text:
## Stream: Is there code for X?

### Topic: gsmul and smul commute

#### Eric Wieser (Nov 27 2020 at 18:11):

example {R : Type*} {L : Type*} {ι : Type*}
[semiring R] [add_comm_group L] [semimodule R L]
(y : ℤ) (c : R) (x : L) :
y •ℤ c • x = c • (y •ℤ x) :=
begin
rw [gsmul_eq_smul, gsmul_eq_smul],  -- does this help? Which is more canonical?
sorry
end

#### Eric Wieser (Nov 27 2020 at 18:25):

instance nsmul.smul_comm [semiring R] [add_comm_monoid M] [semimodule R M] : smul_comm_class ℕ R M :=
{ smul_comm := λ n r m, begin
dsimp [(•)],
induction n with n ih,
{ simp },
{ simp [succ_nsmul, ih] },
end }

instance gsmul.smul_comm [semiring R] [add_comm_group L] [semimodule R L] : smul_comm_class ℤ R L :=
{ smul_comm := λ z r l, by cases z; simp [←gsmul_eq_smul, ←nat.smul_def, smul_comm] }

example {R : Type*} {L : Type*} {ι : Type*}
[semiring R] [add_comm_group L] [semimodule R L]
(y : ℤ) (c : R) (x : L) :
y •ℤ c • x = c • (y •ℤ x) :=
begin
rw [gsmul_eq_smul, gsmul_eq_smul],
apply smul_comm y c x,
end

#### Eric Wieser (Nov 27 2020 at 18:27):

@Yury G. Kudryashov, since you wrote smul_comm_class, do you have an opinion on where I should put those instances?

#### Eric Wieser (Nov 27 2020 at 18:37):

#5135

Last updated: May 16 2021 at 05:21 UTC


text:
# lefse

The lefse command is modeled after the LEfSe program written and published by the Huttenhower lab. See make.lefse for formating files for input into the original LEfSe program.

## Default Setting

The shared and design parameters are required.

mothur > lefse(shared=final.an.shared, design=mouse.sex_time2.design)

If you open the final.an.0.03.lefse_summary file, you will see:

OTU     LogMaxMean  Class   LDA     pValue
Otu001  5.21329     -
Otu002  5.06449     -
Otu003  5.08958     -
Otu004  5.06121     -
Otu005  4.82611     -
Otu006  4.76942     -
Otu007  4.81303     -
Otu008  4.69046     -
Otu009  4.71183     -
Otu010  4.77212     -
Otu011  4.63326     -
Otu012  4.80072      F003Late   4.43296     0.0363836
Otu013  4.58877      F003Late   4.02028     0.0363836
Otu014  4.5691  -
Otu015  4.42951     -
...

## Options

### class

The class parameter is used to indicate the which category you would like used as the class. If none is provided first category is used.

### subclass

The subclass parameter is used to indicate the which category you would like used as the subclass. If none is provided the class is used.

### sets

The sets parameter allows you to specify which of the sets in your design file you would like to analyze. The set names are separated by dashes. The default is all sets in the designfile.

### label

The label parameter is used to indicate which distances in the shared file you would like to use. labels are separated by dashes.

### walpha

The walpha parameter is used to set the alpha value for the Wilcoxon test. Default=0.05.

### aalpha

The aalpha parameter is used to set the alpha value for the Krukal Wallis Anova test Default=0.05.

### lda

The lda parameter is used to set the threshold on the absolute value of the logarithmic LDA score. Default=2.0.

### wilc

The wilc parameter is used to indicate whether to perform the Wilcoxon test. Default=T.

### iters

The iters parameter is used to set the number of bootstrap iteration for LDA. Default=30.

### curv

The curv parameter is used to set whether perform the wilcoxon testing the Curtis’s approach [beta version] Default=F.

### fboots

The fboots parameter is used to set the subsampling fraction value for each bootstrap iteration. Default=0.67.

### strict

The strict parameter is used to set the multiple testing correction options. 0 no correction (more strict, default), 1 correction for independent comparisons, 2 correction for independent comparison. Options = 0,1,2. Default=0.

### minc

The minc parameter is used to minimum number of samples per subclass for performing wilcoxon test. Default=10.

### multiclass

The multiclass parameter is used to (for multiclass tasks) set whether the test is performed in a one-against-one ( onevone - more strict!) or in a one-against-all setting ( onevall - less strict). Default=onevall.

### norm

The norm parameter is used to multiply relative abundances by 1000000. Recommended when very low values are present. Default=T.

## Revisions

• 1.32.0 - First Introduced
• 1.37.0 - Fixes “Skipping x iter...” warning.
• 1.38.0 - Adds sets parameter
• 1.40.0 - Speed and memory improvements for shared files. #357 , #347
• 1.40.0 Bug Fix: Fixes seeded random issue. #416

format: plain


text:
# Is it true that for a value of n (n-principal quantum number), the value of m_l (m_l-magnetic quantum number) is equal to n^2? If it is true, can you please explain this to me? Thanks.

Jul 6, 2015

The actual value, no. The number of values, yes.

#### Explanation:

The relationship between the principal quantum number, $n$, and the magnetic quantum number, ${m}_{l}$, actually goes through the angular momentum quantum number, $l$.

The angular momentum quantum number can take any value that ranges from $0$ to $n - 1$

$l = 0 , 1 , 2 , \ldots , \left(n - 1\right)$

The magnetic quantum number can take any value that ranges from $- l$ to $l$

${m}_{l} = - l , \ldots , - 1 , 0 , 1 , \ldots , l$

If you take into account the possible values of $l$, the magnetic quantum number can thus have values that range from

${m}_{l} = - \left(n - 1\right) , \ldots , - 1 , 0 , 1 , \ldots , \left(n - 1\right)$

So, for example, if $n = 2$, ${m}_{l}$ can be equal to

${m}_{l} = 0$ $\to$ for $l = 0$

and

${m}_{l} = \left\{- 1 , 0 , 1\right\}$ $\to$ for $l = 1$

As you can see, the individual values of ${m}_{l}$ don't even come close to the value of ${n}^{2}$.

However, the total number of values ${m}_{l}$ can take, if you take into account the values of $l$, is indeed equal to ${n}^{2}$.

The magnetic quantum number actually tells you how many orbitals a subshell has. In the above example, if $n = 2$, you get

$n = 2$, $l = 0$, ${m}_{l} = 0$ $\to$ one 2s-orbital;

{:(n=2, l=1, m_l = -1->2p_x), (n=2, l=1, m_l = 0->2p_y), (n=2, l=1, m_l = 1->2p_z) :}} $\to$ three 3p-orbitals.

This means that the second energy level has a total of two subshells, the 2s-subshell and the 2p-subshell.

The number of orbitals each subshell contains is given by ${m}_{l}$. The 2s-subshell contains one orbital, since you only have one possible value for ${m}_{l}$.

The 2p-subshell contains 3 orbitals, each denoted by a different value of ${m}_{l}$.

The total number of orbitals in the second energy level is 4, which is equal to ${n}^{2}$.

If you try this with $n = 3$, you'll find that you get a total number of $9$ orbitals

• one 3s-orbital;
• three 3p-orbitals;
• five 3d-orbitals.


text:
WEBSITE > HOME:

#### What we do and why

The Electronic Colloquium on Computational Complexity is a new forum for the rapid and widespread interchange of ideas, techniques, and research in computational complexity. The purpose of this Colloquium is to use electronic media for scientific communication and discussions in the computational complexity community. The Electronic Colloquium on Computational Complexity (ECCC) welcomes papers, short notes and surveys with
• relevance to the theory of computation,
• clear mathematical profile and
• strictly mathematical format.

#### Central topics

• models of computation and their complexity,
• complexity bounds (with the emphasis on lower bounds).
Specific areas including complexity issues are
• combinatorics,
• communication complexity,
• cryptography,
• combinatorial optimization,
• complexity of learning algorithms,
• logic.

Here are some papers on the idea and concept of electronic colloquia and ECCC.
7th April 2014 13:36

#### ECCC Archive DVD 2013

191 reports have been published on ECCC in 2013. The collection of all these reports is now available on DVD. You can order the archive (and also the archive DVDs from earlier years) at the local office. Please email < href="mailto:eccc@eccc.hpi-web.de">eccc@eccc.hpi-web.de for ordering.

4th March 2013 09:03

#### ECCC Archive DVD 2012

In 2012 we had a total count of 186 published reports on ECCC. The collection of all the reports from 2012 is now available on DVD. You can order the archive (and also the archive DVDs from earlier years) at the local office. Please email to eccc@eccc.hpi-web.de for ordering.

6th March 2012 12:04

#### ECCC Archive DVD 2011

In 2011 we had a total count of 174 published reports on ECCC. The collection of all the reports from 2011 is now available on DVD. You can order the archive (and also the archive DVDs from earlier years) at the local office. Please email to eccc@eccc.hpi-web.de for ordering.

-> Older news
TR14-111 | 15th August 2014
Vikraman Arvind, Gaurav Rattan

#### Faster FPT Algorithm for Graph Isomorphism Parameterized by Eigenvalue Multiplicity

We give a $O^*(k^{O(k)})$ time isomorphism testing algorithm for graphs of eigenvalue multiplicity bounded by $k$ which improves on the previous
best running time bound of $O^*(2^{O(k^2/\log k)})$.

more >>>

TR14-110 | 19th August 2014
Uriel Feige, Shlomo Jozeph

#### Separation between Estimation and Approximation

We show (under standard assumptions) that there are NP optimization problems for which estimation is easier than approximation. Namely, one can estimate the value of the optimal solution within a ratio of $\rho$, but it is difficult to find a solution whose value is within $\rho$ of optimal.
As an ... more >>>

TR14-109 | 14th August 2014
Aran Nayebi, Scott Aaronson, Aleksandrs Belovs, Luca Trevisan

#### Quantum lower bound for inverting a permutation with advice

Given a random permutation $f: [N] \to [N]$ as a black box and $y \in [N]$, we want to output $x = f^{-1}(y)$. Supplementary to our input, we are given classical advice in the form of a pre-computed data structure; this advice can depend on the permutation but \emph{not} on ... more >>>

-> Older reports

ISSN 1433-8092 | Imprint


text:
# Installing a font in .NET

This topic is 4610 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

## Recommended Posts

Specifically, VB.NET I want to install a font using VB.NET. I'm not talking about the private install, but rather, installing a font on the system. After some extensive research on this, I can't find a way to do it. It seems, according to the MSDN help, that you need to use AddFontResource, which is part of the GDI. However, I have yet to find how to import GDI functions into .NET applications. Many thanks to anyone who can help.

##### Share on other sites
Try putting the following in one of your classes:
[DllImport("gdi32.dll")]public static extern int AddFontResource(string lpszFilename);
I like to make a Gdi32 class and put all of my GDI calls in there:
public class Gdi32{    [DllImport("gdi32.dll")]    public static extern int AddFontResource(string lpszFilename);}

##### Share on other sites
Thanks for sending me down the right track, Rob. In VB, the formatting is a little different, but I figured it out.

