text:
• ### Momentum-space resummation for transverse observables and the Higgs $p_\perp$ at N$^3$LL+NNLO(1705.09127)

Jan. 5, 2021 hep-ph, hep-ex
We present an approach to the momentum-space resummation of global, recursive infrared and collinear safe observables featuring kinematic zeros away from the Sudakov limit. In the hadro-production of a generic colour singlet, we consider the family of inclusive observables which do not depend on the rapidity of the radiation, prime examples being the transverse momentum of the singlet, and $\phi^*$ in Drell-Yan pair production. We derive a resummation formula valid up to next-to-next-to-next-to-leading-logaritmic accuracy for the considered observables. This formula reduces exactly to the customary resummation performed in impact-parameter space in the known cases, and it also predicts the correct power-behaved scaling of the cross section in the limit of small value of the observable. We show how this formalism is efficiently implemented by means of Monte Carlo techniques in a fully exclusive generator that allows one to apply arbitrary cuts on the Born variables for any colour singlet, as well as to automatically match the resummed results to fixed-order calculations. As a phenomenological application, we present state-of-the-art predictions for the Higgs-boson transverse-momentum spectrum at the LHC at next-to-next-to-next-to-leading-logarithmic accuracy matched to fixed next-to-next-to-leading order.
• ### NNLOPS accurate associated HZ production with NLO decay ${\rm{H}} \to b\bar{b}$(1804.08141)

April 22, 2018 hep-ph, hep-ex
We present a next-to-next-to-leading order (NNLO) accurate description of associated HZ production, followed by the Higgs boson decay into a pair of $b$-quarks treated at next-to-leading order (NLO), consistently matched to a parton shower (PS). The matching is achieved by performing reweighting of the $\texttt{HZJ-MiNLO}$ events, using multi-dimensional distributions that are fully-differential in the HZ Born kinematics, to the NNLO results obtained by using the $\texttt{MCFM-8.0}$ fixed-order calculation. Additionally we include the $gg\to\rm{HZ}$ contribution to the discussed process that appears at the $\mathcal{O}(\alpha_s^2)$. We present phenomenological results obtained for 13 TeV hadronic collisions.
• ### Latest developments in the simulation of final states involving top-pair and heavy bosons(1704.08577)

April 27, 2017 hep-ph
I give an overview of recent progress in the simulation of final states involving top-quarks and vector bosons pair. First I'll discuss the recently found solutions needed to simulate fully differential top pair production ($pp\to b\bar{b}$ + 4 leptons) at NLO+PS accuracy, retaining off-shellness and interference effects exactly. In the second part, I'll review the MiNLO (Multi-scale Improved NLO) method, and then show a recent application, namely the simultaneous NLO+PS description of $W^+W^-$ and $W^+W^- +$ 1 jet production.
• ### Constraining Light-Quark Yukawa Couplings from Higgs Distributions(1606.09253)

April 1, 2017 hep-ph, hep-ex
We propose a novel strategy to constrain the bottom and charm Yukawa couplings by exploiting LHC measurements of transverse momentum distributions in Higgs production. Our method does not rely on the reconstruction of exclusive final states or heavy-flavour tagging. Compared to other proposals it leads to an enhanced sensitivity to the Yukawa couplings due to distortions of the differential Higgs spectra from emissions which either probe quark loops or are associated to quark-initiated production. We derive constraints using data from LHC Run I, and we explore the prospects of our method at future LHC runs. Finally, we comment on the possibility of bounding the strange Yukawa coupling.
• ### NNLOPS accurate associated HW production(1603.01620)

July 7, 2016 hep-ph, hep-ex
We present a next-to-next-to-leading order accurate description of associated HW production consistently matched to a parton shower. The method is based on reweighting events obtained with the HW plus one jet NLO accurate calculation implemented in POWHEG, extended with the MiNLO procedure, to reproduce NNLO accurate Born distributions. Since the Born kinematics is more complex than the cases treated before, we use a parametrization of the Collins-Soper angles to reduce the number of variables required for the reweighting. We present phenomenological results at 13 TeV, with cuts suggested by the Higgs Cross Section Working Group.
• ### Merging WW and WW+jet with MINLO(1606.07062)

June 22, 2016 hep-ph, hep-ex
We present a simulation program for the production of a pair of W bosons in association with a jet, that can be used in conjunction with general-purpose shower Monte Carlo generators, according to the POWHEG method. We have further adapted and implemented the MINLO' method on top of the NLO calculation underlying our WW+jet generator. Thus, the resulting simulation achieves NLO accuracy not only for inclusive distributions in WW+jet production but also WW production, i.e. when the associated jet is not resolved, without the introduction of any unphysical merging scale. This work represents the first extension of the MINLO' method, in its original form, to the case of a genuine underlying 2->2 process, with non-trivial virtual corrections.
• ### A new approach to the Higgs transverse-momentum resummation at NNLL+NNLO(1604.02191)

April 22, 2016 hep-ph, hep-ex
We propose a new approach to the resummation of the transverse-momentum distribution of a high-mass colour-singlet system in hadronic collisions. The resummation is performed in momentum space and is free of kinematic singularities at small transverse momentum. We derive a formula accurate at the next-to-next-to-leading-logarithmic level, and present the first matched predictions to next-to-next-to-leading order for Higgs-boson production in gluon fusion at the LHC. This method can be adapted to all observables featuring kinematic cancellations in the infrared region.
• This document outlines a set of simplified models for dark matter and its interactions with Standard Model particles. It is intended to summarize the main characteristics that these simplified models have when applied to dark matter searches at the LHC, and to provide a number of useful expressions for reference. The list of models includes both s-channel and t-channel scenarios. For s-channel, spin-0 and spin-1 mediation is discussed, and also realizations where the Higgs particle provides a portal between the dark and visible sectors. The guiding principles underpinning the proposed simplified models are spelled out, and some suggestions for implementation are presented.
• ### Latest theory developments for top pair production, generators and showering(1601.03647)

March 18, 2016 hep-ph
I summarize the state of the art of cross-section computations and of available simulation tools for top-quark pair production in hadron collisions. Particular emphasis is put on recent theory developments relevant for LHC phenomenology.
• ### Top-pair production and decay at NLO matched with parton showers(1412.1828)

Jan. 17, 2016 hep-ph, hep-ex
We present a next-to-leading order (NLO) calculation of $t\bar{t}$ production in hadronic collisions interfaced to shower generators according to the POWHEG method. We start from an NLO result from previous work, obtained in the zero width limit, where radiative corrections to both production and decays are included. The POWHEG interface required an extension of the POWHEG BOX framework, in order to deal with radiation from the decay of resonances. This extension is fully general (i.e. it can be applied in principle to any process considered in the zero width limit), and is here applied for the first time. In order to perform a realistic simulation, we introduce finite width effects using different approximations, that we validated by comparing with published exact NLO results. We have interfaced our POWHEG code to the PYTHIA8 shower Monte Carlo generator. At this stage, we dealt with novel issues related to the treatment of resonances, especially with regard to the initial scale for the shower that needs to be set appropriately. This procedure affects, for example, the fragmentation function of the b quark, that we have studied with particular attention. We believe that the tool presented here improves over previous generators for all aspects that have to do with top decays, and especially for the study of issues related to top mass measurements that involve B hadrons or b jets. The work presented here also constitutes a first step towards a fully consistent matching of NLO calculations involving intermediate resonances decaying into coloured particles, with parton showers.
• ### Simplified dark matter top-quark interactions at the LHC(1503.00691)

Aug. 20, 2015 hep-ph, hep-ex
Stringent limits on the interactions between dark matter (DM) and the standard model can be set by studying how initial-state or final-state particles recoil against missing transverse energy (MET). In this work, we improve, extend and correct LHC constraints on the interactions between DM and top quarks that are mediated by the exchange of spin-0 s-channel resonances. A comparison of the LHC run-1 sensitivity of the two main search channels is presented, which shows that mono-jet searches are typically more restrictive than the MET + tbar t searches. We furthermore explore the reach of the 14 TeV LHC. The collider constraints are compared to the restrictions arising from direct and indirect detection as well as the DM relic abundance, and we also reflect on effective field theory interpretations of the LHC exclusions.
• This document is the final report of the ATLAS-CMS Dark Matter Forum, a forum organized by the ATLAS and CMS collaborations with the participation of experts on theories of Dark Matter, to select a minimal basis set of dark matter simplified models that should support the design of the early LHC Run-2 searches. A prioritized, compact set of benchmark models is proposed, accompanied by studies of the parameter space of these models and a repository of generator implementations. This report also addresses how to apply the Effective Field Theory formalism for collider searches and present the results of such interpretations.
• ### Beauty-quark and charm-quark pair production asymmetries at LHCb(1505.02429)

May 10, 2015 hep-ph, hep-ex
The LHCb collaboration has recently performed a first measurement of the angular production asymmetry in the distribution of beauty quarks and anti-quarks at a hadron collider. We calculate the corresponding standard model prediction for this asymmetry at fixed-order in perturbation theory. Our results show good agreement with the data, which is provided differentially for three bins in the invariant mass of the $b \bar b$ system. We also present similar predictions for both beauty-quark and charm-quark final states within the LHCb acceptance for a collision energy of $\sqrt{s} = 13 \, {\rm TeV}$. We finally point out that a measurement of the ratio of the $b \bar b$ and $c \bar c$ cross sections may be useful for experimentally validating charm-tagging efficiencies.
• ### NNLOPS accurate Drell-Yan production(1407.2940)

July 10, 2014 hep-ph, hep-ex
We present a next-to-next-to-leading order accurate description of Drell-Yan lepton pair production processes through $\gamma^*/Z$ or $W$ exchange that includes consistently parton shower effects. Results are obtained by upgrading the vector-boson plus one jet NLO calculation in POWHEG with the MiNLO procedure and by applying an appropriate reweighting procedure making use of the DYNNLO program. We compare to existing data and to accurate resummed calculations.
• ### Higgs production at NNLOPS(1405.7079)

May 27, 2014 hep-ph
We describe the method used to build a simulation of Higgs boson production accurate at next-to-next-to-leading order and matched to a parton shower. The adopted procedure makes use of a combination of the POWHEG and MiNLO methods. We also use results from HNNLO as final input to reach the claimed accuracy. Results for typical observables are shown.
• ### Bounding the top Yukawa with Higgs-associated single-top production(1402.0445)

May 9, 2014 hep-ph, hep-ex
After the discovery of the 125 GeV scalar boson with gauge properties similar to the Standard Model Higgs, the search for beyond the SM interactions will focus on studying the discovered particles' coupling properties more precisely and shedding light on the relation of fermion masses with the electroweak vacuum. The large mass of the top quark and the SM-predicted order one top Yukawa coupling is a natural candidate for BSM physics, though experimentally challenging to constrain. In this paper, we argue that investigating angular correlations in $pp\to tHj$ production provides an excellent handle to constrain the top Yukawa coupling $y_t$ via direct measurements, even when we focus on rare exclusive final states. We perform a hadron-level analysis and show that we may expect to constrain $y_t\gtrsim 0.5\, y_t^{\text{SM}}$ at 95%-99% confidence level at the high luminosity LHC using semi-leptonic top decays and $H\to \gamma \gamma$ alone, by employing a two-channel measurement approach.
• ### Determining the structure of dark-matter couplings at the LHC(1311.7131)

Jan. 17, 2014 hep-ph, hep-ex
The latest LHC mono-jet searches place stringent bounds on the pp -> chibar chi cross section of dark matter. Further properties such as the dark matter mass or the precise structure of the interactions between dark matter and the standard model can however not be determined in this manner. We point out that measurements of the azimuthal angle correlations between the two jets in 2 j + chibar chi events may be used to disentangle whether dark matter pair production proceeds dominantly through tree or loop diagrams. Our general observation is illustrated by considering theories in which dark matter interacts predominantly with the top quark. We show explicitly that in this case the jet-jet azimuthal angle difference is a gold-plated observable to probe the Lorentz structure of the couplings of dark matter to top quarks, thus testing the CP nature of the particle mediating these interactions.
• ### Reaching NNLOPS accuracy with POWHEG and MiNLO(1401.2944)

