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# Schedule for: 19w2257 - Mathematics and Computer Science in Modeling and Understanding of Structure and Dynamics of Biomolecules

Arriving in Banff, Alberta on Friday, August 9 and departing Sunday August 11, 2019
Friday, August 9
16:00 - 19:30 Check-in begins (Front Desk – Professional Development Centre - open 24 hours)
Note: the Lecture rooms are available after 16:00.
(Front Desk – Professional Development Centre)
19:30 - 20:15 Robert Jernigan: The importance of correlations in biology (opening lecture) (chairperson: Adam Liwo)
The availability of large genome-related data provides the opportunity to extract sequence correlations. Already there have been huge advances in structure prediction and predictions of interactions by using information about the correlated pairs of amino acids in multiple sequence alignments. We have also been using this same type of information to improve sequence matching for the purpose of annotating genes and proteins for their function. We observe major gains in the specificities of function, estimated at nearly half of genes, in nearly all cases confirming the function assignments from BLAST/Blosum62 but providing significantly more useful information.

The denseness of biological systems would immediately suggest the importance of these pairs, and higher order correlations. Reliably extracting higher order correlations requires a larger set of data. Although the data is growing rapidly, success in obtaining the higher order correlations at present will require the integration of diverse sets of data.

The ultimate goal for such efforts is to understand the origin of specific phenomes and disease states, as well as the effects of mutations.
(TCPL 201)
20:30 - 23:00 Informal gathering in 2nd floor lounge, Corbett Hall
Beverages and a small assortment of snacks are available in the lounge on a cash honour system.
(TCPL or Corbett Hall Lounge (CH 2110))
Saturday, August 10
07:00 - 08:45 Breakfast
A buffet breakfast is served daily between 7:00am and 9:00am in the Vistas Dining Room, the top floor of the Sally Borden Building. Note that BIRS does not pay for meals for 2-day workshops.
(Vistas Dining Room)
08:45 - 09:00 Welcome Talk by BIRS Staff
A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.
(TCPL 201)
09:00 - 11:30 Session 1: Protein structure and function. Chairperson: Marek Cieplak (TCPL 201)
09:00 - 09:30 Daisuke Kihara: Computational protein tertiary structure modeling from cryo-EM maps of intermediate resolution
The significant progress of the cryo-EM poses a pressing need for software for structural interpretation of EM maps. Particularly, protein structure modeling tools are needed for EM maps determined at a resolution around 4 Å or lower, where finding main-chain structure and assigning the amino acid sequence into EM map is challenging. In this seminar, we discuss computational protein structure modeling tools we have been developing and future directions, opportunities, and challenges. We have developed a de novo modeling tool named MAINMAST (MAINchain Model trAcing from Spanning Tree) for EM maps for this resolution range [1, 2, 3]. MAINMAST builds main-chain traces of a protein in an EM map from a tree structure constructed by connecting high-density points without referring to known protein structures or fragments. The method has substantial advantages over the existing methods. MAINMAST showed better modeling performance than existing methods. The method is further enhanced recently to be able to model symmetric protein complexes and ligand (drug) molecules that bind to a protein in a map. Moreover, to provide structure information for maps determined at a lower resolution (5~10 Angstroms), we have recently developed a new tool, Emap2sec, which uses convolutional deep learning for detecting secondary structures of proteins [3]. Emap2sec scans an EM map with a voxel and assigns a secondary structure, i.e. alpha helix, beta strand, or coil, from density patterns of the voxel and its neighbors.

Acknowledgments: This research was partly supported by NIH (R01GM123055), NSF (DMS1614777, CMMI1825941), and Purdue Institute of Drug Discovery.

[1] G. Terashi, D. Kihara, Nature Communications, 9, 1618, 2018
[2] G. Terashi, D. Kihara, J. Struct. Biol., 204, 351-359, 2018.
[3] S. R. M. V. Subramanya, G. Terashi, D. Kihara, accepted, 2019.
[4] http://kiharalab.org/mainmast/
(TCPL 201)
09:30 - 10:00 Ilya Vakser: Comparative Modeling of Protein Complexes
Comparative modeling of protein complexes is a technique complementary to free docking. Accounting for similarity of protein-protein interfaces in the docking algorithms is important for generating near-native docking models. Traditionally, template-based modeling of protein-protein complexes has relied on similarity of the entire proteins (targets) to known structures of protein-protein complexes (templates). However, similarity can be also inferred between the target protein surface and the template protein-protein interface. Earlier studies indicated that interface alignment can also be used to refine docking models generated by the full structure alignment. We performed a comprehensive comparative benchmarking of these two approaches (full structure alignment and interface alignment) based on 223 bound and unbound complexes from the DOCKGROUND benchmark set 4. The results showed that docking performance for the unbound structures compared to that for the bound structures decreases only slightly for both methods. Overall, the interface alignment performs marginally better than the full structure alignment in selecting templates which provide models with low i-RMSD and good quality. If a common template is selected in both interface and full structure alignments, the docking success rate increases significantly. Interface templates perform better than full structures for targets with highly specific interactions, when the full structures are different (e.g. in case of multidomain targets with a different relative orientation of domains than in the templates). Full structure alignment performs better for proteins with different binding sites for different functions. The interface-based predictions require less refinement for all high-quality models irrespective of the rank. In general, matching CATH annotations for target and template yield correct solutions. However, the success can be affected by alternative binding sites. Combining templates from sequence and structure homology increases the success rate to ~87%. Structural refinement of modeled protein-protein complexes is an essential step in protein docking. Such refinement becomes increasingly challenging when the interacting proteins undergo significant conformational changes upon binding (unbound/bound interface RMSD >= 3 Å). We developed a flexible interface refinement protocol and benchmarked it on a set of top 100 docking models, which are inside the docking funnel, generated for binary protein-protein complexes from the DOCKGROUND benchmark set 4. The refinement of the docking models was performed by a systematic local conformational search using previously developed contact potentials. The results showed that the procedure performs better than other widely used refined protocols benchmarked on the same set of protein-protein complexes.
(TCPL 201)
10:00 - 10:30 Coffee Break (TCPL Foyer)
10:30 - 11:00 Gregory Chirikjian: Mathematical Methods for Biomolecular Structure Determination
This talk discusses advances in modelling data acquisition and information fusion in biomolecular structure determination from x-ray crystallography, SAXS, and EM. Whereas the configuration space of a single rigid body is the group of orientation-preserving Euclidean motions, the configuration space of a collection of rigid bodies that move in lock step constrained by a discrete symmetry group is actually a coset space (or quotient space) of the configuration space of a single body by the discrete symmetry group. This quotient space, called a motion space', is the configuration space in the molecular replacement method in macromolecular crystallography. Of particular importance is the characterization of those coordinated motions which place symmetry mates in collision, since only the complement of this space is physically viable. This talk also discusses how SAXS and single-particle EM provide complementary information which can be fused to obtain better electron density estimates than each one can obtain independently.

Acknowledgments: This work was funded under grants NSF CCF-1640970, NSF IIS-1619050, NIH R01GM113240. The collaborations with the co-authors on the papers listed below are greatly appreciated.

References
Chirikjian, GS. "Mathematical aspects of molecular replacement. I. Algebraic properties of motion spaces." Acta Crystallographica Section A: Foundations of Crystallography v67, no. 5 (2011): 435-446.
Chirikjian, GS. "Kinematics Meets Crystallography: The Concept of a Motion Space." Journal of Computing and Information Science in Engineering 15, no. 1 (2015): 011012.
Chirikjian, GS., and B Shiffman. "Collision-free configuration-spaces in macromolecular crystals." Robotica 34, no. 8 (2016): 1679-1704.
Chirikjian GS, Sajjadi S, Shiffman B, Zucker SM. Mathematical aspects of molecular replacement. IV. Measure-theoretic decompositions of motion spaces. Acta Crystallographica Section A: Foundations and Advances. 2017 Sep 1;73(5):387-402.
Kim, J.S., Afsari, B., Chirikjian, G.S., Cross-Validation of Data Compatibility Between SAXS and Cryo-EM,''
Journal of Computational Biology, 24(1):13--30, 2017.
Dong, H., Kim, J.S., Chirikjian, G.S., `Computational Analysis of SAXS Data Acquisition,'' Journal of Computational Biology, 22(9): 787-805, 2015.
(TCPL 201)
11:00 - 11:30 Changbong Hyeon: Cost-precision trade-off and transport efficiency of molecular motors
An efficient molecular motor would deliver cargo to the target site at a high speed and in a punctual manner while consuming a minimal amount of energy. However, according to a recently formulated thermodynamic principle, known as the thermodynamic uncertainty relation, the travel distance of a motor and its variance are constrained by the free energy being consumed. Here we use the principle underlying the uncertainty relation to quantify the transport efficiency of molecular motors for varying ATP concentration ([ATP]) and applied load (f). Our analyses of experimental data find that transport efficiencies of the motors studied here are semi-optimized under the cellular condition. The efficiency is significantly deteriorated for a kinesin-1 mutant that has a longer neck-linker, which underscores the importance of molecular structure. It is remarkable to recognize that, among many possible directions for optimization, biological motors have evolved to optimize the transport efficiency in particular.

[1] "Energetic Costs, Precision, and Transport Efficiency of Molecular Motors" W. Hwang, C. Hyeon, J. Phys. Chem. Lett. (2018) 9, 513-520
[2] "Physical insight into the thermodynamic uncertainty relation using Brownian motion in tilted periodic potentials" C. Hyeon, W. Hwang, Phys. Rev. E. (2017) 96, 012156
[3] "Quantifying the Heat Dissipation from a Molecular Motor's Transport Properties in Nonequilibrium Steady States" W. Hwang, C. Hyeon J. Phys. Chem. Lett. (2017) 8, 250-256
(TCPL 201)
11:30 - 13:00 Lunch
A buffet lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. Note that BIRS does not pay for meals for 2-day workshops.
(Vistas Dining Room)
13:00 - 13:20 Group Photo
Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!
(TCPL Foyer)
13:20 - 15:05 Session 2: Topology, disorder and allostery. Chairperson: Gabriel del Rio (TCPL 201)
13:20 - 14:05 Marek Cieplak: Emergence of knots in intrinsically disordered proteins
Transient knotted structures are expected to arise during the volatile evolution of intrinsically disordered peptide chains. We show that this is indeed the case for sufficiently long polyglutamine tracts and α-synuclein. The polyglutamine tracts are fused within huntingtin protein that is associated with the Huntington neurodegenerative disease. We show that the presence of knots in the tracts hinders and sometimes even jams translocation, especially when the knots are deep. The knots in polyglutamine may form in tracts exceeding about 40 residues. This fact explains the existence of a similarly sized length treshold above which there is an experimentally observed toxicity at the monomeric level. We also discuss emergence of knots in α-synuclein. We show that these knots are either shallow or deep and last for about 3 – 5 µs, as inferred from an all-atom explicit-solvent 30 µs trajectories. We discuss conformational biasses that take place in α-synuclein and contact formation during aggregation of two chains of this protein. We then discuss several aspects of dynamics of knotted structured proteins as assessed within a Go-like model. In particular, we argue that folding under the nascent conditions is essential to fold to a structure that is deeply knotted. In collaboration with: M. Chwastyk, Ł. Mioduszewski, A. Gomez-Sicilia, M. Carrion-Vazquez, P. Robustelli, and Y. Zhao.
(TCPL 201)
14:05 - 14:35 Banu Ozkan: Nature utilizes dynamic allostery for evolution
The discovery of the protein structure proved to be one of the major scientific achievements in biological sciences and marked a milestone in the field, spawning the sequence-structure-function paradigm. That is, how can the 1-dimensional set of information from the amino acid sequence give rise to a unique 3-dimensional structure which, in turn, determines the function of a given protein. However, it quickly became apparent that it was not so simple as assigning a single 3-dimensional structure to an amino acid sequence. Some proteins which bind to ligands could take on multiple, distinct conformations depending on the binding event or environmental conditions. Furthermore, there are proteins sharing the similar 3-D structure despite the significant difference in amino acid sequences or even specific domains of proteins present across protein families. Thus, all these studies gave rise to the ensemble picture of proteins. That is, proteins are not simple static objects but rather dynamic entities, that sample many conformational states even in the absence of ligand binding. Variations in amino acid sequences can alter ligand recognition, binding rates, and other biophysical, thermodynamic and kinetic properties of homologous enzymes while still maintaining similar 3-dimensional folds. Differing functions between structural homologues gave rise to a view of protein evolution which proceeds through conformational dynamics and functional promiscuity, where a relationship may exist between active site flexibility and amino acid evolvability. Here we introduce two tools, the Dynamic Flexibility Index (DFI)[1] and the Dynamic Coupling Index (DCI) [2]which can quantify structural flexibility and dynamic coupling at a site-specific, single amino acid level. We show that it is possible to relate evolutionary conservation corelates with the flexibility of a given position. Through conformational dynamics analysis of ancestral proteins, we present that conformational ensemble of a protein is modified to adopt to a new environment and/or to emerge a new function by modifying rigidity and flexibility of its positions [3]. Finally, we discuss how Nature can modulate change through allosteric mutations which alter the internal interaction network of proteins, and how changes in allosteric regulation can result in disease phenotypes[4].