Also, for reference, here's how to install a font from a ttf file in VB.net

Before your class declaration, you need to put:
<Assembly: RegistryPermissionAttribute( _    SecurityAction.RequestMinimum, All:="HKEY_LOCAL_MACHINE")>

Then, inside the class, import some API functions:
Private Const HWND_BROADCAST = &HFFFF&Private Const WM_FONTCHANGE As Long = &H1D<DllImport("gdi32.dll")> _Public Shared Function AddFontResource(ByVal lpszFilename As String) As IntegerEnd Function<DllImport("user32.dll")> _Public Shared Function SendMessage(ByVal hWnd As Integer, ByVal wMsg As Integer, ByVal wParam As Integer, ByVal lParam As Integer) As IntegerEnd Function

Then, is the actual installation. I put it in a button press.
'let's assume that filePath is a string with the full path of the ttf filePrivate Sub cmdAddFont_Click(ByVal sender As System.Object, ByVal e As System.EventArgs) Handles cmdAddFont.Click    Dim regKey As RegistryKey = Registry.LocalMachine.OpenSubKey("Software", True).OpenSubKey("Microsoft", True).OpenSubKey("Windows NT", True).OpenSubKey("CurrentVersion", True).OpenSubKey("Fonts", True)    Dim fileName As String    Dim tempFonts As New PrivateFontCollection    Dim fontName As String    fileName = GetFileName(filePath)    tempFonts.AddFontFile(filePath)    fontName = tempFonts.Families(0).Name    If Not System.IO.File.Exists(FONTS_FOLDER & fileName) Then        'copy the font into the windows font directory        System.IO.File.Copy(filePath, FONTS_FOLDER & fileName)        'add it to the system font table        AddFontResource(FONTS_FOLDER & fileName)        'write the change in the registry        regKey.SetValue(fontName, fileName)        'alert all top-level programs about the change        SendMessage(HWND_BROADCAST, WM_FONTCHANGE, 0, 0)    End If    tempFonts.Dispose()End SubPrivate Function GetFileName(ByRef fullPath As String) As String    Return fullPath.Substring(fullPath.LastIndexOf("\") + 1)End FunctionPrivate Function GetFontName(ByRef fileName As String)    Return fileName.Substring(0, fileName.IndexOf("."))End Function

Note: this only works on NT machines (this includes XP). To make it work on machines before then, change "Windows NT" in the registry key to just "Windows"

Also, for some reason, I cannot get the newly installed font to show up on an InstalledFontsCollection until I restart the program. I'm working on that one. Post here if you have any ideas.

Hope this was helpful.

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# Asymptotic preserving schemes for the FitzHugh-Nagumo transport equation with strong local interactions

This paper is devoted to the numerical approximation of the spatially-extended FitzHugh-Nagumo transport equation with strong local interactions based on a particle method. In this regime, the time step can be subject to stability constraints related to the interaction kernel. To avoid this limitation, our approach is based on higher-order implicit-explicit numerical schemes. Thus, when the magnitude of the interactions becomes large, this method provides a consistent discretization of the macroscopic reaction-diffusion FitzHugh-Nagumo system. We carry out some theoretical proofs and perform several numerical experiments that establish a solid validation of the method and its underlying concepts.

## Authors

• 1 publication
• 7 publications
03/18/2020

### Convergence Analysis of Asymptotic Preserving Schemes for Strongly Mangnetized Plasmas

The present paper is devoted to the convergence analysis of a class of a...
03/18/2020

### Convergence Analysis of Asymptotic Preserving Schemes for Strongly Magnetized Plasmas

The present paper is devoted to the convergence analysis of a class of a...
10/29/2021

### An Assessment of Solvers for Algebraically Stabilized Discretizations of Convection-Diffusion-Reaction Equations

We consider flux-corrected finite element discretizations of 3D convecti...
01/20/2022

### Numerical simulation of singularity propagation modeled by linear convection equations with spatially heterogeneous nonlocal interactions

We study the propagation of singularities in solutions of linear convect...
06/12/2020

### Asymptotic preserving IMEX-DG-S schemes for linear kinetic transport equations based on Schur complement

We consider a linear kinetic transport equation under a diffusive scalin...
03/29/2021

### Translating Numerical Concepts for PDEs into Neural Architectures

We investigate what can be learned from translating numerical algorithms...
01/02/2020

### Asymptotically compatible reproducing kernel collocation and meshfree integration for the peridynamic Navier equation

In this work, we study the reproducing kernel (RK) collocation method fo...
##### This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

## 1 Introduction

The FitzHugh–Nagumo (FHN) system [17, 27]

, models the pulse transmission in animal nerve axons and allows to describe complicated interactions of neurons in large neural networks. More precisely, we consider a network composed of

neurons interacting with each other, where each one is labeled by , and endowed with a parameter for standing for the constant spatial position in the network. The FHN system accounts for the variations of the membrane potential of a neuron coupled to an auxiliary variable called the adaptation variable. It can be written as follows for all ,

 (1.1)

where and are given constants, with a fixed parameter whereas is a scaling small parameter describing the intensity of local interactions between neurons. For all and for all , the connectivity kernel only depends on the relative distance between neurons and is given by

 Ψε(∥y∥):=1εdΨ(∥y∥ε),y∈Rd,

where . This scaling with respect to means that when goes to zero, space interactions are highly dominated by local ones compared to long range correlations. In the rest of this article, we assume that the connectivity kernel is nonnegative and rapidly vanishing at infinity, hence we introduce the following quantities,

 (1.2)

which will play an important role later. A typical example for is a Gaussian function, or the indicator function in a compact set.

In [8], we proved that as the number of neurons goes to infinity and for , the set of neurons at time and position can be described by a distribution function solution to a mean-field equation,

 (1.3)

with and given by

 ⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩Kε[fε](t,x,v)=1ε2∫Rd∫R2Ψε(∥x−x′∥)(v′−v)fε(t,x′,dv′,dw′)dx′,A(v,w)=τ(v−γw). (1.4)

Here, we want to construct numerical solutions to (1.3)–(1.4) using particle methods, which consist in approximating the distribution function by a finite number of macro-particles. The trajectories of these particles are determined from the characteristic curves corresponding to the (1.3). Indeed, for any initial data

with finite second moments in

and , the solution to (1.3)–(1.4) is uniquely defined as the push-forward of by the flow of the characteristic system of equations associated to (1.3)–(1.4), which can be written for and as

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩dVεdt=N(Vε)−Wε+Kε[fε](t,x,Vε),dWεdt=A(Vε,Wε),Vε(0)=v,Wε(0)=w. (1.5)

Then we denote by the flow , hence the solution to (1.3)–(1.4) is given by

 fε(t,x,.)=Φt,x#fε0(x,.), (1.6)

that is, for any test-function and ,

 ∫Bφ(v,w)fε(t,x,dv,dw)=∫Φ−1t,x(B)φ∘Φt,xfε0(x,dv,dw).

We also define for all and the following macroscopic quantities,

 ρε⎛⎜⎝1VεWε⎞⎟⎠(t,x):=∫R2⎛⎜⎝1vw⎞⎟⎠fε(t,x,dv,dw), (1.7)

so that is the average neuron density in the network at time and location , and is the average pair membrane potential - adaptation variable. Therefore, we observe that may be written with respect to the macroscopic quantities and as

 Kε[fε](.,v)=1ε2[Ψε⋆(ρεVε)−Ψε⋆ρεv], (1.8)

where denotes the standard convolution product in .

Before describing and analyzing a class of numerical methods for (1.3)–(1.4) in the presence of strong local space interactions (), we first briefly expound what may be expected from the continuous model in the limit .

On the one hand, by integrating (1.3) with respect to , we observe that for all

 ρε(t,x)=ρε0(x),x∈Rd

and moreover we suppose that it does not depend neither on , so that with

 ρ0≥0,ρ0∈L∞(Rd). (1.9)

On the other hand, using (1.8), we observe that

 ∫R2Kε[fε](t,x,v)fε(t,x,dvdw)=ρ0ε2[Ψε⋆(ρ0Vε)−(Ψε⋆ρ0)Vε]. (1.10)

Hence, multiplying (1.3) by (resp. ) and integrating with respect to and using (1.10), we get a time evolution equation for the macroscopic quantities as

 ⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩∂t(ρ0Vε)−ρ0ε2[Ψε⋆(ρ0Vε)−(Ψε⋆ρ0)Vε]=∫R2N(v)fε(.,dv,dw)−ρ0Wε,∂t(ρ0Wε)=ρ0A(Vε,Wε). (1.11)

Of course, this system is not closed since the right hand side of the equation on again depends on the distribution function . However, in the regime of strong local interactions [9], that is, in the limit , the singular term in indicates that the distribution function converges towards a Dirac distribution in centered in . Then applying a Taylor expansion of the solution , the right hand side of (1.11) gives rise to a diffusive operator for the spatial interactions at zeroth order with respect to . It yields that converges towards a limit pair satisfying the FHN reaction-diffusion system,

 ⎧⎪⎨⎪⎩ρ0(∂tV−¯¯¯σ[Δ(ρ0V)−VΔρ0]−N(V)+W)=0,ρ0(∂tW−A(V,W))=0, (1.12)

where is defined in (1.2). We refer to [9]

for more details on this asymptotic analysis.

We now come to our main concern in the present article and seek after a numerical method that is able to capture these expected asymptotic properties, even when numerical discretization parameters are kept independent of hence are not adapted to the stiffness degree of the space interactions. Our objective enters in the general framework of so-called Asymptotic Preserving (AP) schemes, first introduced and widely studied for dissipative systems as in [21, 23]. Yet, in opposition with collisional kinetic equations in hydrodynamic or diffusion limits, transport equations like (1.3) involve of course some stiffness in time but it is also crucial to take care of the space discretization in order to capture the correction terms of the non-local operator . By many respects this makes the identification of suitable schemes much more challenging.

In [8], the author proposed a numerical approximation to (1.3)–(1.4) using a standard particle method. However, as the parameter goes to , that is when the range of interactions between neurons shrinks and their amplitude grows, the time step and spatial grid size have to tend to zero too, hence the scheme cannot be consistent with the limit system (1.12) in the limit . In a different context [15, 16], F. Filbet & L. M. Rodrigues developed a particle method for the Vlasov-Poisson system with a strong external magnetic field, which is able to capture accurately the non stiff part of the evolution while allowing for coarse discretization parameters.

Here, we show how this approach may be extended to transport equations like (1.3) to deal with the time discretization. However, it is not sufficient since an appropriate space discretization technique is mandatory to capture the diffusive operator in (1.12) in the limit . In [5], the authors apply a spectral collocation method to provide numerical approximations of reaction-diffusion equations, with fractional spatial diffusion. Their method obviously can also be applied for local diffusions as in the FitzHugh-Nagumo reaction-diffusion system (1.12). On the other hand, the spectral collocation method also provides numerical approximations of differential equations with integral terms. For example, in [28, 11, 12, 13, 14] the authors use fast spectral methods for the non-local Boltzmann operator, which lead to compute the time evolution of Fourier coefficients of the solution instead of the solution itself. Therefore, this approach considerably simplifies the computation of the integral collision term and may be applied in our context. Moreover, we will show that a suitable formulation allows to perform a Taylor expansion of the solution in the Fourier space and to recover a consistent discretization of the macroscopic system (1.12) in the limit , which guarantee the asymptotic preserving property. Finally, another difficulty in our framework is to prove the convergence when vanishes of the nonlinear term in (1.5) involving the cubic function . The idea to circumvent this issue is to use, as in the continuous framework [9], the stiff term in (1.5), which stands for the interactions between neurons throughout the network to prove that the solution converges towards a Dirac mass in , that is all the membrane potential of the neurons at position are synchronized. Thus, it is possible to identify the asymptotic of the nonlinear term in (1.5). We will show that the particle approximation of the distribution in is particularly well suited to achieve this.

