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# All the Signs point to the Second Prime…Figure out the Missing Digits

$$Given$$

The six Interrelated Equations A, B, C, D, E, F

All equal to 3.

Allowed Digits.. 1 to 9.

All the missing 9 are distinct...9?s...9 Different Digits

Fillin all the digits that satisfy all these Relations.

• On the ordering of operations, do we assume multiplication and division are performed before addition and subtraction? Also, is the bottom row and rightmost column "(?-?)+?" or "?-(?+?)" – hexomino Jun 13 at 9:30
• Is there a unique solution and are all the values unique? – Bee Jun 13 at 9:31
• Order of operations ..A, B, C ...3 Horizontal. D, E, F...3 Vertical...Bottom right 4 squares...blank..like other black cells. – Uvc Jun 13 at 9:36
• Based on what I could figure out, it is unique – Uvc Jun 13 at 9:38
• Order of operations as given would be..3+3=6,,,6x2=12....follow the operations in the order given in both horizontal and vertical sets. – Uvc Jun 13 at 9:50

I've found a solution, which is unique given the constraints.

 1 * 9 - 6  = 3
*   -   -
7 + 8 / 5  = 3
-   *   +
4 - 3 + 2  = 3
=   =   =
3   3   3

Proof

The digit in the bottom middle must be either $$1$$ or $$3$$ since the product of the second column is $$3$$.
If it is $$1$$ then the other entries in the bottom row are either $$2,2$$ or $$1,3$$ but we are not allowed to repeat digits. Hence, it must be $$3$$.

This means the other digits in the bottom row will be $$2,4$$ or $$1,5$$.
Now, suppose $$1$$ is on the left in the bottom row. This means the first product on the left-hand column is $$4$$ which cannot be constructed without repeating digits.
Alternatively, suppose $$5$$ is on the left in the bottom. Then the other digits in the left-hand column must be $$2$$ and $$4$$ since $$1$$ has already been used. This means the last entry in the second row is at least $$6$$ which is not permissible since the sum cannot be greater than $$17$$.
Hence, the other two entries in the bottom row must be $$2$$ and $$4$$.

Now suppose $$2$$ is on the left. This means that the other entries in the left-hand column are $$1$$ and $$5$$ which forces the last entry in the second row to be at least $$6$$, not allowed. Hence the bottom row must be $$4,3,2$$.

From there, we know that the other entries in the left-hand column are $$1$$ and $$7$$.
This means the last entry in the second row must be $$5$$ since it is the only option available that could possibly work. Hence the sum of the first two digits in the second row is $$15$$ which makes the first entry $$7$$ and the second $$8$$.
This means the top left-hand corner must be $$1$$ and we are left with $$9$$ and $$6$$ to complete the grid.

• This morning, my mind is not sharp..forgot to mention that there are no repeating digits...will edit to reflect that.. – Uvc Jun 13 at 9:55
• Got it !!...sorry about all those edits – Uvc Jun 13 at 10:04
• @Uvc No, that's fine. Nice puzzle! – hexomino Jun 13 at 10:10

Here's a solution:

 1 * 3 - 0  = 3
*   -   -
3 + 0 / 1  = 3
-   *   +
0 - 1 + 4  = 3
=   =   =
3   3   3

• I forgot to mention..that only 1 to 9 are allowed...will edit it now..sorry about that – Uvc Jun 13 at 9:46


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## The Annals of Applied Probability

### First-order Euler scheme for SDEs driven by fractional Brownian motions: The rough case

#### Abstract

In this article, we consider the so-called modified Euler scheme for stochastic differential equations (SDEs) driven by fractional Brownian motions (fBm) with Hurst parameter $\frac{1}{3}<H<\frac{1}{2}$. This is a first-order time-discrete numerical approximation scheme, and has been introduced in [Ann. Appl. Probab. 26 (2016) 1147–1207] recently in order to generalize the classical Euler scheme for Itô SDEs to the case $H>\frac{1}{2}$. The current contribution generalizes the modified Euler scheme to the rough case $\frac{1}{3}<H<\frac{1}{2}$. Namely, we show a convergence rate of order $n^{\frac{1}{2}-2H}$ for the scheme, and we argue that this rate is exact. We also derive a central limit theorem for the renormalized error of the scheme, thanks to some new techniques for asymptotics of weighted random sums. Our main idea is based on the following observation: the triple of processes obtained by considering the fBm, the scheme process and the normalized error process, can be lifted to a new rough path. In addition, the Hölder norm of this new rough path has an estimate which is independent of the step-size of the scheme.

#### Article information

Source
Ann. Appl. Probab., Volume 29, Number 2 (2019), 758-826.

Dates
Revised: October 2017
First available in Project Euclid: 24 January 2019

https://projecteuclid.org/euclid.aoap/1548298930

Digital Object Identifier
doi:10.1214/17-AAP1374

Mathematical Reviews number (MathSciNet)
MR3910017

Zentralblatt MATH identifier
07047438

#### Citation

Liu, Yanghui; Tindel, Samy. First-order Euler scheme for SDEs driven by fractional Brownian motions: The rough case. Ann. Appl. Probab. 29 (2019), no. 2, 758--826. doi:10.1214/17-AAP1374. https://projecteuclid.org/euclid.aoap/1548298930

#### References

• [1] Bayer, C., Friz, P. K., Riedel, S. and Schoenmakers, J. (2016). From rough path estimates to multilevel Monte Carlo. SIAM J. Numer. Anal. 54 1449–1483.
• [2] Billingsley, P. (2013). Convergence of Probability Measures. Wiley, New York.
• [3] Breuer, P. and Major, P. (1983). Central limit theorems for nonlinear functionals of Gaussian fields. J. Multivariate Anal. 13 425–441.
• [4] Cass, T., Hairer, M., Litterer, C. and Tindel, S. (2015). Smoothness of the density for solutions to Gaussian rough differential equations. Ann. Probab. 43 188–239.
• [5] Cass, T., Litterer, C. and Lyons, T. (2013). Integrability and tail estimates for Gaussian rough differential equations. Ann. Probab. 41 3026–3050.
• [6] Corcuera, J. M., Nualart, D. and Podolskij, M. (2014). Asymptotics of weighted random sums. Commun. Appl. Ind. Math. 6 e-486.
• [7] Davie, A. M. (2007). Differential equations driven by rough paths: An approach via discrete approximation. Appl. Math. Res. Express. AMRX 2 Article ID abm009.
• [8] Deya, A., Neuenkirch, A. and Tindel, S. (2012). A Milstein-type scheme without Lévy area terms for SDEs driven by fractional Brownian motion. Ann. Inst. Henri Poincaré Probab. Stat. 48 518–550.
• [9] Deya, A., Panloup, F. and Tindel, S. (2016). Rate of convergence to equilibrium of fractional driven stochastic differential equations with rough multiplicative noise. Available at arXiv:1605.00880.
• [10] Dieker, T. (2004). Simulation of fractional Brownian motion. Master’s thesis, Univ. Twente.
• [11] Friz, P. and Riedel, S. (2014). Convergence rates for the full Gaussian rough paths. Ann. Inst. Henri Poincaré Probab. Stat. 50 154–194.
• [12] Friz, P. K., Gess, B., Gulisashvili, A. and Riedel, S. (2016). The Jain–Monrad criterion for rough paths and applications to random Fourier series and non-Markovian Hörmander theory. Ann. Probab. 44 684–738.
• [13] Friz, P. K. and Hairer, M. (2014). A Course on Rough Paths: With an Introduction to Regularity Structures. Springer, Cham.
• [14] Friz, P. K. and Victoir, N. B. (2010). Multidimensional Stochastic Processes as Rough Paths: Theory and Applications. Cambridge Studies in Advanced Mathematics 120. Cambridge Univ. Press, Cambridge.
• [15] Garsia, A. M., Rodemich, E. and Rumsey, H. Jr. (1970/1971). A real variable lemma and the continuity of paths of some Gaussian processes. Indiana Univ. Math. J. 20 565–578.
• [16] Giraitis, L. and Surgailis, D. (1985). CLT and other limit theorems for functionals of Gaussian processes. Z. Wahrsch. Verw. Gebiete 70 191–212.
• [17] Gubinelli, M. (2004). Controlling rough paths. J. Funct. Anal. 216 86–140.
• [18] Hu, Y., Liu, Y. and Nualart, D. (2016). Rate of convergence and asymptotic error distribution of Euler approximation schemes for fractional diffusions. Ann. Appl. Probab. 26 1147–1207.
• [19] Hu, Y., Liu, Y. and Nualart, D. (2016). Taylor schemes for rough differential equations and fractional diffusions. Discrete Contin. Dyn. Syst. Ser. B 21 3115–3162.
• [20] Hu, Y., Liu, Y. and Nualart, D. (2016). Crank–Nicolson scheme for stochastic differential equations driven by fractional Brownian motions with $H>\frac{1}{2}$. Available at arXiv:1709.01614.
• [21] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer, Berlin.
• [22] Kurtz, T. G. and Protter, P. (1991). Wong–Zakai corrections, random evolutions, and simulation schemes for SDEs. In Stochastic Analysis 331–346. Academic Press, Boston, MA.
• [23] León, J. and Ludeña, C. (2007). Limits for weighted $p$-variations and likewise functionals of fractional diffusions with drift. Stochastic Process. Appl. 117 271–296.
• [24] Neuenkirch, A., Tindel, S. and Unterberger, J. (2010). Discretizing the fractional Lévy area. Stochastic Process. Appl. 120 223–254.
• [25] Nourdin, I. (2008). Asymptotic behavior of weighted quadratic and cubic variations of fractional Brownian motion. Ann. Probab. 36 2159–2175.
• [26] Nourdin, I. and Nualart, D. (2010). Central limit theorems for multiple Skorokhod integrals. J. Theoret. Probab. 23 39–64.
• [27] Nourdin, I., Nualart, D. and Tudor, C. A. (2010). Central and non-central limit theorems for weighted power variations of fractional Brownian motion. Ann. Inst. Henri Poincaré Probab. Stat. 46 1055–1079.
• [28] Nourdin, I. and Peccati, G. (2008). Weighted power variations of iterated Brownian motion. Electron. J. Probab. 13 1229–1256.
• [29] Nourdin, I. and Peccati, G. (2012). Normal Approximations with Malliavin Calculus: From Stein’s Method to Universality. Cambridge Tracts in Mathematics 192. Cambridge Univ. Press, Cambridge.
• [30] Nourdin, I. and Réveillac, A. (2009). Asymptotic behavior of weighted quadratic variations of fractional Brownian motion: The critical case $H=1/4$. Ann. Probab. 37 2200–2230.
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# 3rd Users' Conference of IT4Innovations

Europe/Prague
atrium (IT4Innovations)

### atrium

#### IT4Innovations

Studentská 1B 708 33 Ostrava - Poruba
Description

3rd Users' Conference of IT4Innovations will take place on 5 and 6 November 2019. All of our users as well as research and project partners from various organisations, research institutions and industry are welcome to attend the Conference.