Acknowledgments: This research was supported by NSF

[1] Z.N. Gerek, S. Kumar, S.B. Ozkan Evol Appl 6, 423–433, 2013.
[2] A.Kumar, T.J. Glembo, S.B. Ozkan Biophys J 109:1273–1281, 2015
[3] V.A. Risso, J.M. Sanchez-Ruiz, S.B. Ozkan Curr. Opp. Struct. Biol. 51,106-115, 2018.
[4] A. Kumar, B.M. Butler, S. Kumar, S.B.Ozkan Curr. Opp. Struct. Biol. 51,135-142, 2015
(TCPL 201)
14:35 - 15:05 Nina Pastor: Effect of Phosphorylation and Mutation to Asp and Glu on the Conformational Landscape of an Intrinsically Disordered Region
Intrinsically disordered proteins are notorious for their conformational flexibility and capacity for interaction with different targets by acquiring distinct conformations depending on the specifics of the binding site. They can also engage in specific interactions without loosing conformational freedom, forming fuzzy complexes. The particular conformations favored by these proteins can be tuned by posttranslational modifications, such as phosphorylation. A common experimental strategy to study the effect of phosphorylation is to perform mutations of the modified residues by aspartate (D) or glutamate (E), under the assumption that the main effect of phosphorylation is the inclusion of negative charge. Whether the mutation to D or E is equivalent to phosphorylation is case dependent. In this work we explore the conformational landscape of an intrinsically disordered region at the C-terminus of adenoviral protein E1B55kDa, which is regulated by phosphorylation at three residues at the C-terminus, with the aim of establishing whether mutation to D or E is equivalent to modification by phosphorylation. In the context of the complete virus, the triple mutant with Ds produces a more efficient virus compared to wild type, and mutation to alanine (A), which cannot be phosphorylated, is equivalent to not having the full protein. We chose the last 20 residues of E1B55kDa as our reference peptide, as multiple disorder predictors consider it to be disordered. This peptide has only one cationic residue (arginine 9), and the C-terminal half is enriched in negative residues. We submitted the wild type sequence to Pepfold3, and obtained 100 different structures for it. Taking these as a reference, we built versions with three phosphorylated residues (two serines and one threonine), three Ds, three Es and three As. We placed each peptide in a water box with 0.15M NaCl using Charmm-gui, and ran it in NAMD in the NPT ensemble at 298K and 1 atm with the Charmm36m forcefield for 50 ns, achieving a total simulated time of 5 µs for each peptide variant. The distribution of the radius of gyration shows the prevalence of extended structures, with a slightly expanded ensemble for the triple D and triple E variants, and a slightly compressed one for the phosphorylated version, compared to the wild type. This is reflected in almost saturated hydration for the peptide in all its residues, except for a small decrease in hydration number for arginine 9 in the phosphorylated version. A closer look at intrapeptide hydrogen bonds reveals that there are few interactions in general, but arginine 9 engages in many more interactions with the phosphorylated residues than with the other charges in the peptide; the interaction with phosphorylated threonine 19 is preferred above all. This interaction leads to the formation of a loop that prefers to adopt disordered conformations, so we propose that it engages in fuzzy complexes with other proteins. In general, phosphorylation leads to an increase in alpha helix formation in the peptide, while substitution for Ds and Es leads to a loss of this structure. We conclude that phosphorylation and the mutation to D and E are not equivalent.

Acknowledgments: This research was supported by a CONACYT scholarship for MARM and supercomputing time at the Laboratorio Nacional de Supercómputo del Sureste (LNS), LANCAD in México City, and the Laboratorio de Dinámica de Proteínas at UAEM.
(TCPL 201)
15:05 - 15:30 Coffee Break (TCPL Foyer)
15:30 - 16:50 Session 3. Structure/function prediction. Chairperson: Banu Ozkan (TCPL 201)
15:30 - 15:50 Yi He: Chemical shifts based structural ensemble generation for intrinsically disordered proteins: A case study
About a third of the proteome consists of intrinsically disordered proteins (IDPs) that fold, whether fully or partially, upon binding to their partners [1]. IDPs use their inherent flexibility to play key regulatory roles in many biological processes [2]. Such flexibility makes their structural analysis extremely challenging, being nuclear magnetic resonance (NMR) the most suitable high-resolution technique. However, conventional NMR structure determination methods, which seek to determine a single high-resolution structure [3], are inadequate for IDPs. There are several tools available for the structural analysis of IDPs using NMR data and primarily Chemical Shifts (CS) [4-6]. However, a persistent problem is how to effectively sample the extensive, but not random, conformational space of IDPs. We have implemented a novel relational database, termed Glutton, that links all existing CS data with corresponding protein 3D structures with the goal of enabling the conformational analysis of IDPs directly from their experimental CS. Glutton’s uniqueness is in its focus on dihedral angle distributions consistent with a given set of CS rather than with unique structures. Such dihedral distributions define how native-like is the ensemble and lead to the effective calculation of large ensembles of structures that efficiently sample the available conformational space. With Glutton, we examined Nuclear Coactivator Binding Domain (NCBD), an IDP with NMR structure obtained using osmolyte stabilizers that is partly disordered in native conditions [7]. As means of comparison, we produced a 60s long MD simulation of NCBD in explicit solvent starting from the NMR structure and using the CHARMM36m force field with modified TIP3P water which was suggested as a good combination to explore the conformational space of IDPs [8]. The structural ensembles obtained from Glutton are based only on geometric considerations and CS restraints, but they can be further refined using additional computational (force field) and/or experimental (distance restraints) information.

Acknowledgments: This work was supported by grants: the startup fund at the University of New Mexico, the W.M. Keck Foundation, the National Science Foundation [NSF-MCB-161759 and NSF-CREST-1547848] and the European Research Council [ERC-2012-AdG-323059].

[1] H.J. Dyson, P.E. Wright, Chem. Rev., 104, 3607–3622, 2004

[2] M.M. Babu, Biochem. Soc. Trans., 44, 1185–1200, 2016

[3] A.M., Gronenborn, G.M. Clore, Anal. Chem., 62, 2–15, 1990

[4] V. Ozenne, F. Bauer, L. Salmon, J. Huang, M.R. Jensen, S. Segard, P. Bernadó, C. Charavay, M. Blackledge, Bioinformatics, 28, 1463–1470, 2012

[5] M. Krzeminski, J.A. Marsh, C. Neale, W. Choy, J.D. Forman-Kay, Bioinformatics, 29, 398–399, 2013

[6] D.H. Brookes, T. Head-Gordon, J. Am. Chem. Soc., 138, 4530–4538, 2016

[7] A. Naganathan, M. Orozco, J. Am. Chem. Soc., 133, 12154–12161, 2011

[8] J. Huang, S. Rauscher, G. Nawrocki, T. Ran, M. Feig, B. L de Groot, H. Grubmüller, A.D. MacKerell Jr, Nat Methods, 14, 71–73, 2017

(TCPL 201)
15:50 - 16:10 Michał Boniecki: SimRNA: a coarse-grained method for RNA 3D structure modeling - new ideas accounting for on non-canonical base pairing
The molecules of the ribonucleic acid (RNA) perform a variety of vital roles in all living cells. Their biological function depends on their structure and dynamics, both of which are difficult to experimentally determine but can be theoretically inferred based on the RNA sequence. SimRNA [1] is one of the computational methods for molecular simulations of RNA 3D structure formation. The method is based on a simplified (coarse-grained) representation of nucleotide chains, a statistically derived model of interactions (statistical potential), and the Monte Carlo method as a conformational sampling scheme. In SimRNA, the backbone of the RNA chain is represented by two atoms per nucleotide, whereas nucleotide bases are represented by three atoms each. In fact, these three atoms are used to calculate a system of local coordinates that allows for positioning of a 3D grid - the actual representation of the base. The 3D grid contains information about the interactions of the entire base moiety (not only the three atoms explicitly included in the SimRNA representation). The current version of SimRNA (3.22) is able to predict basic topologies of RNA molecules with sizes up to about 50-70 nucleotides, based on their sequences only, and larger molecules if supplied with appropriate distance restraints. However, it should be noted that the current version of SimRNA, as well as other methods for RNA 3D structure prediction, exhibit a number of limitations, which reduce the accuracy of RNA 3D structure models obtained. One of the biggest challenges is the prediction of non-canonical base pairs, which are crucial for the formation of functional motifs in RNA structure. Current studies and developments are focused on a new version of SimRNA, which will overcome the key limitations that exist in the current version of the program, as well as general limitations in current methods for RNA 3D structure prediction. The major idea is to split all the contacts corresponding to base-base interactions into classes that describe specific types of base-base interactions (canonical and non-canonical), while derivation of the statistical potential.

Acknowledgments: This work was supported by the Polish National Science Center Poland (NCN) (grant 2016/23/B/ST6/03433 to M.J.B.)

