soc.,
Providence, R. I., Contemp.
Math. 627 (2014), 111122]

On the work of Igor Frenkel (jointly with
John Duncan,
Pavel Etingof,
Ivan Ip,
Mikhail Khovanov,
Matvei Libine,
Anthony Licata,
and
Alistair Savage)
[in Perspectives in Representation Theory
(P. Etingof,
M. Khovanov, and
A. Savage, eds.),
Amer. Math. Soc.,
Providence, R. I., Contemp.
Math. 610 (2014), 121]

Multiple hypergeometric series  Appell series and beyond
[in Computer Algebra in Quantum Field Theory:
Integration, Summation and Special Functions
(J. Blümlein and
C. Schneider, eds.),
Texts & Monographs in
Symbolic Computation,
SpringerVerlag,
Heidelberg/Vienna, 2013; pp. 305324]

On an identity by Chaundy and Bullard. II. More history
(jointly with
Tom H. Koornwinder)
[Indag.
Math. (N.S.) 24 (1) (2013), 174180]

New curious bilateral qseries identities
(jointly with
Frédéric
Jouhet)
[Axioms
2012, 1 (3), 365371]

Recurrence formulas for Macdonald polynomials of type A
(jointly with
Michel Lassalle)
[J.
Algebraic Combin. 32 (1) (2010), 113131]

Noncommutative hypergeometric and basic hypergeometric equations
(jointly with
Alessandro Conflitti)
[J.
Nonlinear Math. Phys. 17 (4) (2010), 429443]

Theta functions, elliptic hypergeometric series, and Kawanaka's Macdonald
polynomial conjecture (jointly with Robin Langer and
S. Ole Warnaar)
[SIGMA
5 (2009), 055, 20 pp.]

Multilateral inversion of A_{r}, C_{r}
and D_{r} basic hypergeometric series
[Ann.
Comb. 13 (2009), 341363]

On an identity by Chaundy and Bullard. I
(jointly with
Tom H. Koornwinder)
[Indag.
Math. (N.S.) 19 (2) (2008), 239261]

A Taylor expansion theorem for an elliptic extension
of the AskeyWilson operator
[in Special
Functions and Orthogonal Polynomials
(D. Dominici and
R. S. Maier, eds.),
Amer. Math. Soc.,
Providence, R. I., Contemp.
Math. 471 (2008), 175186]

A new multivariable _{6}ψ_{6}
summation formula
[Ramanujan
J. 17 (3) (2008), 305319]

Curious extensions of Ramanujan's _{1}ψ_{1}
summation formula
(jointly with
Victor J. W. Guo)
[J. Math. Anal. Appl. 334 (1) (2007),
393403]
 Macdonald
symmetric functions for partitions with complex parts
[FPSAC'07,
Nankai University, Tianjin, China, 10 pp.]

Macdonald polynomials and multivariable basic hypergeometric series
[Vadim
Kuznetsov Memorial Issue on Integrable Systems and Related
Topics,
SIGMA
3 (2007), 056, 30 pp.]

Elliptic enumeration of nonintersecting lattice paths
[J. Combin. Theory Ser. A 114 (3) (2007),
505521]

Some curious qseries expansions and beta integral
evaluations
(jointly with
George Gasper)
[Ramanujan
J. 13 (13) (2007), 229242]

Elliptic determinant evaluations and the Macdonald identities
for affine root systems
(jointly with
Hjalmar Rosengren)
[Compos. Math.
142 (4) (2006), 937961]

Noncommutative extensions of Ramanujan's
_{1}ψ_{1} summation
[Electron. Trans.
Numer. Anal. 24 (2006), 94102]

Explicit computation of the q,tLittlewoodRichardson
coefficients
[Proceedings
of the Workshop on Jack,
HallLittlewood and Macdonald polynomials (V. Kuznetsov
and S. Sahi, eds.) Edinburgh, ICMS, Sept. 2326, 2003;
Contemp.
Math. 417 (2006), 335343]

Inversion of the Pieri formula for Macdonald polynomials
(jointly with
Michel Lassalle)
[Adv.
Math. 202 (2) (2006), 289325]

Summation, transformation, and expansion formulas for
multibasic theta hypergeometric series
(jointly with
George Gasper)
[Adv. Stud. Contemp. Math. (Kyungshang) 11 (2005), no. 1, 6784;
also in
Proceedings
of the Workshop Elliptic
Integrable Systems (M. Noumi
and K. Takasaki,
eds.) Kyoto,
RIMS, 2004;
Rokko Lectures
in Mathematics 18 (2005), 117]

Another proof of Bailey's _{6}ψ_{6} summation
(jointly with
Frédéric
Jouhet)
[Aequationes
Math. 70 (12) (2005), 4350]

On Warnaar's elliptic matrix inversion and KarlssonMintontype elliptic
hypergeometric series
(jointly with
Hjalmar Rosengren)
[J.
Comput. Appl. Math. 178 (2005), 377391]

AbelRothe type generalizations of Jacobi's triple product identity
[in Theory
and Applications of Special Functions. A Volume Dedicated to Mizan Rahman
(M. E. H. Ismail and
E. Koelink, eds.),
Dev.
Math. 13 (2005), 383400]

qAnalogues of the sums of consecutive integers, squares,
cubes, quarts and quints
[Electron. J. Combin.
11 (1) (2004), #R71, 12 pp.]