Registration is open until 30 October 2019.

Attendees will discover more about our future upgrade plans, listen to talks given by our prominent users and can engage in discussions during the Users' Council meeting and poster session.

Contribution types: Users' talks/Posters

Users' talks

Selected talks by our prominent users will be presented at scheduled times during the whole conference. Each talk is expected to take max. 15 mins (with discussion included).

Posters

Please note that the required poster size is A1 portrait orientation

The poster session will take place on 5 November during the lunch and dinner.

Participants
• Alena Ješko
• Alexej Kolcun
• Andreas Erlebach
• Andrzej Piotr Kądzielawa
• Branislav Jansík
• Cemal Köprülüoğlu
• David Wagenknecht
• Dominik Legut
• Dominika Mašlárová
• Dominique Geffroy
• Ekaterina Grakova
• Gabriel Bordovský
• Hana Dohnalová
• Ilia Ponomarev
• Ivana Paidarova
• Jakub Beránek
• Jakub Smutek
• Jakub Šafařík
• Jan Heyda
• Jan Křenek
• Jan Martinovič
• Jan Zapletal
• Jan Řezáč
• Jana Pavlíková Přecechtělová
• Jana Pavlů
• Jiri Jaros
• Jiri Klimes
• Kateřina Janurová
• Kateřina Slaninová
• Krystof Brezina
• Kývala Lukáš
• Libuše Horáčková
• Lubomír Rulíšek
• Lubomír Říha
• Lukas Krupcik
• Lukas Maly
• Marek Chrastina
• Marek Lampart
• Marta Jaros
• Martin Beseda
• Martin Bouda
• Martin Culka
• Martin Friák
• Martin Hasal
• Martin Matys
• Martin Matys
• Martin Melčák
• Martin Mokrejš
• Martin Šviček
• Michael Owen
• Michal H. Kolar
• Michal Kravčenko
• Michal Merta
• Milan Lazecky
• Miroslav Rubes
• Monika Všianská
• Ondrej Vysocky
• Ondřej Fikar
• Ondřej Jakl
• Pablo Nieves Cordones
• Paolo Nicolini
• Peter Arbenz
• Petr Kulhanek
• Petr Pexa
• Petr Strakos
• Petr Řehák
• Petr Šesták
• Saltuk Mustafa Eyrilmez
• shihao zhang
• Stepan Sklenak
• Tomáš Karásek
• Tomáš Kozubek
• Tomáš Martinovič
• Tugba Dogan
• Vaclav Skala
• Victor Claerbout
• Victor Montagud-Camps
• Vit Ptosek
• Václav Svatoň
• Vít Vondrák
• Zuzana Červenková
Support
• Tuesday, 5 November
• 10:00 11:00
Registration and networking
• 11:00 11:30
Welcome and IT4Innovations news - Branislav Jansík, IT4Innovations
• 11:30 12:00
Keynote I
• 11:30
2D magnetic semiconductors - a power of HPC to predict 30m