[1] M.J. Boniecki, G. Lach, W.K. Dawson, K. Tomala, P. Lukasz, T. Soltysinski, K.M. Rother, J.M. Bujnicki, Nucleic Acids Res. 2016 Apr 20;44(7):e63
(TCPL 201)
16:10 - 16:30 Agnieszka Karczynska: Structure prediction of mono- and oligomeric tarets using the physics-based coarse-grained UNRES force field and information from databases – CASP13/CAPRI46
The results of blind prediction of the structures of monomeric and oligomeric proteins obtained in the recent CASP/CAPRI experiment by the KIAS-Gdansk/Czaplewski groups, by using the physics-based coarse-grained UNRES force field [1] and information from databases are presented. For both monomeric and oligomeric targets, the methodology of the KIAS-Gdansk/Czaplewski groups included extensive conformational search by means of the Multiplexed Replica Exchange Molecular Dynamics (MREMD) simulations [2] with the UNRES force field [3], with geometry restraints from server models [4,5]. For the monomeric targets, the restraints were derived from fragments with similar geometry that occured in the server models (the consensus fragments), while monomer geometries except for the terminal, flexible loop, and linker regions, were restrained in oligomer simulations. The server models were selected mainly based on DeepQA score [6] ranking; when the score was below 0.5, models from the servers which performed well in previous CASP exercises: Zhang, Quark, and BAKER-ROSETTASERVER were selected. For oligomeric targers, monomers were modeled first and, subsequently, initial oligomeric structures were modeled based with the aid of the packing proposed by the HHpred server [7]; for smaller targets the monomers were oriented randomly. The results of MREMD simulations were processed by using the Weighted Histogram Analysis Method (WHAM) [8] to obtain the probabilities of conformations and subsequently subjected to cluster analysis to obtain the 5 (CASP) or 10 (CAPRI) families of conformations, from which the conformations closest to the mean conformations were, in turn, selected as candidate predictions [1], which were subsequently converted to all-atom conformations submitted to CASP/CAPRI. The obtained models were ranked solely based on the computed probabilities of the families obtained by summing up the probabilities of the constituent conformations computed by WHAM based on the UNRES effective function [1].

Acknowledgments: This research was supported by grant UMO-2017/26/M/ST4/00044 from the National Science Centre of Poland (Narodowe Centrum Nauki).

[1] A. Liwo et al., J. Mol. Model. 20 (2014): 2306.
[2] Y.M. Rhee et al., Biophys. J. 84 (2003): 775-786.
[3] C. Czaplewski et al., J. Chem. Theor. Comput. 5 (2009): 627-640.
[4] P. Krupa et al., J. Chem. Inf. Model. 55 (2015): 1271-1281.
[5] M. Mozolewska et al., J. Chem. Inf. Model. 56 (2016): 2263-2279.
[6] R. Cao et al., BMC Bioinformatics. 17 (2016): 495. [7] L. Zimmermann et al., J Mol Biol. 430 (2018): 2237-2243.
[8] S. Kumar et al., J. Comput. Chem., (2001) 8, 1011-1021.
(TCPL 201)
16:30 - 16:50 Marcelino Arciniega: Pruning false positives cases from small molecule docking results using machine learning techniques
Over the last three decades, Computer Aided Drug Design (CADD) has positioned as one of the more useful approaches aiding the research at early stages of drug discovery process [1]. Particularly, small molecule docking algorithms have been employed exhaustively to identify the possible atomic interactions, between the protein target and a suggested small molecule, that support the formation of the protein-ligand complex. This evaluation is performed by employing a scoring function that relates geometric patterns of the interacting molecules to free energy values. However, the accurate and exact prediction of the binding free energy, along with the complex conformation, remains as an open problem [2]. As consequence, docking results present a high rate of false positive cases. The high complexity of the physicochemical process, together with the vast amount of structural experimental information available, renders the use of machine learning algorithms an attractive possibility [3, 4, 5]. In the present work, we briefly describe the problems associated with current docking scoring functions and posit the idea of pruning the false positive cases using machine learning algorithms. Then, we expose the limitations the docking algorithm (not only of the scoring function) of AutodockVina [6] by analyzing a set of approximately 15000 crystallographic complexes retrieved from Protein Data Bank [7]. Finally, we present preliminary results obtained with our tools designed for false positive identification. Specifically, we show how a relatively simple Bayesian Network, based on interaction fingerprints, can be used to infer the badly placed fragment molecules (with molecular weights in the range of 150-350 Da). Additionally, we present the results obtained of a Convolutional Neural Network to analyze docking poses (molecules with molecular weights in the range of 150-850 Da). Both networks show promising results by improving Receiver Operating Characteristic metrics as compared with the use of the docking protocol alone.

Acknowledgments: This project was supported by Dirección General de Asuntos del Personal Académico at Universidad Nacional Autónoma de México (PAPIIT-IA202917). The authors thank Dirección General de Cómputo y de Tecnologías de Información y Comunicación at Universidad Nacional Autónoma de México for granting the use of the supercomputer Miztli (LANCAD-UNAM-DGTIC-320).

[1] G. Sliwoski, S. Kothiwale, J. Meiler, E. W. Lowe-Jr. Pharmacol. Rev., 66, 334-395, 2014.
[2] HA Carlson, et. al. J. Chem. Inf. Model., 56, 1063-1077, 2016.
[3] M. Arciniega, O. F. Lange. J. Chem. Inf. Model., 54, 1401-1411, 2014.
[4] J. C. Pereira, et. al. J. Chem. Inf. Model., 56, 2495-2506, 2016.
[5] M. Wójcikowski, P. J. Ballester, P. Siedlecki. Sci. Rep. 7, 46710, 2017.
[6] O. Trott, A. J. Olson. J. Compt. Chem. 31, 455-461, 2010.
[7] www.rcsb.org.
(TCPL 201)
17:30 - 19:30 Dinner
A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. Note that BIRS does not pay for meals for 2-day workshops.
(Vistas Dining Room)
19:30 - 22:00 Poster session + discussion (TCPL Foyer, TPCL 201)
Sunday, August 11
07:00 - 09:00 Breakfast (Vistas Dining Room)
09:00 - 11:10 Session 4. Models and algorithms. Chairperson: Changbong Hyeon (TCPL 201)
09:00 - 09:30 Carlos Brizuela: Limits of performance for protein side chain packers
To date, it is possible to design proteins with an improved function starting from known scaffolds. This design applies to the enhancement of enzymatic capabilities, inhibitors of protein-protein interactions, among others [1]. The next generation for the design of functional proteins will be guided by the approach known as template-free design. The goal of this approach is to design a sequence of amino acids that will have a predefined function. A more conservative approach seeks to find a chain of amino acids that will fold into a predefined backbone geometry. A central challenge to the latter approach is the side chain packing problem (SCPP) that aims to find a set of rotamers that minimizes a given scoring function, for a fixed backbone geometry associated to a candidate sequence. In this talk, we will define the computational model for the SCPP, analyze the results achieved by state-of-the-art packers, and determine a lower bound for the maximum achievable accuracy of a simple rotamer library [2]. We also show that a strong limitation to reduce the gap between state-of-the-art results and the maximum attainable accuracy is the scoring function. Furthermore, we show that the limitation in the scoring function is not related to an incorrect weighting of its components nor to the constrained geometry of the crystal [3].
[1]. P.S. Huang, S.E. Boyken, and D. Baker. ¨The coming of age of de novo protein design¨. Nature 537 (7620): 320 – 327, 2016.
[2]. J. Colbes, R.I. Corona, C. Lezcano, D. Rodriguez, C.A. Brizuela. “Protein aide-chain packing problem: is there still room for improvement?.” Briefings in Bioinformatics, doi:10.1093/bib/bbw079, 2016.
[3]. J. Colbes, S. Aguila, C.A. Brizuela. “Scoring of side-chain packings: An analysis of weight factors and molecular dynamics structures”. Journal of Chemical Information and Modeling, 58 (2), 443-452, 2018.
(TCPL 201)
09:30 - 10:00 Zhiyun Wu: A Global Subspace Optimization Algorithm for Minimum Energy Molecular Cluster Conformation
We consider the problem to obtain the optimal conformation of a given molecular cluster with the lowest possible potential energy. This problem has been studied as a test case for global optimization algorithms, and considered as a starting point for the study of more complicated conformational problems such as protein folding through potential energy minimization. Here we propose a global subspace optimization algorithm for the solution of the problem, with the variables of the energy function divided into subgroups and optimization performed successively in the subspaces corresponding to the subgroups of variables. The idea behind the algorithm comes from the study of group behaviors of biological populations, where species compete for resources yet find strategies to co-exist and co-evolve. We show that such behaviors can be modeled as a multi-player evolutionary game, and the potential energy minimization problem can be reduced to such a game, with each subgroup of variables considered as a strategy set to be determined by a subpopulation of species. Thus, the successive subspace minimization of an energy function proceeds like a game played among subgroups of species in a biological population. We show that a Nash-equilibrium of the game is equivalent to a KKT point of the energy minimization problem subject to a set of linear and nonnegative constraints, and an evolutionary stable equilibrium corresponds to a strict energy minimizer. We describe the implementation of the algorithm and present some preliminary test results for a small group of clusters.
(TCPL 201)
10:00 - 10:30 Coffee Break (TCPL Foyer)
10:30 - 10:50 Tamara Bidone: Computational model of kinetochore-microtubule attachments
The ability of cells to separate chromosomes during mitosis is critical to several phases of their physiology. Chromosome segregation is mediated by spindle microtubules that attach to mitotic kinetochores via a dynamic protein interface, which includes Ndc80 and its accessory proteins, Ska, Cdt1 and ch-TOG [1-3]. The Ndc80 complex forms the core component of the attachment sites while Ska, Cdt1 and ch-TOG binds kinetochores via the Ndc80 complex. From prometaphase to metaphase, the kinetochore levels of Ska and Cdt1 increase in Hela cells, while that of Ndc80 remains constant. This suggests a correlation between concentration of proteins at the kinetochore-microtubule (kMT) interface and increasing amounts of load during mitosis. Interestingly, while being dynamic, the kMT interface ensures stability of the connection between chromosomes and kinetochore microtubules. How the various interface proteins interplay to ensure a dynamic yet stable connection is not known because their exact roles in this process are still elusive. An interesting hypothesis is that the Ndc80-accessory proteins Ska, Cdt1 and ch-TOG directly strengthen the kinetochore-microtubule interface by forming additional connections between kinetochore-bound Ndc80 and spindle microtubules. However, since Ska, Cdt1 and ch-TOG dynamically form and break their connections with microtubules, a synergy between them is likely to exist. Here, in order to characterize the synergy between Ska, Cdt1 and ch-TOG, we developed a new computational model, based on a kinetic Monte Carlo approach. The model allowed us to explicitly incorporate Ndc80, Ska1, Cdt1 and ch-TOG, isolate their contributions, and characterize their synergistic effects on the stability of the interface. Each protein is defined by a position along a tubulin protofilament, and exists in two states, bound or unbound, while undergoing biased diffusion, as observed in experiments. The model also incorporates tension-dependent unbinding rates for each protein, including catch bond kinetics for ch-TOG, as detected experimentally [2]. As for the output, the model evaluates: (i) displacement of the kMT interface along the tubulin protofilament; (ii) time of kMT attachment under tension; and (iii) kMT attachment rupture force, corresponding to the force that detaches all proteins. We find that combining Ndc80, Ska and Cdt1 enhances kMT attachment strength with respect to individual components. Ch-TOG further strengthens the complex because of its catch bond kinetics. In addition, the model shows that the rupture force, corresponding to the load under which no protein is bound, increases in proportion to the number of simulated microtubules. Taken together, our results provide important mechanistic insights into how kMT proteins coordinate with each other to withstand tension and ensure accurate chromosome segregation.

