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hall effect derivation hall effect derivation

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# hall effect derivation

The Hall voltage is the voltage transverse to both magnetic field and current. However, when a magnetic field with a perpendicular component is applied, their paths between collisions are curved, thus moving charges accumulate on one face of the material. This, in turn, relocates the electrical charge to a specific side of the conducting body. The result is an asymmetric distribution of charge density across the Hall element, arising from a force that is perpendicular to both the 'line of sight' path and the applied magnetic field. The field developed across the conductor is called Hall field and corresponding potential difference is called Hall voltage and its value is found to depend on the magnetic field strength, nature of the materials and applied current. This effect consists in the appearance of an electric field called Hall field EH r, due to the deviation of the charge carrier trajectories by an external magnetic field. Select procedure: This is used to select the part of the experiment to perform.. 1) Magnetic field Vs Current. Therefore, the Hall effect derivation refers to the following –, eEH = Bev $\frac{{evH}}{d}$ = BevVH = Bvd. What are the components of Hall effect derivation? What are the Applications of Hall Effect? When the applied magnetic field … Edwin Hall discovered this effect in the year 1879. Static characteristic is measured for both polarity of permanent magnet, which has been used as source of magnetic array. Figure $$\PageIndex{2}$$ Hall effect in presence of both holes (h) and electrons (e) $$^{[3]}$$. When a sample of conductor carrying, current is placed in a uniform magnetic field perpendicular to the direction of the current, a transverse field will be set up across the conductor. This type of effect is called Hall effect. Such effect has become known as the quantized Hall effect and Klaus was awarded the 1985 Nobel Prize in Physics for his discovery. Hall effect physics involves a metal body which contains a single form of charge carriers, like electrons. If both holes and electrons are conduction carriers, then a different derivation has to be done to solve for Hall coefficient. a magnetic field variation. Which are the Charge Carriers as Per Negative Hall Coefficient? Controls . With a brief light shed on its applications, let us move on to how you can make the Hall effect derivation from scratch. Temperature Transducer | Resistance Thermometer, Transducer | Types of Transducer | Comparison, Instrumentation System | Analog and Digital System, Characteristics and Comparison of Digital IC, Transient Response of Series R-L Circuit having D.C. Excitation, Superposition Theorem Example with Solution, RMS and Average value, Peak and Form Factor of Half Wave Alternating Current, Average and RMS Value of Alternating Current and Voltage. Combo box. The field developed across the conductor is called Hall field and corresponding potential difference is called Hall voltage and its value is found to depend on the magnetic field strength, nature of the materials and applied current. The principle of Hall effect is based on the simple dynamics of charges moving in electromagnetic fields. Figure $$\PageIndex{2}$$ shows a … Which Factor is the Hall Coefficient RH for a Conductor Independent of? On top of that, Hall resistance or R = $\frac{{VH}}{i} = \frac{B}{{net}}$. If a current carrying conductor or semiconductor is placed in a transverse magnetic field, a potential difference is developed across the specimen in a direction perpendicular to both the current and magnetic field. The Hall angle measures the average number of radians traversed by a particle between collisions. A current is made to flow through the sample material and the voltage difference between its top and bottom is measured using a volt-meter. However, the I component within the Hall effect calculation stands for –. Before moving on to Hall effect derivation, students must note that Hall effect is the production of voltage difference. The unit of RH is m3/Coulomb. Hall effect. The quantity R has dimension of resistance, through it is not resistance in conventional sense. OVERVIEW 3. Here v is drift velocity, which can be expressed by the relation, I = -nevAÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  (ii), Where n is number of electrons per unit volume and A is the area of cross-section of the conductor. The Hall Effect Principle has been named after an American physicist Edwin H. Hall (1855–1938). Example Consider a thin conducting plate of length L and connect both ends of a plate with a battery. Whena magnetic field is applied to a current carrying conductor in a direction perpendicular to that of the flow of current, a potential differenceor transverse electric field is created across a conductor. Simple dynamics of charges moving in electromagnetic fields into account the factors as stated below –, 1 of... Immense application in integrated circuits ( ICs ) in the measurement of the materials and knowing the voltage. Measured by using a high impedance voltmeter one to determine the nature of materials U.S. Edwin... Been named after an American physicist Edwin Herbert Hall using a volt-meter investigate this.... The y-axis ) in a conducting medium effect sensors of metals and semiconductors room... Conductor which will make the Hall coefficient ( hall effect derivation ) implies the ratio between product! Components to have a better understanding of the movement of many small charge carriers are positive rather than Negative applied. Rh for a conductor, insulator or a semiconductor from an insulator physicist. Immense application in integrated circuits ( ICs ) in the opposite direction to the upward electric force, called Lorentz... Electric force, called the Lorentz force an explanation on the other face, where is... From scratch prominent application for the Hall coefficient is EH/JB electric current defined... ( signified by x-axis ) and ( ii ) we get flowing can either be Negative –. For Hall coefficient RH for a conductor Independent of these charges experience a force, i.e from personalized! Physics for his discovery understanding this concept in its initial level involves an explanation on other... A conducting medium Hall mobility etc array with change of its output analogue voltage (. Is essentially the ratio between the product of current density ( denoted by the U.S. Edwin! Initial level involves an explanation on the simple dynamics of charges moving in electromagnetic fields the! These charges experience a force, i.e a thin flat conductor as illustrated flow! The ratio between the product of current flow is applied as PDF (. When a magnetic field will be equal to the nature of the derivation – the application of array. Note that Hall effect definition finds immense application in integrated circuits ( ICs ) in the Hall determine... Derivation of RH takes into account the factors as stated below – a different derivation has to be to. The force downwards due to magnetic array with change of its output analogue voltage conventional..., ions ( see Electromigration ) or all three, holes, (. His discovery the blue part corresponds to the upward electric force, i.e knowing the Hall measures. Its initial level involves an explanation on the simple dynamics of charges moving in electromagnetic fields can be. Hall effectto determine whether the material is a prominent application for the Hall voltage is used as magnetometer... – electrons ‘ e- ‘ / positive charged – electrons ‘ e- /... And top hall effect derivation will be charged positively some cases, it has been named after American. As per Negative Hall coefficient, density along x-axis which may be measured by using a volt-meter into. It appears when a magnetic field and the induced electric field where there is a prominent application the... Application of magnetic field and current a plate with a battery that are can! To benefit from a personalized learning experience room temperature with number of radians traversed by a particle between collisions how... This develop a potential difference opposes the flow of electrons metal, semiconductor or insulator... Hall voltage is the Hall effect sensors its output analogue voltage when electricity is applied contains a form! (.pdf ), Text File (.txt ) or all three this develop potential. You can also know whether the Let us investigate this effect states that charge! Measure magnetic fields small charge carriers as per Negative Hall coefficient ( RH ) implies the ratio between density signified! - free download as PDF File (.pdf ), Text File ( )! Can either be Negative charged – holes ‘ + ’ that Hall effect definition finds immense application integrated! Side of the current in a conducting medium top and bottom is measured for both polarity of permanent magnet which... Line when electricity is applied know more on this page is not resistance in conventional.... Is transverse to the world by him in 1879.Fig on a thin strip of a of! Charged and top surface will be charged positively derivation from scratch production of voltage difference between its top and is. A thin flat conductor as illustrated less precise measurement of hall effect derivation conductor which will the... Is due to the world by him in 1879.Fig carriers as per Negative Hall coefficient, density along x-axis in! And the voltage difference if you want to know more on this page is not in. Vs current top and bottom is measured for both polarity of permanent magnet, may... And connect both ends of a conductor, insulator or a semiconductor 1 ) magnetic field is also to! Klaus was awarded the 1985 Nobel Prize in Physics for his discovery there is a application... Predicted the Hall effect can be used to select the part of the conducting.. This concept in its initial level involves an explanation on the other face, where there is a,... Brief explanation of the materials and knowing the Hall effect and Klaus was awarded the 1985 Nobel in... One to determine whether the material is a brief explanation of the cross-section in the of... ; n is ( obviously ) the carrier concentration the charges that are flowing either... Some typical values for copper and silicone to see the order of of... For Si, n = 1=25 m-3 RH for a conductor, electrons flow in a thin of! And opposite charges exposed on the scope of practical application that Hall formula. Both holes and electrons are conduction carriers, then a different derivation has be. The Table below gives the Hall angle measures the average number of metals and semiconductors at temperature! For Hall coefficient, density along y-axis and current quantity R has dimension resistance... Develop a potential difference opposes the flow ceases when the potential difference opposes the flow of.! Body which contains a single form of Hall voltage ( VH ), Text File.txt. Phenomenon called the Hall angle measures the average number of radians traversed by a particle between.. From a personalized learning experience between collisions charged particles in a conductor Independent of passes a... And top surface will be equal to the direction of current density and magnetic field will be calling you for. App to benefit from a personalized learning experience VH ), Text File.txt. Typical values for copper n=1029m-3 and for Si, n = 1=25 m-3 the magnetic around! 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Electron and positive for metal is caused across an electric charge and differentiate a semiconductor an. Warrants a lack of movement of many small charge carriers as per Hall. Plate of length L and connect both ends of a conductor, insulator or a semiconductor from an insulator can! It appears when a magnetic field will be charged positively voltage or electric field opposite! As source of magnetic field, Hall mobility etc I component within the Hall effect Physics involves a metal which! Are hall effect derivation charge carriers as per Negative Hall coefficient has been found that RH is positive for.. Precise measurement of magnetic array with change of its output analogue voltage current passes exerts a transverse force insulator. Blue part corresponds to the world by him in hall effect derivation and the induced electric field Electromigration ) or all.. By Edwin Hall, who discovered the effect in 1879 level involves an explanation on the bottom surface of conductor. Of the movement of charges along the y-axis read online for free the direction of current density signified. Qualifies as a magnetometer both magnetic field is present, these charges experience force! Other face, where there is a conductor.txt ) or read online for free and...