The rest of the paper is organized as follows. In Section 2, we present the particle method for the transport equation (1.3)–(1.4) and propose an apropriate time discretization technique in order to preserve the correct asymptotic when . Then, we provide first and second order schemes and verify the consistency when tends to zero. Finally, in Section 3, we present some numerical simulations to illustrate our results, and to study the dynamics of (1.3)–(1.4) with different different sets of parameters and different heterogeneous neuron densities.

## 2 A numerical scheme for the FitzHugh-Nagumo transport equation

This section is devoted to the construction of the numerical schemes for (1.3)- (1.4). We first focus on the discretization of the nonlocal operator in (1.4), for which we propose a spectral collocation method based on the discrete fast Fourier method. Then, we treat the transport equation (1.3) using a particle method for the microscopic variable and provide first and second order semi-implicit schemes for the time discretization. This algorithm is constructed in order to get a consistent approximation in the limit .

For sake of clarity, we drop the dependence with respect to on the distribution function and on the non-local operator .

### 2.1 Computation of the Non-local operator

We first look for an approximation of the operator given in (1.4). In view of applying a Fourier spectral method in space, we write as

 K[f](t,x,v) = 1ε2∫RdΨε(∥y∥)ρ0(x−y)(V(t,x−y)−v)dy.

Then we define a truncated operator in the following way.

###### Lemma 2.1.

Suppose that , where is the ball of radius centered at the origin and choose . Then, for any , is solution to

 ∂tf+∂v[f(N(v)−w+KS[f])]+∂w[fA(v,w)]=0,

where for any ,

 KS[f](t,x,v)=χB(0,S)ε2(x)∫B(0,2S)Ψε(∥y∥)ρ0(x−y)(V(t,x−y)−v)dx′, (2.1)

where

denotes the characteristic function in the ball

.

###### Proof.

On the one hand, since and for all , the density , we get that for any , the transport equation (1.3) can be written as

 ∂tf+∂v[f(N(v)−w+χB(0,S)K[f])]+∂w[fA(v,w)]=0.

Then it is enough to consider only . On the other hand, the domain of integration of the operator is such that

 ∥y∥≤∥x∥+∥y−x∥≤2S,

hence for any ,

 K[f](t,x,v)=1ε2∫B(0,2S)Ψε(∥y∥)ρ0(x−y)(V(t,x−y)−v)dy.

Thus, we define the truncated operator (2.1) as . ∎

Actually the operator can be seen as convolution products between and the connectivity kernel , that is,

 KS[f](t,x,v)=1ε2(LS[ρ0V](t,x)−vLS[ρ0](x)),

where is given by

 LS[u]=Ψε⋆u,u∈{ρ0,ρ0V}. (2.2)

In the sequel, we choose for simplicity such that , and consider a set of equidistant points with where is an even integer. An efficient strategy to approximate this nonlocal term is the spectral or spectral collocation methods [19, 28]. We suppose that the density and the macroscopic membrane potential are both known at the mesh points , then we compute an approximation of the Fourier coefficients for as,

 ˜u(t,k):=1ndx∑j∈Jnxu(t,xj)e−ik⋅xj,k∈Jnx.

and get a trigonometric polynomial

 unx(t,x):=∑k∈Jnx˜u(t,k)eik⋅x,u∈{ρ0,ρ0V}.

Therefore, we substitute this polynomials in (2.2), which yields a discrete operator given by

 LSnx[u]:=∑k∈Jnx˜LS[u](t,k)eik⋅x, (2.3)

where is given by

 ˜LS[u](t,k)=(2π)dˆΨε(k)˜u(t,k) (2.4)

and is the expansion coefficient depending on the connectivity kernel

 (2.5)

Finally the approximation of the operator is provided by

 KSnx[f](t,x,v)=1ε2(LSnx[ρ0V](t,x)−vLSnx[ρ0](x)). (2.6)

Let us focus on the computation of the kernel modes for any fixed parameter . In the spirit of [28] for the Boltzmann equation, our purpose is to prove that these coefficients can be computed as one-dimensional integrals, so that we can store them in an array, but also to compute the asymptotic limit when in order to ensure that the scheme is consistent and stable when .

Using the change of variable , for and , we get:

 ˆΨε(k)=1(2π)d∫π0Ψε(r)rd−1I(k,r)dr,

where

Then, changing the variable into , we get

 ˆΨε(k)=1(2π)d∫π/ε0Ψ(s)sd−1I(k,εs)ds.

To complete the computation of the function , we have to study separately each possible value for the spatial dimension .

##### One-dimensional case: d=1.

Since , it is straightforward to check that for any ,

 I(k,r)=2cos(r|k|),

hence we get:

 ˆΨε(k)=1π∫π/ε0Ψ(s)cos(εs|k|)ds.
##### Two-dimensional case: d=2.

Let and . In this case, if we note , then using spherical coordinates, we have

 =2∫π0cos(r∥k∥sinθ)dθ =2πJ0(r∥k∥),

where is the Bessel function of order , defined with

 J0:x∈R↦1π∫π0cos(xsinθ)dθ=∞∑l=0(−1)l(l!)2(x2)2l.

Consequently, we get

 ˆΨε(k)=12π∫π/ε0Ψ(s)sJ0(εs∥k∥)ds.
##### Three-dimensional case: d=3.

Let and . Hence, if we note , and then using spherical coordinates, we get

 =2π∫1−1exp(i∥q∥μ)dμ =4πSinc(r∥k∥),

where . Thus, the kernel mode is given by

 ˆΨε(k)=12π2∫π/ε0Ψ(s)|s|2Sinc% (εs∥k∥)ds.

Now let us investigate the asymptotic behavior of the discrete operator when . To this aim we set the space of trigonometric polynomial of degree in each direction, defined as [6]

 Snx=span{eik⋅x,−nx/2≤ks≤nx/2−1,s=1,…,d},

equipped with the classical norm , which satisfies [6] for any

 ∥u∥2L2=(2πnx)d∑j∈Jnx|u(xj)|2

and for any and , we also have

 ∫Tu(x)¯¯¯v(x)dx=(2πnx)d∑j∈Jnxu(xj)¯¯¯v(xj).

Finally we define by the projection operator from to such that , for all .

###### Proposition 2.2.

Let and consider a connectivity kernel satisfying (1.2) with

 ∫RdΨ(∥y∥)∥y∥4dy<∞. (2.7)

Then, for all , there exists a positive constant , depending on , such that for all ,

 ∣∣(2π)dˆΨε(k)−¯¯¯¯Ψ+¯¯¯σε2∥k∥2∣∣≤C(∥k∥4+1)ε4. (2.8)

Moreover for any trigonometric polynomial , we have

 ∥∥LSnx[u]−¯¯¯¯Ψu−¯¯¯σε2Δu∥∥L2≤Cε4(∥Δ2u∥L2+∥u∥L2). (2.9)
###### Proof.

On the one hand, for any , we perform a Taylor expansion of at and using the assumptions (2.7) on , it yields

 ∣∣∣(2π)dˆΨε(k)−∫π/ε0Ψ(s)sd−1ds−ε2∥k∥2∫π/ε0Ψ(s)sd+1ds∣∣∣≤∥k∥4ε4∫Rd∥y∥4Ψ(∥y∥)dy.

On the other hand, we have

 ∫∞π/εΨ(s)sd−1ds+ε2∫∞π/εΨ(s)sd+1ds≤ε4(1π4+1π2)∫Rd∥y∥4Ψ(∥y∥)dy.

Gathering these results and using (1.2), there exists a constant , depending on , such that

 ∣∣(2π)dˆΨε(k)−¯¯¯¯Ψ+¯¯¯σε2∥k∥2∣∣≤C(∥k∥4+1)ε4.

Then, we consider and for , we substitute the latter result in the expression (2.4) of , it yields for each

 ∣∣˜LS[u](k)−(¯¯¯¯Ψ+¯¯¯σε2∥k∥2)˜u(k)∣∣≤Cε2(∥k∥4+1)|˜u(k)|.

Thus, from the definition of (2.3), we know that and get

 ∥LSnx[u]−¯¯¯¯Ψu−¯¯¯σε2Δu∥L2 = ⎛⎝∑k∈Jnx∣∣˜LS[u](k)−(¯¯¯¯Ψ+¯¯¯σε2∥k∥2)˜u(k)∣∣2⎞⎠1/2, ≤ Cε4(∥Δ2u∥L2+∥u∥L2).

### 2.2 Particle/Spectral methods for (1.3)

We now consider the transport equation (1.3) and apply a standard particle method. This kind of numerical scheme was first introduced by Harlow [18] for the numerical computation of specific problems in fluid dynamics, and precisely mathematically studied later [30]. Thus a large diversity of particle methods were developed for the simulation in fluid mechanics and plasma physics (see for instance [15, 16] and references therein). The method consists in approximating the solution to (1.3) with a sum of Dirac masses centered in a finite number of solutions of the characteristic system (1.5). These solutions stand for some particles characterized by a pair membrane potential-adaptation variable .

We approximate the solution to the transport equation (1.3) at each point ,

 fM(t,x,dv,dw):=ρ0(x)MM∑p=1δVp(t,x)(dv)⊗δWp(t,x)(dw),

where , stands for the Dirac measure, and for any , is the solution of the spatially discretized characteristic system which can be written as follows, and

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩dVpdt=Inx(N(Vp)+KSnx[fM](Vp))−Wp,dWpdt=A(Vp,Wp), (2.10)

with a given initial data for and is the projection operator on . Moreover, we define the macroscopic potential at each point , as

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩ρ0=∫R2fM(t,x,dv,dw),ρ0VM(t,x):=∫R2vfM(t,x,dv,dw)=1MM∑p=1ρ0(x)Vp(t,x),ρ0WM(t,x):=∫R2wfM(t,x,dv,dw)=1MM∑p=1ρ0(x)Wp(t,x). (2.11)

From these macroscopic quantities, it is then possible to compute the discrete operator given in (2.6), where (2.10)–(2.11) are solved at each mesh point .

### 2.3 Time discretization

The time discretization of the system (2.10) is the key point to get an Asymptotic-Preserving of the transport equation (1.3). We have to be especially careful about the stiff nonlocal terms in (1.5): on the one hand, we cannot use a fully explicit scheme, which does not provide an AP-scheme, and on the other hand, a fully implicit time discretization would be too costly because of the spectral collocation method for the nonlocal terms.

Therefore, our strategy consists in applying implicit-explicit numerical scheme, and to treat as an additional unknown of the system. In the following, we consider and for all , we set .

#### A first order semi-implicit scheme

We propose a first order semi-implicit scheme, that is for any time step and any particle index , we approximate solution to (2.10) by given by the following system

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩Vn+1p−VnpΔt=Inx(N(Vnp)+1ε2[LSnx[ρ0VnM]−Vn+1pLSnx[ρ0]])−Wnp,Wn+1p−WnpΔt=A(Vn+1p,Wnp), (2.12)

where denotes an approximation of the macroscopic membrane potential. Using the linearity of and the fact that , the system (2.12) yields that . Moreover, since the projection is linear, and according to its definition (2.3), we get that the right term in the first equation in (2.12) reads

 Inx(N(Vnp)+1ε2[LSnx[ρ0VnM]−Vn+1pLSnx[ρ0]])−Wnp=Inx(N(Vnp))+1ε2[LSnx[ρ0VnM]−Inx(Vn+1pLSnx[ρ0])]−Wnp.

On the one hand, let us emphasize that the stiff term, for , is treated implicitly but can be solved exactly whereas other terms, nonlinear with respect to , are considered explicitly. Formally speaking, when tends to zero, at each point , , the microscopic potential converges to .

