text:
# How should the Calculus of Variations deal with $\delta(t-t_0)$ variations?

I'm familiar with using the Calculus of variations to find the condition for which first order variations of a functional wrt a function are zero:

1. We start with a functional $J[x]= \int_{t_f}^{t_i}L(x(t),\dot x(t),t)dt$, where $t_i, t_f$ are constants and $\dot x(t)= dx/dt$

2. If $J[f]$ is stationary at $f$, we add to $f$ another continuous function $\epsilon\eta(t)$ where $\eta(t)$ is arbitrary and $\epsilon$ is a positive number

3. For any small number $\epsilon$ close to 0, the first order variation of the functional is zero around $f$, giving the Euler-Lagrange equations as the necessary condition.

If $\delta(t)$ is what physicists/engineers call the Dirac-Delta function, even though mathematicians wouldn't define it as a function:

Can the above procedure be applied to $\delta(t-t_0)$ variations of f at $t = t_0$ to again yield the Euler-Lagrange equations?

I've had a go by adding $\epsilon\delta(t-t_0)$ to $f$ as the variation, getting the differential of $L$ in terms of the usual partial derivatives and differentials of the independent variables. But the differential of $x$, for example, then becomes $dx = \epsilon\delta(t-t_o)$ which appears to be infinite for any $\epsilon$, making my expression for $dL$ nonsense.

Again, according to an answer given on PSE that motivated me to ask my question here, the change in $L$ to an $\epsilon\delta(t-t_0)$ variation in $\dot x$ is:

$$dL= {\partial L \over \partial \dot{x_1}} \delta \dot{x_1} = {\partial L \over \partial \dot{x_1}}\epsilon\delta(t-t_0)$$

Is this correct?

You can't proceed naively because $\epsilon \delta(x)$ is not continuos for any finite $\epsilon$.


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# For any positive integer $n$, prove the following inequality. [duplicate]

Prove that for any positive integer n, the following inequality is true. $$\left(1+ \frac{1}{n}\right)^n < \left(1+\frac{1}{n+1}\right)^{n+1}$$

Attempt

Not a good attempt but this is my thinking

It will not be a problem when checking whether this is true for smaller integers. Only for larger integers, it will be difficult to prove. So, I took the equation LHS-RHS < $$0$$. I took limit for n tending to infinity and then applied L'Hospital Rule( though the equation was difficult to handle) & therefore could not do anything further.

Adding graph for @Rebellos 's last equation

## marked as duplicate by Jack D'Aurizio, Martin R, Don Thousand, Leucippus, user10354138Nov 6 '18 at 0:19

• Any hints and tips are welcome. – jayant98 Nov 5 '18 at 18:17
• Try the AM-GM inequality. This has been asked uncountably times, by the way. – Jack D'Aurizio Nov 5 '18 at 18:24
• – Martin R Nov 5 '18 at 20:24

If you are allowed to take derivatives, then letting $$f(x)=\left(1+\frac{1}{x}\right)^x$$, we have $$f'(x)=f(x)\left(\frac{(x+1)\log\left(1+\frac{1}{x}\right)-1}{x+1}\right)$$

And $$f'(x)>0$$ as long as $$x>0$$.

An alternative approach :

Assume it indeed is :

$$\bigg(1+ \frac{1}{n}\bigg)^n < \bigg(1+\frac{1}{n+1}\bigg)^{n+1}$$

Then, since the natural logarithm is a strictly increasing function, it is :

$$\ln\bigg(1+ \frac{1}{n}\bigg)^n < \ln\bigg(1+\frac{1}{n+1}\bigg)^{n+1} \Leftrightarrow n\ln\bigg(1+\frac{1}{n}\bigg) < (n+1)\ln\bigg(1+\frac{1}{n+1}\bigg)$$

$$n\bigg[\ln\bigg(1 + \frac{1}{n+1}\bigg)-\ln\bigg(1+\frac{1}{n}\bigg)\bigg]+\ln\bigg(1+\frac{1}{n+1}\bigg) > 0$$

$$\Leftrightarrow$$

$$\ln\Bigg(\frac{1+\frac{1}{n+1}}{1+\frac{1}{n}}\Bigg)^n + \ln\bigg(1+ \frac{1}{n+1}\bigg) >0$$

But for $$n \in \mathbb N^+$$ we can clearly see that this is true (why ?). Thus the initial inequality holds, since the logical $$\Leftrightarrow$$ implies it.

• @cansomeonehelpmeout Thanks for the typo correction ! – Rebellos Nov 5 '18 at 18:38
• Thanks for helping me but I have a doubt from 3rd step onwards. The 3rd term in the 3rd step shouldn't be positive and hence for the further steps also? – jayant98 Nov 5 '18 at 19:07
• @jayant98 Yes and this just made our life easier! – Rebellos Nov 5 '18 at 19:10
• @jayant98 Did you manage to prove why the last expression holds ? – Rebellos Nov 5 '18 at 19:24
• Nope,still trying. :| – jayant98 Nov 5 '18 at 19:26

I know this question is already answered, but I want to give an answer that only uses the binomium of Newton. This is certainly not the easiest way to show the inequality, but it is an "artisanal" way.

By the binomium we get \begin{align*} \left(1+ \frac{1}{n}\right)^n &= \sum_{k=0}^n \binom{n}{k} \frac{1}{n^k} \\ &= 1 + \frac{n}{1!}\frac{1}{n} + \frac{n(n-1)}{2!}\frac{1}{n^2} + \frac{n(n-1)(n-2)}{3!}\frac{1}{n^3}+ \ldots + \frac{n(n-1)(n-2)\cdots 1}{n!}\frac{1}{n^n} \\ &= 2 + \frac{1}{2!}\left(1-\frac{1}{n}\right)+ \frac{1}{3!}\left(1-\frac{1}{n}\right)\left(1-\frac{2}{n}\right)+ \cdots + \frac{1}{n!}\left(1-\frac{1}{n}\right)\left(1-\frac{2}{n}\right)\cdots\left(1-\frac{n-1}{n}\right). \end{align*} Similarly \begin{align*} \left(1 + \frac{1}{n+1}\right)^{n+1} &= 2 + \frac{1}{2!}\left(1-\frac{1}{n+1}\right)+ \frac{1}{3!}\left(1-\frac{1}{n+1}\right)\left(1-\frac{2}{n+1}\right)+ \cdots \\ & + \frac{1}{n!}\left(1-\frac{1}{n+1}\right)\left(1-\frac{2}{n+1}\right)\cdots\left(1-\frac{n-1}{n+1}\right) \\ &+ \frac{1}{(n+1)!}\left(1-\frac{1}{n+1}\right)\left(1-\frac{2}{n+1}\right)\cdots\left(1-\frac{n}{n+1}\right). \end{align*}

For every term in the expression for $$\left(1+\tfrac{1}{n}\right)^n$$ there is a similar term in the expression for $$\left(1+\tfrac{1}{n+1}\right)^{n+1}$$ that is equal of bigger. Furthermore, the last expression contains one positive term more. Hence the latter expression is bigger.

Since $$\log$$ function is strictly increasing, we have that

$$(1+ \frac{1}{n})^n < (1+\frac{1}{n+1})^{n+1} \iff n\log\left(1+\frac1n\right)<(n+1)\log\left(1+\frac1{n+1}\right)$$

and for $$x>1$$

$$f(x)=x\log x \implies f'(x)=\log x+1>0$$

format: plain


text:
# What does non-contact forcemean?

###### Question:

What does non-contact forcemean?

### What’s an algebraic expression for 25 less than a number

what’s an algebraic expression for 25 less than a number...

### Think about the narrator's role in any story. He or she is the person telling the story and providing the descriptive details. What do the details suggest about the narrator's point of view?

Think about the narrator's role in any story. He or she is the person telling the story and providing the descriptive details. What do the details suggest about the narrator's point of view?...

