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# Beyond the Standard Model, Progress in Lattice QCD, Heavy Ion physics

(last updated: February, 7th 2014)

Contacts:

• Lorenzo Feligioni (CPPM) – lorenzo[at]cppm.in2p3.fr
• Jean Loïc Kneur (L2C) – jean-loic.kneur[at]univ-montp2.fr

## Scope of the conference

The standard model (SM) of the strong, weak, and electromagnetic interactions gives a complete description of the elementary particles and their interactions, accurately describing present data up to the highest presently available TeV scale energies at the LHC. This success culminated with the 2012 discovery of a Higgs-like boson by the LHC ATLAS and CMS collaborations. However, the SM has serious limitations, and the particle physics community is now confronted with the need to identify the correct extension of the SM above the electroweak scale, allowing for a more natural electroweak symmetry breaking (EWSB).

Indeed, the scalar sector effective potential could be the manifestation of a more fundamental mechanism with a dynamical explanation for the EWSB. The two main scenarios for a more natural EWSB mechanism, addressing the hierarchy and naturalness problems, are either low energy supersymmetry (SUSY) or a TeV scale strongly interacting sector. Also of crucial importance to any BSM scenario are the quark and lepton flavor issues, namely the consistency and possible dynamical explanation of the observed pattern of quark masses and mixing as well as neutrino masses and mixing. The direct searches of SUSY signals at the LHC will continue to be crucial to constrain the many SUSY scenarios presently considered. Alternatively, the precise measurements of all the available Higgs decay modes at the LHC will be crucial to disentangle the many strongly interacting scenarios.

On the other side, the dark matter (DM) problem is still unsolved. There are strong theoretical arguments in favor of a "particle physics" solution within the previous context beyond the SM. A generic DM particle candidate in these theories is the so-called WIMP (weakly interacting massive particle). Complementarily to direct and indirect detection experiments, the LHC also gives hope that if a DM particle is discovered with parameters accessible to it, it will be possible to place this particle into a coherent particle physics scenario.

The symposium will also aim to discuss recent progress in lattice QCD and, more generally, lattice field theory computations. These play a critical role in our understanding of important nonlinear phenomena, ranging from phase transitions in the early universe and in very dense stars to the binding of quarks and gluons into hadrons and nuclei, or from a possible substructure for the Higgs boson to very small effects that may reveal new fundamental physics in experiments.

Another topics aimed to be addressed is Heavy Ion physics, after the first LHC results from the ALICE collaboration and recent theoretical developments on the status of the quark gluon plasma and the deconfinement transition.

## Topics

• Beyond the Standard Model (scenarios, LHC constraints and prospects)
• Supersymmetric Models
• Dynamics of Electroweak Symmetry Breaking
• Quark and Lepton Flavor physics and issues
• Neutrino Physics and Models
• Progress in Lattice QCD (zero and finite temperature)
• Heavy Ion physics
• Dark matter (candidates and models)

## Schedule

You don't need to print this webpage: pdf files of the program (for printing or reading on screen) are available in the Download zone.

The rooms mentionned in the program are located on this map of the campus (pdf file). This map will be distributed at the beginning of the Symposium.

DayJuly, 14Before the symposium…
Event19h00, Welcome Party: French National Day, fireworks…No conference associated
DayJuly, 15First day of the symposium…
Break08h00 – 08h45Conference Registration, Room “Salle de Conférences”
Parallel Sessions08h45 – 10h30Plenary session
Break10h30 – 11h00Coffee break
Parallel Sessions11h00 – 12h30Morning parallel plenary sessions
Break12h30 – 14h30Lunch
Parallel Sessions14h30 – 16h30First afternoon parallel sessions
PlaceAmphi “Massiani”Standard Model and EW precision measurementHigh Energy Physics
Chairman: Cristi Diaconou
Talk14h30 – 15h00July, 15
speaker-5398579c2ca34 Alain Blondel (DPNC University of Geneva) Precision Electroweak Measurements at Future High Energy Colliders

The prospects for precision measurements of the properties of the Z, W, Higgs and top particles will be reviewed. The machines considered will be the LHC and its upgrade, the future e+e- colliders, both linear and circular, and the 100 TeV Future Hadron Collider. The importance of improvements in some other experimental inputs and in the theoretical calculations will be addressed.

Talk15h00 – 15h30July, 15
speaker-539ce658f1b51 Satoshi Mishima (University of Rome, La Sapienza) Constraints on Higgs physics from EW precision measurements

The electroweak precision measurements offer a very powerful handle on the underlying mechanism of electroweak symmetry breaking and on the nature of the Higgs boson. In this talk, I will present current constraints on new physics beyond the Standard Model from the electroweak precision fit, and discuss the interplay between the electroweak precision fit and the Higgs measurements at the Tevatron and LHC.

Talk15h30 – 16h00July, 15
speaker-53917902212b9 Marco Zaro (LPTHE, UPMC) (Getting ready for) precision physics at hadron colliders

I will review recent progresses in the field of higher-order predictions at hadron colliders, with focus on the LHC. The inclusion of higher order corrections, in particular of those corrections related with QCD, is crucial in order to get accurate and reliable predictions which are needed both to validate the Standard Model of fundamental interaction and to seek for yet unknown particles, but leads to a huge growth of the computational complexity. Recent works have lead to the possibility of computing the first subleading order (Next-to-Leading Order, NLO) corrections in a fully automated manner for any process, hiding all the computational complexity to the user. For what concerns higher orders (next-to-next-to Leading Order, NNLO and beyond ), the most relevant processes for LHC physics in the SM have been covered at NNLO in the last few years, with huge efforts from different groups.

Talk16h00 – 16h30July, 15
speaker-539edcc616045 Kiran Joshi (UoM) Measurement of EW production of $Z+2j$ at the LHC

The electroweak production of a $Z$-boson in association with dijets ($Z+2j$), which includes the vector boson fusion process, is an important background to the vector boson fusion production of a Higgs boson in association with dijets. Both ATLAS [1] and CMS [2,3] have made measurements of the electroweak production of $Z+2j$. In this talk the methods and results of extracting the electroweak component of the cross section are presented. Detector-corrected distributions of hadronic jets are also presented and show sensitivity to the electroweak production process. The distributions can be used to probe the different approaches to generating both QCD and electroweak $Z+2j$ events.

References

[1] ATLAS Collaboration, Measurement of the electroweak production of dijets in association with a Z-boson and distributions sensitive to vector boson fusion in proton-proton collisions at $\sqrt{s} = 8$ TeV using the ATLAS detector, JHEP 1404 (2014) 031 [arXiv:1401.7610].

[2] CMS Collaboration, Measurement of the hadronic activity in events with a Z and two jets and extraction of the cross section for the electroweak production of a Z with two jets in pp collisions at $\sqrt{s} = 7$ TeV, JHEP 1310 (2013) 062 [arXiv:1305.7389].

[3] CMS Collaboration, Measurement of pure electroweak production of a Z boson in association with two forward/backward jets in proton-proton collisions at 8 TeV, CMS-PAS-FSQ-12-035.

Break16h30 – 17h00Coffee break
Parallel Sessions17h00 – 18h30Second afternoon parallel sessions
PlaceAmphi “Massiani”Standard Model and EW precision measurementHigh Energy Physics
Chairman: Gilbert Moultaka
Talk17h00 – 17h30July, 15
speaker-53b4515a682d3 Jordi Nadal (II. Physikalisches Institut) Top production at the LHC

Latest LHC results of the top quark production are presented. With the focus on the ATLAS and CMS experiment I will present top quark production cross-section in the different channels and also the latest results on the single top production, giving the state-of-art measurements. I also include differential cross-section as well as top quark pair production in association with W/Z bosons.

Talk17h30 – 18h00July, 15
speaker-538c3071820cb Gordon Watts (UW, Seattle) Top quark mass measurements at hadron colliders

The top quark mass is the best known quark mass. This talk will be a quick review of the measurements of the top mass at the four main collider experiments, CDF and DZERO from the Tevatron accelerator located at Fermilab, and ATLAS and CMS from the Large Hadron Collider located at CERN. Besides results at brief survey of techniques and future prospects will be given.

Talk18h00 – 18h30July, 15
speaker-538f292f887a1 Susanne Westhoff (PITT PACC) The Top-Quark Charge Asymmetry – Testing Strong Interactions and More

An overview of the charge asymmetry in top-antitop quark production at hadron colliders is presented. I discuss the origin of the charge asymmetry in the Standard Model and potential contributions from new physics [1,2] in the light of the enhanced asymmetry observed by the CDF collaboration at Tevatron. A measurement of the charge asymmetry at the LHC requires tailored observables, which cope with the large symmetric background [3].

References

[1] M. Bauer, F. Goertz, U. Haisch, T. Pfoh and S. Westhoff, Top-Quark Forward-Backward Asymmetry in Randall-Sundrum Models Beyond the Leading Order, JHEP 1011, 039 (2010) [arXiv:1008.0742 [hep-ph]].

[2] U. Haisch and S. Westhoff, Massive Color-Octet Bosons: Bounds on Effects in Top-Quark Pair Production, JHEP 1108, 088 (2011) [arXiv:1106.0529 [hep-ph]].

[3] S. Berge and S. Westhoff, Top-Quark Charge Asymmetry Goes Forward: Two New Observables for Hadron Colliders, JHEP 1307, 179 (2013) [arXiv:1305.3272 [hep-ph]].

Event19h00, Public Lecture, Jean-Pierre Luminet : Les bâtisseurs du ciel, de Copernic à demainNo conference associated
DayJuly, 16Second day of the symposium…
Parallel Sessions09h00 – 10h30Plenary session
Break10h30 – 11h00Coffee break
Parallel Sessions11h00 – 12h30Morning parallel plenary sessions
Break12h30 – 14h30Lunch
Parallel Sessions14h30 – 16h30First afternoon parallel sessions
Place (alias)Amphi “Sciences Naturelles”Neutrinos (common with Astroparticle Physics)High Energy Physics
Chairman: Jose W.F Valle
Talk14h30 – 15h00July, 16
speaker-53a92c65d26d4 Antonio Palazzo (MPI) Global status of neutrino oscillations

I will present the current status of the global neutrino data analysis, pointing out its unique role in constraining the two crucial (still) unknown parameters: the CP-violating phase $\delta$ and the $\theta_{23}$ octant. In this context, I will discuss the slight overall preference for $\theta_{23}$ in the first octant and for non-zero CP violation with $\sin \delta < 0$. The (in-)stability of such intriguing indications within different combinations of data sets will be considered in detail. Finally, I will discuss the robustness of the 3-flavor analysis in the presence of new light sterile neutrinos, whose existence is suggested by very-short-baseline experiments and cosmological data.