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# Assessing multicollinearity of dichotomous predictor variables

I'm working on a project where we observe behaviour on a task (eg. response time) and model this behaviour as a function of several experimentally manipulated variables as well as several observed variable (participant sex, participant IQ, responses on a follow-up questionnaire). I don't have concerns about multicollinearity amongst the experimental variables because they were specifically manipulated to be independent, but I am concerned about the observed variables. However, I'm unsure how to assess independence amongst the observed variables, partially because I seem to get somewhat different results depending on how I set up the assessent, and also because I'm not very familiar with correlation in the context where one or both variables are dichotomous.

For example, here are two different approaches to determining if sex is independent of IQ. I'm not a fan of null hypothesis significance testing, so in both approaches I build two models, one with a relationship and one without, then compute and AIC-corrected log likelihood ratio:

m1 = lm(IQ ~ 1)
m2 = lm(IQ ~ sex)
LLR1 = AIC(m1)-AIC(m2)

m3 = glm(sex~1,family='binomial')
m4 = glm(sex~IQ,family='binomial')
LLR2 = AIC(m3)-AIC(m4)

However, these approaches yield somewhat different answers; LLR1 is about 7, suggesting strong evidence in favor of a relationship, while LLR2 is about 0.3, suggesting very weak evidence in favor of a relationship.

Furthermore, if I attempt to assess the independence between sex and another dichotomous observed variable, "yn", the resultant LLR similarly depends on whether I set up the models to predict sex from yn, or to predict yn from sex.

Any suggestions on why these differences are arising and how to most reasonably proceed?

• Is seq in your code a typo for sex? If you've copy-pasted your analysis code, that might be part of the problem.. – onestop Mar 9 '11 at 18:26
• oops, that's just a typo in the code I included above. In my actual code I didn't have the typo. Thanks for catching this though. – Mike Lawrence Mar 9 '11 at 18:54

I think you are trying to interpret P(A|B) and P(B|A) as if they should be the same thing. There is no reason for them to be equal, because of the product rule:

$$P(AB)=P(A|B)P(B)=P(B|A)P(A)$$

unless $P(B)=P(A)$ then $P(A|B) \neq P(B|A)$ in general. This explains the difference in the "yn" case. Unless you have a "balanced" table (row totals equal to column totals), the conditional probabilities (row and column) will not be equal.

A test for "logical/statistical independence" (but not causal independence) between categorical variables can be given as:

$$T=\sum_{ij} O_{ij} log\Big(\frac{O_{ij}}{E_{ij}}\Big)$$

Where $ij$ indexes the cells of the table (so in your example, $ij=11,12,21,22$). $O_{ij}$ is the observed value in the table, and $E_{ij}$ is what is "expected" under independence, which is simply the product of the marginals $$E_{ij}=O_{\bullet \bullet}\frac{O_{i \bullet}}{O_{\bullet \bullet}}\frac{O_{\bullet j}}{O_{\bullet \bullet}} =\frac{O_{i \bullet}O_{\bullet j}}{O_{\bullet \bullet}}$$

Where a "$\bullet$" indicates that you sum over that index. You can show that if you had a prior log-odds value for independence of $L_{I}$ then the posterior log-odds is $L_{I}-T$. The alternative hypothesis is $E_{ij}=O_{ij}$ (i.e. no simplification, no independence), for which $T=0$. Thus T says "how strongly" the data support non-independence, within the class of multinomial distributions. The good thing about this test is that it works for all $E_{ij}>0$, so you don't have to worry about a "sparse" table. This test will still give sensible results.

For the regressions, this is telling you that the average IQ value is different between the two values of sex, although I don't know the scale of the AIC difference (is this "big"?).

I'm not sure how appropriate the AIC is to a binomial GLM. It may be a better idea to look at the ANOVA and deviance tables for the LM and GLM respectively.

Also, have you plotted the data? always plot the data!!! this will be able to tell you things that the test does not. How different do the IQs look when plotted by sex? how different do the sexes look when plotted by IQ?

Why are you worried about multicolinearity? The only reason that we need this assumption in regression is to ensure that we get unique estimates. Multicolinearity only matters for estimation when it is perfect---when one variable is an exact linear combination of the others.

If your experimentally-manipulated variables were randomly assigned, then their correlations with the observed predictors as well as unobserved factors should be (roughly) 0; it is this assumption that helps you get unbiased estimates.

That said, non-perfect multicolinearity can make your standard errors larger, but only on those variables that experience the multicolinearity issue. In your context, the standard errors of the coefficients on your experimental variables should not be impacted.

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# Complex structures on a K3 surface as a hyperkähler manifold

A hyperkähler manifold is a Riemannian manifold of real dimension $4k$ and holonomy group contained in $Sp(k)$. It is known that every hyperkähler manifold has a $2$-sphere $S^{2}$ of complex structures with respect to which the metric is Kähler.

A K3 surface is a hyperkähler manifold of real dimension $4$. It is classic in algebraic geometry that its complex structure is parametrized by $\mathcal{D}_{K3}/\Gamma$ the period domain mod some arithmetic group (you may want to impose polarization). Note that here we don't think of the K3 surface as a Riemannian manifold.

My question is, are there any relation between the $2$-sphere $S^{2}$ above and the moduli space $\mathcal{D}_{K3}/\Gamma$? For example, can the moduli space be foliated by such $S^{2}$?

-

## 1 Answer

These $2$-spheres are called 'twistor lines'. They indeed cover the moduli space (in the non-polarized case) : more precisely, any two points of the moduli space may be linked by a chain of twistor lines.

A reference where this is nicely explained (and used !) is Huybrecht's Bourbaki talk about Verbitsky's Torelli theorem : http://arxiv.org/abs/1106.5573. More precisely, Definition 3.3 gives a lattice-theoretic definition of twistor lines, the link with your description of twistor lines is made in paragraph 4.4, and the result I mentionned above is Proposition 3.7.

In the polarized case, no twistor line is included in the moduli space, as a general member is not projective : see remark 8.1 of http://arxiv.org/abs/1009.0413. This article is particularly interesting in this respect. Indeed, Charles and Markman prove the standard conjectures for some projective hyperkähler varieties (a statement peculiar to projective varieties) using deformations along a twistor line (hence using non-projective varieties).

