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Detecting objects in a video is a compute-intensive task. In this paper we propose CaTDet, a system to speedup object detection by leveraging the temporal correlation in video. CaTDet consists of two DNN models that form a cascaded detector, and an additional tracker to predict regions of interests based on historic detections. We also propose a new metric, mean Delay(mD), which is designed for latency-critical video applications. Experiments on the KITTI dataset show that CaTDet reduces operation count by 5.1-8.7x with the same mean Average Precision(mAP) as the single-model Faster R-CNN detector and incurs additional delay of 0.3 frame. On CityPersons dataset, CaTDet achieves 13.0x reduction in operations with 0.8% mAP loss.

We consider the problem of learning from sparse and underspecified rewards, where an agent receives a complex input, such as a natural language instruction, and needs to generate a complex response, such as an action sequence, while only receiving binary success-failure feedback. Such success-failure rewards are often underspecified: they do not distinguish between purposeful and accidental success. Generalization from underspecified rewards hinges on discounting spurious trajectories that attain accidental success, while learning from sparse feedback requires effective exploration. We address exploration by using a mode covering direction of KL divergence to collect a diverse set of successful trajectories, followed by a mode seeking KL divergence to train a robust policy. We propose Meta Reward Learning (MeRL) to construct an auxiliary reward function that provides more refined feedback for learning. The parameters of the auxiliary reward function are optimized with respect to the validation performance of a trained policy. The MeRL approach outperforms our alternative reward learning technique based on Bayesian Optimization, and achieves the state-of-the-art on weakly-supervised semantic parsing. It improves previous work by 1.2% and 2.4% on WikiTableQuestions and WikiSQL datasets respectively.

We provide a framework to approximate the 2-Wasserstein distance and the optimal transport map, amenable to efficient training as well as statistical and geometric analysis. With the quadratic cost and considering the Kantorovich dual form of the optimal transportation problem, the Brenier theorem states that the optimal potential function is convex and the optimal transport map is the gradient of the optimal potential function. Using this geometric structure, we restrict the optimization problem to different parametrized classes of convex functions and pay special attention to the class of input-convex neural networks. We analyze the statistical generalization and the discriminative power of the resulting approximate metric, and we prove a restricted moment-matching property for the approximate optimal map. Finally, we discuss a numerical algorithm to solve the restricted optimization problem and provide numerical experiments to illustrate and compare the proposed approach with the established regularization-based approaches. We further discuss practical implications of our proposal in a modular and interpretable design for GANs which connects the generator training with discriminator computations to allow for learning an overall composite generator.

A major tenet in theoretical neuroscience is that cognitive and behavioral processes are ultimately implemented in terms of the neural system dynamics. Accordingly, a major aim for the analysis of neurophysiological measurements should lie in the identification of the computational dynamics underlying task processing. Here we advance a state space model (SSM) based on generative piecewise-linear recurrent neural networks (PLRNN) to assess dynamics from neuroimaging data. In contrast to many other nonlinear time series models which have been proposed for reconstructing latent dynamics, our model is easily interpretable in neural terms, amenable to systematic dynamical systems analysis of the resulting set of equations, and can straightforwardly be transformed into an equivalent continuous-time dynamical system. The major contributions of this paper are the introduction of a new observation model suitable for functional magnetic resonance imaging (fMRI) coupled to the latent PLRNN, an efficient stepwise training procedure that forces the latent model to capture the 'true' underlying dynamics rather than just fitting (or predicting) the observations, and of an empirical measure based on the Kullback-Leibler divergence to evaluate from empirical time series how well this goal of approximating the underlying dynamics has been achieved. We validate and illustrate the power of our approach on simulated 'ground-truth' dynamical (benchmark) systems as well as on actual experimental fMRI time series. Given that fMRI is one of the most common techniques for measuring brain activity non-invasively in human subjects, this approach may provide a novel step toward analyzing aberrant (nonlinear) dynamics for clinical assessment or neuroscientific research.