For the spintronic applications like large data storages (high capacity HDD) the
industry search for ferromagnetic insulators and at the same time a very thin materials. Recently the discovery of Bi$_2$O$_2$Se/Te phases could exist as 2D material and still be semiconducting. Here we investigate these novel layer-structured materials with high electron mobility, while its efficiency could be greatly improved by doping different elements to introduce a magnetic spin order utilizing quite demanding quantum-mechanical calculations. We explore the electronic and magnetic properties of various ferromagnetic (e.g. Fe) or antiferromagnetic (e.g. Mn) transitional metals doped Bi$_2$O$_2$Se/Te phases within the framework of density functional theory based electronic structure calculations. We start with the magnetic order of the bulk phase in which the magnetic atoms form interlayer coupling that vary with the type and concentration of doped atoms and go towards the nanoscale dimension, i.e. 2D materials. As a result of the competitions of magnetic interactions the magnetic anisotropy energy is a crucial quantity leading to a potential spintronic material at the nano-scale dimensions. In combinations with Monte Carlo simulations we are able to solve the exchange interaction constants for the Heisenberg model and therefore evaluate another important quantity, the transition (Curie temperature) among the order and paramagnetic state of Bi$_{2-n}$X_n$O$_2$Se/Te to see if these types of materials are suitable to become novel dilute magnetic semiconductors for spintronic applications at room and above temperatures. Speaker: Dominik Legut (IT4I) • 12:00 13:00 Users Talks I - Chairman: Branislav Jansík - IT4Innovations • 12:00 Effects of liquid water on the stability and reactivity of zeolites 15m Zeolites are traditionally used in high-temperature applications at low water vapor pressures, such as heterogeneous catalysis and gas separation, where the zeolite framework is generally considered to be stable and static. Increasingly, zeolites are being considered for applications under milder aqueous conditions, in emerging fields such as biomass conversion, low-temperature oxidation catalysis, and medicine. However, it has not yet been established how neutral liquid water at mild conditions affects the stability and reactivity of the zeolite framework at the atomistic level. Therefore, we carried out (biased) ab initio molecular dynamics (MD) calculations that predict novel, energetically viable reaction mechanisms by which Al-O, Si-O, Ge-O bonds rapidly and reversibly break [1]. Even at ambient conditions, zeolites show significant, fast lability of their bonds when in contact with liquid water. In addition, solvation of the extra-framework cations compensating the negative charge of the framework are solvated in the nanoporous channel system, influencing the catalytic activity of the material [2]. However, ab initio MD simulations are computationally too demanding for the statistically significant sampling of reaction mechanisms of larger systems at several different synthesis/operating conditions. While available empirical force fields allow such large scale sampling of the configuration space, their accuracy is too low to provide a reliable atomistic description of the water-zeolite interface. Therefore, we develop a computational tool providing the best compromise of accuracy and computational costs by combining state-of-the-art artificial neural networks and ab initio (MD) simulations. Such neural network potentials (NNP) are capable of modeling interatomic interactions with considerably reduced computational effort and retaining the accuracy of first-principles calculations [3]. Consequently, atomistic simulations employing NNP are expected to provide a deeper understanding of zeolite hydrolysis mechanisms at realistic conditions relevant for, e.g., the targeted synthesis of novel zeolites via the ADOR process [4]. [1] C. J. Heard, L. Grajciar, C. M. Rice, S. M. Pugh, P. Nachtigall, S. Ashbrook, and R. E. Morris, “Fast room temperature lability of aluminosilicate zeolites”, Nat. Comm., 2019, under review. [2] C. J. Heard, L. Grajciar, and P. Nachtigall, “The effect of water on the validity of Lowenstein’s rule”, Chem. Sci., 2019, DOI: 10.1039/C9SC00725C. [3] J. Behler, “First Principles Neural Network Potentials for Reactive Simulations of Large Molecular and Condensed Systems”, Angew. Chem. Int. Ed., 2017, 56, 12828-12840. [4] W. J. Roth, P. Nachtigall, R. E. Morris, P. S. Wheatley, V. R. Seymour, S. E. Ashbrook, P. Chlubná, L. Grajciar, M. Položij, A. Zukal, O. Shvets, and J. Cejka, “A family of zeolites with controlled pore size prepared using a top-down method.”, Nat. Chem., 2013, 5, 628–633. Speaker: Mr Andreas Erlebach (Department of Physical and Macromolecular Chemistry, Charles University, Prague) • 12:15 HPC aspects of microscale urban climate modelling 15m PALM is an open-source large-eddy simulation model with strong focus on parallelization and HPC computing. The members of the atmospheric modelling group at the Institute of Computer Science (ICS) of the Czech Academy of Sciences successfully extended this model with the Urban Surface Module (USM) and Radiative Transfer Module (RTM), which simulate processes necessary for microscale urban climate modelling (3-D multi-reflective radiation, surface energy balance, 3-D resolved vegetation, basic air quality modelling and others). The extended model has been verified in a pilot study [1]. The PALM-4U [2] is a new complex urban climate modelling system developed by multiple research institutions from Germany, Czech Republic and other European countries, which is built on PALM-USM for urban layer radiation and surface energy processing. The ICS team is a core member of the PALM-4U development team. Modelling multi-reflective radiation within fully resolved 3-D terrain, buildings and vegetation is a complex task which brings many issues that are not normally encountered in atmospheric modelling. One of the crucial problems is the fact that in a highly parallelized atmospheric model, each process simulates only a small part (subdomain) of the whole modelled area, it has no direct access to values outside it and a typical data exchange pattern involves only neighbouring subdomains. With 3-D radiation, there are direct interactions among arbitrarily distant surfaces and vegetation. Another significant problem presents determining mutual visibility between individual surface and plant canopy elements. The Radiation Transfer Model (RTM) within PALM-4U uses specially designed raytracing algorithm to establish information about mutual visibility. This differs significantly from a typical raytracing algorithm used in computer graphics due to different data structures and parallelism used in atmospheric modelling. The raytracing process, although highly optimized, is computationally demanding and data-transfer intensive, therefore it is performed as part of model initialization and its results are stored in the form of view factors. During model time-stepping, the precalcuated view factors are used to calculate the actual radiative processes with much smaller demands on CPU time, memory size and data transfers. We will present the basic principles of the RTM and its implementation and demonstrate its performance, efficiency of parallelization and its scalability on dedicated performance testing simulations. Speaker: Dr Pavel Krc (Institute of Computer Science, CAS) • 12:30 Benzene radical anion in the context of the Birch reduction: when solvation is the key 15m The symmetry of the molecular structure of the benzene radical anion ($C_6H_6^{\bullet -}$) leads to a rich and intriguing palette of quantum behavior manifestations. Moreover, this simplest aromatic anion possesses a great importance in organic chemistry. In particular, one of the most well-known appearances in this field is as the first reactive intermediate in the Birch reduction used to reduce benzene to 1,4-cyclohexadiene using the blue solution of alkali metals in liquid ammonia. In this work, we perform and analyze ab initio molecular dynamics (AIMD) simulations of the benzene radical anion in liquid ammonia to shed light on its condensed phase behavior never explored before. As the title of this contribution suggests, this solvation is crucial for the existence and stability of the anion. In fact, it was demonstrated in previous works that in the gas phase (i.e., without solvent) the excess electron of the radical anion is unbound. However, it was suggested and, in this work, we show that as soon as it is submerged in a solvent such as liquid ammonia, it becomes a true bound state. In the context of this phenomenon, we discuss the methodological aspects necessary to capture a bound species in the simulation: we conclude that an expensive hybrid DFT electronic structure calculation is needed to obtain a "well-behaved" radical anion. With these computational settings, we obtained 100 ps of production dynamics of the radical anion and a neutral benzene reference in liquid ammonia. Based on these trajectories, we are able to discuss structural features of the radical anion itself such as the dynamic Jahn-Teller distortions and the structure of the surrounding solvent including the actively discussed phenomenon of$\pi$-hydrogen bonding. We show that benzene and its anion both form a characteristic hydrogen bond with the solvent which possibly correlates with macroscopic effects such as the elevated benzene solubility in protic solvents such as ammonia and further extending into liquid water. Speaker: Krystof Brezina (Faculty of Mathematics and Physics, Charles University) • 12:45 Modelling of lattice defects in advanced materials 15m These days, the effective development of new materials is a priority of research and industry in modern society. However, the basic description of material structure is insufficient to achieve detailed understanding of behaviour of many advanced materials. Here the studies of structure defects become more and more important. At this level of research, the theoretical modelling, such as ab initio calculations, serves as very effective approach as it provides information which is experimentally inaccessible. In our group, we use the VASP code [1–3] with projector-augmented plane wave (PAW) potentials [4, 5] which, in combination with the IT4I computational resources, forms very effective tool for material research. In our presentation, we will present several case studies of lattice defects including the disorder in sublattices in Fe-Al-Co system and grain boundaries in fcc Ni containing vacancies and segregated impurities. We will show typical problems and challenges we face (optimization of cell size and shape, appropriate choice of defect position or correct choice of structure relaxation method) and solutions we use. Also, some typical material characteristics that can be obtained by our calculations will be presented, such as heats of formation, equilibrium structure configurations, segregation energies of impurities at grain boundaries or energies of vacancy formation. Acknowledgement This research was supported by the Grant Agency of the Czech Republic (Project No. GA16-24711S, GA17-22139S), by the Ministry of Education, Youth and Sports of the Czech Republic (Project CEITEC 2020 (LQ1601)) and by the Academy of Sciences of the Czech Republic (Institutional Project No. RVO:68081723). Additional resources were provided by the Academy of Sciences of the Czech Republic through the Fellowship of J. E. Purkyně (M.F.). Computational resources were provided by the Ministry of Education, Youth and Sports of the Czech Republic under the Projects CESNET (Project No. LM2015042), CERIT-Scientific Cloud (Project No. LM2015085) and IT4Innovations National Supercomputing Center (Project No. LM2015070). References [1] Kresse G, Hafner J. Phys Rev B 1993;47:558; ibid. 1994;49:14251. [2] Kresse G, Furthmüller J. Phys Rev B 1996;54:11169. [3] Kresse G, Furthmüller J. Comput Mater Sci 1996;6:15. [4] Blöchl PE. Phys Rev B 1994;50:17953. [5] Kresse G, Joubert D. Phys Rev B 1999;59:1758. Speaker: Jana Pavlů (1 Department of Chemistry, Faculty of Science, Masaryk University, Kotlářská 2, CZ-611 37 Brno, Czech Republic; 2 CEITEC - Central European Institute of Technology, Masaryk University, Kamenice 753/5, CZ-625 00 Brno, Czech Republic; 3 Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Žižkova 22, CZ-616 62 Brno, Czech Republic) • 13:00 14:00 Lunch: Lunch & Poster Session • 14:00 14:45 Users' Talks II - Chairman Paolo Nicolini - Czech Technical University in Prague • 14:00 Theoretical study of effect of segregated impurities on the structure and properties of grain boundaries in Ni3Fe 15m Introduction. Grain boundaries (GB) represent an important class of two-dimensional extended defects and macroscopic strength of polycrystalline materials depends strongly on GB cohesion. It was found that the impurities in ppm concentration can drastically change material properties. For example, in magnetic materials they can significantly change magnetic moments in the GB region. Intergranular embrittlement which is usually associated with segregation of impurities on the GB can result in a dramatic reduction of the ductility and strength. Here we present an ab initio study of$\Sigma$5(210) grain boundary in Ni$_3$Fe compound with two different interface stoichiometries,$\Sigma5(210)^{\mathrm{Fe,Ni}}$with both Fe and Ni atoms at the GB and$\Sigma5(210)^{\mathrm{Ni,Ni}}$with Ni atoms only. We consider both clean GB and the GB with segregated Al and Si. Summary. Our calculations show that the$\Sigma5(210)^{\mathrm{Fe,Ni}}$has the GB energy of 1.34 J/m$^2$and an additional volume per unit GB area of 0.50$\unicode{xC5}$while the$\Sigma5(210)^{\mathrm{Ni,Ni}}$exhibits a little bit higher GB energy of 1.43 J/m$^2$and an additional volume per unit GB area of 0.51$\unicode{xC5}$. Here we are not able to decide which interface stoichiometry,$\Sigma5(210)^{\mathrm{Fe,Ni}}$or$\Sigma5(210)^{\mathrm{Ni,Ni}}$, is more stable. The energy difference between them is only 4.5 meV/atom in favor of the$\Sigma5(210)^{\mathrm{Fe,Ni}}$configuration. Whereas magnetic moments of Ni atoms at the clean GBs are slightly decreased with respect to the bulk (less than by 0.1$\mu_{\mathrm{B}}$), there is a slight enhancement (by 0.11$\mu_{\mathrm{B}}$) of magnetic moment of Fe atoms at the clean$\Sigma5(210)^{\mathrm{Fe,Ni}}$GB. On the other hand, in the neighborhood of$\Sigma5(210)^{\mathrm{Ni,Ni}}$GB the magnetic moments of Fe atoms are decreased by about 0.18$\mu_{\mathrm{B}}$. Segregated Al and Si impurities reduce the magnetic moments of Fe and Ni atoms in the region of GB up to 0.4$\mu_{\mathrm{B}}$with respect to the bulk. Acknowledgement. This research has been financially supported by the Ministry of Education, Youth and Sports of the Czech Republic under the Project CEITEC 2020 (Project No. LQ1601), by the Czech Science Foundation (Project No. GA16-24711S) and by the Academy of Sciences of the Czech Republic (Institutional Project No. RVO:68081723). Computational resources were provided by the Ministry of Education, Youth and Sports of the Czech Republic under the Project IT4Innovations National Supercomputer Center (Project No. LM2015070) within the program Projects of Large Research, Development and Innovations Infrastructures. Speaker: Monika Všianská (Masaryk University, Faculty of Science, Department of Chemistry) • 14:15 Accurate calculation of the binding strength of methane clathrate 15m Methane clathrates are solid materials ocurring in the depths of oceans. Their structure composes of water cage-like framework which surrounds methane molecules. One of the basic properties which decides the stability of the structure is the strength of the binding between methane molecules and the water framework. Interestingly, this property is very difficult to calculate accurately using modelling methods based on quantum mechanics [1,2]. We have used two clathrate models to understand the origin of the difficulties and to assess the quality of promising quantum mechanical methods. The first structure is a single cage formed by twenty molecules and a single methane molecule [1], second is a solid clathrate [2]. I will discuss some of the interesting observations that we made while studying these systems. For example, decomposing the binding energy of the single cage into contributions of dimers and trimers of molecules clearly showed that the reference binding energy is not very precise and we were able to improve it. Moreover, I will discuss the transferability of the observations made for the cluster to the bulk clathrate. [1] M. J. Deible, O. Tuguldur, K. D. Jordan: J. Phys. Chem. B 118, 8257 (2014) [2] S. J. Cox, M. D. Towler, D. Alfe, A. Michaelides: J. Chem. Phys. 140, 174703 (2014) Speaker: Jiri Klimes (Faculty of Mathematics and Physics, Charles University) • 14:30 Alloy analogy model: Finite-temperature description of magnetic solids 15m For a purpose of novel electronic and spintronic applications, it is desired to describe realistic behavior of magnetic materials from first principles. Ab inito approaches can save a lot of experimental expenses; however, it is extremely complicated to compute material properties influenced by real-life phenomena. For a design of new devices, treatment of chemical impurities and temperature-induced disorder (phonons and magnons) is essential because, e.g., computers should be reliable not only at temperature of$T=0$K. We will present implementation of the alloy analogy model (AAM) within the tight-binding linear muffin tin orbital method and the coherent potential approximation, which was successfully used to described electrical transport of transition metals and random alloys [1, 2]. Within this formalism, effects of finite temperature and their combination with impurities can be treated. The technique is both robust and numerical effective; therefore, it can be employed also for complex multi-sublattice materials. Especially recent comprehensive study of half-Heusler NiMnSb [3, 4] will be shown with a focus on spintronic applications and experimentally hardly-accessible quantities such as spin polarization of the electrical current$P$. For example, influences of atomic vibrations, spin fluctuations, and alloying on electrical transport and the polarization$P$can be separated in our calculations in order to show that the spin disorder has the highest influence on$P$, which is supposed to be above$90 \%\$ at ambient conditions (room temperature, realistic impurities) anyway. Last but not least, perfect agreement with experimental literature will be presented and aspects of the AAM for the high-performance computing discussed.

[1] D. Wagenknecht et al. IEEE 53, 11, 1700205, (2017)
[2] D. Wagenknecht et al. Proc. SPIE 10357, Spintronics X, 103572W (2017)
[3] D. Wagenknecht et al. JMMM 474, 517 (2019)
[4] D. Wagenknecht et al. PRB 99, 174433 (2019)

Speaker: Dr David Wagenknecht (Department of Condensed Matter Physics, Faculty of Mathematics and Physics, Charles University)
• 14:45 15:30
Users' Council
• 15:30 16:00
Coffee break
• 16:00 16:30
Keynote II
• 16:00
Ultrasound Simulation for Transcranial Neurostimulation 30m

Transcranial transmission of ultrasound is increasingly used in a variety of clinical and research applications, including high intensity ablation, opening the blood brain barrier, and neural stimulation. Large scale numerical simulations of ultrasound propagation in the head are used to enable effective transcranial focusing and predict intracranial fields.
This talk will present the k-Stim tool chain covering the complete process from the preoperative patient CT/MR screening to the application the ultrasound in a custom 3D printed ultrasound device. After the screening, the preoperative treatment planning continues in k-Plan, a graphical user interface where the target area and ultrasound transducer parameters are defined by a clinician. The treatment plan is then submit for validation using the HPC services via an in-house k-Dispatch server. k-Dispatch manages several HPC facilities, optimizes the execution configuration, monitors the execution and provide accounting services. The k-Wave toolbox is then used to execute the simulation using one of available implementations optimized for processors, accelerators or GPUs. Once the plan has been validated, it is sent back to the k-Dispatch and then delivered to the clinician for approval. If approved, the treatment plans parameters are extracted and uploaded into the operating device. Finally, the treatment is applied in a MR guided environment.