[1] D. Varma, and E. D. Salmon. J. Cell. Sci., 2013.
[2] M. P. Miller, C.L. Asbury, and S. Biggins. Cell, 2016.
[3] S. Agarwal, K.P. Smith, Y. Zhou, A. Suzuki, R.J. McKenney, and D. Varma. J. Cell Biol., 2018
(TCPL 201)
10:50 - 11:10 Maribel Hernández Rosales: Graph Theory in Orthology Detection
During evolution genes go throw many events, such as duplication, speciation, loss, horizontal gene transfer, among others. Two genes are said to be paralogs if they are the product of a duplication event, and orthologs if they are the product of a speciation event. The distinction of paralogs and orthologs is an important problem in comparative and evolutionary genomics. Moreover, orthology detection is a first step towards any functional annotation study. The evolutionary history of a set of genes can be represented as a phylogenetic tree where leaves represent genes and internal nodes evolutionary events. In this work, we investigate a graph theory-based method for the prediction of large-scale orthologous genes and the reconstruction of their evolutionary history [1,2,3]. We represent genes as vertices of a graph and place an edge between two genes if their sequence similarity is high. We characterize mathematically the topological properties of this graph in order to have only valid orthology relations. Surprisingly, graphs that represent valid orthology relations are P4-free, i.e. graphs which do not contain induced paths of length four. These graphs have been studied earlier and have been characterized as cographs [4]. We further investigate a set of induced subgraphs that give us evidence of noise in the data set or of wrong orthology predictions. In order to remove those induced subgraphs, we need to come up with a solution for the cograph editing problem, which has been found to be NP-complete [5]. Here we also present a work-in-process heuristic for the cograph edting problem that will help us to induce valid orthology relations.

Acknowledgments: This research was supported by Conacyt Mexico and DAAD Germany.

[1] Marcus Lechner, Maribel Hernandez-Rosales, Daniel Doerr, Nicolas Wieseke, An- nelyse Thevenin, Jens Stoye, Sonja J. Prohaska and Peter F. Stadler. Orthology Detection Combining Clustering and Synteny for Very Large Data Sets. PlosONE, 9(8):e105015, (2014).
[2] Marc Hellmuth, Maribel Hernandez-Rosales, Katharina T. Huber, Vincent Moul- ton, Peter F. Stadler, and Nicolas Wieseke. Orthology relations, symbolic ul- trametrics, and cographs. J. Math. Biol. 66(1-2):399-420, (2013).
[3] Maribel Hernandez-Rosales, Marc Hellmuth, Nicolas Wieseke, Katharina Huber, Vincent Moulton, and Peter F. Stadler. From event-labeled gene trees to species trees. BMC Bioinformatics 13(Suppl. 19):S6, (2012)
[4] Corneil DG, Lerchs H, Stewart Burlingham LK. Complement reducible graphs. Discrete Appl Math 3:163–174, (1981).
[5] Liu Y, Wang J, Guo J, Chen J. Cographs editing: complexity and parametrized algorithms. In: Fu B, Du DZ (eds) COCOON 2011. Lecture notes computer science, vol 6842. Springer, Berlin, pp 110–121, (2011).
(TCPL 201)
11:10 - 11:30 Closing remarks (TCPL 201)
11:30 - 12:00 Checkout by Noon
2-day workshop participants are welcome to use BIRS facilities (Corbett Hall Lounge, TCPL, Reading Room) until 15:00 on Sunday, although participants are still required to checkout of the guest rooms by 12 noon. There is no coffee break service on Sunday afternoon, but self-serve coffee and tea are always available in the 2nd floor lounge, Corbett Hall.
(Front Desk – Professional Development Centre)


text:
# Revision history [back]

### Equivalent of Polynomial.list() for expression involving generator of GaloisField

I know that it is possible to use the method list() to get a list with the coefficients of a polynomial. For instance:

sage: S.<x> = PolynomialRing(ZZ, 'x')
sage: (1 - 5*x + 3*x**2 + 2*x**3).list()
[1, -5, 3, 2]

I would like to do something like that with an expression involving a generator of a Galois Field.

For example:

sage: q = 5
sage: m = 2
sage: F.<a> = GF(q**m)
sage: a**9
3*a + 1

So, ideally, I would like to do the following

(a**9).list()

and get

[1, 3]

Is there any simple way to that?

format: plain


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New interface

Positive solutions for large random linear systems

Abstract : Consider a large linear system where $A_n$ is a $n\times n$ matrix with independent real standard Gaussian entries, $\boldsymbol{1}_n$ is a $n\times 1$ vector of ones and with unknown the $n\times 1$ vector $\boldsymbol{x}_n$ satisfying $\boldsymbol{x}_n = \boldsymbol{1}_n +\frac 1{\alpha_n\sqrt{n}} A_n \boldsymbol{x}_n\, .$ We investigate the (componentwise) positivity of the solution $\boldsymbol{x}_n$ depending on the scaling factor $\alpha_n$ as the dimension $n$ goes to $\infty$. We prove that there is a sharp phase transition at the threshold $\alpha^*_n =\sqrt{2\log n}$: below the threshold ($\alpha_n\ll \sqrt{2\log n}$), $\boldsymbol{x}_n$ has negative components with probability tending to 1 while above ($\alpha_n\gg \sqrt{2\log n}$), all the vector's components are eventually positive with probability tending to 1. At the critical scaling $\alpha^*_n$, we provide a heuristics to evaluate the probability that $\boldsymbol{x}_n$ is positive. Such linear systems arise as solutions at equilibrium of large Lotka-Volterra systems of differential equations, widely used to describe large biological communities with interactions such as foodwebs for instance. In the domaine of positivity of the solution $\boldsymbol{x}_n$, that is when $\alpha_n\gg \sqrt{2\log n}$, we establish that the Lotka-Volterra system of differential equations whose solution at equilibrium is precisely $\x_n$ is stable in the sense that its jacobian ${\mathcal J}(\boldsymbol{x}_n) = \mathrm{diag}(\boldsymbol{x}_n)\left(-I_n + \frac {A_n}{\alpha_n\sqrt{n}}\right)$ has all its eigenvalues with negative real part with probability tending to one. Our results shed a new light and complement the understanding of feasibility and stability issues for large biological communities with interaction.
Document type :
Journal articles
Domain :
Complete list of metadata

https://hal.archives-ouvertes.fr/hal-02093170
Contributor : Jamal Najim Connect in order to contact the contributor
Submitted on : Monday, January 4, 2021 - 4:33:05 PM
Last modification on : Thursday, September 29, 2022 - 10:47:11 AM

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Citation

Pierre Bizeul, Jamal Najim. Positive solutions for large random linear systems. Proceedings of the American Mathematical Society, In press, pp.1. ⟨10.1090/proc/15383⟩. ⟨hal-02093170v2⟩

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Dimensional formula for thermal conductivity is (here $K$ denotes the temperature :
Question:

Dimensional formula for thermal conductivity is (here $K$ denotes the temperature :

1. $\mathrm{MLT}^{-2} \mathrm{~K}$

2. $\mathrm{MLT}^{-2} \mathrm{~K}^{-2}$

3. $\mathrm{MLT}^{-3} \mathrm{~K}$

4. $\mathrm{MLT}^{-3} \mathrm{~K}^{-1}$

Correct Option: , 4

Solution:

(4) From formula, $\frac{d Q}{d t}=k A \frac{d T}{d x}$

$\Rightarrow k=\frac{\left(\frac{d Q}{d t}\right)}{A\left(\frac{d T}{d x}\right)}$

$[k]=\frac{\left[\mathrm{ML}^{2} \mathrm{~T}^{-3}\right]}{\left[\mathrm{L}^{2}\right]\left[\mathrm{KL}^{-1}\right]}=\left[\mathrm{MLT}^{-3} \mathrm{~K}^{-1}\right]$


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# Thread: Help with this problems!asap

1. ## Help with this problems!asap

New here so be nice

How do you find the exact value of these expressions?

sin^2 15(degrees) cot^2 75(degrees) + cos^2 15(degrees) tan^2 75(degrees)

and

1-tan(pi/6)+2csc(pi/3)

Thanks a TONNE

2. I have tryed to solve them the best I could...

Im assuming that the first one is 0 because they are the inverse of eachother.

The second one I got 2.73 by converting tan(pi/6) to sqr3/3 and csc to 2/sqr3, then solving accordingly.

Does this sound correct?

3. Originally Posted by gabrie30

How do you find the exact value of these expressions?

sin^2 15(degrees) cot^2 75(degrees) + cos^2 15(degrees) tan^2 75(degrees)
Note that:
$\displaystyle sin(15^{\circ}) = sin(45^{\circ}-30^{\circ})$
$\displaystyle cot(75^{\circ}) = cot (45^{\circ} + 30^{\circ})$

Use the sum-difference formulas to solve them. (i.e. sin(a-b) = sin(a)cos(b)-cos(a)sin(b))

1-tan(pi/6)+2csc(pi/3)
$\displaystyle tan \frac{\pi}{6}$ has exact value $\displaystyle \frac{1}{\sqrt3}$

$\displaystyle cosec \frac{\pi}{3} = \frac{1}{sin \frac{\pi}{3}}\Rightarrow sin\frac{\pi}{3}$ has exact value $\displaystyle \frac{\sqrt3}{2}$

4. Hello, gabrie30!

Welcome aboard!

Find the exact value: .$\displaystyle \sin^2\!15^o\cot^2\!75^o + \cos^2\!15^o \tan^2\!75^o$
Use the Double Angle identities:

. . $\displaystyle \begin{array}{ccccccc}\sin^2\!\theta &=& \dfrac{1-\cos2\theta}{2} & & \cos^2\!\theta &=& \dfrac{1 + \cos2\theta}{2} \\ \\[-3mm] \tan^2\!\theta &=& \dfrac{1-\cos2\theta}{1 + \cos2\theta} & & \cot^2\theta &=& \dfrac{1 + \cos2\theta}{1-\cos2\theta} \end{array}$

We have: .$\displaystyle \frac{1-\cos30^o}{2}\cdot\frac{1 + \cos150^o}{1-\cos150^o} + \frac{1+\cos30^o}{2}\cdot\frac{1 - \cos150^o}{1 + \cos150^o}$

Since $\displaystyle \cos150^o \:=\:\text{-}\cos30^p$, we have: .$\displaystyle \frac{1-\cos30^o}{2}\cdot\frac{1-\cos30^o}{1+\cos30^o} + \frac{1+\cos30^o}{2}\cdot\frac{1+\cos30^o}{1-\cos30^o}$

. . which simplfies to: .$\displaystyle \frac{(1-\cos30^o)^3 + (1+\cos30^o)^3}{2(1-\cos^230^o)} \;=\;\frac{1+3\cos^2\!30^o}{\sin^2\!30^o}$

Therefore, we have: .$\displaystyle \frac{1 + 3\left(\frac{\sqrt{3}}{2}\right)^2}{\left(\frac{1} {2}\right)^2} \;=\;\frac{1 + 3\cdot\frac{3}{4}}{\frac{1}{4}} \;=\;\boxed{13}$

$\displaystyle 1-\tan\left(\frac{\pi}{6}\right) + 2\csc\left(\frac{\pi}{3}\right)$
These are standard angles . . .
We are expected to know that: .$\displaystyle \begin{array}{ccc}\tan\left(\dfrac{\pi}{6}\right) &=& \dfrac{1}{\sqrt{3}} \\ \\[-3mm] \csc\left(\dfrac{\pi}{3}\right) &=&\dfrac{2}{\sqrt{3}} \end{array}$


text:
View Single Post
P: 10

## Capacitance of a Fractal Surface

You said, "The concept of capacitance does not really make sense if there is only one conductor," however it was in that context that I was asking the question. I'm actually pretty comfortable and familiar with real electronic circuit capacitors and their "two plates" and how they work. But what I was asking about was situations where you have just a single object and you want to know how much charge it can hold at a given potential. For example, a VanDegraff generator with a spherical top will have some particular charge (in microcoulombs) at some particular potential (in killovolts) for some particular sphere radius. Now I'm a bit fuzzy on the following statement, but here it is: I think you pretend the other plate is another sphere of infinite radius at an infinite distance, and it becomes a math problem with some value that you take the limit as it tends to infinity or some such approach.