### A 29-year-old G1P0 woman at 41 weeks gestation presents for a prenatal visit. Her prenatal course is complicated by tobacco abuse and intermittent prenatal care. Her last visit was at 35 weeks. Prenatal labs are unremarkable except cervical DNA probe positive for Chlamydia, which was treated, and a Pap smear with low-grade squamous intraepithelial lesion. Ultrasound at 21 weeks was consistent with gestational age based on her certain regular LMP. Her vitals reveal a blood pressure of 128/76; pul

A 29-year-old G1P0 woman at 41 weeks gestation presents for a prenatal visit. Her prenatal course is complicated by tobacco abuse and intermittent prenatal care. Her last visit was at 35 weeks. Prenatal labs are unremarkable except cervical DNA probe positive for Chlamydia, which was treated, and a ...

### You are the owner of a Comfort Inn and would like to hire a new general manager. In terms of preventing possible agency problems, what should you include in your offer?

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### What is the part on an enzyme where a substrate molecule attaches

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Which behavior is an example of how animals defend their territory? O A. Rhinoceros males urinate to mark the area where they live. OB. Several dolphins swim around a school of fish and take turns feeding. OC. Hyenas and lions compete for prey. O D. A bee shares the location of food with other bees....

### Proteins work as buffers to prevent acidosisOO False

Proteins work as buffers to prevent acidosisOO False...

### Tim counted 12 nails to fix his wagin he used only 1/2 of them how many did he use\

tim counted 12 nails to fix his wagin he used only 1/2 of them how many did he use\...

### Sodium is in group 1 of the periodic table. A sodium ion (Na+) has a charge of +1. Which of the following would be most likely to combine with an sodium ion to form a new compound? A.a helium atom that has no charge B.a calcium ion with a charge of +2 C.a fluoride ion with a charge of -1 D.a lithium ion with a charge of +1

Sodium is in group 1 of the periodic table. A sodium ion (Na+) has a charge of +1. Which of the following would be most likely to combine with an sodium ion to form a new compound? A.a helium atom that has no charge B.a calcium ion with a charge of +2 C.a fluoride ion with a charge of -1 D.a lithium...

### What three things most directly affect carrying capacity?

What three things most directly affect carrying capacity?...

### What is hydrogen ?,,,

what is hydrogen ?,,,...

### 2 of 4 The total volume of the diluted solution is 212 mL. What is the concentration of KMnO4 in the diluted solution? 2.6 x 10-6 mol KMnO4 M

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### In the diagram below, AB and BC are tangent to 0. What is the measure of ADC?A.252°B.180°C.216°D.144°

In the diagram below, AB and BC are tangent to 0. What is the measure of ADC?A.252°B.180°C.216°D.144°...

### What is the difference between a certification and a license? Certifications give people specific privileges to enhance their careers and licenses are more general. Certifications and licenses are not different because they both show that a person has completed advanced training. Certifications reflect a person's skills and knowledge and licenses give people special privileges. Certifications only apply to biomedical careers and licenses apply to all medical professional careers.

What is the difference between a certification and a license? Certifications give people specific privileges to enhance their careers and licenses are more general. Certifications and licenses are not different because they both show that a person has completed advanced training. Certifications refl...

### Which of these units should be used to measure the mass of a small dog? A. milligram B. centigram C. gram D. kilogram

Which of these units should be used to measure the mass of a small dog? A. milligram B. centigram C. gram D. kilogram...

### How might life today be different without sailboats?

How might life today be different without sailboats?...

### 4. What best describes the role of mutations in evolution? a. Mutations are not necessary for evolution because they are not herita b. Mutations are useful for evolution to occur because they promote ger uniformity within a population. c. Mutations are essential for evolution to occur because they produce variations within a population. d. Mutations are essential for evolution to occur because they are respo environmental changes.

4. What best describes the role of mutations in evolution? a. Mutations are not necessary for evolution because they are not herita b. Mutations are useful for evolution to occur because they promote ger uniformity within a population. c. Mutations are essential for evolution to occur because they p...

### Write an equation in slope intercept form of the line with slope 1/2 and y-intercept of 8. Then graph the line

write an equation in slope intercept form of the line with slope 1/2 and y-intercept of 8. Then graph the line...

-- 0.011990--


text:
## Jun-Yong Park (朴俊勇)

School of Mathematics
University of Minnesota
127 Vincent Hall
206 Church St. SE
Minneapolis, MN 55455

Office: 454 Vincent Hall

Email: junepark "at" math.umn.edu

I'm a Ph.D. student in the School of Mathematics at University of Minnesota.
My advisor is Craig Westerland, and my interests are in algebraic geometry & topology of moduli spaces.

### Publications and Preprints

• Lantern substitution and new symplectic 4-manifolds with $${b_{2}}^{+} = 3$$ (with A. Akhmedov)
Mathematical Research Letters, Vol. 21, No. 1 (2014), P.1-17.
• (Preprint, Journal)

• Unique fiber sum decomposability of genus 2 Lefschetz fibrations
• (Preprint)

• Planar line arrangements and hyperelliptic Lefschetz fibrations
• (Preprint)

• Topology and arithmetic geometry of hyperelliptic Lefschetz fibration moduli space
• (Preprint)

### Miscellaneous

Last updated: January 29th, 2015.

The views and opinions expressed in this page are strictly those of the page author.
The contents of this page have not been reviewed or approved by the University of Minnesota.