Speaker: Jiri Jaros (Brno University of Technology)
• 16:30 18:00
Users' Talks III - Chairman Tomas Kozubek - IT4Innovations
• 16:30
Propene adsorption in alkali metal-exchanged FER zeolite 15m

Adsorption of propene on Li-, Na- and K-FER zeolites was investigated using a combination of IR spectroscopy, calorimetric measurements of adsorption heats and DFT calculations employing DFT/CC scheme based on PBE density functional. On the basis of good agreement between experimental and theoretical results, following adsorption complexes of propene in investigated M-FER zeolites can be distinguished: (i) propene interacting with zeolitic framework via dispersion interactions populated mainly in the zeolites with high Si/Al ratio and characterized by νC=C vibrational band at 1646 cm-1 and adsorption heat around 48 kJ/mol, (ii) propene interacting with cations coordinated in 6- rings characterized by the IR band at 1637 cm-1 (Li-FER) and 1636 cm-1 (Na-FER), (iii) propene adsorbed on remaining cationic positions excluding cationic positions in 6-rings characterized by IR band at 1630 cm-1 (Li-FER), 1633 cm-1 (Na-FER) and 1639 cm-1 (K-FER) (iv) propene bridging two nearby sodium or potassium cations in dual-cation sites characterized by vibrational band at 1626 cm-1 (Na-FER) and 1633 cm-1 (K-FER). Population of bridged complexes in Na-FER was significantly lower compared with K-FER zeolites due to preference of potassium cations for 8-rings, which is more suitable for creation of dual-cation sites then 6-rings where sodium cations are preferentially coordinated. No bridged complexes were found in the case of Li-FER due to the fact that Li+ is too close to the framework oxygen atoms and thus relatively long distance from each other.

Speaker: Miroslav Rubes (IOCB, AS, CR)
• 16:45
IT4I Satellite SAR Interferometry Monitoring System 15m

The partially autonomous satellite SAR interferometry (InSAR) system is being developed at IT4Innovations ADAS since 2016. It is based on open-source tools such as ISCE (NASA JPL), STAMPS (Stanford University), LiCSInfo (University of Leeds) or SALSIT (insar.cz). The processing package shared as IT4S1 offers a specific preprocessing of all Sentinel-1 data over the Czech Republic, stored at CESNET Czech Copernicus Collaborative Segment. It allows for InSAR time series processing over Czech areas with a high processing performance: ~5 km area is processed on-demand typically in less than a half-hour at the HPC. The whole map of displacements over Czech Republic based on Persistent Scatterers (PS) InSAR technique has been generated in 2018. PS InSAR technique allows to identify infrastructure displacements or terrain deformation of coherent areas in the sensitivity of few mm/year. In this contribution, an updated map is to be introduced and few interesting areas of IT4S1 application are to be presented - topics such as mining-induced subsidence, infrastructure displacements, landslide detection but also ongoing approach to detect hurricane-induced forest damage in Czech Republic will be presented.

Speaker: Milan Lazecky (IT4Innovations)
• 17:00
Application of quantum-mechanical computational methods to computer-aided drug design 15m

The development of new drugs and the study of their mechanisms of action can benefit from computational modelling. The need to use large model systems, combined with virtual screening of databases of compounds, severely limits the complexity of the computational methods applicable. The field is thus dominated by either statistical ("knowledge based") methods or molecular mechanics.

We have pioneered the application of semiempirical quantum-mechanical methods to this field. Besides higher accuracy, these methods have also one important practical advantage - they are based on fundamental physical principles so they do not need any system-specific parameterization. They are computationally more demanding, but with the recent developments in the algorithms, and with increasingly larger resources available, these methods can now be applied to many problems in this field.

I have already presented some results of this project, and this time I will focus on the methodology. We are developing our own corrections to the semiempirical QM methods, without which it will not be possible to achieve the required accuracy. This development starts with very accurate quantum-mechanical calculations to be used as a benchmark for parametrization and testing of the approximate methods.

Speaker: Prof. Jan Řezáč (UOCHB)
• 17:15
Sparse Quantum Monte Carlo sampling for two-particle Green’s functions 15m

We present a dynamical mean-field study of dynamical susceptibility in the two-orbital Hubbard model across the exciton condensation transition. We observe the appearance of Goldstone modes consistent with the broken symmetries. Reducing continuously the symmetry of the Hamiltonian we open gaps in the Goldstone modes. Upon increasing the amplitude of the symmetry breaking terms the character of the gapped modes smoothly changes from gapped Goldstone modes to Higgs amplitude fluctuations. Another notable observation is a qualitative change of the dynamical spin structure factor upon exciton condensation, which may be used as an experimentally accessible probe of the spin-triplet exciton condensate.

Speaker: Dr Dominique Geffroy (Masaryk University, Brno)
• 17:30
Periodic DFT study of siting of divalent cations in silicon-rich zeolites 15m

Zeolites are crystalline microporous aluminosilicates widely used as molecular sieves and catalysts in industrial chemical processes. Silicon-rich zeolites (Si/Al > 8), such as ZSM-5, mordenite, ferrierite, and beta zeolite, represent catalytic materials with wide industrial applications due to their high stability and unique properties of cationic species in these matrices. The positively charged extra-framework divalent cations, which are the active sites in catalysis, balance the negative charge of two framework AlO4- tetrahedra located in 6-rings and 8-rings in dehydrated zeolite matrixes. The determination of the local structure of divalent cations is of great importance because the local structure affects the catalytic activity and selectivity. In the case of silicon-rich zeolites, the application of diffraction methods for the analysis of the siting of extra-framework cationic species in zeolite frameworks is significantly limited. In this contribution, we will present our results regarding the siting of Co(II),1-5 Fe(II),5-6 and Cu(II)1 in the zeolites of the FER,1, 5-6 BEA,2 TNU-9,3 and SSZ-13.4 Furthermore, the results of the investigation of the formation of the very reactive Alpha oxygen on Co(II) and Fe(II) cations,5 which is able to selectively oxidize methane to methanol at room temperature, will be presented.

Reference

1. Sklenak, S.; Andrikopoulos, P. C.; Whittleton, S. R.; Jirglova, H.; Sazama, P.; Benco, L.; Bucko, T.; Hafner, J.; Sobalik, Z. J. Phys. Chem. C 2013, 117, 3958-3968.
2. Sazama, P.; Tabor, E.; Klein, P.; Wichterlova, B.; Sklenak, S.; Mokrzycki, L.; Pashkkova, V.; Ogura, M.; Dedecek, J. J. Catal. 2016, 333, 102-114.
3. Karcz, R.; Dedecek, J.; Supronowicz, B.; Thomas, H. M.; Klein, P.; Tabor, E.; Sazama, P.; Pashkova, V.; Sklenak, S. Chem. Eur. J. 2017, 23, 8857-8870.
4. Mlekodaj, K.; Dedecek, J.; Pashkova, V.; Tabor, E.; Klein, P.; Urbanova, M.; Karcz, R.; Sazama, P.; Whittleton, S. R.; Thomas, H. M.; Fishchuk, A. V.; Sklenak, S. J. Phys. Chem. C 2019, 123, 7968-7987.
5. Tabor, E.; Lemishka, M.; Sobalik, Z.; Mlekodaj, K.; Andrikopoulos, P. C.; Dedecek, J.; Sklenak, S. Commun. Chem. 2019, 2.
6. Sklenak, S.; Andrikopoulos, P. C.; Boekfa, B.; Jansang, B.; Novakova, J.; Benco, L.; Bucko, T.; Hafner, J.; Dedecek, J.; Sobalik, Z. J. Catal. 2010, 272, 262-274.
Speaker: Dr Stepan Sklenak (J. Heyrovsky Institute of Physical Chemistry)
• 17:45
Reaction Mechanism of Non-Heme Diiron Δ9-Desaturase: Why Does it Desaturate and Not Hydroxylate? 15m

Mono- and binuclear non-heme iron sites in proteins serve as efficient catalysts of a broad set of oxidation reactions, including activation of unreactive C–H bonds of organic substrates for subsequent desaturation, hydroxylation, halogenation, peroxidation, etc. A prominent example of such class of enzymes is the soluble O2-dependent binuclear non-heme iron enzyme Δ9-desaturase (Δ9D), evolved for the conversion of stearoyl acid to oleic acid as a part of the fatty-acid metabolic pathway of plants. Herein, we build on a detailed investigation of initial stages of O2 activation in the Δ9D active site reported earlier by our group[1,2] to probe several reaction pathways to complete the catalytic cycle, employing the hybrid quantum mechanics/molecular mechanics (QM/MM) calculations. As a result, the activation of P intermediate via electron-proton transfer (Figure 1) is the most likely mechanism with respect to C10‒H and C9‒H bonds cleavages. At last, the energetics of key reaction steps were carefully examined at the level of advanced multi-configurational methods, allowing for the clarification of Δ9D selectivity toward desaturation (at the expense of thermodynamically more favored hydroxylation).

Speaker: Dr Lubomir Rulisek
• 18:00 21:00
Conference Dinner & Poster Session
• Wednesday, 6 November
• 09:00 09:30
Keynote 3: Keynote III
• 09:00
Double layer target with interface modulations for laser acceleration of collimated ion beams 30m

With the advent of multi-petawatt laser systems like the ELI-Beamlines (Czech Republic),
APOLLON (France) and SEL (China) the laser-driven ion accelerators will enter the acceleration regimes dominated by radiation pressure [1]. High quality ion beams with low emittance and narrow energy spectrum will be generated when these lasers irradiate tailored targets. Below we present the results of studying the effects of the interface modulations in double layer targets. The numerical particle-in-cell simulations with the code EPOCH [2] are used. We show that the pre-modulated targets can undergo relativistic Rayleigh-Taylor [3] and Richtmyer-Meshkov instabilities. Their use can improve the properties of generated ion beams [4].
It is shown that small perturbations originated from the interface modulation grow during the laser-target interaction. This leads to the formation of low-density regions and high-density ion bunches between them at the positions determined by the pre-modulation geometry. The ion bunches are then accelerated by the laser radiation pressure. The collimated central bunch of proton beam has the average energy in the multi-GeV range with narrow energy spread [5]. The laser accelerated ion beams from composite targets will find applications in nuclear physics research [6].