Talk15h00 – 15h30July, 16
speaker-53a8a879656dc Anatael Cabrera (APC) Neutrino Oscillations: Experimental Review

The Neutrino Oscillation phenomenon has been fully established manifestation of both the non-trivial mixing in the leptonic sector, characterised by the PMNS (3x3 matrix in the 3 neutrino type scenario) and the non-degenerate mass state spectrum of neutrinos. As a consequence, we know that mixing is part of the phenomenology of both quarks and leptons, including CP violation, although their mixing patterns have so far been measured to be very different and the possible physics behind remains unknown. Evidence for neutrino CP violation has not yet been observed, due to lack of experimental sensitivity. Also, thanks to the observation of neutrino oscillations, neutrinos are known to be massive. The absolute mass value is still unknown, being much lower than all other known fermions. In my talk, I will briefly review the experimental status of our knowledge as of the Neutrino 2014 (June 2014 at Boston, USA) conference. Improvements due to foreseen experiments will be also highlighted.

Talk15h30 – 16h00July, 16
speaker-538dd36b8303d Pasquale Di Bari (UoS) Leptogenesis and low energy neutrino data

I will review the status of thermal leptogenesis and how this, in combination with low energy neutrino data, can be used as a powerful tool to probe models of new Physics. I will discuss how the possibility to realise independence of the initial conditions (strong thermal leptogenesis) relies on a special right-handed neutrino mass pattern and require quite special conditions, in particular a deviation of light neutrino masses from the hierarchical limit that can be tested in future years with cosmological observations. Intriguingly the set of conditions for strong thermal leptogenesis can be realise within SO(10)-inspired leptogenesis and in this case one obtains a set of definite predictions on future low energy neutrino measurements. Interestingly this solution requires non vanishing reactor mixing angle, now robustly established in reactor neutrino experiments, and a negative neutrino oscillation CP violation invariant parameter, as hinted by recent neutrino oscillation experimental results.

Talk16h00 – 16h30July, 16
speaker-539efa9def4d4 Xavier Sarazin (LAL) Review of neutrinoless double beta decay search

The observation of a neutrinoless double beta decay would be the proof that the neutrino is a Majorana particle, identical to its own anti-particle, and would correspond to a process violating the leptonic number. In this talk, I will give a review of the various experiments which are searching for a neutrinoless double beta decay.

Break16h30 – 17h00Coffee break
Parallel Sessions17h00 – 18h30Second afternoon parallel sessions
Place (alias)Amphi “Sciences Naturelles”Neutrinos (common with Astroparticle Physics)High Energy Physics
Chairman: Jurgen Brunner
Talk17h00 – 17h30July, 16
speaker-539077cddc2fd Paschal Coyle (CPPM) Mediterranean Neutrino Telescopes

ANTARES is a deep sea neutrino telescope located in the deep sea 40km off the southern coast of France. The results of searches for cosmic neutrinos produced in astrophysical sources and from annhilation of dark matter in the Sun and Galactic centre are presented.

The European neutrino astronomy community has recently started the construction of KM3NeT, a several cubic kilometre neutrino telescope. The R&D for this new research infrastructure are described. Finally, the potential for a measurement of the neutrino mass hierarchy, with a densely instrumented detector configuration (ORCA) is discussed.

Talk17h30 – 18h00July, 16
speaker-538e3b729d4a2 Cristina Volpe (APC) Recent advances in neutrino astrophysics

Neutrinos of astrophysical origin are messengers produced in stars, in explosive phenomena like core-collapse supernovae, in the accretion disks around black holes, or in the Earth's atmosphere. Their fluxes and spectra encode information both on their sources, the environments they traverse and on key unknown neutrino properties. We will discuss the status and key open questions in our current understanding of neutrino flavour conversion in media and its impact on astrophysical observations. We will describe some of the important recent advances in the field of neutrino astrophysics.

References

[1] C. Volpe, Open issues in neutrino astrophysics, Annalen Phys. 8-9 (2013) [arXiv:1303.1681].

Talk18h00 – 18h30July, 16
speaker-53a1c3b253304 Eric Baussan (IPHC) Future of neutrino based reactor experiments

The last angle of the PMNS mixing matrix has been measured by the neutrino reactor experiments. This important result opens the door to the precision era in the neutrino oscillation landscape. In this context, the next generation of reactor experiments at the kilo ton scale will significantly improve the measurements on the oscillation parameters and will give an answeron the mass hierarchy in the next decades. After a brief summary of the last results, these experiments will be presented with their technological challenges to reach the required sensitivity.

EventBanquet at Palais du Pharo — 19h00No conference associated
DayJuly, 17Third day of the symposium…
Parallel Sessions09h00 – 10h30Plenary session
Break10h30 – 11h00Coffee break
Parallel Sessions11h00 – 12h30Morning parallel plenary sessions
Break12h30 – 14h30Lunch
Parallel Sessions14h30 – 16h30First afternoon parallel sessions
PlaceAmphi “Massiani”Higgs measurement and BSMHigh Energy Physics
Chairman: Cyril Hugonie
Talk14h30 – 15h00July, 17
speaker-539b34898862e Jérémie Quevillon (LPT Orsay) Constraints on BSM physics through the Higgs couplings

We will discuss the implications from the measured mass and production/decay rates of the observed Higgs boson on several well motivated extensions of the Standard Model.

Firstly, we will show that in the Minimal Supersymmetric Standard Model, to a good approximation, the phenomenology of the lighter Higgs state can be described by its mass and three couplings: those to massive gauge bosons and to top and bottom quarks. We will then demonstrate that the value $M_{h} \simeq125$~GeV together with the non-observation of superparticles at the LHC, indicates that the SUSY-breaking scale $M_{S}$ is rather high, $M_{S}>1$~TeV, leading to a Higgs sector that can be described, to a good approximation, by only two free parameters.

Secondly, we will discuss how the LHC limits on the invisible Higgs branching fraction impose strong constraints on Higgs portal models of dark matter.

Talk15h00 – 15h30July, 17
speaker-5385cea8d3b55 Christopher Neu (UVa) Search for ttH in Run 1 at the LHC

With the discovery in 2012 of a Higgs boson at the LHC, the focus of the Higgs physics campaigns at the ATLAS and CMS experiments has transitioned from the search for the Higgs to the pursuit of the complete characterization of this new particle. This is an essential step that is necessary in the determination of whether this is the Higgs boson of the standard model, or a component of some other, more exotic explanation of the fundamental world. Principal among these characterizations is the understanding of the interaction strength, or coupling, between this Higgs boson and the other known fundamental particles; among these couplings, the largest and arguably the most important is that between the Higgs and the top quark, the most massive of the known fundamental particles. The only direct probe of this coupling is through the observation of Higgs boson production in association with top quarks at the LHC. In this talk, the search strategy and current status will be described, focusing on the results from the 2010-2012 Run 1 of the LHC, with also an eye toward future studies in Run 2. The observation of this process is one of the highest-priority aspects of the current LHC physics program.

Talk15h30 – 16h00July, 17
speaker-539ea2b7f0f6a Wolfgang Mader (IKTP) Searches for BSM Higgs bosons at LHC

Since the discovery of the Higgs-like boson at the LHC by the two experiments ATLAS and CMS, properties of the new boson are now being measured with increasing precision, restricting the phase space for Beyond the Standard Model physics. In this talk, the re-interpretation of these measurements by ATLAS and CMS in BSM scenarios like e.g. the Two-Higgs-Doublet Model are reviewed.

Talk16h00 – 16h30July, 17
speaker-53995bc005170 Nikolas Kauer (RHUL) Off-shell Higgs signal and total width determination at the LHC

A substantial far-off-shell contribution to the Higgs boson signal in the $H\to ZZ$ decay mode [1] facilitates a novel, complementary approach to constraining the Higgs width $\Gamma_H$ at the Large Hadron Collider (LHC) [2]. A first experimental analysis [3] obtained $\Gamma_H < 22$ MeV (95% CL), which is $5.4$ times the expected value in the Standard Model at the measured Higgs boson mass. I review the theoretical basis of the new approach and discuss its significance and potential in comparison to other methods to bound and measure the Higgs width at the LHC and future colliders.

References

[1] N. Kauer and G. Passarino, Inadequacy of zero-width approximation for a light Higgs boson signal, JHEP 1208 (2012) 116 [arXiv:1206.4803 [hep-ph]].

[2] F. Caola and K. Melnikov, Constraining the Higgs boson width with ZZ production at the LHC, Phys. Rev. D 88 (2013) 054024 [arXiv:1307.4935 [hep-ph]]; J. M. Campbell, R. K. Ellis and C. Williams, Bounding the Higgs width at the LHC using full analytic results for $gg \to e^- e^+ \mu^- \mu^+$, JHEP 1404 (2014) 060 [arXiv:1311.3589 [hep-ph]].

[3] V. Khachatryan et al. [CMS Collaboration], Constraints on the Higgs boson width from off-shell production and decay to $Z$-boson pairs, CERN-PH-EP-2014-078, arXiv:1405.3455 [hep-ex].

Break16h30 – 17h00Coffee break
Parallel Sessions17h00 – 18h30Second afternoon parallel sessions
PlaceAmphi “Massiani”Higgs measurement and BSMHigh Energy Physics
Chairman: Sara Diglio
Talk17h00 – 17h30July, 17
speaker-5384baecd7bfa Heidi Rzehak (Uni Freiburg) SUSY after LHC run 1

The successful data taking of the LHC experiments during run 1 allowed already for a detailed investigation of possible models with underlying supersymmetry as well as for the discovery of a Higgs boson. In this talk, the status of the experimental results will be briefly summarized and implications of the findings of LHC run 1 for viable models based on supersymmetry will be discussed. Constraints from direct searches as well as from indirect measurements, in particular the Higgs boson mass, will be taken into account.

Talk17h30 – 18h00July, 17
speaker-538b16e21b5f1 Sabine Crépé-Renaudin (LPSC) Searches for heavy resonances at the LHC

Resonances decaying into a pair of particles are an obvious place to look for phenomena beyond the Standard Model. This talk summarizes recent results from the ATLAS and CMS experiments on searches for resonances in pairs of jets, leptons, bosons or a mix of those objects. Various models are considered from Z' and W' to the ADD large extra dimension scenario, from excited quarks or technicolor to quantum black holes as well as contact interactions. Results from $\sqrt{s} = 8$ TeV are presented.

EventCocktail at the City Hall — 19h00No conference associated
DayJuly, 18Fourth day of the symposium…
Parallel Sessions09h00 – 10h30Plenary session
Break10h30 – 11h00Coffee break
Parallel Sessions11h00 – 12h30Morning parallel plenary sessions
Break12h30 – 14h30Lunch
Parallel Sessions14h30 – 16h30First afternoon parallel sessions
PlaceAmphi “Massiani”QCD, Heavy Flavors and Heavy IonsHigh Energy Physics
Chairman: Olivier Leroy
Talk14h30 – 15h00July, 18
speaker-5370fc74df131 Jeremy Dalseno (UOB) “CP violation effects in multibody B decays” on behalf of the LHCb Collaboration

$CP$ Violation is one of the necessary ingredients to produce the matter anti-matter asymmetry we observe in the Universe today. The LHCb experiment is a general purpose forward-spectrometer located along the LHC proton-proton collider at CERN and is ideally suited for the investigation of such phenomena. We present the latest results of 3-body charmless $B$ decays where large local $CP$ violating effects have been observed across various regions of the phase space [1,2]. These results are discussed in the light of recent theoretical developments that attempt to understand the origin of the large asymmetries [3] and their impact on future amplitude analyses.