Jan. 13, 2014 hep-ph, hep-ex
We describe how a simulation of Higgs boson production accurate at next-to-next-to-leading order and matched to a parton shower can be built by combining the POWHEG and MiNLO methods and using HNNLO results as input.
• ### QCD effects in mono-jet searches for dark matter(1310.4491)

Dec. 13, 2013 hep-ph, hep-ex
LHC searches for missing transverse energy in association with a jet allow to place strong bounds on the interactions between dark matter and quarks. In this article, we present an extension of the POWHEG BOX capable of calculating the underlying cross sections at the next-to-leading order level. This approach enables us to consistently include the effects of parton showering and to apply realistic experimental cuts. We find significant differences from a fixed-order analysis that neglects parton showering effects. In particular, next-to-leading order corrections do not lead to a significant enhancement of the mono-jet cross section once a veto on additional jets is imposed. Nevertheless, these corrections reduce the theoretical uncertainties of the signal prediction and therefore improve the reliability of the derived bounds. We present our results in terms of simple rescaling factors, which can be directly applied to existing experimental analyses and discuss the impact of changing experimental cuts.
• ### NNLOPS simulation of Higgs boson production(1309.0017)

Oct. 8, 2013 hep-ph, hep-ex
We detail a simulation of Higgs boson production via gluon fusion, accurate at next-to-next-to-leading order in the strong coupling, including matching to a parton shower, yielding a fully exclusive, hadron-level description of the final-state. The approach relies on the POWHEG method for merging the NLO Higgs plus jet cross-section with the parton shower, and on the MINLO method to simultaneously achieve NLO accuracy for inclusive Higgs boson production. The NNLO accuracy is reached by a reweighting procedure making use of the HNNLO program.
• ### Pinning down Higgs triplets at the LHC(1306.6228)

Aug. 29, 2013 hep-ph, hep-ex
Extensions of the Standard Model Higgs sector involving weak isotriplet scalars are not only benchmark candidates to reconcile observed anomalies of the recently discovered Higgs-like particle, but also exhibit a vast parameter space, for which the lightest Higgs' phenomenology turns out to be very similar to the Standard Model one. A generic prediction of this model class is the appearance of exotic doubly charged scalar particles. In this paper we adapt existing dilepton+missing energy+jets measurements in the context of SUSY searches to the dominant decay mode $H^{\pm\pm}\to W^\pm W^\pm$ and find that the LHC already starts probing the model's parameter space. A simple modification towards signatures typical of weak boson fusion searches allows us to formulate even tighter constraints with the 7 TeV LHC data set. A corresponding analysis of this channel performed at 14 TeV center of mass energy will constrain the model over the entire parameter space and facilitate potential $H^{\pm\pm}\to W^\pm W^\pm$ discoveries.
• ### Triplet Higgs boson collider phenomenology after the LHC(1302.6505)

May 28, 2013 hep-ph
ATLAS and CMS have discovered a Standard Model Higgs-like particle. One of the main discovery channels is the Higgs decay to two photons, which, at the moment, seems to be considerably enhanced over the Standard Model expectation. Models with additional charged matter coupling to the Higgs sector can enhance or decrease the Higgs to two photons branching ratio. We take this as motivation to confront the so-called Georgi-Machacek model of Higgs triplets with the results of recent searches for a SM Higgs boson performed at the LHC. We also investigate the model in regions of the allowed parameter space relevant for a SM-like phenomenology. The Georgi-Machacek model avoids tree-level issues of the T parameter, while offering a vastly modified Higgs phenomenology compared to the Standard Model. This comes at the price of introducing another fine-tuning problem related to electroweak precision measurements. We investigate the collider phenomenology of the Georgi-Machacek model in the light of existing collider constraints beyond any effective field theory approximation and contextualize our findings with electroweak precision constraints.
• ### NLO corrections merged with parton showers for Z+2 jets production using the POWHEG method(1204.5433)

Oct. 4, 2012 hep-ph
We present results for the QCD production of Z/{\gamma} + 2 jets matched with parton showers using the POWHEG method. Some technicalities relevant for the merging of NLO corrections for this process with parton showers are discussed, and results for typical distributions are shown, in presence of different sets of cuts. A comparison with ATLAS data is also presented, and good agreement is found.
• ### Single-top t-channel hadroproduction in the four-flavour scheme with POWHEG and aMC@NLO(1207.5391)

Oct. 2, 2012 hep-ph
We present results for the QCD next-to-leading order (NLO) calculation of single-top t-channel production in the 4-flavour scheme, interfaced to Parton Shower (PS) Monte Carlo programs according to the POWHEG and MC@NLO methods. Comparisons between the two methods, as well as with the corresponding process in the 5-flavour scheme are presented. For the first time results for typical kinematic distributions of the spectator-b jet are presented in an NLO+PS approach.
• ### Probing higher-order corrections in dijet production at the LHC(1202.1475)

June 16, 2012 hep-ph, hep-ex
Both the ATLAS and CMS Collaborations have sought for effects beyond pure next-to-leading order in dijet observables, with the goal to distinguish between the perturbative descriptions provided by a next-to-leading order plus collinear-resummation calculation and by the resummation of wide-angle, hard emissions. In this paper we identify regions of phase space in dijet production where some observables receive large corrections beyond next-to-leading order and study their theoretical description with two tools that perform these two different resummations: the POWHEG BOX and HEJ. Furthermore, we suggest analyses where the predictions from POWHEG and HEJ can be clearly distinguished experimentally.


text:
# Is there a formula for the sputtering yield of compound materials for ion beam milling?

I am aware of the theory of sputtering by Sigmund and of the formula he derived for calculating the sputtering yield in focused ion beam milling. The modern version (modified by Matsunami and Yamamura) goes as follow:

$Y(E)=\frac{3.56}{U_0} \frac{Z_t Z_p}{\sqrt{Z_t^{2/3} + Z_p^{2/3}}} \frac{M_p}{M_t+M_p} \alpha(M_t/M_p) S_n(E/E_{tp})$

where $p$ refers to "projectile" and $t$ to "target", $U_0$ is the surface binding energy, $Z$ are the atomic numbers of the materials, $M$ the atomic masses, $\alpha$ is an experimental formula and $S_n$ is the nuclear stopping power with $E_{tp}$ being a constant related to both projectile and target.

However I would love to be able to estimate the sputtering yield for compound material. I'm aware of the difficulty of such a thing, but it would be really helpful in my work if I could predict at least the range of the sputtering yield. If there is no such formula then my other thought would be to use Sigmund's formula but with the average atomic number of the compound instead of $Z_t$ and the average atomic mass of the compound instead of $M_t$.

Can I rely on that as a vague approximation or am I completely off? As an example a prediction I get calculating that way is that the sputtering yield of neon ions on silicon is 5 times higher than on gallium oxide.

• Compounds get complicated, since the energy transfer to the various elements are different, the surface binding energies are different, etc. Plus, you get differential sputtering, so the surface composition changes until the various sputtering rates all work out nicely. – Jon Custer Feb 15 '18 at 22:43
• So you think that it’s quicker to just go experimentally instead of trying to predict the range of the sputtering yield? As a side note I forgot to mention that I took the surface binding energy of the compound, that is about 21eV for Ga$_2$O$_3$ if I believe the source that I found on google. – Jxx Feb 15 '18 at 23:11

format: plain


text:
# Goodstein's theorem without transfinite induction

Is it possible to prove Goodstein's theorem without transfinite induction? Is there such a proof?

-
You might in general be interested in reverse mathematics, which studies the amount of induction (and other axioms) needed in many theorems. – Noah Schweber Jul 1 '15 at 2:44

I'm pretty sure the short answer is "no" (if you mean, can you prove Goodstein's theorem without invoking apparatus as strong as a transfinite induction which can't be reduced to an ordinary induction). For if I recall correctly, Goodstein's theorem is actually equivalent (over a weak base theory) to transfinite induction up to $\varepsilon_0$. My first port of call to check this would be Kirby and Paris's paper on 'Accessible independence results for Peano Arithmetic'.


text:
# Get Answers to all your Questions

#### Please solve RD Sharma class 12 Chapter Functions exercise 2.1 question 2 sub question (i)  maths textbook solution.

$f_1$ is one-one and onto function.

Given:

$f_{1}=\{(1,3),(2,5),(3,7)\} ; A=\{1,2,3\} ; B=\{3,5,7\}$

Hint:

One-one function means every element in the domain has a distinct image in the co-domain.

Onto function means every element in the codomain has at least one pre image in the domain of function.

Solution:

We need to find the one-one function

Injectivity: (one-one)

\begin{aligned} &f_{1}(1)=3 \\ &f_{1}(2)=5 \\ &f_{1}(3)=7 \end{aligned}

Every element of A has different images in B.

$\therefore$ Function $f_1$ is one-one.

Surjectivity: (onto)

Co-domain of$f_1= \left \{ 3,5,7 \right \}$

Range of $f_1$ set of image$\left \{ 3,5,7 \right \}$

Co-domain=Range

So, $f_1$ is onto.


text:
Corpus ID: 220265387

# Delay Violation Probability and Effective Rate of Downlink NOMA over $\alpha$-$\mu$ Fading Channels

@inproceedings{Kumar2020DelayVP,
title={Delay Violation Probability and Effective Rate of Downlink NOMA over \$\alpha\$-\$\mu\$ Fading Channels},
author={Vaibhav Kumar and Barry Cardiff and Shankar Prakriya and Mark F. Flanagan},
year={2020}
}
• Vaibhav Kumar, +1 author Mark F. Flanagan
• Published 2020
• Computer Science, Engineering, Mathematics
• Non-orthogonal multiple access (NOMA) is a potential candidate to further enhance the spectrum utilization efficiency in beyond fifth-generation (B5G) standards. However, there has been little attention on the quantification of the delay-limited performance of downlink NOMA systems. In this paper, we analyze the performance of a two-user downlink NOMA system over generalized α-μ fading in terms of delay violation probability (DVP) and effective rate (ER). In particular, we derive an analytical… CONTINUE READING

#### References

##### Publications referenced by this paper.
SHOWING 1-10 OF 37 REFERENCES

## Delay Guarantee and Effective Capacity of Downlink NOMA Fading Channels

• Chiyang Xiao, +3 authors Jing Wang
• Computer Science
• IEEE Journal of Selected Topics in Signal Processing
• 2019
VIEW 7 EXCERPTS
HIGHLY INFLUENTIAL

## Effective Rate Analysis in Weibull Fading Channels

• Computer Science, Mathematics
• IEEE Wireless Communications Letters
• 2016
VIEW 5 EXCERPTS
HIGHLY INFLUENTIAL

## A New Formula for the Average Bit Error Probability of Dual-Hop Amplify-and-Forward Relaying Systems over Generalized Shadowed Fading Channels

• Kostas P. Peppas
• Computer Science, Mathematics
• IEEE Wireless Communications Letters
• 2012
VIEW 1 EXCERPT

## An integral involving generalized function of two variables

• Mathematics
• 1972

## Cross-Layer Power Allocation in Nonorthogonal Multiple Access Systems for Statistical QoS Provisioning

VIEW 2 EXCERPTS

## Cross-Layer Power Control for Uplink NOMA in IoT Applications with Statistical Delay Constraints

• Computer Science
• 2018 IEEE Global Communications Conference (GLOBECOM)
• 2018
VIEW 2 EXCERPTS