text:
# zbMATH — the first resource for mathematics

Smooth four-manifolds and complex surfaces. (English) Zbl 0817.14017
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 27. Berlin: Springer-Verlag. x, 520 p. (1994).
In the early 1960’s there were significant breakthroughs in two branches in mathematics. On the one hand, after the rigorous foundation of algebraic geometry and complex-analytic geometry by sheaf-theoretic and cohomological methods in the foregoing decade, K. Kodaira succeeded in giving the classical theory of complex algebraic surfaces the necessary precision and solid base. He not only re-established and extended the Enriques’ classification of surfaces, including also compact non- algebraic surfaces, but he also created (together with D. C. Spencer) the deformation-theoretic framework for the moduli theory of special classes of complex surfaces. In particular, he did the first steps towards the classification of K3-surfaces and began the systematic study of (non-) algebraic elliptic surfaces, a topic that had not been taken up by the great old Italian geometers.
On the other hand, and almost simultaneously, S. Smale made a great progress towards the old and difficult problem of classifying all possible smooth and/or topological structures on a given differentiable manifold. He established the generalized Poincaré conjecture for manifolds of dimension at least five and, in the sequel, proved the famous $$h$$-cobordism theorem. Based on these results, the main classification scheme for manifolds of dimension $$\geq 5$$ could be developed, in the following decade, and by the mid 1970’s it had reached its nearly complete shape. The major effort then shifted to the remaining cases of three-manifolds and four-manifolds. Due to the work of W. Thurston in the early 1980’s, there is now a possible classification scheme in dimension three, too, at least in a concrete conjectural form, which may serve as the guiding idea for completing the classification theory of three-manifolds. The case of four-manifolds, however, still remains exceptional and fairly mysterious, even with regard to the existence of guiding conjectural outlines as in the three-dimensional situation. – One possible approach to a better understanding of four- manifolds is provided by the fact that some of them occur as the topological objects underlying smooth complex surfaces, for which the well-developed Kodaira classification theory is available. On the other hand, many still unsolved problems on the topology and differential geometry of algebraic surfaces lead to the methods of real four-manifold theory.
This link between smooth complex surfaces and real four-manifolds is the topic of the present book. The basic method used by the authors is S. Donaldson’s recent approach of studying four-manifolds by introducing a suitable gauge theory for them and, along this way, defining new smooth invariants for four-manifolds, which possibly are appropriate substitutes for the usual homotopy or homeomorphism invariants that definitely fail in the exceptional case of dimension four. Donaldson’s so-called “polynomial invariants” are not easily accessible to computation, and in general very little is known about their structure and classifying significance. On the other hand, Donaldson’s gauge theory methods are particularly well-suited to study four-manifolds underlying complex algebraic surfaces. Namely, according to a theorem S. K. Donaldson [Proc. Lond. Math. Soc., III. Ser. 50, 1-26 (1985; Zbl 0529.53018)], for an algebraic surface the (gauge-theoretic) moduli space of anti-self-dual connections is equivalent to the moduli space of certain stable holomorphic vector bundles on the surface, and that makes it possible to study Donaldson’s polynomial invariants of complex algebraic surfaces by algebro-geometric techniques.
According to the aim of this book, the authors have devoted much of the text to study Donaldson’s invariants for special types of algebraic surfaces and, as a consequence from their results, to discuss new informations about the differential geometry of those kinds of surfaces. Altogether, the central part of this work represents a far-reaching continuation of the (generally very intensive) study of the Donaldson invariants of algebraic surfaces, based upon a subtle combination of methods from the algebro-geometric classification theory and the computation of polynomial invariants. The main result, derived in the course of the text, shows that the smooth classification of algebraic surfaces, unlike their topological classification or the case of manifolds of dimension different from four, is deeply related to their algebro-geometric structure. This unique phenomenon has been proved by the authors in a previous original article [cf. R. Friedman and J. W. Morgan, J. Differ. Geom. 27, No. 2, 297-369 and No. 3, 371- 398 (1988; Zbl 0669.57016 and Zbl 0669.57017)], independently also by C. Okonek and A. Van de Ven [Invent. Math. 86, 357-370 (1986; Zbl 0613.14018)], and these results comprise most of the last two chapters of this book, together with furthergoing results on the topological characterization of algebraic K3-surfaces and (blow-ups of) nonrational elliptic surfaces.
In this regard the present book represents both a research monograph on algebro-geometric methods in the theory of smooth four manifolds, on the one hand, and a textbook introducing the reader into the classification theory of algebraic surfaces and into Donaldson’s gauge theory of smooth four-manifolds, on the other hand. The text consists of seven chapters which are arranged as follows.
Chapter 1 resembles, in a fairly self-contained manner and for the convenience of the reader, the Kodaira classification of algebraic surfaces. The authors have put special emphasis on elliptic surfaces, since those play a crucial role in the sequel, and because the literature on elliptic surfaces is much too vast as for reasonable references. This chapter also contains some new results on the deformation behavior of elliptic surfaces. – Chapter 2 discusses the topology of elliptic surfaces and provides, in this way, the perspective of the later differentiable classification of algebraic surfaces. – Chapter 3 deals with the basics of Donaldson’s gauge theory for four-manifolds and, in particular, with the definition of his polynomial invariants. This includes material on the space of connections, moduli of anti-self-dual connections, and various kinds of Donaldson polynomial invariants. – Chapter 4 is devoted to the moduli spaces of stable holomorphic vector bundles over complex algebraic surfaces and the gauge theory corresponding to the Kähler differential geometry of surfaces. – Then, in chapter 5, the authors develop explicit methods for computing Donaldson polynomials of algebraic surfaces, in terms of algebraic cohomology classes. In an appendix, this method is compared to different technical approaches based on various notions of determinants of line bundles (due to Atiyah-Singer, Bismut-Gillet-Soulé, and others). – Chapter 6 switches back to four-manifold theory. The objects under consideration are particular four-manifolds, namely those with so-called big diffeomorphism groups with respect to a given cohomology class, and the structure of their Donaldson polynomial invariants. – The concluding chapter 7 provides explicit computations of Donaldson polynomials for elliptic surfaces. Using the techniques developed in chapters 5 and 6, together with some standard algebraic geometry, the authors derive their main results (explained above) and some conjectural hints for further research in this area.
The authors have done their very best to present a comprehensive, detailed and self-contained combination of a research monograph and a textbook on this fascinating, utmost actual topic in mathematics today. Requiring only the basics of algebraic geometry and gauge theory in differential geometry, the content of the book leads the reader to the forefront of contemporary research in both complex surface theory and the topology of smooth four-manifolds. Already the introduction is written in a very careful, motivating and enlightening way, and each section comes with its own detailed introduction. The style of writing is masterly lucid, rigorous and comprehensible. As for more background material, the reader should be referred to the classics “Compact complex surfaces” by W. Barth, C. Peters and A. Van de Ven [Ergeb. Math. Grenzgeb., 3. Folge, Bd. 4 (1984; Zbl 0718.14023)] and “The geometry of four-manifolds” by S. K. Donaldson and P. B. Kronheimer (1990), but the essential basic material is actually contained in the beautiful book under review.

##### MSC:
 14J15 Moduli, classification: analytic theory; relations with modular forms 14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations 32G13 Complex-analytic moduli problems 57R55 Differentiable structures in differential topology 58D27 Moduli problems for differential geometric structures 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)


text:
# Inverse dft and dft question

1. Oct 25, 2011

### seamie456

How does one edit the input sequence and the results so as to be able to calculate the inverse dft with the dft algorthm?

2. Oct 27, 2011

### jasonRF

if you define the DFT by:
$$X_k = \sum_{n=0}^{N-1} x_n e^{-i 2 \pi \frac{k}{N} n} = DFT\left(x\right)_k$$
and it inverse DFT by
$$x_n = \frac{1}{N} \sum_{k=0}^{N-1} X_k e^{+i 2 \pi \frac{k}{N} n} = IDFT\left(X\right)_n,$$
then the way to get $x_n$ from $X_k$ using a DFT would be something like,
$$x_n = \frac{1}{N} \left(\sum_{k=0}^{N-1} X_k^* e^{-i 2 \pi \frac{k}{N} n}\right)^* = \frac{1}{N} \left( DFT \left( X^* \right) \right)_n^*$$.
where the asterix represents conjugation. Does that make sense?

jason


text:
# Is Gibbs free energy affected by a catalyst?

No, because a catalyst will change the activation energy, but not the overallenthalpy change of a reaction, $\Delta H$.
(The equation for finding Gibb's free energy is $\Delta G = \Delta H - T \Delta S$)


text:
poly

Polynomial Coefficients

poly(a)

• a should be a vector or a square matrix.
• If a is a vector, it returns cofficients of the polynomial with roots given by a. The first coefficient corresponds to the highest order term.
• If a is a square matrix, it returns coefficients of the characteristic polynomial of a, i.e., $$\det(a-I\lambda)=0.$$
• a should not contain inf or nan values.
• If a is a logical or character array, it will be converted to a double array before executing poly(a).
• If a is empty, it returns 1.

Example 1: Plotting the polynomial having roots [3 4 5].

r=[3 4 5];
p=poly(r);
x=linspace(2.8,5.5);
plot(x,polyval(p,x))

p =
1.000  -12.00   47.00  -60.00

Plot is generated.

Example 2: Coefficients of characteristic polynomial of a matrix.

A=[1 2;3 4];
p=poly(A)

p =
1.000  -5.000  -2.000


text:
### Marginal sequential Monte Carlo for doubly intractable models

Bayesian inference for models that have an intractable partition function is known as a doubly intractable problem, where standard Monte Carlo methods are not applicable. The past decade has seen the development of auxiliary variable Monte Carlo techniques (M{\o}ller et al., 2006; Murray et al., 2006) for tackling this problem; these approaches being members of the more general class of pseudo-marginal, or exact-approximate, Monte Carlo algorithms (Andrieu and Roberts, 2009), which make use of unbiased estimates of intractable posteriors. Everitt et al. (2017) investigated the use of exact-approximate importance sampling (IS) and sequential Monte Carlo (SMC) in doubly intractable problems, but focussed only on SMC algorithms that used data-point tempering. This paper describes SMC samplers that may use alternative sequences of distributions, and describes ways in which likelihood estimates may be improved adaptively as the algorithm progresses, building on ideas from Moores et al. (2015). This approach is compared with a number of alternative algorithms for doubly intractable problems, including approximate Bayesian computation (ABC), which we show is closely related to the method of M{\o}ller et al. (2006).