Some curious extensions of the classical beta integral evaluation
[in Mathematics
and Computer Science, III (M. Drmota,
P. Flajolet, D. Gardy,
B. Gittenberger, eds.)
Vienna, 2004; Trends
Math., Birkhäuser, Basel, 2004, pp. 5968]

An analytic formula for Macdonald polynomials
(jointly with
Michel Lassalle)
[C.
R. Math. Acad. Sci. Paris 337 (9) (2003), 569574]

Summations and transformations for multiple basic and
elliptic hypergeometric series by determinant evaluations
(jointly with
Hjalmar Rosengren)
[Indag.
Math. (N.S.) 14 (2003), 483514]

A nonterminating _{8}φ_{7} summation for the
root system C_{r}
[J.
Comput. Appl. Math. 160 (2003), 283296]

Inversion of bilateral basic hypergeometric series
[Electron. J. Combin.
10 (2003), #R10, 27 pp.]

A multidimensional generalization of Shukla's
_{8}ψ_{8} summation
[Constr.
Approx. 19 (2003), 163178]

Elementary derivations of identities for bilateral basic
hypergeometric series
[Selecta
Math. (N.S.) 9 (2003) 1, 119159]

A new A_{n} extension of Ramanujan's
_{1}ψ_{1} summation with applications to
multilateral basic hypergeometric series (jointly with
Stephen C. Milne)
[in Special Functions 2000
(J. Bustoz and
S. K. Suslov, eds.),
Rocky Mount. J. Math. 32 (2) (2002), 759792]

A simple proof of Bailey's verywellpoised
_{6}ψ_{6} summation
[Proc.
Amer. Math. Soc. 130 (2002), 11131123]

Multilateral transformations of qseries with quotients of
parameters that are nonnegative integral powers of q
[in qSeries
with Applications to Combinatorics, Number Theory, and Physics
(B. C. Berndt and
K. Ono, eds.),
Amer. Math. Soc.,
Providence, R. I., Contemp.
Math. 291 (2001), 203227]

A new multidimensional matrix inversion in A_{r}
[in qSeries
from a Contemporary Perspective
(M. E. H. Ismail and
D. Stanton, eds.),
Amer. Math. Soc.,
Providence, R. I., Contemp.
Math. 254 (2000), 413432]

Summation theorems for multidimensional basic
hypergeometric series by determinant evaluations
[Discrete
Math. 210 (2000), 151169]

Some new applications of matrix inversions in A_{r}
[Ramanujan
J. 3 (1999), 405461]

A new multidimensional matrix inverse with applications to multiple
qseries (jointly with
Christian
Krattenthaler)
[(H. W. Gould Anniversary Volume)
Discrete
Math. 204 (1999), 249279; selected to be included in
Discrete
Math., spec. vol. ``Editors' Choice, Edition 1999'']

Multidimensional matrix inversions and multiple basic hypergeometric
series associated to root systems
[in Special
functions and differential equations
(K. Srinivasa Rao,
R. Jagannathan,
G. Vanden Berghe, and
J. Van der Jeugt, eds.),
Proceedings of a workshop, WSSF '97, Madras, India, January 1324, 1997;
Allied Publ., New Delhi (1998), 2530]

C_{n} and D_{n} verywellpoised
_{10}φ_{9} transformations
(jointly with Gaurav Bhatnagar)
[Constr.
Approx. 14 (1998), 531567]

Multidimensional matrix inversions and A_{r} and
D_{r} basic hypergeometric series
[Ramanujan
J. 1 (1997), 243274]

Multidimensional
matrix inversions
[Sém. Lothar. Combin.
35 (1995), B35g, 24 pp.]
 Books and special issues edited:
 Basic
Hypergeometric Series Associated with Root Systems and Applications in
honor of Stephen C. Milne's 75th birthday
(jointly with G. Bhatnagar,
Frank Garvan, and
Christian Krattenthaler),
Special Issue of
SIGMA,
under completion.
 The Krattenthaler Festschrift (jointly with
V. Strehl and
J.Y. Thibon), Issue 81 of
Sém. Lothar. Combin.,
completed in April 2020.
 Elliptic
Hypergeometric Functions and Their Applications
(jointly with S. V. Spiridonov and
S. Ole Warnaar),
Special Issue of
SIGMA,
completed in December 2018.