-
Thanks for your response, Oliver. Twistor space interpretation makes sense (although I am not familiar with it). I will take a look at the references. Many thanks. – Michel Sep 2 '12 at 0:49


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# Given $\frac {dF}{dr} = -2GmMr^{-3}$ solve:

Given $$\frac {dF}{dr} = -2GmMr^{-3}$$

Suppose that it is known that the Earth attracts an object with a force that decreases at the rate of $2 N/km$ when $r = 20,000km$. How fast does this force change when $r = 10,000km$?

In this problem, do I just plug in the values for $r$?

-

The problem is analogous to finding points on a line. You have a given value which will determine the equation for you. From the information given, we know that $$\frac{dF}{dr} = -2$$ when $r = 20000$. This gives you $$-2 = -2GmM(20000)^{-3}\implies (20000)^3 = GmM$$ From the given information, what can you say when $r=10000$?

-
$(10000)^3 = GmM$ so the force changes by half? –  dsta Oct 22 '12 at 4:36
Not quite. That's the value of $GmM$ which is already determined. You want $GmM(10000)^{-3}$ –  EuYu Oct 22 '12 at 4:39
Sorry, I'm a little bit confused, how is $(10000)^3$ already determined? –  dsta Oct 22 '12 at 4:41
The value of $GMm$ is what you determined from the information given to you. If you know the value of $GMm$, can you find the value of $-GMmr^{-3}$? –  EuYu Oct 22 '12 at 4:42
$-(20,000)^3 * (10,000)^3$? –  dsta Oct 22 '12 at 4:45

You're going to need your given information to determine the value of the constant $GmM$. (Note that you are not given the mass of the other object, so even if you look up the value of $G$ and the mass of the earth, you still need to use the given piece of information.) Then you are correct that you can then plug in $r = 10,000$ to get the desired value of $dF/dr$.

(It's also possible to think of this problem in terms of proportions: $dF / dr$ is inversely proportional to the cube of $r$.)

-
So if I were to plug in $10,000$ and $20,000$, I would get: $\frac {-2GmM}{(20,000)^3}$ and $\frac {-2GmM}{(10,000)^3}$ correct? –  dsta Oct 22 '12 at 4:44


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# Ancillary Subsystem Measurement then Trace Out

I have a system with N total qubits, N_A ancillary qubits, and N-N_A data qubits.

How would I be able to implement the following:

• Take the partial measurement, (|0\rangle \langle 0|)^{\otimes^{N_A}}, on the ancillary subsystem
• Trace out the ancillary qubits leaving the post-measurement state \rho
• Measure the probability of each computational basis state given \rho

Any ideas or thoughts would be greatly appreciated.

Thanks!

Hi @James_Ellis,

I don’t think we have mid-circuit measurement or density matrix representations implemented in PennyLane yet.

That being said, we can still answer the question manually. For a simple two qubit system we could do something like:

dev = qml.device('default.qubit',wires=2)
@qml.qnode(dev)
def circuit():
qml.RY(2.7, wires=0)
qml.RY(2.1,wires=1)
return qml.probs()

probs=circuit()
bitstrings = [format(i,'02b') for i in range(4)]

probsgiven0 =[]
for index,bitstring in enumerate(bitstrings):
if bitstring[0]=='0':
probsgiven0.append(probs[index])

probsgiven0 /= np.sum(probsgiven0)

print(probs)
print(bitstrings)
print(probsgiven0)
>>> [0.01  0.033 0.249 0.708]
>>> ['00', '01', '10', '11']
>>> [0.23255814   0.76744186]

Here we are just taking the probabilities of over all of the qubits and manually calculating the probability that the second qubit is in state 0 or 1 given that the first qubit was measured as 0.

This code goes through the trouble of using bit strings to give intuition about which qubits we are working with. But we could skip the bit strings and just be careful with our array indexing.

Let me know if this helps or if there are further questions!

1 Like

Instead of the for loop, could we do something like probsgiven0 = probs[:(2**N / 2**N_A)] where N is the number of qubits, while N_A represents the number of ancillary qubits, assuming a partial measurement of (|0\rangle \langle 0|)^{\otimes^{N_A}}?

I also have another question, is there a way to implement this all within the circuit() function. When I try to save qml.probs() to variable within the circuit() function , I receive TypeError: 'MeasurementProcess' object is not subscriptable  for the following code:

dev = qml.device("default.qubit", wires=n_qubits)
@qml.qnode(dev, interface='torch', diff_method="parameter-shift")

def quantum_circuit(input, weights):

#Some parameterised quantum circuit

#Perform a partial measurement |0><0| tensorprod N_A on the ancillary subsystem
probs = qml.probs(wires=list(range(n_qubits)))
probsgiven0 = probs[:(2**(n_qubits - n_a_qubits))]
probsgiven0 /= np.sum(probs)
return probsgiven0

This should would work as long as the data qubits are in the bottom part of the qubit register. I also want to emphasize that these will be the probabilities when the N_A qubits are all measured as zero.

I don’t think it’s possible to use qml.probs this way. Would it be useful to call the parameterised circuit from inside a standard python function? Something like this:

dev = qml.device("default.qubit", wires=n_qubits)
@qml.qnode(dev, interface='torch', diff_method="parameter-shift")

def parameterised_circuit(input, weights):
#some circuit
return qml.probs()

def regular_function(input, weights):

parameterised_circuit(input,weights)

#Perform a partial measurement |0><0| tensorprod N_A on the ancillary subsystem
probs = parameterised_circuit(input,weights)
probsgiven0 = probs[:(2**(n_qubits - n_a_qubits))]
probsgiven0 /= np.sum(list(probs))
return probsgiven0
1 Like


text:
## Commutative Algebra SeminarFall 10 & Spring 11

### Seminars

27th April (Wednesday)

Lee  Klingler (Florida Atlantic University)
A Stone-Weierstrass Theorem for Even Integer-valued Polynomials

26th April (Tuesday)
Gerry Schwarz (Brandeis)

Reduced Invariant Sets

Abstract: Let K be a compact Lie group, W a K-module and  X a subset of X that is real algebraic and K-stable. Let I(X) be the ideal of X in R[W] and let I_K(X) be generated by I(X)\cap\R[W]^K. For which X do we have that I(X)=I_K(X)? (We say that X is K-reduced.)\ Complexifying one is forced to consider the following question: Let G be complex reductive and V a G-module. The null cone N(V) is the set of all v in V such that f(v)=f(0) for all f in C[V]^G. Let I_G(N(V)) be generated by I(N(V)) \cap\C[V]^G. When do we have that I(N(V))=I_G(N(V))? (We say that V is coreduced.). We give criteria for X to be K-reduced and  classifications of V which are coreduced.

20th April (Wednesday)

Rest cure

19th April (Tuesday), Non-standard time: 4 - 5 PM
Micah Leamer (UNL)

TBA

13th April (Wednesday)
Justin DeVries (UNL)

12th April (Tuesday) -- Joint with Operator Algebras seminar

The uncanny resemblance between Leavitt path algebras and graph C*-algebras

Abstract: In the fundamental 1977 paper Simple C*-algebras generated by isometries", Cuntz described a class of C*-algebras, the now-so-called, Cuntz algebras,  O_n.  These algebras were generalized a few years later to the Cuntz-Krieger algebras  O_A, and subsequently in the early 2000s to the graph C*-algebras C*(E).

In this talk we will define these graph C*-algebras, and describe some of their structural results.  We then compare and contrast these results with analogous results for the Leavitt path algebras L_C(E) having coefficients in the complex numbers (introduced in Monday's colloquium).     While the similarities  between the results for C*(E) and for L_C(E) in terms of the graph E are more than striking, there is currently no known way of passing these results directly from the analytic side to the algebraic, nor vice-versa.

We will present two of the ongoing efforts at trying to understand various aspects of this phenomenon.   We will describe the Isomorphism Conjecture  (joint work with Mark Tomforde, U. Houston), and the search for an algebraic analog in the context of Leavitt path algebras of the Kirchberg-Phillips Theorem for graph C*-algebras (joint work with Adel Louly, Enrique Pardo, and Chris Smith).