References

1] R. Aaij et al. (LHCb Collaboration), Measurement of $CP$ Violation in the Phase Space of $B^\pm \to K^\pm \pi^+ \pi^-$ and $B^\pm \to K^\pm K^+ K^-$ Decays, Phys. Rev. Lett. 111 (2013) 101801.

[2] R. Aaij et al. (LHCb Collaboration), Measurement of $CP$ Violation in the Phase Space of $B^\pm \to K^+K^- \pi^\pm$ and $B^\pm \to \pi^+\pi^-\pi^\pm$ Decays, Phys. Rev. Lett. 112 (2014) 011801.

[3] Z.-H. Zhang, X.-H. Guo, and Y.-D. Yang, Phys. Rev. D 87 (2013) 076007; B. Bhattacharya, M. Gronau, and J. L. Rosner, Phys. Lett. B 726 (2013) 337; I. Bediaga, O. Lourenço, and T. Frederico, Phys. Rev. D 89 (2014) 094013; D. Xu, G.-N. Li, and X.-G. He, arXiv:1307.7186 (2013).

Talk15h00 – 15h30July, 18
speaker-5374c9d89a4a8 Sam Cunliffe (IC) Observables and anomalies in $B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$ decays

Recent angular analyses of ${B^0\rightarrow K^{*0}\mu^{+}\mu^{-}}$ decays at LHCb are in tension with the standard model predictions. Phenomenological interpretations of these results imply hints of a contribution from a new vector particle. These results are difficult to account for within supersymmetric models. This talk will cover the experimental aspects of performing such an angular analysis using the LHCb detector as well as the theoretical motivation for making such a measurement and the rationale behind the choices of observables. I will devote some time to reviewing the phenomenological interpretations.

Talk15h30 – 16h00July, 18
speaker-5375915b221f3 Agostino Patella (CERN) Lattice gauge theories beyond QCD

As the number of flavours in QCD is increased beyond some threshold, chiral symmetry is restored and long-distance physics becomes scale invariant. The region of parameter space in which this happens is called “conformal window”. The phenomenology of theories in the conformal window or close to it is very different from real-world QCD, and clearly shows how rich gauge theories can be. Near-conformal gauge theories have been conjectured to be viable models for physics beyond the Standard Model, and as they might naturally incorporate a Higgs-like scalar, and in this sense they are very different from traditional techicolor theories. As these theories are inherently non-perturbative, lattice simulation provide a unique setup to investigate them from first principles. In this talk I will review the main results and open questions in this field, the challenges ahead, and the techniques used.

Talk16h00 – 16h30July, 18
speaker-539178972493c John Bulava (TCD) Exotic hadrons in Experiment and on the Lattice

Although the constituent quark model correctly predicts the quantum numbers of nearly all known hadrons, it fails to account for so-called 'exotic' states. After briefly reviewing the current experimental status of some candidate exotic hadrons, I will discuss prospects for elucidation of their properties using numerical Lattice QCD simulations. Recent theoretical advances have greatly improved lattice calculations of hadronic resonances, but some of the most interesting systems (such as exotic charmonium states) remain a challenge.

Break16h30 – 17h00Coffee break
Parallel Sessions17h00 – 18h30Second afternoon parallel sessions
PlaceAmphi “Massiani”QCD, Heavy Flavors and Heavy IonsHigh Energy Physics
Chairman: Jérôme Charles
Talk17h00 – 17h30July, 18
speaker-53b174df0f436 Kalman Szabo (ISC-JSC) Recent progress in Lattice QCD thermodynamics

I review some of the most important recent results from finite temperature lattice QCD based on the contributions of the Budapest-Wuppertal collaboration. In particular I'll cover the equation of state [1], the role of the charm quark, and the fluctuation observables also measured in heavy ion physics [2]. I will comment on the the reliability of the hadron resonance gas model for low temperatures and the hard thermal loop approximation at high temperatures [3].

References

[1] S. Borsanyi, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg and K. K. Szabo, %“Full result for the QCD equation of state with 2+1 flavors,” Phys.Lett.B 730 (2014) 99 [arXiv:1309.5258 [hep-lat]].

[2] S. Borsanyi, Z. Fodor, S. D. Katz, S. Krieg, C. Ratti and K. K. Szabo, %“Freeze-out parameters from electric charge and baryon number fluctuations: is there consistency?,” arXiv:1403.4576 [hep-lat].

[3] N. Haque, A. Bandyopadhyay, J. O. Andersen, M. G. Mustafa, M. Strickland and N. Su, %“Three-loop HTLpt thermodynamics at finite temperature and chemical potential,” JHEP 1405 (2014) 027 [arXiv:1402.6907 [hep-ph]].

Talk17h30 – 18h00July, 18
speaker-5399bee0b6f5b Diego Stocco (Subatech) ALICE results in p–Pb collisions at the LHC

ALICE studies the properties of the strongly interacting matter at the extreme energy densities reached in heavy-ion collisions at the LHC. In this context, the measurements in proton-proton and proton-nucleon collisions are mandatory: the former sets the reference, while the latter provides further insight into the effects due to cold nuclear matter, which is crucial in the understanding of heavy-ion collisions. In this talk, a selection of the recent results in p–Pb collisions at the LHC will be presented.


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# Minimal polynomial for a bijective linear transformation

Let $V$ be a vector space over a field $F$ which is not algebraically closed and $T:V \rightarrow V$ be a linear transformation. I am just trying to understand if injectiviity can be used to study the structre of the minimal polynomial of $T$ because none of the theorems from my linear algebra textbook make this notion explicitly clear.

So suppose first that $T$ is a bijection and that $V$ has dimension $1$. Since $T$ is a bijection can we just fix a $v \in Image (T)$ so that $Tv = av$ for some $a \in F$ and therefore the minmal polynomial of T is just $x-a$. (since $V$ has dimension 1).

Question: Suppose $dim(V) \ge 2$ if $T$ is a bijection does it follow that the minmal polynomial of $T$ has a linear factor or can we say something stronger?

It is not clear to me at this point if there is a connection between injectivity of a linear operator and diagonaliziablity .

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Are you working over an arbitrary field, or a specific one? Over $\mathbb{C}$, for example, every polynomial splits into linear factors. (And no, there is no connection between injectivity and diagonalisability.) –  Zhen Lin Aug 9 '11 at 11:29
@Zhen Thank you for the comment. I have edited to the post to rule out the trivial case of algebraically closed fields. –  user7980 Aug 9 '11 at 11:32
There's one minor connection between the two concepts: if $T$ is not injective, then $0$ is an eigenvalue of $T$ and thus the minimal polynomial has a linear factor (since $0$ always lies in the underlying field). Of course, this doesn't guarantee that $T$ is diagonalizable, but merely that it has at least one eigenvalue. –  Mark Aug 9 '11 at 11:51

If you move away from algebraically closed fields, the answer is "not in general". For example, if we work over the real field $\mathbb{R}$, the linear transformation $: \mathbb{R}^2 \to \mathbb{R}^2$ with $T(a,b) = (b,-a)$ has $T^{2} = -I$, and has minimum polynomial $x^{2}+1$, which has no linear factor in $\mathbb{R}[x].$ It is also a bijection, as is easily checked.
@Mathematica: This is possible, though computing that matrix would be the easiest way. For example, in the example above, you can see that $T^{2}(a,b) = (-a,-b)$, so $T^{2} +I =0.$ –  Geoff Robinson Jun 6 '14 at 10:27

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# How do you find the n-th derivative of a power series?

Sep 12, 2014

If $f \left(x\right) = {\sum}_{k = 0}^{\infty} {c}_{k} {x}^{k}$, then
${f}^{\left(n\right)} \left(x\right) = {\sum}_{k = n}^{\infty} k \left(k - 1\right) \left(k - 2\right) \cdots \left(k - n + 1\right) {c}_{k} {x}^{k - n}$

By taking the derivative term by term,
$f ' \left(x\right) = {\sum}_{k = 1}^{\infty} k {c}_{k} {x}^{k - 1}$
$f ' ' \left(x\right) = {\sum}_{k = 2}^{\infty} k \left(k - 1\right) {c}_{k} {x}^{k - 2}$
$f ' ' ' \left(x\right) = {\sum}_{k = 3}^{\infty} k \left(k - 1\right) \left(k - 2\right) {c}_{k} {x}^{k - 3}$
.
.
.
${f}^{\left(n\right)} \left(x\right) = {\sum}_{k = n}^{\infty} k \left(k - 1\right) \left(k - 2\right) \cdots \left(k - n + 1\right) {c}_{k} {x}^{k - n}$


text:
#### Question

Which of the following describes the placebo effect? a. Students in art class are not told that their work will be evaluated for a scholarship. b. participants in an experiment do not know if they are in the experimental or control group. c. Participants in a drug study are given an inert pill instead of the drug and behave as though they are given the drug. d. Only females are chosen for a study, even though the population included males. e. Experiments manipulate one variable in a study but not any others.

Verified

1 of 3

C.