Learning low-dimensional embeddings of knowledge graphs is a powerful approach used to predict unobserved or missing edges between entities. However, an open challenge in this area is developing techniques that can go beyond simple edge prediction and handle more complex logical queries, which might involve multiple unobserved edges, entities, and variables. For instance, given an incomplete biological knowledge graph, we might want to predict "em what drugs are likely to target proteins involved with both diseases X and Y?" -- a query that requires reasoning about all possible proteins that {\em might} interact with diseases X and Y. Here we introduce a framework to efficiently make predictions about conjunctive logical queries -- a flexible but tractable subset of first-order logic -- on incomplete knowledge graphs. In our approach, we embed graph nodes in a low-dimensional space and represent logical operators as learned geometric operations (e.g., translation, rotation) in this embedding space. By performing logical operations within a low-dimensional embedding space, our approach achieves a time complexity that is linear in the number of query variables, compared to the exponential complexity required by a naive enumeration-based approach. We demonstrate the utility of this framework in two application studies on real-world datasets with millions of relations: predicting logical relationships in a network of drug-gene-disease interactions and in a graph-based representation of social interactions derived from a popular web forum.

We introduce the problem of learning mixtures of $k$ subcubes over $\{0,1\}^n$, which contains many classic learning theory problems as a special case (and is itself a special case of others). We give a surprising $n^{O(\log k)}$-time learning algorithm based on higher-order multilinear moments. It is not possible to learn the parameters because the same distribution can be represented by quite different models. Instead, we develop a framework for reasoning about how multilinear moments can pinpoint essential features of the mixture, like the number of components. We also give applications of our algorithm to learning decision trees with stochastic transitions (which also capture interesting scenarios where the transitions are deterministic but there are latent variables). Using our algorithm for learning mixtures of subcubes, we can approximate the Bayes optimal classifier within additive error $\epsilon$ on $k$-leaf decision trees with at most $s$ stochastic transitions on any root-to-leaf path in $n^{O(s + \log k)}\cdot\text{poly}(1/\epsilon)$ time. In this stochastic setting, the classic Occam algorithms for learning decision trees with zero stochastic transitions break down, while the low-degree algorithm of Linial et al. inherently has a quasipolynomial dependence on $1/\epsilon$. In contrast, as we will show, mixtures of $k$ subcubes are uniquely determined by their degree $2 \log k$ moments and hence provide a useful abstraction for simultaneously achieving the polynomial dependence on $1/\epsilon$ of the classic Occam algorithms for decision trees and the flexibility of the low-degree algorithm in being able to accommodate stochastic transitions. Using our multilinear moment techniques, we also give the first improved upper and lower bounds since the work of Feldman et al. for the related but harder problem of learning mixtures of binary product distributions.

In this paper, we study the problem of minimizing a sum of smooth and strongly convex functions split over the nodes of a network in a decentralized fashion. We propose the algorithm $ESDACD$, a decentralized accelerated algorithm that only requires local synchrony. Its rate depends on the condition number $\kappa$ of the local functions as well as the network topology and delays. Under mild assumptions on the topology of the graph, $ESDACD$ takes a time $O((\tau_{\max} + \Delta_{\max})\sqrt{{\kappa}/{\gamma}}\ln(\epsilon^{-1}))$ to reach a precision $\epsilon$ where $\gamma$ is the spectral gap of the graph, $\tau_{\max}$ the maximum communication delay and $\Delta_{\max}$ the maximum computation time. Therefore, it matches the rate of $SSDA$, which is optimal when $\tau_{\max} = \Omega\left(\Delta_{\max}\right)$. Applying $ESDACD$ to quadratic local functions leads to an accelerated randomized gossip algorithm of rate $O( \sqrt{\theta_{\rm gossip}/n})$ where $\theta_{\rm gossip}$ is the rate of the standard randomized gossip. To the best of our knowledge, it is the first asynchronous gossip algorithm with a provably improved rate of convergence of the second moment of the error. We illustrate these results with experiments in idealized settings.