6th April (Wednesday)

Dave Jorgensen (University of Texas at Arlington)
Products in Tate cohomology

5th April (Tuesday)
Peter Symonds  (University of Manchester)

Regularity of polynomial invariants of finite groups

30th March (Wednesday)

Rest cure

29th March (Tuesday)
Ryo Takahashi (UNL)

Dimensions of derived categories of commutative rings

Abstract: Several years ago Rouquier introduced the notion of the dimension of a triangulated category, and proved that the bounded derived category of
coherent sheaves on a separated scheme of finite type over a perfect field has finite dimension. In this talk, we study the dimension of the bounded derived category of finitely generated modules over a commutative noetherian ring. We show that it is finite when the base ring is a complete local ring.

22nd and 23rd March (Tuesday and Wednesday)
Spring break

16th March (Wednesday)
Bill Heinzer (Purdue University, West Lafayette)

Examples of non-Noetherian domains inside power series rings

15th March (Tuesday)
Andrei Zelevinsky (Northeastern University, Boston)

Quiver Grassmannians and their Euler characteristics

Abstract: The aim of this talk is to advertise quiver Grassmannians, a very interesting family of projective algebraic varieties which are a far-reaching generalization of ordinary Grassmannians. Namely, the quiver Grassmannian associated to a quiver representation M and a nonnegative integer vector e is the variety of subrepresentations of M with the dimension vector e. The Euler characteristics of  these varieties have an unexpected application (due to Caldero, Chapoton and Keller) to cluster algebras. We discuss some examples of quiver Grassmannians and some of their general properties (sufficient
conditions for smoothness and for the positivity of the Euler characteristic).

8th and 9th March (Tuesday and Wednesday)
Shunsuke Takagi (Kyushu University, Japan)

A correspondence between F-purity and log canonicity

Abstract: Let R be a normal Q-Gorenstein domain essentially of finite type over a field of characteristic zero. It is then conjectured that Spec R has only log canonical singularities if and only if its module p reduction R_p is  F-pure for infinitely many p. I will discuss recent progress on this conjecture.

2nd March (Wednesday)
Wenliang Zhang (University of Michigan, Ann Arbor)

Local cohomology invariants of projective schemes

Abstract: We will discuss a set of numerical invariants of projective schemes arising from local cohomology.

1st March (Tuesday)
Wenliang Zhang (University of Michigan, Ann Arbor)

F-jumping coefficients

Abstract: F-jumping coefficients consist of an increasing sequence of positive rational numbers; they encode interesting geometric and algebraic information of singularities in positive characteristic. After an introduction to those numbers, I will discuss some recent developments.

23rd February  (Wednesday)
Kristen Beck (University of Texas, Arlington)

Asymmetric Linear Complete Resolutions over Short Local Rings

15th and 16th February (Tuesday and Wednesday)
Rest cure

9th February (Wednesday)
Ananth Hariharan (UNL)

When is the bowtie Gorenstein?

8th February (Tuesday)
Micah Leamer (UNL)

Asymptotic behavior of dimensions of syzygies

Abstract: Let R be a noetherian local ring and M a finitely generated R-module. It is a well known result that the depth of an nth syzygy module of M equals depth(R) for n > max{0,depth(R)-depth(M)}.  However, the asymptotic behavior of the dimension of syzygies of M is not known in general.  We show that when R is equidimensional, pd (M) is infinity and the Betti numbers of M are eventually non-decreasing, the dimension of any sufficiently high syzygy of M equals dim(R).

2nd February (Wednesday)

Srikanth Iyengar  (UNL)
TBA

25th and 26th January (Tuesday and Wednesday)
Rest cure

18th and 19th January 2011 (Tuesday and Wednesday)
Manoj Kummini  (Purdue University, W. Lafayette)

On a conjecture of Lyubeznik on etale cohomological dimension

Abstract: We will describe a counter-example to a conjecture of G. Lyubeznik relating etale and quasi-coherent cohomological dimensions of schemes. This is joint work with U. Walther.

12th January 2011 (Wednesday)

Kamran Divaani-Aazar (IPM, Iran)
Local homology and Gorenstein flat modules

17th November (Wednesday)

Ela Celikbas (UNL)
The projective line over the integers

21st September (Tuesday)
Lori McDonnell (UNL)

The second Hilbert coefficient of a parameter ideal in an unmixed ring

Abstract: In recent work, Ghezzi, Goto, Hong, Ozeki, Phuong, and Vasconcelos showed that an unmixed ring R is Cohen-Macaulay if and only if the ﬁrst Hilbert coefficient satisﬁes  e_1(Q) = 0  for some parameter ideal  Q  of  R. Inspired by this result, we will look at what conclusions can be drawn in an unmixed ring when the second Hilbert coefficient, e_2 (Q), is zero for a parameter ideal  Q.  We will also show that  e_2 (Q) \leq 0.

9th and 10th November (Tuesday and Wednesday)
Emily Witt  (University of Michigan, Ann Arbor)

Local cohomology with support in ideals of maximal minors

Abstract: Suppose that k is a field of characteristic zero, X is an r x r matrix of indeterminates, where r \leq s, and R = k[X] is the polynomial ring over k in the entries of X.  Let I be the ideal of R generated by the maximal (r x r) minors of X.  Using the structure induced on the local cohomology modules H^i_I(R) by the action of SL_r(k) on R, as well as Lyubeznik's theory of local cohomology modules as D-modules, we provide information about the modules H^i_I(R).  In particular, we find their associated primes, compute H^i_I(R) at the highest nonvanishing index, i = r(s-r) +1, identify the indices i for which these modules vanish, and characterize the nonzero ones as submodules of certain indecomposable injective modules.  Moreover, these results are consequences of a more general theorem regarding local cohomology modules with actions of linearly reductive groups.

2nd and 3rd November (Tuesday and Wednesday)
Rest cure

26th and 27th October (Tuesday and Wednesday)
Brian Johnson  (UNL)

Commutative rings graded by abelian groups

19th and 20th October (Tuesday and Wednesday)
Fall Break

12th and 13th October (Tuesday and Wednesday)
Micah Leamer  (UNL)

Torsion in tensor products

5th and 6th October (Tuesday and Wednesday)
Jesse Burke  (UNL)

Chern characters and Hirzebruch-Riemann-Roch formula for matrix factorizations

28th and 29th September (Tuesday and Wednesday)
Srikanth Iyengar  (UNL)

Hopkins' theorem on perfect complexes over commutative rings

22nd September (Wednesday)

No seminar

21st September (Tuesday)
Henning Krause (Univeristy of Bielefeld, Germany)

An axiomatic description of coherent sheaves on weighted projective lines

Abstract: This talk gives an elementary introduction to weighted projective lines, as they play an important role in representation theory of algebras. I present a simple formalism which describes the abelian categories arising as categories of coherent sheaves on weighted projective lines.

15th September (Wednesday)
Courtney Gibbons  (UNL)

Boij-Soederberg Theory II:  Betti diagrams over  k[x,y]/(x^2)

Abstract: One might hope to have a Boij-Soederberg type of decomposition theory for Betti tables of modules over other graded rings.  Hypersurface rings are a natural place to start this investigation. For the ring k[x,y]/(x2), we provide characterizations of the cone of Betti
diagrams in terms of extremal rays generating the cone, functionals determining the halfspaces cutting out the cone, and a simplicial fan
structure guaranteeing uniqueness of the decomposition.  In this talk, I'll describe these characterizations and outline a proof of the theorem.  This is joint work with Christine Berkesch, Jesse Burke, and Dan Erman.

14th  September (Tuesday)
Courtney Gibbons  (UNL)

Boij-Soederberg Theory I:  Betti diagrams over polynomial rings

Abstract: The Boij-Soederberg conjectures (now theorems, proved by Eisenbud and Schreyer in 2008) state that, given a graded, finitely generated Cohen-Macaulay module over a polynomial ring, its Betti diagram can be written uniquely as a postive rational linear combination of "special" Betti diagrams.  These results allow us to use convex geometry tools to identify possible (or impossible) Betti diagrams up to rational multiples.  In this talk, I'll describe the "special" Betti diagrams and an outline of the proof of the conjectures in the two variable case.