### Bayesian model comparison with un-normalised likelihoods

Models for which the likelihood function can be evaluated only up to a parameter-dependent unknown normalising constant, such as Markov random field models, are used widely in computer science, statistical physics, spatial statistics, and network analysis. However, Bayesian analysis of these models using standard Monte Carlo methods is not possible due to the intractability of their likelihood functions. Several methods that permit exact, or close to exact, simulation from the posterior distribution have recently been developed. However, estimating the evidence and Bayes' factors (BFs) for these models remains challenging in general. This paper describes new random weight importance sampling and sequential Monte Carlo methods for estimating BFs that use simulation to circumvent the evaluation of the intractable likelihood, and compares them to existing methods. In some cases we observe an advantage in the use of biased weight estimates. An initial investigation into the theoretical and empirical properties of this class of methods is presented. Some support for the use of biased estimates is presented, but we advocate caution in the use of such estimates.

### Bootstrapped synthetic likelihood

Approximate Bayesian computation (ABC) and synthetic likelihood (SL) techniques have enabled the use of Bayesian inference for models that may be simulated, but for which the likelihood cannot be evaluated pointwise at values of an unknown parameter $\theta$. The main idea in ABC and SL is to, for different values of $\theta$ (usually chosen using a Monte Carlo algorithm), build estimates of the likelihood based on simulations from the model conditional on $\theta$. The quality of these estimates determines the efficiency of an ABC/SL algorithm. In standard ABC/SL, the only means to improve an estimated likelihood at $\theta$ is to simulate more times from the model conditional on $\theta$, which is infeasible in cases where the simulator is computationally expensive. In this paper we describe how to use bootstrapping as a means for improving SL estimates whilst using fewer simulations from the model, and also investigate its use in ABC. Further, we investigate the use of the bag of little bootstraps as a means for applying this approach to large datasets, yielding Monte Carlo algorithms that accurately approximate posterior distributions whilst only simulating subsamples of the full data. Examples of the approach applied to i.i.d., temporal and spatial data are given.

### Kernel Sequential Monte Carlo

We propose kernel sequential Monte Carlo (KSMC), a framework for sampling from static target densities. KSMC is a family of sequential Monte Carlo algorithms that are based on building emulator models of the current particle system in a reproducing kernel Hilbert space. We here focus on modelling nonlinear covariance structure and gradients of the target. The emulator's geometry is adaptively updated and subsequently used to inform local proposals. Unlike in adaptive Markov chain Monte Carlo, continuous adaptation does not compromise convergence of the sampler. KSMC combines the strengths of sequental Monte Carlo and kernel methods: superior performance for multimodal targets and the ability to estimate model evidence as compared to Markov chain Monte Carlo, and the emulator's ability to represent targets that exhibit high degrees of nonlinearity. As KSMC does not require access to target gradients, it is particularly applicable on targets whose gradients are unknown or prohibitively expensive. We describe necessary tuning details and demonstrate the benefits of the the proposed methodology on a series of challenging synthetic and real-world examples.

### Fast Approximate Bayesian Computation for Estimating Parameters in Differential Equations

Approximate Bayesian computation (ABC) using a sequential Monte Carlo method provides a comprehensive platform for parameter estimation, model selection and sensitivity analysis in differential equations. However, this method, like other Monte Carlo methods, incurs a significant computational cost as it requires explicit numerical integration of differential equations to carry out inference. In this paper we propose a novel method for circumventing the requirement of explicit integration by using derivatives of Gaussian processes to smooth the observations from which parameters are estimated. We evaluate our methods using synthetic data generated from model biological systems described by ordinary and delay differential equations. Upon comparing the performance of our method to existing ABC techniques, we demonstrate that it produces comparably reliable parameter estimates at a significantly reduced execution time.


text:
Friday, April 23, 2004
Plamen Stefanov
(Purdue University)

Boundary rigidity of Riemannian manifolds
2:30 pm / CH 445
Abstract. In this talk, I will present the recent progress in solving the boundary rigidity problem. Let (M,g) be a compact Riemannian manifold with boundary. Assume that for each pair of boundary points (x,y), we know the distance function d(x,y) . The manifold M is called boundary rigid, if it is determined uniquely by d(x,y) (known on the boundary). At present, boundary rigidity is known only for some classes of manifolds. The distance between boundary points measures also the travel times of light rays through the domain. This information can be extracted from the scattering kernel or from the associated hyperbolic Dirichlet-to-Neumann map.
We study this problem for simple manifolds, i.e., strictly convex ones with no caustics inside. The associated linearized problem is recovery of the so-called solenoidal part of a tensor from its X-ray transform If along geodesics. We show that N=I^*I is a pseudodifferential operator, analyze its principal symbol and invert it microlocally. We prove an a priori stability estimates for N. Using analytic pseudodifferential calculus, we show that for analytic metrics, the linearized problem is invertible. For the non-linear one, we prove generic local uniqueness and stability.

Friday, April 16, 2004
Yanyan Li (Rutgers University)
On a fully nonlinear version of the Yamabe problem
2:30 pm / CH 445
Abstract. We present some recent results on a fully nonlinear version of the Yamabe problem which includes existence and compactness of solutions and some Liouville type theorems.

Friday, April 2, 2004
Robert Sims (Princeton University)
A variational principle for the Sherrington-Kirkpatrick model
2:30 pm / CH 445
Abstract. Ising systems are models of "simple" magnetic structures composed of particles (called spins) whose values are restricted to be +1 or -1. For a finite region of such a system, one may ask: given a particular energy function, describing the interaction between these particles, is there a well-defined "infinite volume limit". For extensive systems, (roughly speaking: systems where the range of the interactions is assumed to be finite) such an infinite volume limit was known to exist. For the Sherrington-Kirkpatrick model, in which the range of the interactions is not restricted, the existence of such a well-defined limit was only recently proven. To make matters slightly more interesting, not only is the Sherrington-Kirkpatrick model one of infinite range, the couplings between spins (i.e. the interactions) are also random variables. I will discuss a recent result which characterizes this infinite volume limit in terms of a variational principle over "random overlap structures". I will also begin with a basic introduction to the methods of statistical mechanics.TBA

Friday, March 19, 2004
Tomio Umeda (Himeji Institute of Technology)
Generalized eigenfunctions of relativistic Schroedinger operators, the massless case
2:30 pm / CH 445
Abstract. It is well-known that generalized eigenfunctions of Schroedinger operators are superposition of plane waves and spherical waves satisfying the Sommerfeld radiation conditions. This fact is very natural from the view points of physics as well as mathematics. Unlike the Schroedinger operators, it is highly nontrivial from the view point of mathematics that generalized eigenfunctions of relativistic Schroedinger operators are superposition of plane waves and spherical waves satisfying the Sommerfeld radiation conditions, even though it is natural from the physics view point. In this talk, I will describe how to prove that generalized eigenfunctions of relativistic Schroedinger operators are superposition of plane waves and spherical waves, how different it is from that for the Schroedinger operators, and why it must be different.

Friday, March 12, 2004
Marco Merkli (McGill University)
Some recent developments in the theory of open quantum systems
2:30 pm / CH 445
Abstract. Over the last ten years, the mathematical theory of open quantum systems has experienced significant progress. The aim of this talk is to give an overview of phenomena, results and techniques involved in this branch of rigorous non-equilibrium quantum statistical mechanics. Systems considered consist either of a single heat reservoir (modelled by free Bosons or Fermions) interacting with a "small" quantum system possessing finitely many degrees of freedom (e.g. a spin), or several heat reservoirs interacting with each other, possibly via small systems. For the former class of systems, one is interested in the existence and stability of equilibria ("return to equilibrium", "thermal ionization"). For models where several reservoirs interact with each other, the goal is to show that the system approaches, in the large time limit, a "non-equilibrium stationary state", and to characterize that state by its thermodynamic properties, such as the strict positivity of entropy production, or the presence of energy or matter flows between the reservoirs. We formulate the theory of open quantum systems in the language of C* or W* dynamical systems. The dynamics can be described by a so-called Liouville operator (positive temperature Hamiltonian), whose spectral properties reflect the asymptotic behavior of the system. Many recent investigations focus on developing tools for spectral analysis of Liouville-type operators (complex deformation, Mourre theory). Another approach, based on algebraic scattering techniques, allows for an explicit and detailed description of limiting states for relatively simple, but physically relevant models.

