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# Implicit Normalizing Flows

Normalizing flows define a probability distribution by an explicit invertible transformation 𝐳=f(𝐱). In this work, we present implicit normalizing flows (ImpFlows), which generalize normalizing flows by allowing the mapping to be implicitly defined by the roots of an equation F(𝐳, 𝐱)= 0. ImpFlows build on residual flows (ResFlows) with a proper balance between expressiveness and tractability. Through theoretical analysis, we show that the function space of ImpFlow is strictly richer than that of ResFlows. Furthermore, for any ResFlow with a fixed number of blocks, there exists some function that ResFlow has a non-negligible approximation error. However, the function is exactly representable by a single-block ImpFlow. We propose a scalable algorithm to train and draw samples from ImpFlows. Empirically, we evaluate ImpFlow on several classification and density modeling tasks, and ImpFlow outperforms ResFlow with a comparable amount of parameters on all the benchmarks.

## Authors

• 9 publications
• 15 publications
• 19 publications
• 2 publications
• 150 publications
• ### Equivariant Flows: sampling configurations for multi-body systems with symmetric energies

Flows are exact-likelihood generative neural networks that transform sam...
10/02/2019 ∙ by Jonas Köhler, et al. ∙ 0

• ### Universal Approximation of Residual Flows in Maximum Mean Discrepancy

Normalizing flows are a class of flexible deep generative models that of...
03/10/2021 ∙ by Zhifeng Kong, et al. ∙ 0

• ### Quasi-Autoregressive Residual (QuAR) Flows

Normalizing Flows are a powerful technique for learning and modeling pro...
09/16/2020 ∙ by Achintya Gopal, et al. ∙ 0

• ### The Convolution Exponential and Generalized Sylvester Flows

This paper introduces a new method to build linear flows, by taking the ...
06/02/2020 ∙ by Emiel Hoogeboom, et al. ∙ 10

• ### Gravitational-wave parameter estimation with autoregressive neural network flows

We introduce the use of autoregressive normalizing flows for rapid likel...
02/18/2020 ∙ by Stephen R. Green, et al. ∙ 0

• ### Jacobian Determinant of Normalizing Flows

Normalizing flows learn a diffeomorphic mapping between the target and b...
02/12/2021 ∙ by Huadong Liao, et al. ∙ 24

• ### Parallel multilevel restricted Schwarz preconditioners for implicit simulation of subsurface flows with Peng-Robinson equation of state

Parallel algorithms and simulators with good scalabilities are particula...
07/28/2020 ∙ by Rui Li, et al. ∙ 0

## Code Repositories

### implicit-normalizing-flows

Code for "Implicit Normalizing Flows" (ICLR 2021 spotlight)

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## 1 Introduction

Normalizing flows (NFs) (rezende2015variational; nice) are promising methods for density modeling. NFs define a model distribution by specifying an invertible transformation from

to another random variable

. By change-of-variable formula, the model density is

 lnpx(x)=lnpz(f(x))+ln∣∣det(Jf(x))∣∣, (1)

where follows a simple distribution, such as Gaussian. NFs are particularly attractive due to their tractability, i.e., the model density can be directly evaluated as Eqn. (1). To achieve such tractability, NF models should satisfy two requirements: (i) the mapping between and is invertible; (ii) the log-determinant of the Jacobian is tractable. Searching for rich model families that satisfy these tractability constraints is crucial for the advance of normalizing flow research. For the second requirement, earlier works such as inverse autoregressive flow (kingma2016improved) and RealNVP (realnvp) restrict the model family to those with triangular Jacobian matrices.

More recently, there emerge some free-form Jacobian approaches, such as Residual Flows (ResFlows) (behrmann2019invertible; chen2019residual)

. They relax the triangular Jacobian constraint by utilizing a stochastic estimator of the log-determinant, enriching the model family. However, the Lipschitz constant of each transformation block is constrained for invertibility. In general, this is not preferable because mapping a simple prior distribution to a potentially complex data distribution may require a transformation with a very large Lipschitz constant (See Fig.

3 for a 2D example). Moreover, all the aforementioned methods assume that there exists an explicit forward mapping . Bijections with explicit forward mapping only covers a fraction of the broad class of invertible functions suggested by the first requirement, which may limit the model capacity.

In this paper, we propose implicit flows (ImpFlows) to generalize NFs, allowing the transformation to be implicitly defined by an equation . Given (or ), the other variable can be computed by an implicit root-finding procedure . An explicit mapping used in prior NFs can viewed as a special case of ImpFlows in the form of . To balance between expressiveness and tractability, we present a specific from of ImpFlows, where each block is the composition of a ResFlow block and the inverse of another ResFlow block. We theoretically study the model capacity of ResFlows and ImpFlows in the function space. We show that the function family of single-block ImpFlows is strictly richer than that of two-block ResFlows by relaxing the Lipschitz constraints. Furthermore, for any ResFlow with a fixed number of blocks, there exists some invertible function that ResFlow has non-negligible approximation error, but ImpFlow can exactly model.

On the practical side, we develop a scalable algorithm to estimate the probability density and its gradients, and draw samples from ImpFlows. The algorithm leverages the implicit differentiation formula. Despite being more powerful, the gradient computation of ImpFlow is mostly similar with that of ResFlows, except some additional overhead on root finding. We test the effectiveness of ImpFlow on several classification and generative modeling tasks. ImpFlow outperforms ResFlow on all the benchmarks, with comparable model sizes and computational cost.

## 2 Related Work

Expressive Normalizing Flows There are many works focusing on improving the capacity of NFs. For example, nice; realnvp; kingma2018glow; ho2019flow++; song2019mintnet; hoogeboom2019emerging; de2020block; durkan2019neural design dedicated model architectures with tractable Jacobian. More recently, grathwohl2018ffjord; behrmann2019invertible; chen2019residual propose NFs with free-form Jacobian, which approximate the determinant with stochastic estimators. In parallel with architecture design, chen2020vflow; huang2020augmented; cornish2019relaxing; nielsen2020survae improve the capacity of NFs by operating in a higher-dimensional space. As mentioned in the introduction, all these existing works adopt explicit forward mappings, which is only a subset of the broad class of invertible functions. In contrast, the implicit function family we consider is richer. While we primarily discuss the implicit generalization of ResFlows (chen2019residual) in this paper, the general idea of utilizing implicit invertible functions could be potentially applied to other models as well. Finally, zhang2020approximation formally prove that the model capacity of ResFlows is restricted by the dimension of the residual blocks. In comparison, we study another limitation of ResFlows in terms of the bounded Lipschitz constant, and compare the function family of ResFlows and ImpFlows with a comparable depth.

Continuous Time Flows (CTFs) (chen2018neural; grathwohl2018ffjord; chen2018continuous)

are flexible alternative to discrete time flows for generative modeling. They typically treat the invertible transformation as a dynamical system, which is approximately simulated by ordinary differential equation (ODE) solvers. In contrast, the implicit function family considered in this paper does not contain differential equations, and only requires fixed point solvers. Moreover, the theoretical guarantee is different. While CTFs typically study the universal approximation capacity under the continuous time case (i.e., “

infinite depth” limit), we consider the model capacity of ImpFlows and ResFlows under a finite number of transformation steps. Finally, while CTFs are flexible, their learning is challenging due to instability (liu2020does; massaroli2020stable) and exceedingly many ODE solver steps (finlay2020train), making their large-scale application still an open problem.

Implicit Deep Learning

Utilizing implicit functions enhances the flexibility of neural networks, enabling the design of network layers in a problem-specific way. For instance,

bai2019deep propose a deep equilibrium model as a compact replacement of recurrent networks; amos2017optnet generalize each layer to solve an optimization problem; wang2019satnet integrate logical reasoning into neural networks; reshniak2019robust utilize the implicit Euler method to improve the stability of both forward and backward processes for residual blocks; and sitzmann2020implicit incorporate periodic functions for representation learning. Different from these works, which consider implicit functions as a replacement to feed-forward networks, we develop invertible implicit functions for normalizing flows, discuss the conditions of the existence of such functions, and theoretically study the model capacity of our proposed ImpFlow in the function space.

## 3 Implicit Normalizing Flows

We now present implicit normalizing flows, by starting with a brief overview of existing work.

### 3.1 Normalizing Flows

As shown in Eqn. (1), a normalizing flow is an invertible function that defines a probability distribution with the change-of-variable formula. The modeling capacity of normalizing flows depends on the expressiveness of the invertible function . Residual flows (ResFlows) (chen2019residual; behrmann2019invertible) are a particular powerful class of NFs due to their free-form Jacobian. ResFlows use to construct the invertible mapping, where each layer is an invertible residual network with Lipschitz constraints bounded by a fixed constant :

 fl(x)=x+gl(x),Lip(gl)≤κ<1, (2)

where is the Lipschitz constant of a function (see Sec. 4.1 for details). Despite their free-form Jacobian, the model capacity of ResFlows is still limited by the Lipschitz constant of the invertible function. The Lipschitz constant of each ResFlow block cannot exceed 2 (behrmann2019invertible), so the Lipschitz constant of an -block ResFlow cannot exceed . However, to transfer a simple prior distribution to a potentially complex data distribution, the Lipschitz constant of the transformation can be required to be sufficiently large in general. Therefore, ResFlows can be undesirably deep simply to meet the Lipschitz constraints (see Fig. 3 for a 2D example). Below, we present implicit flows (ImpFlows) to relax the Lipschitz constraints.

### 3.2 Model Specification

In general, an implicit flow (ImpFlow) is defined as an invertible mapping between random variables and of dimension by finding the roots of , where is a function from to . In particular, the explicit mappings used in prior flow instances (chen2019residual; kingma2018glow) can be expressed as an implicit function in the form . While ImpFlows are a powerful family to explore, generally they are not guaranteed to satisfy the invertibility and the tractability of the log-determinant as required by NFs. In this paper, we focus on the following specific form, which achieves a good balance between expressiveness and tractability, and leave other possibilities for future studies.

###### Definition 1.

Let and be two functions such that and , where is the Lipschitz constant of a function . A specific form of ImpFlows is defined by

 F(z,x)=0, where F(z,x)=gx(x)−gz(z)+x−z. (3)

The root pairs of Eqn. (3) form a subset in , which actually defines the assignment rule of a unique invertible function . To see this, for any , according to Definition 1, we can construct a contraction with a unique fixed point, which corresponds to a unique root (w.r.t. ) of , denoted by . Similarly, in the reverse process, given a , the root (w.r.t. ) of also exists and is unique, denoted by . These two properties are sufficient to ensure the existence and the invertibility of , as summarized in the following theorem.

###### Theorem 1.

Eqn.(3) defines a unique mapping , and is invertible.

See proof in Appendix A.1. Theorem 1 characterizes the validness of the ImpFlows introduced in Definition 1. In fact, a single ImpFlow is a stack of a single ResFlow and the inverse of another single ResFlow, which will be formally stated in Sec 4. We will investigate the expressiveness of the function family of the ImpFlows in Sec 4, and present a scalable algorithm to learn a deep generative model built upon ImpFlows in Sec. 5.