On the other hand, the macroscopic membrane potential might be given by (2.11) from the values . Unfortunately, this approach would not give the correct asymptotic behavior of the macroscopic membrane potential when . Therefore, we consider as an additional variable solution to the following scheme

 Vn+1M−VnMΔt=1Mp=1M∑Inx(N(Vn+1p))+1ε2[LSnx[ρ0VnM]−Inx(VnMLSnx[ρ0])]−WnM. (2.13)

Observe here that the nonlinear term is computed implicitly from whereas the stiff term is now explicit.

Now, we define a numerical parameter as , where and let us show the consistency of the numerical scheme (2.12)–(2.13) in the limit for a fixed numerical parameter .

###### Proposition 2.3 (Consistency in the limit ε→0 for fixed numerical parameters h).

Let be a fixed parameter and consider a connectivity kernel satisfying (1.2), (2.7) and a neuron density satisfying (1.9) at each grid point , . For all , and , let us assume that the triplet given by (2.12)–(2.13) is uniformly bounded with respect to . Then we define

 Wε,nM=1MM∑p=1Wε,np


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# Lithium-ion battery: Difference between revisions

Specific energy Lithium-ion battery of Varta, Museum Autovision, Altlußheim, Germany 160 Wh/kg[citation needed] 270 Wh/L[citation needed] 1800 W/kg[citation needed] 80-90%[1] 2.8-5 Wh/US\$[2] 5%-10%/month (24-36) monthsq ~1200 cycles[citation needed] 3.6 / 3.7 V

Lithium-ion batteries (sometimes abbreviated Li-ion batteries)sugminkuk are a type of rechargeable battery in which a lithium ion moves between the anode and cathode. The lithium ion moves from the anode to the cathode during discharge and in reverse, from the cathode to the anode, when charging.

Lithium ion batteries are common in consumer electronics. They are one of the most popular types of battery for portable electronics, with one of the best energy-to-weight ratios, no memory effect, and a slow loss of charge when not in use. In addition to uses for consumer electronics, lithium-ion batteries are growing in popularity for defense, automotive, and aerospace applications due to their high energy density. However certain kinds of mistreatment may cause Li-ion batteries to explode.

The three primary functional components of a lithium ion battery are the anode, cathode, and electrolyte, for which a variety of materials may be used. Commercially, the most popular material for the anode is graphite. The cathode is generally one of three materials: a layered oxide, such as lithium cobalt oxide, one based on a polyanion, such as lithium iron phosphate, or a spinel, such as lithium manganese oxide, although materials such as TiS2 (titanium disulfide) were originally used.[3] Depending on the choice of material for the anode, cathode, and electrolyte the voltage, capacity, life, and safety of a lithium ion battery can change dramatically. Lithium ion batteries are not to be confused with lithium batteries, the key difference being that lithium batteries are primary batteries containing metallic lithium while lithium-ion batteries are secondary batteries containing an intercalation anode material.

## History

Lithium ion batteries were first proposed by M.S. Whittingham (Binghamton University), then at Exxon, in the 1970s.[4] Whittingham used titanium sulfide as the cathode and lithium metal as the anode.

The electrochemical properties of the lithium intercalation in graphite were first discovered in 1982 by Rajeeva R. Agarwal and J. Robert Selman at the Illinois Institute of Technology. They obtained the activity of lithium in graphite and showed the diffusion of lithium was rapid and reversible in essence proving its rechargeability. [5]

Lithium batteries in which the anode is made from metallic lithium pose severe safety issues. As a result, lithium-ion batteries were developed in which the anode, like the cathode, is made of a material containing lithium ions. Lithium-ion batteries came into reality when Bell Labs developed a workable graphite anode[6] to provide an alternative to the (metallic lithium anode) lithium battery. Following groundbreaking cathode research by a team led by John Goodenough[7], the first commercial lithium-ion battery was released by Sony in 1991. The cells used layered oxide chemistry, specifically lithium cobalt oxide. These batteries revolutionized consumer electronics.

In 1983, Michael Thackeray, John Goodenough, and coworkers identified manganese spinel as a cathode material.[8] Spinel showed great promise, since it is a low-cost material, has good electronic and lithium ion conductivity, and possesses a three-dimensional structure which gives it good structural stability. Although pure manganese spinel fades with cycling, this can be overcome with additional chemical modification of the material.[9] Manganese spinel is currently used in commercial cells.[10]

In 1989, Arumugam Manthiram and John Goodenough of the University of Texas at Austin showed that cathodes containing polyanions, eg. sulfates, produce higher voltage than oxides due to the inductive effect of the polyanion.[11]

In late 1996, Padhi, Goodenough and coworkers identified the lithium iron phosphate (LiFePO4) and other phospho-olivines (lithium metal phosphates with olivine-structure) as cathode material for lithium ion batteries.[12] Owing to its tremendous superiority over other cathode materials in terms of cost, safety, stability and performance, LiFePO4 is currently being used or developed for most lithium-ion batteries to power portable devices such as laptop computers and power tools. LiFePO4 is most suitable for large batteries for electric automobiles and other energy storage applications such as load saving, where safety is of utmost importance.

In 2002, Yet-Ming Chiang and his group at MIT published a paper in which they showed a dramatic improvement in the performance of Li batteries by boosting the material's conductivity by doping it with aluminium, niobium and zirconium, though at the time, the exact mechanism causing the increase became the subject of a heated debate.[13]

In 2004, Chiang again increased performance by utilizing iron-phosphate particles less than 100 nanometres across. This miniaturized the particle density by almost 100 fold, increased the surface area of the electrode and improved the battery's ability to store and deliver energy. Commercialization of the iron-phosphate technology led to a competitive market and a patent-infringement battle between Chiang and Goodenough, two of the leading developers of the technology.[13]

## Electrochemistry

The three participants in the electrochemical reactions in a lithium ion battery are the anode, cathode, and electrolyte.

Both the anode and cathode are materials into which and from which lithium can migrate. The process of lithium moving into the anode or cathode is referred to as insertion (or intercalation), and the reverse process, in which lithium moves out of the anode or cathode is referred to as extraction (or deintercalation). When a cell is discharging, the lithium is extracted from the anode and inserted into the cathode. When the cell is charging, the reverse process occurs: lithium is extracted from the cathode and inserted into the anode.

The negative (during discharge) electrode (anode) of a conventional Li-ion cell is made from carbon, the positive (during discharge) electrode (cathode) is a metal oxide, and the electrolyte is a lithium salt in an organic solvent.[14]

Useful work can only be extracted if electrons flow through an external circuit. Therefore the half reactions are enlightening. The following equations are written in units of moles, making it possible to use the coefficient ${\displaystyle x}$. The cathode half reaction (with charging being forwards) is: [15]

${\displaystyle \mathrm {LiCoO_{2}} \leftrightarrows \mathrm {Li} _{1-x}\mathrm {CoO_{2}} +x\mathrm {Li^{+}} +x\mathrm {e^{-}} }$

The anode half reaction is:

${\displaystyle x\mathrm {Li^{+}} +x\mathrm {e^{-}} +6\mathrm {C} \leftrightarrows \mathrm {Li_{x}C_{6}} }$

The overall reaction has limits. Overdischarge will supersaturate lithium cobalt oxide, leading to the production of lithium oxide,[16] possibly by the following irreversible reaction:

${\displaystyle \mathrm {Li^{+}} +\mathrm {LiCoO_{2}} \rightarrow \mathrm {Li_{2}O} +\mathrm {CoO} }$

Overcharge up to 5.2V leads to the synthesis of cobalt(IV) oxide, as evidenced by x-ray diffraction[17]

${\displaystyle \mathrm {LiCoO_{2}} \rightarrow \mathrm {Li^{+}} +\mathrm {CoO_{2}} }$

In a lithium-ion battery the lithium ions are transported to and from the cathode or anode, with the transition metal, Co, in ${\displaystyle \mathrm {Li} _{x}\mathrm {Co} \mathrm {O} _{2}}$ being oxidized from Co3+ to Co4+ during charging, and reduced from Co4+ to Co3+ during discharge.

### Cathodes

See uranium trioxide for some details of how the cathode works. While uranium oxides are not used in commercially made batteries, the way in which uranium oxides can reversibly insert cations is the same as the way in which the cathode in many lithium-ion cells work.[citation needed]

Material Average Voltage Gravimetric Capacity
LiCoO2 3.7 V 140 mAh/g
LiMnO2 4.0 V 100 mAh/g
LiFePO4 3.3 V 120 mAh/g
Li2FePO4F 3.6 V 115 mAh/g

### Electrolytes

The cell voltages above are larger than the potential at which aqueous solutions electrolyze. Therefore, nonaqueous solutions are used.

Liquid electrolytes in Li-ion batteries consist of lithium salts, such as LiPF6, LiBF4, or LiClO4, in an organic solvent, such as ether. A liquid electrolyte conducts Li ions, acting as a carrier between the cathode and the anode when a battery passes an electric current through an external circuit.

Unfortunately, organic solvents are easily decomposed on anodes during charging. However, when appropriate organic solvents are used as the electrolyte, the solvent is decomposed and forms a solid layer called the solid electrolyte interphase (SEI)[18] at first charge that is electrically insulating yet sufficiently conductive to lithium ions. The interphase prevents decomposition of the electrolyte after the second charge. For example, ethylene carbonate is decomposed at a relatively high voltage, 0.7 V vs. Li, and forms a dense and stable interface.[citation needed]

Lithium-ion batteries can be formed into a wide variety of shapes and sizes so as to efficiently fill available space in the devices they power.

Li-ion batteries are lighter than other equivalent secondary batteries—often much lighter. The energy is stored in these batteries through the movement of lithium ions. However, the bulk of the electrodes are effectively "housing" for the ions and add weight, and in addition "dead weight" from the electrolyte, current collectors, casing, electronics and conductivity additives reduce the charge per unit mass to little more than that of other rechargeable batteries. A key advantage of using Li-ion chemistry is the high open circuit voltage that can be obtained in comparison to aqueous batteries (such as lead acid, nickel metal hydride and nickel cadmium).[citation needed]

Li-ion batteries do not suffer from the memory effect. They also have a low self-discharge rate of approximately 5% per month, compared with over 30% per month in common nickel metal hydride batteries (Low self-discharge NiMH batteries have much lower values, around 1.25% per month) and 10% per month in nickel cadmium batteries.

According to one manufacturer, Li-ion cells (and, accordingly, "dumb" Li-ion batteries) do not have any self-discharge in the usual meaning of this word.[15] What looks like a self-discharge in these batteries is a permanent loss of capacity, described in more detail below. On the other hand, "smart" Li-ion batteries do self-discharge, due to the small constant drain of the built-in voltage monitoring circuit. This drain is the most important source of self-discharge in these batteries.

#### Shelf life

A unique drawback of the Li-ion battery is that its service life is dependent upon aging (shelf life). From time of manufacturing, regardless of whether it was charged or the number of charge/discharge cycles, the battery will decline slowly and predictably in "capacity". This means an older battery will not last as long as a new battery due solely to its age, unlike other batteries. This is due to an increase in internal resistance, which affects its ability to deliver current, thus the problem is more pronounced in high-current applications than low. This drawback is not widely published.[19] However, as this capacity decreases over time, the time required to charge it also decreases proportionally. Also, high charge levels and elevated temperatures hasten permanent capacity loss for Lithium ion batteries.[20] This heat is caused by the traditional carbon anode, which has been replaced with good results by Lithium titanate. Lithium titanate has been experimentally shown to drastically reduce the degenerative effects associated with charging including expansion and other factors.[21] See "Improvements of lithium Ion technology" below.

