**This page contains a list of publications which contain results
of research supported from the grant (in an arbitrary order).**

**6.** M. Krčál, P. Pilarczyk,
Computation of Cubical Steenrod Squares.
In: A. Bac and J.-L. Mari (Eds.): Computational Topology in Image Context:
6th International Workshop, CTIC 2016, Marseille, France, June 15-17, 2016,
Proceedings. LNCS 9667, pp. 140-151, 2016.
DOI: 10.1007/978-3-319-39441-1_13.

[*Lecture Notes in Computer Science*
is a book series indexed in ISI, ISSN 0302-9743.]

Abstract:
Bitmap images of arbitrary dimension may be formally perceived
as unions of *m*-dimensional boxes
aligned with respect to a rectangular grid in **R**^{m}.
Cohomology and homology groups
are well known topological invariants of such sets.
Cohomological operations, such as the cup product,
provide higher-order algebraic topological invariants,
especially important for digital images of dimension higher than 3.
If such an operation is determined at the level of simplicial chains
[see e.g. Gonzalez-Diaz, Real,
Homology, Homotopy Appl, 2003, 83-93],
then it is effectively computable.
However, decomposing a cubical complex into a simplicial one
deleteriously affects the efficiency of such an approach.
In order to avoid this overhead,
a direct cubical approach was applied in
[Pilarczyk, Real, Adv. Comput. Math., 2015, 253-275]
for the cup product in cohomology,
and implemented in the
ChainCon
software package.

We establish a formula for the Steenrod square operations
[see Steenrod, Annals of Mathematics. Second Series, 1947, 290-320]
directly at the level of cubical chains,
and we prove the correctness of this formula.
An implementation of this formula is programmed in C++
within the ChainCon software framework.
We provide a few examples and discuss the effectiveness of this approach.

One specific application follows from the fact
that Steenrod squares yield tests for the topological extension
problem: Can a given map from a set *A*
to a sphere S^{d}
be extended to a given super-complex *X* of *A*?
In particular, the Robust Satisfiability Problem (ROB-SAT)
reduces to the extension problem.

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**5.** T. Miyaji, P. Pilarczyk, M. Gameiro, H. Kokubu, K. Mischaikow,
A study of rigorous ODE integrators for multi-scale set-oriented computations,
Appl. Numer. Math., Vol. 107 (2016), 34-47.
DOI: 10.1016/j.apnum.2016.04.005.

[*Applied Numerical Mathematics*
is indexed in ISI, ISSN 0168-9274,
Impact Factor (2014): 1.221,
MathSciNet MCQ (2014): 0.85.]

Abstract: We study the usefulness of two most prominent publicly
available rigorous ODE integrators: one provided by the
CAPD group, the other based
on the COSY Infinity project.
Both integrators are capable of handling entire sets of initial conditions
and provide tight rigorous outer enclosures of the images under a time-T
map. We conduct extensive benchmark computations using the well-known
Lorenz system, and compare the computation time against the final
accuracy achieved. We also discuss the effect of a few technical
parameters, such as the order of the numerical integration method, the
value of T, and the phase space resolution. We conclude that COSY may
provide more precise results due to its ability of avoiding the variable
dependency problem. However, the overall cost of computations conducted
using CAPD is typically lower, especially when intervals of parameters
are involved. Moreover, access to COSY is limited (registration required)
and the rigorous ODE integrators are not publicly available, while CAPD
is an open source free software project. Therefore, we recommend the
latter integrator for this kind of computations. Nevertheless, proper
choice of the various integration parameters turns out to be of even
greater importance than the choice of the integrator itself.

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**4.** P. Pilarczyk, A space-efficient algorithm
for computing the minimum cycle mean in a directed graph,
submitted.

Abstract: An algorithm is introduced
for computing the minimum cycle mean
in a strongly connected directed graph
with *n* vertices and *m* arcs
that requires *O* (*n*) working space.
This is a considerable improvement for sparse graphs
in comparison to the classical algorithms
that require *O* (*n*^{2}) working space.
The time complexity of the algorithm
is still *O* (*n m*).
An implementation in C++ is made publicly available
at http://www.pawelpilarczyk.com/cymealg/.