Graph Convolutional Networks (GCNs) and their variants have experienced significant attention and have become the de facto methods for learning graph representations. GCNs derive inspiration primarily from recent deep learning approaches, and as a result, may inherit unnecessary complexity and redundant computation. In this paper, we reduce this excess complexity through successively removing nonlinearities and collapsing weight matrices between consecutive layers. We theoretically analyze the resulting linear model and show that it corresponds to a fixed low-pass filter followed by a linear classifier. Notably, our experimental evaluation demonstrates that these simplifications do not negatively impact accuracy in many downstream applications. Moreover, the resulting model scales to larger datasets, is naturally interpretable, and yields up to two orders of magnitude speedup over FastGCN.

In this paper, we propose a generalization of the Batch Normalization (BN) algorithm, diminishing batch normalization (DBN), where we update the BN parameters in a diminishing moving average way. BN is very effective in accelerating the convergence of a neural network training phase that it has become a common practice. Our proposed DBN algorithm remains the overall structure of the original BN algorithm while introduces a weighted averaging update to some trainable parameters. We provide an analysis of the convergence of the DBN algorithm that converges to a stationary point with respect to trainable parameters. Our analysis can be easily generalized for original BN algorithm by setting some parameters to constant. To the best knowledge of authors, this analysis is the first of its kind for convergence with Batch Normalization introduced. We analyze a two-layer model with arbitrary activation function. The primary challenge of the analysis is the fact that some parameters are updated by gradient while others are not. The convergence analysis applies to any activation function that satisfies our common assumptions. In the numerical experiments, we test the proposed algorithm on complex modern CNN models with stochastic gradients and ReLU activation. We observe that DBN outperforms the original BN algorithm on MNIST, NI and CIFAR-10 datasets with reasonable complex FNN and CNN models.

Mobility in an effective and socially-compliant manner is an essential yet challenging task for robots operating in crowded spaces. Recent works have shown the power of deep reinforcement learning techniques to learn socially cooperative policies. However, their cooperation ability deteriorates as the crowd grows since they typically relax the problem as a one-way Human-Robot interaction problem. In this work, we want to go beyond first-order Human-Robot interaction and more explicitly model Crowd-Robot Interaction (CRI). We propose to (i) rethink pairwise interactions with a self-attention mechanism, and (ii) jointly model Human-Robot as well as Human-Human interactions in the deep reinforcement learning framework. Our model captures the Human-Human interactions occurring in dense crowds that indirectly affects the robot's anticipation capability. Our proposed attentive pooling mechanism learns the collective importance of neighboring humans with respect to their future states. Various experiments demonstrate that our model can anticipate human dynamics and navigate in crowds with time efficiency, outperforming state-of-the-art methods.

Sum-of-norms clustering is a method for assigning $n$ points in $\mathbb{R}^d$ to $K$ clusters, $1\le K\le n$, using convex optimization. Recently, Panahi et al.\ proved that sum-of-norms clustering is guaranteed to recover a mixture of Gaussians under the restriction that the number of samples is not too large. The purpose of this note is to lift this restriction, i.e., show that sum-of-norms clustering with equal weights can recover a mixture of Gaussians even as the number of samples tends to infinity. Our proof relies on an interesting characterization of clusters computed by sum-of-norms clustering that was developed inside a proof of the agglomeration conjecture by Chiquet et al. Because we believe this theorem has independent interest, we restate and reprove the Chiquet et al.\ result herein.

In this paper, we propose a fully supervised speaker diarization approach, named unbounded interleaved-state recurrent neural networks (UIS-RNN). Given extracted speaker-discriminative embeddings (a.k.a. d-vectors) from input utterances, each individual speaker is modeled by a parameter-sharing RNN, while the RNN states for different speakers interleave in the time domain. This RNN is naturally integrated with a distance-dependent Chinese restaurant process (ddCRP) to accommodate an unknown number of speakers. Our system is fully supervised and is able to learn from examples where time-stamped speaker labels are annotated. We achieved a 7.6% diarization error rate on NIST SRE 2000 CALLHOME, which is better than the state-of-the-art method using spectral clustering. Moreover, our method decodes in an online fashion while most state-of-the-art systems rely on offline clustering.