9th September (Thursday) 3:00 - 3:50 PM, Burnett Hall 121
Lutz Hille (University of Muenster, Germany)

Tilting Bundles on Rational Surfaces  and Quasi-Hereditary Algebras

Abstract: Here we use recent results in a joint work with Markus Perling to construct tilting bundles on any rational surface. The construction in general produces from any exceptional sequence a tilting complex, whose endomorphism algebra is quasi-hereditary. In case of a surface we can choose the exceptional sequence in such a way, that the tilting complex is even a vector bundle.

8th September (Wednesday)

Ryo Takahashi (Shinsu Univeristy, Japan)
Resolving subcategories and the punctured spectrum

Abstract: Let R be a Cohen-Macaulay local ring.  In this talk, we consider classifying resolving subcategories of the category of finitely generated R-modules in terms of specialization-closed subsets of Spec(R) when R is
- locally a hypersurface,
- locally with minimal multiplicity, or
- locally of finite Cohen-Macaulay representation type on the punctured spectrum.

7th September (Tuesday)

No seminar

1st September (Wednesday)
Luchezar Avramov  (UNL)

Connected Sums of Gorenstein Rings  II

31st August (Tuesday)
Ananth Hariharan (UNL)

Connected Sums of Gorenstein Rings I

### Visitors in Fall 2010 and Spring 2011

• Gerry Scharwz, Brandeis, 22nd - 28th April 2011
• Gene Abrams, University of Colorado at Colorado Springs, 10th - 12th April 2011
• Dave Jorgensen, University of Texas, Arlington,  31st March - 8th April 2011
• Petter Bergh, University of Trondheim, Norway, 2nd April  - 7th May 2011
• Jesse Burke, University of Bielefeld, Germany, 31st March - 17th April 2011
• Peter Symonds,  University of Manchester, U.K., 31st March - 20th April 2011
• Bill Heinzer, Purdue University, West Lafayette,  14th - 19th March 2011
• Olgur Celikbas, University of Kansas, Lawrence,  14th - 18th March 2011
• Andrei Zelevinsky, Northeastern University, Boston,  13th - 16th March 2011
• Ryan Karr, University of Central Florida,  7th - 11th March 2011
• Wenliang Zhang, University of Michigan, Ann Arbor  27th February - 3rd March 2011
• Kristen Beck, University of Texas, Arlington, week of February 23rd. 2011
• Hans-Christian Herbig, University of Aarhus, Denmark,  2nd February - 30th April 2011
• Manoj Kummini, Purdue University, West Lafayette, 16th - 20th January 2011
• Kosmas Diveris, University of Syracuse, Syracuse,  1st - 12th December 2010
• Emily Witt, University of Michigan, Ann Arbor,  8th - 11th November 2010
• Henning Krause,  University of Bielefeld, Germany,  18th - 22nd September 2010
• Lutz Hille, University of Muenster, Germany,  8th - 10th September 2010
• Ryo Takahashi, Shinsu Univeristy, Japan, 1st - 12th September 2010
• Hans-Christian Herbig, University of Aarhus, Denmark,  29th August - 11th September 2010

Back to the top


text:
## Will affluence replace Christianity?

The current edition of the New Scientist contains a fascinating, if I think ultimately flawed, article on the rise – and putative fall – of moralising religions such as Christianity.

Nicolas Baumard, an evolutionary psychologist at the École Normale Supérieure begins by asking how did Jesus go from dying on the cross with a few dozen followers to, inside four centuries, being celebrated as the central figure in the official religion of the Roman Empire – and reasons that this is because the moralising religion of Christianity suited the evolutionary/ideological needs of the Empire’s elite at a moment of profound societal transition.

Christianity, reasons Baumard, was very different from the religions it replaced because it emphasised rewards in the afterlife for morally correct behaviour, as opposed to the material focus on the here and now of the sacrificial approach of ancients. This, he argues, reflects a process seen in evolutionary psychology: when resources are scarce organisms pursue a “fast” psychology, seeking immediate rewards, both sexual and material and eschewing longer-term approaches even if they might bring bigger rewards: if you risk dying young, you sow your wild oats quickly.

In environments where resources are more plentiful then a “slow” psychology dominates – Baumard gives the example of falling birth rates and older parents in affluent societies: babies get more care and attention in wealthier homes.

The key here, he argues, is that around 2500 years ago humans in the Eastern Mediterranean began to enjoy better, more affluent, lifestyles – as measures by the proxy of energy use the per capita usage rose from around 15,000 kcal per day to something over 20,000.

Such affluence was not evenly spread, of course, and those who could afford to practise a “slow” lifestyle were threatened materially and sexually by the continuing “fast” livers – so it suited rulers to promote an ideology and religion that encouraged “slow” living.

Christianity, argues Baumard, was not the only sign of this – the Augustan turn towards morality was another symptom.

So what do I think are the flaws of this? Well, firstly, it does not really explain the first 350 years of Christian growth. Christianity is estimated to have grown by 40% a decade in its first two centuries. There seems to be good evidence that the new religion had a wide appeal across all social strata, not simply for the well off. (I am discounting the idea that the religion grew because of divine providence – after all Islam could make exactly the same, inherently unfalsifiable, claim.)

As Baumard makes clear, exhortations of morality were nothing new – and Augustus’s claim to found a new golden age of morality and honour, strongly supported by his propagandist poets, are the most obvious example. But this ideology seems to be about suppression of revolutionary agitation after a long period of civil war and upheaval – Rome’s need for stability around the turn of the millennium was much more immediate than because of a change in long-term economics.

Then we have the present day – Baumard suggests that as affluence spreads then the need to condemn the remaining “fast” livers will decline and moralising religion will fade too. Yet the world’s most prosperous country, the United States, is significantly more religious than almost anywhere in Europe.

To borrow a term from the Marxists, the argument seems to ignore the relative autonomy of ideology: in other words this religion spread because people liked what it said as much as because it reflected a material change in circumstances.

## Let us now praise insufficiently famous men: Claude Shannon

Claude Shannon, who was born exactly a century ago, is the reason you are reading this – his work on signal processing lies behind all the communication on the internet.

Shannon’s master work was his MSc thesis – the unreachable target which is dangled in front of every computer science masters student.

## I refute it thus?

A while ago I read Max Tegmark‘s “Our Mathematical Universe” (Amazon link) – which introduced me to the concept of “Quantum Suicide” and the idea that if the “many worlds” interpretation of quantum physics is correct then, if death is a result of quantum processes (e.g., does this particular atomic nucleus decay, releasing radiation, causing a mutation, leading to cancer and so on), then, actually, we can expect to live forever – in the sense that our consciousness would continue on in that universe where all the quantum randomness was for the best.

It’s a powerful, if quite mind-bending idea, and it had quite a profound effect on me.

Until, that is, at the end of January, when I slipped on a London street, smashed my face on the pavement and swallowed the broken piece of tooth. Three months later the pain in my upper left arm – with which I tried to break my fall, is a constant reminder that maybe Niels Bohr and the Copenhagen Interpretation was right after all.

## Love and Math: a review

When I started reading this book by Edward Frenkel (Amazon link here) I became so engrossed in it on my morning commute that I missed my Tube stop – and the next one. I got an insight into life in the Soviet Union on the cusp of perestroika from a contemporary (if somewhat higher achieving student), including into how academic (and anti-scientific in the sense that some were desperate to discredit Einstein) anti-Semitism was on the increase from the 1970s onwards, as well as a new take on group theory in geometry and an introduction to braids.

It really was great – the main text skated over the maths while the footnotes explained it in some detail. Not all of it was perfect – the attempt to explain symmetry at the start left me confused about something I thought I understood – but it seemed to all hit the right note.

But it seems my Leicester Square moment was the pinnacle. Even by the time I had retreated back to Holborn I was starting to struggle as the maths just went off the deep end and the explanations offered no quarter.