## Delay Analysis for Wireless Fading Channels with Finite Blocklength Channel Coding

• Computer Science
• MSWiM '15
• 2015
VIEW 3 EXCERPTS

## Delay Performance of Wireless Communications With Imperfect CSI and Finite-Length Coding

• Mathematics, Computer Science
• IEEE Transactions on Communications
• 2018
VIEW 2 EXCERPTS

## Delay Performance of the Multiuser MISO Downlink Under Imperfect CSI and Finite-Length Coding

• Computer Science, Mathematics
• IEEE Journal on Selected Areas in Communications
• 2019
VIEW 1 EXCERPT

## Delay Violation Probability and Age of Information Interplay in the Two-user Multiple Access Channel

• Computer Science, Mathematics
• 2019 IEEE 20th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC)
• 2019
VIEW 2 EXCERPTS


text:
## Stream: general

### Topic: decidable_eq problem

#### Heather Macbeth (Jan 12 2021 at 23:24):

I would like some help fixing an error. Apologies for the length, this was a little hard to minimize.

import analysis.normed_space.inner_product

open_locale classical

variables (𝕜 : Type*) {E : Type*} [is_R_or_C 𝕜] [inner_product_space 𝕜 E]

local notation ⟪x, y⟫ := @inner 𝕜 _ _ x y

/-- Example with no errors -/
example (k : ℕ) {i j : fin (k + 1)} (h : ¬i = 0) (h' : ¬j = 0) :
ite (i = j) (1:𝕜) 0 = ite (i.pred h = j.pred h') 1 0 :=
begin
refine if_congr _ rfl rfl, -- no errors here

end

/-- build up to the (apparent) same goal state, with error -/

instance submodule_inner_product_space {W : submodule 𝕜 E} : inner_product_space 𝕜 W :=
{ inner             := λ x y, ⟪(x:E), ↑y⟫,
conj_sym          := λ _ _, inner_conj_sym _ _ ,
nonneg_im         := λ _, inner_self_nonneg_im,
norm_sq_eq_inner  := λ _, norm_sq_eq_inner _,
smul_left         := λ _ _ _, inner_smul_left,
..submodule.normed_space W }

def orthonormal {ι : Type*} (v : ι → E) : Prop :=
∀ i j, ⟪v i, v j⟫ = if i = j then (1:𝕜) else (0:𝕜)

example [finite_dimensional 𝕜 E] {k : ℕ} (hk : finite_dimensional.findim 𝕜 E = k + 1)
{x : E} (hx : ∥x∥ = 1) {w : fin k → (𝕜 ∙ x)ᗮ} (hw : orthonormal 𝕜 w) :
orthonormal 𝕜 (λ i : fin (k + 1), if h : i = 0 then x else coe (w (i.pred h))) :=
begin
intros i j,
by_cases h : i = 0, { sorry },
by_cases h' : j = 0, { sorry },
convert hw (i.pred h) (j.pred h') using 1, { sorry },
refine if_congr _ rfl rfl, -- error here

end

#### Heather Macbeth (Jan 12 2021 at 23:24):

The error is

synthesized type class instance is not definitionally equal to expression inferred by typing rules, synthesized
subtype.decidable_eq i j
inferred
classical.prop_decidable (i = j)

#### Yakov Pechersky (Jan 12 2021 at 23:31):

Do you have to have open_locale classical at the top?

#### Yakov Pechersky (Jan 12 2021 at 23:32):

Your orthonormal doesn't require decidable_eq on the {ι : Type*}

#### Yakov Pechersky (Jan 12 2021 at 23:33):

I would just require that, which will make the if then else happy, and then rely in other things to choose whether to use a type which is truly decidable_eq like fin n, or conjure up such an instance with classical.

#### Yakov Pechersky (Jan 12 2021 at 23:34):

Those suggestions are without actually trying it, I might be wrong.

#### Heather Macbeth (Jan 12 2021 at 23:35):

Yup, requiring decidable_eq for orthonormal seems to work!

#### Heather Macbeth (Jan 12 2021 at 23:36):

Yakov Pechersky said:

Do you have to have open_locale classical at the top?

Is this a problem? I thought that in mathlib it was supposed to be safe to make everything classical everywhere :)

#### Yakov Pechersky (Jan 12 2021 at 23:37):

In any case, whenever you do an ite (a = b), you're making the claim you can actually decide that. And what happens downstream is that fin n knows it can supply decidable_eq from the fact that nat is decidable_eq and fin is a subtype.

#### Yakov Pechersky (Jan 12 2021 at 23:37):

And borks, because it's not the more general and not computable classical.prop_decidable

#### Yakov Pechersky (Jan 12 2021 at 23:38):

We ran into this problem recently with equiv.swaps, which should only work on decidable_eqtypes. So some lemmas about them were stated with classical instead of just general variables ... [decidable_eq ...]

#### Yakov Pechersky (Jan 12 2021 at 23:39):

Because later on, proving concrete specialized lemmas over fin or option ι or what have you hits the same issue you're hitting now.

#### Yakov Pechersky (Jan 12 2021 at 23:40):

Someone that understand this better than me can comment -- but should defs state explicitly the decidable_eq constraints, and downstream, uses of the def can satisfy that with classical?

#### Heather Macbeth (Jan 12 2021 at 23:41):

Yakov Pechersky said:

Someone that understand this better than me can comment -- but should defs state explicitly the decidable_eq constraints, and downstream, uses of the def can satisfy that with classical?

If this is correct, it seems like a useful rule of thumb.

#### Yakov Pechersky (Jan 12 2021 at 23:44):

For your (λ i : fin (k + 1), if h : i = 0 then x else coe (w (i.pred h))) you might prefer fin.cases _ _ i

#### Heather Macbeth (Jan 12 2021 at 23:58):

Could you indicate the specific syntax? I had a quick look but didn't see how to use it.

#### Yakov Pechersky (Jan 13 2021 at 00:00):

fin.cases x (λ j, coe (w j)) i

#### Yakov Pechersky (Jan 13 2021 at 00:01):

So no need to juggle hypotheses about positivity or pred. And the simp lemmas for fin.cases are clean.

#### Yakov Pechersky (Jan 13 2021 at 00:02):

Otherwise, you'd be proving things about how ite i.succ = 0 is always false by having to introduce the relevant inequality, then i.succ.pred = i etc etc

#### Heather Macbeth (Jan 13 2021 at 00:04):

Yakov Pechersky said:

Otherwise, you'd be proving things about how ite i.succ = 0 is always false by having to introduce the relevant inequality, then i.succ.pred = i etc etc

Hmm, I didn't need to do anything like this in my proof, maybe it's a slightly different situation.

#### Yakov Pechersky (Jan 13 2021 at 00:08):

With fin (n + 1) it's either casing on if it is 0 or positive (I like rcases eq_or_lt_of_le (fin.zero_le i)) or casing on if it is 0 or the i.succ for some i : fin n.

#### Kevin Buzzard (Jan 13 2021 at 00:09):

Isn't there some rule of thumb about when to use decidable_eq, which I've never managed to internalise because it's "not mathematics"?

#### Heather Macbeth (Jan 13 2021 at 00:16):

Yakov Pechersky said:

With fin (n + 1) it's either casing on if it is 0 or positive (I like rcases eq_or_lt_of_le (fin.zero_le i)) or casing on if it is 0 or the i.succ for some i : fin n.

I suppose the former is closer to the style of the proof I was writing. I hadn't found it awkward, but let me experiment with fin.cases. The syntax you suggested gives an error, any ideas?

type mismatch at application
fin.cases x
term
x
has type
E : Type u_2
but is expected to have type
?m_1 0 : Sort ?

This is for my original code, modified to have

  orthonormal 𝕜 (fin.cases x (λ j, coe (w j))) :=

as line 34.

#### Heather Macbeth (Jan 13 2021 at 00:21):

Yakov Pechersky said:

With fin (n + 1) it's either casing on if it is 0 or positive (I like rcases eq_or_lt_of_le (fin.zero_le i)) or casing on if it is 0 or the i.succ for some i : fin n.

More precisely, my proof before was casing using by_cases h : i = 0, and in the second (nonzero) case working with i.pred h.

#### Yakov Pechersky (Jan 13 2021 at 01:17):

For fin.cases, you'll still need the 'fun i, fin.cases _ _ i'. I think your way is great and clear.

#### Heather Macbeth (Jan 13 2021 at 01:25):

Hmm, with the modification

  orthonormal 𝕜 (λ i, (fin.cases x (λ j, coe (w j)) i)) :=

I get a different error,

invalid 'fin.cases' application, elaborator has special support for this kind of application (it is handled as an "eliminator"), but expected type must not contain metavariables
?m_1

#### Yakov Pechersky (Jan 13 2021 at 02:03):

Well then, I guess the ite is the best choice :-)

#### Anne Baanen (Jan 13 2021 at 10:02):

Heather Macbeth said:

Yakov Pechersky said:

Someone that understand this better than me can comment -- but should defs state explicitly the decidable_eq constraints, and downstream, uses of the def can satisfy that with classical?

If this is correct, it seems like a useful rule of thumb.

This is more or less the design rule I follow: classical is for internal use only. Therefore, unfolding a definition or applying a lemma should not introduce classical. So not only definitions, but also lemma statements, should avoid the use of classical.

In practice, I declare variables [deι : decidable_eq ι] if I need decidability. Since named parameters won't get included in the hypotheses if they are not used in a definition or lemma statement, there should be no complaints from the classical linter.