In practice, I believe such measurements are with respect to ground which we assume is at a neutral charge and zero potential. So when there is 350kV wrt ground on the sphere of a VanDegraff generator, we can calculate not only the "capacitance" of the sphere, but can use Q=CV to solve for the charge on the sphere. And I've seen such "solved problems" in many text books, so I know the idea of calculating the capacitance of a free object is not unusual.

Now I know traditionally capacitors were constructed as long strips of almost a sort of tinfoil and wax paper and were all rolled up to cram a huge area into a small volume, and I'm good with that, however when you do that, the surfaces of the plates remain parallel to one another (with the advantage that BOTH sides of BOTH plates are "looking" at each other thereby doubling the capacitance). If you "fractalled" them all up in a convoluted mess, but ensured that the surfaces were not allowed to cross the original boundary, would the capacitance change? i.e. the wax paper must remain stretched tight as original, but the surfaces of the plates can have ruts and textures and places where the material opens into huge "caverns" with tiny openings, etc. etc. I'm not certain how I'd model that. But I have a suspicion that it wouldn't make more "capacitance".

Let me suggest a particular configuration as an example to make my point. Lets say we have a two (identical) plate capacitor, where each plate is described as follows: Each plate is a thin rectangle, but the inside has been "hollowed out." Clearly this fact would be immaterial if that were the end of it all, and the capacitance would be calculated the same as if the plate was solid. But now, lets say that we cut a sliver out in the surface of the plates so that the hollowed interior is "accessible" and each plate can "see inside" the hollowed portion of the other through this "slit". I have a lot of trouble seeing that suddenly the capacitance will now double or triple now, even though we now have the original surface, the back of the surface, and the inside surface of the opposite side, now all "accessible" because of the slit. Clearly the surface area is four times more than the area of the facing side of the plate, and three quarters of this surface area now all "face towards" the other plate. One quarter of it is on the opposite side and everyone would agree it remains excluded from discussion just as when the plate is solid.

Now when I look at this, I tend to think that because all this new surface area is not really "line of sight" to the other plate, it doesn't really count. Its a hollow, cavernous region inside the plate with a little slit leading to the outside world.

So that's in the context of a two plate capacitor. I could be wrong, but I think it makes no difference. However, in the context of a single free object in space that we are calculating the charge on, I'm not as convinced, because it DOES represent an area that free charge can occupy, because even if its "inside", its still "on the surface" ... i.e. the surface is still a surface, even if you have to go through a little slit to reach it. But I really don't know, which is why I'm posting this ...


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# The number of marshmallows in a 24 oz. box of Lucky Charms Cereal is normally distributed...

###### Question:

The number of marshmallows in a 24 oz. box of Lucky Charms Cereal is normally distributed with a mean of 100 marshmallows and a standard deviation of 15 marshmallows.

A) What is the probability that a box of cereal will have no more than 110 marshmallows in it? a. 0.75 b. 0.68 c. 0.83 d. 0

B) What is the probability that a box of cereal will have between 80 to 120 marshmallows? a. 0.32 b. 0.67 c. 0.82 d. 0.5

C) How many marshmallows must be in a box so that the box is in the top 3% with respect to marshmallow content? a. >106 b. >112 c. >128 d. >142

D) The manager wants 0.9 probability that a box of cereal has 60 or more marshmallows. The process remains normally distributed and the standard deviation of process remains the same (i.e., 15 marshmallows). What should be the mean of the number of marshmallows in a box of cereal? a. 85 b. 79 c. 105 d. 105

In this process the speed at which a box of cereal is filled and sealed is uniformly distributed and takes between 30 to 45 seconds.

A) What is the probability that a box can be filled and sealed in less than 42 seconds? a. 0.08 b. 0.6 c. 0.5 d. 0.25

B) What is the probability that the time it takes to fill and seal a box is between 40 to 50 seconds? a. 0.33 b. 0.50 c. 0.67 d. 0.25

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text:
# [Math] permutations with the English alphabet

combinatoricspermutations

How many four-letter words, using the English alphabet, are possible if letters if only vowels may be repeated? How many four-letter words are there if at most one repetition of any letter is allowed? Or if exactly one repetition is allowed?

I am trying to answer these questions. I know there are 26 letters in the english alphabet. if no letters can be repeated, it would be 26*25*24*23. however, my problem is if only one repetition is allowed, i don't know which letter would be repeated. 26*26*25*24 is different than 26*25*25*24. how do i get around this? I was wondering if there would be a way to subtract from the total possible number of words(26^4). Or add the four letter words with no repetitions (26*25*24*23) plus the ones with one repetition. Unfortunately, it seems like im still stuck with that one-repetition part. any help would be greatly appreciated!

If we have exactly one repetition is allowed, we start by choosing our three letters: $\binom{26}{3}$. We then choose a letter to be repeated in $\binom{3}{1} = 3$ ways. Finally, we permute our letters in $4!/2!$ ways by the multinomial distribution. By rule of product, we multiply, to get:
$$\binom{26}{3} \cdot 3 \cdot 4!/2!$$
This is correct thinking. Notice that the words with no repetition are disjoint from words with exactly one repetition. So by rule of sum, you add $26!/(26-4)!$ with the quantity I noted above.


text:
Home Board Paper Solutions Text Book Solutions Articles NCERT Exemplar Solutions
Question 10

# Let A = R – {3} and B = R – {1}. Consider the function  f : A → B defined by. Is f one-one and onto? Justify your answer.

Answer

A = R - {3}, B = R - {1}

f: A → B is defined as.

.

∴ f is one-one.

Let y ∈B = R - {1}. Then, y ≠ 1.

The function f is onto if there exists x ∈A such that f(x) = y.

Now,

Thus, for any y ∈ B, there existssuch that

Hence, function f is one-one and onto.

#### Write a Comment:

• NCERT Chapter


text:
# Non-linear Regression Demo¶

☝Before moving on with this demo you might want to take a look at:

Polynomial regression is a form of regression analysis in which the relationship between the independent variable x and the dependent variable y is modelled as an nth degree polynomial in x. Although polynomial regression fits a nonlinear model to the data, as a statistical estimation problem it is linear, in the sense that the regression function E(y|x) is linear in the unknown parameters that are estimated from the data. For this reason, polynomial regression is considered to be a special case of multiple linear regression.

Demo Project: In this example we will train our model to imitate an artificial non-linear equation.

In [1]:
# To make debugging of linear_regression module easier we enable imported modules autoreloading feature.
# By doing this you may change the code of linear_regression library and all these changes will be available here.

import sys
sys.path.append('../..')

### Import Dependencies¶

• pandas - library that we will use for loading and displaying the data in a table
• numpy - library that we will use for linear algebra operations
• matplotlib - library that we will use for plotting the data
• linear_regression - custom implementation of linear regression
In [2]:
# Import 3rd party dependencies.
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

# Import custom linear regression implementation.

In this demo we will use artificial non-linear data set.

In [3]:
# Load the data.

# Fetch traingin set and labels.
x = data['x'].values.reshape((data.shape[0], 1))
y = data['y'].values.reshape((data.shape[0], 1))

# Print the data table.

Out[3]:
y x
0 97.58776 1.0
1 97.76344 2.0
2 96.56705 3.0
3 92.52037 4.0
4 91.15097 5.0
5 95.21728 6.0
6 90.21355 7.0
7 89.29235 8.0
8 91.51479 9.0
9 89.60966 10.0

### Plot the Data¶

Let's visualize the training and test datasets to see the shape of the data.

In [4]:
plt.plot(x, y)
plt.show()

### Init and Train Linear Regression Model¶

☝🏻This is the place where you might want to play with model configuration.

• polynomial_degree - this parameter will allow you to add additional polynomial features of certain degree. More features - more curved the line will be.
• num_iterations - this is the number of iterations that gradient descent algorithm will use to find the minimum of a cost function. Low numbers may prevent gradient descent from reaching the minimum. High numbers will make the algorithm work longer without improving its accuracy.
• learning_rate - this is the size of the gradient descent step. Small learning step will make algorithm work longer and will probably require more iterations to reach the minimum of the cost function. Big learning steps may couse missing the minimum and growth of the cost function value with new iterations.
• regularization_param - parameter that will fight overfitting. The higher the parameter, the simplier is the model will be.
• polynomial_degree - the degree of additional polynomial features (x1^2 * x2, x1^2 * x2^2, ...). This will allow you to curve the predictions.
• sinusoid_degree - the degree of sinusoid parameter multipliers of additional features (sin(x), sin(2*x), ...). This will allow you to curve the predictions by adding sinusoidal component to the prediction curve.
In [5]:
# Set up linear regression parameters.
num_iterations = 50000  # Number of gradient descent iterations.
regularization_param = 0  # Helps to fight model overfitting.
learning_rate = 0.02  # The size of the gradient descent step.
polynomial_degree = 15  # The degree of additional polynomial features.
sinusoid_degree = 15  # The degree of sinusoid parameter multipliers of additional features.
normalize_data = True  # Flag that indicates that data needs to be normalized before training.

# Init linear regression instance.
linear_regression = LinearRegression(x, y, polynomial_degree, sinusoid_degree, normalize_data)

# Train linear regression.
(theta, cost_history) = linear_regression.train(
learning_rate,
regularization_param,
num_iterations
)

# Print training results.
print('Initial cost: {:.2f}'.format(cost_history[0]))
print('Optimized cost: {:.2f}'.format(cost_history[-1]))

# Print model parameters
theta_table = pd.DataFrame({'Model Parameters': theta.flatten()})
theta_table

Initial cost: 137723281.24
Optimized cost: 2189884.51

Out[5]:
Model Parameters
0 60.531874
1 -3.730982
2 -3.265008
3 -4.293987
4 -3.048819
5 -1.617722
6 -1.801005
7 -3.069723
8 -3.551792
9 -2.751712
10 -2.019922
11 -2.086688
12 -1.761378
13 -0.340730
14 0.214782
15 -1.184080
16 0.614349
17 -3.730982
18 -3.730982
19 -40.780573
20 -40.780573
21 -40.780573
22 -1.502418
23 -1.502418
24 -1.502418
25 -1.502418
26 23.068044
27 23.068044
28 23.068044
29 23.068044
... ...
122 6.423522
123 6.423522
124 6.423522
125 6.423522
126 6.423522
127 6.423522
128 6.423522
129 6.423522
130 6.423522
131 6.423522
132 6.423522
133 6.423522
134 6.423522
135 6.423522
136 0.052387
137 0.052387
138 0.052387
139 0.052387
140 0.052387
141 0.052387
142 0.052387
143 0.052387
144 0.052387
145 0.052387
146 0.052387
147 0.052387
148 0.052387
149 0.052387
150 0.052387
151 0.052387

152 rows × 1 columns

The plot below illustrates how the cost function value changes over each iteration. You should see it decreasing.