Our results were visualized in collaboration with our colleagues from the Virtual Beamline (VBL) team [7] at ELI-Beamlines.

Our work is supported by projects High Field Initiative (CZ.02.1.01/0.0/0.0/15 003/0000449) and Extreme Light Infrastructure Tools for Advanced Simulation (CZ.02.1.01/0.0/0.0/16_013/0001793) from the European Regional Development Fund and by Czech Science Foundation (18-09560S).

References:
[1] T. Esirkepov, M. Borghesi, S. V. Bulanov et al., Phys. Rev. Lett. 92, 175003 (2004)
[2] T. D. Arber, K. Bennett, C. S. Brady et al., Plasma Phys. Control. Fusion 57, 113001 (2015)
[3] F. Pegoraro and S. V. Bulanov, Phys. Rev. Lett. 99, 065002 (2007)
[4] S. V. Bulanov, E. Y. Echkina, T. Z. Esirkepov et al, Phys. Rev. Lett. 104, 135003 (2010)
[5] M. Matys, K. Nishihara, M. Danielova et al., Proc. SPIE 11037, 110370Z (2019)
[6] M. Nishiuchi, H. Sakaki, T. Z. Esirkepov et al, Plasma Phys. Rep. 42, 327 (2016)
[7] https://www.eli-beams.eu/en/facility/computing-simulations/virtual-beamline/

Speaker: Mr Martin Matys (ELI Beamlines / CTU in Prague)
• 09:30 10:15
Users Talks IV - Chairman Jiří Jaroš - Brno University of Technology
• 09:30
Toward ab initio protein folding 15m

Computer modeling has been quite successful in predicting protein fold from amino acid sequence using similarity patterns in the growing amount of data from structural biology experiments. Moreover, the dynamics of the folding process itself has been simulated in the case of small proteins. The energy landscape of complex protein systems has been usually described by empirical physical models (force fields). In contrast, more accurate quantum-chemical (QM) models have been extensively used to study chemistry in the truncated active sites of the enzymes. In would be desirable to use the accurate QM methods to study protein folding, however, such models are still computationally very demanding. In our recent study, we came with an idea to divide the protein into capped tripeptides, which represent a reasonable model of an amino acid in its local environment that can be used for extensive conformational sampling and QM free energy calculations. When we compare conformer ensembles of tripeptide sequences that are found predominantly in one type of secondary structure (alpha-helix or beta-sheet), we see an energetic preference to form hydrogen bonds typical for the respective secondary structure type. This propensity is further modulated by the polarity of the environment. In addition, we have shown that tripeptide conformers extracted from experimental structures must be stabilized by significant amount of energy coming from the protein context. When an empirical force field is used for energy evaluation, the outcome is far less conclusive. These results led us to the central idea of the present study – can we identify the folding initiation sites by dividing the protein of interest into tripeptides and calculating their relative free energy within the unbiased conformer ensemble? We test this idea on well-studied stable miniprotein folds. In line with existing experimental and theoretical studies of protein folding mechanisms we see that the tripeptides with the lowest energy relative to the most stable conformer of the respective isolated tripeptide lie within alpha-helices and in beta-turns. This implies that the protein folding is initiated either by alpha-helix formation, or by aligning two extended segments connected by the beta-turn toward beta-sheet formation. The tripeptides within the beta-sheets show the highest interaction energy with the rest of the protein. Furthermore, we observe that these highly strained but highlt interaction regions are those conserved by evolution. Our results indicate that the evolution has conserved the residues contributing for overall thermodynamic stability at the cost of lower local stability. In contrast, the segments connecting these crucial regions were evolutionary optimized to facilitate the folding dynamics.

Speaker: Dr Martin Culka (Institute of Organic Chemistry and Biochemistry, Czech Academy of Sciences, Prague)
• 09:45
Helium-based plasma interactions with air molecules 15m

Modeling of cold rare-gas plasmas is currently one of the popular topics in the field of quantum chemistry because of their possible applications in many other fields, e.g. surface treatment, food industry or plasma medicine, with the last being the most interesting to our team. It was shown previously that rare-gas plasmas are well-working in such applications, which resulted in broad research in this field. To understand the healing properties of cold rare-gas plasmas, detailed knowledge of processes on the microscopic level is of crucial importance. This talk will be focused on interactions between rare-gas ions and air molecules with the main focus on the simplest rare-gas (He) and the most abundant air molecule (N2). However, both the ab initio computations of potential energy surfaces and the semiclassical nonadiabatic molecular dynamics are very computationally demanding and, in some cases, even the modern computation methods show themselves unstable. The one way to overcome these obstacles is the utilization of machine learning methods, especially artificial neural networks (ANNs). Thus, this talk is going to cover our work both in ab initio computations of potential energy surfaces and their ANN representations.

Speaker: Martin Beseda (IT4Innovations, VSB-TUO)
• 10:00
How Do Atomistic Simulations of Huge Biomolecular Complexes Help Us Understand Protein Birth and Death? 15m

Life is defined by proteins. These key biomolecules have countless functions – from the structural, through signaling to catalytic. A vast majority of proteins in living organisms is synthesized in a biomolecular complex called ribosome. In higher organisms, most of the unneeded proteins are then degraded in other biomolecular complex called proteasome. The size of both complexes is about a few tens of nanometres. Thus, they are tiny according to our daily experience, but huge for atomistic computer simulations.

![Two biomolecular complexes responsible for protein synthesis and degradation.][https://pbs.twimg.com/media/EBs74IBX4AEzxWG?format=jpg&name=large]

Here, I will briefly introduce two projects, which have run in the past four years, using resources of German and Czech supercomputing centres (HLRS, LRZ, IT4I). The projects involve massive all-atom molecular dynamics simulations of up to 2 million particles. The audience will learn what may happen with the nascent protein before it exits the ribosome, i.e. before it is finally born, and how this relates to antibiotics. On the other hand, handling protein death in proteasome is a promising anti-cancer strategy, and I will explain how a small drug binding affects the choreography of the proteasome dynamics.

Speaker: Michal H. Kolar (University of Chemistry and Technology)
• 10:15 10:30
Coffee Break 15m
• 10:30 11:30
Users` Talk V - Chairman Martin Matys - Czech Technical University in Prague / Czech Academy of Sciences
• 10:30
Continuum and atomistic assessment of crack stability in nanocomponents 15m

At large length scales any material behaviour is described using continuum approaches. Nowadays, when nanotechnology is already capable of producing new materials and devices on submicrometer scale, it has become indispensable to understand possible mechanical instabilities such as nanocrack initiation and the underlying physics at this scale. Brittle fracture through cleavage of crystal lattice is the simplest and most typical fracture associated with bond breaking. However, at the nanoscale, continuum methods based on linear elasticity start to fail in description of the lattice distortions and in predictions of its stability. Recently developed strain gradient elasticity theory (SGET) can help us to overcome this problem. In this work, we combined SGET with atomistic calculations in a model for prediction of crack stability and crack shape in selected brittle materials. This model has already been successfully tested for a number of crystals (such as Si and W [1]).

[1] M. Kotoul et al.; Mechanics of Materials 136 (2019) 103074

Speaker: Dr Petr Šesták (Central European Institute of Technology, Brno University of Technology)
• 10:45
Thermal conductivity of nuclear fuels from first principles 15m

Thermal conductivity is one of the most important property of nuclear fuel materials affecting many processes such as swelling, grain growth, and fission gas release and limits the transfer of the linear power [1].

However, the measurement of thermal conductivity is difficult even for simple elements like silicon. Loss of heat through convection, conduction, and radiation coexist at the same time. Measuring radioactive elements is even more difficult because of the sample self-heating. This makes the theoretical calculations appreciated and a valuable tool.

Here we demonstrate to determine not only the total conductivity but their electronic and phononic (lattice vibrations) contributions as a function of temperature just based on quantum-mechanical calculations. The forces on atoms to the third order are accurately calculated and derived results are presented for the thorium metal and thorium monocarbide. In addition, the analysis which vibration modes transfer most of the heat here was performed.

[1] T. R. G. Kutty, J. Banerjee, and A. Kumar, in Thoria-based Nuclear Fuels (Springer London, 2013) pp. 11–70.

Speaker: Mr Lukáš Kývala (IT4Innovation)
• 11:00
Nanoscale frictional properties of ordered and disordered molybdenum disulfide 15m

One third of energy produced by industrial countries is lost as friction. High wear caused by friction means approx. 35% of industrial production is used to replace degraded products, whilst causing the breakdown of machinery, resulting in safety risks and environmental pollution. Controlling and reducing friction is a fundamental step in attaining the sustainable development of our society, as detailed in the Brundtland report. In this contribution I will present the results of a study on the nanometric sliding of molybdenum disulfide against itself both from the experimental and from the computational point of view. The differences between ordered material (single crystal) and disordered material (sputtered coating) in both humid and dry environments were investigated. Tribological experiments were performed using lateral force microscopy. Atomic force microscopy tips modified by sputter deposition of molybdenum disulfide were used for the first time. This feature opened up the possibility for a close comparison with classical molecular dynamics simulations. In both cases, the coefficient of friction for the ordered system in inert conditions was found to be smaller than for disordered system. This result demonstrates the impact of morphology at the nanoscale and highlights the importance of molecular dynamics as a diagnostic and predictive tool in nano-friction. Furthermore, experiments show that the effects of the environment on nanoscale friction are reduced with respect to the macroscale case. These findings can expedite the process of fabricating molybdenum disulfide-based coatings with superior tribological properties, with the ultimate aim of reducing the energy dissipation due to friction.

Speaker: Paolo Nicolini (Czech Technical University in Prague)
• 11:15
Rendering on an HPC cluster with R-a-a-S 15m

In our contribution we describe a service that has been recently developed at IT4Innovations. The service specializes on image rendering and utilizes performance of an HPC cluster for this kind of task. The abbreviated name is R-a-a-S and the meaning is Rendering as a Service.