## 4 Expressiveness Power

We first present some preliminaries on Lipschitz continuous functions in Sec. 4.1 and then formally study the expressiveness power of ImpFlows, especially in comparison to ResFlows. In particular, we prove that the function space of ImpFlows is strictly richer than that of ResFlows in Sec. 4.2 (see an illustration in Fig. 1 (a)). Furthermore, for any ResFlow with a fixed number of blocks, there exists some function that ResFlow has a non-negligible approximation error. However, the function is exactly representable by a single-block ImpFlow. The results are illustrated in Fig. 1 (b) and formally presented in Sec. 4.3.

### 4.1 Lipschitz Continuous Functions

For any differentiable function and any , we denote the Jacobian matrix of at as .

###### Definition 2.

A function is called Lipschitz continuous if there exists a constant , s.t.

 ∥f(x1)−f(x2)∥≤L∥x1−x2∥, ∀x1,x2∈Rd.

The smallest that satisfies the inequality is called the Lipschitz constant of , denoted as .

Generally, the definition of depends on the choice of the norm , while we use -norm by default in this paper for simplicity.

###### Definition 3.

A function is called bi-Lipschitz continuous if it is Lipschitz continuous and has an inverse mapping which is also Lipschitz continuous.

It is useful to consider an equivalent definition of the Lipschitz constant in our following analysis.

###### Proposition 1.

(Rademacher (federer1969grundlehren, Theorem 3.1.6)) If is Lipschitz continuous, then is differentiable almost everywhere, and

 Lip(f)=supx∈Rd∥Jf(x)∥2,

where is the operator norm of the matrix .

### 4.2 Comparison to two-block ResFlows

We formally compare the expressive power of a single-block ImpFlow and a two-block ResFlow. We highlight the structure of the theoretical results in this subsection in Fig. 1 (a) and present a 1D motivating example in Fig. 2. All the proofs can be found in Appendix. A.

On the one hand, according to the definition of ResFlow, the function family of the single-block ResFlow is

 R \coloneqq{f:f=g+Id, g∈C1(Rd,Rd),Lip(g)<1}, (4)

where consists of all functions from to with continuous derivatives and denotes the identity map. Besides, the function family of -block ResFlows is defined by composition:

 Rℓ\coloneqq{f:f=fℓ∘⋯∘f1 for some% f1,⋯,fℓ∈R}. (5)

By definition of Eqn. (4) and Eqn. (5), .

On the other hand, according to the definition of the ImpFlow in Eqn. (3), we can obtain where denotes the composition of functions. Equivalently, we have which implies the function family of the single-block ImpFlow is

 I={f:f=f−12∘f1 for some f1,f2∈R}. (6)

Intuitively, a single-block ImpFlow can be interpreted as the composition of a ResFlow block and the inverse function of another ResFlow block, which may not have an explicit form (see Fig. 2 (c) and (d) for a 1D example). Therefore, it is natural to investigate the relationship between and . Before that, we first introduce a family of “monotonically increasing functions” that does not have an explicit Lipschitz constraint, and show that it is strictly larger than .

###### Lemma 1.
 R⫋F\coloneqq{f∈D:infx∈Rd,v∈Rd,∥v∥2=1vTJf(x)v>0}, (7)

where is the set of all bi-Lipschitz -diffeomorphisms from to , and means is a proper subset of .

Note that it follows from behrmann2019invertible that all functions in are bi-Lipschitz, so . In the 1D input case, we can get , and . In the high dimensional cases, and are hard to illustrate. Nevertheless, the Lipschitz constants of the functions in is less than  (behrmann2019invertible), but those of the functions in can be arbitrarily large. Based on Lemma 1, we prove that the function family of ImpFlows consists of the compositions of two functions in , and therefore is a strictly larger than , as summarized in the following theorem.

###### Theorem 2.

(Equivalent form of the function family of a single-block ImpFlow).

 I=F2\coloneqq{f:f=f2∘f1 for some f1,f2∈F}. (8)

Note that the identity mapping , and it is easy to get . Thus, the Lipschitz constant of a single ImpFlow (and its reverse) can be arbitrarily large. Because and there exists some functions in (see a constructed example in Sec. 4.3), we can get the following corollary.

###### Corollary 1.

.

The results on the 1D example in Fig. 2 (b) and (c) accord with Corollary 1. Besides, Corollary 1 can be generalized to the cases with -block ResFlows and -block ImpFlows, which strongly motivates the usage of implicit layers in normalizing flows.

### 4.3 Comparison with multi-block ResFlows

We further investigate the relationship between for and , as illustrated in Fig. 1 (b). For a fixed , the Lipschitz constant of functions in is still bounded, and there exist infinite functions that are not in but in . We construct one such function family: for any , define

 P(L,r)={f:f∈F,∃ Br⊂Rd,∀x,y∈Br,∥f(x)−f(y)∥2≥L∥x−y∥2}, (9)

where is an -dimensional ball with radius of . Obviously, is an infinite set. Below, we will show that , has a non-negligible approximation error for functions in . However, they are exactly representable by functions in .

###### Theorem 3.

Given and , we have

• .

• , . Moreover, for any with -dimensional ball , the minimal error for fitting in by functions in satisfies

 infg∈Rℓsupx∈Br∥f(x)−g(x)∥2≥r2(L−2ℓ) (10)

It follows Theorem 3 that to model , we need only a single-block ImpFlow but at least a -block ResFlow. In Fig. 2 (b), we show a 1D case where a 3-block ResFlow cannot fit a function that is exactly representable by a single-block ImpFlow. In addition, we also prove some other properties of ImpFlows. In particular, . We formally present the results in Appendix B.

## 5 Generative Modeling with ImpFlows

ImpFlows can be parameterized by neural networks and stacked to form a deep generative model to model high-dimensional data distributions. We develop a scalable algorithm to perform inference, sampling and learning in such models. For simplicity, we focus on a single-block during derivation.

Formally, a parametric ImpFlow block is defined by

 (11)

and , . Let denote all the parameters in and (which does NOT mean and share parameters). Note that refers to the input of the layer, not the input data.

The inference process to compute given in a single ImpFlow block is solved by finding the root of w.r.t. , which cannot be explicitly computed because of the implicit formulation. Instead, we adopt a quasi-Newton method (i.e. Broyden’s method (broyden1965class)) to solve this problem iteratively, as follows:

 z[i+1]=z[i]−αBF(z[i],x;θ), for i=0,1,⋯, (12)

where is a low-rank approximation of the Jacobian inverse111We refer readers to broyden1965class for the calculation details for . and is the step size which we use line search method to dynamically compute. The stop criterion is , where

is a hyperparameter that balances the computation time and precision. As Theorem

1 guarantees the existence and uniqueness of the root, the convergence of the Broyden’s method is also guaranteed, which is typically faster than a linear rate.

Another inference problem is to estimate the log-likelihood. Assume that where is a simple prior distribution (e.g. standard Gaussian). The log-likelihood of can be written by

 lnp(x)=lnp(z)+lndet(I+Jgx(x))−lndet(I+Jgz(z)), (13)

where denotes the Jacobian matrix of a function at . See Appendix. A.4 for the detailed derivation. Exact calculation of the log-determinant term requires

time cost and is hard to scale up to high-dimensional data. Instead, we propose the following unbiased estimator of

using the same technique in chen2019residual with Skilling-Hutchinson trace estimator (skilling1989eigenvalues; hutchinson1989stochastic):

 lnp(x)=lnp(z)+En∼p(N),v∼N(0,I)[n∑k=1(−1)k+1k(vT[Jgx(x)k]v−vT[Jgz(z)k]v)P(N≥k)], (14)

where is a distribution supported over the positive integers.

The sampling process to compute given can also be solved by the Broyden’s method, and the hyperparameters are shared with the inference process.

In the learning

process, we perform stochastic gradient descent to minimize the negative log-likelihood of the data, denoted as

. For efficiency, we estimate the gradient w.r.t. the model parameters in the backpropagation manner. According to the chain rule and the additivity of the log-determinant, in each layer we need to estimate the gradients w.r.t.

and of Eqn. (13). In particular, the gradients computation involves two terms: one is and the other is , where is a function satisfying and denotes or . On the one hand, for the log-determinant term, we can use the same technique as chen2019residual, and obtain an unbiased gradient estimator as follows.

 \pdvlndet(I+Jg(x;θ))(⋅)=En∼p(N),v∼N(0,I)[(n∑k=0(−1)kP(N≥k)vTJg(x;θ)k)\pdvJg(x;θ)(⋅)v], (15)

where is a distribution supported over the positive integers. On the other hand, can be computed according to the implicit function theorem as follows (See details in Appendix A.5):

 \pdvLz\pdvz(⋅)=\pdvLzJ−1G(z)\pdvF(z,x;θ)(⋅), where G(z;θ)=gz(z;θ)+z. (16)

In comparision to directly calculate the gradient through the quasi-Newton iterations of the forward pass, the implicit gradient above is simple and memory-efficient, treating the root solvers as a black-box. Following bai2019deep, we compute by solving a linear system iteratively, as detailed in Appendix C.1. The training algorithm is formally presented in Appendix C.4.

## 6 Experiments

We demonstrate the model capacity of ImpFlows on the classification and density modeling tasks222 See https://github.com/thu-ml/implicit-normalizing-flows for details.. In all experiments, we use spectral normalization (miyato2018spectral) to enforce the Lipschitz constrants, where the Lipschitz constant upper bound of each layer (called Lipschitz coefficient) is denoted as . For the Broyden’s method, we use and for training and testing to numerically ensure the invertibility and the stability during training. Please see other detailed settings including the method of estimating the log-determinant, the network architecture, learning rate, batch size, and so on in Appendix D.

### 6.1 Verifying Capacity on Classification

We first empirically compare ResFlows and ImpFlows on classification tasks. Compared with generative modeling, classification is a more direct measure of the richness of the functional family, because it isolates the function fitting from generative modeling subtleties, such as log-determinant estimation. We train both models in the same settings on CIFAR10 and CIFAR100 (krizhevsky2009learning). Specifically, we use an architecture similar to ResNet-18 (he2016deep). Overall, the amount of parameters of ResNet-18 with vanilla ResBlocks, ResFlows and ImpFlows are the same of M. The detailed network structure can be found in Appendix D. The classification results are shown in Table 1. To see the impact of the Lipschitz constraints, we vary the Lipschitz coefficient to show the difference between ResFlows and ImpFlows under the condition of a fixed Lipschitz upper bound. Given different values of , the classification results of ImpFlows are consistently better than those of ResFlows. These results empirically validate Corollary 1, which claims that the functional family of ImpFlows is richer than ResFlows. Besides, for a large Lipschitz constant upper bound , ImpFlow blocks are comparable with the vanilla ResBlocks in terms of classification.

### 6.2 Density Modeling On 2D Toy Data

For the density modeling tasks, we first evaluate ImpFlows on the Checkerboard data whose density is multi-modal, as shown in Fig. 3 (a). For fairness, we follow the same experiment settings as chen2019residual (which are specified in Appendix D), except that we adopt a Sine (sitzmann2020implicit)activation function for all models. We note that the data distribution has a bounded support while we want to fit a transformation

mapping it to the standard Gaussian distribution, whose support is unbounded. A perfect

requires a sufficiently large for some mapped far from the mean of the Gaussian. Therefore, the Lipschtiz constant of such is too large to be fitted by a ResFlow with 8 blocks (See Fig. 3 (b)). A -block ImpFlow can achieve a result of bits, which outperforms the bits of a -block ResFlow with the same number of parameters. Such results accord with our theoretical results in Theorem 2 and strongly motivate ImpFlows.