In case if cost function value increases it may mean that gradient descent missed the cost function minimum and with each step it goes further away from it. In this case you might want to reduce the learning rate parameter (the size of the gradient step).

From this plot you may also get an understanding of how many iterations you need to get an optimal value of the cost function. In current example you may see that there is no much sense to increase the number of gradient descent iterations over 500 since it will not reduce cost function significantly.

In [6]:
# Plot gradient descent progress.
plt.plot(range(num_iterations), cost_history)
plt.xlabel('Iterations')
plt.ylabel('Cost')
plt.show()

### Plot the Model Predictions¶

Since our model is trained now we may plot its predictions over the training and test datasets to see how well it fits the data.

In [7]:
# Get model predictions for the trainint set.
predictions_num = 1000
x_predictions = np.linspace(x.min(), x.max(), predictions_num).reshape(predictions_num, 1);
y_predictions = linear_regression.predict(x_predictions)

# Plot training data with predictions.
plt.scatter(x, y, label='Training Dataset')
plt.plot(x_predictions, y_predictions, 'r', label='Prediction')
plt.show()

You may see from the plot how well our model predicts the training set examples.


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# Lower Semicontinuity Equivalence

Let be $$(X , ||\cdot||)$$ a normed space and $$f:X \longrightarrow \mathbb{R}$$ a function. $$f$$ is lower semicontinuous if $$\{x \in X:f(x) \leq c \}$$ is closed $$\forall c \in \mathbb{R}$$, and is said to be sequentially lower semicontinuous at $$x_0$$ if for each sequence $$\{x_n\}_{n \in \mathbb{R}}$$ that converges to $$x_0$$ is verified that: $$\begin{equation*} f(x_0) \leq \liminf_{n \to \infty}f(x_n) \end{equation*}$$ How to prove that in a normed space (the statement is valid for a first countable space) both conditions are equivalent, (f is a lower semicontinuous function if and only if is sequantially lower semicontinuous at each $$x_0 \in X$$).

Necessity: Suppose $$f$$ is lower semicontinuous and let $$\{x_n:n\in \mathbb{N}\}$$ be a sequence that converges to $$x$$. For any $$\alpha>f(x)$$ the set $$V=\{f>\alpha\}$$ is an open neighborhood of $$x$$. Hence there is $$n_0\in \mathbb{N}$$ such that $$n\geq n_0$$ implies that $$f(x_n)>\alpha$$; this implies that $$\alpha\leq\liminf_nf(x_n)$$. The conclusion follows by letting $$\alpha\rightarrow f(x)$$.

Sufficiency: It is enough to show that $$F_\alpha:=\{f\leq \alpha\}$$ is closed for any $$\alpha\in\mathbb{R}$$. Let $$\{x_n:n\in \mathbb{N}\}$$ be a sequence in $$F_\alpha$$ that converges to a point $$x\in X$$. Then $$f(x_n)\leq \alpha$$ for all $$n\in \mathbb{N}$$, and so \begin{aligned} f(x)\leq \liminf_nf(x_n)= \sup_{n\in \mathbb{N}}\inf_{m\in \mathbb{N}: m\geq n}f(x_m)\leq \alpha. \end{aligned} Therefore $$x\in F_\alpha$$.

Note: For general Hausdorff topological spaces, the statement holds after substituting sequences by nets.

• Ok, but if the sequentially lower semicontinuous definition remains (not with nets or filters) the sufficiency fails in the $T_2$ topological spaces right? Apr 27 '20 at 20:25
• Not quite sure follow your question. If the space (besides Hausdorff separability) is also first countable, then it is enough to consider sequences; otherwise nets are a good device to probe continuity-like properties. In you case, sequences are enough because your space id a normed space. Apr 27 '20 at 23:36
• I mean, only the Hausdorff separability is not enough for use sequences instead nest or filters. Apr 28 '20 at 14:57
• That is what I said earlier. For general Hausdorff spaces, if one used nets in place of sequences, the statement holds true. Apr 28 '20 at 17:03

I'll call $$f$$ "set-lower semicontinuous" if $$f^{-1}(-\infty,c]$$ is closed for all $$c\in\mathbb{R}$$, and "net-lower semicontinuous" if $$f(x_0)\leq\liminf f(x_n)$$ whenever $$x_n\to x_0$$.

First assume that $$f$$ is set-lower semicontinuous, and suppose $$x_n\to x_0$$. Given $$\epsilon>0$$, let $$c_\epsilon=f(x_0)-\epsilon$$. Then $$f^{-1}(-\infty,c_\epsilon]$$ is closed, thus $$f^{-1}(c_\epsilon,\infty)$$ is open, and hence contains a tail $$\left\{x_n:n\geq N\right\}$$ of the sequence $$\left\{x_n\right\}$$. Thus $$x_n\in f^{-1}(c_\epsilon,\infty)$$ for $$n\geq N$$, i.e. $$c_\epsilon\leq\inf_{n\geq N}f(x_n)\leq\liminf f(x_n).$$

Taking $$\epsilon\to 0$$ we have $$c_\epsilon\to f(x_0)$$, so we conclude that $$f$$ is net-lower semicontinuous.

In the other direction, suppose $$f$$ is net lower-semicontinuous. Let $$c\in\mathbb{R}$$ and $$x_n\to x_0$$ with $$x_n\in f^{-1}(-\infty,c]$$. We need to prove that $$x_0\in f^{-1}(-\infty,c]$$. The net-lower semicontinuous condition yields $$f(x_0)\leq\liminf_n f(x_n)=\sup_N\inf_{n\geq N}f(x_n)\leq\sup_N c=c,$$ so $$x_0\in f^{-1}(-\infty,c]$$.


text:
CAT Quant Practice Problems

Question: If x = – 0.5, then which of the following has the smallest value?

 ${2^{\frac{1}{x}}}$ $\frac{1}{x}$ $\frac{1}{{{x^2}}}$ ${2^x}$ $\frac{1}{{\sqrt { - x} }}$

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text:
10.02.2018

# Research

### Time-dependent density-functional theory for strongly interacting electrons

We consider an analytically solvable model of two interacting electrons that allows for the calculation of the exact exchange-correlation kernel of time-dependent density functional theory. This kernel, as well as the corresponding density response function, is studied in the limit of large repulsive interactions between the electrons and we give analytical results for these quantities as an asymptotic expansion in powers of the square root of the interaction strength. We find that in the strong interaction limit the three leading terms in the expansion of the kernel act instantaneously while memory terms only appear in the next orders. We further derive an alternative expansion for the kernel in the strong interaction limit on the basis of the theory developed previously [Phys. Chem. Chem. Phys. 18, 21092 (2016)] using the formalism of strictly correlated electrons in the adiabatic approximation. We find that the first two leading terms in this series, corresponding to the strictly correlated limit and its zero-point vibration correction, coincide with the two leading terms of the exact expansion. We finally analyze the spatial nonlocality of these terms and show when the adiabatic approximation breaks down. The ability to reproduce the exact kernel in the strong interaction limit indicates that the adiabatic strictly correlated electron formalism is useful for studying the density response and excitation properties of other systems with strong electronic interactions.

Luis Cort, Daniel Karlsson, Giovanna Lani, and Robert van Leeuwen, Physical Review A95, 042505 (2017)

### Partial self-consistency and analyticity in many-body perturbation theory: Particle numberconservation and a generalized sum rule

We consider a general class of approximations which guarantees the conservation of particle number in many-body perturbation theory. To do this we extend the concept of derivability for the self-energy to a larger class of diagrammatic terms in which only some of the Green’s function lines contain the fully dressed Green’s function G. We call the corresponding approximations for partially derivable. A special subclass of such approximations, which are gauge invariant, is obtained by dressing loops in the diagrammatic expansion of consistently with G. These approximations are number conserving but do not have to fulfill other conservation laws, such as the conservation of energy and momentum. From our formalism we can easily deduce whether commonly used approximations will fulfill the continuity equation, which implies particle number conservation. We further show how the concept of partial derivability plays an important role in the derivation of a generalized sum rule for the particle number, which reduces to the Luttinger-Ward theorem in the case of a homogeneous electron gas, and the Friedel sum rule in the case of the Anderson model. To do this we need to ensure that the Green’s function has certain complex analytic properties, which can be guaranteed if the spectral function is positive-semidefinite. The latter property can be ensured for a subset of partially -derivable approximations for the self-energy, namely those that can be constructed from squares of so-called half diagrams. For the case in which the analytic requirements are not fulfilled we highlight a number of subtle issues related to branch cuts, pole structure, and multivaluedness. We also show that various schemes of computing the particle number are consistent for particle number conserving approximations.

Daniel Karlsson and Robert van Leeuwen, Physical Review B94, 125124 (2016)

### Vertex Corrections for Positive-Definite Spectral Functions of Simple Metals

We present a systematic study of vertex corrections in a homogeneous electron gas at metallic densities. The vertex diagrams are built using a recently proposed positive-definite diagrammatic expansion for the spectral function. The vertex function not only provides corrections to the well known plasmon and particle-hole scatterings, but also gives rise to new physical processes such as the generation of two plasmon excitations or the decay of the one-particle state into a two-particle–one-hole state. By an efficient Monte Carlo momentum integration we are able to show that the additional scattering channels are responsible for a reduction of the bandwidth, the appearance of a secondary plasmon satellite below the Fermi level, and a substantial redistribution of spectral weights. The feasibility of the approach for first-principles band-structure calculations is also discussed.

Yaroslav Pavlyukh, Anna-Maija Uimobnen, Gianluca Stefanucci, Robert van Leeuwen, Physical Review Letters 117, 206402, (2016)

#### Transient dynamics in correlated  quantum transport

We study quantum transport of interacting many-electron systems in the time-domain, with a focus on transient processes.