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#### Psychology

12th EditionC. Nathan DeWall, David G Myers
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3rd EditionC. Nathan DeWall, David G Myers
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2nd EditionDavid G Myers
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#### Psychology

11th EditionC. Nathan DeWall, David G Myers
359 solutions


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# Introductory Real Analysis (Dover Books on Mathematics)1st Edition

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## Editorial Reviews

### Language Notes

Text: English, Russian (translation)
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## Product Details

• Series: Dover Books on Mathematics
• Paperback: 416 pages
• Publisher: Dover Publications; 1st edition (June 1, 1975)
• Language: English
• ISBN-10: 0486612260
• ISBN-13: 978-0486612263
• Product Dimensions: 1 x 5.8 x 8.5 inches
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## Customer Reviews

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### Top Customer Reviews

By A Customer on May 26, 2004
Format: Paperback
First, let us be precise in reviewing this book. It is NOT a book by Kolmogorov/Fomin, but rather an edited version by Silverman. So, if you read the first lines in the Editor's Preface, it states, "The present course is a freely revised and restyled version of ... the Russian original". Further down it continues, "...As in the other volumes of this series, I have not hesitated to make a number of pedagogical and mathematical improvements that occurred to me...". Read it as a big red warning flag. Alas, I would have to agree with the reader from Rio de Janeiro. I've been working through this book to rehash my knowledge of measure theory and Lebesgue integration as a prerequisite for stochastic calculus. And I've encountered many results of "mathematical improvements" that occurred to the esteemed "translator". Things are fine when topics/theorems are not too sophisticated (I guess not much room for "improvements"). Not so when you work through some more subtle proofs. Most mistakes I discovered are relatively easy to rectify (and I'm ignoring typos). But the latest is rather egregious. The proof of theorem 1 from ch. 9 (p.344-345) (about the Hahn decomposition induced on X by a signed measure F) contains such a blatant error, I am very hard pressed to believe it comes from the original. That book survived generations of math students at Moscow State, and believe me, they would go through each letter of the proofs. Astounded by such an obvious nonsense, I grabed the only other reference book on the subject I had at hand, "Measure Theory" by Halmos. The equivalent there is theorem A, sec. 29 (p.121 of Springer-Verlag edition), which has a correct proof.
For those interested in details, Silverman's proof states that positive integers are strictly ordered: k1<k2<...<kn<...
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By Kevin C. on February 5, 2001
Format: Paperback
Overall, this book is a very strong "introduction" (I use the word grudgingly, see below) to real analysis. Topics range from the basics of set theory through metrics, operators, and Lebesgue measures and integrals. Particularly well done are the section on linear maps and operators, which include excellent generalizations as well as the usual concrete examples. The book usually includes a large number of examples and exercises on each topic which aid in the understanding of the material (though in a few instances, most notably the introduction to measure, it would have been more helpful to have examples as the theory was being developed instead of spending 20 pages getting through the theorems and only then giving a few examples).
The main problem for this book, however, is that it is located at an awkward level in terms of its assumptions of what students have seen before. Most of the material covered is that of a first analysis course, and the book is probably usually used as such. The authors, however, sometimes make assumptions that students have had exposure to some of the concepts before, claiming that "the reader has probably already encountered the familiar Heine-Borel theorem", for example. One particularly annoying case was when the authors gave as an example that the set of polynomials with rational coefficients is dense in the set of continuous functions, and left it at that. Are we supposed to have encountered Weierstrass's theorem before we take our first analysis course?
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on March 14, 2008
Format: Paperback
The advantages of this text have been pointed out by other readers, so I will attempt to exhibit the problems of this book.

There are a lot of mistakes. And by 'a lot', I mean that the careful reader should be able to find at least 5 mathematical mistakes in each chapter. I used this text mainly as a supplement to a fairly advanced analysis course, and we'd often have problems from it used in our problem sets. At first, it appeared as if this were a very well-written text, but once we started with our problem sets, there were at least 2 e-mails sent out per week addressing a concern a student had pointed out. After a while, students stopped e-mailing the professor with their concerns, instead just assuming that they were correct whenever they spotted something weird.

Let's take an example:

Problem 1, pg. 137: Let M be the set of all points x = (x1, x2, ..., xn, ...) in l2 satisfying the condition \sum^{\infty}_{n=1} (n^2) (x_n)^2 \le 1. Prove that M is a convex set, but not a convex body.

The problem with this is that M IS easily a convex body, precisely because x = (0,0,...) is in M.

There are many more big mistakes and little mistakes throughout the exercises, oftentimes destroying the entire POINT of the problem. Take, for example, Problem 1 of pg. 76: Let A be a mapping of a metric space R into itself. Prove that the condition p(Ax,Ay) < p(x,y) (x\ne y) is insufficient for the existence of a fixed point of A.

Now, a counterexample here can be easily produced, even by the most elementary reader. But the exercise quickly becomes worthwhile if we make R complete. It's the little things that count in mathematics, and the small errors like these are clearly detrimental to the student.
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# What molar quantity of carbon dioxide will result, if a six mole quantity of pentane are completely combusted?

Dec 3, 2016

Approx. $30 \cdot m o l$ of carbon dioxide will evolve from combustion.

#### Explanation:

For all problems of this kind, our starting point is a stoichiometrically balanced equation:

${C}_{5} {H}_{12} \left(l\right) + 8 {O}_{2} \left(g\right) \rightarrow 5 C {O}_{2} \left(g\right) + 6 {H}_{2} O \left(l\right)$

The equation is stoichiometrically balanced: $\text{garbage in equals garbage out}$.

It tells us unequivocally that if $72 \cdot g$ of pentane react with stoichiometric, i.e. $256 \cdot g$ of dioxygen, we are going to get $220 \cdot g$ of carbon dioxide gas, and $108 \cdot g$ of water vapour.

Here we start with, $5.79 \cdot m o l$ pentane, and thus $5 \times 5.79 \cdot m o l$ $C {O}_{2}$ will evolve, all calculated on the basis of the given equation. What are (i) the mass of carbon dioxide, and (ii) the mass of water evolved from this quantity of pentane?


text:
this function produces an interactive view of the specified raster object(s) on a plain grey background but for any CRS.

## Usage

# S4 method for RasterLayer
plainView(
x,
maxpixels = 1e+08,
col.regions = viridisLite::inferno,
at,
na.color = "#BEBEBE",
legend = TRUE,
verbose = FALSE,
layer.name = deparse(substitute(x, env = parent.frame())),
gdal = TRUE,
...
)

# S4 method for RasterStackBrick
plainView(
x,
r = 3,
g = 2,
b = 1,
na.color = "#BEBEBE",
maxpixels = 1e+08,
layer.name = deparse(substitute(x, env = parent.frame())),
...
)

# S4 method for SpatialPixelsDataFrame
plainView(x, zcol = 1, ...)

# S4 method for ANY
plainview(...)

## Arguments

x

a raster* object

maxpixels

integer > 0. Maximum number of cells to use for the plot. If maxpixels < ncell(x), sampleRegular is used before plotting.

col.regions

color (palette).See levelplot for details.

at

the breakpoints used for the visualisation. See levelplot for details.

na.color

color for missing values.

legend

either logical or a list specifying any of the components decribed in the colorkey section of levelplot.

verbose

should some details be printed during the process

layer.name

the name of the layer to be shown on the map

gdal

logical. If TRUE (default) gdal_translate is used to create the png file for display when possible. See details for further information.

...

arguments passed on to respective methods

r

integer. Index of the Red channel, between 1 and nlayers(x)

g

integer. Index of the Green channel, between 1 and nlayers(x)

b

integer. Index of the Blue channel, between 1 and nlayers(x)

zcol

attribute name or column number in attribute table of the column to be rendered

## Details

If the raster object is not in memory (i.e. if raster::inMemory is FLASE) and argument gdal is set to TRUE (default) gdal_translate is used to translate the rsater object to a png file to be rendered in the viewer/browser. This is fast for large rasters. In this case, argument maxpixels is not used, instead the image is rendered in original resolution. However, this means that RasterLayers will be shown in greyscale. If you want to set a color palette manually, use gdal = FALSE and (optionally provide) col.regions.

For plainView there are a few keyboard shortcuts defined:

• plus/minus - zoom in/out

• space - toggle antialiasing

• esc - zoom to layer extent

• enter - set zoom to 1

• ctrl - increase panning speed by 10

## Methods (by class)

• RasterStackBrick: stack / brick

• SpatialPixelsDataFrame: SpatialPixelsDataFrame

• ANY: alias for ease of typing

## Author

Stephan Woellauer

Tim Appelhans

## Examples

if (interactive()) {

# RasterLayer
plainView(poppendorf[[4]])

# RasterStack
plainview(poppendorf, r = 4, g = 3, b = 2) # true color
plainview(poppendorf, r = 5, g = 4, b = 3) # false color

}


text:
chapter  8
16 Pages

## ■ Jumps and Microstructure Noise in Stock Price Volatility

ByRITUPARNA SEN

Hence, there is a need for accurate model free measures of volatility. Note that the volatility of a price process is fairly model free. “Any log-price process subject to a no-arbitrage condition and weak auxiliary assumptions will constitute a semi-martingale that may be decomposed into a locally predictable mean component and a martingale with finite second moment” (Andersen et al., 2005). The predictable quadratic variation of the martingale is the volatility.


text:
J Electromagn Eng Sci Search

CLOSE

 J Electromagn Eng Sci > Volume 14(1); 2014 > Article
 Journal of Electromagnetic Engineering and Science 2014;14(1):1-8. DOI: https://doi.org/10.5515/JKIEES.2014.14.1.1
 CMOS Linear Power Amplifier with Envelope Tracking Operation (Invited Paper) Byungjoon Park1, Jooseung Kim2, Yunsung Cho1, Sangsu Jin2, Daehyun Kang3, Bumman Kim2 1Division of Information Technology Convergence Engineering, Pohang University of Science and Technology (POSTECH)2Department of Electrical Engineering, POSTECH3Broadcom Corporation Abstract A differential-cascode CMOS power amplifier (PA) with a supply modulator for envelope tracking (ET) has been implemented by 0.18 ${mu}m$ RF CMOS technology. The loss at the output is minimized by implementing the output transformer on a FR-4 printed circuit board (PCB). The CMOS PA utilizes the $2^{nd}$ harmonic short at the input to enhance the linearity. The measurement was done by the 10MHz bandwidth 16QAM 6.88 dB peak-to-average power ratio long-term evolution (LTE) signal at 1.85 GHz. The ET operation of the CMOS PA with the supply modulator enhances the power-added efficiency (PAE) by 2.5, to 10% over the stand-alone CMOS PA for the LTE signal. The ET PA achieves a PAE of 36.5% and an $ACLR_{E-UTRA}$ of -32.7 dBc at an average output power of 27 dBm. Key words: CMOS, Envelope Tracking, Long-Term Evolution, Power Amplifier, Supply Modulator
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text:
# B Faraday's law of electromagnetism

1. Aug 17, 2016

### Nimbus2000

I have a confusion regarding Faraday's law of electromagnetism. Consider this diagram

At this instant, the velocity of the rotor is parallel to the field lines, hence, no emf is induced in the rotor. Altetnatively, we can say that the rate of change of flux associated with the rotor is 0 at this instant, hence emf induced is 0.
Also, motional emf is given as E=BlvsinΘ and magnetic flux Φ=BAcosΘ. Now, E=BlvsinΘ=Bl(dx/dt)sinΘ=d(B*l*dx*sinΘ)/dt=d(BAcosΘ)/dt=dΦ/dt. This shows that if we use motional emf formula or the dΦ/dt formula, we arrive at the same result. But what if the conductor is not cutting any magnetic field and there is a change in flux with respect to the loop?

Suppose the middle limb is excited using a dc voltage. All the flux will be enclosed in the limb. Now consider the red loop on the middle limb. If the loop is tilted by some angle, flux through the loop will change since Θ changed. But the loop doesn't cut any flux since all the flux is enclosed in the limb. Will there be emf induced in the loop without cutting of flux lines?

2. Aug 17, 2016

### Hesch

A small amount of flux will be present outside the limb.