### 6.3 Density Modeling On Real Data

We also train ImpFlows on some real density modeling datasets, including the tabular datasets (used by papamakarios2017masked), CIFAR10 and 5-bit CelebA (kingma2018glow). For all the real datasets, we use the scalable algorithm proposed in Sec. 5.

We test our performance on five tabular datasets: POWER (), GAS (), HEPMASS (), MINIBOONE () and BSDS300 () from the UCI repository (Dua:2019), where is the data dimension. For a fair comparison, on each dataset we use a 10-block ResFlow and a 5-block ImpFlow with the same amount of parameters, and a 20-block ImpFlow for a better result. The detailed network architecture and hyperparameters can be found in Appendix D. Table 2 shows the average test log-likelihood for ResFlows and ImpFlows. ImpFlows achieves better density estimation performance than ResFlow consistently on all datasets. Again, the results demonstrate the effectiveness of ImpFlows.

Then we test ImpFlows on the CIFAR10 dataset. We train a multi-scale convolutional version for both ImpFlows and ResFlows, following the same settings as chen2019residual except that we use a smaller network of 5.5M parameters for both ImpFlows and ResFlows (see details in Appendix D). As shown in Table 3, Impflow achieves better results than ResFlow consistently given different values of the Lipschitz coefficient . Moreover, the computation time of ImpFlow is comparable to that of ResFlow. See Appendix C.2

for detailed results. Besides, there is a trade-off between the expressiveness and the numerical optimization of ImpFlows in larger models. Based on the above experiments, we believe that advances including an lower-variance estimate of the log-determinant can benefit ImpFlows in larger models, which is left for future work.

We also train ImpFlows on the 5-bit CelebA. For a fair comparison, we use the same settings as chen2019residual. The samples from our model are shown in Appendix E.

## 7 Conclusions

We propose implicit normalizing flows (ImpFlows), which generalize normalizing flows via utilizing an implicit invertible mapping defined by the roots of the equation . ImpFlows build on Residual Flows (ResFlows) with a good balance between tractability and expressiveness. We show that the functional family of ImpFlows is richer than that of ResFlows, particularly for modeling functions with large Lipschitz constants. Based on the implicit differentiation formula, we present a scalable algorithm to train and evaluate ImpFlows. Empirically, ImpFlows outperform ResFlows on several classification and density modeling benchmarks. Finally, while this paper mostly focuses on the implicit generalization of ResFlows, the general idea of utilizing implicit functions for NFs could be extended to a wider scope. We leave it as a future work.

## Acknowledgement

We thank Yuhao Zhou, Shuyu Cheng, Jiaming Li, Kun Xu, Fan Bao, Shihong Song and Qi’An Fu for proofreading. This work was supported by the National Key Research and Development Program of China (Nos. 2020AAA0104304), NSFC Projects (Nos. 61620106010, 62061136001, U19B2034, U181146, 62076145), Beijing NSF Project (No. JQ19016), Beijing Academy of Artificial Intelligence (BAAI), Tsinghua-Huawei Joint Research Program, Huawei Hisilicon Kirin Intelligence Engineering Development, the MindSpore team, a grant from Tsinghua Institute for Guo Qiang, Tiangong Institute for Intelligent Computing, and the NVIDIA NVAIL Program with GPU/DGX Acceleration. C. Li was supported by the fellowship of China postdoctoral Science Foundation (2020M680572), and the fellowship of China national postdoctoral program for innovative talents (BX20190172) and Shuimu Tsinghua Scholar. J. Chen was supported by Shuimu Tsinghua Scholar.

## Appendix A Additional Lemmas and Proofs

### a.1 Proof For Theorem 1

###### Proof.

(Theorem 1)

Firstly, , the mapping

is a contrative mapping, which can be shown by Lipschitz condition of :

 ∥(F(z1,x0)+z1)−(F(z2,x0)+z2)∥=∥gz(z1)−gz(z2)∥<∥z1−z2∥.

Therefore, has an unique fixed point, denoted by :

 hx0(f(x0))=f(x0)⇔F(f(x0),x0)=0

Similarly, we also have: , there exists an unique satisfying .

Moreover, Let , we have . By the uniqueness, we have . Similarly, . Therefore, is unique and invertible. ∎

### a.2 Proof For Theorem 2

We denote as the set of all bi-Lipschitz -diffeomorphisms from to .

Firstly, we prove Lemma 1 in the main text.

###### Proof.

(Lemma 1). , we have

 supx∈Rd∥Jf(x)−I∥22<1,

which is equivalent to

 supx∈Rd,v∈Rd,∥v∥2=1∥(Jf(x)−I)v∥22<1 (% Definition of operator norm.) supx∈Rd,v∈Rd,∥v∥2=1vT(JTf(x)−I)(Jf(x)−I)v<1

Note that is nonsingular, so , we have . Thus,

So we have

 infx∈Rd,v∈Rd,∥v∥2=1vTJf(x)v>0.

Note that the converse is not true, because does not restrict the upper bound of Lipschitz constant of . For example, when where is a positive real number, we have

 infx∈Rd,v∈Rd,∥v∥2=1vTJf(x)v=infx∈Rd,v∈Rd,∥v∥2=1vT(mI)v=m>0

However, can be any large positive number. So we have . ∎

###### Lemma 2.

, if

 infx∈Rd,v∈Rd,∥v∥2=1vTJf(x)v>0, (17)

then

 infx∈Rd,v∈Rd,∥v∥2=1vTJf−1(x)v>0, (18)
###### Proof.

(Proof of Lemma 2). By Inverse Function Theorem,

 Jf−1(x)=J−1f(f−1(x)).

Because is from to , we have

 infx∈Rd,v∈Rd,∥v∥2=1vTJf−1(x)v =infx∈Rd,v∈Rd,∥v∥2=1vTJ−1f(f−1(x))v =infx∈Rd,v∈Rd,∥v∥2=1vTJ−1f(x)v

Let and , we have , and

 vTJ−1f(x)v=uTJTf(x)u=uTJf(x)u=∥u∥22vT0Jf(x)v0.

The above equation uses this fact: for a real matrix , because .

Note that is Lipschitz continuous, . So

which means

 ∥u∥2≥1Lip(f).

Thus,

 infx∈Rd,v∈Rd,∥v∥2=1vTJ−1f(x)v =infx∈Rd,v∈Rd,∥v∥2=1∥u∥22vT0Jf(x)v0 ≥infx∈Rd,u∈Rd,∥Jf(x)u∥2=1∥u∥22infx∈Rd,v0∈Rd,∥v0∥2=1vT0Jf(x)v0 ≥1Lip(f)2infx∈Rd,v∈Rd,∥v∥2=1vTJf(x)v >0

###### Lemma 3.

, if , we have

 infx∈Rd,v∈Rd,∥v∥2=1vTJf(x)v>0. (19)
###### Proof.

(Proof of Lemma 3). , if , then from Lemma 1, we have

 infx∈Rd,v∈Rd,∥v∥2=1vTJf−1(x)v>0.

Note that , from Lemma 2 we have

 infx∈Rd,v∈Rd,∥v∥2=1vTJf(x)v>0.

###### Lemma 4.

, if

 infx∈Rd,v∈Rd,∥v∥2=1vTJf(x)v>0,

then , s.t. ,

 supx∈Rd∥αJf(x)−I∥2<1.
###### Proof.

(Proof of Lemma 4). Note that is Lipschitz continuous, so . Denote

 β=infx∈Rd,v∈Rd,∥v∥2=1vTJf(x)v.

And let

 α0=βLip(f)2>0.

, we have

 supx∈Rd∥αJf(x)−I∥22 =supx∈Rd,v∈Rd,∥v∥2=1vT(αJTf(x)−I)(αJf(x)−I)v =1+supx∈Rd,v∈Rd,∥v∥2=1α2vTJTf(x)Jf(x)v−2αvTJf(x)v ≤1+α2supx∈Rd,v∈Rd,∥v∥2=1vTJTf(x)Jf(x)v +2αsupx∈Rd,v∈Rd,∥v∥2=1(−vTJf(x)v) =1+α2supx∈Rd,v∈Rd,∥v∥2=1vTJTf(x)Jf(x)v −2αinfx∈Rd,v∈Rd,∥v∥2=1vTJf(x)v =1+α2supx∈Rd∥Jf(x)∥22−2αβ =1+α(αLip(f)2−2β) <1+α(α0Lip(f)2−2β) =1−αβ <1.

The above equation uses this fact: for a real matrix , because . ∎

###### Proof.

(Theorem 2) Denote

 P={f∈D |∃f1,f2∈D,f=f2∘f1,where infx∈Rd,v∈Rd,∥v∥2=1vTJf1(x)v>0,infx∈Rd,v∈Rd,∥v∥2=1vTJf2(x)v>0}.

Firstly, we show that . , assume , where and . By Lemma 1 and Lemma 3, we have

 infx∈Rd,v∈Rd,∥v∥2=1vTJf1(x)v >0, infx∈Rd,v∈Rd,∥v∥2=1vTJf2(x)v >0.

Thus, . So .

Next, we show that . , assume , where

 infx∈Rd,v∈Rd,∥v∥2=1vTJf1(x)v >0, infx∈Rd,v∈Rd,∥v∥2=1vTJf2(x)v >0.

From Lemma 2, we have

 infx∈Rd,v∈Rd,∥v∥2=1vTJf−12(x)v >0.

From Lemma 4, , s.t. ,

 supx∈Rd∥αJf1(x)−I∥2<1, supx∈Rd∥αJf−12(x)−I∥2<1.

Let . Let , where

 g1(x) =αf1(x), g2(x) =f2(xα).

We have , and

 Jg1(x) =αJf1(x), g−12(x) =αf−12(x), Jg−12(x) =αJf−12(x).

So we have

 supx∈Rd∥Jg1(x)−I∥2=supx∈Rd∥αJf1(x)−I∥2<1, supx∈Rd∥Jg−12(x)−I∥2=supx∈Rd∥αJf−12(x


text:
# How do you use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region  y = x^3, y = 0, x = 2 rotated about the x axis?

May 23, 2015

The answer is $V = \frac{128 \pi}{7}$.

Firstly, you should graph your three functions to see clearly what is the delimited region :

We are rotating our plane region around the x-axis, which means we have to write our function $f \left(x\right) = {x}^{3}$ in the form $f \left(y\right) = \sqrt[3]{y}$.

The radius of any cylinder in our final volume will be given by $y$.

And the width will be $2 - \sqrt[3]{y}$.

(Look here for more details or diagrams about the cylinders)

Therefore, the cross-sectional area of any cylinder will be :

$A \left(y\right) = 2 \pi y \left(2 - \sqrt[3]{y}\right) = 4 \pi y - 2 \pi {y}^{\frac{4}{3}}$

The radius of the cylinders, according to the delimited region, goes from :

$f \left(x\right) = 0 = y$ (with $x = 0$) to $f \left(x\right) = 8 = y$ (with $x = 2$).