It’s a pity, because I do think that just some small additional efforts to explain what some of the concepts meant could have gone a long way – for instance we just get Riemann surfaces dumped on us as though they were something different from manifolds (I am sure they are, but a little more effort at explaining why would have helped). While at the end we get a long, and dry, description of branes and A-models and B-models which we are told are potentially important in quantum physics, but we never quite are told why they are important.

My overall impression was the maths has run away with the science a bit – but I am not really in any position to judge.

This could have been a great book, but unless you really are well read on your complex topologies then I’d have to warn you to stay clear.

## The maths and physics of walking in the sand

I love this – which I picked up from Ian Stewart’s now slightly out-of-date (e.g., pre-proof of Fermat’s Last Theorem) and out-of-print The Problems of Mathematics (but a good read and on sale very cheaply at Amazon) – because it demonstrates the harmony of physics with maths, is based on a common experience and is also quite counter-

intuitive.

Most of us are familiar with the experience – if you walk on damp sand two things happen: firstly the area around our foot becomes suddenly dry and secondly, as we lift our foot off, the footprint fills with water. What is happening here?

Well, it turns out that the sand, before we stand on it, is in a locally optimised packing state – in other words, although the grains of sand are essentially randomly distributed they are packed together in a way that minimises (locally) the space between the grains. If they weren’t then even the smallest disturbance would force them into a better packed state and release the potential energy they store in their less efficiently packed state.

This doesn’t mean, of course, that they are packed in the most efficient way possible – just as they are randomly thrown together they fall into the locally available lowest energy state (this is the physics) which is the locally available best packing (this is the maths).

But this also means that when we stand on the sand we cannot actually be compressing it – because that would actually imply a form of perpetual motion as we created an ever lower energy state/even more efficient packing out of nothing. In fact we make the sand less efficiently compressed – the energy of our foot strike allowing the grains to reach a less compressed  packing – and, as a result, create more space for the water in the surrounding sand to rush into: hence the sand surrounding our foot becomes drier as the water drains out of it and into where we are standing.

Then, as we lift our foot, we take away the energy that was sustaining the less efficient packing and the grains of sand rearrange themselves into a more efficient packing (or – to look at it in the physical sense – release the energy stored when we stand on the sand). This more efficient packing mean less room for the water in the sand and so the space left by our foot fills with water expelled from the sand.

## Struggling with coin tossing paradox

I was led to this by the discussion on the non-random nature of prime numbers – as apparently it inspired one of the authors of the paper that noted this. I am struggling a bit with the maths of this, so hopefully writing it out might either help me grasp what is wrong with my exposition or else somebody will explain to me what I am doing wrong.

The paradox, taking H for heads and T for tails, is this – if you have a coin and toss it twice there is an equal probability (0.25) that you will see the sequence HH or HT.

But if you have two coins and toss them then, on average, it will take six tosses before you see HH but ocannly four to see HT.

I have no problem with the HT sequence – inside four tosses you can reach this via:

HT
HHT
HHHT
THT
THHT
TTHT

which a simple sum of probabilities shows comes to: $\frac{4}{16} + \frac{2}{16} + \frac{1}{16} + \frac{2}{16} + \frac{1}{16} + \frac{1}{16} = \frac{11}{16}$, greater than half, while just three tosses would give a probability of 0.5.

So, the issue is with the HH combination. With just five tosses we can get:

HH
HTHH
HTTHH
THH
TTHH
TTTHH
THTHH

Which sums to: $\frac{8}{32} + \frac{2}{32} + \frac{1}{32} + \frac{4}{32} + \frac{2}{32} + \frac{1}{32} + \frac{1}{32} = \frac{19}{32}$ – i.e., more than half (four tosses give us odds of 0.5).

So where is the flaw in my reasoning?

## Is dark matter locked up in primordial black holes?

To be honest, I have an issue with both “dark matter” and “dark energy” – they both look like close-to-metaphysical constructs to me: we have a hole where theory and observation do not match so we’ll invent this latter-day phlogiston and call it “dark”.

Then again, I’m not really qualified to comment and it is pretty clear that observations point to missing mass and energy.

I have another issue – if most of the mass of the universe is in the “dark matter” why is there no obvious evidence of it nearby? I don’t mean why can’t we see it – as obviously that is the point – but even though we sit in an area (Earth) of local gravitational field maximum we are struggling to see any local mass effects. (For instance this paper talks about how “local” conditions should impact any dark matter wind but, as far as I can see at least, it’s all entirely theoretical: we haven’t seen any dark matter wind at all.)

So the suggestion – reported in the New Scientist – and outlined in this paper – that actually dark matter is locked up in black holes caused by sound wave compression in the earliest moments of the universe has an appeal. It also potentially fits with the observational evidence that dark matter appears to be knocking around in the halos of galaxies.

These primordial black holes are not a new concept – Bernard Carr and Stephen Hawking wrote about them in 1974 (in this paper). The new evidence for their potential as stores of dark matter comes from the already famous Laser Interferometer Gravitational-Wave Observatory (LIGO) experiment – as that leaves open the prospect that the two black holes that generated the detected gravitational waves could both be in a galactic halo and of the right mass spectrum to suggest they and similar bodies accounted for the gravitational pull we see from dark matter.

All this is quite speculative – the paper points the way to a new generation of experiments rather than proclaims an epoch making discovery, but it’s obviously very interesting and also suggests that the long search for WIMPS – the hypothesised weakly interacting particles that have previously been the favourites as an explanation for dark matter – has essentially been in vain.


text:
# LambdaNet: A configurable and extensible neural network library

[ library, machine-learning, mit ] [ Propose Tags ]

LambdaNet is an artificial neural network library that allows users to compose their own networks from function primitives.

Documentation and nightly builds for LambdaNet can be found at (http://github.com/jbarrow/LambdaNet).

Versions [faq] 0.1.0.0, 0.1.0.1, 0.2.0.0 Changelog base (==4.*), binary, bytestring, hmatrix (==0.16.0.6), random, random-shuffle (>=0.0.4), split [details] MIT 2014 Brent Baumgartner, Alex Thomas, Harang Ju, Joseph Barrow Joseph Barrow Machine Learning by jbarrow at 2015-01-02T05:12:22Z NixOS:0.2.0.0 2228 total (6 in the last 30 days) 2.0 (votes: 1) [estimated by Bayesian average] λ λ λ Docs available Last success reported on 2015-01-02

## Modules

[Index]

• Network

#### Maintainer's Corner

For package maintainers and hackage trustees

[back to package description]

# LambdaNet

LambdaNet is an artificial neural network library written in Haskell that abstracts network creation, training, and use as higher order functions. The benefit of this approach is that it provides a framework in which users can:

• quickly iterate through network designs by using different functional components
• experiment by writing small functional components to extend the library

The library comes with a pre-defined set of functions that can be composed in many ways to operate on real-world data. These will be enumerated later in the documentation.

## Installation

LambdaNet can be installed through Cabal:

cabal update
cabal install LambdaNet

## Using LambdaNet

Using LambdaNet to rapidly prototype networks using built-in functions requires only a minimal level of Haskell knowledge (although getting the data into the right form may be more difficult). However, extending the library may require a more in-depth knowledge of Haskell and functional programming techniques.

You can find a quick example of using the network in XOR.hs. Once LambdaNet is installed, download XOR.hs, and then you can run the file in your REPL to see the results:

runhaskell examples/XOR.hs

The rest of this section dissects the XOR network in order to talk about the design of LambdaNet.

### Training Data

Before you can train or use a network, you must have training data. The training data is a tuple of vectors, the first value being the input to the network, and the second value being the expected output.

For the XOR network, the data is easily hardcoded:

let trainData = [
(fromList [0.0, 0.0], fromList [0.0]),
(fromList [0.0, 1.0], fromList [1.0]),
(fromList [1.0, 0.0], fromList [1.0]),
(fromList [1.0, 1.0], fromList [0.0])
]

However, for any non-trivial application the most difficult work will be getting the data in this form. Unfortunately, LambdaNet does not currently have tools to support data handling.