Ramanujan
Rediscovered (jointly with
N. D. Baruah,
B. C. Berndt,
S. Cooper and
T. Huber),
Proceedings of a Conference on Elliptic Functions, Partitions, and
qSeries in Memory of K. Venkatachaliengar, Bangalore (15 June, 2009);
Ramanujan Mathematical Society
Lecture Notes Series, Vol. 14, 2010
(also published by International Press, Somerville, MA, USA, 2012)
Editorial boards:
Postdocs:
Ph.D. students:
Selected events:

FPSAC'25
(37th International Conference on Formal Power Series and Algebraic
Combinatorics), Hokkaido University, Sapporo, Japan,
July 2125, 2025

Joint meeting of
the NZMS, AustMS and AMS,
University of Auckland,
December 913, 2024

SLC 92
(92nd Séminaire Lotharingien de Combinatoire), Strobl,
September 1518, 2024

ICECA
2024 (International Conference on Enumerative Combinatorics
and Applications), Virtual, August 2628, 2024

FPSAC'24
(36th International Conference on Formal Power Series and Algebraic
Combinatorics), RuhrUniversität Bochum, Germany,
July 2226, 2024

AlCoVE 2024 (An Algebraic Combinatorics Virtual Expedition),
June 1718, 2024

The Legacy of Ramanujan 2024, Celebrating the 85th birthdays of George Andrews & Bruce Berndt,
Penn State University, USA, June 69, 2024

SLC 91
(Séminaire Lotharingien de Combinatoire 91),
Salobreña, Spain, March 1720, 2024

ICECA
2023 (International Conference on Enumerative Combinatorics
and Applications), Virtual, September 46, 2023

Elliptic
Integrable Systems, Representation Theory and Hypergeometric
Functions (MSJSI 2023 Workshop),
University of Tokyo,
Tokyo, Japan, July 31August 4, 2023

18th Annual
International Conference on Srinivasa Ramanujan,
Srinivasa
Ramanujan Centre (SRC), Kumbakonam, India, December 2022, 2022

ICECA
2022 (International Conference on Enumerative Combinatorics
and Applications), Virtual, September 67, 2022

FPSAC'22
(34th International Conference on Formal Power Series and Algebraic
Combinatorics), Indian Institute of
Science, Bangalore, India, July 1822, 2022

Algorithmic
and Enumerative Combinatorics (AEC 2022) conference,
Vienna, Austria, July 48, 2022

OPSFA
2022 (16th International Symposium on Orthogonal Polynomials,
Special Functions and Applications),
CRM
Montreal, Canada, Online, June 1317, 2022

FPSAC'21
(33rd International Conference on Formal Power Series and Algebraic
Combinatorics), BarIlan University, RamatGan, Israel,
July 59, 2021

Lattice Paths,
Combinatorics and Interactions hybrid conference,
CIRM, Luminy, Marseille,
France, June 2125, 2021

International Conference
on Special Functions & Applications (ICSFA2020),
Babu Banarasi Das University, Lucknow, Online,
December 2223, 2020

Hypergeometry, Integrability and Lie Theory, Workshop at the
Lorentz Center,
Leiden, The Netherlands, Online, December 711, 2020

AMS
Fall Eastern Virtual Sectional Meeting,
October 34, 2020

FPSAC'20
(32nd International Conference on Formal Power Series and Algebraic
Combinatorics), Online, July 624, 2020

ÖMG
Conference 2019 (Conference of the
Austrian Mathematical Society),
University of Applied Sciences Vorarlberg, Dornbirn, Austria,
September 1620, 2019

OPSFA
2019 (15th International Symposium on Orthogonal Polynomials,
Special Functions and Applications),
RISC
Hagenberg, Austria, July 2226, 2019

Elliptic integrable systems, special functions and quantum field
theory, Nordita,
Stockholm, Sweden, June 1620, 2019

Analytic and Combinatorial
Number Theory: The Legacy of Ramanujan, A conference in honor of
Bruce Berndt's 80th birthday, University of Illinois at
UrbanaChampaign, IL, USA, June 69, 2019

Topics on
Orthogonal Matters Fest, An informal gathering to celebrate
Tom Koornwinder's 75th birthday, Radboud University, Nijmegen,
Netherlands, December 1011, 2018

SLC 81
(81st Séminaire Lotharingien de Combinatoire
2018/KrattenthalerFest), Strobl, September 912, 2018

Combinatory
Analysis, A conference in honor of George Andrews' 80th
birthday, Penn State University, State College, PA, USA,
June 2124, 2018

International
Conference on Special Functions, Theory, Computation, and
Applications, In Celebration of the 20th Anniversary of the
Lie Bie Ju Centre for Mathematical Sciences, City University of
Hong Kong, June 59, 2017

Elliptic
Hypergeometric Functions in Combinatorics, Integrable Systems and
Physics, ESI, Vienna,
March 2024, 2017

Number Theory
in honor of Krishna Alladi's 60th birthday, University of Florida,
Gainesville, FL, USA, March 1721, 2016
Talks:
Curiosities:

I used to play a lot of chess;
for a few (not so recent) games I played, see
here,
here,
and there.