So the volume of our solid, which is the sum of all the cross-sectional area of the cylinders, is :

$V = {\int}_{0}^{8} A \left(y\right) \mathrm{dy}$.

The antiderivative of $a {y}^{n}$ is given by $a \cdot \frac{1}{n + 1} {y}^{n + 1}$.

Thus, the antiderivative of $A \left(y\right)$ is :

$\frac{1}{2} 4 \pi {y}^{2} - \frac{3}{7} 2 \pi {y}^{\frac{7}{3}} = 2 \pi {y}^{2} \left(1 - \frac{3}{7} \sqrt[3]{y}\right)$

Now we can calculate the integral :

$V = {\int}_{0}^{8} A \left(y\right) \mathrm{dy} = {\left[2 \pi {y}^{2} \left(1 - \frac{3}{7} \sqrt[3]{y}\right)\right]}_{0}^{8}$

$= 2 \pi {\left[{y}^{2} \left(1 - \frac{3}{7} \sqrt[3]{y}\right)\right]}_{0}^{8} = 2 \pi \left({8}^{2} \left(1 - \frac{3 \cdot 2}{7}\right) - 0\right)$

$= 2 \pi \cdot \frac{64}{7} = \frac{128 \pi}{7}$.

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Here's the question you clicked on:

anonymous one year ago Which statement is true about the product square root of 2(3square root of 2 + square root of 14)? It is rational and equal to 12. It is rational and equal to 18. It is irrational and equal to 6 + 2square root of 18. It is irrational and equal to 3 +square root of 18.

• This Question is Closed
1. geerky42

$\sqrt{2}(3\sqrt2 + \sqrt{14})$Right? What do you think so far?

2. geerky42

You can split $$\sqrt{14}$$ into two square roots; $$\sqrt{14} = \sqrt{2\cdot7}=\sqrt2\sqrt7$$

3. geerky42

$\sqrt{2}(3\sqrt2 + \sqrt2\sqrt7)$ Now distribute $$\sqrt2$$

4. anonymous

Um I think I get it but what next?

5. anonymous

simplifying further, you'll get $6+2\sqrt{7}$

6. anonymous

so would that be c ? @aryandecoolest

7. anonymous

@geerky42

8. anonymous

C

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# How is the double slit experiment modeled in contemporary physical theories?

Suppose I have the following double split experiment set up:

• a monochromatic electron source of low intensity, which we can model as emitting a single electron at a time with energy $T$.
• a diffraction screen at a distance $A$ with two macroscopic slits equidistant from the source with width $w$ and distance from the center of the screen $\frac{d}{2}$.
• a detector screen sitting parallel to the diffraction screen at a distance $B$ which consists essentially of a lattice of ammeters each connected with a square surface of side $r$.

Classical quantum mechanics, and experiments, tell us that the following behaviour is experienced:

• Only one random ammeter is disturbed each time an electron is fired
• The probability distribution of the random ammeters being fired is as given by QM.

The interaction of the electron with the detector is not modeled, the electron is essentially modeled as a wave with interference and which ammeter is fired, essentially an effect of particle behaviour, cannot be discussed within this picture.

So, the measurement or decoherence of the electron wave function corresponds to this switch between wave-like and particle-like models of the electron - which is clearly unsatisfying, intellectually, because QM is vague about this - it merely associates mathematical operations with the procedure, without giving any particular physical or mathematical justification of it.

The questions:

Do further refinements of QM (QED or QFT or ST) give any better explanation of what justifies the measurement "recipe" of classical QM by correctly modelling the electron/detector interaction?

How, in layman terms, is the change of picture from wave to particle modeled (if it is at all)?

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Try to look at one dot on the detector screen as at a bit of information about the whole interference pattern. And look at the interference picture as at a picture of something complicated, non trivial, non reducible to one dot. The interference picture (photo) describes the whole experimental set, not one electron. – Vladimir Kalitvianski Jan 30 '11 at 14:25
Sklivvz, To tease out more of the kind of answer I think you might be looking for, consider a relativistic version of the two-slit. Here if D is a distance in the experiment (c=light speed), then what about opening and closing shutters in times small compared to D/c, versus large compared to D/c? The answers might reveal how this is thought to work in a relativistic QM. – Roy Simpson Jan 30 '11 at 16:06
@Roy, maybe you care to elaborate in an answer? – Sklivvz Jan 30 '11 at 19:39
Sklivvz, the dots on the display are OK, they're classical scale thermodynamic events in the detectors, but the dots "traveling" from the source to the detector are prejudicial to the question of which interpretations of QM we might be discussing. While only when the state preparation device is present and powered up do we see events at the screen, nonetheless there might be no particles or nothing at all traveling from one to the other (or both ways in some interpretations). You could draw and think about the experiment without those intermediate dots. – Peter Morgan Mar 14 '11 at 13:06
If you don't have the Feynman Lectures On Physics, volumes 1-3, you might want to get them, though I'm sure there are other equally good sources I don't know about. They're very easy to follow. Even in the first volume, he does a beautiful job of explicating the wave-particle duality. – Mike Dunlavey Dec 4 '11 at 23:27

Dear sklivvz, the very same question was asked a few days ago. Quantum field theory, string theory, or any other viable theory that may supersede quantum mechanics directly reduces to non-relativistic quantum mechanics in the non-relativistic limit and changes nothing about the basic postulates of quantum mechanics.

It means that you may find the corresponding low-energy, low-speed (multi-)particle states in the Hilbert space of QFT or string theory or anything else - essentially creation operators acting on the relevant vacuum - and you may prove that the QFT or stringy Hamiltonian acts on these states exactly as the non-relativistic Hamiltonian does, plus corrections that go like positive powers of $1/c$ and may be neglected in the non-relativistic limit.

So nothing changes about quantum mechanics and the double-slit experiment in QFT or string theory and it's likely that those things will never change. Cheers, LM

-
Hi Lubos, this does not address the question - it's a given that more advanced theories must reduce to QM, as QM models successful experiments. This is just like saying that any gravitational theory must reduce to Newton. However, for example, QED lets us model the detectors so, in theory, one should be able model the measurement in a way that reduces to classical QM. What is this model? How does it reduce to classical QM? – Sklivvz Jan 30 '11 at 12:15
Dear Sklivvz, if you're asking how a silicon plate or iron or glass plate is represented in QED, then you're not asking a question about the double slit experiment; instead, you're asking a question about condensed matter physics. The precise description of a material depends on the material. At any rate, QED or SM or string theory won't say anything substantially different than non-relativistic QM. If you're assuming that relativistic QFT or string theory change anything about the interpretation of QM or the measurement, then your assumption is incorrect. They don't change anything. – Luboš Motl Jan 30 '11 at 16:46
Dear Lubos, I am asking whether modelling the detectors gives us a better understanding on why we observe a particle-like effect (one ammeter spikes per electron), instead of a wave like effect (all ammeters have spikes distributed in interference patterns). Classical QM doesn't explain this at all (or for that matter why we should use a position operator when we are measuring current). – Sklivvz Jan 30 '11 at 19:32

The answer is "yes and no". Important research is being done on modelling, with a Hamiltonian, the joint interaction between a microscopic particle being measured and the macroscopic measurement apparatus doing the measuring. Allahaverdyan, Balian, et al. have done the best, latest work. H.S. Green did a very stylised model long ago. Hannabuss has been doing important work on this.
Bibliographic information for their papers, and others by Collet, Milburn, et al., in Quantum Optics, there is also C. Gardiner and P. Zoller, Quantum Noise, can be found in the bibliography to my Thermodynamic Limits, Non-commutative Probability, and Quantum Entanglement, published, available for free at http://arxiv.org/abs/quant-ph/0507017 and my longer paper at http://arxiv.org/abs/0705.2554

That was the "yes" part. The cited authors are mainstream important researchers.

But. It doesn't do much to change the big picture.

In particular, it does not resolve the controversy about decoherence. The work of these authors is valid whether or not there is decoherence, nor do they address the question of what would produce it and when.
Instead, they address directly the measurement process as a unitary, quantum mechanical process. The electron is still entangled with the slits and ammeter and everything, but they can say something about the precise way in which that entanglement is "negligible for all practical purposes". So it is important work.

More precisely, to answer the last parts of your question, it is not even a good idea to think of the "particle" as sometimes being examined in a a wave picture and sometimes in a a particle mode. That is just sloppy undergraduate thinking, you won't find that in the axioms of QM and you won't find it in these papers. There is no "this switch between wave-like and particle-like models of the electron" in their careful analyses of the measurement process, nor in the axioms of QM. In the axioms there is a switch from using the unitary evolution axioms of the wave function to the axioms for a measurement process, but this has nothing to do with modelling the electron. These cited papers succeed in modelling the measurement process as a unitary deterministic evolution of a quantum combined system. The resulting entangled superposition of states is not the same thing as what the measurement axioms predict, because the entangled state is a superposition of quantum states and the "result of a measurement process" is supposed to be a probability distribution on separable states. But they can show that the difference between these two different things is practically negligible because the "coherence" in the superposition is very very low...If now at this step you, dear reader, take the point of view of the "decoherence" crowd and apply it and go a little further than these results do, you can then say that since the coherence is negligible, this superposition of entangled states will appear the same as a mixed state, and a mixed state is, « as we all know », a probability distribution on its different components.

So yes, progress has been made, but J.S. Bell would never have accepted going from a quantum superposition of states to even a diagonal density matrix since logically these are distinct conceptions, so the controversy continues.

-

An important contemporary approach to the two-slit situation you ask about is decoherence. Many contributors to this forum are fans of decoherence, I am not, but it is very important and worth attention. I couldn't help noticing you still had queries about it even after the explanations of these «fans».

Since one should always be able to include more of the outside world in the box of quantum analysis, always be able to push the boundary out, you are basically asking if any progress has been made by including the slits, and detectors, in the unitary Quantum picture, or if instead any progress has been made by changing the unitary picture even a little. The decoherence approach does not change the unitarity of the evolution, and neither will QFT. (Some have wondered if gravity or other non-linearities will, but we will not go into that right now.)