Say that the loop is just a little wider than the limb ( so that a tilt is possible ), and that the relative permeabilty, μr = 1000 as for the limb, the flux density just outside the limb will become 0.001*( density inside the limb ). So tilting the loop will cut a small amount of flux, and will induce a small amount of emf.

So just to be very accurate: An emf will be induced due to some flux lines being cut.

( Emf = dψ / dt , ψ = flux )

3. Aug 17, 2016

### Simon Bridge

... ie if all the flux is enclosed within the loop ...

In practice this won't happen.
Mathematically we can set up anything - so set up the math and crunch the numbers.

Define a uniform magnetic field in rectangular coords: $\vec B = B\hat k : -a/2 < x,y < a/2$ but 0 everywhere else, and there is a square conducting loop with sides length b > a in the x-y plane, centered on the origin. You want to rotate the loop through angle $\theta$ about the y axis, being careful to keep in bounds: $0 < \theta < \cos^{-1}(a/b)$ (check?) so that the wire never cuts the magnetic field.

You want to know if there is a change in flux due to the component of $\vec B$ perpendicular to the area of the loop changing.
Is this what you are thinking of?

The equation you want is:
$$\Phi = \iint_S \vec B\cdot \text{d}\vec S$$

It needn't be that complicated: a simple situation is to change the magnetic field strength so $\vec B = B(t)\hat k$ and leave the loop in the x-y plane. Now what happens? Is flux cutting happening?

4. Aug 21, 2016

### cnh1995

Thank you @Hesch, @Simon Bridge for your replies. Actually, Nimbus2000 is my friend (and room-mate) and I was facing some technical problems while posting my own thread


text:
## Section2.1Strings: A First Look

Let $$n$$ be a positive integer. Throughout this text, we will use the shorthand notation $$[n]$$ to denote the $$n$$-element set $$\{1,2,\dots,n\}\text{.}$$ Now let $$X$$ be a set. Then a function $$s\colon[n]\rightarrow X$$ is also called an $$X$$-string of length $$n$$. In discussions of $$X$$-strings, it is customary to refer to the elements of $$X$$ as characters, while the element $$s(i)$$ is the $$i^{\text{th} }$$ character of $$s\text{.}$$ Whenever practical, we prefer to denote a string $$s$$ by writing $$s=$$“$$x_1x_2x_3\dots x_n$$”, rather than the more cumbersome notation $$s(1)=x_1\text{,}$$ $$s(2)=x_2\text{,}$$ …, $$s(n)=x_n\text{.}$$

There are a number of alternatives for the notation and terminology associated with strings. First, the characters in a string $$s$$ are frequently written using subscripts as $$s_1,s_2,\dots,s_n\text{,}$$ so the $$i^{\text{th} }$$-term of $$s$$ can be denoted $$s_i$$ rather than $$s(i)\text{.}$$ Strings are also called sequences, especially when $$X$$ is a set of numbers and the function $$s$$ is defined by an algebraic rule. For example, the sequence of odd integers is defined by $$s_i=2i-1\text{.}$$

Alternatively, strings are called words, the set $$X$$ is called the alphabet and the elements of $$X$$ are called letters. For example, $$aababbccabcbb$$ is a $$13$$-letter word on the $$3$$-letter alphabet $$\{a,b,c\}\text{.}$$

In many computing languages, strings are called arrays. Also, when the character $$s(i)$$ is constrained to belong to a subset $$X_i\subseteq X\text{,}$$ a string can be considered as an element of the cartesian product $$X_1\times X_2\times \dots\times X_n\text{,}$$ which is normally viewed as $$n$$-tuples of the form $$(x_1,x_2,\dots,x_n)$$ such that $$x_i\in X_i$$ for all $$i\in [n]\text{.}$$

###### Example2.1.

In the state of Georgia, license plates consist of four digits followed by a space followed by three capital letters. The first digit cannot be a $$0\text{.}$$ How many license plates are possible?

Solution.

Let $$X$$ consist of the digits $$\{0,1,2,\dots,9\}\text{,}$$ let $$Y$$ be the singleton set whose only element is a space, and let $$Z$$ denote the set of capital letters. A valid license plate is just a string from

\begin{equation*} (X-\{0\})\times X\times X\times X\times Y\times Z\times Z\times Z \end{equation*}

so the number of different license plates is $$9\times10^3\times1\times 26^3=158\,184\,000\text{,}$$ since the size of a product of sets is the product of the sets' sizes. We can get a feel for why this is the case by focusing just on the digit part of the string here. We can think about the digits portion as being four blanks that need to be filled. The first blank has $$9$$ options (the digits $$1$$ through $$9$$). If we focus on just the digit strings beginning with $$1\text{,}$$ one perspective is that they range from $$1000$$ to $$1999\text{,}$$ so there are $$1000$$ of them. However, we could also think about there being $$10$$ options for the second spot, $$10$$ options for the third spot, and $$10$$ options for the fourth. Multiplying $$10\times 10\times 10$$ gives $$1000\text{.}$$ Since our analysis of filling the remaining digit blanks didn't depend on our choice of a $$1$$ for the first position, we see that each of the $$9$$ choices of initial digit gives $$1\, 000$$ strings, for a total of $$9\,000 = 9\times 10^3\text{.}$$

In the case that $$X=\{0,1\}\text{,}$$ an $$X$$-string is called a $$0$$–$$1$$ string (also a binary string or bit string.). When $$X=\{0,1,2\}\text{,}$$ an $$X$$-string is also called a ternary string.

###### Example2.2.

A machine instruction in a $$32$$-bit operating system is just a bit string of length $$32\text{.}$$ Thus, there are $$2$$ options for each of $$32$$ positions to fill, making the number of such strings $$2^{32} = 4\,294\,967\,296\text{.}$$ In general, the number of bit strings of length $$n$$ is $$2^n\text{.}$$

###### Example2.3.

Suppose that a website allows its users to pick their own usernames for accounts, but imposes some restrictions. The first character must be an upper-case letter in the English alphabet. The second through sixth characters can be letters (both upper-case and lower-case allowed) in the English alphabet or decimal digits ($$0$$–$$9$$). The seventh position must be ‘@’ or ‘.’. The eighth through twelfth positions allow lower-case English letters, ‘*’, ‘%’, and ‘#’. The thirteenth position must be a digit. How many users can the website accept registrations from?

Solution.

We can visualize the options by thinking of the $$13$$ positions in the string as blanks that need to be filled in and putting the options for that blank above. In Figure 2.4, we've used U to denote the set of upper-case letters, L for the set of lower-case letters, and D for the set of digits.

Below each position in the string, we've written the number of options for that position. (For example, there are 62 options for the second position, since there are $$52$$ letters once both cases are accounted for and $$10$$ digits. We then multiply these possibilities together, since each choice is independent of the others. Therefore, we have

\begin{equation*} 26\times 62^5 \times 2 \times 29^5\times 10 = 9\,771\,287\,250\,890\,863\,360 \end{equation*}


text:
# How would you summarize the Law of Conservation of Mass when dealing with a chemical change?

$\text{Mass is conserved in all chemical reactions..........}$
And what does it mean? It means that if you start with $10 \cdot g$ of reactant, from all sources, AT MOST YOU CAN GET $10 \cdot g$ of product. This conservation is unequivocal and universal in all chemical reactions.


text:
Math Help - Random sample from a continuous type dist

1. Random sample from a continuous type dist

Let X1,X2,,,,Xn be a random sample from a continuous type distribution
a) find P(X1<=X2),P(X1<=X2,X1<=X3),...,P(X1<=Xi, i=2,3,...,n).

(The answer for this is 1/n but I am not sure the reasoning behind of that.
Can anybody explain this?)

b) Suppose the sampling continues until X1 is no longer the smallest observation, (i.e., Xj < X1 <= Xi, i=2,3,...,j-1). Let Y equal the number of trials until X1 is no longer the smallest observation, (i.e., Y=j-1). Show that the distribution of Y is P(Y=y) = 1 / y(y+1), y=1,2,3,...
(????????)

c) Compute the mean and variance of Y if they exist.
(?????????)

2. Since $1=P(X>Y)+P(X.
I would think you just have to show that these probabilities are equal,
giving you .5.

As for (c), if P(Y=y) = 1 / y(y+1), y=1,2,3,..., then

$E(Y)=\sum_{y=1}^{\infty}{1\over y+1}=\sum_{n=2}^{\infty}{1\over n}=\infty$.
Thus the second moment and variance are also infinite.


text:
# American Institute of Mathematical Sciences

March  2022, 14(1): 131-150. doi: 10.3934/jgm.2021032

## Modeling student engagement using optimal control theory

 Mathematics Department, University of California Santa Cruz, Santa Cruz, CA 95064, USA

* Corresponding author: Debra Lewis

Dedicated to Professor Anthony Bloch on the occasion of his 65th birthday.

Received  June 2021 Revised  November 2021 Published  March 2022 Early access  January 2022

Student engagement in learning a prescribed body of knowledge can be modeled using optimal control theory, with a scalar state variable representing mastery, or self-perceived mastery, of the material and control representing the instantaneous cognitive effort devoted to the learning task. The relevant costs include emotional and external penalties for incomplete mastery, reduced availability of cognitive resources for other activities, and psychological stresses related to engagement with the learning task. Application of Pontryagin's maximum principle to some simple models of engagement yields solutions of the synthesis problem mimicking familiar behaviors including avoidance, procrastination, and increasing commitment in response to increasing mastery.