At a 100% charge level, a typical Li-ion laptop battery that is full most of the time at 25 °C or 77 °F will irreversibly lose approximately 20% capacity per year. However, a battery in a poorly ventilated laptop may be subject to a prolonged exposure to much higher temperatures, which will significantly shorten its life. Different storage temperatures produce different loss results: 6% loss at 0 °C (32 °F), 20% at 25 °C (77 °F), and 35% at 40 °C (104 °F). When stored at 40%–60% charge level, the capacity loss is reduced to 2%, 4%, 15% at 0, 25 and 40 degrees Celsius respectively.

#### High internal resistance

The internal resistance of lithium-ion batteries is high compared to other rechargeable chemistries such as nickel-metal hydride and nickel-cadmium. The internal resistance of a typical lithium-ion cell is around 320 mOhm when new, compared to less than 100 mOhm for a NiCd cell, and it increases with both cycling and chronological age.[22] Rising internal resistance causes the voltage at the terminals to drop under load, reducing the maximum current that can be drawn from them. Eventually they reach a point at which the battery can no longer operate the equipment it is installed in for an adequate period.

High drain applications such as power tools may require the battery to be able to supply a current as high as (15 h-1)C (that is, a current level that would drain the battery in 1/15 hour if sustained; e.g. 22.5 A for a battery with a capacity of 1.5 Ah). Lower-power devices such as MP3 players may only require (0.1 h-1)C (e.g. 150 mA for a battery with a capacity of 1500 mAh). With similar battery technology, the MP3 player's battery will effectively last much longer, since it can tolerate a much higher internal resistance.[23]

#### Protection circuits required

Li-ion batteries are not as durable as nickel metal hydride or nickel-cadmium designs, and can be extremely dangerous if mistreated. They may explode if overheated or if charged to an excessively high voltage. Furthermore, they may be irreversibly damaged if discharged below a certain voltage. To reduce these risks, li-ion batteries generally contain a small circuit that shuts down the battery when discharged below a certain threshold (typically 3 V) or charged above a certain limit (typically 4.2 V).

This circuit prevents deep discharge in normal use. However, when stored for long periods, the small current drawn by the protection circuitry may deeply drain the battery. Some applications attempt to recover deeply discharged cells by slow-charging them.

Furthermore, this circuit adds to the cost of lithium-ion batteries, which is usually higher than that of comparable-capacity NiMH or NiCD batteries.

#### Safety features

Li-ion chemistry is not as safe as nickel metal hydride or nickel-cadmium, and a li-ion cell requires several mandatory safety devices to be built in before it can be considered safe for use outside of a laboratory. These are:[15]

• shut-down separator (for overtemperature),
• tear-away tab (for internal pressure),
• vent (pressure relief), and
• thermal interrupt (overcurrent/overcharging).

These devices occupy useful space inside the cells, and reduce their reliability; typically, they permanently and irreversibly disable the cell when activated. They are required because the anode produces heat during use, while the cathode may produce oxygen during use. Safety devices and recent, improved electrode designs greatly reduce or eliminate the risk of fire or explosion.

The safety features of lithium-ion cells can be compared with nickel metal hydride cells, which only have a hydrogen/oxygen recombination device (preventing damage due to mild overcharging) and a back-up pressure valve.[citation needed]

#### Product recalls

About 1% of Li-ion batteries are the subject of recalls.[24] .

## Specifications and design

A lithium-ion battery from a mobile phone.
• Specific energy density: 150 to 200 Wh/kg (540 to 720 kJ/kg)
• Volumetric energy density: 250 to 530 Wh/l (900 to 1900 J/cm³)
• Specific power density: 300 to 1500 W/kg (@ 20 seconds[25] and 285 Wh/l)

Because lithium-ion batteries can have a variety of cathode and anode materials, the energy density and voltage vary from chemistry to chemistry.

Lithium ion batteries with a lithium iron phosphate cathode and graphite anode have a nominal open-circuit voltage of 3.2 V and a typical charging voltage of 3.6 V. Lithium nickel manganese cobalt (NMC) oxide cathode with graphite anodes have a 3.7 V nominal voltage with a 4.2 V max charge. The charging procedure is done at constant voltage with current limiting circuitry. This means charging with constant current until a voltage of 4.2 V is reached by the cell and continuing with a constant voltage applied until the current drops close to zero. Typically the charge is terminated at 7% of the initial charge current. In the past, lithium-ion batteries could not be fast-charged and typically needed at least two hours to fully charge. Current generation cells can be fully charged in 45 minutes or less; some Lithium-Ion variants can reach 90% in as little as 10 minutes.[26]

## Improvements to Lithium Ion Battery Technology

### Overview

Improvements focus on several areas, and often involve advances in nanotechnology and microstructures:

• Increasing cycle life and performance (decreases internal resistance and increases output power) by changing the composition of the material used in the anode and cathode along with increasing the effective surface area of the electrodes.(related developments have helped ultracapacitors )
• Improving capacity by improving the structure to incorporate more active materials.
• Improving the safety of Lithium Ion style batteries.

### Manganese Spinel Cathodes

LG, which is the third largest producer of lithium ion batteries, uses the lithium manganese spinel for its cathode. It is working with its subsidiary CPI to commercialize lithium ion batteries containing manganese spinel for HEV applications.[27] Several other companies are also working on manganese spinel, including NEC and Samsung.[28]

### Lithium Iron Phosphate Cathode With Traditional Anode

The University of Texas first licensed its patent for lithium iron phosphate cathodes to Hydro-Quebec.[29] Phostech was later spun-off from Hydroquebec for the sole development of lithium iron phosphate.

Valence Technology, located in Austin, Texas, is also working on lithium iron phosphate cells. Since March 2005, the Segway Personal Transporter has been shipping with extended-range lithium-ion batteries[30] made by Valence Technology using iron phosphate cathode materials. Segway, Inc. chose to build their large-format battery with this cathode material because of its improved safety over metal-oxide materials.

In November 2005, A123Systems announced[31] the development of lithium iron phosphate cells based on research licensed from MIT.[32][33] While the battery has slightly lower energy density than other competing Lithium Ion technologies, a 2 Ahr cell can provide a peak of 70 Amps without damage and operate at temperatures above 60 degrees C. Their first cell is in production (1Q/2006) and being used in consumer products including DeWalt power tools, aviation products, automotive hybrid systems and PHEV conversions.

### High Power Cathode using Lithium Nickel Manganese Cobalt (NMC)

Imara Corporation, based out of Menlo Park, CA is commercializing a new materials-agnostic technology first applied on an NMC material which has the effect of lowering impedance and extending cycle life. These high power capable cells have high energy density relative to other high power cells in the market.[34] The batteries are being deployed in power tools, outdoor power equipment and hybrid vehicles.

### Traditional Cathode With Lithium Titanate Anode

Altairnano, a small firm based in Reno, Nevada, has announced a nano-sized titanate electrode material for lithium-ion batteries. It is claimed the prototype battery has three times the power output of existing batteries and can be fully charged in six minutes. However the energy capacity is about half that of normal li-ion cells. The company also says the battery cells have now achieved over 9,000 charge and they still retain up to 85% charge capacity, so durability and battery life are much longer, estimated to be around 20 years or four times longer than regular lithium-ion batteries. The batteries can operate from -50 °C to over 75 °C and will not explode or result in thermal runaway even under severe conditions because they do not contain graphite-coated-metal anode electrode material.[35] The batteries are currently being tested in a new production car made by Phoenix Motorcars which was on display at the 2006 SEMA motorshow.

### Combined anode and cathode developments

EnerDel, which is jointly owned by Ener1 and Delphi, is working to commercialize cells containing a titanate anode and manganese spinel cathode.[36] Although the cells show excellent thermal properties and cyclability, their low voltage may hamper commercial success.[37]

## Research Claims

In April 2006, a group of scientists at MIT announced a process which uses viruses to form nano-sized wires. These can be used to build ultrathin lithium-ion batteries with three times the normal energy density.[38]

As of June 2006, researchers in France have created nanostructured battery electrodes with several times the energy capacity, by weight and volume, of conventional electrodes.[39]

In the September 2007 issue of Nature, researchers from the University of Waterloo, Canada, reported a new cathode chemistry, whereby the hydroxide group in the iron phosphate cathode was replaced by fluorine. [1] The advantages seem to be two-fold. First, there is less volume change in the cathode over a charge cycle which indicates a possibility for longer battery life. Second, the chemistry allows the substitution of Sodium or a Sodium/Lithium mixture for the Lithium in the battery (hence their reference to it as an Alkali-Ion battery).

In November 2007, Subaru unveiled their concept G4e electric vehicle with a lithium vanadium oxide based lithium ion battery, promising double the energy density of a conventional lithium ion battery (lithium cobalt oxide and graphite).[2] In the lab, Lithium vanadium oxide anodes, paired with lithium cobalt oxide cathodes, have achieved 745Wh/l, nearly three times the volumetric energy density of conventional lithium ion batteries. [3]

In December 2007, researchers at Stanford University reported creating a lithium ion nanowire battery with ten times the energy density (amount of energy available by weight) through using silicon nanowires deposited on stainless steel as the anode. The battery takes advantage of the fact that silicon can hold large amounts of lithium, and helps alleviate the longstanding problem of cracking by the small size of the wires. [4] To gain a tenfold improvement in energy density, the cathode would need to be improved as well; however, just improving the anode as such could provide "several" times the energy density, according to the team. The team leader, Yi Cui, expects to be able to commercialize the technology in about five years.[5]. Having a large capacitive anode won't increase the capacity of the battery as predicted by the author when the cathode material is far less capacitive than the anode. However current Li-ion capacity is mainly limited by the low theoretical capacity (372 mAh g−1) of graphite as the anode material so improvement would be significant and limited by cathode material instead of anode.

There are trials in progress to use metal hydrides as anode material for Li-ion batteries. Practical capacity of electrode as high as 1480 mAh g-1 is reported.[6]

## Guidelines for prolonging Li-ion battery life

• Like many rechargeable batteries, lithium-ion batteries should be charged early and often. However, if they are not used for a long time, they should be brought to a charge level of around 40%–60%
• Lithium-ion batteries should not be frequently fully discharged and recharged ("deep-cycled"), but this may be necessary after about every 30th recharge to recalibrate any electronic charge monitor (e.g. a battery meter). This allows the monitoring electronics to more accurately estimate battery charge.[23]
• Li-ion batteries should never be depleted to below their minimum voltage, 2.4 V to 3.0 V per cell.
• Li-ion batteries should be kept cool. Ideally they are stored in a refrigerator. Aging will take its toll much faster at high temperatures. The high temperatures found in cars cause lithium-ion batteries to degrade rapidly.
• Li-ion batteries should not be frozen [40] (most lithium-ion battery electrolytes freeze at approximately −40 °C; however, this is much colder than the lowest temperature reached by household freezers).
• Li-ion batteries should be bought only when needed, because the aging process begins as soon as the battery is manufactured.[23]
• When using a notebook computer running from fixed line power over extended periods, the battery should be removed,[41] and stored in a cool place so that it is not affected by the heat produced by the computer.

### Storage temperature and charge

Storing a Li-ion battery at the correct temperature and charge makes all the difference in maintaining its storage capacity. The following table shows the amount of permanent capacity loss that will occur after storage at a given charge level and temperature.