Last updated: May 13 2021 at 23:16 UTC


text:
List of important publications in statistics
... emphasizing hypothesis testing via likelihood ratio tests and the properties of power functions: Admissiblity, unbiasedness and ... Statistical Decision Functions Author: Abraham Wald Publication data: 1950. John Wiley & Sons. Description: Exposition of ... JSTOR 2281868 Description: First description of the now ubiquitous Kaplan-Meier estimator of survival functions from data with ... Aldrich, John (1997). "R.A. Fisher and the making of maximum likelihood 1912-1922". Statistical Science. 12 (3): 162-176. doi: ...
Likelihoodfunction
When considered a function of N for fixed n, this is a likelihood function. The maximum likelihood estimate for N is n (by ... In statistics, a likelihood function (often simply the likelihood) is a function of the parameters of a statistical model given ... In such a situation, the likelihood function factors into a product of individual likelihood functions. The logarithm of this ... The relative likelihood is closely related to the likelihood ratio used in the likelihood-ratio test. The likelihood ratio is ...
NumXL
Mixed model - likelihood function. Goodness of fit - LLF, AICc, and model's diagnosis. Interpolation functions - Flat forward- ... and partial autocorrelation function (PACF). Cross-correlation functions - XCF and EWXCF MD, RMD, MAD Statistical functions - ... Goodness of fit - Likelihood function (LLF), Akaike information criterion (AICc) and model's diagnosis Simple linear regression ... Likelihood function (LLF), Akaike information criterion (AICc) and model's diagnosis. Forecast and back-testing Simulation ...
Thompson sampling
The elements of Thompson sampling are as follows: a likelihood function P ( r , θ , a , x ) {\displaystyle P(r,\theta ,a,x)} ; ... is the likelihood function. Thompson sampling consists in playing the action a ∗ ∈ A {\displaystyle a^{\ast }\in {\mathcal {A ... likelihoods of the actions a 1 , a 2 , … , a T {\displaystyle a_{1},a_{2},\ldots ,a_{T}} , and then by sampling the action a T ... likelihoods of the observations o 1 , o 2 , … , o T {\displaystyle o_{1},o_{2},\ldots ,o_{T}} and ignoring the (causal) ...
Deviance information criterion
... is the likelihood function. C {\displaystyle C\,} is a constant that cancels out in all calculations that compare different ... AIC and BIC require calculating the likelihood at its maximum over θ {\displaystyle \theta \,} , which is not readily available ... Note, that the p in this expression is the predictive distribution rather than the likelihood above. Akaike information ...
Whittle likelihood
In statistics, Whittle likelihood is an approximation to the likelihood function of a stationary Gaussian time series. It is ... In a stationary Gaussian time series model, the likelihood function is (as usual in Gaussian models) a function of the ... likelihood function weighted least squares matched filter discrete Fourier transform power spectral density coloured noise ... Hurvich, C. (2002). "Whittle's approximation to the likelihood function" (PDF). NYU Stern. Calder, M.; Davis, R. A. (1997), "An ...
Fisher information
... or probability mass function) for X conditional on the value of θ. This is also the likelihood function for θ. It describes the ... the likelihood function is a probability density function, and therefore ∫ f d x = 1 {\displaystyle \int f\,dx=1} . A basic ... Formally, the partial derivative with respect to θ of the natural logarithm of the likelihood function is called the "score". ... Statistical systems of a scientific nature (physical, biological, etc.) whose likelihood functions obey shift invariance have ...
Probability
These data are incorporated in a likelihood function. The product of the prior and the likelihood, normalized, results in a ... The first law was published in 1774 and stated that the frequency of an error could be expressed as an exponential function of ... The second law of error was proposed in 1778 by Laplace and stated that the frequency of the error is an exponential function ... The objective wave function evolves deterministically but, according to the Copenhagen interpretation, it deals with ...
Oscar Kempthorne
... neo-Fisherian statistics emphasizes likelihood functions of parameters. Second, Kempthorne was skeptical of Bayesian statistics ... which use not only likelihoods but also probability distributions on parameters. Nonetheless, while subjective probability and ...
Gamma distribution
The likelihood function for N iid observations (x1, ..., xN) is L ( k , θ ) = ∏ i = 1 N f ( x i ; k , θ ) {\displaystyle L(k,\ ... is the gamma function evaluated at k. The cumulative distribution function is the regularized gamma function: F ( x ; k , θ ... The cumulative distribution function is the regularized gamma function: F ( x ; α , β ) = ∫ 0 x f ( u ; α , β ) d u = γ ( α , β ... from which we calculate the log-likelihood function ℓ ( k , θ ) = ( k − 1 ) ∑ i = 1 N ln  ( x i ) − ∑ i = 1 N x i θ − N k ln ...
Negative binomial distribution
The likelihood function for N iid observations (k1, ..., kN) is L ( r , p ) = ∏ i = 1 N f ( k i ; r , p ) {\displaystyle L(r,p ... prod _{i=1}^{N}f(k_{i};r,p)\,\!} from which we calculate the log-likelihood function ℓ ( r , p ) = ∑ i = 1 N ln  ( Γ ( k i + r ... mass function and obtain the following mass function of the distribution of houses (for n ≥ 5): f ( n ) = ( ( n − 5 ) + 5 − 1 n ... The cumulative distribution function can be expressed in terms of the regularized incomplete beta function: f ( k ; r , p ) ≡ ...
Approximate Bayesian computation
ABC methods bypass the evaluation of the likelihood function. In this way, ABC methods widen the realm of models for which ... All ABC based methods approximate the likelihood function by simulations, the outcomes of which are compared with the observed ... For simple models, an analytical formula for the likelihood function can typically be derived. However, for more complex models ... In all model-based statistical inference, the likelihood function is of central importance, since it expresses the probability ...
Bayesian inference in marketing
This is done through the calculation shown below, where P ( D , H ) {\displaystyle P(D,H)} is the likelihood function. This ... which states that models or inferences for datasets leading to the same likelihood function should generate the same ... "Statistical Decision Functions", in: Kotz, S. And Johnson, N. L. (Eds.) (1992). Breakthroughs in Statistics: Foundations and ... is misread by judging A's likelihood by how well the evidence X matches A, but crucially without considering the prior ...
Bayesian linear regression
The prior belief about the parameters is combined with the data's likelihood function according to Bayes theorem to yield the ... Here, the model is defined by the likelihood function p ( y , X , β , σ ) {\displaystyle p(\mathbf {y} ,\mathbf {X} ,{\ ... A prior ρ ( β , σ 2 ) {\displaystyle \rho ({\boldsymbol {\beta }},\sigma ^{2})} is conjugate to this likelihood function if it ... denotes the gamma function. Because we have chosen a conjugate prior, the marginal likelihood can also be easily computed by ...
Unified neutral theory of biodiversity
For this purpose, the approptiate likelihood function should be used. For the metacommunity this was given above. For the local ... Then the formula above would allow us to assess the likelihood of different values of θ. There are thus S = 3 species and ϕ 1 ... is the gamma function, and γ = ( J − 1 ) m / ( 1 − m ) {\displaystyle \gamma =(J-1)m/(1-m)} . This formula is however an ... The maximum likelihood estimate for θ is about 1.1478. We could have labelled the species another way and counted the ...
Monotone likelihood ratio
Monotone likelihood functions are used to construct uniformly most powerful tests, according to the Karlin-Rubin theorem. ... This task is simplified if the family has the monotone likelihood ratio property (MLRP). A family of density functions { f θ ( ... In statistics, the monotone likelihood ratio property is a property of the ratio of two probability density functions (PDFs). ... H_{1}:\theta >\theta _{0}.} Monotone likelihood-functions are used to construct median-unbiased estimators, using methods ...
Theta
The statistical parameter frequently used in writing the likelihood function. The Watterson estimator for the population ... A special function of several complex variables. One of the Chebyshev functions in prime number theory. The potential ... The ordinal collapsing function developed by Solomon Feferman The upper-case letter Θ is used as a symbol for: Quantity or ...
Bayesian information criterion
It is based, in part, on the likelihood function and it is closely related to the Akaike information criterion (AIC). When ... That is, the integral of the likelihood function p ( x , θ , M ) {\displaystyle p(x,\theta ,M)} times the prior probability ... The BIC is an increasing function of the error variance σ e 2 {\displaystyle \sigma _{e}^{2}} and an increasing function of k. ... the maximized value of the likelihood function of the model M {\displaystyle M} , i.e. L ^ = p ( x , θ ^ , M ) {\displaystyle ...
Median
Such constructions exist for probability distributions having monotone likelihood-functions. One such procedure is an analogue ... A C function is a real valued function, defined on the set of real numbers R, with the property that for any real t f − 1 ... The first and third inequalities come from Jensen's inequality applied to the absolute-value function and the square function, ... In practice, the function f ( v ) {\displaystyle f(v)} will often not be known but it can be estimated from an observed ...
Maximum likelihood estimation
Implementing MLE for your own likelihood function using R A selection of likelihood functions in R Myung, I. J. (2003). " ... A maximum likelihood estimator is an extremum estimator obtained by maximizing, as a function of θ, the objective function (c.f ... Since the logarithm function itself is a continuous strictly increasing function over the range of the likelihood, the values ... The method of maximum likelihood is based on the likelihood function. We are given a statistical model, i.e. a family of ...
Robert Wedderburn (statistician)
Wedderburn, RWM (1974). "Quasi-likelihood functions, generalized linear models, and the Gauss-Newton method". Biometrika. 61 (3 ... and then expanded this subject to develop the idea of quasi-likelihood. Wedderburn was born in Edinburgh, where he attended ...
Multiple-try Metropolis
... is the likelihood function. Define w ( x , y ) = π ( x ) Q ( x , y ) λ ( x , y ) {\displaystyle w(\mathbf {x} ,\mathbf {y} )=\ ... Suppose Q ( x , y ) {\displaystyle Q(\mathbf {x} ,\mathbf {y} )} is an arbitrary proposal function. We require that Q ( x , y ... is a non-negative symmetric function in x {\displaystyle \mathbf {x} } and y {\displaystyle \mathbf {y} } that can be chosen by ... "A multi-point Metropolis scheme with generic weight functions". Statistics & Probability Letters. 82 (7): 1445-1453. doi: ...
Mode choice
... this is our likelihood function. The likelihood function for n independent observations in a logit model is L ∗ = ∏ n = 1 N P i ... The MNL approach is to make a maximum likelihood estimate of this functional form. The likelihood function is: L ∗ = ∏ n = 1 N ... The log-likelihood function is maximized setting the partial derivatives to zero: ∂ ℓ ∂ β = ∑ i = 1 n ( Y i − P ^ i ) = 0 {\ ... As noted above, we think of observable utility as being a function: v A = β 0 + β 1 ( c A − c T ) + β 2 ( t A − t T ) + β 3 I ...
Variance function
With a specified variance function and link function we can develop, as alternatives to the log-likelihood function, the score ... In general, maximum likelihood estimation requires that a likelihood function be defined. This requirement then implies that ... In addition, we describe the applications and use of variance functions in maximum likelihood estimation and quasi-likelihood ... In statistics, the variance function is a smooth function which depicts the variance of a random quantity as a function of its ...
Wrapped Cauchy distribution
page needed] J. Copas (1975). "On the unimodality of the likelihood function for the Cauchy distribution". Biometrika. 62 (3): ... Expressing the above pdf in terms of the characteristic function of the Cauchy distribution yields: f W C ( θ ; μ , γ ) = 1 2 π ... The maximum-likelihood estimate for the median ( μ ^ {\displaystyle {\hat {\mu }}} ) and scale parameter ( γ ^ {\displaystyle ... Ferguson, Thomas S. (1978). "Maximum Likelihood Estimates of the Parameters of the Cauchy Distribution for Samples of Size 3 ...
Uncertainty quantification
Module 2: Gaussian process modeling for the discrepancy function Similarly with the first module, the discrepancy function is ... one can derive the maximum likelihood estimates for { β δ , σ δ , ω k δ , k = 1 , … , d } {\displaystyle \left\{{\boldsymbol {\ ... denotes the additive discrepancy function (aka bias function), and ε {\displaystyle \varepsilon } denotes the experimental ... The objective is to estimate the discrepancy function δ ( x ) {\displaystyle \delta (\mathbf {x} )} , and as a by-product, the ...
CUSUM
... represents the likelihood function, but this is common usage. Note that this differs from SPRT by always using zero function as ... CUSUM does not require the use of the likelihood function. As a means of assessing CUSUM's performance, Page defined the ...


text:
# On Uniformization of Complex Manifolds: The Role of Connections. (MN-22)

ROBERT C. GUNNING
Pages: 150
https://www.jstor.org/stable/j.ctt130hk2c

1. Front Matter
(pp. None)
2. PREFACE
(pp. i-i)
R. C. Gunning
(pp. ii-ii)
4. §1. Introduction
(pp. 1-5)

The general uniformization theorem for Riemann surfaces is one of the most remarkable results in complex analysis, and is at the center of a circle of problems which are still very actively being investigated. An interest in extending this theorem to complex manifolds of higher dimensions has long been manifest, and indeed there have been several extensions of one or another aspect of the general uniformization theorem. As has been observed in other cases, some theorems in classical complex analysis appear as the accidental concurrence in the one-dimensional special case of rather separate phenomena in the general case; so a...

5. Part I: Description of the pseudogroups
• §2. The group of k-jets and its Lie algebra.
(pp. 6-15)

Consider the set of all germs of complex analytic mappings from the origin to the origin in the space ℂnof n complex variables. The k-jet of such a germ f, denoted by jkf, is defined to consist of the terms of order ≤ k in the Taylor expansion of the germ f ; but since all these germs are assumed to take the origin to the origin the conventional usage will be slightly modified in that the constant terms in the Taylor expansion, the terms of order = 0, will not be considered as part of the k-jet. Upon...