## Brief sketch of decoherence

For simplicity, assume there are only two ammeters behind the slits. As always with unitary evolutions, the electron, after being diffracted by the slits, and when it reaches the plane of the ammeters, is in a superposition of two states belonging to the different locations on that plane of the two ammeters (normally it will be in a superposition of many states but we can simplify here, too). $c_1\psi_1 + c_2\psi_2$. (We will simply neglect the occurrences where neither ammeter fires.) The state space is effectively C$^2$. Now, the ammeters are being modelled quantum-ly, too, so the set of two ammeters has a Hamiltonian and a Hilbert Space for the state vectors (wave functions) of the system of two ammeters, $H_{amm}$. The state space for the combined system of electron--being--measured and measurement apparatus (two ammeters) is then C$^2\otimes H_{amm}$. The decoherence approach makes more or less the same key assumption Mott, London, Wigner, and many others make down to today (but which I query, elsewhere, see my other posts and links), which is that whatever dictionary or correspondence there is between the macroscopic idea of «ammeter 1 fires» or, parallellly, «ammeter 2 fired», is to be modelled by a collection of quantum states in $H_{amm}$ (this is what I usually complain about, but not here), or, since we can pass to their closed span and simplify to assume there is one state, label them $\phi_1$ and $\phi_2$. (A key point in this regard will be expanded upon in a minute...) Now since this means that $\psi_1 \otimes \phi_o$ evolves unitarily to $\psi_1 \otimes \phi_1$ (Here, $\phi_o$ is the initial state of the ammeter-lattice while it waits to discharge or fire) but that $\psi_2 \otimes \phi_o$ would evolve to $\psi_2 \otimes \phi_2$, by linearity (no progress has been made, IMHO, by tinkering with the assumption of linearity), what actually happens is that our electron and ammeter as a combined system evolve to $$c_1 \psi_1 \otimes \phi_1 + c_2 \psi_2 \otimes \phi_2,$$ which is an entangled state: neither the ammeter nor the electron can be considered as a separate system anymore.

At this point, decoherence, depending on which flavour is administered, points out something undeniable: in fact, there are many more degrees of freedom within the ammeters: our simplification (which is the same as that of Wigner, EPR, and many others) overlooks something important. It is, according to this theory, important that the ammeters are at least weakly coupled to the environment. It is not quite a closed system, so the analysis above is only approximate.

All decoherence approaches (as far as I know) use density matrices. (I would be interested in a current reference to one that works only with pure states and not density matrices.) It can be shown in theory, rigorously, that this coupling with the environment leads to a further thermodynamic-like evolution to a density matrix which is very nearly diagonal and so can be regarded as a classical (or Bayesian) probability distribution on the two states, which are each obviously separable: $$\psi_1 \otimes \phi_1$$ and $$\psi_2 \otimes \phi_2.$$

## The Coleman--Hepp model and Bell's response

In my opinion, the grand-daddy of all decoherence theories is the so-called Coleman--Hepp model. I learned it in Bell's famous paper, a freely available copy of it is here: http://www.mast.queensu.ca/~jjohnson/Bellagainst.html in which he tranlsates it from the language of QFT in C*algebras to the Schroedinger picture and Heisenberg picture. Not on Los Alamos archive. Of course Coleman and Hepp are two most distinguished physicists. Briefly, the criticism, which I agree with, is that a density matrix is not a state, so this is really no better from a logical or foundational point of view than the open system approach and suffers from the same question-begging. (Which is more than thiry years old now, so it's not progress.)

### The Physics of these models

These models all use a kind of thermodynamics, and this is surely right, as Peter Morgan (hi peter) said in a previous post, the possible detections are thermodynamic events...and the way to make that precise is to take some sort of limit as the number of degrees of freedom goes to infinity. But none of these models reflects in the model that measurement is a kind of amplification, none of the thermodynamic limits involved uses negative temperature, and this is surely wrong. Feynman's opinion was that this was decisive, see Feynman and Hibbs, Quantum Mechanics and Path Integrals, New York, 1965, p. 22. So these models do not incorporate Feynamn's insight. The models of Balian et al. cited above do incorporate this, and study phase transitions induced by tiny disturbances from an unstable equilibrium which is indeed the physics of bubble chambers and photographic emulsions. Bohr thought that the apparatus had to be classical, and decoherence models do not use a limiting procedure that introduces a classical approximation. So they do not incorporate Bohr's insight.

There is experimental evidence that interaction with the environment induces decoherence, but not yet in a way relevant to measurement. There is also experimental evidence , the so-called spin echoes, that mesoscopic systems can recover their coherence after losing it, which is what Wigner always assumed.

-

So, the measurement or decoherence of the electron wave function corresponds to this switch between wave-like and particle-like models of the electron - which is clearly unsatisfying, intellectually, because QM is vague about this - it merely associates mathematical operations with the procedure, without giving any particular physical or mathematical justification of it.

Quantum mechanics doesn't have two models of the electron. For intuitive purposes it can be useful to visualize very broad wave packets (that are almost a momentum eigenstate) as "waves" and very localized wave packets (that are almost a position eigenstate) as "particles", but the QM description is unique.

The Copenhagen interpretation assumes that a measurement of position "collapses" the (position) wavefunction to a very localized wave packet with a width in the order of the measurement uncertainty. Decoherence explains the apparent collapse of the wavefunction by an entanglement between the system and environmental degrees of freedom due to the measurement process.

-
 "Decoherence explains the apparent collapse of the wavefunction by an entanglement between the system and environmental degrees of freedom due to the measurement process.": Ok can you be more clear about this by avoiding jargon? I don't really understand what you are saying. – Sklivvz♦ Jan 30 '11 at 13:48 If you are measuring a system, you can't avoid entangling the system with macroscopic objects (measurement device, observer, ...) because the state of this objects must be modified by the measurement. So you won't observe a superposition of measurements because your state will be entangled with the measurement (in the MWI the different measurement outcomes are in "different branches"). – mmc Jan 30 '11 at 14:15 All "mainstream" interpretations of quantum mechanics assume that the results of measurements will be non-deterministic. In the Copenhagen Interpretation it's just a postulate, while in the Many Wolrds Interpretation it comes from you being unable to know the "branch" where you will end. I don't know enough about the Consistent Histories Interpretation to explain it in detail, but I know it doesn't deterministically select an outcome (I think that it just discards unobservable entanglements to get a "classical superposition".) – mmc Jan 30 '11 at 14:34 My question is not about which interpretation of QM is correct, but about why QM is correct as-is. – Sklivvz♦ Jan 30 '11 at 19:38 @Skli Because it agrees with experiments – Holowitz Jan 30 '11 at 20:41

I find it strange that even after 85 years since its inception, quantum theory has not been fully appreciated! I don't know what you meant by the term "classical quantum mechanics". I guess, it reflects your desire to "explain" quantum theory in terms of classical intuition. But what one should do is really the opposite. As far as your question is concerned, quantum theory in its current formulation has no problem. No agonizing scrutiny of the basic postulates are at all required. "Measurement" in QT has always confused some people and doubtless will continue to do so. But once decoherence is understood, it is not such a big deal.

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By classical QM I mean QM as it is taught in 99% of all courses (e.g. Copenhagen, not relativistic, Schrödinger, etc). – Sklivvz Jan 30 '11 at 13:44
Exactly, sb1! 85 years is a pretty long time. People often humiliate historical figures who didn't accept heliocentrism or something like that 20 years after they were established in science, but sometimes the very same people deny key insights of physics that were found 85 years ago. – Luboš Motl Jan 30 '11 at 16:48
I fully appreciate QM. That said, it doesn't mean that it explains everything or that it is a complete theory or that I can't have a question on it - comparing having doubts about something and not believing in heliocentrism is quite frankly demeaning and laughable. Especially because what is the correct interpretation of QM is still an open problem in physics. – Sklivvz Jan 30 '11 at 19:37
@skli not in physics, in philosophy – Holowitz Jan 30 '11 at 20:43
@kake, that is completely debatable. – Sklivvz Jan 30 '11 at 22:35
show 1 more comment

I am always amazed that no one ever points out the obvious in these discussion: it is impossible experimentally to verify a one-to-one correspondence between the emission of an electron and the firing of a detector. The reason is that we do not yet have a pea-shooter for electrons. We never quite know when one has been fired. All we can say experimentally is that the rate of firing detectors is in some way proportional to the average flux of electrons.

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A thin layer of a weak beta-emitter on a wire or as a dot on a metal sheet is such a "one electron at a time gun. A multiplier arrangement hit by such an electron is able to detect it clearly. – Georg Dec 5 '11 at 10:22
arxiv.org/abs/1103.2260 "A new type of electron sources has emerged which permits to inject particles in a controllable manner, one at a time, into an electronic circuit." – anna v Dec 5 '11 at 14:39
It is interesting that the authors of the paper argue that until March of this year, my position was the correct one: that we did not yet have a pea-shooter for electrons. The authors claim to have recently gotten around this constraint. I'm betting that they haven't. – Marty Green Dec 5 '11 at 15:32


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# Domain optimization for the Navier-Stokes equation by an embedding domain method

## Inst. Mathematik

Domain optimization problems for the two-dimensional stationary flow of incompressible linear-viscous fluids, i.e. the Navier-Stokes equations, are studied. An embedding domain technique which provides an equivalent formulation of the problem on a fixed domain is introduced. Existence of a solution to the domain optimization problem and Fréchet differentiability with respect to the variation of the domain of tracking type cost functionals for the velocity field are proved. A simply computable formula for the derivative of the cost functional is presented. Numerical examples show the advantages of the embedding domain method and the reliability of the derivative formula.


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# SciPy 2015: Cython Tutorial

written by Eric J. Ma on 2015-07-06

This morning, I sat in the Cython tutorial by Kurt Smith, of Enthought, at SciPy 2015. We’re only halfway in at the time of starting the writing, but it has been a really useful tutorial!

### Tutorial Content

Kurt has designed this tutorial to be an intermediate tutorial, where participants are expected to have had some experience with Python before.

During this tutorial, we covered two practical ways to speed up Python code. One is to declare Python variables, functions or objects as C types using the cdef statement, and the other is to simply wrap existing C code.

### Tutorial Pace

It was a very well-paced tutorial, and Kurt did a great job of structuring the content such that each exercise followed roughly the same pattern. We spent approximately 20-30 minutes on each section, partly because of the really advanced questions (from my point of view) that other participants asked.

### Coding Practice

We got four activities to get used to doing things. Each practice activity was small but not trivial - basically adding the cdef declarations or writing the wrapper code, not needing to write complex algorithms.

### Reflections

Already in the first half of the tutorial, I think I have gained two superpowers. The first is that I can speed up my own code without needing to write C, if I need to, simply by adding cdef and cpdef statements where needed. The second is that I can use pre-existing C code that others have written if I need to. Both ways, I can easily speed up my computation with minimal effort. Having Python under my belt to do scientific computing was already a superpower; now, faster Python is also in the repertoire!

From an education standpoint, I think it is also true that knowing the “howto” steps is really important in learning programming. For the majority of scientists, why certain things work on the computer might not be as important as knowing how to make certain things work - as computational scientists, we’re most interested in getting answers, and preferably getting them fast. The simple and structured examples, helped a lot with this.


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# Quantum Field Theory

1. Mar 7, 2003

### Tom Mattson

Staff Emeritus
Hello everyone!

This is the rebirth of my thread in PF v2.0 entitled "Do you know QM and SR?" Since I started that thread, a 2nd edition of the book (Warren Siegel's Fields) has been released. The url is:

http://xxx.lanl.gov/pdf/hep-th/9912205

I'll post some of the more useful comments from the old thread shortly.