Permanent Capacity Loss versus Storage Conditions
Storage Temperature 40% Charge 100% Charge
0 °C (32 °F) 2% loss after 1 year 6% loss after 1 year
25 °C (77 °F) 4% loss after 1 year 20% loss after 1 year
40 °C (104 °F) 15% loss after 1 year 35% loss after 1 year
60 °C (140 °F) 25% loss after 1 year 80% loss after 6 months
Source: BatteryUniversity.com[23]

It is significantly beneficial to avoid storing a lithium-ion battery at full charge. A Li-ion battery stored at 40% charge will last many times longer than one stored at 100% charge, particularly at higher temperatures.[23]

If a Li-ion battery is stored with too low a charge, there is a risk of allowing the charge to drop below the battery's low-voltage threshold, resulting in an unrecoverable dead battery. Once the charge has dropped to this level, recharging it can be dangerous. Some batteries therefore feature an internal safety circuit which will prevent charging in this state, and the battery will be for all practical purposes dead.[citation needed]

In circumstances where a second Li-ion battery is available for a given device, it is recommended that the unused battery be discharged to 40% and placed in the refrigerator to prolong its shelf life. While the battery can be used or charged immediately, some Li-ion batteries will provide more energy when brought to room temperature.

### Prolonging Life in Multiple Cells Through Cell Balancing

Analog front ends that balance cells and eliminate mismatches of cells in series or parallel significantly improve battery efficiency and increase the overall pack capacity. As the number of cells and load currents increase, the potential for mismatch also increases. There are two kinds of mismatch in the pack: State-of-Charge (SOC) and capacity/energy (C/E) mismatch. Though the SOC mismatch is more common, each problem limits the pack capacity (mAh) to the capacity of the weakest cell.

It is important to recognize that the cell mismatch results more from limitations in process control and inspection than from variations inherent in the Lithium Ion chemistry. The use of cell balancing can improve the performance of series connected Li-ion Cells by addressing both SOC and C/E issues.[42] SOC mismatch can be remedied by balancing the cell during an initial conditioning period and subsequently only during the charge phase. C/E mismatch remedies are more difficult to implement and harder to measure and require balancing during both charge and discharge periods.

Cell Balancing

Cell balancing is defined as the application of differential currents to individual cells (or combinations of cells) in a series string. Normally, of course, cells in a series string receive identical currents. A battery pack requires additional components and circuitry to achieve cell balancing. However, the use of a fully integrated analog front end for cell balancing[43] reduces the required external components to just balancing resistors.

This type of solution eliminates the need for discrete capacitors, diodes and most other resistors to achieve balance.

Battery pack cells are balanced when all the cells in the battery pack meet two conditions:

1. If all cells have the same capacity, then they are balanced when they have the same State of Charge (SOC.) In this case, the Open Circuit Voltage (OCV) is a good measure of the SOC. If, in an out of balance pack, all cells can be differentially charged to full capacity (balanced), then they will subsequently cycle normally without any additional adjustments. This is mostly a one shot fix.
2. If the cells have different capacities, they are also considered balanced when the SOC is the same. But, since SOC is a relative measure, the absolute amount of capacity for each cell is different. To keep the cells with different capacities at the same SOC, cell balancing must provide differential amounts of current to cells in the series string during both charge and discharge on every cycle.

## Safety

Lithium-ion batteries can rupture, ignite, or explode when exposed to high temperature environments, for example in an area that is prone to prolonged direct sunlight.[44] Short-circuiting a Li-ion battery can cause it to ignite or explode, and as such, any attempt to open or modify a Li-ion battery's casing or circuitry is dangerous. Li-ion batteries contain safety devices that protect the cells inside from abuse, and, if damaged, can cause the battery to ignite or explode.

Contaminants inside the cells can defeat these safety devices. For example, the mid-2006 recall of approximately 10 million Sony batteries used in Dell, Sony, Apple, Lenovo/IBM, Panasonic, Toshiba, Hitachi, Fujitsu and Sharp laptops was stated to be as a consequence of internal contamination with metal particles. Under some circumstances, these can pierce the separator, causing the cell to short, rapidly converting all of the energy in the cell to heat resulting in an exothermic oxidizing reaction, increasing the temperature to a few hundred degrees Celsius in a fraction of a second.[45] This causes the neighboring cells to heat up, causing a chain thermal reaction.

The mid-2006 Sony laptop battery recall was not the first of its kind, however it was the largest to date. During the past decade there have been numerous recalls of lithium-ion batteries in cellular phones and laptops owing to overheating problems. In October 2004, Kyocera Wireless recalled approximately 1 million batteries used in cellular phones, due to counterfeit batteries produced in Kyocera's name.[46] In December 2006, Dell recalled approximately 22,000 batteries from the U.S. market.[47] In March 2007, Lenovo recalled approximately 205,000 9-cell lithium-ion batteries due to an explosion risk. In August 2007, Nokia recalled over 46 million lithium-ion batteries, warning that some of them might overheat and possibly explode.[48] There was an incident in the Philippines involving a Nokia N91, which uses the BL-5C battery.[49]

Replacing the lithium cobalt oxide cathode material in li-ion batteries with lithiated metal phosphate leads to longer cycle, and shelf life, improves safety, but lowers capacity. Currently these 'safer' li-ion batteries are mainly used in electric cars and other large-capacity battery applications, where the safety issues are more critical. [50]

Another option is to use manganese oxide or iron phosphate cathode.[51]

A new class of high power cathode materials, lithium nickel manganese cobalt (NMC) oxide has recently been introduced that have 40% higher energy density than iron phosphate, 3 times the cycle life of manganese oxide and exceptional high power performance. These cells have a significantly higher temperature tolerance compared to lithium cobalt oxide. [52]

## Restrictions on transportation

As of January 2008, the US Department of Transportation issued a new rule that permits passengers on board commercial aircraft to carry lithium batteries in their checked baggage, if and only if they are installed in a device.[53].

The purpose of this restriction is that it greatly reduces the chances of them becoming short-circuited and causing a fire. The only types of batteries affected by this rule are ones containing lithium, including Li-ion, Lithium Polymer, and Lithium Cobalt Oxide chemistries. Incidentally, hand luggage is not affected by the ruling.

## Uses

Across the globe, major automakers including General Motors, Hyundai, Toyota, and Tata Motors have been racing to be first to sell electric vehicles powered by lithium-ion batteries.[54] A US company Tesla Motors is already delivering their lithium-ion-powered BEV Tesla Roadster.[55] In the aftermarket, models such as the Prius have been fitted with lithium-ion batteries.[56]

## References

1. ^ The effect of PHEV and HEV duty cycles on battery and battery pack performance, http://www.pluginhighway.ca/PHEV2007/proceedings/PluginHwy_PHEV2007_PaperReviewed_Valoen.pdf
2. ^
3. ^ MRS Website : Theme Article - Science and Applications of Mixed Conductors for Lithium Batteries
4. ^ Electrical Energy Storage and Intercalation Chemistry - WHITTINGHAM 192 (4244): 1126 - Science
5. ^ Activity and Diffusivity of Lithium Intercalated in Graphite - Dept. of Energy Citation
6. ^ US 4304825, "Rechargeable battery", issued 1981-12-08
7. ^ USPTO search for inventions by "Goodenough, John"
8. ^ http://dx.doi.org/10.1016/0025-5408(83)90138-1
10. ^ IEEE Spectrum: Lithium Batteries Take to the Road
11. ^ http://dx.doi.org/10.1016/0378-7753(89)80153-3
12. ^
13. ^ a b "In search of the perfect battery". The Economist. 2008-03-06. Retrieved 2008-08-24. Check date values in: |date= (help)
14. ^ Silberberg, M. 2006. Chemistry: The Molecular Nature of Matter and Change, 4th Ed. New York (NY): McGraw-Hill Education. p 935.
15. ^ a b c
16. ^ H.C. Choi et al., J. Phys. Chem. B 107 p5806(2003)
17. ^ G.G. Amatucci, J.M. Tarascon, L.C. Kein J. Electrochemical Society 143 p1114 1996
18. ^ Balbuena, P.B., Wang, Y.X., eds. Lithium Ion Batteries: Solid Electrolyte Interphase 2004 Imperial College Press, London
19. ^ Buchmann, Isidor. "Will Lithium-Ion batteries power the new millennium?". Isidor Buchmann (CEO of Cadex Electronics Inc.).
20. ^ Aging - capacity loss
21. ^ http://www.b2i.us/profiles/investor/fullpage.asp?f=1&BzID=546&to=cp&Nav=0&LangID=1&s=236&ID=9307
22. ^ Buchmann, Isidor. "Choosing a battery that will last". Isidor Buchmann (CEO of Cadex Electronics Inc.). Unknown parameter |pubdate= ignored (help)
23. Buchmann, Isidor (2006). "BatteryUniversity.com: How to prolong lithium-based batteries". Cadex Electronics Inc. Unknown parameter |month= ignored (help)
24. ^ Lewis, Leo (August 21, 2007). "Japanese experts demand change to make phones and laptops safe". The Times.
25. ^ http://www.e-one.com.tw/News_2005_e.htm
26. ^ AeroVironment Achieves Electric Vehicle Fast Charge Milestone Test Rapidly Recharges a Battery Pack Designed for Use in Passenger Vehicles. 10 Minute Re-Charge Restores Enough Energy to Run Electric Vehicle for Two Hours at 60 Miles Per Hour
27. ^ http://www.aei-online.org/automag/techbriefs/10-2006/1-114-10-16.pdf
28. ^ IEEE Spectrum: Lithium Batteries Take to the Road
29. ^ Bickel & Brewer - A law firm devoted exclusively to the resolution of complex commercial disputes. Bickel & Brewer was formed with a singular goal: to serve clients in significant, disputed matters that involve substantial dollar or business exposures...: Detail
30. ^ http://www.segway.com/personal-transporter/lithium_ion.html
31. ^ http://www.a123systems.com/html/news/articles/051102_news.html
32. ^ Green Car Congress: A123Systems Launches New Higher-Power, Faster Recharging Li-Ion Battery Systems
33. ^ Yahoo! Groups
34. ^
35. ^ Microsoft PowerPoint - 061125 Altair EDTA Presentation
36. ^ Welcome to Ener1
37. ^ Microsoft PowerPoint - EnerDel Technical Presentation.ppt [Read-Only]
38. ^ Science Express (preprint) http://www.sciencemag.org/cgi/content/abstract/1122716
39. ^ Technology Review: Higher-Capacity Lithium-Ion Batteries
40. ^ L.M. Cristo, T. B. Atwater. Characteristics and Behavior of 1M LiPF6 1EC:1DMC Electrolyte at Low Temperatures. Fort Monmouth, NJ: U.S. Army Research.
41. ^ How to prolong lithium-based batteries
42. ^ AN1333
43. ^ ISL9208 Multi-Cell Li-ion Battery Pack OCP/Analog Front End
44. ^ http://www.tayloredge.com/museum/mymuseum/sciencefun/li-ion_003.mov
45. ^ Dell laptop explodes at Japanese conference - The INQUIRER
46. ^ "Kyocera Launches Precautionary Battery Recall, Pursues Supplier of Counterfeit Batteries" (Press release). Kyocera Wireless. 2004-10-28. Check date values in: |date= (help)
47. ^ Tullo, Alex. "Dell Recalls Lithium Batteries." Chemical and Engineering News 21 August 2006: 11.
48. ^ http://en.wikinews.org/wiki/Nokia_issues_BL-5C_battery_warning%2C_offers_replacement
49. ^ Nokia N91 cell phone explodes
50. ^ Safety Last - New York Times
51. ^ Technology Review: New Batteries Readied for GM's Electric Vehicle
52. ^ http://www.imaracorp.com
53. ^ http://www.atbatt.com/blog/31.asp
54. ^ http://www.cleantech.com/news/3694/electrovaya-tata-motors-make-electric-indica
55. ^ http://www.teslamotors.com
56. ^ http://www.calcars.org/priusplus.html


text:
### Bailey and Daum's q-Kummer Theorem and Extensions

By means of the linearization method, we establish four analytical formulae for the $q$-Kummer sum extended by two integer parameters. Ten closed formulae are presented as examples.


text:
# [texhax] where are latex functions defined?