• §3. The pseudogroups defined by partial differential equations.
(pp. 16-20)

The definition and classification of the pseudogroups defined by families of partial differential equations are rather straightforward matters once the preceding general machinery has been developed. An analytic family of partial differential equations of order k in the analytic local homeomorphisms from ℂnto ℂncan be thought of merely as being an analytic subvariety A ⊆ Gk(n,ℂ). Of course this is a somewhat restrictive definition, since such families of partial differential equations do not involve the actual values of the mappings but only the derivatives of orders 1 through k of the component functions of the mappings, and the...

• §4. The classification of tangentially transitive pseudogroups: algebraic aspects
(pp. 21-38)

The detailed classification of pseudogroups will only be attempted here for the special case of the tangentially transitive Lie pseudogroups, those for which all the defining groups A ⊆ Gk(n,ℂ) have the property that j1A = G1(n,ℂ); these are the pseudogroups for which there are no restrictions imposed on the values of the Jacobian matrices of the mappings. The classification apparently involves determining all the integrable subgroups A ⊆ Gk(n,ℂ) with j1A = G1(n,ℂ), for all k ≥ 1, and then determining which of these subgroups describe the same pseudogroups; but it is actually a considerably simpler matter than might...

• §5. The classification of tangentially transitive pseudogroups: analytic aspects
(pp. 39-52)

There are the four general classes of possible defining groups for tangentially transitive Lie pseudogroups, described by the four classes of equations (51), (52), (53), and (54) respectively; and within each class the defining groups are parametrized by a linear space of tensors A. It is useful to introduce a notion of equivalence among the defining groups in each class separately; but the definition and elementary properties of this relation are formally almost the same in the different cases, so for convenience will only be discussed in detail for the class given by equation (51). In that case the defining...

6. Part II: Description of the connections
• §6. Pseudogroup structures and their associated connections
(pp. 53-67)

As is of course well known, an m-dimensional topological manifold M is a Hausdorff space each point of which has an open neighborhood homeomorphic to an open subset of ℝm. A coordinate covering [symbol] = {U2} of such a manifold is a covering of M by open subsets Uα⊆ M, for each of which there is a homeomorphism zα: Uα→ Vαbetween Uαand an open subset Vα⊆ ℝm; the sets Uαare called coordinate neighborhoods, and the homeomorphisms zαare called local coordinates. The compositions${{\text{f}}_{\alpha\beta}}={{\text{z}}_{\alpha}}\circ \text{z}_{\beta}^{-1}:{{\text{z}}_{\beta}}({{\text{U}}_{\alpha}}\cap {{\text{U}}_{\beta}})\to {{\text{z}}_{\alpha}}({{\text{U}}_{\alpha}}\cap {{\text{U}}_{\beta}})$are then homeomorphisms between subsets of Vβand...

• §7. Complex analytic affine connections
(pp. 68-78)

To begin the more detailed discussion of some properties and applications of the complex analytic connections associated with the various pseudogroup structures, consider the complex analytic affine connections. As might be expected from the terminology, these are essentially just the complex analytic analogues of the classical affine connections in differential geometry; but there is one point of difference which must be kept in mind. If$\{{{\text{s}}_{\alpha}}\}=\{\text{s}_{\alpha{{\text{j}}_{1}}{{\text{j}}_{2}}}^{\text{i}}\}$is a complex analytic affine connection then recalling (24) and (44) the defining equation (77) can be written out explicitly in the form$\caption {(89)} {{\Sigma }_{\text{k}}}\frac{\partial {{\text{z}}_{\beta\text{i}}}}{\partial {{\text{z}}_{\alpha\text{k}}}}\frac{{{\partial }^{2}}{{\text{z}}_{\alpha\text{k}}}}{\partial {{\text{z}}_{\beta{{\text{j}}_{\text{1}}}}}\partial {{\text{z}}_{\beta{{\text{j}}_{\text{2}}}}}}=\text{s}_{\beta{{\text{j}}_{\text{1}}}{{\text{j}}_{2}}}^{\text{i}}-{{\Sigma }_{\text{k}}}\frac{\partial {{\text{z}}_{\beta\text{i}}}}{\partial {{\text{z}}_{\alpha\text{k}}}}\text{s}_{\alpha{{\text{k}}_{\text{1}}}{{\text{k}}_{\text{2}}}}^{\text{k}}\frac{\partial {{\text{z}}_{\alpha{{\text{k}}_{1}}}}}{\partial {{\text{z}}_{\beta{{\text{j}}_{1}}}}}\frac{\partial {{\text{z}}_{\alpha{{\text{k}}_{2}}}}}{\partial {{\text{z}}_{\beta{{\text{j}}_{2}}}}};$and that is the complex analytic analogue of the familiar...

• §8. Complex analytic projective connections.
(pp. 79-94)

Turning next to the complex analytic projective connections, there is a well developed but perhaps not so well known classical theory of projective connections [11]; a particularly readable recent survey of that theory can be found in [28]. Here too the complex analytic projective connections are just the complex analytic analogues of the classical projective connections, indeed more so than in the case of the complex analytic affine connections since for the projective connections symmetry is normally presumed. For the case n = 1 complex analytic projective connections were discussed in [20], and it was shown there that all such...

• §9. Complex analytic canonical connections
(pp. 95-100)

The complex analytic canonical connections are in many ways the simplest of the three classes of connections considered here. As was already essentially noted, a complex analytic canonical connection$\{\text{s}_{\alpha}^{'}\}=\{\text{s}_{\alpha{{\text{j}}_{1}}{{\text{j}}_{2}}}^{{{'}^{\text{i}}}}\}$can be written in the form (116) in each coordinate neighborhood Uαwhere the functions${{\text{x}}_{\alpha\text{j}}}={{\Sigma }_{\text{k}}}\text{s}_{\alpha\text{kj}}^{{{'}^{\text{k}}}}$are complex analytic in Uα; and the condition that$\text{s}_{\alpha}^{'}$is a canonical connection is equivalent to (117). Introducing the complex analytic 1-form φαin the coordinate neighborhood Uαdefined by$\caption {(132)} {{\varphi}_{\alpha}}({{\text{z}}_{\alpha}})={{\Sigma }_{\text{k}}}{{\text{ x}}_{\alpha\text{k}}}({{\text{z}}_{\alpha}})\text{d}{{\text{z}}_{\alpha}},$the condition (117) can be rewritten$\caption {(138)} \text{d}\ \log \,{{\vartriangle }_{\alpha\beta}}={{\varphi}_{\beta}}-{{\varphi}_{\alpha}}\quad \text{in}\ {{\text{U}}_{\alpha}}\cap {{\text{U}}_{\beta}}$where Δαβ= det{∂zαi/∂zβj} ; thus a complex analytic canonical connection can be described equivalently...

7. Part III: Complex analytic surfaces
• §10. Complex flat canonical structures on surfaces
(pp. 101-108)

A discussion of complex analytic pseudogroup structures on one-dimensional compact complex manifolds can be found in [20] ; although there remain several open problems in that case, they mostly have to do with more detailed properties of complex projective structures, and will not be considered here. Turning next to two-dimensional compact complex manifolds, it is convenient to change from the order in which the various pseudogroups have been discussed and to begin with the simplest case, that of the complex flat canonical structures.

If the two-dimensional compact complex manifold M admits a complex flat canonical structure then as already noted...

• §11. Complex affine structures on surfaces.
(pp. 109-122)

Although complex affine structures are more complicated than flat canonical structures, in part because of the nonlinearity of the defining partial differential equations, there is nonetheless a great deal known about such structures. Complex affine structures are of course subordinate to complex flat canonical structures, so that to determine which complex manifolds admit complex flat canonical structures it is only necessary to run through the list of complex manifolds with complex flat canonical structures and see which admit this finer structure. Thus the only compact complex surfaces that can possibly admit complex affine structures are those with Chern classes c1...

• §12. Complex projective structures on surfaces
(pp. 123-136)

Complex projective structures, the general structures in the case of one-dimensional manifolds, have been the least investigated of the structures considered here in the two-dimensional case. Turning now to the discussion of these structures, recall from Theorem 5 that on a two-dimensional manifold which admits a complex analytic projective connection the Chern classes must satisfy$\caption {(152)} {{\text{c}}_{2}}=\frac{1}{3}{{\text{c}}_{1}}^{2}.$

The first stage of the discussion is to determine in general terms which two-dimensional compact complex manifolds M satisfy this topological restriction (152). Unlike the conditions considered previously this topological restriction does not preclude the possibility that the surface M contains exceptional curves of...

8. Bibliography
(pp. 137-142)
9. Back Matter
(pp. 143-143)


text:
# Why Drifts are not in the Black Scholes Formula

This question has puzzled me for a while.

We all know geometric brownian motions have drifts $\mu$:

$dS / S = \mu dt + \sigma dW$

and different stocks have different drifts of $\mu$. Why would the drifts go away in Black Scholes? Intuitively, everything else being equal, if a stock has higher drift, shouldn't it have higher probability of finishing in-the-money (and higher probability of having higher payoff), and the call option should be worth more?

Is there an intuitive and easy-to-understand answer? thanks.

-
amazon.com/Heard-Street-Quantitative-Questions-Interviews/dp/… look at question 2.49. You will get an explanation without any mathematics. (Girsanov theorem might be overkill here) –  lmorin Jun 18 '13 at 23:02

Here couple pointers that may make it clearer:

• Drift can be replaced by the risk-free rate through a mathematical construct called risk-neutral probability pricing.

• Why can we get away with that without introducing errors? The reason lies in the ability to setup a hedge portfolio, thus the market will not compensate us for the drift above and beyond the risk free rate under risk-neutral probability pricing.

• As long as such hedge exists and couple other conditions are met (please look up Girsanov's Theorem) we can introduce a risk-neutral measure so that when applying it to the differential equation and through application of Ito Calculus the drift term vanishes, which greatly simplifies the underlying math.

• It is not easy to come up with a way to reliably measure drift as it is not well known and hard to estimate in reality. Similarly, real probabilities are different for every underlying asset and are difficult to estimate because investors require different risk premiums for each and every asset. Hence, being able to take the detour through risk neutral pricing greatly simplifies the derivation not only mathematically but also intuitively. Please note that the volatility of the risk neutral paths and paths under real-world probabilities are identical, what differs is the drift term vs risk-free rate.

• Keep in mind that not all derivative functions can be derived through risk-neutral pricing, in fact I venture to estimate that less than 20% of all traded derivatives by investment banks can be priced through risk-neutral pricing (in terms of number of different instruments not trading volume).

• In order to understand the technical reason of the elimination of drift through risk-neutral pricing you need to walk through the mathematics and one of the cleanest and most intuitive approaches I have seen was in Steven Shreve's book, Stochastic Calculus for Finance II, page 218-220 (2004 edition)

-

Being on the sell side and selling options you can intuitively think of it like this:

An option is like any other product that is being produced out of ingredients and because of the competitive situation of the producer this is done by the cheapest possible production process.

The ingredients are in a simple (Black Scholes) setting a stock and and a risk free bond. So this is where the name "derivative" comes from: It is derived from other, simpler products (the underlying).

Now to make it even simpler think of a call with a zero strike on the underlying. How would you price such a call? Well, you would price it like the stock (up to a correction for the risk free interest rate)! Here the same question could be asked: Shouldn't it be priced higher with the underlying having higher drift? The answer is no, because this is already included in the price of the underlying! You would kind of double count this effect.

With "real" options the reason is the same, the only thing that changes is that you have to adapt the proportion of the stock when the price of the underlying changes. But the original reasoning stays the same: The drift does not enter into the price of the derivative because it is already included in the price of the underlying.

A very readable intuitive paper on this issue is by a giant of the field, Emanuel Derman. It can be found here: The Boy's Guide to Pricing & Hedging

See also my answer to a similar question here: How does the "risk-neutral pricing framework" work?