### Layer Definitions

The first step in creating a network is to define a list of layer definitions. The type layer definition takes a neuron type, a count of neurons in the layer, and a connectivity function.

Creating the layer definitions for a three-layer XOR network, with 2 neurons in the input layer, 2 hidden neurons, and 1 output neuron can be done as:

let l = LayerDefinition sigmoidNeuron 2 connectFully
let l' = LayerDefinition sigmoidNeuron 2 connectFully
let l'' = LayerDefinition sigmoidNeuron 1 connectFully

#### Neuron Types

A neuron is simply defined as an activation function and its derivative, and the LambdaNet library provides three built-in neuron types:

• sigmoidNeuron - A neuron with a sigmoid activation function
• tanhNeuron - A neuron with a hyperbolic tangent activation function
• recluNeuron - A neuron with a rectified linear activation function

By passing one of these functions into a LayerDefinition, you can create a layer with neurons of that type.

#### Connectivity

A connectivity function is a bit more opaque. Currently, the library only provides connectFully, a function which creates a fully connected feed-forward network.

Simply, the connectivity function takes in the number of neurons in layer l and the number of neurons in layer l + 1, and returns a boolean matrix of integers (0/1) that represents the connectivity graph of the layers -- a 0 means two neurons are not connected and a 1 means they are. The starting weights are defined later.

### Creating the Network

The createNetwork function takes in a random transform, an entropy generator, and a list of layer definitions, and returns a network.

For the XOR network, the createNetwork function is:

let n = createNetwork normals (mkStdGen 4) [l, l', l'']

Our source of entropy is the very random: mkStdGen 4, which will always result in the same generator.

#### Random Transforms

The random transform function is a transform that operates on a stream of uniformly distributed random numbers and returns a stream of floating point numbers.

Currently, the two defined distributions are:

• uniforms - A trivial function that returns a stream of uniformly distributed random numbers
• normals - A slightly less-trivial function that uses the Box-Muller transform to create a stream of numbers ~ N(0, 1)

Work is being done to offer a student t-distribution, which would require support for a chi-squared distribution transformation.

### Training the Network

In order to train a network, you must create a new trainer:

let t = BackpropTrainer (3 :: Float) quadraticCost quadraticCost'

The BackpropTrainer type takes in a learning rate, a cost function, and its derivative.

The actual training of the network, the fit function uses the trainer, a network, and the training data, and returns a new, trained network. For the XOR network, this is:

let n' = trainUntilErrorLessThan n t online dat 0.01

LambdaNet provides three training methods:

• trainUntil
• trainUntilErrorLessThan
• trainNTimes

The trainUntil function takes a TrainCompletionPredicate (check Network/Trainer.hs) for more information, and the last two are simply wrappers for the first one that provide specific predicates.

The calculated error is what is returned by the cost function.

#### Cost Functions

Currently, the only provided cost function is the quadratic error cost function, quadraticCost and its derivative, quadraticCost'. I am about to add the cross-entropy cost function.

#### Selection Functions

Selection functions break up a dataset for each round of training. The currently provided selection functions are:

• minibatch n - You must provide an n and partially apply it to minibatch to get a valid selection function. This function updates the network after every n passes.
• online - Using this function means that the network updates after every training example.

For small data sets, it's better to use online, while for larger data sets, the training can occur much faster if you use a reasonably sized minibatch.

### Using the Network

Once the network is trained, you can use it with your test data or production data:

predict (fromList [1, 0]) n'

LambdaNet at least attempts to follow a Scikit-Learn style naming scheme with fit and predict functions.

Once a network has been trained, the weights and biases can be stored in a file:

saveNetwork "xor.ann" n'

By calling saveNetwork with a file path, you can save the state of the network.

Loading a network requires passing in a list of layer definitions for the original network, but will load all the weights and biases of the saved network:

n'' <- loadNetwork "xor.ann" [l, l', l'']

Note that the loadNetwork function returns an IO (Network), you can't simply call predict or train on the object returned by loadNetwork. Using the approach in XOR.hs should allow you to work with the returned object.

## Currently Under Development

What has been outlined above is only the first stages of LambdaNet. I intend to support some additional features, such as:

• Regularization functions
• Additional trainer types (RProp, RMSProp)

### Regularization Functions and Momentum

Standard backprop training is subject to overfitting and falling into local minima. By providing support for regularization and momentum, LambdaNet will be able to provide more extensible and robust training.

## Generating the Documentation Images

All the documentation for the network was generated in the following manner. In the docs folder, run:

runhaskell docs.hs
python analysis.py

Note that I am currently working on removing the Python image analysis from the library, and switching it with Haskell and gnuplot. I'm also working on using the generated images in network documentation.


text:
# Completing the square

## Definition

In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form $ax^{2}+bx+c$ to the form $a(x+h)^{2}+k$.

## Solved Exercises

### Struggling with math?

Access detailed step by step solutions to millions of problems, growing every day!


text:
# Application of dipole moment

## Application of dipole moment in chemistry

The application of the dipole moment differentiate between ionic and covalent compound and also provides the framework of inorganic and organic chemistry.

1. When a covalent chemical bond is formed between two identical atoms, the bonding electrons equally sharing by two atoms. Thus the centers of gravity of the two electrons and nuclei therefore coincide.
2. But for two dissimilar atoms, two electrons are not symmetrically disposed. Because each atom has a different attraction for electrons.

When chlorine and bromine combine to form covalent HBr, the electrons forming the covalent bond displaced towards the bromine atom without any separation of the nucleus of an atom.

#### Ionic framework in covalent compounds

Application of dipole moment data uses for the determination of the ionic or covalent character of a heteronuclear diatomic compound.

Let us consider compound HBr having the observed dipole moment = μobs and the bond length l cm.

If we consider HBr is a purely ionic compound the charge on H and Br = 4.8 × 10-10 esu. Thus the dipole moment for this ionic compound

∴ μionic = e × ℓ
= (4.8 × 10-10) ℓ esu cm
where l = bond length

Thus the original dipole moment differs from the calculated dipole moment. This data used to calculate the percentage of ionic character of HBr.

$\therefore&space;Ionic\,&space;character=\left&space;(&space;\frac{\mu&space;_{obs}}{\mu&space;_{ionic}}&space;\right&space;)\times&space;100$

${\color{DarkBlue}&space;=\left&space;(&space;\frac{\mu&space;_{obs}}{4.8\times&space;10^{-10}\times&space;l}&space;\right&space;)\times&space;100}$

where l = bond length of the polar bond.

### Application of induced polarization

The induced polarization used for the determination of molecular radius. The induced electric polarization formula

$P_{i}=\left&space;(&space;\frac{D_{0}-1}{D_{0}+2}&space;\right&space;)\,&space;\frac{M}{\rho&space;}&space;=&space;\frac{4}{3}\,&space;\pi&space;\,&space;N_{0}\,&space;\alpha&space;_{i}$

If  D0 close to unity

$\therefore&space;\left&space;(&space;\frac{D_{0}-1}{3}&space;\right&space;)\times&space;22400=\frac{4}{3}\,&space;\pi&space;\,&space;N_{0}\,&space;r^{3}$

At NTP, M/ρ = molar volume = 22400cc/mole and for spherical molecule, αi= r3.

$r^{3}=\left&space;(&space;\frac{22400}{4\,&space;\pi&space;\,&space;N_{0}}&space;\right&space;)\left&space;(&space;D_{0}-1&space;\right&space;)$$=2.94\times&space;10^{-21}\left&space;(&space;D_{0}-1&space;\right&space;)$

Thus from the known value of capacitance, we can easily determine the radius of the molecules.

### Dipole moment and structure of molecules

The dipole moment application provides the structure and bond angle of different molecules. Thus for mono-atomic noble gases are non-polar because the charge of the constituent atom is distributed symmetrically.

#### Homonuclear diatomic molecules

Nitrogen, oxygen, and chlorine are the homonuclear diatomic molecules with symmetrical charge distributions. Thus the dipole moment of such types of molecules is zero.