Citation: Debra Lewis. Modeling student engagement using optimal control theory. Journal of Geometric Mechanics, 2022, 14 (1) : 131-150. doi: 10.3934/jgm.2021032
##### References:
 [1] M. C. Anderson, R. A. Bjork and E. L. Bjork, Remembering can cause forgetting: Retrieval dynamics in long-term memory, Journal of Experimental Psychology. Learning, Memory, and Cognition, 20 (1994), 1063-1087.  doi: 10.1037/0278-7393.20.5.1063. [2] L. Averell and A. Heathcote, The form of the forgetting curve and the fate of memories, Journal of Mathematical Psychology, 55 (2011), 25-35.  doi: 10.1016/j.jmp.2010.08.009. [3] J. Baillieul and J. Willems, Mathematical Control Theory, Springer, 1999. doi: 10.1007/978-1-4612-1416-8. [4] A. Bandura, Self-Efficacy in Changing Societies, Cambridge University Press, 1995. [5] A. M. Bloch, Nonholonomic Mechanics and Control, Springer, 2003. doi: 10.1007/b97376. [6] H. Chang and S. L. Beilock, The math anxiety-math performance link and its relation to individual and environmental factors: a review of current behavioral and psychophysiological research, Current Opinion in Behavioral Sciences, 10 (2016), 33-38.  doi: 10.1016/j.cobeha.2016.04.011. [7] K. Choe, J. Jenifer, C. Rozek, M. Berman and S. Beilock, Calculated avoidance: Math anxiety predicts math avoidance in effort-based decision-making, Science Advances, 5 (2019).  doi: 10.1126/sciadv.aay1062. [8] R. Dorfman, An economic interpretation of optimal control theory, The American Economic Review, 59 (1969), 817-831. [9] H. Ebbinghaus, Über das Gedächtnis, Dunker, 1885. [10] E. O. Finkenbinder, The curve of forgetting, The American Journal of Psychology, 24 (1913), 8-32.  doi: 10.2307/1413271. [11] T. L. Griffiths, F. Lieder and N. D. Goodman, Rational use of cognitive resources: Levels of analysis between the computational and the algorithmic, Topics in Cognitive Science, 11 (2015), 217-229.  doi: 10.1111/tops.12142. [12] K. Hall, E. Fawcett, K. Hourihan and J. Fawcett, Emotional memories are (usually) harder to forget: A meta-analysis of the item-method directed forgetting literature, Psychonomic Bulletin & Review, 28 (2021), 1313-1326.  doi: 10.3758/s13423-021-01914-z. [13] D. Kirk, Optimal Control Theory: An Introduction, Dover Publications, Inc., 2004. [14] S. Koessler, H. Engler, C. Riether and J. Kissler, No retrieval-induced forgetting under stress, Psychological Science, 20 (2009), 1356-1363.  doi: 10.1111/j.1467-9280.2009.02450.x. [15] D. Lewis, A soothing invisible hand: Moderation potentials in optimal control, in Geometry, Mechanics, and Dynamics: The Legacy of Jerry Marsden, Springer, 2015,257–284. doi: 10.1007/978-1-4939-2441-7_12. [16] G. R. Loftus, Evaluating forgetting curves, Journal of Experimental Psychology, 11 (1985), 397-406.  doi: 10.1037/0278-7393.11.2.397. [17] I. Lyons and S. Beilock, When math hurts: Math anxiety predicts pain network activation in anticipation of doing math, PLOS ONE, 7. doi: 10.1371/journal.pone.0048076. [18] E. Maloney, M. Schaeffer and S. Beilock, Mathematics anxiety and stereotype threat: Shared mechanisms, negative consequences, and promising interventions, Research in Mathematics Education, 15 (2013), 115-128.  doi: 10.1080/14794802.2013.797744. [19] A. Mattarella-Micke, J. Mateo, M. Kozak, K. Foster and S. Beilock, Choke or thrive? The relation between salivary cortisol and math performance depends on individual differences in working memory and math anxiety, Emotion, 11 (2011), 1000-1005.  doi: 10.1037/a0023224. [20] H. Melville, Bartleby, the scrivener: A story of Wall Street, Putnam's Magazine, November and December (1853), 446–457 and 609–615. doi: 10.1093/oseo/instance.00209193. [21] R. Montgomery, Optimal control of deformable bodies and its relation to gauge theory, Math. Sci. Res. Inst. Publ., 22 (1991), 403-438.  doi: 10.1007/978-1-4613-9725-0_15. [22] J. Murre and J. Dros, Replication and analysis of Ebbinghaus' forgetting curve, PLOS ONE, 10 (2015).  doi: 10.1371/journal.pone.0120644. [23] L. Pontryagin, V. Boltyanskii, R. Gamkrelidze and E. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience Publishers, 1962. [24] R. Rydell, A. McConnell and S. Beilock, Multiple social identities and stereotype threat: Imbalance, accessibility, and working memory, Journal of Personality and Social Psychology, 96 (2009), 949-966.  doi: 10.1037/a0014846. [25] A. Shenhav, S. Musslick, F. Lieder, W. Kool, T. Griffiths, J. Cohen and M. Botvinick, Toward a rational and mechanistic account of mental effort, Annual Review of Neuroscience, 40 (2017), 99-124.  doi: 10.1146/annurev-neuro-072116-031526. [26] F. Sirois and T. Pychyl, Procrastination and the priority of short-term mood regulation: Consequences for future self, Social and Personality Psychology Compass, 7 (2013), 115-127.  doi: 10.1111/spc3.12011. [27] N. Slamecka, On comparing rates of forgetting: Comment on Loftus (1985), Journal of Experimental Psychology. Learning, Memory, and Cognition, 11 (1985), 812-816.  doi: 10.1037/0278-7393.11.1-4.812. [28] E. Sontag, Integrability of certain distributions associated with actions on manifolds and applications to control problems, in Nonlinear Controlability and Optimal Control, Marcel Dekker, Inc., 133 (1990), 81–131.

show all references

##### References:
 [1] M. C. Anderson, R. A. Bjork and E. L. Bjork, Remembering can cause forgetting: Retrieval dynamics in long-term memory, Journal of Experimental Psychology. Learning, Memory, and Cognition, 20 (1994), 1063-1087.  doi: 10.1037/0278-7393.20.5.1063. [2] L. Averell and A. Heathcote, The form of the forgetting curve and the fate of memories, Journal of Mathematical Psychology, 55 (2011), 25-35.  doi: 10.1016/j.jmp.2010.08.009. [3] J. Baillieul and J. Willems, Mathematical Control Theory, Springer, 1999. doi: 10.1007/978-1-4612-1416-8. [4] A. Bandura, Self-Efficacy in Changing Societies, Cambridge University Press, 1995. [5] A. M. Bloch, Nonholonomic Mechanics and Control, Springer, 2003. doi: 10.1007/b97376. [6] H. Chang and S. L. Beilock, The math anxiety-math performance link and its relation to individual and environmental factors: a review of current behavioral and psychophysiological research, Current Opinion in Behavioral Sciences, 10 (2016), 33-38.  doi: 10.1016/j.cobeha.2016.04.011. [7] K. Choe, J. Jenifer, C. Rozek, M. Berman and S. Beilock, Calculated avoidance: Math anxiety predicts math avoidance in effort-based decision-making, Science Advances, 5 (2019).  doi: 10.1126/sciadv.aay1062. [8] R. Dorfman, An economic interpretation of optimal control theory, The American Economic Review, 59 (1969), 817-831. [9] H. Ebbinghaus, Über das Gedächtnis, Dunker, 1885. [10] E. O. Finkenbinder, The curve of forgetting, The American Journal of Psychology, 24 (1913), 8-32.  doi: 10.2307/1413271. [11] T. L. Griffiths, F. Lieder and N. D. Goodman, Rational use of cognitive resources: Levels of analysis between the computational and the algorithmic, Topics in Cognitive Science, 11 (2015), 217-229.  doi: 10.1111/tops.12142. [12] K. Hall, E. Fawcett, K. Hourihan and J. Fawcett, Emotional memories are (usually) harder to forget: A meta-analysis of the item-method directed forgetting literature, Psychonomic Bulletin & Review, 28 (2021), 1313-1326.  doi: 10.3758/s13423-021-01914-z. [13] D. Kirk, Optimal Control Theory: An Introduction, Dover Publications, Inc., 2004. [14] S. Koessler, H. Engler, C. Riether and J. Kissler, No retrieval-induced forgetting under stress, Psychological Science, 20 (2009), 1356-1363.  doi: 10.1111/j.1467-9280.2009.02450.x. [15] D. Lewis, A soothing invisible hand: Moderation potentials in optimal control, in Geometry, Mechanics, and Dynamics: The Legacy of Jerry Marsden, Springer, 2015,257–284. doi: 10.1007/978-1-4939-2441-7_12. [16] G. R. Loftus, Evaluating forgetting curves, Journal of Experimental Psychology, 11 (1985), 397-406.  doi: 10.1037/0278-7393.11.2.397. [17] I. Lyons and S. Beilock, When math hurts: Math anxiety predicts pain network activation in anticipation of doing math, PLOS ONE, 7. doi: 10.1371/journal.pone.0048076. [18] E. Maloney, M. Schaeffer and S. Beilock, Mathematics anxiety and stereotype threat: Shared mechanisms, negative consequences, and promising interventions, Research in Mathematics Education, 15 (2013), 115-128.  doi: 10.1080/14794802.2013.797744. [19] A. Mattarella-Micke, J. Mateo, M. Kozak, K. Foster and S. Beilock, Choke or thrive? The relation between salivary cortisol and math performance depends on individual differences in working memory and math anxiety, Emotion, 11 (2011), 1000-1005.  doi: 10.1037/a0023224. [20] H. Melville, Bartleby, the scrivener: A story of Wall Street, Putnam's Magazine, November and December (1853), 446–457 and 609–615. doi: 10.1093/oseo/instance.00209193. [21] R. Montgomery, Optimal control of deformable bodies and its relation to gauge theory, Math. Sci. Res. Inst. Publ., 22 (1991), 403-438.  doi: 10.1007/978-1-4613-9725-0_15. [22] J. Murre and J. Dros, Replication and analysis of Ebbinghaus' forgetting curve, PLOS ONE, 10 (2015).  doi: 10.1371/journal.pone.0120644. [23] L. Pontryagin, V. Boltyanskii, R. Gamkrelidze and E. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience Publishers, 1962. [24] R. Rydell, A. McConnell and S. Beilock, Multiple social identities and stereotype threat: Imbalance, accessibility, and working memory, Journal of Personality and Social Psychology, 96 (2009), 949-966.  doi: 10.1037/a0014846. [25] A. Shenhav, S. Musslick, F. Lieder, W. Kool, T. Griffiths, J. Cohen and M. Botvinick, Toward a rational and mechanistic account of mental effort, Annual Review of Neuroscience, 40 (2017), 99-124.  doi: 10.1146/annurev-neuro-072116-031526. [26] F. Sirois and T. Pychyl, Procrastination and the priority of short-term mood regulation: Consequences for future self, Social and Personality Psychology Compass, 7 (2013), 115-127.  doi: 10.1111/spc3.12011. [27] N. Slamecka, On comparing rates of forgetting: Comment on Loftus (1985), Journal of Experimental Psychology. Learning, Memory, and Cognition, 11 (1985), 812-816.  doi: 10.1037/0278-7393.11.1-4.812. [28] E. Sontag, Integrability of certain distributions associated with actions on manifolds and applications to control problems, in Nonlinear Controlability and Optimal Control, Marcel Dekker, Inc., 133 (1990), 81–131.
Left: graphs of $u_c(\mu, \cdot )$ for sample values of $\mu$; the lowest curve is the graph of $u_c(1, \cdot )$. Right: contour plot of $u_{\rm opt}$, with contour values $\frac j {10}, j = 0, \ldots, 10$. Black: u = 0; white: u = 1
Left: $K_{\frac \pi {10}}(r)$ for $r$ near $\tan \frac \pi {10}$, right: $\upsilon_\circ(r)$ associated to $\xi_\circ(u) = \sqrt{1 - u^2}$
Graphs of $X_{h_0}^\pm$ for $\mathcal{S} = [0, 1]$, $\gamma(m) = (1 - m)^2$, $\mu(m) = {1 \over 2} \left ( 1 - \frac m 2 \right )^2$, constant functions $\psi_{\rm pe} = {1 \over 2}$ and $\phi = \frac \pi {12}$, and representative values of $h_0$ for which $\breve C_{h_0}$ has a strict global minimum ${c_{h_0}}$. Solid curves: $X_{h_0}^+$; dashed curves: $X_{h_0}^-$
Representative vertically aligned graphs of $K_\phi(r)$ and $\breve C_{h_0}(m)$ for constant $\phi$ and three values of $h_0$. Solid: projections of the trajectories with initial data $(m_0, r_0)$; dotted: points on the graphs outside those projections. Upper row: $r_0 < r_\phi$ and $m_0 > m_c$; initially, $m$ decreases and $r$ increases. Lower row: $r_0 > r_\phi$ and $m_0 < m_c$; initially, $m$ increases and $r$ decreases. Left: $m'$ changes sign as $r$ passes through $r_\phi$; center: $m$ and $r$ asymptotically approach the equilibrium $(m_c, r_\phi)$; right: $r'$ changes sign as $m$ passes through $m_c$
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## Algebra and Trigonometry 10th Edition

The equation for the velocity of the edge of the wheel given it is fixed at an angular velocity $\omega$ is $v=\omega r$. Thus, when the diameter (and thus the radius) increase, the velocity at the edge increases as well.


text:
# Can bidirectional RNN use variable sequence length?