It is common in recommendation systems that users both consume and produce information as they make strategic choices under uncertainty. While a social planner would balance "exploration" and "exploitation" using a multi-armed bandit algorithm, users' incentives may tilt this balance in favor of exploitation. We consider Bayesian Exploration: a simple model in which the recommendation system (the "principal") controls the information flow to the users (the "agents") and strives to incentivize exploration via information asymmetry. A single round of this model is a version of a well-known "Bayesian Persuasion game" from [Kamenica and Gentzkow]. We allow heterogeneous users, relaxing a major assumption from prior work that users have the same preferences from one time step to another. The goal is now to learn the best personalized recommendations. One particular challenge is that it may be impossible to incentivize some of the user types to take some of the actions, no matter what the principal does or how much time she has. We consider several versions of the model, depending on whether and when the user types are reported to the principal, and design a near-optimal "recommendation policy" for each version. We also investigate how the model choice and the diversity of user types impact the set of actions that can possibly be "explored" by each type.

Health data are generally complex in type and small in sample size. Such domain-specific challenges make it difficult to capture information reliably and contribute further to the issue of generalization. To assist the analytics of healthcare datasets, we develop a feature selection method based on the concept of Coverage Adjusted Standardized Mutual Information (CASMI). The main advantages of the proposed method are: 1) it selects features more efficiently with the help of an improved entropy estimator, particularly when the sample size is small, and 2) it automatically learns the number of features to be selected based on the information from sample data. Additionally, the proposed method handles feature redundancy from the perspective of joint-distribution. The proposed method focuses on non-ordinal data, while it works with numerical data with an appropriate binning method. A simulation study comparing the proposed method to six widely cited feature selection methods shows that the proposed method performs better when measured by the Information Recovery Ratio, particularly when the sample size is small.

Adaptive gradient methods like AdaGrad are widely used in optimizing neural networks. Yet, existing convergence guarantees for adaptive gradient methods require either convexity or smoothness, and, in the smooth setting, only guarantee convergence to a stationary point. We propose an adaptive gradient method and show that for two-layer over-parameterized neural networks -- if the width is sufficiently large (polynomially) -- then the proposed method converges \emph{to the global minimum} in polynomial time, and convergence is robust, \emph{ without the need to fine-tune hyper-parameters such as the step-size schedule and with the level of over-parametrization independent of the training error}. Our analysis indicates in particular that over-parametrization is crucial for the harnessing the full potential of adaptive gradient methods in the setting of neural networks.

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text:
# Algebra Examples

,
Distribute the value across the operation.
Evaluate .
Replace the variable with in the expression.
Simplify the result.
Remove parentheses.
The final answer is .
Evaluate .
Replace the variable with in the expression.
Simplify the result.
Multiply by to get .
The final answer is .
Compose the result function for by replacing the function designators with the actual functions.
Rewrite the division as a fraction.

format: plain


text:
Definition:Derivative/Notation/Leibniz Notation

Leibniz's Notation for Derivative

Leibniz's notation for the derivative of a function $y = f \left({x}\right)$ with respect to the independent variable $x$ is:

$\dfrac {\mathrm d y} {\mathrm d x}$

Historical Note

Leibniz's notation for a derivative came at about the same time that the manuscript dated $29$th October $1675$ in which the notation for the integral had been devised.

At the same time he introduced the differential symbol $\d$.

Thus he was soon writing $\d x$, $\d y$, and $\dfrac {\d y} {\d x}$ soon followed.