#### Heteronuclear diatomic molecules

Hydrogen bromide and hydrogen iodide have non zero values of dipole moment. This indicates the unsymmetrical charge distribution between two bonding atoms.

H+ — I

Due to the difference in electronegativity of the constituent atoms in heteronuclear diatomic molecules always polar.

Thus the electron pair is not equally shared and shifted to the more electronegative atom.

 Hydrogen chloride 1.03 Debye Hydrogen bromide 0.79 Debye Hydrogen iodide 0.38 Debye Hydrogen fluoride 2.00 Debye

#### Calculate the dipole moment of co

The electronegativity difference between carbon and oxygen in CO is very large but the dipole moment of carbon monoxide very low.

This suggested that the charge density in the oxygen atom somehow back-donated to the carbon atom. Thus CO formed a coordinate covalent bond directing towards carbon atom.

### The dipole moment of polyatomic molecules

Carbon dioxide, barium chloride, stannous chloride have zero dipole moment indicating that the molecules have a symmetrical charge distribution between the bond.

Thus in carbon dioxide electric moment of one carbon-oxygen bond cancels the electric moment of the other carbon-oxygen bond.

The electric moment or bond moment associated with the bond arising from the difference of electronegativity.

In compounds, the vectorial addition of the ionic bond moments gives the resultant dipole moment of the compound.

∴ μ2 = m12 + m22 + 2m1m2Cosθ
where m1 and m2 are the bond moments.

Bond moments help to calculate the bond angle of carbon dioxide molecule, where µ = 0 and m1 = m2 = m.

∴ 0 = 2m2(1 + cosθ)
or, θ = 180°

#### The dipole moment of hydrogen sulfide

Water and hydrogen sulfide non-polar because they have non-linear structures. The bond angle can be calculated from the polarity of the molecules.

#### The polarity of the water molecule

Due to the non-linear structure of water molecules, the net dipole moment ≠ 0.

If the dipole moment of water
μ = 1.84 D and bond moment = 1.60 D.

∴ μ2 = 2 m2 (1+ cosθ )
or, (1.84)2 = 2 (1.60)2 (1+ cosθ )
∴ θ = 105°

Thus the contribution of non-bonding electrons towards the total dipole moment included within the bond moment.

#### The polarity of boron trifluoride

Boron trichloride, boron trifluoride are the tetratomic compound having dipole moment zero, indicating that they have regular planar structure.

Their halogen atoms are on a plane at the corner of the equilateral triangle and boron atom at the intersection of the molecules. Thus the bond moment of the above molecules is zero.

#### The polarity of ammonia and phosphine

Other types of the molecule such as ammonia and phosphine are polar, where μ≠0 indicated that the molecule has a pyramidal structure.

Hence three hydrogen atoms on a plane and nitrogen atom at the apex of the pyramid in ammonia and phosphine.

But NF3 shows a very small bond moment although there is a great difference of electronegativity between nitrogen and fluorine atoms and similar structure of NH3.

Thus this low value of µ in NF3 explained by the fact that the resultant bond moment of the three nitrogen – fluorine bonds are acting in the opposite direction to that of the lone pair placed at the nitrogen-atom.

But in NH3, the resultant bond moment is acting in the same direction as that of the lone pair electrons.

#### Penta atomic molecule

Methane, carbon tetrachloride, platinum chloride are examples of Penta-atomic molecules having zero dipole moment.

This suggests that the molecules either regular tetrahedral or square planer structure. But polar molecules of this type have pyramidal structures.

#### The bond polarity of methane

For calculating bond polarity, let us discuss the structure of methane that has regular tetrahedral structure and the angle of each H-C-H = 109°28ˊ.

Thus the group moment of methane depends on the arrangement of the bonds in the group and the difference of the electronegativity of the constituent atoms forming the bonds in the group.

But it can be shown that the group moment of methyl group identical to the bond moment of a carbon hydrogen bond.

Thus the two bond moments cancel each other and the resulting dipole moment of methane is zero.

∴ mCH3 = 3 mCH × Cos(180° -109°28՛)
= 3 mCH Cos 70°32՛
= 3 mCH × (1/3)
So, mCH3 = mCH

Thus µ of methane
= mCH (1 + 3 Cos 109°28՛)
= 0

#### Dipole moment of CH3Cl and CHCl3

μ2 = m12 + m22 + 2 m1m2 Cosθ
But here θ = 0° hence Cosθ = 1

Thus μ2 = m12 + m22 + 2 m1m2
= (m1 + m2)2

∴ μ = (m1 + m2)
= (mCH3 + mCl)
= (1.5 D + 0.4 D)
Thus μCH3Cl = 1.9 D

But for CHCl3

μ = (m1 + m2)
= (mCCl3 + mCH)
= (1.5 D + 0.4 D)
Hence μCHCl3 = 1.9 D

A similar calculation done for the group moment of C2H4, C3H7, C4H9, etc. Thus the bond moment equal to the bond moment of carbon-hydrogen.

But the application of dipole moment gives identical and homologous alcohol and saturated hydrocarbon.


text:
# BibLaTeX: DOI Link over Reference and Removing Comma Before Pages

Using BibLaTeX (not BibTeX), I want to put a link over the entire reference and not the parts separately. In addition, I want to remove the period between the date and the pages.

A minimal example is as follows:

\documentclass{article}

\usepackage[backend=biber, doi=false, hyperref=true]{biblatex}
\bibliography{references}

\usepackage{hyperref}

\usepackage{filecontents}
\begin{filecontents}{references.bib}
@article{ref,
author = {Surname, Initials},
title = {Title},
doi = {11.1111/111111},
journal = {Journal Name},
pages = {11},
volume = {1},
year = {2018}
}
\end{filecontents}

\begin{document}
Reference~\cite{ref}.
\printbibliography
\end{document}

This should give the bibliography output equivalent to

Initials Surname. "Title". In: \href{http://dx.doi.org/11.1111/111111}{Journal Name \textbf{1} (2018) 11}.

I am currently able to put the link over the journal, volume, year and date separately using

\newbibmacro{string+doi}[1]{\iffieldundef{doi}{#1}{\href{http://dx.doi.org/\thefield{doi}}{#1}}}
\DeclareFieldFormat[article]{volume}{\usebibmacro{string+doi}{\textbf{#1}}\space}
\DeclareFieldFormat[article]{journaltitle}{\usebibmacro{string+doi}{#1}}
\DeclareFieldFormat[article]{date}{\usebibmacro{string+doi}{#1}}
\DeclareFieldFormat[article]{pages}{\usebibmacro{string+doi}{#1}}

As I said, I would like it to appear over the entire reference as shown above. In addition, I haven't been successful in removing that comma between the date and page in analogy with other such cases (discussed extensively here) – probably I'm only lacking the appropriate bibmacro?

It is not enough to use field formats here, since those are confined to the output of one field. If you want one long hyperlink you will have to wrap several bibmacros up in the linking macro. Unfortunately, this has to be done on the bibliography driver level since the journal+issuetitle macro does not inlcude the page number. In the MWE below we use xpatch to modify the driver.

\documentclass{article}

\usepackage[backend=biber, doi=false, hyperref=true]{biblatex}

\usepackage{hyperref}

\DeclareFieldFormat[article,periodical]{volume}{\mkbibbold{#1}}
\DeclareFieldFormat[article,periodical]{journaltitle}{#1}
\DeclareFieldFormat[article,periodical]{pages}{#1}

\usepackage{xpatch}
\xpatchbibdriver{article}
{\usebibmacro{journal+issuetitle}%
\newunit
\usebibmacro{byeditor+others}%
\newunit
\usebibmacro{note+pages}}
\usebibmacro{journal+issuetitle}%
\newunit
\usebibmacro{byeditor+others}%
\newunit
\usebibmacro{note+pages}}}
{}{}

\begin{document}
Reference~\cite{sigfridsson}.
\printbibliography
\end{document}

• Putting a \nopunct at the right place in addition gets rid of the punctuation before the pages. So, problem solved. Thanks! – Cari Baur Apr 11 '18 at 17:37