2. Mar 17, 2003

### Tom Mattson

Staff Emeritus
OK, first off my notes on the first subsection of Chapter 1 are on the web, here:

Second, once it came to light that group theory is a central mathematical theme in QFT, the following links were provided:

http://members.tripod.com/~dogschool
http://www.wspc.com/books/physics/0097.html [Broken]
http://www.wspc.com/books/physics/1279.html [Broken]

Third, the subject of the physical meaning of the commutator came up. Here was my attempt to explain it:

in classical mechanics, a quantity 'A' that does not explicitly depend on time and whose Poisson bracket with the Hamiltonian {A,H}=0, and for every conserved quantity there is an associated symmetry. The result carries over to quantum mechanics if one takes the commutator [A,H]=AH-HA instead of the Poisson bracket. [A,H]=0 implies that A is conserved (again, provided that A does not depend explicitly on time).

Commutators are also involved in the uncertainty principle. When [A,B]=0, then A and B are simultaneously diagonalizable. That means that there are simultaneous eigenfunctions of both operators, meaning that both observables can be measured simultaneously. If [A,B] does not equal zero, then there is an uncertainty relation. An example is [x,p]=i(hbar).

Fourth, the subject of Hermiticity of operators came up. Again, here is what I had to say on the subject:

The dagger signifies the Hermitian conjugate. For a matrix A, A+ is the complex conjugate of the transpose. An operator is Hermitian if A=A(+). The Hermitian conjugate is important for (at least) two reasons:
1. A=A(+)-->A has real eigenvalues, so all physical observables correspond to Hermitian operators.
2. An operator is unitary if A(+)=A-1. (Someone correct me if this is wrong): All operators that form a group with Hermitian generators are unitary. Unitarity is necessary to guarantee probability conservation.

A(+) is "A-dagger". It seems that superscripts are not done in the same way here as they were in the last version of PF.

The following link on brackets was supplied:

I'll post some more of the relevant discussion when I get a chance.

Last edited by a moderator: May 1, 2017
3. Mar 17, 2003

### Hurkyl

Staff Emeritus
Hrm, did you mean to say "unitary generators" instead of "hermitian generators"?

BTW that last link is a nice one! The section on generators makes more sense now; the way it was presented by Siegel seemed fairly arbitrary: it worked, but there was no motivation for it.

Hurkyl

Last edited: Mar 17, 2003
4. Mar 18, 2003

### Tom Mattson

Staff Emeritus
Nope.

Take the time evolution operator, for instance:

U(t,t0)=exp[-iH(t-t0)/hbar]

U is unitary, and it is generated by H, a Hermitian operator.

5. Mar 18, 2003

### arivero

Peskin

Aside, let me to note that in superstringtheory.com people is doing (for free) a reading of the Peskin Schroeder. This week they are starting chapter nine.

6. Mar 18, 2003

### Hurkyl

Staff Emeritus
Oh right, lie generator. (bonks self) Ok I withdraw my objection.

Hurkyl

7. Mar 18, 2003

### Tom Mattson

Staff Emeritus
OK, back to my recap:

Fifth, the subject of symmetry in the context of indices came up. My remarks:

He's talking about symmetry/antisymmetry under an exchange of indices.
Take Sij=xiyj+xjyi
That combination is symmetric under the exchange i<-->j, because Sij=Sji.
Now take Aij=xiyj-xjyi
That combination is antisymmetric because Aij=-Aji.

Sixth, there was the ever-so-troublesome issue of correspondence. Specifically, what the heck are those terms of order (hbar)^2 in the correspondence between the commutator and the Poisson bracket?

A guy (lethe) at another forum (sciforums) helped me out here:

Tom, you have written:

the correspondence principle that you learned in QM does not always work like this.

a reminder:
the correspondence principle says turn your classical poison bracket into your quantum commutator. and then write your classical function with the classical canonical variables replaced by the corresponding quantum operators.

for example, if the classical function is qp, the quantum operator cannot be QP, but rather must be 1/2(QP+PQ). for functions of degree 3 or higher it gets more complicated. you have to also include some quantum anomoly terms, when taking the poison pracket to the commutator.

for example, one can show that the correspondence principle takes {pq2,p2q} = 3p2q2 --> i\hbar/3[P3,Q3] + \hbar^2

that term on the end there means that we have to add a quantum anomoly term when we quantize the system, in order to be consistent with the assumptions of correspondence principle. but notice that the anomoly term is of order hbar^2.

you can also get quantum anomoly terms in your hamiltonian, when you, for example, try to quantize a noncartesian system. here again, the anomoly term is of order hbar^2.

so in the classical limit, hbar --> 0, and all variables commute.

in the semiclassical limit, divide by hbar, then take hbar --> 0, and the anomoly terms disappear, and the commutators simply become the poisson brackets, the hamiltonian becomes the classical hamiltonian.

i believe that their is no axiomatic way that is self consistent to specify the rules for quantization.

tom, i haven t looked at any of those documents you listed as references, do they treat this issue?

is it clear what s going on here? i m getting this mostly from ticciati. also shankar has a bit about this.

i think this is what s going on with siegel, but i ain t positive, so i would appreciate any feedback.

let me explain that in a little more detail:

let s say that i have a classical system that i want to quantize. this means that i want to write down a mapping z that, at the very least satisfies the following conditions:

z(p) = P
z(q) = Q

z({f(p,q),g(p,q)}) = i*hbar [z(f(p,q)),z(g(p,q))]

where f and g are any functions.

naively, you might want the mapping to be a little stronger. you might want it to take

z({f(p,q),g(p,q)}) = i*hbar [f(P,Q),g(P,Q)]

but we will see that even the weaker condition above is not possible. using just these three assumptions, you can discover by considering all the commutators that z(p2)=P2 + k, for some constant number k, and similarly for Q. then taking the poisson bracket of those two functions immediately shows that

z(pq)=1/2(PQ+QP)

so my second stronger condition above is violated. if i wanted to preserve the functional form of the observable, i would have to require z(pq) = PQ.

this fact is not so distressing however, because the classical variables commute, there is some ordering ambiguity. and furthermore, this symmetric sum is required to make the quantum observable hermitian.

however, the problem gets worse when you go to third degree functions.

you can calculate the poisson bracket for those third degree terms i mentioned above, and then plug in the mapping for that poisson bracket, and compare it with the commutator, you will see that the mapping no longer preserves the bracket to commutator isomorphism, except in the hbar --> 0 limit.

according to ticciati though, this is only of academic interest, since in QFT we never have to quantize systems with cross terms like that.

Pellman and Alis also had some helpful comments, but unfortunately I did not copy them in time.

8. Mar 18, 2003

### Tom Mattson

Staff Emeritus
On to Fermions...

This section is written in anticipation of later studying Supersymmetry (SUSY). Looking at the superstringtheory.com site, I see that the two main mathematical topics unique to SUSY are *drumroll*

Anticommuting variables and Graded Lie Algebras

So, it's not surprising that we find both of those in Chapter 1 of a book written by a string theorist.

Hurkyl's thoughts:

So, I imagine then that the vector space involved in the problems is either a Grassman algebra, or is implicitly extended to one, which leaves the |x> notation for a vector kinda moot because the vector is identified with the operator x, but I suppose kets are still useful for clarity's sake.
I imagine then that |0> is not the zero vector, but is instead the multiplicative identity for the algebra? That would explain how the one homework problem could possibly work:
|x> = e^(xa+)|0>

I also imagine then that each "anticommuting variable" is simply a generator of the algebra? So then any vector v in the algebra of a single anticommuting variable x can be written:
v = a 1 + b x = a|0> + b|x>
?
And then for two anticommuting variables x & y:
v = a 1 + b x + c y + d xy = a|0> + b|x> + c|y> + d|xy>
?

The notation of Grassman numbers comes from old days, when the notation of c- and q-numbers to write complex numbers and operators in a Hilbert space was common. Now these numbers are in fact elements of the exterior algebra, and the anticommutation rule is nothing more but the statement of the elementary property of the wedge product of linear forms.

A little addendum. All these commutation anticommutation relations indeed show that we will have to enlarge the concept of Poisson bracket or Lie bracket to a more general structure that covers them. If we write the anticommutation as the ordinary bracket, the operators will be symmetric, and therefore do not belong to ordinary structures. However, since bosons behave like ordinary algebras and the action boson-fermion gives fermion, the fermions will have a module structure with respect to the bosons. Now it follows that all these are the ingredients to define a supersymmetry algebra.

since Tom made me curious with the question, I have had a look on the book and seen that brackets are formally introduced after the question of commutation anticommutation of bosons and fermions. This is bad, as I see it, since finally all will be reduced to these or similar brackets and the isolated presentation leads to confusion.

And Heumpje was quite helpful here, too:

For instance:
[int]dc 1 = 0
[int]dc c = 1
or
(d/dc)(c*c)=-c*
while
(d/dc)(cc*)= c*
where c is a grassman number and c* its complex conjugate.
When calculating a path integral you have the problem that because of the introduced minus signs in the exponent your integral becomes indefinite. To prevent this Grassman numbers are introduced. It doesn't solve anything in the end, since your now stuck on a set of, although nice from a mathematical viewpoint, ununderstandable numbers.
There is a nice introduction to the use of Grassman numbers in:
"Quantum Many-Particle systems" By Negele & Orland, Frontiers in Physics series from Adisson-Wesley, 1988

That's all I want to bring over from PF v2.0 right now. I have a little bit on Lie groups (the 3rd subsection), but I want to try to pick it up here with Fermions next time, if for no other reason than to figure out what I am doing wrong on those exercises.

9. Mar 20, 2003

### lethe

did you guys figure out what the half bracket half brace: [A,B} notation means? that was kind of confusing to me...

10. Mar 20, 2003

### rutwig

This notation wants to point out that you have two types of commutators, the ordinary Lie bracket [] and the symmetric bracket {} corresponding to the fermion action. But this is the worst possible notation, and leads to confusion. It can all be expressed by using only the bracket [] and requiring the Jacobi superidentity.

regards,

11. Mar 31, 2003

### pellman

Anybody get exercise IA2.2 on pg 45? Here's how I see it. (I'm going to use P instead of psi).

The delta function is P '- P, the most general function of P is a + bP, so the integrand is

(P' - P)(a - bP') = aP' - aP -bP'^2 + bPP' (P'^2 term vanishes)
= (a + bP)P' - aP

The anti-derivative of this is (a + bP)P'^2/2 - aPP' = aPP', but what are the limits of integration for P'? All "P-space?" What space is that anyway? And does it even have the necessary characteristics (e.g., the correct topology) to allow integration to be done with anti-derivatives?

I don't see how we can get a + bP out of this.

- Todd

12. Apr 4, 2003

### Tom Mattson

Staff Emeritus

I, too, am stumped here. Oh, Ruuuutwiiiig!

Whoa, are you saying that the delta function is the simple difference of P and P'? I thought it was a distribution defined similarly to the delta function for "normal" (commuting) variables?

Siegel explicitly states that the integration is indefinite.

I found a paper at the LANL arXiv that gives a rundown this Grassman calculus. I will dig up the link and post it here, as well as take another crack at this one.

13. Apr 5, 2003

### rutwig

Ok, here is another assertion of Siegel which is not vey clear. The intention here is to generalize the known analysis to "superanalytic" functions, that is, functions having commuting an anticommuting variables. Let v be an anticommuting variable and let be given a displacement (infinitesimal) by an (infinitesimal) odd supernumber dv. If f: F--> G (F odd supernumbers, G Grassmann algebra) is a function, then f(v) must have also a displacement with:

df(v)=dv[(d/dv)f(v)]=[f(v)(d/dv)]dv (**)

(d/dv) is called the left derivative operator, and ...the right...