Friday, March 5, 2004
Ben Chow (University of California at San Diego)
Monotonicity formulas for geometric flows
2:30 pm / CH 445
Abstract. Geometric flows deform geometric structures on manifolds such as metrics, submanifolds, and maps. The processes we will discuss are like heat equations where a smoothing process occurs. Monotonicity formulas tell us that the geometric structures are improving and are used to understand how the structures evolve. We shall focus on the heat equation on manifolds and the Ricci flow of metrics where a metric is evolved in the direction of its Ricci curvature. This talk will be aimed at graduate students and nonspecialists.

Friday, February 27, 2004
Dirk Hundertmark (University of Illinois at Urbana-Champaign)
Some results on Lieb-Thirring inequalities
2:30 pm / CH 445
Abstract. Non-relativistic quantum mechanics, or the spectral theory of Schr"odinger operators, is a fascinating theory for real matter. Up to date it is the best theory physicists came up with and, as a bonus, it also makes sense for mathematicians! However, it is a fairly abstract theory and getting an intuition for it is hard. This has mainly to do with the fact that we live in a classical world and, from our day to day life, are used to classical concepts, but certainly are not at all familiar with the quantum world.
So how does one bridge the gap between real matter and our classical intuition? Often one takes limits, for example, noticing that the Planck constant is really small, one takes it to zero, thereby ending up with the so-called semi-classical theories. Another one is trying to bound quantum mechanical expressions by the corresponding classical ones, not necessarily for a small Planck constant. A particularly useful type of inequalities, which does exactly this, are the so-called Lieb-Thirring inequalities. We will discuss the motivation behind these inequalities and present some of the recent results. The talk will be mainly non-technical, some understanding of convexity is helpful.

Friday, February 6, 2004
Robert L. Devaney (Boston University)
The Mandelbrot Set, the Farey Tree, and the Fibonacci Sequence
2:30 pm / CH 445
Abstract. In this lecture we describe several folk theorems concerning the Mandelbrot set. While this set is extremely complicated from a geometric point of view, we will show that, as long as you know how to add and how to count, you can understand this geometry completely. We will encounter many famous mathematical objects in the Mandelbrot set, like the Farey tree and the Fibonacci sequence. And we will find many soon-to-be-famous objects as well, like the "Devaney" sequence. There might even be a joke or two in the talk.

Friday, January 30, 2004
Robert Kauffman (University of Alabama at Birmingham)
Discovery Teaching at the Graduate Level
2:30 pm / CH 445
Abstract. Many people in this department have used a method of teaching where students prove theorems at the board, in class, instead of listening to lectures. This is standard in advanced calculus here, and has been used in other courses as well. Could this method of teaching be used in more graduate courses? Is it possible to cover a reasonable syllabus this way? What are the virtues and flaws of such a method?
This colloquium will feature a short presentation by Bob Kauffman, followed by comments from a panel consisting of Gunter Stolz, Lee Goswick, Gary Russell and Mitch Wyatt, and then by an open discussion between the audience and the panel and Bob.

Friday, January 23, 2004
A homotopy along $p$ for a system of PDE with a vector $p$-Laplace operator, the radial case.
2:30 pm / CH 445
Abstract. We consider the system of PDE containing a vector p-Laplace operator {\rm div} ( | \nabla u |^{p-2} \nabla u)+ \lambda |u|^{p-2}u=0,\quad x\in B(0,R) u(x)=0,\quad x\in \partial B(0,R),\\ p>1, under radial symmetry. Here B(0,R) denotes the ball center 0 and radius R in R^N. Also u(x)=(u_1,\cdots,u_n)(x) in R^n for each x and |\cdot| denotes the Euclidean norm in R^{nN}. For this problem we extend a method that uses a suitable homotopy along p to evaluate an associated Leray-Schauder degree. We apply this result to prove existence of nontrivial radial solutions to a system of the form {\rm div} ( a(| \nabla u |) \nabla u)+\lambda a(| u |) u=0,\quad x\in B(0,R)\\ u(x)=0,\quad x\in \partial B(0,R).

Friday, January 16, 2003
Jan Dijkstra (the Free University of Amsterdam)
The (in)stability of (complete) Erd\H os space
2:30 pm / CH 445
Abstract. It is proved that the countable infinite power of complete Erd\H os space E_c is not homeomorphic to E_c. The method by which this result is obtained consists of showing that E_c does not contain arbitrarily small closed subsets that are one-dimensional at every point. This observation also produces solutions to several problems that were posed by Aarts, Kawamura, Oversteegen, and Tymchatyn. In contrast, we can also show that the original (rational) Erd\H os space is homeomorphic to its infinite power. We obtain this last result by developing topological characterizations of Erd\H os space.

Friday, December 5, 2003
Avy Soffer (Rutgers University)
Multichannel Nonlinear Dynamics and Scattering
2:30 pm / CH 445
Abstract. The Nonlinear Schrodinger equation, which results from the Hartree Fock approximation in Condensed Matter theory ,and in nonlinear optics, is an example of a dispersive wave equation which has many different asymptotic states depending on the initial data. Such time dependent equations play a central role in many recent scientific advances,such as Bose-Einstein condensates and optical devices . I will discuss the solutions of such equations,including the large time behavior. Asymptotic Stability of N-Soliton solutions will be described. Furthermore, for nonhomogenous equations, rigorous results have shown, for the first time, the phenomena of ground state selection,asymptotic instability of the excited states and more. These results are obtained by deriving a novel Nonlinear Master equation and multitime scale analysis of its properties. The talk will be introductory for general audience.

Friday, November 21, 2003
Alexander Blokh and Lex Oversteegen
(University of Alabama at Birmingham)

Wandering triangles exist
2:30 pm / CH 445
Abstract. In his preprint On the combinatorics and dynamics of iterated rational maps, Princeton, 1985, W. P. Thurston developed the theory of quadratic invariant laminations. He studied connections between the dynamics of complex polynomials on their Julia sets and the dynamics of the map Pd(z) = zd on the unit circle in the complex plane. The following notion was introduced in the preprint: a set A0 = {x0,y0,z0} on the unit circle is said to be a wandering triangle (for f = Pd) if for the sets fn(A0) = An, n = 0, 1, ... we have that:
(1) each An consists of three distinct points for all n = 0, 1, ...;
(2) the convex hulls of all sets An in the unit disk are pairwise disjoint.
The theorem below was proven by Thurston in his preprint.
No Wandering Triangle Theorem. There exist no wandering triangles for P2.
Recently we constructed an example which shows that wandering triangles exist for P3 and actually for all Pd with d greater than 2. Thus, Thurston's No Wandering Triangle Theorem fails for d greater than 2.
In this talk we will describe our example. We will try to make the presentation accessible to all members of the department, including all students.

Friday, November 14, 2003
Lennie Friedlander (University of Arizona)
Witten deformation of the deRham complex
2:30 pm / CH 445
Abstract. Twenty years ago, Witten found an analytical proof of Morse inequalities. This proof is based on the study of the spectrum of the Laplacians that are accosiated to the family of deformed deRham differentials $\exp{-th}d\exp{th}$ where $h(x)$ is a Morse function on a manifold. Later, Helffer and Sj"ostrand applied methods of micro-local analysis to study in details the asymptotic behavior of the deRham differential when $t\to\infty$. This analysis made possible to apply the Witten deformation to the study of analytic torsion. I am planning the talk to be as elemetary as possible, and most of it will be accessible to graduate students. Part of the talk is based on the work done by me in collaboration with Dan Burghelea and Thomas Kappeler.