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**3.** A. Golmakani, S. Luzzatto, P. Pilarczyk,
Uniform expansivity outside a critical neighborhood
in the quadratic family, Exp. Math., Vol. 25, No. 2 (2016), 116-124.

DOI: 10.1080/10586458.2015.1048011.

[*Experimental Mathematics*
is indexed in ISI, 1058-6458 (Print), 1944-950X (Online),
Impact Factor (2013): 1,
MathSciNet MCQ (2013): 0.64.]

Abstract: We use rigorous numerical techniques to
compute a lower bound for the exponent of expansivity outside a
neighborhood of the critical point for thousands of intervals of
parameter values in the quadratic family. We first compute a radius of
the critical neighborhood outside which the map is uniformly expanding.
This radius is taken as small as possible, yet large enough for our
numerical procedure to succeed in proving that the expansivity exponent
outside this neighborhood is positive. Then, for each of the intervals,
we compute a lower bound for this expansivity exponent, valid for all
the parameters in that interval. We illustrate and study the
distribution of the radii and the expansivity exponents. The results of
our computations are mathematically rigorous. The source code of the
software and the results of the computations are made publicly available
at http://www.pawelpilarczyk.com/quadratic/.

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**2.** S. Harker, H. Kokubu, K. Mischaikow, P. Pilarczyk,
Inducing a map on homology from a correspondence,
Proc. Amer. Math. Soc., Vol. 144, No. 4 (2016), 1787-1801.

DOI: 10.1090/proc/12812.

[*Proceedings of the American
Mathematical Society (Proceedings of the AMS)*
are indexed in ISI, ISSN 1088-6826 (online), ISSN 0002-9939 (print),
Impact Factor (2013): 0.627,
MathSciNet MCQ (2013): 0.64.]

Abstract: We study the homomorphism induced in homology
by a closed correspondence between topological spaces,
using projections from the graph of the correspondence
to its domain and codomain.
We provide assumptions under which the homomorphism
induced by an outer approximation of a continuous map
coincides with the homomorphism induced in homology by the map.
In contrast to more classical results we do not require
that the projection to the domain have acyclic preimages.
Moreover, we show that it is possible to retrieve
correct homological information from a correspondence
even if some data is missing or perturbed.
Finally, we describe an application to combinatorial maps
that are either outer approximations of continuous maps
or reconstructions of such maps from a finite set of data points.

[PDF]
[arXiv:1411.7563] [math.AT]

**1.** D.H. Knipl, P. Pilarczyk, G. Röst,
Rich bifurcation structure in a two-patch vaccination model,
*SIAM J. Appl. Dyn. Syst.*,
Vol. 14, No. 2 (2015), 980-1017.

DOI: 10.1137/140993934.

[*SIAM Journal on Applied
Dynamical Systems (SIADS)*
is indexed in ISI, ISSN 1536-0040, Impact Factor (2013): 1.245,
MathSciNet MCQ (2013): 0.88.]

Abstract: We show that incorporating spatial dispersal of
individuals into a simple vaccination epidemic model may give rise to a
model that exhibits rich dynamical behavior. Using an SIVS (susceptible
– infected – vaccinated – susceptible) model as a
basis, we describe the spread of an infectious disease in a population
split into two regions. In each sub-population, both forward and
backward bifurcations can occur. This implies that for disconnected
regions, the two-patch system may admit several steady states. We
consider traveling between the regions, and investigate the impact of
spatial dispersal of individuals on the model dynamics. We establish
conditions for the existence of multiple non-trivial steady states in
the system, and we study the structure of the equilibria. The
mathematical analysis reveals an unusually rich dynamical behavior, not
normally found in the simple epidemic models. In addition to the
disease free equilibrium, eight endemic equilibria emerge from backward
transcritical and saddle-node bifurcation points, forming an interesting
bifurcation diagram. Stability of steady states, their bifurcations and
the global dynamics are investigated with analytical tools, numerical
simulations, and rigorous set-oriented numerical computations.

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