-
I love this short read you referenced, +1 alone for this one. Shows me time and again that the top researchers in this and related fields all strive for simplicity because reputation is earned by explaining stochastic calculus to an ape rather than how to peel a banana to an astronaut. –  Matt Wolf Jun 19 '13 at 1:39
@MattWolf: Thank you, Matt. I agree, the great minds always try to purify the core ideas. They are the ones who are also able to always go back to the basics and connect them to a bigger picture. (BTW I also upvoted your answer because it is really a nice overview of the relevant ideas :-) –  vonjd Jun 19 '13 at 7:31
I take a different view from yours: For me there is a fundamental difference between the price of a derivative and the value of the same. What risk neutral pricing does is assisting in finding the price of a derivative. The value may vastly differ depending on the inputs to the pricing model. That differential for me explains why some people buy, others sell. Of course each participant has a slightly different pricing model though most are surprisingly similar. What differs is the estimation of value. That for me is where buy and sell side guys differ in their view –  Matt Wolf Jun 20 '13 at 5:09
...and which ultimately causes each individual to buy when he thinks price < value and sell if price > value. The pricing algorithm itself is equally applied to buy and sell side demand and supply. Given I traded today on the sell side, and tomorrow I was trading on the buy side, I would apply the exact same pricing models and derivations, nothing changes there. What changes is my risk transfer needs and thus where I see value. As risk seller I see value in the spread I can provide, as risk buyer I see value in the alpha generated from underlying assets vs hedge risk protection. –  Matt Wolf Jun 20 '13 at 5:10
In a sense yes. Though Q != P, the pricing in both, Q and P, must agree because of the no-arbitrage requirement. But assigning a specific probability measure to either pricing or valuation could lead down a dangerous path. Purely intuitively speaking though I like your comparison and agree. –  Matt Wolf Jun 20 '13 at 7:03

Recently I came across an interesting intuitive explanation:

Suppose driftless market. Market price is 105, strike price is 100. Call option costs 8, put option 3. (intrinsic value of call is 5, time value of both is 3)

Now the market starts drifting upwards massively. You say, that you would probably price call higher, e.g. at 10. Would you also price put higher? Probably not, since it's less probable that it will end up in the money. You would probably price put even lower, e.g. at 2.

But in that case, you clearly violate put/call parity, and it's possible for me to buy the call, and completely hedge it with the put, some cash and some stocks and earn riskless profit from you. Due to put/call parity, the price of put must rise together with price of call (they must have the same time value), which is inconsistent with dependence on drift.

I disregarded interest rates for clarity.

-
This is basically Derman-Taleb's argument that put-call parity implies the BS equations. I like it! But Merton apparently did not, in fact he sicced some attack dogs on DT. Anyone more knowledgeable care to step in and arbitrate? –  Andrew Dabrowski Nov 5 '13 at 14:09

$$dS / S = \mu dt + \sigma dW \\ \\ dS / S -r dt= \mu dt - rdt + \sigma dW \\ \\ dS / S -r dt= [\frac{(\mu - r)}{\sigma}dt + dW]\sigma \\$$ Then, Girsanov tells us that, as long as the risk premium is bounded from below, we can write $[\frac{(\mu - r)}{\sigma}dt + dW]\sigma$ as $\sigma d\tilde{W}$ where $\tilde{W}$ is simply another brownian motion with mean 0 and variance equal to its time step. Hence, $$dS / S = rdt + \sigma d\tilde{W} \\$$

Another way to think about this is that, in the Black-Scholes world of option pricing, markets are assumed to be friction-less and hence, all assets can be perfectly hedged. If we can perfectly hedge our underlying asset, then in theory, it should have no volatility and should grow at the risk free rate, r.

-
+1 Great answer. –  SRKX Jun 18 '13 at 15:49
Did you mean to say that in a world where we can perfectly hedge the risk premium has to be equal to the risk free rate? I am asking because the asset clearly has volatility even after changing measure to a risk neutral probability measure. In fact volatility is identical before and after. –  Matt Wolf Jun 19 '13 at 0:45
I'm puzzled by something: unless $\mu=r$ the mean of ${(μ−r)\over σ}dt+dW$ is not 0, so how can it be equivalent to anything with mean 0? –  Andrew Dabrowski Nov 5 '13 at 14:12
@AndrewDabrowski: Check out the Girsanov Theorem. It lets you transform one brownian motion into another. –  fbt Apr 22 at 22:56


text:
## mrs.perkins Group Title If the radius of a circle is multiplied by 4, then how many times larger is the circumference of the new circle? 2 years ago 2 years ago

1. Monkeyball Group Title

The circumference of a circle is given by $2*\pi*r$

2. Monkeyball Group Title

and lets say we pick r=1 we get $2*\pi$

3. Monkeyball Group Title

now lets say that we multiply the radius by 4 so now r=4

4. Monkeyball Group Title

$8*\pi$

5. Monkeyball Group Title

8 pi is 4 times bigger than 2 pi

6. mrs.perkins Group Title

7. Monkeyball Group Title

Look at my last post

8. mrs.perkins Group Title

so what do i do next

9. mrs.perkins Group Title

so is 32

10. Monkeyball Group Title

wait can you tell me what the original radius equals?

11. mrs.perkins Group Title

i really dont know

12. Champs Group Title


text:
# Incidence Categories - Mathematics > Quantum Algebra

Incidence Categories - Mathematics > Quantum Algebra - Descarga este documento en PDF. Documentación en PDF para descargar gratis. Disponible también para leer online.

Abstract: Given a family $\F$ of posets closed under disjoint unions and the operationof taking convex subposets, we construct a category $\C {\F}$ called the\emph{incidence category of $\F$}. This category is -nearly abelian- in thesense that all morphisms have kernels-cokernels, and possesses a symmetricmonoidal structure akin to direct sum. The Ringel-Hall algebra of $\C {\F}$ isisomorphic to the incidence Hopf algebra of the collection $\P\F$ of orderideals of posets in $\F$. This construction generalizes the categoriesintroduced by K. Kremnizer and the author In the case when $\F$ is thecollection of posets coming from rooted forests or Feynman graphs.

Autor: Matt Szczesny

Fuente: https://arxiv.org/


text:
# Aspect ratio in LibGdx

I'm learning LibGdx and I need some help with the camera.

Let's see, I'm making a Chu Chu Rocket! clon to start. I've loaded a TiledMap (10x10), and I've created an OrthographicCamera to show the map. This is my camera code, written in the Map class constructor, which is invoked in the create() method:

map = new TmxMapLoader().load("data/maps/" + filename);
renderer = new OrthogonalTiledMapRenderer(map, 1.0f/24.0f);
camera = new OrthographicCamera();
camera.setToOrtho(false, 10, 10);

And this is the result, when I compile

The problem is that the tiles I used in Tiled were squares. When I load, as you can see, they're rectangles. That's a problem for me.

I've read some things about resizing, changing aspect ratio, etc, but everithing was about doing it to fit various devices resolutions / screen sizes. So, my questions are:

• How to define correctly the aspect ratio for a tiled map using a ortographic camera?
• When I do that, the ratio will be correct for every device?
• The elements which aren't the map (like the mice) would need to be also adjusted? How?


text:
# LiIon battery discharging

I have a question regarding LiIon battery discharging. Lets say I have a battery which is connected to a dc bus via a battery charger which has a bypass contactor in parallel.

The charger is capable to supply some maximal discharging current Imax01. In case I need greater discharging current than $$\I_{max_{01}}\$$ I can shortcircuit the battery charger via the bypass contactor. In that case I can discharge the battery with discharging current $$\I_{max_{02}} > I_{max_{01}}\$$.

The desired discharging current is calculated by the a dc bus voltage PI controller based on the dc bus reference voltage and dc bus actual voltage. The discharging current is controlled by a current PI controller in the charger based on the reference value from dc bus voltage controller and actual value measured.

The problem is that I don't know how to determine the time instant when the bypass contactor shall be closed which will cause the charger to be "removed" from the discharging process (question is whether this "by the load controlled" discharging is good idea at all and what influence will this action have onto the dc bus voltage controller). My first idea was to determine the bypass contactor closing based on magnitude of control error value i.e. $$\U_{dc\_bus\_ref} - U_{dc\_bus\_act}\$$. In case it is greater than some threshold that the bypass contactor is closed and otherwise it is opened. Does anybody have any idea how to realize quick discharging of a battery in my conditions i.e. battery connected to the dc bus via battery charger capable of carring current less than nominal battery discharging current? Thanks for any ideas.


text:
# pgfplots: legendentrys are wrong

I'm using pgfplots to plot a lot of data as you can see below. For some of the plots I'v specified a legendentry, but the marks for these plots are not shown in the legend. does anyone know where the mistake is?

\documentclass[a4paper]{article}
\usepackage{pgfplots}
\usepgflibrary{shapes.geometric}
\usetikzlibrary{patterns,arrows}
\begin{document}

\begin{tikzpicture}
\centering
\begin{axis}[
height=10cm,
width=\columnwidth,
enlarge y limits=0.07,
enlarge x limits=0.03,
legend style={at={(0.5,-0.1)}, anchor=north,legend columns=-1},
%ylabel={Bezugsphoneme: regspr. Vnt. (Hn)},
symbolic x coords={a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w},
xtick=data
]