Now, f(v)=av+b is the most general soltuion to (**). In fact, if you expand v as a series in the Grassmann algebra, and similarly f(v), regard the coefficients of the latter as functions of the coefficients of v and vary them infinitesimally, so that the expression of dv is obtained. Now look at the conditions which the coefficients of f(v) must satisfy in order to agrre with (**).
Since the f is linear, it suffices to give sense to [inte]dv and [inte]v dv. Now, if the equation (known for distributions)

[inte] [(d/dx)f(x)]dx=0

shall also hold for anticommuting beasts, then

[inte]dv= 0 and [inte]vdv=y (y is a supernumber). Thus

[inte] f(v+a)dv=[inte] f(v)dv and

[inte]f(v)(d/dv)g(v)dv= [inte] f(v)(d/dv)g(v)dv

14. Apr 7, 2003

### pellman

That's the problem. We're trying to show that δ(ψ) = ψ. (I've got this math-symbol thing figured out now.)

Why should it hold at all? For a plain old commuting scalar variable, for instance,

∫ [(d/dx)f(x)] dx = f(b) - f(a)

if b and a are the endpoints. In any case, it would only be special cases that vanish right?

15. Apr 7, 2003

### pellman

Easier(?) question

I'm really more interested in Ex IA2.3 on page 47. (Note: there is a correction to this exercise on the errata page.)

Can someone please show me how

{a,a*} = {ζ,-∂/∂ ζ}
= - ζ∂/∂ ζ - (∂/∂ ζ)ζ
= 1 ?

If ζ were a usual commuting variable, then [ζ,-∂/∂ ζ] = - ζ∂/∂ ζ + (∂/∂ ζ)ζ = 1 is easy to show. How does the anticommuting nature of ζ change the behavior of the derivative here?

(Edited for more)

1.
{a,a*}ψ(ζ) = {ζ,-∂/∂ ζ}ψ(ζ)
= - ζ(∂/∂ ζ)ψ(ζ) - (∂/∂ ζ)[ζψ(ζ)]
= - ζ(∂/∂ ζ)ψ(ζ) + ζ (∂/∂ ζ)ψ(ζ) - 1 x ψ(ζ)
(the plus sign arise because I guess the zeta and derivative anticommute, although I don't really see it.)
= - ψ(ζ)
=> {a,a*} = -1

2. Another approach
Let ψ(ζ) = A + Bζ. Then..

{a,a*}ψ(ζ) = - ζ(∂/∂ ζ)(A + Bζ) - (∂/∂ ζ)[ζ(A + Bζ)]
= - Bζ - (∂/∂ ζ)(Aζ) (ζ2 = 0)
= - Bζ - A
= - ψ(ζ)
=> {a,a*} = -1

(even more)

Okay. The reason 1 and 2 give {a,a*} = -1 is because I am working in the <ζ|ψ> representation and the operators are acting on the bra instead of the ket and the negative derivative becomes a positive derivative in that case. That is,

<ζ|a = (a*|ζ>)* = (-∂/∂ζ |ζ>)*
= + ∂/∂ζ<ζ|

and

<ζ|a* = <ζ|ζ

Why we get the plus sign is still a mystery to me.

Last edited: Apr 8, 2003
16. Apr 9, 2003

### rutwig

Re: Easier(?) question

Again there is a mistake of signs in the book due to an unconventional choice. The definition of fermionic oscillator is fixed, and set for convenience (see the realization of the Heisenberg algebra). Now, if we set a|[the]>=[the] |[the]>. Thus |[the]>=exp([the]a*)|0> and we obtain a*|[the]>= ([pard]/[pard][the])[the].

Strong recommendation: take another book, Siegel has not been revised carefully. The text I used to recommend can be found at:

http://arturo.fi.infn.it/casalbuoni/lezioni99.pdf [Broken]

Last edited by a moderator: May 1, 2017
17. Apr 9, 2003

### rutwig

for antocommuting variables the comparison with ordinary ones leads inevitably to confusion. Conventional prejudices must be given up, since here no measure theory plays a role. And it is required to hold.

18. Apr 9, 2003

### pellman

Re: Re: Easier(?) question

Thanks, rutwig!

Last edited by a moderator: May 1, 2017
19. Apr 11, 2003

### pellman

I have corresponded with Siegel about ExIA2.3 and he has added another comment to the errata page. However, his comment is brief and may still be confusing.

If any of you are interested, just let me know and I will type up the exercise as it should correctly appear and post it here.

- Todd

20. Apr 15, 2003

### Tom Mattson

Staff Emeritus
Could ya?

Thanks.

Rutwig--I am printing out that other book. It looks much more conventional, which is good, but I want to ask you something. What would it take for us to be able to get through Siegel's book? I ask because he has a trilogy of books (QFT, SUSY, and String Theory) online, and I was hoping to get through them all.


text:
# Vector integration by parts

## Main Question or Discussion Point

I was trying to derive the following results from 4B.8 as suggested by using the vector triple product identity but have been unsuccessful in deriving $\vec{L_R}$ and $\vec{S_R}$ in the end. After using the identity and finding the integrand to be $\vec{E}(\vec{r}\cdot\vec{B}) - \vec{B} (\vec{r} \cdot \vec{E})$ what would be the next best step to take to re-derive the 4B.10 and 4B.11?

After using the identity and finding the integrand to be $\vec{E}(\vec{r}\cdot\vec{B}) - \vec{B} (\vec{r} \cdot \vec{E})$
Okay. I've looked into that identity as well now, and after using it, still don't quite get any further since it reduces back to the case of the integrand equalling $\vec{E}(\vec{r}\cdot\vec{B}) - \vec{B} (\vec{r} \cdot \vec{E})$.


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# Untitled

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LOL POINTER TRUNCATION!!1

Ok, so what is the correct way to use SetWindowLongPtr? Doing this:
SetWindowLongPtr(m_hWnd, GWLP_USERDATA, (LONG_PTR)this);
works fine in x64, but gives the warning "warning C4244: 'argument' : conversion from 'LONG_PTR' to 'LONG', possible loss of data" in Win32.
So, I changed it to:
SetWindowLongPtr(m_hWnd, GWLP_USERDATA, (LONG)(LONG_PTR)this);
Expecting LONG to be 64 bits in x64. Wrong.

It worked fine (surprisingly) till shutdown, when I got a couple of access violations (Which didn't crash the app, however), sue to the pointer getting truncated to 32 bits.

So, my fix was:
// Fucking Microsoft headers#ifdef _WIN64#	define CAST_TYPE	(LONG_PTR)#else#	define CAST_TYPE	(LONG)(LONG_PTR)#endif// Later on...SetWindowLongPtr(m_hWnd, GWLP_USERDATA, CAST_TYPE this);

Surely there's a nicer way?

On the plus side, x64 works fine now. Yaaaaaay [smile].

lol steve you resorted to macro hax!!

[sick] @ syntax!

I'm a fan of C++ style casts, static_cast<type>(expr) for example, but your current macro looks like non-code... without knowing what your macro's definition is I'd read that code as being uncompilably broken [oh]

Jack

Quote:
 Original post by jollyjeffers [lol] @ code comments [sick] @ syntax! I'm a fan of C++ style casts, static_cast(expr) for example, but your current macro looks like non-code... without knowing what your macro's definition is I'd read that code as being uncompilably broken [oh] Jack
Yeah, the comment is because it all resolves down to something that won't compile in Win32 mode, which is somewhat retarded.

I should really learn to use the C++ casts. I'm all self taught, and a lot of it is from reading other peoples code - I taught myself C from Quake C, Allegro stuff, and online tutorials before I actually got a proper C++ book. I think I know how to use C++ casts (static_cast is a cast from related types, reinterpret_cast is mainly int->pointer conversion [unrelated types], and dynamic_cast is the work of the devil. I'll implement RTTI for the classes I need, not everything...)

## Create an account

Register a new account


text:
# Revision history [back]

This may not be the "right way" to do it, but here is what I do:

In my home directory, in the .sage subdirectory I have the following "init.sage" file:

CODE = '/home/mike/Projects/Sage/'
attach CODE+'defaults.sage'

Everything that I always want is in defaults.sage. Then in a particular sage program, I do something like:

attach CODE+'Differential_Equations/series.sage'
attach CODE+'Differential_Equations/pde_misc.sage'
attach /home/mike/Projects/Science/Electrodynamics/Electrostatics.sage

Note that while a couple of the things I needed were in my default "CODE" place, one of them was just in a random directory, so I had to include the whole path.

This certainly isn't perfect, but it's the best thing that I have found to do. I would also be interested in hearing other ideas. Figuring out the best was to "organize" my sage code has been an ongoing issue for me. I have never found a reasonable way to do it for sage notebooks.


text:
# Estimating the number of 'defectives' in a sample.

A container holds 20 bars of chocolate with raisins and nuts.A bar of chocolate is picked at random from the container.The probability of picking a bar chocolate with raisins is $\frac{3}{5}$.How many bars of chocolate with nuts are there in the container..?
A container holds 20 bars of chocolate with raisins and nuts.A bar of chocolate is picked at random from the container.The probability of picking a bar chocolate with raisins is $\frac{3}{5}$.How many bars of chocolate with nuts are there in the container..?


text:
predict.bam() produces different results when data is subset

Apologies in advance since I cannot provide a reproducible example due to the immense size of my model. I'll do my best to describe my situation fully, hopefully this will be sufficient.

My model looks like this:

model <- binary_outcome ~ s(height, side, bottom, top, by=level) + s(side_angle, height_angle, by=level) + fixed_eff + level

I'm trying to use splines to model non-linear relationships in the data, splitting them by level. I am fairly confident that this model structure makes sense. Due to data set size issues, I estimate a different model for each year.

I then fit the model:

fit <- bam(model, data=d, family=binomial(link="probit"), gc.level=1, nthreads=14, control=gam.control(trace=TRUE))

The first thing I've noticed is that the size of fit does not vary in proportion by d. The largest data sets sometimes create the smallest model sizes. Inspecting this further it seems to be that fit$smooths has an object X0 that drives the size of fit. The largest objects have nrow(X0) == nrow(d) whereas the smaller objects have nrow(X0) << nrow(d). This leads me to the biggest issue which is that the smaller fit objects tend to produce much worse predictions (measuring by RMSE) than the larger objects when I do: d$pred <- predict.bam(fit, newdata=d, type="response")

Now I've found a solution which seems to produce predictions in line with what I would expect, but it seems strange that it should work, and I would love to know why it's working. It involves subsetting d by level, applying d$$level <- factor(d$$level) and then running predict.bam() again on the subsetted data:

d_l <- d[d$$level == level,] d_l$$level <- factor(d$$level) d_l$$pred <- predict.bam(fit, newdata=d_l, type="response")

This seems odd since the data is essentially the same, the only difference really is that the level factor now only has one level since the data was subset in line 1.

Happy to elaborate or test out any suggestions/ideas. Thanks!