In his $1684$ article Nova Methodus pro Maximis et Minimis, published in Acta Eruditorum, he casually drops the notation in place with very little explanation.


text:
Volume XVIII , issue 1 ( 2015 ) back
 Von der Klassifikation der Vierecke zum System der Vierecke 1$-$15 Joachim T. Haug and Carolin Haug Ludwig-Maximilians-Universität München, Biozentrum -- Department Biologie II, Groß-haderner Str. 2, 82152 Planegg-Martinsried, Deutschland

Abstract

MSC Subject Classification: G44

MathEduc Subject Classification: 97G40

 On Catalan numbers 16$-$24 Tanja Stojadinović Faculty of Mathematics, University of Belgrade, Belgrade, Serbia

Abstract

In this article we introduce one of the most frequently encountered sequences in mathematics, the sequence of Catalan numbers. The numerous and various occurrences and applications in combinatorial problems as well as their relations with other famous numerical sequences make Catalan numbers very suitable for creative teaching of mathematics at all levels.

Keywords: Catalan numbers; binary trees; associahedron; permutations.

MSC Subject Classification: 97K20

MathEduc Subject Classification: K25

 A simple proof of the change of variable theorem for the Riemann integral 25$-$28 Haryono Tandra

Abstract

In this note, we present a simple proof of the change of variable theorem for the Riemann integral. The proof gives an alternate approach of how a class material on this topic can sistematically be carried out through a simple, succinct, and convenient way.

Keywords: Change of variable theorem; Riemann integral.

MSC Subject Classification: 97I55

MathEduc Subject Classification: I50

 On-line compendium of triangle construction problems with automatically generated solutions 29$-$44 Vesna Marinković Faculty of Mathematics, University of Belgrade, Serbia

Abstract

In this paper an on-line compendium of solutions to a class of triangle construction problems using straightedge and compass is presented. It consists of: informal description of constructions in natural-language form, formal description of constructions, illustrations of generated constructions, as well as step-by-step animations of generated constructions. This collection was generated completely automatically, using the system for automated solving of construction problems ArgoTriCS. To our knowledge this is the first automatically generated compendium of construction problems which offers solutions to almost all solvable problems from one large corpus of problems.

Keywords: Triangle construction; automatically generated solutions.

MSC Subject Classification: 97P50

MathEduc Subject Classification: P55


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# Binomial coefficient equality

QUESTION: \begin{equation*} \dbinom{n+m+1}{n}=\sum_{k=0}^n \dbinom{r+k}{k} \dbinom{m+n-r-k}{n-k} \end{equation*} where $n, m, r \geq 0$

I tried proving using binomial coefficient formula $\big[\dbinom{n}{k}=\frac{n!}{k!(n-k)!}\big]$, but dont think its possible with summation, however if it is would it work? Maybe lattice paths would work better, but my understanding of lattice squares is minimal.

So i attempted the proof using binomial theorems and exponent combination laws and i get stuck

\begin{equation*} \sum_{k=0}^n \dbinom{r+k}{k} \dbinom{m+n-r-k}{n-k}=\dbinom{n+m+1}{n} \end{equation*} working with the right side . \begin{equation*}\dbinom{n+m+1}{n}= \sum_n\dbinom{n+m+1}{n}x^n=(1+x)^{n+m+1}\end{equation*} \begin{equation*}=(1+x)^n(1+x)^m(1+x)\end{equation*} or would the next step be \begin{equation*}(1+x)^n(1+x)^{m+1}\end{equation*}

• I think the LHS should be $\dbinom{n+m}n$, in which case this is just a simple variation of Vandermonde's identity. – Prasun Biswas Nov 3 '17 at 2:00
• @PrasunBiswas no, the LHS is correct: this a "double convolution" (upper and lower terms), in which case the upper term gets a +1. – G Cab Nov 3 '17 at 9:24
• @MahlissaJayde: The right-hand side $(1+x)^{n+m+1}$ of your calculation is not valid, since $n$ is the index of the sum and so $n$ can't occur on the right-hand side. – Markus Scheuer Nov 3 '17 at 22:32
• i just realized that. so this approach will not work then. – Mahlissa LECKY Nov 3 '17 at 22:39

We obtain \begin{align*} \color{blue}{\sum_{k=0}^n}&\color{blue}{\binom{r+k}{k}\binom{m+n-r-k}{n-k}}\\ &=\sum_{k=0}^n\binom{-r-1}{k}(-1)^k\binom{-m+r-1}{n-k}(-1)^{n-k}\tag{1}\\ &=(-1)^n\binom{-m-2}{n}\tag{2}\\ &\color{blue}{=\binom{n+m+1}{n}}\tag{3} \end{align*}

Comment:

• In (1) we apply the binomial identity $\binom{-p}{q}=\binom{p+q-1}{q}(-1)^q$ twice.