\addplot+[sharp plot, only marks, mark options={mark=-, scale=0.8, draw=black}] table[x=vpid, y=-] {data.dat};
\addplot+[sharp plot, only marks, mark options={mark=-, scale=0.8, draw=black}] table[x=vpid, y=a] {data.dat};
\addplot+[sharp plot, only marks, mark options={mark=-, scale=0.8, draw=black}] table[x=vpid, y=aa] {data.dat};
\addplot+[sharp plot, only marks, mark options={mark=-, scale=0.8, draw=black}] table[x=vpid, y=ai] {data.dat};
\addplot+[sharp plot, only marks, mark options={mark=-, scale=0.8, draw=black}] table[x=vpid, y=au] {data.dat};
\addplot+[sharp plot, only marks, mark options={mark=-, scale=0.8, draw=black}] table[x=vpid, y=b] {data.dat};
\addplot+[sharp plot, only marks, mark options={mark=o, scale=0.7, draw=black, solid}] table[x=vpid, y=c] {data.dat}; \addlegendentry{c}
\addplot+[sharp plot, only marks, mark options={mark=-, scale=0.8, draw=black}] table[x=vpid, y=d] {data.dat};
\addplot+[sharp plot, only marks, mark options={mark=triangle, scale=0.8, draw=black, solid}] table[x=vpid, y=ee] {data.dat}; \addlegendentry{ee}
\addplot+[sharp plot, only marks, mark options={mark=-, scale=0.8, draw=black}] table[x=vpid, y=schwa] {data.dat};
\addplot+[sharp plot, only marks, mark options={mark=-, scale=0.8, draw=black}] table[x=vpid, y=E] {data.dat};
\addplot+[sharp plot, only marks, mark options={mark=-, scale=0.8, draw=black}] table[x=vpid, y=EE] {data.dat};
\addplot+[sharp plot, only marks, mark options={mark=-, scale=0.8, draw=black}] table[x=vpid, y=f] {data.dat};
\addplot+[sharp plot, only marks, mark options={mark=-, scale=0.8, draw=black}] table[x=vpid, y=g] {data.dat};
\addplot+[sharp plot, only marks, mark options={mark=-, scale=0.8, draw=black}] table[x=vpid, y=ii] {data.dat};
\addplot+[sharp plot, only marks, mark options={mark=-, scale=0.8, draw=black}] table[x=vpid, y=i] {data.dat};
\addplot+[sharp plot, only marks, mark options={mark=-, scale=0.8, draw=black}] table[x=vpid, y=k] {data.dat};
\addplot+[sharp plot, only marks, mark options={mark=-, scale=0.8, draw=black}] table[x=vpid, y=m] {data.dat};
\addplot+[sharp plot, only marks, mark options={mark=-, scale=0.8, draw=black}] table[x=vpid, y=n] {data.dat};
\addplot+[sharp plot, only marks, mark options={mark=-, scale=0.8, draw=black}] table[x=vpid, y=ng] {data.dat};
\addplot+[sharp plot, only marks, mark options={mark=-, scale=0.8, draw=black}] table[x=vpid, y=oo] {data.dat};
\addplot+[sharp plot, only marks, mark options={mark=-, scale=0.8, draw=black}] table[x=vpid, y=oe] {data.dat};
\addplot+[sharp plot, only marks, mark options={mark=-, scale=0.8, draw=black}] table[x=vpid, y=oee] {data.dat};
\addplot+[sharp plot, only marks, mark options={mark=-, scale=0.8, draw=black}] table[x=vpid, y=o] {data.dat};
\addplot+[sharp plot, only marks, mark options={mark=-, scale=0.8, draw=black}] table[x=vpid, y=oi] {data.dat};
\addplot+[sharp plot, only marks, mark options={mark=-, scale=0.8, draw=black}] table[x=vpid, y=p] {data.dat};
\addplot+[sharp plot, only marks, mark options={mark=-, scale=0.8, draw=black}] table[x=vpid, y=pf] {data.dat};
\addplot+[sharp plot, only marks, mark options={mark=square, scale=0.7, draw=black, solid}] table[x=vpid, y=r] {data.dat}; \addlegendentry{r}
\addplot+[sharp plot, only marks, mark options={mark=diamond, scale=0.8, draw=black, solid}] table[x=vpid, y=s] {data.dat}; \addlegendentry{s}
\addplot+[sharp plot, only marks, mark options={mark=pentagon, scale=0.8, draw=black,solid}] table[x=vpid, y=sch] {data.dat}; \addlegendentry{sch}
\addplot+[sharp plot, only marks, mark options={mark=-, scale=0.8, draw=black}] table[x=vpid, y=t] {data.dat};
\addplot+[sharp plot, only marks, mark options={mark=-, scale=0.8, draw=black}] table[x=vpid, y=ts] {data.dat};
\addplot+[sharp plot, only marks, mark options={mark=-, scale=0.8, draw=black}] table[x=vpid, y=uu] {data.dat};
\addplot+[sharp plot, only marks, mark options={mark=-, scale=0.8, draw=black}] table[x=vpid, y=u] {data.dat};
\addplot+[sharp plot, only marks, mark options={mark=-, scale=0.8, draw=black}] table[x=vpid, y=v] {data.dat};
\addplot+[sharp plot, only marks, mark options={mark=-, scale=0.8, draw=black}] table[x=vpid, y=uee] {data.dat};
\addplot+[sharp plot, only marks, mark options={mark=-, scale=0.8, draw=black}] table[x=vpid, y=ue] {data.dat};
\addplot+[sharp plot, only marks, mark options={mark=-, scale=0.8, draw=black}] table[x=vpid, y=z] {data.dat};
\addplot+[sharp plot, only marks, mark options={mark=-, scale=0.8, draw=black}] table[x=vpid, y=sth_sch] {data.dat};

\end{axis}
\end{tikzpicture}
\end{document}

The data table:

vpid    -   a   aa  ai  au  b   c   d   ee  schwa   E   EE  f   g   ii  i   k   m   n   ng  oo  oe  oee o   oi  p   pf  r   s   sch t   ts  uu  u   v   uee ue  z   sth_sch
a   4   17  6   8   2   10  28  10  33  6   0   0   3   17  23  4   2   1   14  0   5   0   0   20  0   0   0   79  1   18  10  0   0   3   7   1   0   13  0
b   0   11  2   4   1   2   33  0   24  3   2   0   0   2   12  6   5   0   62  0   5   0   0   1   0   7   2   50  2   22  36  0   0   4   10  0   0   2   0
c   1   12  2   3   1   5   11  1   33  3   4   0   0   6   12  11  2   0   17  0   7   1   0   1   0   6   0   56  3   7   17  0   0   16  28  1   1   3   0
d   0   18  1   6   4   6   56  7   58  6   3   0   4   15  18  15  4   1   27  0   1   1   0   3   0   12  0   74  45  29  29  0   0   8   17  1   0   24  2
e   0   16  1   3   1   8   46  9   35  6   6   0   3   18  11  5   0   0   21  0   2   0   1   22  0   5   2   78  2   47  14  0   1   9   1   0   0   3   2
f   0   1   1   10  2   10  68  7   35  2   5   0   2   7   10  9   0   0   15  0   0   0   0   4   0   3   2   65  24  30  40  0   0   10  1   0   0   1   0
g   4   28  16  40  3   26  49  16  27  10  3   0   5   19  40  40  4   0   24  1   8   0   0   7   0   3   3   71  52  24  19  0   2   5   15  2   2   14  0
h   0   4   1   1   4   5   56  4   29  1   3   0   5   7   19  9   1   1   10  0   0   1   0   2   0   4   2   35  15  35  26  0   2   5   3   0   0   5   0
i   1   10  2   5   5   2   111 8   27  0   2   0   0   11  16  17  1   1   13  0   4   0   0   7   0   1   2   48  30  37  16  0   2   8   5   0   0   2   0
j   0   13  1   5   5   2   95  5   36  2   1   0   1   3   21  20  0   6   11  0   1   0   0   6   0   4   1   43  21  15  15  0   0   4   7   0   0   8   0
k   0   9   9   7   9   12  76  7   41  4   2   0   8   11  31  15  4   0   16  0   12  0   0   4   0   7   2   82  27  43  10  0   0   3   10  0   0   5   0
l   0   12  6   11  8   6   64  9   35  2   4   0   11  15  33  19  1   1   17  0   18  0   0   6   0   3   1   63  44  15  23  0   0   9   27  0   0   31  0
m   0   30  4   3   4   10  64  7   48  6   2   0   3   8   20  8   0   0   26  0   11  1   0   9   0   2   0   115 0   66  13  0   0   14  6   0   0   15  0
n   0   17  0   8   4   6   65  7   33  3   4   0   4   9   22  6   5   3   9   0   7   0   1   4   0   2   2   59  24  44  18  0   1   2   3   0   0   6   0
o   0   1   1   0   3   12  77  5   30  2   3   0   1   10  15  6   0   0   7   0   2   0   0   1   0   4   0   51  0   45  21  0   1   5   3   1   0   2   0
p   0   28  4   6   5   6   41  8   38  1   0   0   2   4   43  3   1   0   17  0   3   0   0   4   0   4   0   65  2   35  26  0   1   2   21  0   0   3   3
q   0   14  3   4   6   2   60  2   30  2   3   0   6   13  29  2   0   1   9   0   3   0   0   3   0   4   2   66  0   38  10  0   0   6   14  0   0   8   0
r   0   40  7   5   5   5   109 13  43  3   2   0   5   17  30  25  1   1   14  0   3   0   0   13  0   4   0   74  71  32  22  0   6   2   22  0   0   4   0
s   1   12  1   2   2   17  41  14  26  2   0   0   3   18  22  5   0   0   11  0   3   0   0   7   0   1   1   62  4   41  8   0   2   4   1   0   0   6   1
t   6   81  44  16  6   20  111 23  31  29  6   0   3   34  70  61  3   0   8   0   24  0   0   13  3   6   3   95  102 52  56  0   18  5   35  6   1   8   0
u   0   19  5   4   17  15  114 22  38  2   8   0   5   8   24  16  1   0   10  0   13  0   0   7   0   2   0   82  25  61  25  0   0   0   11  0   0   13  0
v   0   25  7   6   11  20  142 7   21  6   4   0   7   21  45  15  3   1   12  0   13  0   0   5   0   0   0   52  24  23  37  0   2   6   11  0   0   16  0
w   0   24  1   6   14  6   74  14  15  3   3   0   11  12  22  2   5   1   3   0   7   0   0   6   0   2   0   46  0   35  17  0   1   2   8   0   0   14  0

-

Instead of reordering the plots, you can also supply forget plot to those series that will not get an entry in the legend.

For complicated plots like yours, it's best to work with styles. I've adjusted your example a bit:

\documentclass[a4paper]{article}
\usepackage{pgfplots}
\usepgflibrary{shapes.geometric}
\begin{document}

\begin{tikzpicture}
\centering
\begin{axis}[
height=10cm,
width=\columnwidth,
enlarge y limits={upper, value=0.07},
enlarge x limits=0.03,
legend style={
at={(0.5,-0.1)},
anchor=north, %Proper alignment
nodes={anchor=mid},
legend columns=-1,
/tikz/every even column/.append style={column sep=4pt} %Increase spacing in legend
},
symbolic x coords={a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w},
xtick=data,
xticklabel style={text height=1.5ex}, % To get proper alignment
only marks,
every axis plot post/.style={draw=black}, %Common options
other/.style={
forget plot,
mark options={
mark=-, scale=0.8
}
},
c plot/.style={
mark options={mark=o, scale=0.7, solid}
},
ee plot/.style={
mark options={mark=triangle, scale=0.8, solid}
},
r plot/.style={
mark options={mark=square, scale=0.7, solid}
},
s plot/.style={
mark options={mark=diamond, scale=0.8, solid}
},
sch plot/.style={
mark options={mark=pentagon, scale=0.8,solid}
}
]

\end{axis}
\end{tikzpicture}
\end{document}

-
Hi Jake! Thanks again for your really nice work, I'm learning a lot from your examples. I had migrated the styles to my original plot, but now I the effect of \addlegendimage{other} is nothing. The "other"-style is not plotted in the legend. Do you have some ideas on that? –  MaKa TaBo Dec 8 '11 at 22:39
Glad it helps! Hm, not sure about the missing entry for other. Probably best to post a new question with an example. –  Jake Dec 8 '11 at 22:50
I tried Jake's example and the entry for other is present and seems to be ok.... are you sure you added both \addlegendimage and \addlegendentry? –  Christian Feuersänger Dec 10 '11 at 9:04

That is because \addlegendentry makes a list which is constructed from the number of calls to that command. This means that the first call to \addlegendentry applies to the first \addplot, the second call of \addlegendentry applies to the second \addplot. No matter the location of the command itself.
What you need to do is assemble all your \addplots to the \addlegendentry calls.

Effectively:

\begin{axis}
\addplot+[sharp plot, only marks, mark options={mark=o, scale=0.7, draw=black, solid}] table[x=vpid, y=c] {data.dat}; \addlegendentry{c}
\addplot+[sharp plot, only marks, mark options={mark=triangle, scale=0.8, draw=black, solid}] table[x=vpid, y=ee] {data.dat}; \addlegendentry{ee}
\addplot+[sharp plot, only marks, mark options={mark=square, scale=0.7, draw=black, solid}] table[x=vpid, y=r] {data.dat}; \addlegendentry{r}
\addplot+[sharp plot, only marks, mark options={mark=diamond, scale=0.8, draw=black, solid}] table[x=vpid, y=s] {data.dat}; \addlegendentry{s}
\addplot+[sharp plot, only marks, mark options={mark=pentagon, scale=0.8, draw=black,solid}] table[x=vpid, y=sch] {data.dat}; \addlegendentry{sch}
....
<other>
\end{axis}

-


text:
# Model -- a type used to store a statistical model and its parameters

## Description

In order to generate and study random algebraic objects formally, one should define a statistical model for such objects. A model captures the random generating process of the objects. It consists of a name, a set of parameters, and a generating function, stored under the corresponding keys in the hash table.

 i1 : ER(3,2,0.2) o1 = Model{Generate => -*Function[/usr/share/Macaulay2/RandomMonomialIdeals.m2:130:23-130:47]*-} Name => Erdos-Renyi Parameters => (3, 2, .2) o1 : Model

Combined with the type Sample, the type Model defined here stores such information and allows for a streamlined way to sample random objects, store the data as a proper statistical sample, and study their algebraic properties under the probabilistic regime.