Friday, November 7, 2003
Marcel Griesemer (University of Alabama at Birmingham)
Electron-Photon Scattering
2:30 pm / CH 445
Abstract. An atom or molecule in an excited state with energy below the ionization threshold will eventually relax to its ground state by dissipating excess energy in the form of radiation. This process of relaxation to the ground state is one of the basic phenomena responsible for the production of visible light.
I will introduce a simple mathematical model of atoms interacting with quantized radiation (photons) and give a mathematical formulation of the phenomenon of relaxation to the ground state. The proof is based on methods and ideas from the scattering theory of N-particle quantum systems; in particular the positivity of certain commutators and propagation estimates play an important role.

Friday, October 31, 2003
Sumio Yamada (University of Alabama at Birmingham)
On Penrose Inequality with Electric Charge.
2:30 pm / CH 445
Abstract. There has been much progress in the subject of Penrose inequality which provides a lower bound on the total mass of asymptotically flat space-time in terms of the geometry of the black holes it contains. In this collaborative work with Gilbert Weinstein, we seek a larger bound for the total mass, by taking into account the electric charges inside the black holes. In particular we provide a counter example to a version of generalized Penrose inequalities, and discuss the implications of this counter example in further understanding of the subject.

Friday, October 24, 2003
Robert Gulliver (University of Minnesota)
Curves of Small Curvature and Embedded Minimal Surfaces
2:30 pm / CH 445
Abstract. Every closed curve in Euclidean space is the boundary of a minimal surface: think of a soap film which spans an arbitrarily curvy wire. Some mathematical formulations of this phenomenon produce a surface of the topological type of the disk, but allowing the surface to intersect itself (Rado-Douglas, 1930). On the other hand, it is known that a curve with total curvature less than 4 pi is unknotted (Fary-Milnor, 1949). A new result (Ekholm-White-Wienholtz 2002) proves this unknottedness by showing that any minimal surface of the type of the disk is embedded. Their result is extended to the case where the ambient space has nonpositive curvatures. There are some fascinating open questions, even for Euclidean space. This is joint work with Jaigyoung Choe.

Friday, September 26, 2003
Jan Lang (Ohio State University)
p-Laplacian in dimension one and Sobolev embeddings
2:30 pm / CH 445
Abstract. We will show the relation between eigenvalues of the p-Laplacian and exact approximative numbers of the Sobolev embedding and also relation between eigenfunctions of the p-Laplacian and extreme functions for the best n-dimensional approximation of the Sobolev embedding. Exact value of approximative numbers and the corresponding best n-dimensional linear approximation of the Sobolev Embedding are going to be shown. Difference between L^2 case (Hilbert space) and L^p, p not equal to 2 (non-Hilbert spaces) will be discussed.

Friday, September 19, 2003
Anthony Skjellum (University of Alabama at Birmingham)
Polyalgorithms for Matrix Multiplication: Parallel and Cache-Friendly Methodologies
2:30 pm / CH 445
Abstract. Although dense matrix multiplication is a straightforward mathematical operation, both cache-memory hierarchy and related distributed memory parallel computing create a large space of potential algorithms, data structures, and performance differentials that vary with problem size, matrix shape, and data layout. This richness can be described in terms of polyalgorithms for the O(N^3) algorithm, and some of the methods that evolve help with Strassen-type algorithms as well. A taxonomy for 2D decompositions of matrix algorithms is given, based on the work of Li, Falgout, and the speaker; a discussion of efficient 2D decompositions inside a single memory space based on modified morton ordering, and due to several researchers, is also discussed, including recent results by Wu and the speaker. Among the goals of this research are to always use the highest performing mathematical operation, to be able to predictive decide what will be highest performance, and to be able to smooth performance as a function of problem size, memory hierarchy, and concurrency.

Friday, September 12, 2003
W.D. Evans (University of Wales, Cardiff)
Inequalities associated with magnetic fields
2:30 pm / CH 445
Abstract. Hardy and Sobolev-type inequalities in L^p(R^n), 1 < p < \infty, n > 1, associated with magnetic fields, will be discussed, with particular emphasis on the sharpness, or otherwise, of constants. Similar inequalities in the case p=2, n=2 for Aharonov-Bohm-type magnetic fields have already been determined by Laptev/Weidl and Balinsky/Evans/Lewis. The lecture will report on recent joint work with Christer Bennewitz.

Friday, August 29, 2003
Ken McLaughlin (University of North Carolina)
Introduction to random matrix theory
2:30 pm / CH 445
Abstract. There has been a number of developments of late connected to eigenvalues statistics of random matrices from the so-called "invariant ensembles". Most of these results have to do with the statistics of eigenvalues when the size of the matrices in question grows to infinity. This presentation will be accessible to graduate students and should serve as an introduction to some aspects of this field of research. Bring fruits! If at the end of the presentation viewers have not seen (1) basic questions about eigenvalue statistics connected directly to (2) explicit probability measures on spaces of matrices and (3) a discussion of the behavior of said statistics using orthogonal polynomials, then these fruits may be freely hurled at the speaker. In the time remaining, the speaker will either partake of the fruits of his labor, or discuss some more recent connections between random matrix theory and other combinatorial problems.


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Step-by-step Solution

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Step-by-step explanation

Problem to solve:

$\int4x\:\sec^2x\:dx$

Learn how to solve trigonometric integrals problems step by step online.

$4\int x\sec\left(x\right)^2dx$

Learn how to solve trigonometric integrals problems step by step online. Solve the trigonometric integral int(4*x*sec(x)^2)dx. The integral of a constant by a function is equal to the constant multiplied by the integral of the function. We can solve the integral \int x\sec\left(x\right)^2dx by applying integration by parts method to calculate the integral of the product of two functions, using the following formula. First, identify u and calculate du. Now, identify dv and calculate v.

$4x\tan\left(x\right)+4\ln\left|\cos\left(x\right)\right|+C_0$

Problem Analysis

$\int4x\:\sec^2x\:dx$

Main topic:

Trigonometric integrals

~ 0.08 seconds


text:
# Parameter testing¶

This Notebook accompanies an Agile Geoscience blog post on the same subject.

A quick look at a progression of ways to parameterize a seismic attribute. We'll use the Canny filter, since I was playing with it anyway (see Sobel_filtering_horizons.ipynb).

Some ways to parameterize, in roughly increasing order of sophistication:

### In a software tool (Petrel, Landmark, etc)¶

• Use the default. Unlikely to be optimal.
• Try some settings, and inspect the result. Also unlikely to be optimal — depends how patient you are, and how good your memory is.
• Get more systematic. Run the test with various settings, saving results with cunning names (test_16ms, test_20ms, etc) to remember what you did. Take screenshots. Put them in a PowerPoint. Compare and contrast. Choose a winner. This method still fails to really explore the parameter space, and if there are multiple parameters, you are stuffed.

### In an IPython Notebook (like this one!)¶

We could use all of the above approaches in a Notebook too — but why would we? There are better ways.

## Preliminaries¶

In [1]:
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline

Next we need some data.

We will use the open-access Penobscot dataset from offshore Nova Scotia, quite near my house. This is a great dataset for lots of reasons, but especially because it is freely available and can be shared in almost any way you like. You can download it from the Open Seismic Repository. The data is owned by the Nova Scotia Deptartment of Energy and licensed CC-BY.

I saved this horizon as a NumPy object in the Notebook Sobel_filtering_horizons.ipynb.

In [2]:
horizon = np.load('Penobscot_HorB.npy')

In [3]:
plt.figure()
plt.imshow(-1 * horizon, aspect=0.5)
plt.show()

In [4]:
vmin = np.percentile(horizon[horizon > 0], 1)
vmax = np.percentile(horizon[horizon > 0], 99)
print 'min {0}, max {1}'.format(vmin, vmax)

min 891.804814339, max 1065.95427275

## Sobel filtering¶

We will compare our new attribute with this one later.