This way we get for example $\binom{r+k}{k}=\binom{-(-r-k)}{k}=\binom{(-r-k)+k-1}{k}(-1)^k=\binom{-r-1}{k}(-1)^k$ with $-r-k=p$ and $k=q$.

• In (2) we apply the Chu-Vandermonde identity.

• In (3) we apply again the binomial identity as in (1).

• perfect (as always!) – G Cab Nov 3 '17 at 9:27
• @GCab: You're too friendly, thanks. But sometimes there are nice miniatures, which I don't see straightaway. This post is a nice example where SangchulLee provided a very nice answer to my hint. :-) – Markus Scheuer Nov 3 '17 at 9:32
• @MahlissaJayde: There's always more than one way to do it. Not so clear what you want to know. Some more information to Chu Vandermonde or alternative proofs of your problem? – Markus Scheuer Nov 3 '17 at 22:14
• @MahlissaJayde: I've extended my comment somewhat which might be useful. – Markus Scheuer Nov 3 '17 at 22:38
• i understood that step, thank you its just the elimination of k in step 2 – Mahlissa LECKY Nov 3 '17 at 22:41


text:
This is from Sanjoy Mahajan's The Art of Insight Problem 1.4

As a bank robber sitting in the vault planning your getaway, do you fill your suitcase with gold bars or $100 bills? Assume first that how much you can carry is a fixed weight. Then redo your analysis assuming that how much you can carry is a fixed volume. As I estimated in suitcase of money the mass of a paper note is about 1 gram, and the volume is about 1 cm³. So the mass density of an$100 note is $100 per gram, and the volume density is$100 per cubic centimeter.

As I estimated in Value of Gold the volume density of gold is about $2000 per cubic centimeter. So clearly for a fixed volume you are much better off filling a suitcase with gold bars. However gold is quite dense, with a density of about 20 grams per cubic centimeter (about twice the "typical" amount of 10 grams per cubic centimeter for many metals). So the mass density is about$100 per gram, similar to a money note. I actually overestimated the value density slightly, so in fact for a fixed weight you would be better off with paper money.

Note that a small suitcase of money weighs around 50 kilograms. Gold is about 20 times denser, so would weigh 1 tonne. In practice weight is more likely to be the limiting factor and you're better off with a suitcase full of money.


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```
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text:
+0

# help geometry

0
54
1

A sector of a circle has a central angle of 80 degrees. If the area of the sector is 35pi, what is the radius of the circle?

Dec 8, 2021

#1
+13025
+1

What is the radius of the circle?

Hello Guest!

$$A=\pi r^2\cdot \frac{80°}{360°}\\ 35\pi =\pi r^2\cdot \frac{80°}{360°}\\ r=\sqrt{\frac{35\cdot 360}{80}}$$

$$r=12.55$$

!

Dec 8, 2021


text:
# A confusion about Church's simple theory of types and the Curry-Howard isomorphism

I am confused about Church's simply typed lambda calculus and the Curry-Howard isomorphism.

Church's simply typed lambda calculus in the paper cited above is given a classical proof theory, in that on p. 4 of the PDF above (p. 58 of the original article) we have the classical definitions of disjunction, conjunction and existential quantification.

Later authors (Henkin in Completeness in the Theory of Types and A Theory of Propositional Types) follow suit; section §4 of the latter article building up the connectives via $\top$ and $\bot$ with a classical disjunction (see 4.5 of section §4). Henkin's type theory is classical in its proof rules and its semantics.

However, the simply typed lambda calculus is usually taken to correspond syntactically via the Curry Howard isomorphism to intuitionistic propositional logic.