In the figure we show a chain of four atoms coupled to quasi two-dimensional electronic leads. The electrons are pushed through the four-atom chain by a bias voltage, where they interact with each other. These interactions are described by diagrammatic method in terms of Feynman diagrams. On the upper right hand side you can see the electronic density waves that appear in the electrodes in the course of time, whereas in the lower panels you can see how the currents develop in time for different voltages applied to the electrodes. The lower left figure is a mean field approach whereas the lower right one describes many-body electrons that go beyond mean field. As you can see, many-body interactions, considerably affect the properties of the molecular junction. On all the quantum transport work we have a wonderful collaboration with Gianluca Stefanucci of the University of Rome Tor Vergata, Italy. More information on the method and the applications can be found in our papers:

• N. E. Dahlen and R. van Leeuwen, Solving the Kadanoff-Baym equations for inhomogeneous systems: Application to atoms and molecules, Phys. Rev. Lett. 98, 153004 (2007)
• P. Myöhänen, A. Stan, G. Stefanucci, R. van Leeuwen, Conserving approximations in time-dependent quantum transport: Initial correlations and memory effects Europhys. Lett. 84, 67001 (2008)
• P. Myöhänen, A. Stan, G. Stefanucci, R. van Leeuwen, Kadanoff-Baym approach to quantum transport through interaction nanoscale systems: From the transient to the steady-state regime Phys. Rev. B 80, 115107 (2009)
• A.-M. Uimonen, E. Khosravi, A. Stan, G. Stefanucci, S. Kurth, R. van Leeuwen, Comparative study of many-body perturbation theory and time-dependent density functional theory in the out-of-equilibrium Anderson model Phys. Rev. B 84, 115103 (2011)

#### Coupled systems of electrons and phonons

Another application of Green's function methods is the study of electron-phonon coupled systems. Electrons in a crystal lattice or in a molecule do not experience a static field from the atomic nuclei but the nuclei move themselves as well. The nuclear motion is much slower than the electronic motion and therefore it is often a good approximation to use the Born-Oppenheimer approximation. However, there are many other situations in which nuclear motion leads to drastically new physics, the most well-known is probably the phenomenon of superconductivity. Also in many other situations the coupling between electrons and nuclear vibrations (also known as phonons) is of key importance. Electronic charges deform the local lattice or molecular structure and electrons can be dressed by lattice deformations to form a new quasi-particle known as the polaron that can considerably change the excitation properties of materials. In some organic molecules, such as polyacetylene, the polaron can have very special features and become a travelling soliton.

The molecular or lattice vibrations are quantized and can be theoretically described as bosonic field in which the number of bosons (excitations of the phonon modes) is not conserved (and hence the system does not Bose condense). The bosons interact with the electrons which, being fermions, are described by an anti-commuting field. This system of coupled fields can be studied by diagrammatic methods and the dynamics of coupled electron-phonon systems can be studied using time-propagation of the nonequilibrium Green's functions. In the figure we display how the properties of a single electron (described by the operators $\hat{b}$ and $\hat{b}^\dagger$) change when the coupling to a single phonon-mode is (described by the operators $\hat{a}$ and $\hat{a}^\dagger$) suddenly switched on. The energy of the electron (being $\epsilon_0=0$ initially) is lowered by the dressing of phonons to form a new polaronic ground state. Also visible are phonon side bands that describe excited states of the polaron. On this topic we collaborate with Heiko Appel and Yang Peng of the Fritz-Haber Institute in Berlin, Germany. More information on our research in this topic is here:

• N. Säkkinen and R. van Leeuwen, Electrons and Phonons with the Kadanoff-Baym formalism, poster Lausanne workshop on vibrational coupling, 2012

#### Nonlinear phenomena and bistability

The equations of motion for the particle and hole propagators in many-body theory are nonlinear integro-differential equations. The reason for the nonlinearity is that the Feynman diagrams are polynomials of the particle and hole propagators as a consequence of the various particle-particle scattering processes that they describe.

When we carry out re-summations to obtain dressed propagators we obtain consequently nonlinear self-consistent equations. It is well known that nonlinear differential equations can lead to important physical phenomena (e.g. solitons, chaotic systems). However, much less is known about nonlinear integro-differential equations. An important physical characteristic of the integral kernels that appear in such equations is that they carry memory. A simple physical picture that goes with this memory is that the way two particles interact or collide depends on how they collided earlier. When we completely neglect memory effects we are in the mean-field approximation. We can therefore ask whether the combination of nonlinearity and memory can lead to new non-equilibrium phenomena. We have investigated this for the case of a quantum transport system. We have been investigating this for the creation of steady-state currents in the out-of-equilibrium Anderson model (essentially a single interacting atom coupled to leads). We find that within the mean-field approximation we can push the system in multple steady states depending on how fast we switch on the bias voltage. In the figure we show the non-equilibrium spectral function in the time-dependent Hartree-Fock approxmation that illustrates how the atomic level switches in time to another energy state, while passing through a third unstable state. We found, however, that more sophisticated approach with memory kernels destroy the bistability. The project was carried out in collaboration with Gianluca Stefanucci of the University of Rome Tor Vergata, Italy, Stefan Kurth of the University of San Sebastian, Spain and Elham Khosravi and Hardy Gross of the Max-Planck Institute in Halle, Germany.

• A.-M. Uimonen, E. Khosravi, G. Stefanucci, S. Kurth, R. van Leeuwen and E. K. U. Gross, Real-time switching between multiple steady-states in quantum transport, J. Phys. Conf. Ser. 220, 012018 (2010)
• E. Khosravi, A. - M. Uimonen, A. Stan, G. Stefanucci, S. Kurth, R. van Leeuwen and E. K. U. Gross, Correlation effects in bistability at the nanoscale: steady state and beyond, Phys. Rev. B 85, 075103 (2012)

#### Theory of general initial states

The dynamics of a many-particle quantum state is an initial state problem. In experiment initial quantum states are often prepared such that they are eigenstates of specific operators. There are, however, also situations in which more general initial states are studied. This happens, for instance, in the relaxation dynamics of ultracold gases, response of nanoscale systems in nonequilibrium states and in optimal control theory. The solution of the initial state problem is obvious from the viewpoint of the time-dependent Schrödinger equation for which the evolution of the state is determined by time-propagation of a linear partial differential equation once we specific the initial state. The problem is that, in practice, we can not solve the time-dependent Schrödinger equation due to the large number of degrees of freedom in most many-body systems.

However, we have a number of methods based on reduced quantities, such as many-body Green's function theory. We can therefore as ourselves the question how to describe the initial state in this approach. If the initial state is a correlated ground state in thermodynamic equilibrium there is a special technique based on imaginary time propagation to calculate it (the so-called Matsubara technique). However, for more general initial states this is not possible. To incorporate such general initial states in the Green's function formalism one needs to specify initial n-body correlations by means of n-body density matrices. These density matrices appear as correlation blocks in the Feynman diagrams. The exact mathematical prescription for doing this was derived in our group in collaboration with Gianluca Stefanucci of the University of Rome Tor Vergata, Italy. The formalism will be applied to study systems in which the initial correlated state is hard to calculate in standard many-body theory. For more information see

• R. van Leeuwen and G. Stefanucci, Wick theorem for general initial states, Phys. Rev. B 85, 115119 (2012)

#### Vertex corrections

The many-body approaches that we have been developing are based on expansion of the self-energy operator (which provides an effective field for added or removed particles) in powers of the particle and hole propagators (the Green's function) and the interaction. The self-energy can alternatively be expanded in terms of the Green's function G and the screened interaction W (the renormalized interaction between the electrons). This is of special importance in extended systems where bare interactions lead to infinities and the screened interactions presents a natural and physically interpretable quantity to expand in instead. In most of our calculations we have been using the lowest orders in this expansion (the so-called GW- approximation).

It is, however, known that a self-consistent solution of the GW equations in an extended system such as the electron gas can give unphysical spectral properties (washing out of plasmon structure). The natural step to go beyond GW and to possibly cure this problem would be to take the diagram to second order in the screened interaction (the so-called vertex diagram). Apart from being a higher order diagram, it also contains new physics. In the electron gas it would contain multiple electron-hole excitations and plasmon satellites that also have been observed in experiments on metals. Work on this project is in progress in collaboration with Yaroslav Pavlyukh of the University of Kaiserslautern, Germany.

#### Mathematical foundations of time-dependent density functional theory

The foundations of time-dependent density functional theory rest on two important assumptions

• The uniqueness of a density-potential mapping for a given initial state
• The existence of a Kohn-Sham system

Both assumptions can be proven under various mathematical conditions. The uniqueness was proven by Runge and Gross under the assumption of Taylor-expandability (and the subsequent assumption of convergence of the Taylor series) of the external potential around the initial time. The existence of the Kohn-Sham system requires that a given density can be reproduced in a different system with different two-body interactions. This was proven under the same assumptions by our own research group. Recently we have been developing a new proof that widens the range of validity of the uniqueness and existence assumption. The proof is on a fixed point mapping between normed (Banach) spaces of potentials, and does not need the assumption of Taylor expandability. We have been further elucidating the structure of the (singular) Sturm-Liouville problem related to the spatial boundary conditions of the density-potential mapping. We further aim to elucidate for which class of potentials (appropriate Sobolev spaces) the uniqueness and existence theorems can be applied. Strong forces behind this project are our collaborators Michael Ruggenthaler and Markus Penz of the University of Innsbruck, Austria. For more information see:

• R. van Leeuwen, Mapping from densities to potentials in time-dependent density-functional theory Phys. Rev. Lett. 82, 3863 (1999)
• M. Penz and M. Ruggenthaler, Domains of time-dependent density-potential mappings J. Phys. A: Math. Theor. 44, 335208 (2011)
• M. Ruggenthaler and R. van Leeuwen, Global fixed-point proof of time-dependent density-functional theory Europhys. Lett. 95, 13001 (2011)
• M. Ruggenthaler, K. J. H. Giesbertz, M. Penz and R. van Leeuwen, Density-potential mappings in quantum dynamics Phys. Rev. A 85, 052504 (2012)

#### Numerical construction of density-potential mappings

The mathematical work on the foundations of time-dependent density functional theory has also provided a constructive scheme to construct the external potential that produces a given a time-dependent density. The method therefore solves the optimal control problem which potential produces a given time-dependent density profile for a given initial state. The method has the advantage that it can be applied to general initial states and systems with arbitrary interactions. Furthermore it is numerically stable even for large density changes. When applied to non-interacting systems it allows to construct the Kohn-Sham potential corresponding to a given density.

An example of our construction scheme is given in the figure. We consider two non-interacting particles on a one-dimensional ring. The initial state is a correlation two-body state which is not an eigenstate of the system. As a consequence this two-particle state develops in time with a strong time-dependence (see the figure in the left panel). We require, however, that the density is stationary and remains identical to its value at the initial time (see the inset panel in left hand side of the figure). To achieve this the external potential (which can not be stationary as the initial state is not an eigenstate) must have a very non-trivial time-dependence. We have therefore constructed a counter intuitive situation. Although the two-body correlations in the wave function have a strong time-dependence, this information is integrated out in the one-body density. The strong forces between this project are our collaborators Søren Nielsen of the University of Aarhus, Denmark and Michael Ruggenthaler of the University of Innsbruck, Austria. For more information see:

• S. E. B. Nielsen, M. Ruggenthaler and R. van Leeuwen, Many-body quantum dynamics from the density, Europhys. Lett. 101, 33001 (2013)
• M. Ruggenthaler, S. E. B. Nielsen, and R. van Leeuwen, Exact density functionals for correlated dynamics on a quantum ring, arXiv:1209.2949v2.


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# Reynolds averaging

(Difference between revisions)
 Revision as of 09:57, 7 September 2005 (view source)Jola (Talk | contribs)← Older edit Latest revision as of 09:59, 7 September 2005 (view source)Jola (Talk | contribs) Line 9: Line 9: - Where $T$ is a long enough time to average out the typical fluctuations in $\Phi$. + Where $T$ is a long enough time to average out the fluctuations in $\Phi$. - Reynolds averaging is often used in fluid dynamics to separate turbulent fluctuations from the mean-flow. The term "Reynolds averaging" originates from [[Osborne Reynolds]], who was the first to use this type of averaging in fluid dynamics. + Reynolds averaging is often used in fluid dynamics to separate turbulent fluctuations from the mean-flow. The term Reynolds averaging originates from [[Osborne Reynolds]], who was the first to use this type of averaging in fluid dynamics.

