### My Refereed Publications (published/accepted - 22, submitted - 1)

#### 2014 (probably), i.e. just submitted papers

1. Journal: Factorization of Darboux Transformations of Arbitrary Order for Two-dimensional Schr\"odinger operator, 2014.

#### 2013

2. Journal: Invertible Darboux Transformations , SIGMA 9 (2013), 002, special issue “Symmetries of Differential Equations: Frames, Invariants and Applications” in honor of the 60th Birthday of Peter Olver. Editors: N.Kamran, G.M.Beffa, W. Miller, G. Sapiro, 2013.
Informal abstract. For operators of many different kinds Darboux transformations can be built using Wronskian formulas. These transformations are not invertible in the sense that the corresponding mappings of the operator kernels are not invertible. The only known invertible ones were Laplace transformations (and their compositions), which are special cases of Darboux transformations for hyperbolic bivariate operators of order 2. Here we find a criteria for a bivariate linear partial differential operator of an arbitrary order d to have an invertible Darboux transformation. Give explicit example when Wronkians do not work. Find conditions for Wronkians to work.
3. Refereed conference procs: Invariants for Darboux transformations of Arbitrary Order for $D_x D_y +aD_x + bD_y +c$ , Geometric Methods in Physics. XXXI Workshop, Bialowieza, Poland, June 30 to July 6, 2013, P. Kielanowski, S. T. Ali, A.Odesski, A. Odzijewicz, M. Schlichenmaier, Th. Voronov (editors), Trends in Mathematics. Springer, Basel, 2013. bibtex .
4. Journal: Proof of the Completeness of Darboux Wronskian Formulae for Order Two, Canadian Journal of Mathematics 65(2013), no. 3, 655-674, http://dx.doi.org/10.4153/CJM-2012-026-7, 2013. Get it here or at arxiv
Informal abstract. Wronskian formulas allow one to construct Darboux Transformations (DTs). Laplace transformations (DT of order one) cannot be represented in this way. Proved before: among DTs of total order 1 - NO exceptions, other than Laplace transformations. Here: for DTs of total order 2 - NO exceptions. Here also: simple invariant description of all possible DTs of total order 2.

#### 2012

5. Journal: Package LPDO for MAPLE, accepted to Programming and Computer Software, special issue Computer Algebra, (Russian Academy of Science, eds. S. Abramov, S.Tsarev), number 2, 2013.
Informal abstract. Help file for my LPDO package .
6. Journal: Laplace Transformations as the Only Degenerate Darboux Transformations of First Order, Programming and Computer Software, special issue Computer Algebra, (Russian Academy of Science, eds. S. Abramov, S.Tsarev), volume 38, number 2, 2012, bibtex .
Informal abstract. Wronskian formulas allow one to construct Darboux Transformations (DTs). Laplace transformations (DT of order one) cannot be represented in this way. Here: among DTs of total order 1 - NO exceptions, other than Laplace transformations.

#### 2011

7. Journal: X- and Y-invariants of partial differential operators in the plane , Programming and Computer Software, special issue devoted to Computer Algebra 2011, (Russian Academy of Science, eds. S. Abramov, S.Tsarev), (37), no.4, pp.192-196, 2011, bibtex.
8. Book chapter: with F. Winkler, Linear Partial Differential Equations and Linear Partial Differential Operators in Computer Algebra, in Monographs in Symbolic Computation, Springer (book chapter),
editors: P. Paule et al., vol. Progress and Prospects in Numerical and Symbolic Scientific Computing, 2011.

#### 2010

9. Journal: Refinement of Two-Factor Factorizations of a Linear Partial Differential Operator of Arbitrary Order and Dimension, Mathematics in Computer Science, (4), no.2-3, pp. 223-230, 2010, bibtex .
10. Journal: with S.I.Khashin, and D.J.Jeffrey, Conjecture concerning a completely monotonic function, Computers & Mathematics with Applications, vol. 60, issue 5, pp.1360-1363, 2010. ScienceDirect , bibtex .

#### 2009

11. Refereed conference procs: : Multiple factorizations of bivariate linear partial differential operators, Lecture Notes in Computer Science, vol. 5743, pp. 299--309, 2009, bibtex .
12. Refereed conference procs: : On the invariant properties of non-hyperbolic third-order linear partial differential operators, Conferences on Intelligent Computer Mathematics, vol.5625, pp.154--169, 2009, bibtex .
13. Journal: with S.Tsarev Differential transformations of parabolic second-order operators in the plane, Proceedings Steklov Inst. Math. (Moscow), vol.266, pp.219--227, 2009, bibtex .

#### 2008

14. Refereed conference procs: : with E.Mansfield, Moving frames for Laplace invariants, Proceedings of ISSAC 2008 (The International Symposium on Symbolic and Algebraic Computation), pp.295--302, 2008, bibtex .

#### 2007 (PhD degree obtained in 2007)

15. Refereed conference procs: : with F.Winkler, On the invariant properties of hyperbolic bivariate third-order linear partial differential operators, Lecture Notes in Artificial Intelligence, vol.5081, pp.199--212, 2007, bibtex .
16. Refereed conference procs: : with F.Winkler, A full system of invariants for third-order linear partial differential operators in general form, Lecture Notes in Comput. Sci., vol.4770, pp.360--369, 2007, bibtex .
17. Journal: The Parametric Factorizations of Second-, Third- and Fourth-Order Linear Partial Differential Operators on the Plane , Mathematics in Computer Science, vol.1, no.2, pp.225--237, 2007, bibtex .
18. Journal: with F.Winkler, Obstacles to the Factorization of Linear Partial Differential Operators into Several Factors , Programming and Computer Software, vol.33, no.2, pp.67--73, 2007, bibtex .
19. Refereed conference procs: : with F.Winkler, Symbolic and Algebraic Methods for Linear Partial Differential Operators , Lecture Notes in Computer Science, vol.4770, 2007, bibtex .

#### 2006

20. Refereed conference procs: : with F.Winkler, Obstacle to Factorization of LPDOs, in Proc. Transgressive Computing 2006 (J.-G. Dumas, ed.), pp.435--441, 2006.
21. Refereed conference procs: : A full system of invariants for third-order linear partial differential operators, Lecture Notes in Computer Science, vol.4120, pp.360--369, 2006, bibtex .

#### 2004 (while 5th year undergraduate student)

22. Journal: Involutive divisions. Graphs., Programming and Computer Software, vol.30, no.2, pp.68--74, 2004.

#### 2003 (while 4th year undergraduate student)

23. Journal: Involutive divisions for effective involutive algorithms, Fundam. Prikl. Mat., vol.9, no.3, pp.237--253, 2003.

## Minor Refereed Publications

1. Abstract of the PhD thesis, M.Giesbrecht (eds.), ACM Communications in Computer Algebra, 41, N.3, issue 161, 2007.
2. with F. Winkler. Extended abstract: Algebraic Methods for Linear Partial Differential Operators. M.Giesbrecht, I.Kotsireas, A.Lobo (eds.), ACM Communications in Computer Algebra, 41, N.2, issue 160, 2007.
3. with F. Winkler. Extended abstract: Approximate Factorization of Linear Partial Differential Operators. Full System of Invariants for Order Three. Zh. Wan, A.Lobo(eds.), ACM Communications in Computer Algebra, 40, N.2, issue 156, 2006.

## Technical Reports

1. with J. Middeke, F. Winkler, Proceedings of DEAM (Workshop for Differential Equations by Algebraic Methods), 2009, RISC Report Series, University of Linz, Austria.