• Sample -- a type used to store a data sample from a statistical model
• statistics -- generate statistics for a sample

## Functions and methods returning an object of class Model :

• ER -- model for sampling from Erdos-Renyi type distributions on monomials
• model -- creates a new model for random objects with a given list of parameters and generating function

## Methods that use an object of class Model :

• "sample(Model,ZZ)" -- see sample -- generates a sample from the given model

## For the programmer

The object Model is a type, with ancestor classes HashTable < Thing.


text:
# ieeetran caption question

In the ieeetran how to, they are using caption with their examples; however, latex tells me not to use caption.

So if I use subfloat, all my figures are labeled as such (a), (b) and don't have a main label Figure:1. In order to get the main label, I would need to use caption.

How can I change the label from (a) to figure 1 without using caption?

\documentclass[journal]{IEEEtran}
\uspackage{subfig, float, wrapfig}
\begin{document}
\begin{figure}
\centering
\subfloat[caption]{picture}
\label{}
\end{figure}
\end{document}

-
LaTeX tells you not to use \caption? You can, even in ieeetran. When I add \caption{A caption} to your MWE, I obtain Fig 1. A caption. Or is it that you don't want to use \caption but \subfloat instead? –  Werner May 9 at 16:08
It seems to me you are mixing the caption package and the \caption command. Of course you can use \caption but should not use the caption package since it is incompatible to the IEEEtran document class. So better replace \usepackage{subfig} with \usepackage[caption=false]{subfig} to prevent loading the caption package. –  Axel Sommerfeldt May 9 at 16:09
@AxelSommerfeldt I am not using the caption package and I get the latex warning with \caption –  dustin May 9 at 16:13
@dustin You are using the caption package since \usepackage{subfig} does load it unless you say \usepackage[caption=false]{subfig}. The warning you get does not come from LaTeX itself but from the caption package. –  Axel Sommerfeldt May 9 at 17:34
@AxelSommerfeldt post your comment as a solution. You fixed the problem. –  dustin May 10 at 18:03
add comment

## 1 Answer

I have mentioned it a few times before but IEEE does not directly use your .tex file. It's there to supply the bare minimal so if it looks like an article it's enough. It doesn't have to be following the standards internally.

They will recode your article with their own terrible terrible workflow, rasterize your images and ruin your math after acceptance. I've been using the following for a couple of years without any issues. It uses subcaption and does a superb job even though it has a command clash due to the redefinitions.

\documentclass[final]{ieeetran}
\usepackage{tikz,lipsum}
\usepackage[labelformat=simple]{subcaption}

% Training subcaption package to comply with
% IEEE standards. We can ignore the warning
% generated by caption.sty which is due to
% the redefinition of \@makecaption
\DeclareCaptionLabelSeparator{periodspace}{.\quad}
\captionsetup{font=footnotesize,labelsep=periodspace,singlelinecheck=false}
\captionsetup[sub]{font=footnotesize,singlelinecheck=true}
\renewcommand\thesubfigure{(\alph{subfigure})}

\title{My title}
\author{my author}

\begin{document}
\maketitle
\begin{abstract}
\lipsum*[1]
\end{abstract}

\lipsum[2-5]
\begin{figure}
\begin{subfigure}[b]{.6\linewidth}
\centering%
\begin{tikzpicture}
\draw[help lines] (-1,-1) grid (1,2);
\end{tikzpicture}
\caption{A caption}
\label{fig:a}
\end{subfigure}%
\begin{subfigure}[b]{.4\linewidth}
\centering%
\begin{tikzpicture}
\draw[help lines] (-1,-1) grid (1,2);
\end{tikzpicture}
\caption{B cap}
\label{fig:b}
\end{subfigure}
\caption{Two figures.}\label{fig:1}
\end{figure}
\end{document}

If you only want the caption labels leave the sub \caption{} definitions empty and describe the subfigures in the main caption. If you want to remove also the (a),(b) use \phantomcaption commands as described in the manual.

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I have used your above code to redefine the captions for figures. The captions for tables –  J Spen Oct 25 at 7:02
I have used your above code to be able to use the subcation class with IEEEtran and get the same figure captions. The subcaption class also redefines the table caption so can you provide the code for the correct table caption or inform me how I can find it? I have all my other documents using subcaption package so much easier than changing everything over to use the subfig package. –  J Spen Oct 25 at 7:05
if you need an example let me know but can just look at any IEEE journal with a table caption and what you get is a table caption that is the same type as the figure caption. –  J Spen Oct 25 at 7:06
This seems to be the right settings for the table caption: \captionsetup[table]{format=plain,labelformat=simple,justification=centering, labelsep=newline, singlelinecheck=false, textfont={sc}}. Do you have something similar?? Also I noticed another issue of the table and figures not lining up to text at the top of the pages. They align maybe 0.2 or 0.3 cms above the text. This is not supposed to be the case as it should line up evenly with the text. Do you know how to set the margins so the figures line up correctly? –  J Spen Oct 25 at 7:41
@JSpen That's a known issue. you don't need to worry about it because IEEE will fix stuff by themselves. I'll check the table settings when I have the chance in the meantime I have this answer for a reasoning about why you don't need to make it perfect tex.stackexchange.com/a/89790/3235 –  percusse Oct 25 at 9:10
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text:
zbMATH — the first resource for mathematics

Chaos control of a fractional order modified coupled dynamos system. (English) Zbl 1187.34080
This paper analyzes some Routh-Hurwitz stability conditions generalized to the fractional order case, and discusses the stability region of the fractional order system. The authors analyze the chaotic behavior of the fractional order modified coupled dynamos system concretely, and provide conditions suppressing chaos to unstable equilibrium points. They use the feedback control method to control chaos in the fractional order modified coupled dynamos system. Numerical simulations show the effectiveness of the method. The stability results in this paper have been generalized to 4-dimensional systems in the following reference: [A. E. Matouk, Phys. Lett., A 373, 2166–2173 (2009)].

MSC:
 34H10 Chaos control for problems involving ordinary differential equations 34A08 Fractional ordinary differential equations and fractional differential inclusions 34D20 Stability of solutions to ordinary differential equations
Full Text:
References:
 [1] Podlubny, I., Fractional differential equations, (1999), Academic Press New York · Zbl 0918.34010 [2] Hilfer, R., Applications of fractional calculus in physics, (2001), World Scientific New Jersey · Zbl 0998.26002 [3] Bagley, R.L.; Calico, R.A., Fractional order state equations for the control of viscoelastically damped structures, Journal of guidance, control and dynamics, 14, 2, 304-311, (1991) [4] Koeller, R.C., Application of fractional calculus to the theory of viscoelasticity, Journal of applied mechanics, 51, 2, 294-298, (1984) · Zbl 0544.73052 [5] Koeller, R.C., Polynomial operators, Stieltjes convolution, and fractional calculus in hereditary mechanics, Acta mechanica, 58, 3-4, 251-264, (1986) · Zbl 0578.73040 [6] Heaviside, O., Electromagnetic theory, (1971), Chelsea New York · JFM 30.0801.03 [7] Ahmad, W.M.; El-Khazali, R.; Al-Assaf, Y., Stabilization of generalized fractional order chaotic systems using state feedback control, Chaos, solitons & fractals, 22, 1, 141-150, (2004) · Zbl 1060.93515 [8] Ahmed, E.; El-Sayed, A.M.A.; El-Saka, H.A.A., On some routh – hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rössler, Chua and Chen systems, Physics letters A, 358, 1, 1-4, (2006) · Zbl 1142.30303 [9] Matignon, D., Stability results for fractional differential equations with application to control processing, (), 963-968 [10] Singer, J.; Wang, Y.Z.; Bau, H.H., Controlling a chaotic system, Physical review letters, 66, 1123-1125, (1991) [11] Chen, G.; Dong, X., Controlling chua’s circuit, Circuits, systems and computers, 3, 139-149, (1993) [12] Chen, G.; Dong, X., On feedback control of chaotic dynamic systems, International journal of bifurcation and chaos, 2, 2, 407-411, (1992) · Zbl 0875.93176 [13] Chen, G., On some controllability conditions for chaotic dynamics control, Chaos, solitons & fractals, 8, 9, 1461-1470, (1997) [14] Pyragas, K., Control of chaos via extended delay feedback, Physica A, 206, 323-330, (1995) · Zbl 0963.93523 [15] Vassiliadis, D., Parametric adaptive control and parameter identification of low-dimensional chaotic systems, Physica D, 71, 319-341, (1994) · Zbl 0825.93365 [16] Li, Y.X.; Tang, W.K.S.; Chen, G.R., Generating hyperchaos via state feedback control, International journal of bifurcation and chaos, 15, 10, 3367-3375, (2005) [17] Wang, X.Y.; Liu, M., Sliding mode control the synchronization of master-slave chaotic systems with sector nonlinear inputs, Acta physica sinica, 54, 6, 2584-2589, (2005) [18] Chen, A.M.; Lu, J.A.; Lü, J.H.; Yu, S.M., Generating hyperchaotic Lü attractor via state feedback control, Physica A, 364, 103-110, (2006) [19] Tian, Y.; Gao, F., Adaptive control of chaotic continuous-time systems with delay, Physica D, 117, 1-12, (1998) · Zbl 0941.93534 [20] Pavlica, V.; Petrovacki, D., About simple fuzzy control and fuzzy control based on fuzzy relational equations, Fuzzy sets & systems, 101, 1, 41-47, (1999) [21] Wang, X.; Chen, G.; Yu, X., Anticontrol of chaos in continuous-time systems via time-delay feedback, Chaos, 10, 4, 771-779, (2000) · Zbl 0967.93045 [22] Liu, F.; Ren, Y.; Shan, X.M., A linear feedback synchronization theorem for a class of chaotic system, Chaos, solitons & fractals, 13, 4, 723-730, (2002) · Zbl 1032.34045 [23] Ömer, M.; Ercan, S., Observer based synchronization of chaotic systems, Physical review E, 54, 5, 4803-4811, (1996) [24] Wang, Y.W.; Guan, Z.H.; Wang, H.O., LMI-based fuzzy stability and synchronization of chen’s system, Physics letters A, 320, 2-3, 154-159, (2003) · Zbl 1065.37503 [25] Ahmed, E.; El-Sayed, A.M.A.; Elsaka, H.A.A., Equilibrium points, stability and numerical solutions of fractional-order predator – prey and rabies models, Journal of mathematical analysis and applications, 325, 1, 542-553, (2007) · Zbl 1105.65122 [26] Caputo, M., Linear models of dissipation whose $$Q$$ is almost frequency independent, The geophysical journal of the royal astronomical society, 13, 529-539, (1967) [27] Agiza, H.N., Controlling chaos for the dynamical system of coupled dynamos, Chaos, solitons & fractals, 13, 2, 341-352, (2002) · Zbl 0994.37047 [28] Agiza, H.N., Chaos synchronization of two coupled dynamos systems with unknown system parameters, International journal of modern physics C, 15, 6, 873-883, (2004) · Zbl 1082.34041 [29] Wang, X.Y.; Wu, X.J., Chaos control of a modified coupled dynamos system, Acta physica sinica, 55, 10, 5083-5093, (2006) · Zbl 1202.93141 [30] Vaněček, A.; Čelikovský, S., Control systems from linear analysis to synthesis of chaos, (1996), Prentice-Hall London, pp. 108-262 · Zbl 0874.93006 [31] Wang, X.Y., Chaos in the complex nonlinearity system, (2003), Electronics Industry Press Beijing, pp. 28-32 [32] Benettin, G.; Galgani, L.; Strelcyn, J.M., Kolmogorov entropy and numerical experiments, Physical review A, 14, 2338-2345, (1976)
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