A bidirectional RNN consists of two RNNs, one for the forward and another for the backward sequential directions, which outcome is concatenated at each time step. Would this configuration restrict the model to always use a fixed sequence length? Or would it still work as the unidirectional RNN, which can be applied to any sequence length?

This question was raised because the bidirectional architecture merges the output of both forward and backward RNNs at each time step. Thus, if the sequence length was 4, the outputs of both forward and backward RNNs would merge this way: 1th forward with 4th backward, 2nd forward with 3rd backward,... 4th forward with 1st backward. However, if a different sequence length was used this merge order would be modified:

Let's say the network was trained with sequence length 4, but at test time a sequence length of 5 was used. Merge would be: 1th forward with 5th backward, 2nd forward with 4th backward... 5th forward with 1th backward. Would this shift in merge order negatively affect the bidirectional RNN performance?

• – Valentas Jun 2 at 7:02

The short answer is no, a bidirectional architecture will still take in a variable sequence length. To understand why, you should understand how padding works.

For example, let's say you are implementing a bidirectional LSTM-RNN in tensorflow on variable length time series data for multiple subjects. The input is a 3D array with shape: [n_subjects, [n_features, [n_timesteps...] ...] ...] so to ensure that the array has consistent dimensions, you pad the other subject's features up to the length of the subject with features measured for the longest period of time.

Let's say subject 1 has one feature with values = [22,20,19,21,33,22,44,21,19,26,27] measured at times = [0,1,2,3,4,5,6,7,8,9,10]. subject 2 has one feature with values = [21,12,22,30,13,42,20] measured at times = [0,1,2,3,4,5,6]. You would pad features for Subject 2 by extending the array so that the padded_values = [21,12,22,30,13,42,20,0,0,0,0] at times = [0,1,2,3,4,5,6,7,8,9,10], then do the same thing for every subsequent subject.

This means the number of timesteps for each subject can be variable, and the merge you refer to occurs with the dimension for that particular subject.

Below is an example of a bidirectional LSTM-RNN architecture for a model that predicts sleep stages for different subjects using biometric features measured over variable lengths of time.

• Does this mean we always need a padding in bidirectional RNN if we want to use variable sequence length? – DiMorten - Jorge Chamorro Jun 11 '20 at 16:58
• Yes you need padding for any RNN with variable sequence length, at least when you feed the 3D data array into a model built in Keras or Tensorflow. – Derek O Jun 11 '20 at 19:06

Yes, there is a way with TF 2.0, which is to use Ragged Tensors as below:

# Task: predict whether each sentence is a question or not.
sentences = tf.constant(
['What makes you think she is a witch?',
'She turned me into a newt.',
'A newt?',
'Well, I got better.'])
is_question = tf.constant([True, False, True, False])

# Preprocess the input strings.
hash_buckets = 1000
words = tf.strings.split(sentences, ' ')
hashed_words = tf.strings.to_hash_bucket_fast(words, hash_buckets)

# Build the Keras model.
keras_model = tf.keras.Sequential([
tf.keras.layers.Input(shape=[None], dtype=tf.int64, ragged=True),
tf.keras.layers.Embedding(hash_buckets, 16),
tf.keras.layers.LSTM(32, use_bias=False),
tf.keras.layers.Dense(32),
tf.keras.layers.Activation(tf.nn.relu),
tf.keras.layers.Dense(1)
])

keras_model.compile(loss='binary_crossentropy', optimizer='rmsprop')
keras_model.fit(hashed_words, is_question, epochs=5)
print(keras_model.predict(hashed_words))

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text:
Absolute Value Functions (Part 2)

ID: vomub-tokol
Illustrative Mathematics, CC BY 4.0
Subject: Algebra, Algebra 2
Grade: 8-9
Standards: HSF-BF.A.1.aHSF-IF.C.7.bHSF-IF.CHSF-BF.A.1HSF-BF.B.3
PreviewAnswer Key

22 questions

Absolute Value Functions (Part 2)

Classroom:
Due:
Student Name:
Date Submitted:
Temperature in Toronto

Toronto is a city at the border of the United States and Canada, just north of Buffalo, New York. Here are twelve guesses of the average temperature of Toronto, in degrees Celsius, in February 2017.

5, 2, -5, 3, 0, -1, 1.5, 4, -2.5, 6, 4, -0.5

The actual average temperature of Toronto in February 2017 was 0 degree Celsius.

1) Use this information to sketch a scatter plot representing the guesses, $x$, and the corresponding absolute guessing errors, $y$.

2) What rule can you write to find the output given the input?

The Distance Function

The function $A$ gives the distance of $x$ from 0 on the number line.

Complete the table. Write the values for the labeled cells in the answer boxes below.

$\begin{array}{|c|c|} \hline \\[-1em] x & A(x) \\[-1em] \\ \hline \\[-1em] 8 & \text{A} \\[-1em] \\ \hline \\[-1em] \text{B} & 5.6 \\[-1em] \\ \hline \\[-1em] \pi & \text{C} \\[-1em] \\ \hline \\[-1em] \frac{1}{2} & \text{D} \\[-1em] \\ \hline \\[-1em] \text{E} & 1 \\[-1em] \\ \hline \\[-1em] 0 & \text{F} \\[-1em] \\ \hline \\[-1em] -\frac{1}{2} & \text{G} \\[-1em] \\ \hline \\[-1em] -1 & \text{H} \\[-1em] \\ \hline \\[-1em] -5.6 & \text{I} \\[-1em] \\ \hline \\[-1em] \text{J} & 8 \\[-1em] \\ \hline \end{array}$

3) Cell A

4) Cell B

5) Cell C

6) Cell D

7) Cell E

8) Cell F

9) Cell G

10) Cell H

11) Cell I

12) Cell J

13) Sketch a graph of function $A$.

Andre and Elena are trying to write a rule for this function.

$\boldsymbol{\cdot}$ Andre writes: $\begin{cases} x, & x \geq 0 \\ -x, & x < 0 \end{cases}$

$\boldsymbol{\cdot}$ Elena writes: $A(x) = |x|$

14) Explain why both equations correctly represent the function $A$.

Moving Graphs Around

Graph the functions $f(x) = |x + a|$ and $g(x) = |x| + b$. Experiment with different values of $a$ and $b$ and observe what happens to the graphs.

15) How does changing $a$ change the graph?

16) How does changing $b$ change the graph?

More Moving Graphs Around

Here are five equations and four graphs.

Equation 1: $y = |x - 3| \quad \quad$ Equation 2: $y = |x - 9| + 3 \quad \quad$ Equation 3: $y = |x| - 6$$\quad \quad$ Equation 4: $y = |x + 3| \quad \quad$ Equation 5: $y = |x + 3| - 6$

Match each equation with a graph that represents it. Write the letter of the graph that corresponds to each equation. One equation has no match. (Write "No match" for this equation.)

17) Equation 1

18) Equation 2

19) Equation 3

20) Equation 4

21) Equation 5

22) For the equation without a match, sketch a graph on the blank coordinate plane.


text:
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# Outdoor level rendering/management

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I''ve got this neat idea for a game in the back of my head, but it requires me to render a big outdoor level. I have never written a landscape generator or heightmap loader or anything like that, so I''d like to inform what techniques are often used in rendering and managing outdoor levels. I''d like to know what I''m getting myself into before I start cursing and hitting my computer screen if it doesn''t work I''ve just read the "is BSP dead?" thread and the term quadtree caught my eye. I have a basic idea what it is, but is it used a lot in outdoor levels? If so what for? For rendering or actor-management? Are the quadtree sections a static size? What techniques are used for occluded terrain culling? Or do you render everything up to a certain distance and have the API handle the culling? Any tips/linkage/answers are very appreciated Thank you!

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I''m at school atm and shouldn''t b here so here goes a quick answer, but there''s lots more about it really...

First of all, before I forget, let me mention some techniques you should master to build a big terrain:

1. Make a x-z grid with the sizes of your map. Most of the time the delta x (so how much the x changes, this is also the size of your polygons) is 32. It''s fast (the smaller the polygons the slower) but it allows great detail. (So it looks smooth)
2. Read in the y (height) values from a file. This could be a file you made with some sort of map editor, but if you don''t have one you could use the heightmap loading technique too (loading height values out of a RAW data file, technique #1 look it up)
3. Make the octree. (technique #2 An octree is basically just a division of the landscape. See it as this: First, you draw a cube around the whole map. Then, you make 1 subdivision. So you split the x-axis into 2 parts, and the z and y-axis too. That''d leave you with 8 different cubes now, each 1/8th the size of the original cube. Now, if you''d check which of the cubes are in your viewing frustum (check if they collide with the shape that''s your view, this technique is called frustum culling (technique #3 enters)) you can only draw the cubes that are visible. This might leave A LOT out if your landscape is big (and your view is optimal). BTW, if you DRAW one of the cubes, you draw the polygons inside that cube. I think www.gametutorials.com has one of the best octree tutorials.
So now you have your octree. There''s a lot more to it than this, but you''ll see.
4. Use LOD. (Technique 4) LOD means level of detail, and it basically just merges polygons where the detail level (so the size of the polygon) is higher than required. If, for example, the polygons are far far away from you, or if the height delta between some polygons is (almost) the same, you could decrease the amount of polygons a lot by merging the small polygons together in less, bigger polygons (thereby decreasing the detail level, but this goes (almost) unnoticed) This saves polygons, which saves speed.