In [5]:
import scipy.ndimage

In [6]:
def do_sobel(image):
dx = scipy.ndimage.sobel(image, 0)  # horizontal derivative
dy = scipy.ndimage.sobel(image, 1)  # vertical derivative
mag = np.hypot(dx, dy)      # magnitude
mag *= 255.0 / np.max(mag)  # normalize
return mag

In [7]:
mag = do_sobel(horizon)

In [8]:
plt.imshow(mag, aspect=0.5, cmap=plt.get_cmap('Greys'), vmin=0, vmax=3)
plt.show()

## Canny edge detection¶

In [9]:
plt.figure(figsize=(15,10))
plt.imshow(-1*horizon, aspect=0.5, cmap="cubehelix", vmin=-vmax, vmax=-vmin)
plt.colorbar(shrink=0.75)  # shrink makes the colourbar a bit shorter
plt.show()

The nicest way to figure out how it works is to look at the example from SciKits website. Copy and paste and it runs perfectly...

In [10]:
# You might need to install this first: pip install scikit-image
from skimage import filter

In [11]:
# Generate noisy image of a square
im = np.zeros((128, 128))
im[32:-32, 32:-32] = 1

im = scipy.ndimage.rotate(im, 15, mode='constant')
im = scipy.ndimage.gaussian_filter(im, 4)
im += 0.2 * np.random.random(im.shape)

# Compute the Canny filter for two values of sigma
edges1 = filter.canny(im)
edges2 = filter.canny(im, sigma=3)

# display results
fig, (ax1, ax2, ax3) = plt.subplots(nrows=1, ncols=3, figsize=(12, 5))

ax1.imshow(im, cmap=plt.cm.jet)
ax1.axis('off')
ax1.set_title('noisy image', fontsize=20)

ax2.imshow(edges1, cmap=plt.cm.gray)
ax2.axis('off')
ax2.set_title('Canny filter, $\sigma=1$', fontsize=20)

ax3.imshow(edges2, cmap=plt.cm.gray)
ax3.axis('off')
ax3.set_title('Canny filter, $\sigma=3$', fontsize=20)

bottom=0.02, left=0.02, right=0.98)

plt.show()

Now we can get to work. Let's apply some different ways to investigate the parameters of skimage.filter.canny.

## Default¶

In [31]:
edge = filter.canny(horizon)

plt.figure(figsize=(15, 5))
plt.subplot(121)
plt.imshow(horizon, aspect=0.5, cmap='cubehelix_r', vmin=vmin, vmax=vmax)
plt.subplot(122)
plt.imshow(edge, aspect=0.5, cmap='Greys', vmin=0, vmax=1)
plt.show()


text:
## Average node depth in a Full Tree

While doing something else I stumbled upon the interesting problem of computing the average depth of nodes in a tree. The depth of a node is the distance that separates that node from the root. You can either decide that the root is at depth 1, or you can decide that it is at depth zero, but let’s decide on depth 1. So an immediate child of the root is at depth two, and its children at depth 3, and so on until you reach leaves, nodes with no children.

So the calculation of the average node depth (including leaves) in a tree comes interesting when we want to know how far a constructed tree is from the ideal full tree, as a measure of (application-specific) performance. After searching a bit on the web, I found only incomplete or incorrect formulas, or stated with proof. This week, let us see how we can derive the result without (too much) pain.

Let us start with a tree with $m$ nodes (in what follows, the distinction between internal nodes and leaves noted where irrelevant, so we’ll just say ‘nodes’ when it does not matter). First, we find integers $k\leqslant{0}$ and $n\leqslant{0}$ such that $m \sim 2^n+k$.

Then we reflect on what $m$ imposes on the shape of the tree. If $m\sim2^n-1$ (or, from the decomposition above, $m\sim 2^{n-1}+(2^{n-1}-1)$ with $k=2^{n-1}-1$) the tree can be balanced in such a way that it has all leaves on the same level. The tree is perfectly triangular:

In that case, the average depth is given by

$\displaystyle \bar{d}=\frac{1}{2^n-1}\sum_{d=1}^n d 2^{d-1}$

because there are $2^{d-1}$ nodes at depth $d$ (indeed: at depth $d=1$, the root, there is $2^{d-1}=2^0=1$ node. At depth $d=2$, we have $2^{d-1}=2^1=2$ nodes, etc.) At depth $d$, all $2^{d-1}$ nodes have depth $d$, contributing $d 2^{d-1}$ to the total. Since there are $m=2^n-1$ nodes, the average is given by the formula above.

But $\sum d 2^{d}$ should look familiar to you. But we have $\sum d 2^{d-1}$, so we need to transform it a bit so that we fall back again on the form $\sum d 2^d$. First:

$\displaystyle \sum_{d=1}^2 d 2^{d-1}=\sum_{d=1}^n d \left(\frac{1}{2}2^d\right)=\frac{1}{2}\sum_{d=1}^n d 2^d$

and we know that

$\displaystyle \sum_{d=1}^n d 2^d = 2^{n+1}(n-1)+2$

Finally,

$\displaystyle \frac{1}{2^n-1}\sum_{d=1}^n d 2^{d-1}=\left(\frac{1}{2^n-1}\right)\left(\frac{1}{2}\sum_{d=1}^n d 2^d\right)$

$\displaystyle =\left(\frac{1}{2^n-1}\right)\left(\frac{1}{2}\left(2^{n+1}(n-1)+2\right)\right)=\frac{2^n(n-1)+1}{2^n-1}$

and $\displaystyle \frac{2^n(n-1)+1}{2^n-1}$ is approximately $n$.

*
* *

However, if $m\not\sim 2^n-1$, then we have a tree with $k+1$ leaves on one depth more than the main tree, and the tree looks something like:

The depth equation becomes now

$\frac{1}{2^n+k}\left(\sum_{d=1}^n d 2^{d-1}+(k+1)(n+1)\right)$

which simplifies to

$\displaystyle \bar{d}=\frac{2^n(n-1)+1+(k+1)(n+1)}{2^n+k}$

*
* *

The hard—or rather inconvenient—part is to find the decomposition $m=2^n+k$ which needs either a call to log-base-2 (gasp!) or $O(\lg m)$ steps using an integer method. Other than that, now that you know how the formula is derived (with the root starting at depth one, not zero, because you still need 1 operation to look at it), there shouldn’t be any more problems.

Also,the formula applies to full trees, not any kind of trees. If you have a Fibonacci tree, or some other kind of tree, then you will have to find another formula for the average depth, and it may not be trivial.

### 2 Responses to Average node depth in a Full Tree

Without ever stating so, it is clear the ‘n’ is the depth of the tree. But the meaning of ‘k’ is also never explicitly stated and I’m finding it’s relation to any part of the tree difficult to understand.

• Ah. Well, $m=2^n+k$: any number can be written uniquely that way provided that $k<2^n$. For example, $11=2^3+3$, $17=2^4+1$.


text:
Home > Research > Publications & Outputs > Embedded Eigenvalues for operators and Fredholm...

### Electronic data

• 2019Nifeenphd

Final published version, 1.47 MB, PDF document

## Embedded Eigenvalues for operators and Fredholm properties of pencils

Research output: ThesisDoctoral Thesis

Published

### Standard

Lancaster University, 2021. 197 p.