Probably all data that are processed on a computer sooner or later come to the fact that it is necessary to visually represent them. To obtain high visual quality of such outputs image rendering is employed. It synthetizes images from 2D or 3D models in a computer scene. Such task is handled by specialized software called renderer. To achieve photorealistic quality from synthetically generated scenes physically based renderers are used. We have used Cycles renderer from open-source Blender as a main starting point on the way to rendering service. We have modified Cycles core and created CyclesPhi to support efficient rendering on a cluster. It enables rendering on a classical CPU and also MIC accelerated nodes of a cluster.

To continue on the way of creating the rendering service it was necessary to build some sort of intermediate layer that converts user demands into computation tasks for a cluster. We have conveniently used HEAppE middleware that has been also developed at IT4Innovations. It manages and provides information about jobs on the cluster. It implements necessary job management tools together with the user authentication and provides appropriate feedback for the client. We have extended the HEAppE functionality by job arrays to better suit specific requirements typical for rendering jobs.

To complete the pipeline of the rendering service a client application that communicates with the middleware on one side and directly with the user on the other side was necessary. Decision was made to create the client directly in Blender. It is a popular and dynamically growing 3D open-source solution with a broad user base. Therefore, direct implementation of the client in Blender can deliver HPC rendering capability right at hands of a creator or a user of the 3D content. Our rendering client was created as a Blender addon and tested on main user platforms (Win/Mac/Linux). Our rendering service is ready and waiting for users.

Speaker: Petr Strakos (IT4Innovations)
• 11:30 13:00
Lunch


text:
# A Question Regarding Asymptotic Notations

$g(n) = \Theta(n^2)$, $f(n) = g(n) + g(n-1) + ... + g(2) + f(1)$

Given the conditions above, is it suitable for me to conclude that $f(n) = \Theta(ng(n))$?

As the number of the $g(n)$ terms in $f(n)$ seems to grow with $n$... .

$g(n)$ and $f(n)$ are asymptotically positive functions.

• How is the Big Theta notation defined? If you answer this question, the answer to your HW problem is not too hard either. Oct 13, 2014 at 19:05
• Well, I know that $g(n)$ is bounded by two $n^2$s multiplied by some constants, but I don't really see how that gets me anywhere... .
– user182048
Oct 13, 2014 at 19:24
• Note that this is true only because of the specific conditions on $g(n)$. If e.g. $g(n) \in \Theta(\frac1n)$ then the conclusion doesn't hold. Feb 1, 2018 at 23:08

$g$ is bounded above and below asymptotically, i.e., there exist constants $c, C$ and there exists $N \in \mathbb{N}$ such that $$cn^2 \leq g(n) \leq Cn^2$$ for all $n \geq N$.
In what follows I assume that you meant $g(1)$ instead of $f(1)$ in your question. We almost have the following inequalities: $$c \cdot 1^2 + \ldots + c \cdot n^2 \leq f(n) = g(1) + \ldots + g(n) \leq C \cdot 1^2 + \ldots + C \cdot n^2.$$ "Almost" in the sense that this is only true up to a constant tail end (for the numbers from $1$ to $N$).


text:
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#### Could someone tell me if this is correct?

J is filling his gas tank.  After 10 seconds, his gas tank has 5 gallons in it.  After 25 seconds, his tank has 9.5 gallons. Suppose the amount of gas in the tank depends linearly on time. Write... more
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#### drew is 3yrs younger than his brother jimmy the sum of the brothers ages is 21

have to use variables
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how do i go about answering this question ? I know the asnwer is 1/3, but how do i get that and not 1/6
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#### algebra help

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#### At what speeds are the two trains traveling?he northbound train travels 12 miles per hour faster than the southbound train. After 2.5 hours, they are 330 miles

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Word Problem

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#### Word Problem

Penny sells shoes at the local store and recently set up a display to help sell shoes. One third of the shoes in the display are high-tops. One third of the shoes are boots, and one sixth of the... more

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text:
# Software

For my research and duties at the Spinoza Centre for Neuroimaging (location Roeterseiland), I have developed several software packages (of which most using Python). You can find a selection below, but see my personal Github and the Spinoza Centre’s Github for a complete overview.

## bidsify

Widespread sharing of neuroimaging data is, unfortunately, not common in the neuroscience community. In the past years, some researchers have proposed a common format for MRI datasets - the Brain Imaging Data Structure (BIDS) - which aims at improving data sharing, reproducibility, and transparency through providing a guidelines on how to organize your data (which can be, subsequently, be uploaded to openneuro.org very easily).

Image from https://www.nature.com/articles/sdata201644

Organizing your raw data into the required BIDS format, however, can be painstakingly time-consuming: you’ve to convert your files in the right format (e.g. Philips PAR/REC or DICOMs to compressed nifti), rename your files, and extract the necessary meta-data. I’ve written a small package that automatically converts your “raw” and unstructured dataset to BIDS, given a config file that you provide (such that the software “knows” which how to convert/format/rename your files). Recently, I also provide a Docker image so that you can run bidsify without worrying about dependencies. The package has been tested on several datasets recorded at the Philips 3T and 7T scanners from the Spinoza Centre for Neuroimaging, and is currenlty deployed at our MRI center to automatically convert all raw scanner data to BIDS (as part of the nitools package, see below).

The code can be found on Github. The README should contain enough information to get started with bidsify.

## scanphyslog2bids

Ever tried to distill volume triggers from Philips physiology (“SCANPHYSLOG”) files? Trust me, it’s frustrating. I created a small package to do this using either trigger markers, or based on the scanner’s gradients (if they’re logged), or, if everything else fails, to interpolate volume triggers “counting backwards” from the offset of the acquisition. Check it out here.

## exptools2

Many experimental paradigms in psychology/neuroscience are created programmatically in order to have strict control over the order, timing, and logging of stimuli and responses. Together with people from the lab of Tomas Knapen, I developed the exptools2 package, which is basically a wrapper around the software package PsychoPy. With exptools, we provide a set of routines that ensure precise timing and logging of stimulus onset and responses (as well as Eyetracker integration), which is crucial for many neuroimaging experiments. Check out the repository (and especially the README) to get started!

## skbold: Utilities and tools for machine learning on BOLD-fMRI data

Inspired by the scikit-learn (often abbreviated as sklearn) machine learning package in Python, I created skbold - a package aimed at complementing scikit-learn in the organization, representation, and (pre)processing of fMRI data for machine learning (“decoding”) analyses. The package aims for flexibility in the sense that the annoying part of building decoding pipelines (such as getting it in the right format, keeping track of model performance) is taken care of, while the users can still build the actual pipelines using scikit-learn functionality.

The code can be found on Github and the documentation on ReadTheDocs.

## Porcupine

While working on Spynoza during the BrainHack organized by the Spinoza centre, I got to know Tim van Mourik from the Donders Institute, who was working on an open-source pipelining package with a graphical user interface named “Porcupine”, which is short for “PORcupine Creates Ur PipelINE” - in Tim’s words “the worst self-referencing acronym with bad capitalisation and annoying use of slang.” Together with Tomas Knapen, we started working together on Porcupine.

In short, Porcupine is a graphical application that allows you build processing pipelines by defining “nodes” - that implement some function (such as an MRI-processing step) - and connections between them, and importantly then is able to generate the code that would be needed to actually run the pipeline as defined in Porcupine! For example, one framework Porcupine supports is the amazing Nipype framework. So, you could for example build a Porcupine-pipeline implementing an MRI-preprocessing pipeline based on “nodes” from different software packages (such as FSL, AFNI, and Freesurfer) and subsequenly let Porcupine generate the (Nipype-based) code that you can actually run!


text:
# How to create new custom symbols like \[Wolf]?

In Mathematica, we can often get very special symbols by typing esc ... esc or \[...], such as the Wolf or Mathematica's logo. Is it possible to define new special symbols like those and then use them as \[Wolf]?

Any Unicode character in the basic multilingual plane can be entered by typing \:xxxx, where xxxx is its code point in hexadecimal. You can of course define a more memorable input alias for it.
• @Itai May I ask you what encoding is used by the input form \xxx? For example, "\041" corresponds to the character "!" with code 33. Aug 27, 2017 at 3:37


text:
0
2013
Impact Factor

# S. Dobrokhotov

prosp. Vernadskogo 101, Moscow, 119526 Russia
Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences (IPMech RAS)

## Publications:

 Dobrokhotov S. Y., Minenkov D. S., Neishtadt A. I., Shlosman S. B. Classical and Quantum Dynamics of a Particle in a Narrow Angle 2019, vol. 24, no. 6, pp.  704-716 Abstract We consider the 2D Schrödinger equation with variable potential in the narrow domain diffeomorphic to the wedge with the Dirichlet boundary condition. The corresponding classical problem is the billiard in this domain. In general, the corresponding dynamical system is not integrable. The small angle is a small parameter which allows one to make the averaging and reduce the classical dynamical system to an integrable one modulo exponential small correction. We use the quantum adiabatic approximation (operator separation of variables) to construct the asymptotic eigenfunctions (quasi-modes) of the Schr¨odinger operator. We discuss the relation between classical averaging and constructed quasi-modes. The behavior of quasi-modes in the neighborhood of the cusp is studied. We also discuss the relation between Bessel and Airy functions that follows from different representations of asymptotics near the cusp. Keywords: potential well, stationary Schrödinger equation, KAM theory, operator separation of variables, semiclassical asymptotics, Airy function, Bessel function Citation: Dobrokhotov S. Y., Minenkov D. S., Neishtadt A. I., Shlosman S. B.,  Classical and Quantum Dynamics of a Particle in a Narrow Angle, Regular and Chaotic Dynamics, 2019, vol. 24, no. 6, pp. 704-716 DOI:10.1134/S156035471906008X
 Dobrokhotov S. Y., Minenkov D. S. On various averaging methods for a nonlinear oscillator with slow time-dependent potential and a nonconservative perturbation 2010, vol. 15, no. 2-3, pp.  285-299 Abstract The main aim of the paper is to compare various averaging methods for constructing asymptotic solutions of the Cauchy problem for the one-dimensional anharmonic oscillator with potential $V(x, \tau)$ depending on the slow time $\tau=\varepsilon t$ and with a small nonconservative term $\varepsilon g(\dot x, x, \tau)$, $\varepsilon \ll 1$. This problem was discussed in numerous papers, and in some sense the present paper looks like a "methodological" one. Nevertheless, it seems that we present the definitive result in a form useful for many nonlinear problems as well. Namely, it is well known that the leading term of the asymptotic solution can be represented in the form $X(\frac{S(\tau)+\varepsilon \phi(\tau))}{\varepsilon}, I(\tau), \tau)$, where the phase $S$, the "slow" parameter $I$, and the so-called phase shift $\phi$ are found from the system of "averaged" equations. The pragmatic result is that one can take into account the phase shift $\phi$ by considering it as a part of $S$ and by simultaneously changing the initial data for the equation for $I$ in an appropriate way. Keywords: nonlinear oscillator, averaging, asymptotics, phase shift Citation: Dobrokhotov S. Y., Minenkov D. S.,  On various averaging methods for a nonlinear oscillator with slow time-dependent potential and a nonconservative perturbation, Regular and Chaotic Dynamics, 2010, vol. 15, no. 2-3, pp. 285-299 DOI:10.1134/S1560354710020152
 Bruening J., Dobrokhotov S. Y., Semenov E. S. Unstable closed trajectories, librations and splitting of the lowest eigenvalues in quantum double well problem 2006, vol. 11, no. 2, pp.  167-180 Abstract We discuss the structure of asymptotic splitting formula for the lowest eigenvalues of multidimensional quantum double well problem. We show that the change of instanton by closed unstable trajectory of appropriate Hamiltonian system gives more natural and simpler preexponential factor (amplitude) in splitting formula. The projection of this trajectories onto configuration space are well know librations in classical mechanics. Keywords: quantum double well problem, librations, splitting value, tunneling effect, normal forms, complex phase space Citation: Bruening J., Dobrokhotov S. Y., Semenov E. S.,  Unstable closed trajectories, librations and splitting of the lowest eigenvalues in quantum double well problem , Regular and Chaotic Dynamics, 2006, vol. 11, no. 2, pp. 167-180 DOI: 10.1070/RD2006v011n02ABEH000343