1 Answer

It appears when I do

d\$pred <- predict.gam(fit, newdata=d, type="response")

The predictions are in line with doing

d_l <- d[d$$level == level,] d_l$$level <- factor(d$$level) d_l$$pred <- predict.bam(fit, newdata=d_l, type="response")

Still unclear why this would be different.


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# Pentagrammic retroprism

Jump to navigation Jump to search
Pentagrammic retroprism
Rank3
TypeUniform
SpaceSpherical
Bowers style acronymStarp
Info
Coxeter diagrams2s10/3o
SymmetryI2(10)×A1+, order 20
ArmyPap
RegimentStarp
Elements
Vertex figureCrossed isosceles trapezoid, edge lengths 1, 1, 1, (5–1)/2
Faces10 triangles, 2 pentagrams
Edges10+10
Vertices10
Measures (edge length 1)
Circumradius${\displaystyle \sqrt{\frac{5-\sqrt5}8} ≈ 0.58779}$
Volume${\displaystyle \frac{5-2\sqrt5}6 ≈ 0.087977}$
Dihedral angles3–3: ${\displaystyle \arccos\left(\frac{\sqrt5}3\right) ≈ 41.81031°}$
5/2–3: ${\displaystyle \arccos\left(\sqrt{\frac{5+2\sqrt5}{15}}\right) ≈ 37.37737°}$
Height${\displaystyle \sqrt{\frac{5-\sqrt5}{10}} ≈ 0.52573}$
Central density3
Euler characteristic2
Related polytopes
DualPentagrammic concave trapezohedron
ConjugatePentagonal antiprism
Convex corePentagonal antitegum
Properties
ConvexNo
OrientableYes
NatureTame

The pentagrammic retroprism, or starp, also called the pentagrammic crossed antiprism, is a prismatic uniform polyhedron. It consists of 10 triangles and 2 pentagrams. Each vertex joins one pentagram and three triangles. As the name suggests, it is a crossed antiprism based on a pentagram, seen as a 5/3-gon rather than 5/2.

Similar to how the pentagonal antiprism can be vertex-inscribed into the regular icosahedron, the pentagrammic retroprism can be vertex-inscribed into a great icosahedron.

## Vertex coordinates

A pentagrammic retroprism of edge length 1 has vertex coordinates given by:

• ${\displaystyle \left(±\frac{\sqrt5-1}2,\,±\frac12,\,0\right),}$
• ${\displaystyle \left(0,\,±\frac{\sqrt5-1}2,\,±\frac12\right),}$
• ${\displaystyle \left(\frac12,\,0,\,\frac{\sqrt5-1}2\right),}$
• ${\displaystyle \left(-\frac12,\,0,\,-\frac{\sqrt5-1}2\right).}$

These coordinates are obtained by removing two opposite vertices from a great icosahedron.

An alternative set of coordinates can be constructed in a similar way to other polygonal antiprisms, giving the vertices as the following points along with their central inversions:

• ${\displaystyle \left(±\frac12,\,-\sqrt{\frac{5-2\sqrt5}{20}},\,\sqrt{\frac{5-\sqrt5}{40}}\right),}$
• ${\displaystyle \left(±\frac{\sqrt5-1}4,\,\sqrt{\frac{5+\sqrt5}{40}},\,\sqrt{\frac{5-\sqrt5}{40}}\right),}$
• ${\displaystyle \left(0,\,-\sqrt{\frac{5-\sqrt5}{10}},\,\sqrt{\frac{5-\sqrt5}{40}}\right).}$

## Related polyhedra

Two non-prismatic uniform polyhedron compounds are composed of pentagrammic retroprisms:

There are also an infinite amount of prismatic uniform compounds that are the crossed antiprisms of compounds of pentagrams.


text:
# Info on the locale of surjections from the Natural Numbers to the Real Numbers

On the nlab page for locales, it states that there is locale for the surjections from the Naturals to the Reals. This locale has no points (i.e. elements), since there are no such surjections, but the locale is still nontrivial.

What are some of the properties of this locale? Where can I learn more about it? The nlab page only mentioned it in passing.

Disclaimer: I'm not completely sure what I've done is the correct way to go about it, since I wasn't able to find whatever the nlab's source was, but the idea seems reasonable and this question deserves an answer. If someone comes by who knows better comes along, please feel free to correct me.

The usual trick when struggling with contradictory properties of functions is to study the poset of partial functions. The problem is that it's not clear what it means for a partial function to be surjective. The key idea is to use locale theory itself.

A function from $\mathbb{N}$ to $\mathbb{R}$ can be thought of as a continuous map of topological spaces, where $\mathbb{N}$ has the discrete topology. Thus a "surjection" from $\mathbb{N}$ to $\mathbb{R}$ is a continuous "surjection". Using locales, these "surjections" correspond to injective maps of frames $\mathcal{O}(\mathbb{R}) \rightarrow \mathcal{P}(\mathbb{N})$.

None of these "injections" exist either. But the point is that we are now able to consider partial functions: it makes a lot more sense to ask for a partial function to be injective than it does to ask for it to be surjective. We are therefore now able to define the locale of surjections.

The objects of the locale of surjections are injective maps of frames from some subframe of $\mathcal{O}(\mathbb{R})$ to $\mathcal{P}(\mathbb{N})$. Nontrivial such maps exist: for example, for any open set $U$ of $\mathbb{R}$ there are many maps of frames with domain $\{\bot < U < \top \}$. All that needs to be done now is to understand the ordering of these elements and confirm that it is a frame.

Edit: There's also almost certainly some order reversal going on here: i.e. the locale of surjections has the opposite order of the order given by the partial functions. The way to see this is to consider the case where the sets are chosen so that there are surjections. In that case a total map of frames (i.e. a maximal element of the poset of partial maps) corresponds to a single given surjection (a minimal element in the poset of sets of surjections).

• "None of these 'injections' exist either." Are you sure? $\mathbb R$ and $\mathbb N$ have the same number of open sets. – PyRulez Nov 27 '17 at 2:43
• The point would be that an injective total map of frames would correspond to a continuous surjection $\mathbb{N}$ to $\mathbb{R}$, exactly the thing that we know can't exist and is causing the confusion. – Chessanator Nov 27 '17 at 10:32

Ingo Blechschmidt gives a presentation of the locale in a comment on this MO question. I reproduce the comment here, almost word by word.

The locale of surjections $$\mathbb N\to\mathbb R$$ is freely generated by opens $$U_{n,x}$$ for $$n\in\mathbb N$$ and $$x\in\mathbb R$$ ($$U_{n,x}$$ is meant to be thought of as the open $$U_{n,x}\subseteq L$$ of surjections $$f:\mathbb N\to\mathbb R$$ with $$f(n)=x$$), subject to the following relations:

• for all $$n\in\mathbb N$$ set $$\bigvee_x U_{n,x}=\top$$ (read as: "$$f(n)$$ has a value"),
• for all $$n\in\mathbb N$$, $$x,y\in\mathbb R$$ set $$U_{n,x}\wedge U_{n,y}=\bot$$ whenever $$x\neq y$$, (read as: "$$f(n)$$ has a single value"),
• for all $$x\in\mathbb R$$ set $$\bigvee_n U_{n,x}=\top$$ for all $$x\in\mathbb R$$, (read as: "$$f$$ is surjective").

This construction can be found in Peter Johnstone's Sketches of an elephant, C1.2, example 1.2.8.


text:
Evaluation Framework for DAVIS Interactive Segmentation

## Project description

# DAVIS Interactive Evaluation Framework

This is a framework to evaluate interactive segmentation models over the [DAVIS 2017](http://davischallenge.org/index.html) dataset. The code aims to provide an easy-to-use interface to test and validate interactive segmentation models.

This is the tool that will be used to evaluate the interactive track on the DAVIS Challenge on Video Object Segmentation 2018. More info about the challenge on the [website](http://davischallenge.org/challenge2018/interactive.html).

You can find an example of how to use the package in the following repository:

* [Scribble-OSVOS](https://github.com/kmaninis/Scribble-OSVOS)

## DAVIS Scribbles

In the DAVIS **Main** Challenge track, the task consists on object segmentation in a *semi-supervised* manner, i.e. the given input is the ground truth mask of the first frame. In the DAVIS **Interactive** Challenge, in contrast, the user input is in form of scribbles, which can be drawn faster by humans and thus is a more realistic type of input.

The interactive annotation and segmentation consist in an iterative loop which is going to be evaluated as follows:

* On the first iteration, a human-annotated scribble will be provided to the segmentation model. All the scribbles are annotated over the DAVIS 2017 dataset and the objects annotated will be the same as the ground truth masks.
 **Note**: the annotated frame can be any of the sequence as the humans where asked to annotate the frames that found most relevant and meaningfull to annotate.
* During the rest of the iterations, once the predicted masks have been submitted, a scribble is simulated by the server. The new annotation will be performed on a single frame and this frame will be chosen as the one on which the current result is the worst.

**Evaluation**: For now, the evaluation metric will be the Jaccard similarity $\mathcal{J}$.

## Citation

tex
@article{Caelles_arXiv_2018,
author = {Sergi Caelles and Alberto Montes and Kevis-Kokitsi Maninis and Yuhua Chen and Luc {Van Gool} and Federico Perazzi and Jordi Pont-Tuset},
title = {The 2018 DAVIS Challenge on Video Object Segmentation},
journal = {arXiv:1803.00557},
year = {2018}
}

latex
@article{Pont-Tuset_arXiv_2017,
author = {Jordi Pont-Tuset and Federico Perazzi and Sergi Caelles and Pablo Arbel\'aez and Alexander Sorkine-Hornung and Luc {Van Gool}},
title = {The 2017 DAVIS Challenge on Video Object Segmentation},
journal = {arXiv:1704.00675},
year = {2017}
}


text:
### A CM construction for curves of genus 2 with p-rank 1

Laura Hitt O'Connor, Gary McGuire, Michael Naehrig, and Marco Streng

##### Abstract

We construct Weil numbers corresponding to genus-2 curves with $p$-rank 1 over the finite field $\F_{p^2}$ of $p^2$ elements. The corresponding curves can be constructed using explicit CM constructions. In one of our algorithms, the group of $\F_{p^2}$-valued points of the Jacobian has prime order, while another allows for a prescribed embedding degree with respect to a subgroup of prescribed order. The curves are defined over $\F_{p^2}$ out of necessity: we show that curves of $p$-rank 1 over $\F_p$ for large $p$ cannot be efficiently constructed using explicit CM constructions.

Available format(s)
Category
Foundations
Publication info
Published elsewhere. Unknown where it was published
Keywords
hyperelliptic curve cryptography
Contact author(s)
gary mcguire @ ucd ie
History
2010-05-11: last of 2 revisions
See all versions
Short URL
https://ia.cr/2008/491

CC BY

BibTeX

@misc{cryptoeprint:2008/491,
author = {Laura Hitt O'Connor and Gary McGuire and Michael Naehrig and Marco Streng},
title = {A CM construction for curves of genus 2 with p-rank 1},
howpublished = {Cryptology ePrint Archive, Paper 2008/491},
year = {2008},
note = {\url{https://eprint.iacr.org/2008/491}},
url = {https://eprint.iacr.org/2008/491}
}

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