Research output: ThesisDoctoral Thesis

### Vancouver

Altaweel N. Embedded Eigenvalues for operators and Fredholm properties of pencils. Lancaster University, 2021. 197 p. doi: 10.17635/lancaster/thesis/1353

### Bibtex

@phdthesis{e8564b798c98498cb87aa8046c2106a6,
title = "Embedded Eigenvalues for operators and Fredholm properties of pencils",
abstract = "The perturbation problem for operators is considered one of the differential equations with operator coefficients; a possible example of this problem is embedded eigenvalues, which serves as a prototype of this problem.My research is concerned with two main tasks; first, highlighting the idea of the existence of embedded eigenvalues (trapped modes) of different operators. These include the stability of the embedded eigenvalues within the spectrum for the operator on a cylindrical domain. Common threads will be taken from these problems to subsequently develop a more generalised understanding of the existence of embedded eigenvalues. The second task is to study the Fredholm properties of an operator pencil. In particular, we detect and approximate the spectra of the Fredholm operator pencils via a Green{\textquoteright}s kernel (contour integral) by considering exponential solutions of differential equations with operator coefficients. The arguments for this task act on a class of weighted function spaces which can be modelled on Sobolev spaces.One of the main motivation behind this research is to gain a deeper understanding the development of aspects of the theory of ordinary differential equations with operator coefficients by concentrating on some specific examplesof trapped modes.The results of our first task showed that, in different cases, for sufficiently small potential functions our operator has an eigenvalue which is contained in the essential spectrum, and hence is an embedded eigenvalue. According to the result of the second task, it was directly established that Fredholm operator pencil and the index could be calculated without the need to consider the adjoint operator. Also, we leveraged certain concepts to go from the semi-Fredholm property to the Fredholm property using some of the results of the current thesis.",
author = "Nifeen Altaweel",
year = "2021",
month = jun,
day = "28",
doi = "10.17635/lancaster/thesis/1353",
language = "English",
publisher = "Lancaster University",
school = "Lancaster University",

}

### RIS

TY - THES

T1 - Embedded Eigenvalues for operators and Fredholm properties of pencils

AU - Altaweel, Nifeen

PY - 2021/6/28

Y1 - 2021/6/28

N2 - The perturbation problem for operators is considered one of the differential equations with operator coefficients; a possible example of this problem is embedded eigenvalues, which serves as a prototype of this problem.My research is concerned with two main tasks; first, highlighting the idea of the existence of embedded eigenvalues (trapped modes) of different operators. These include the stability of the embedded eigenvalues within the spectrum for the operator on a cylindrical domain. Common threads will be taken from these problems to subsequently develop a more generalised understanding of the existence of embedded eigenvalues. The second task is to study the Fredholm properties of an operator pencil. In particular, we detect and approximate the spectra of the Fredholm operator pencils via a Green’s kernel (contour integral) by considering exponential solutions of differential equations with operator coefficients. The arguments for this task act on a class of weighted function spaces which can be modelled on Sobolev spaces.One of the main motivation behind this research is to gain a deeper understanding the development of aspects of the theory of ordinary differential equations with operator coefficients by concentrating on some specific examplesof trapped modes.The results of our first task showed that, in different cases, for sufficiently small potential functions our operator has an eigenvalue which is contained in the essential spectrum, and hence is an embedded eigenvalue. According to the result of the second task, it was directly established that Fredholm operator pencil and the index could be calculated without the need to consider the adjoint operator. Also, we leveraged certain concepts to go from the semi-Fredholm property to the Fredholm property using some of the results of the current thesis.

AB - The perturbation problem for operators is considered one of the differential equations with operator coefficients; a possible example of this problem is embedded eigenvalues, which serves as a prototype of this problem.My research is concerned with two main tasks; first, highlighting the idea of the existence of embedded eigenvalues (trapped modes) of different operators. These include the stability of the embedded eigenvalues within the spectrum for the operator on a cylindrical domain. Common threads will be taken from these problems to subsequently develop a more generalised understanding of the existence of embedded eigenvalues. The second task is to study the Fredholm properties of an operator pencil. In particular, we detect and approximate the spectra of the Fredholm operator pencils via a Green’s kernel (contour integral) by considering exponential solutions of differential equations with operator coefficients. The arguments for this task act on a class of weighted function spaces which can be modelled on Sobolev spaces.One of the main motivation behind this research is to gain a deeper understanding the development of aspects of the theory of ordinary differential equations with operator coefficients by concentrating on some specific examplesof trapped modes.The results of our first task showed that, in different cases, for sufficiently small potential functions our operator has an eigenvalue which is contained in the essential spectrum, and hence is an embedded eigenvalue. According to the result of the second task, it was directly established that Fredholm operator pencil and the index could be calculated without the need to consider the adjoint operator. Also, we leveraged certain concepts to go from the semi-Fredholm property to the Fredholm property using some of the results of the current thesis.

U2 - 10.17635/lancaster/thesis/1353

DO - 10.17635/lancaster/thesis/1353

M3 - Doctoral Thesis

PB - Lancaster University

ER -


text:
# !LaTeX Error: Cannot determine size of graphic in simLinkError.pdf (no Bounding Box)

I'm trying to put a .pdf graphic in my LaTeX file. This is usually no problem for me. I use pdfLaTeX in TeXworks. The only difference this time seems to be that I'm using the style files from the journal Statistical Science (http://www.e-publications.org/ims/support/sts-instructions.html).

I've tried several solutions. This one and converting the pdf to .eps or .ps all make the error go away, but then there is just a big blank space where the image is supposed to be. Any ideas?

Edit: Here is the preamble:

\documentclass[dvips,sts]{imsart}
\usepackage{graphicx}
\usepackage{float}
\begin{document}
\begin{figure}[h!]
\centering
\caption{Blah}
\label{fig:sim1}
\end{figure}
\end{document}

• "Cannot determine size of graphic #1 (no Bounding Box)" error occour when we compile latex on a TeX file with non-eps images such as PDF/JPG/PNG etc. To fix this error: You need to run pdflatex on the TeX file. Otherwise you need to convert your image to EPS using convert and run latex on the TeX file. – Jagath Jul 17 '13 at 3:49
• @JagathAR, as I mentioned, I am using pdfLaTeX on my TeX file. Is this different than pdflatex? – zkurtz Jul 17 '13 at 13:11
• @dustin, no, [demo] is not there ... – zkurtz Jul 17 '13 at 13:12
• Can you post a small example of your problem which has the preamble? – dustin Jul 17 '13 at 13:16
• Remove the dvips option to the class. This is all is needed. – egreg Jul 17 '13 at 13:58

Options to \documentclass are examined by all packages. So you're loading graphicx with the dvips option, which allows only EPS graphics to be imported and is nonsense when the engine is pdflatex. So remove it.

\documentclass[sts]{imsart}

In general dvips, pdftex or xetex should not be used even with packages that know them, such as

• graphicx
• hyperref
• geometry

that can deduce the needed driver by themselves.

• Thanks again! I'd vote this up if I had enough reputation to do so. – zkurtz Jul 17 '13 at 14:17
• @zkurtz Don't worry, you'll get it. And I really don't need more reputation, do I? ;-) – egreg Jul 17 '13 at 14:20
• Using Texlipse under Project/Properties/Latex Project Properties output format pdf and build commands pdflatex have to be used instead of the corresponding dvi parameters. – Torsten May 8 '14 at 13:46
gs -o /dev/null -sDEVICE=bbox Input.pdf

yields

%%BoundingBox: 0 0 702 399

\includegraphics[bb=0 0 702 399]{Input.pdf}

• Please state in which sense this posting is an answer to the original query. E.g., do state how the OP might use the information you've provided to help solve his/her issues. – Mico Oct 14 '14 at 19:45
• Welcome to TeX.SX! I'm not sure this is an answer to the question. – egreg Oct 14 '14 at 20:15
• It helped me. I could not change teh documentclass command, so this workaround worked. – piffy Sep 4 '16 at 17:40


text:
# Chapter 6 - Section 6.3 - Proportions and Problem Solving - Exercise Set: 2

She can word process 36 pages.

#### Work Step by Step

First, we translate the question into an equation: $\frac{30}{4} = \frac{270}{x}$ (270 minutes = 4.5 hours) Now, we solve for $x$ by cross multiplying, which states that if $\frac{a}{b} = \frac{c}{d}$, then $ad = bc$. This result can be achieved by multiplying both sides by $cd$. $30x = 1080 \rightarrow x = 36$

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
# How to prevent numbers being changed to exponential form in Python Matplotlib?

Using style='plain' in the ticklabel_format() method, we can restrict the value being changed into exponential form.

## Steps

• Pass two lists to draw a line using plot() method.

• Use ticklabel_format() method with style='plain'. If a parameter is not set, the corresponding property of the formatter is left unchanged. Style='plain' turns off scientific notation.

• To show the figure, use plt.show() method.

## Example

from matplotlib import pyplot as plt
plt.plot([1, 2, 3, 4, 5], [11, 12, 13, 14, 15])
plt.ticklabel_format(style='plain')    # to prevent scientific notation.
plt.show()