If you use all these methods, you get a quite fast terrain engine. I believe there''s another technique for checking which polygons are hidden behind others, to skip those too. I haven''t looked into that though. It''s called ray tracing I believe, or DirectX calls it culling I think. Look that up as well. If you''ve done this, I think you got one of the most speedy terrains possible. It''s a really big job though, so start out working on the techniques one by one, and make sure the previous technique is implemented perfectly before you move on to the next. To help you out starting, the best way in technique #1 is to store the vertices in a vertex buffer or make some sort of vertex description and save the data into an array of these vertices. Like this:

class CVERTEX{   float x, y, z;   float u, v; // texture coords   // Add what else you need};

You could also make more classes like one for polygons
class CPOLY{   CVERTEX a, b, c;   texturefile texture;};Then just draw all the vertices (or polygons) and find out how to texture them. There are many of these texturing techniques as well: none is ''best'', it depends on what you want (I always stretch a texture over the landscape, and add a detail texture with multitexturing that''s repeated for higher texture detail)Oh one last note www.gametutorials.com does not only have octree tutorials, but also heightmap tutorials. The way of loading heightmaps is used in the octree code too, so it''s wise to use both if you want to use one of the 2...

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Hey there. I differ slightly from the last poster''s thoughts but only in detail.
Here are some thoughts...
1)A basic landscape is easy to render - just a 2d array of vertices at different heights, where you store the heights as a bitmap or some other way.
2)The sizes of the polygons are NOT normally 32 - in fact the spacing between vertices makes no difference as you can scale them to whatever you want - I use the distance as a value in metres and have points at 1m spacing.
3)Both quadtrees and Octrees are popular. A quadtree divides the map into 4, then each part divides into 4 and so on until you get to a region of the map which is some size e.g 16x16, 32x32, called a leaf - the leaf stores a small section of the map. This means when you render you quickly find out which sections need drawing. An octree is the same idea but you divide a cube into 8 so you divide the map by volume rather than area. I assume which is best depends on how flat your maps are.
4)Typically the quadtree sections (I''d recommend this not octreee personally) are static size but they don''t have to be. For instance you don''t have to have each section having 4 children - if there''s nothing in one section (ie your terrain is not square) you can leave it empty. You can also keep subdividing different sections to different levels so not all leaves are equally big, if you can think of a good reason to (I''d avoid this initially though but maybe leave the design open to this possibility)
5)LOD is typically used to decide for each leaf in view, what detail to draw it at. This choice can be done in different ways, the main problem is where two leaves join it must be at the same detail or you get nasty gaps, so you need to make the rendering code more complex.

I''d advise initially making a quadtree but drawing everything, then do quadtree culling, then lastly implement LOD.
Other occlusion techniques exist but they are a lot more complicated and you''ll probably find you don''t need them. You can also stop drawing leaves which are far enough away - it''s noticeable but not if you introduce a bit of fog.

NEW (18th December)2 new screenshots, one from the engine and one from the level editor

John 3:16

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Thanks, I got quite a few answers here...
A bit of a technical question:
How do you split up the leaves of the quadtree? Do still keep your 2D array, and do you determine each leaf by the index of the array (leaf 1 starting at [0][0], leaf two at [0][10], etc) of do you sort them in a particular way? each leaf in it''s own array for example.

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Thanks d000gh...
I shall continue my questioning
I have also been looking at collision detection with terrain.
I have read on OpenGL.org that a common way of detecting an objects collision with the terrain is checking a point of the object against the y-value of the array entry that the point is over. Now, to smoothen this detection I read about interpolating the points around the collision point, and checking it then. How accurate is this?
More over, how do I interpolate? I do know what it is, but do not know a formula.

Another thing, is it common to smoothen the terrain amongst points between the array entries? Or does is each entry exactly one vertex?

To summarize:

1) Is interpolating for collision detection accurate?
2) what is the formula for interpolating four (or three) points?
3) Is it common a terrain is smoothed between the array entries?

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Well, I''ve not implemented collisions yet so on accuracy I''m not sure. Also, do you mean linear interpolation (drawing a plane between the 3 points of a triangle) or higher order (finding a curved surface that fits well over the points and smooths out the boundaries)?

The latter is more taxing but certainly possible, it''s much more computationally expensive too. For the linear version you just find the plane eqn for the 3 points of the triangle and stuff in the x/z coord to get the height.

Feel free to email me if you have any further questions.

NEW (18th December)2 new screenshots, one from the engine and one from the level editor

John 3:16

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17 Generators and relators of groups

17.1

17.1-1 CayleyGraphOfGroupDisplay
 ‣ CayleyGraphOfGroupDisplay( G, X ) ( function )
 ‣ CayleyGraphOfGroupDisplay( G, X, str ) ( function )

Inputs a finite group G together with a subset X of G. It displays the corresponding Cayley graph as a .gif file. It uses the Mozilla web browser as a default to view the diagram. An alternative browser can be set using a second argument str="mozilla".

The argument G can also be a finite set of elements in a (possibly infinite) group containing X. The edges of the graph are coloured according to which element of X they are labelled by. The list X corresponds to the list of colours [blue, red, green, yellow, brown, black] in that order.

This function requires Graphviz software.

17.1-2 IdentityAmongRelatorsDisplay
 ‣ IdentityAmongRelatorsDisplay( R, n ) ( function )
 ‣ IdentityAmongRelatorsDisplay( R, n, str ) ( function )

Inputs a free ZG-resolution R and an integer n. It displays the boundary R!.boundary(3,n) as a tessellation of a sphere. It displays the tessellation as a .gif file and uses the Mozilla web browser as a default display mechanism. An alternative browser can be set using the second argument str="mozilla". (The resolution R should be reduced and, preferably, in dimension 1 it should correspond to a Cayley graph for G. )

This function uses GraphViz software.

1 , 2

17.1-3 IsAspherical
 ‣ IsAspherical( F, R ) ( function )

Inputs a free group F and a set R of words in F. It performs a test on the 2-dimensional CW-space K associated to this presentation for the group G=F/<R>^F.

The function returns "true" if K has trivial second homotopy group. In this case it prints: Presentation is aspherical.

Otherwise it returns "fail" and prints: Presentation is NOT piece-wise Euclidean non-positively curved. (In this case K may or may not have trivial second homotopy group. But it is NOT possible to impose a metric on K which restricts to a Euclidean metric on each 2-cell.)

The function uses Polymake software.

1 , 2 , 3

17.1-4 PresentationOfResolution
 ‣ PresentationOfResolution( R ) ( function )

Inputs at least two terms of a reduced ZG-resolution R and returns a record P with components

• P.freeGroup is a free group F,

• P.relators is a list S of words in F,

• P.gens is a list of positive integers such that the i-th generator of the presentation corresponds to the group element R!.elts[P[i]] .

where G is isomorphic to F modulo the normal closure of S. This presentation for G corresponds to the 2-skeleton of the classifying CW-space from which R was constructed. The resolution R requires no contracting homotopy.

1 , 2 , 3 , 4

17.1-5 TorsionGeneratorsAbelianGroup
 ‣ TorsionGeneratorsAbelianGroup( G ) ( function )

Inputs an abelian group G and returns a generating set [x_1, ... ,x_n] where no pair of generators have coprime orders.

generated by GAPDoc2HTML


text:
# Vectors: Find w such that dv/ds = w x v

YayMathYay

## Homework Statement

T (tangent), B (binormal), N (normal) are an orthogonal triad of unit vectors of a curve in R3.

Given:
• dT/ds = kN
• dN/ds = -kT + tB
• dB/ds = -tN

Find vector w so that these equations may be written in the form:

dv/ds = w x v, where v = T + N + B

Given above!

## The Attempt at a Solution

I tried splitting v (T + N + B) into its components, as well as w, and putting them into a matrix to find the determinant (for the cross product).

The matrix consists of <i, j, k>, w = <w1, w2, w3>, and v = <(T1 + N1 + B1), (T2 + N2 + B2), (T3 + N3 + B3)>.

However, taking the determinant/cross product gives mismatching components (ie. j and k components supposedly add up to an i component). I'm not too sure where to go from here :(

Note: This is for a Multivariate Calculus course!

Last edited:

Homework Helper
Gold Member

## Homework Statement

T (tangent), B (binormal), N (normal) are an orthogonal triad of unit vectors of a curve in R3.

Given:
• dT/ds = kN
• dN/ds = -kT + tB
• dB/ds = -tN

Find vector w so that these equations may be written in the form:

dv/ds = w x v, where v = T + N + B

Given above!

## The Attempt at a Solution

I tried splitting v (T + N + B) into its components, as well as w, and putting them into a matrix to find the determinant (for the cross product).

The matrix consists of <i, j, k>, w = <w1, w2, w3>, and v = <(T1 + N1 + B1), (T2 + N2 + B2), (T3 + N3 + B3)>.

However, taking the determinant/cross product gives mismatching components (ie. j and k components supposedly add up to an i component). I'm not too sure where to go from here :(

Note: This is for a Multivariate Calculus course!

Hi YayMathYay, welcome to PF!

There's a much easier way to do this than by looking at the Cartesian components of the vectors.

Hint: What is $\mathbf{v}\times ( \mathbf{w} \times \mathbf{v} )$?

YayMathYay
Hi YayMathYay, welcome to PF!

There's a much easier way to do this than by looking at the Cartesian components of the vectors.

Hint: What is $\mathbf{v}\times ( \mathbf{w} \times \mathbf{v} )$?

First of all, thanks for the welcome! :)

I know the hint you gave me means $\left(\mathbf{v} \bullet \mathbf{v}\right)\mathbf{w} - \left(\mathbf{v} \bullet \mathbf{w}\right)\mathbf{v}$. I've tried this method before, but I wasn't too sure where to go from here..

Homework Helper
Gold Member
First of all, thanks for the welcome! :)

I know the hint you gave me means $\left(\mathbf{v} \bullet \mathbf{v}\right)\mathbf{w} - \left(\mathbf{v} \bullet \mathbf{w}\right)\mathbf{v}$. I've tried this method before, but I wasn't too sure where to go from here..

For starters, what is $\mathbf{v} \cdot \mathbf{v}$ ? Next since $\mathbf{T}$, $\mathbf{B}$ & $\mathbf{N}$ are an orthogonal triad in $\mathbb{R}^3$, they span $\mathbb{R}^3$, and so any vector can be decomposed into a linear combination of those unit vectors. So, why not write $\mathbf{w} = w_{T}\mathbf{T} + w_{N}\mathbf{N} + w_{B}\mathbf{B}$... what does that make $\mathbf{w} ( \mathbf{v} \cdot \mathbf{v} ) - \mathbf{v} ( \mathbf{w} \cdot \mathbf{v} )$?


text:
# Problem: Label each variable in the ideal gas law with the appropriate, commonly used units

###### FREE Expert Solution

We are asked to label each variable in the ideal gas law with the appropriate, commonly used units

To easily remember the units we simply need to know the units of R.

96% (431 ratings)
###### Problem Details

Label each variable in the ideal gas law with the appropriate, commonly used units