Manoj Kummini kummini at math.ukans.edu
Mon Dec 22 23:41:21 CET 2003

```On Mon, Dec 22, 2003 at 16:10:15hrs -0600, Joe Corneli wrote:

> For example, I'd like to look at the definition of
> `\alph'. But would also like to look at the definition
> of pretty much everything. Can you tell me what files to
> look in?

The document class files that you are using and latex.ltx. If you are on a
Unix system, you can get the path by

#`kpsewhich latex.ltx'

Also, source2e.tex, which you can download from CTAN and compile.

Manoj.

--
Manoj Kummini
Graduate Student, Dept. of Mathematics, The Univ. of Kansas,
Lawrence KS 66045 USA.  38 deg 55 min N, 95 deg 14 min W.
http://www.math.ukans.edu/~kummini/index.html
```


text:
# Lecture 13

Note If a conic in $\mathbb{R}P^{2}$ contains a line, then it is degenerate.

Theorem. Through five points $P_{1},P_{2},P_{3},P_{4},P_{5} \in \mathbb{R}P^{2}$ there exists a conic in $\mathbb{R}P^{2}$.

$i)$ If no four points lie on a line, then the conic is unique.

$ii)$ If no three lie on one line (five points in general position), then the conic is non-degenerate.

Lemma. If a conic contains three collinear points, then it contains the line through these points. Continue reading

# Lecture 12

Definition.
Let $V$ be a vector space over $\mathbb{R}$ (or $\mathbb{C}$). A map
$b\colon V \times V \to \mathbb{R}$ is a symmetric bilinear form, if

1. $b(v,w)=b(w,v) \quad \forall v,w \in V$
2. $b(\alpha_1 v_1 + \alpha_2 v_2,w) = \alpha_1 b(v_1,w) + \alpha_2 b(v_2,w)$ for all $\alpha_1,\alpha_2 \in \mathbb{R}$, $v_1, v_2, w \in V$.

$b$ is non-degenerate, if
$b(v,w)=0 \quad \forall w \in V \Rightarrow v=0\,.$
The corresponding quadratic form is defined by
$q(v)=b(v,v) \quad \forall v \in V\,.$

# Exercise sheet 5

In exercise 2 on sheet 5, the claim to be proven ($\textrm{cr}(y,p,x,q) = -1$) of course refers to the dual construction. (In the primal construction, these 4 lines are not concurrent).

# Emanuel’s office hours

Please participate in the doodle poll about changing Emanuel’s office hours to Monday.

———————————-

According to the poll, Emanuel’s office hours have been changed to Monday, 10:30 – 12, starting from Monday the 26th on. There is also the option to visit Thilo’s office hours on Tuesdays, 13-14.

# Lecture 11

Cross-ratio of lines through a point (again).

Claim. The map which maps a point $g^{\ast} \in p^{\ast}$ to the intersection point of $g \cap \ell$ is a projective map. Continue reading

# Exercise sheet 4

In exercise 3 about the fundamental theorem of affine geometry, it has to be $n > 1$. Otherwise, the statement is obviously false, as for the fundamental theorem of projective geometry.
(The exercise sheet has been updated accordingly)

# Lecture 9

Fundamental Theorem of real projective geometry: Let $f\colon \RP^n \to \RP^n$, $n \ge 2$, be a bijective map that maps lines to lines. Then $f$ is a projective transformation.

Remark: The theorem does not hold for arbitrary fields. For example $f\colon \CP^n \to \CP^n$ ($n \ge 2$) with
$f\left(\sqvector{z_1\\ \vdots\\ z_{n+1}}\right) = \sqvector{\bar z_1\\ \vdots\\ \bar z_{n+1}}$
is a bijective map, mapping complex projective lines to complex projective lines but it is not a projective transformation of $\CP^n$.

# Lecture 8

Theorem (complete quadrilateral). Consider four lines with intersection points $A$, $B$, $C$, $D$, $P$, and $Q$ as shown in the picture. Then the cross-ratio of the intersection points $X$ and $Y$ of the diagonals with the line $PQ$ and $P$ and $Q$ is:

$\cr(P,X,Q,Y)=−1.$

Proof of theorem on complete quadrilateral (multi-ratio).
Consider the multi-ratio with

$P_{1} = Q_{1} = P$, $P_{3} = Q_{3} = Q$, $Q_{2} = X$, $P_{2} = Y$,

where $p$ is the affine coordinates of $P$, $q$ the ones of $Q$ etc.

\begin{align}
-1 &= \mathrm{m}(P, X, Q, P, Y, Q) \\
&= \frac{p-x}{x-q} \frac{q-p}{p-y} \frac{y-q}{q-p} \\
&= \frac{p-x}{x-q} \frac{q-y}{y-p} \\
&= \mathrm{cr}(P, X, Q, Y). \\
\end{align}

$\square$


text:
+0

+2
123
15
+1164

Cyclic quadrilateral's sides are AB = 11, BC = 4, CD = 12, and AD = 10. Find the length of diagonals AC and BD and prove that (AB*CD) + (BC*AD) = AC*BD.

Mar 6, 2021

#1
+112839
+2

I have shown it but I am sure I have not done it by the intended method

My method is far from elegant.

Find BD in terms of  angle BCD using the cosine rule  and also

Find BD in terms of  angle BAD using the cosine rule

not that angle BCD and angle BAD are supplementary which means  cos

Solve for angle

Then use to find distance  BD

Do the same for AC

Then you can just substitute to show that they are the same.

I am sure this is not quite the intended solution.

Ask questions if you want to :)

Mar 6, 2021
#2
+2

The cosine rule?!?

BD = sqrt(122 + 42 - 2*12*4 * cos(BCD))       and then what?!?

Guest Mar 6, 2021
edited by Guest  Mar 6, 2021
#6
+69
+1

It's called the Law of Cosines,

OofPirate  Mar 7, 2021
#3
+112839
+3

like this

$$BD^2=144+16-96cos\angle BCD\\ BD^2=160-96cos\angle BCD\\~\\ but\;\;also\;\; BD^2=121+100-220cos \angle BAD\\ BD^2=221-220cos \angle BAD\\ BD^2=221+220cos \angle BCD\qquad \text{opposite angles in cyclic quads are supplementary}\\~\\ so\\ 160-96cos\angle BCD=221+220cos \angle BCD\\ -316cos\angle BCD=61\\ cos\angle BCD=\frac{-61}{316}\\ so\\ BD^2=160-96*\frac{-61}{316}\\ BD^2=\frac{14104}{79}\\ BD=\sqrt{\frac{14104}{79}}\\$$

Now do the same for AC

etc

LaTex:

BD^2=144+16-96cos\angle BCD\\
BD^2=160-96cos\angle BCD\\~\\
but\;\;also\;\;
so\\
160-96cos\angle BCD=221+220cos \angle BCD\\
-316cos\angle BCD=61\\
cos\angle BCD=\frac{-61}{316}\\
so\\
BD^2=160-96*\frac{-61}{316}\\
BD^2=\frac{14104}{79}\\
BD=\sqrt{\frac{14104}{79}}\\

Mar 7, 2021
#4
+1470
+1

Excellent work, Melody!!!

Dragan  Mar 7, 2021
edited by Dragan  Mar 7, 2021
#5
+112839
+1

Thanks Dragen

I really do think that there must be a better way to tackle this problem though.

Melody  Mar 7, 2021
#7
+1470
+2

There's more than one way to...

.................................................................

We can use this formula to calculate the circumradius. (R-radius, s-semiperimeter, a,b,c,d-sides)

When we have the length of a circumradius, we can calculate the angles B and C or A and D, and then using the law of cosines we can find diagonals AC and BD.

This is perhaps the longer way, but it would work.

Dragan  Mar 7, 2021
edited by Dragan  Mar 7, 2021
#8
+112839
+1

Mmm

You are using terms and formula that I have not seen before.

But yes, this is probably another way to skin the cat.  (sorry feline lovers - it is just an expression)

Melody  Mar 7, 2021
#11
+2197
+1

aHeM

CalTheGreat  Mar 8, 2021
#12
+112839
0

Hi Cal  :)

Melody  Mar 8, 2021
#13
+2197
+2

Hi Melody. Maybe you can use some other phrases that do not put cats in such a situation, even if it's an expression.

The neighborhood cat Winston is bristling besides me.

CalTheGreat  Mar 8, 2021
#9
+2

is known as Ptolemy's theorem,

dates back around two thousand years.

Mar 7, 2021
#10
+112839
+1

Thanks for the info guest.    It is always good to get interesting background information.

Jugoslav was still asked to prove it though.

He couldn't just say. It is true because Ptolomy said so.

Melody  Mar 7, 2021
#14
+1164
0

Hahahaha...

jugoslav  Mar 8, 2021
#15
+69
0

Ptolemy is always correct: Because the Party and Big Brother says so

In the same logic: Every single math problem has an answer of 53

OofPirate  Mar 13, 2021


text:
# How to Evaluate Recursive Formulas for Sequences

To specify a sequence, like any other function, we specify its domain and rule. In general terms, the sequence rule is called sequence general sentence. The general statement of a sequence is actually a rule through which each member of the domain corresponds to a member of the range set, that is, the general statement produces the statements of the sequence for each value of changes. The general sentence of a sequence is represented by the symbol $$a_n$$.

Evaluating the recursive formula of a sequence uses the recursive relation and the initial value(s) to find the value of each expression in the sequence.

To better understand this issue, let’s use an example. Consider the recursive formula for the sequence {$$a_n$$} where the recursive relation $$a_n=a_{(n-1)}+4$$ and the initial value $$a_1 = 2$$.

To find $$a_2$$, put $$n=2$$ and $$a_1=2$$ into the recursive relation: $$a_2 = a_{(2-1)}+4=a_1+2=2+4=6$$

We continue the same process to find the next terms:

$$a_3 =a_{(3-1)}+4=a_2+2=6+4=10$$

$$a_4= a_{(4-1)}+4=a_3+2 =10+4=14$$

$$a_5=a_{(5-1)}+4=a_4+2=14+4=18$$

Therefore, the order of evaluations will be as follows: {$$2, 6, 10, 14, 18$$}.

Important note: In some cases, evaluating the recursive formula for large values of n may be impractical because it requires a lot of time or computing power.

### What people say about "How to Evaluate Recursive Formulas for Sequences"?

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text:
# scikit-learn score metric on the coefficient of determination $R^2$

I am using scikit-learn in Python and they define a quantity called score. It's defined in the middle of the documentation page.

Reproduced here:

Returns the coefficient of determination R^2 of the prediction. The coefficient R^2 is defined as (1 - u/v), where u is the regression sum of squares ((y_true - y_pred) ** 2).sum() and v is the residual sum of squares ((y_true - y_true.mean()) ** 2).sum(). Best possible score is 1.0, lower values are worse.