text:
# Louise Leung Fung Yee

P1-P2

#### Overall Review:

###### 課堂表現:

出席學生人數:

2

Sat Jun 01 2019 10:58:19 GMT+0000 (Coordinated Universal Time)

Though only 2 students showed up, we are still able to finish the exercises! And the students are interested in learning about location indicators so we learn more about "under", "next to", "near" and "between" relating to the exercise. They are very well behaved and good too. Though a little bit playful, it is still acceptable as they are learning effectively.


text:
e-Statistics > 4480-5480 Probability and Statistics II

## Continuity Correction

When a random variable is discrete, its frequency function is viewed as the height of relative frequency histogram with interval . Then we can approximate its probability by the area under the curve.

Example. Suppose that has a binomial distribution with n = 15 and p = 0.3. We can find the probability on the interval "intval = c(2,7)." The exact probability (in black) and the normal approximation (in red) with continuity correction are shown in the display.

```n = 15
p = 0.3
intval = c(2,7)
nn = 0:n
pmf = dbinom(nn,n,p)
mean = n * p
sd = sqrt(n * p * (1-p))
x = seq(-0.5, n+0.5, length=100);
y = dnorm(x, mean, sd);
plot(x, y, type='l', lwd=2, frame.plot=F, main="Normal Approximation");
prob.hist(nn,pmf,lty=1)
nn = intval[1]:intval[2]
pmf = dbinom(nn,n,p)
prob.hist(nn,pmf,lty=1,col='yellow')
x = seq(intval[1]-0.5, intval[2]+0.5, length=50);
y = dnorm(x, mean, sd);
polygon(c(x,max(x),min(x)), c(y,0,0), lwd=2, col=2, density=20);
prob = pbinom(intval[2],n,p) - pbinom(intval[1]-1,n,p)
text(intval[1], pmf[1], pos=4, round(prob,digits=4))
prob = pnorm(intval[2]+0.5, mean, sd) - pnorm(intval[1]-0.5, mean, sd)
text(intval[1], pmf[1], pos=2, col=2, round(prob,digits=4))
```


text:
# What is $$\sqrt 4 \times \sqrt {64} ?$$

16

Explanation

The square root of 4 is 2 and the square root of 64 is 8 so $$2 \times 8 = 16$$.


text:
# Terminal Speed expression

## Homework Equations

That is the solution:

## The Attempt at a Solution

I think ε≠Bvl because ε = - dø/dt=- Bd(v.t)/dt, v is not constant.
Why ε=Bvl???

CAF123
Gold Member
The induced emf in the loop is related to the change in flux through a surface whose boundary is the loop: $$\epsilon = -\frac{\text{d} \phi}{\text{d}t} = -\frac{\text{d}}{\text{d}t} \int_S \mathbf{B} \cdot \mathbf{dS}$$
In your case, the B field is of constant magnitude and direction and dS instantaneously is Ldx.

1 person
dø/dt = d(B. Area)/dt = d(BLx)/dt = BL dx/dt = BLv.

1 person


text:
# Prove $f$ is integrable on $[a,b]$ if $f$ has finitely many accumulation points on $[a,b]$.

Suppose interval $$I\in[a,b]$$ as finitely many limit points, $$f:[a,b]\to \mathbb{R}$$ is bounded on $$[a,b]$$ and continuous on $$[a,b]\setminus I$$. Use the fact that if $$f$$ is continuous except at finitely many points in an interval, $$f$$ is integrable on that interval.

I can use Darboux and Reimann integration theorems and definitions.

I believe the correct way to start is to partition the accumulation points into intervals, but I'm not sure the correct way to set this up. I also know that given the hypothesis, there are infinitely many discontinuities on $$[a,b]$$, but I m also not sure how to prove that. Any help is appreciated, thank you!

• What about the Dirichlet function? On any interval it has finitely many limit points, yet is not Riemann integrable on any interval by Lebesgue. – Melody Dec 9 '18 at 19:36
• Part of the hypothesis I have is that $f$ is continuous on $[a,b]\setminus I$. It's a homework question, so I'm assuming it is true! – t.perez Dec 11 '18 at 2:53
• If $f$ integral on $[a,b]$, then it is integrable on every subinterval, hence if $f$ so for a counterexample we can assume WLOG $I=[a,b]$. Alternatively, if the subinterval must be proper, just take $f$ to be the Dirichlet function on $I$, and 0 everywhere else. Then $f$ still has finitelt many limit points, but uncountable discontinuities. – Melody Dec 11 '18 at 5:08

The wording of the question is a bit confusing. The correct result is the following

Theorem: Let the function $$f:[a, b]\to\mathbb{R}$$ be bounded on $$[a, b]$$ and $$D$$ be the set of its discontinuities on $$[a, b]$$. If $$D$$ has a finite number of accumulation points then $$f$$ is Riemann integrable on $$[a, b]$$.

For those familiar with basic measure theory note that $$D$$ is a set of measure zero and hence by Lebesgue's criterion $$f$$ is Riemann integrable on $$[a, b]$$.

However the result can be proved using basic theorems on Riemann integration. One can partition the interval $$[a, b]$$ into a finite number of subintervals such that each subinterval contains only one accumulation point of $$D$$ and further the accumulation point is an end point of that subinterval.

This reduces the problem to the case when $$D$$ has only one limit point $$a$$ (or $$b$$ and this case is handled similarly). Let $$\epsilon >0$$ be arbitrary. If $$M$$ is positive upper bound for $$|f|$$ on $$[a, b]$$ then we can choose $$c=\min(b, a+(\epsilon/4M))$$. Since $$a$$ is the only limit point of $$D$$ in $$[a, b]$$ the interval $$[c, b]$$ contains only finitely many points of $$D$$. Then $$f$$ is Riemann integrable on $$[c, b]$$ and hence there is a partition $$P'$$ of $$[c, b]$$ for which $$U(P',f)-L(P',f)<\frac{\epsilon} {2}$$ Let $$P=P'\cup\{a\}$$ so that $$P$$ is a partition of $$[a, b]$$ and then we have $$U(P, f) - L(P, f) <2M\cdot \frac{\epsilon} {4M}+\frac{\epsilon}{2}=\epsilon$$ and therefore $$f$$ is Riemann integrable on $$[a, b]$$.


text:
# Topology, closure definition - well defined?

I came upon the following definition for closure,

Given a subset of a topological space $X$, the closure of $A$ is defined as the intersection of all closed sets containing $A$.

How is this definition well-defined? We need to know if the expression, $$\bar{A} := \bigcap_{U \in S} U$$ where $S$ is the collection of closed sets containing $A$, exists, and is unique?

(We do know $S$ is nonempty as $X \in S$. But why does this guarantee existence of intersection? More generally, is the intersection of the elements of any nonempty collection of sets well-defined?) I think I am now a bit confused on the fundamentals of sets.

May someone explain? Thank you so much.

EDIT: Thank you so much for all the replies. I have now read some basic set theory from Enderton's text. Here is my attempt to prove from scratch (Which is pretty much the same as the comments). Please do tell me if any part is incorrect.

The three axioms which I use:

Power Set Axiom (PSA) $$\forall a, \exists B \forall x ( x \in B \Leftrightarrow x \subseteq a)$$

Exstensionality Axiom (EA) $$\forall A \forall B [ \forall x (x \in A \Leftrightarrow x \in B ) \Rightarrow A = B ]$$

Axiom of Separation (AoS) For each formula $f(x)$ not containing $B$, the following is an axiom: $$\forall t_1 \ldots \forall t_k \forall c \exists B \forall x ( x \in B \Leftrightarrow x \in c, f(x))$$

Well-defineness: Given a Topological Space $(X, \mathcal{T}_X)$ and $A \subseteq X$. Firstly, $\mathcal{T}_X$ is a well-defined set. This is because, by the PSA, set $\mathcal{P}(X)$ exists and is unique by EA.

So by the AoS, $\mathcal{T}_X : = \{ x \in \mathcal {P}(X) : f(x) \}$ (where $f(x)$ is a formula of $x$ for its openess) exists, and is unique by EA.

Define the collection of closed sets which contains $A$ by, $$\mathcal{C} := \{ x \in \mathcal{P}(X) : x^c \in \mathcal{T}_X, A \subseteq x \}$$ which exists by AoS and is unique by EA. Also, $\mathcal{C}$ is nonempty ($X$ is in the set) so the set $$\bigcap \mathcal{C} := \{ x \in X : \forall y \in \mathcal{C}, x \in y \} = \bar{A}$$ again exists by AoS, and is unique by EA, so the closure is well-defined.

• Notice that $A\subseteq U$ for all $U$ that contains $A$ (by the definition of closure), so if $A$ is non-empty then $\overline{A}$ is non-empty as well. And notice that $A\subseteq \overline{A}$ – Masacroso Jul 4 '16 at 6:03

If you're confused about the fundamentals, I'd suggest going over the first chapter in Munkres's book. I'd also recommend the section "Background in set theory" from these notes by Prof. Curtis McMullen, which illuminates the axiomatic foundations of set theory known as the ZFC axioms, the most widely used today.