## Latest revision as of 09:59, 7 September 2005

Reynolds averaging refers to the process of averaging a variable or an equation in time. Let $\Phi$ be any dependent variable that varies in time. This variable can be decomposed into a fluctuating part, $\Phi'$ and an average part $\overline{\Phi}$ in the following way:

 $\overline{\Phi} \equiv \frac{1}{T} \int_T \Phi(t) dt$ (1) $\Phi' \equiv \Phi - \overline{\Phi}$

Where $T$ is a long enough time to average out the fluctuations in $\Phi$.

Reynolds averaging is often used in fluid dynamics to separate turbulent fluctuations from the mean-flow. The term Reynolds averaging originates from Osborne Reynolds, who was the first to use this type of averaging in fluid dynamics.


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BS.univar {changepoints} R Documentation

## Standard binary segmentation for univariate mean change points detection.

### Description

Perform standard binary segmentation for univariate mean change points detection.

### Usage

BS.univar(y, s, e, delta = 2, level = 0)

### Arguments

 y A numeric vector of observations. s A integer scalar of starting index. e A integer scalar of ending index. delta A positive numeric scalar of minimum spacing. level Should be fixed as 0.

### Value

An object of class "BS", which is a list with the following structure:

 S A vector of estimated change point locations (sorted in strictly increasing order). Dval A vector of values of CUSUM statistic. Level A vector representing the levels at which each change point is detected. Parent A matrix with the starting indices on the first row and the ending indices on the second row.

Haotian Xu

### References

Wang, Yu and Rinaldo (2020) <doi:10.1214/20-EJS1710>.

thresholdBS for obtaining change points estimation, tuneBSunivar for a tuning version.

### Examples

set.seed(0)
cpt_true = c(20, 50, 170)
y = rnorm(300) + c(rep(0,20),rep(2,30),rep(0,120),rep(2,130))
temp = BS.univar(y, 1, length(y), delta = 5)
plot.ts(y)
points(x = tail(temp$S[order(temp$Dval)],4),
y = y[tail(temp$S[order(temp$Dval)],4)], col = "red")
BS_result = thresholdBS(temp, tau = 4)
BS_result
print(BS_result$BS_tree, "value") print(BS_result$BS_tree_trimmed, "value")
cpt_hat = sort(BS_result\$cpt_hat[,1]) # the threshold tau is specified to be 4
Hausdorff.dist(cpt_hat, cpt_true)
cpt_LR = local.refine.univar(cpt_hat, y)
Hausdorff.dist(cpt_LR, cpt_true)

[Package changepoints version 1.1.0 Index]


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# Steady state simulation with Raman scattering¶

class SteadyStateSimulationWithRaman

SteadyStateSimulationWithRaman is the main class for running Giles model simulations with Raman scattering. Only one ion population is supported. The class defines the fiber, boundary conditions and optical channels used in the simulation.

__init__()

Initialize self. See help(type(self)) for accurate signature.

add_pulsed_signal(wl, power, f_rep, fwhm_duration, wl_bandwidth=0, mode_shape_parameters=None, label='')

Adds a new forward propagating single-frequency pulsed signal to the simulation. A pulsed signal has a higher peak power resulting in stronger nonlinear effects, in particular spontaneous and stimulated Raman scattering. The pulse shape is assumed to be Gaussian.

Parameters: wl (float) – Wavelength of the signal power (float) – Input power of the signal at the beginning of the fiber f_rep (float) – Repetition frequency of the signal fwhm_duration (float) – Full-width at half-maximum duration of the Gaussian pulses wl_bandwidth (float) – Wavelength bandwidth of the channel. Finite bandwidth means including ASE. mode_shape_parameters (dict) – Defines the mode field shape. Allowed key-value pairs: functional_form -> [‘bessel’, ‘gaussian’, ‘tophat’] mode_diameter -> float overlaps -> list of pre-calculated overlaps between the channel and the ion populations label (str) – Optional label for the channel
add_raman(input_power=1e-14, backward_raman_allowed=True, raman_gain=1e-13)

Adds Raman channels to the simulation.

Parameters: backward_raman_allowed (bool, default True) – Determines if only the forward propagating Raman beam is simulated. input_power (float, default ~0 W) – Input power of the Raman beam(s) raman_gain (float, default 1e-13 m/W) – Raman gain value to be used in the simulation.
add_ase(wl_start, wl_end, n_bins)

Adds amplified spontaneous emission (ASE) channels. Using more channels improves accuracy, but incurs a heavier computational cost to the simulation.

Parameters: wl_start (float) – The shorted wavelength of the ASE band wl_end (float) – The longest wavelength of the ASE band n_bins (positive int) – The number of simulated ASE channels.
add_backward_pump(wl, power, wl_bandwidth=0, mode_shape_parameters=None, label='')

Adds a new backward propagating single-frequency pump to the simulation.

Parameters: wl (float) – Wavelength of the signal power (float) – Input power of the signal at the beginning of the fiber wl_bandwidth (float) – Wavelength bandwidth of the channel. Finite bandwidth means including ASE. mode_shape_parameters (dict) – Defines the mode field shape. Allowed key-value pairs: functional_form -> one of [‘bessel’, ‘gaussian’, ‘tophat’] mode_diameter -> float overlaps -> list of pre-calculated overlaps between the channel and the ion populations label (str) – Optional label for the channel
add_cw_signal(wl, power, wl_bandwidth=0, mode_shape_parameters=None, label='')

Adds a new forward propagating single-frequency CW signal to the simulation.

Parameters: wl (float) – Wavelength of the signal power (float) – Input power of the signal at the beginning of the fiber wl_bandwidth (float) – Wavelength bandwidth of the channel. Finite bandwidth means including ASE. mode_shape_parameters (dict) – Defines the mode field shape. Allowed key-value pairs: functional_form -> one of [‘bessel’, ‘gaussian’, ‘tophat’] mode_diameter -> float overlaps -> list of pre-calculated overlaps between the channel and the ion populations label (str) – Optional label for the channel
add_forward_pump(wl, power, wl_bandwidth=0, mode_shape_parameters=None, label='')

Adds a new forward propagating single-frequency pump to the simulation.

Parameters: wl (float) – Wavelength of the signal power (float) – Input power of the signal at the beginning of the fiber wl_bandwidth (float) – Wavelength bandwidth of the channel. Finite bandwidth means including ASE. mode_shape_parameters (dict) – Defines the mode field shape. Allowed key-value pairs: functional_form -> one of [‘bessel’, ‘gaussian’, ‘tophat’] mode_diameter -> float overlaps -> list of pre-calculated overlaps between the channel and the ion populations label (str) – Optional label for the channel
run(tol=0.001)

Runs the simulation, i.e. calculates the steady state of the defined fiber amplifier. ASE or raman simulations might require higher tolerance than the default value. It is best to decrease the tolerance until the result no longer changes.

Parameters: tol (float) – Target error tolerance of the solver.
set_guess_array(array, force_node_number=None)

Use an existing array as the initial guess. Typically this array is the result of a previous simulation with sligthly different parameters. Note that the number of simulated beams/channels must be the same.

Parameters: array (numpy array) – The initial guess array force_node_number (int, optional) – The new number of columns in the resized array.
set_guess_parameters(guess_parameters)

Overrides the default initial guess parameters.

Parameters: guess_parameters (Instance of GuessParameters class) – Parameters used to create the initial guess array

from pyfiberamp import GuessParameters, GainShapes params = GuessParameters() params.signal.set_gain_shape(GainShapes.LINEAR) params.pump.set_gain_db(-20) simulation.set_guess_parameters(params)

set_number_of_nodes(N)

Override the default number of nodes used by the solver. The solver will increase the number of nodes if necessary.

Parameters: N (int) – New starting number of nodes used by the solver.


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### Theory:

The angle bisector is formed when a line or line segment divides an angle into two equal parts.

Follow the below steps, construct the bisector of the $$∠ABC$$ with the measure $$60°$$.

Step 1: Draw the given angle $$∠ABC$$ with the measure $$60°$$ using the protractor.

Step 2: Taking $$A$$ as centre and convenient radius, draw an arc to cut $$AB$$ and $$AC$$. Mark the points of intersection as $$E$$ on $$AB$$ and $$F$$ on $$AC$$.

With the same radius and $$E$$ as centre, draw an arc in the interior of $$∠CAB$$ and another arc of same measure with centre at $$F$$ to cut the previous arc.

$$AG$$ is the required bisector of the given angle $$∠CAB$$.

Mark the point of intersection as $$G$$. Draw a ray $$AX$$ through $$G$$.


text:
Study on the characteristics of laminar lifted flames using planar laser induced fluorescence technique

Title & Authors
Study on the characteristics of laminar lifted flames using planar laser induced fluorescence technique
Lee, Byeong-Jun; Jeong, Seok-Ho; Han, Jae-Won;

Abstract
Characteristics of the lifted flame which is generated by issuing of the fuel through the miniature nozzle, d = 0.164 mm, are studied using the planar laser induced fluorescence technique. OH radical is excited on the $\small{Q_{1}}$(8) line of the $\small{A^{2}}$.SIGMA.$\small{^{+.leq.X2}}$ .PI.(1, 0) band transition (283.55 nm) and LIF signals are captured at the bands of (0, 0) and (1, 1) transition (306 ~ 326 nm) using the filters and ICCD camera. Hydroxyl radical (OH) profile for nozzle attached flame shows that OH radical populations at the flame sides and flame tip are larger than those at the base. But for the lifted flame (tribrachial flame) case, those are larger at the flame base than at the flame tip and flame sides. The OH radical is more dense near the center line of flame base at the blowing out. This fact proves the Chung and Lee's blowout theory-blowout occurs when the flame is anchored at the flame axis. axis.
Keywords
Nozzle Attached Flame;Lifted Flame;Blowout;PLIF;
Language
Korean
Cited by


text:
## Geometrical Insights for Implicit Generative Modeling

Abstract: Learning algorithms for implicit generative models can optimize a variety of criteria that measure how the data distribution differs from the implicit model distribution, including the Wasserstein distance, the Energy distance, and the Maximum Mean Discrepancy criterion. A careful look at the geometries induced by these distances on the space of probability measures reveals interesting differences. In particular, we can establish surprising approximate global convergence guarantees for the 1-Wasserstein distance, even when the parametric generator has a nonconvex parametrization.

Léon Bottou, Martin Arjovsky, David Lopez-Paz and Maxime Oquab: Geometrical Insights for Implicit Generative Modeling, To appear. ArXiV:1712.07822.
@incollection{bottou-geometry-2018,
author = {Bottou, L{\'e}on and Arjovsky, Martin and Lopez-Paz, David and Oquab, Maxime},
title = {Geometrical Insights for Implicit Generative Modeling},
year = {2018},
note = {To appear. ArXiV:1712.07822},
url = {http://leon.bottou.org/papers/bottou-geometry-2018},
}