A few questions regarding this:

1. What's the intuition behind this metric?

2. What is considered a good score? What is considered bad?

Actually there are two different measures that are called correlations. Let us then call them little $r$, which is the Pearson correlation coefficient, and big $R$, which is what you have; a correlation (usually as $R^2$) adjusted for a generalized residual. Now $|r|=|R|$ only when we restrict ourselves to ordinary least squares linear regression in $Y$. If for example, we restrict our linear regression to slope only and set the intercept to zero, we would then use $R$, not $r$. Little $r$ is still the same, it just won't describe the correlation between the new regression line and the data anymore.

Little r is normalized covariance, i.e., $r= \frac{\operatorname{cov}(X,Y)}{\sigma_X \sigma_Y}=\frac{\sum ^n _{i=1}(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum ^n _{i=1}(x_i - \bar{x})^2} \sqrt{\sum ^n _{i=1}(y_i - \bar{y})^2}}$. Finally, $r^2$ is called the coefficient of determination only for the linear case.

Big $R$ is usually explained using ANOVA intermediary quantities:

• The total sum of squares proportional to the variance of the data: $\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }$ $SS_\text{tot}=\sum_i (y_i-\bar{y})^2,$

• The regression sum of squares, also called the explained sum of squares: $SS_\text{reg}=\sum_i (f_i -\bar{y})^2,$

• The sum of squares of residuals, also called the residual sum of squares: $SS_\text{res}=\sum_i (y_i - f_i)^2=\sum_i e_i^2\,$

$R^2 \equiv 1 - {SS_{\rm res}\over SS_{\rm tot}}.\,$

Now, what is the meaning of this $r^2$ or more generally $R^2$? $R^2$ is the explained fraction and $1-R^2$ is the unexplained fraction of the total variance.

What is a good coefficient and what is bad one? That depends on who says so in what context. Most medical papers say that a correlation is strong when $|r|\geq 0.8$. Since that only explains $0.64$ of the variance, I would call $0.8$ to be moderate correlation, and in most of my work $R^2\geq 0.95$ with $<5\%$ unexplained variance is called good. In some of my biological work $R^2>0.999$ is required for proper results. On the other hand, for experiments with only short $x$-axis data ranges, and copious noise, one is lucky to even get a significant correlation usually circa $0.5$ as a borderline significant (non-zero) result.

Perhaps the best way to communicate how variable the answer is, is to back calculate what the critical $r$ and $r^2$ values are for a $p<0.05$ significance.

First to calculate the t-value from an r-value let us use

$t=\frac{r}{\sqrt{(1-r^2)/(n-2)}}$, where $n\geq 6$

Then $r=\frac{t}{n-2+t^2}$, where $n\geq 6$ and using the t-significance tables

the critical two-tailed values of $r$ for significance are:

n   r       r^2
6   0.9496  0.9018
7   0.8541  0.7296
8   0.7827  0.6125
9   0.7267  0.5281
10  0.6812  0.4640
11  0.6434  0.4140
12  0.6113  0.3737
13  0.5836  0.3405
14  0.5594  0.3129
15  0.5377  0.2891
16  0.5187  0.2690
17  0.5013  0.2513
18  0.4857  0.2359
19  0.4715  0.2223
20  0.4584  0.2101
21  0.4463  0.1992
22  0.4352  0.1894
23  0.4249  0.1806
24  0.4152  0.1724
25  0.4063  0.1650
26  0.3978  0.1582
27  0.3899  0.1520
28  0.3824  0.1462
29  0.3753  0.1408

30  0.3685  0.1358
40  0.3167  0.1003
50  0.2821  0.0796
60  0.2568  0.0659
70  0.2371  0.0562
80  0.2215  0.0491
90  0.2086  0.0435
100 0.1977  0.0391

Note that the explained fraction ($r^2$) need for a significant $r$-value varies from 90% for $n=6$ to 3.9% for $n=100$. Nor does it stop there, the higher the value of $n$, the less explained fraction is needed for significance.

Finally, asking what a 'good' $R^2$ is, is also a bit ambiguous. Unlike $r^2$, $R^2$ can (surprise, shock and awe) actually take on negative values. So, although $R^2$ is more general than $r^2$, it also has problems that never occur with $r^2$. Moreover, like $r$ (see above), $R$ is $n$ biased, and if we adjust $R$ for degrees of freedom using adjusted $R^2$, negative $R^2$ values become even more frequent.

Consider using the precision and recall scores of scikit-learn: http://scikit-learn.org/stable/modules/generated/sklearn.metrics.precision_recall_fscore_support.html. It may give you a more tangible number to consider.

Precision and recall are defined as:

The precision is the ratio tp / (tp + fp) where tp is the number of true positives and fp the number of false positives. The precision is intuitively the ability of the classifier not to label as positive a sample that is negative. The recall is the ratio tp / (tp + fn) where tp is the number of true positives and fn the number of false negatives. The recall is intuitively the ability of the classifier to find all the positive samples.

I often too find the score function of the classifiers to be somewhat abstract/ not applicable to my usecase, but the precision and recall gives you a percentage of how many of the predicted items was actually predicted correctly, and how many did the classifier miss.

• One of my main issues with sklearn is that it emposes a threshold of 0.5 and only 0.5 on the probabilities predicted from the model when doing classification. In my opinon, this makes much of the build in model tuning features like .score useless. – Matthew Drury Mar 7 '17 at 2:14
• @MatthewDrury In that case, do you consider my answer below to be contributory or should I just move it elsewhere to one of the other links listed below the Q above? – Carl Mar 8 '17 at 0:09
• No, your answer is good! I should have clarified, it seems reasonable in the regression case, but in the classification it lulls people into using hard classification based on a naive threshold. The answer above is talking about classification, precision, and recall, which I don't like sklearns handling of. – Matthew Drury Mar 8 '17 at 0:38

The Coefficient of Determination (R^2) generalizes the correlation coefficient (r) to multiple predictors, and is often summarized as the proportion of variance explained by the model. It will be quite comfortable for anyone used to analyzing linear regression models, and will be discussed in any text or course you might have takem.

1.0 is a perfect score, good is relative.

Note that the answer which discusses precision and recall does not answer the question posed, which was about Support Vector Regression, not Support Vector Classification. True Positive and False Positive assume binary (True/False) responses.

• If I had had time I would have written something similar to Carl's answer. I can't comment or downvote (not enough reputation points) but felt it was important to point out that the wrong classifier was discussed in wonderkid2's answer. Maybe someone can downvote wonderkid2 and upvote Carl. – hwrd Mar 7 '17 at 15:01


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CRM-2813Korotkin, D.On some integrable system arising in differential geometry and general relativity
CRM-3053Bertola, M., Eynard, B., Harnad, J.Generalized matrix models with rational potentials and isomonodromic tau functions
CRM-3016Rousseau, C.Divergent series: past, present, future...
CRM-3176Ghidaoui, M. S., Kolyshkin, A. A., Vaillancourt, R.Transient turbulent flow in a pipe
CRM-3168Zhao, M., Ghidaoui, M. S., Kolyshkin, A. A., Vaillancourt, R.On the stability of oscillatory pipe flows
CRM-3125El Boukili, A., Madrane, A., Vaillancourt, R.Multifrontal solution of sparse unsymmetric matrices arising from semiconductor equations
CRM-3179Yoshikawa, M., Gong, Y., Ashino, R., Vaillancourt, R.Case study on SVD multiresolution analysis
CRM-3172Cornea, O., Lalonde, F.A universal Floer theory, localization, and applications
CRM-3173Saidi, A.Consistent testing for independence of two partially nonstationary vector ARMA time series
CRM-3175Nguyen-Ba, T., Kolyshkin, A. A., Vaillancourt, R.Hermite–Birkhoff differential equation solvers
CRM-3178Dryanov, D., Fournier, R.On a discrete variant of Bernstein's polynomial inequality
CRM-3023Angers, J.-F., Merleau, J., Perreault, L.Landmark registration of hydrographs and Bayesian estimation of a mean hydrograph
CRM-3189Bertola, M., Mo, M. Y.Isomonodromic deformation of resonant rational connections
CRM-3127Arminjon, P., Touma, R.Central finite volume methods with constrained transport divergence treatment for ideal MHD
CRM-3185Ratnarajah, T., Vaillancourt, R.Complex singular Wishart matrices and applications
CRM-3177Colin, F., Frigon, M.Systems of singular Poisson equations in unbounded domains
CRM-3169Bertola, M., Eynard, B., Harnad, J.Semiclassical orthogonal polynomials, matrix models and isomonodromic tau functions
CRM-3180Grundland, A. M., Strasburger, A., Zakrzewski, W. J. M.Surfaces immersed in $\mathfrak{su}(N + 1)$ Lie algebras obtained from the $\mathbb{C}\mathrm{P}^N$ sigma models
CRM-3129Mélard, G., Roy, R., Saidi, A.Exact maximum likelihood estimation of structured or unit root multivariate time series models
CRM-3167Rousseau, C., Christopher, C.Modulus of analytic classification for the generic unfolding of a codimension 1 resonant diffeomorphism or resonant saddle
CRM-3182Bertola, M., Gekhtman, M.Biorthogonal Laurent polynomials, Toplitz determinants, minimal Toda orbits and isomonodromic tau functions
CRM-3181Schlomiuk, D., Vulpe, N.Integrals and phase portraits of planar quadratic differential systems with invariant lines with at least five total multiplicity
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# MAC OS - How to open Base reports based on queries in standalone

Hello everybody!

I am using a MAC OS X 10.11.6 and LibreOffice version 6.0.5.2.

I am was reading this link text. I run the macro and start to fill the dropdown fields, but the desktop field is always blank. So I need to select some file to add in that field. (I choose any LibreOffice file in this situation)

After pushing the Create Shortcut button, the shortcut creates an invisible file, and when I tried to open it, it only opens as a note. It is not open the real file.

I appreciate any help. Thank you,

EDIT.

I am editing to better explain the issue.

I am referring to @koolninja comments. I downloaded the DB shortcuts and ran the macro. As you can see, shortcut location is blank, and the only way to include a path is selecting an existing file. So, using an existing file on this, after pushing Create Shortcut button, the file created not work.

C:\fakepath\ScreenShot.png

edit retag close merge delete

Hello,

This needs more information. It appears you are using the macro by @Kridtbandit. Have you modified it as stated in the comments? Also there is nothing mentioned anywhere in that post related to drop downs. What is this drop down stuff all about? Please be as specific as possible.

( 2018-07-21 22:33:05 +0200 )edit

Sort by » oldest newest most voted

Hello,

You need to be a bit more careful on your references. The comments were NOT made by @koolninja but rather by myself. Also the download file (extension is more precise) was DB Doc Shortcut. by Roberto Benitez.

Unfortunately, the referred to post was mode almost two years ago and things don't seem to work the same. The original test was done on a Linux Xubuntu install with an older LO than currently used. Have no idea even at that time if it worked on Mac. Just tried on Mint 18.3 with LO v6.0.5.2 and was able to have the dialog produce the appropriate command but creating the Bash file was a failure - it was blank. Using the created command and generating my own file it worked. However trying to connect to the standalone document was another problem.

I have tried (recently on Mint 18.3) & successfully used the macro on the mentioned post under the answer by @Kridtbandit and had no problem (again look at my comments there as there are a couple of lines to be deleted). This is a much better solution at this point and you should try it.

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