As you said, the intersection is over a nonempty collection, and therefore is non-trivial. Moreover, it is unique, because there's nothing it could possibly depend on; the subset of closed sets that contain $A$ is a well-defined set of subsets of $X$, so the intersection of all of them is well-defined too (this follows by axiom IV from McMullen's notes, Unions, and taking complements).

If $A\in S$, then $\bigcap S \subseteq A$. Thus by the axiom of separation, $\bigcap S$ exists if $S\ne\emptyset$. Because it can be given as an explicit class builder:

$$\bigcap S=\{x\mid\forall y\in S,x\in y\},$$

it is unique. Thus the intersection of any nonempty collection of sets is a well-defined set. (The intersection of the empty set is also well-defined, but equals the universe, $\bigcap\emptyset=V$, which is not a set.) In topology, though, we want more than that: we want to know that the closure is closed, which follows from one of the axioms of a topology - any arbitrary intersection of a nonempty collection of closed sets is closed.

• For the intersection of the empty set, in this context I would understand it to be not the universe of set theory, but that of the current discourse, i.e. the space $X$. But that is of course not important for the OP at this point. – Carsten S Jul 4 '16 at 9:42
• @CarstenS In axiomatic set theory it's a hassle to set this up, because $X$ can't be the universe since it's a set (otherwise we wouldn't even be able to consider subcollections of the powerset of $X$). You can use "relativized intersection", but then that means hidden parameters, which could be a potential cause of confusion for OP's "well-definedness" concerns. I prefer to just use a notation like $X\cap\bigcap S$ to explicitly indicate relativized intersection when relevant. – Mario Carneiro Jul 4 '16 at 12:42
• You are right, I was thinking about this as a hidden parameter. I find it useful in this context, because I can for example check that something is a topology by checking that it is closed under unions and finite intersections, where the latter condition implies that the whole space is open. But it's a matter of taste. – Carsten S Jul 4 '16 at 12:49
• @CarstenS In type theory, your approach is the usual one; in that case $X$ is the universe, but there is no guarantee that it's the only one, and the set of all topologies on $X$ lives in a different (higher) universe. Relativized intersection can also be written as a simple class builder: $$X\cap\bigcap S=\{x\in X\mid\forall y\in S,x\in y\},$$ and it is obviously a set, even if $S$ is empty, since it is a subset of $X$. – Mario Carneiro Jul 4 '16 at 12:53

Sure, let $S = \{ C | C^c \in \tau, A \subset C \}$.

Then $x \in \overline{A}$ iff $x \in C$ for all $C \in S$.

First $S = \{U : U \mbox{ is closed set containing }$A$\}$.

$U\in S$ contains $A$ so $\cap_{U \in S} U$ contains A , So Closure of $A$ is well defined.


text:
# How factorize a lot of “little” numbers?

Several fast algorithms are available to factorize big numbers, but what is the best algorithm to factorize a lot of "little" numbers? I need to factorize a lot of odd numbers $< \mathbf{2^{56}}$. I want to do that only with 64 bits arithmetic.

I use a combination of that:

• A table of all prime $< 2^{16}$.
• A "offset technics" to iterate on possible prime numbers bigger than $2^{16}$.
• The Pollard's rho heuristic to (probably) find a divisor. I implemented the version presented in "Introduction to Algorithms" (Cormen, Leiserson, Rivest and Stein), with added a maximum number of iterations something like $n^{1/2}$ or $n^{1/4}$.

In fact I want to compute the sum of odd divisors $\sigma_{\text{odd}}$ of these numbers. To check that $\forall n \in \mathbb{N}, n > 1, \exists k : \sigma_{\text{odd}}^k(n) < n$. Maybe there is a better way to do that than use the complete factorization in prime numbers?

And what about all these considerations in a parallel context?

Answer: Finally I use an unique big table with all prime numbers $<2^{32}$:

• If you want to factor numbers $< 2^{56}$, the primes $< 2^{28}$ would suffice. Usually, the smaller memory footprint of that table would speed things up (how much depends on various things, like available memory). Also, after trial division by some small primes (perhaps up to $1000$ or $10000$), it may be a good idea to insert a fast primality check like BPSW or Miller-Rabin with fixed bases (small sets of bases that make M-R exact for numbers $< 2^{64}$ are known, BPSW is known to be exact in that range). – Daniel Fischer Dec 21 '17 at 13:10
• Another question is whether you need the individual $\sigma(n)$ or whether you need something like $\sum_{n = 1}^N \sigma(n)$. – Daniel Fischer Dec 21 '17 at 13:13
• – lhf Dec 21 '17 at 13:24

In order to calculate the sum of divisors of definite number you may use the Euler pentagonal number theorem application for divisors sum $\sigma(n)$.

$$\sigma(n)=\sigma(n-1)+\sigma(n-2)-\sigma(n-5)-\sigma(n-7)+...$$ The ${1,2},{5,7},...$ are generalized pentagonal numbers.

$\sigma(n)=0$ if $n<0$

$\sigma(n-n) = 0$ if $n$ is not a generalized pentagonal number and $\sigma(n-n) = n$ otherwise.

The generalized pentagonal numbers can be found:

There are some other articles:

• It is a beautiful formula. Thanks. In term of algorithmic complexity I think it is useless. I think the complexity is something like the naive approach (if $n$ is not a perfect square) : $\sigma(n) = \sum\limits_{\stackrel{d \backslash n}{d \leq \left\lfloor\sqrt(n)\right\rfloor}} \left(d + \frac{n}{d}\right)$ – Olivier Pirson Nov 12 '17 at 15:29
• Yes it is probably useless. But I think that there might be some optimization making this useful. I will try to find some documents describing this. – Gevorg Hmayakyan Nov 12 '17 at 18:05


text:
# Thread: s4.13.5.41 relation of 2 planes

1. $\tiny{{s4}.{13}.{5}.{41}}$

$\textsf{find if planes are$\parallel, \perp$or$\angle$of intersection }\\$
\begin{align}
\displaystyle
{P_1}&={x+z=1}\\
\therefore n_1&=\langle 1,0,1 \rangle\\
\\
{P_2}&={y+z=1}\\
\therefore n_2&=\langle 0,1,1 \rangle\\
\\
\cos(\theta)&=
\frac{n_1\cdot n_2}{|n_1||n_2|}\\
&=\frac{1(0)+0(1)+1(1)}
{\sqrt{1+1}\cdot\sqrt{1+1}}
=\frac{1}{2}\\
\cos^{-1}\left({\frac{1}{2}}\right)&=
\color{red}{60^o}
\end{align}

$\textit{there are 2 more problems like this so presume this is best method.. }\\$
$\textit{didn't know if it is common notation to call a plane$P_1$}$

2. Originally Posted by karush
$\tiny{{s4}.{13}.{5}.{41}}$

$\textsf{find if planes are$\parallel, \perp$or$\angle$of intersection }\\$
\begin{align}
\displaystyle
{P_1}&={x+z=1}\\
\therefore n_1&=\langle 1,0,1 \rangle\\
\\
{P_2}&={y+z=1}\\
\therefore n_2&=\langle 0,1,1 \rangle\\
\\
\cos(\theta)&=
\frac{n_1\cdot n_2}{|n_1||n_2|}\\
&=\frac{1(0)+0(1)+1(1)}
{\sqrt{1+1}\cdot\sqrt{1+1}}
=\frac{1}{2}\\
\cos^{-1}\left({\frac{1}{2}}\right)&=
\color{red}{60^o}
\end{align}

$\textit{there are 2 more problems like this so presume this is best method.. }\\$
$\textit{didn't know if it is common notation to call a plane$P_1$}$
Everything you've posted is fine. You can give a plane any name you like, but I would write something like this:

\displaystyle \begin{align*} P_1 : x + z = 1 \end{align*}

That way we can see that \displaystyle \begin{align*} P_1 \end{align*} is DEFINED as the relationship "the sum of the x and z values needs to be 1", not that it is some variable that has something to do with the equation.

3. The problem asks you to do three things:
1) determine if the planes are parallel.
2) determine if the planes are perpendicular.
3) if neither of those, determine the angle of intersection of the two planes.

Yes, by using that formula to determine the angle, you can then answer all three questions but it should be simpler to determine the first two without using that formula:

The two planes are parallel if and only if the two normal vectors are parallel- if one is a multiple of the other.

The two planes are perpendicular if and only if the two normal vectors are perpendicular: if their dot product is 0.

If neither of those is true, then you can use the dot product you found as the numerator in the formula to determine the angle.

hard to see that in their examples

the next 2 problems are probably
$\displaystyle \parallel , \perp$

$\tiny{s4.854.13.5.43}$
$\textsf{Determine if the 2 given planes are perpendicular, parallel or at an angle to each other}$
\begin{align}
\displaystyle
{p_1}&:{x+4y-3z=1}
\therefore n_1=\langle 1,4,-3 \rangle\\
\nonumber\\
{p_2}&:{-3x+6y+7z=0} \therefore n_2=\langle -3,6,7 \rangle
\end{align}
\begin{align}
\displaystyle
\cos(\theta)&= \frac{n_1\cdot n_2}{|n_1||n_2|}=0\\
\therefore p_1 &\perp p_2
\end{align}

$\tiny{s4.854.13.5.45}$
\begin{align}
\displaystyle
{p_1}&:{2x+4y-2z=1}
\therefore n_1=\langle 2,4,-2 \rangle\\
\nonumber\\
{p_2}&:{-3x+6y+3z=0} \therefore n_2=\langle -3,6,7 \rangle \\
n_1&=-\frac{2}{3} n_2 \\
&\therefore n_1\parallel n_2
\end{align}

$\textit{btw what is the input string to see the 2 plane on a W|A graph?}$

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text:
# If

28 december 2018

If

Informatie:

Lyrics/Muziek: Roger Waters
Zang: Roger Waters

Terug te vinden op:

Atom Heart Mother
The Early Years 1965-1972 Devi/ation

Songtekst:

If I were a swan, I’d be gone
If I were a train, I’d be late
And if I were a good man, I’d talk with you
more often than I do

If I were to sleep, I could dream
If I were afraid, I could hide
If I go Insane,

If I were the moon, I’d be cool
If I were a book, I would bend
If I were a good man,
I’d understand the spaces between friends

If I were alone, I could cry
And if I were with you, I’d be home and dry
And if I go insane,
will you still let me join in with the game

If I were a swan, I’d be gone
If I were a train, I’d be late again
And if I were a good man, I’d talk with you
more often than I do