Given the classical behaviour, proof-theoretically, of Church's original simply typed lambda calculus in the above article, how is this correspondence achieved with intuitionistic logic? Do the simply typed lambda calculi of Henkin and Church correspond to the Intuitionistic propositional calculus? Can we have the simply typed lambda calculus with classical proof rules and a classical semantics corresponding to intuitionistic propositional logic.

What would the law of the excluded middle in the simply typed lambda calculus of the Church or Henkin variety described correspond to on the logic side of the Curry Howard isomorphism?

If someone could clarify where I am going wrong I would be very happy.

• Church's simple type theory, and the Type Theory that arises from the Curry Howard isomorphism are 2 completely different things. It is unfortunate that they share the same name. In Church's simple type theory, all theorems are typed lambda expressions. In C.H. type theory, all proofs are typed lambda expressions. Mar 31, 2018 at 16:52
• Can you give me a reference where this is discussed? As far as I was aware, Church's version of the simply typed lambda calculus does correspond to Intuitionistic propositional logic. What distinction are you drawing between the simply typed lambda calculus and Church's simple type theory? Where can I read about this? Mar 31, 2018 at 17:14
• @user65526 I don't know if I'd say they're 'completely different things' but certainly agree there's an important distinction Daniel is making. The 'intuitionistic propositions' on one end of the CH correspondence correspond to the types in the lambda calculus. The propositions of the embedded classical logic here, on the other hand, are terms of a particular type. What CH says here is that an inhabited type corresponds to a valid proposition. A classic example is the existence of the term Church calls $K_{\alpha\beta\alpha}$ means $A\to (B\to A)$ is an intuitionistically valid prop. Mar 31, 2018 at 18:14
• @user65526 In other words the intuisitionistic implicational logic and the classical logic are "completely different things" in that they play completely different roles here (and of course they also are completely different logics). But the CH correspondence does apply to Church's types here. But to add to what I said above, a proof of one of the valid intuititionistic propositions corresponds to a type inference on a term, not to one of the proofs done in the proof theory described Mar 31, 2018 at 18:23
• I'm not 100% crystal clear I understand. Could you please either give a full answer, or cite some relevant literature where this distinction is made? Mar 31, 2018 at 22:34

In the former view, we think of a type as representing a proposition and that proposition has a proof if and only if there is a lambda term of that type. This is the propositions-as-types/Curry-Howard perspective. We can prove that there is a one-to-one correspondence between the types, terms, and term reductions of the simply typed lambda calculus and the propositions, natural deduction proofs, and proof rewrites of the natural deduction presentation of minimal/intuitionistic propositional logic. $$\beta$$-reduction corresponds to local soundness, and $$\eta$$-expansion to local completeness.
In the latter view, used by e.g. the HOL family of theorem provers and described in The Seven Virtues of Simple Type Theory, the simply typed lambda terms are just terms in an equational logic. In this case, a proof is a proof in this logic which can be classical or intuitionistic or whatever. Typically it is classical and a Boolean type $$\mathbb B$$ is added with axioms to ensure that it has only two values, $$\mathsf{F}$$ and $$\mathsf{T}$$. We can then identify propositions with terms of type $$\mathbb B$$, and more generally, predicates with terms of type $$(\tau_1\to(\tau_2\to\cdots(\tau_n\to\mathbb B)\cdots))$$. Due to the use of higher-order terms (lambda terms), just a handful of rules and axioms for equality and Booleans produces a fairly powerful higher-order logic. See sections 2.3.1 and 2.4.3 of the HOL4 Logic manual for details of the deductive system in the case of HOL4 (which actually uses the polymorphic lambda calculus, not the simply typed lambda calculus as that is much more convenient). So in HOL4, the proofs would be deductions in this deductive system.


text:
# Odd Order Group Element is Square

## Theorem

Let $\struct {G, \circ}$ be a group whose identity is $e$.

Let $x \in G$.

Then:

$\exists y \in G: y^2 = x$
the order $\order x$ is odd

### Corollary

Let $\struct {G, \circ}$ be a finite group.

Then:

$\forall x \in G: \exists y \in G: y^2 = x$

if and only if $\order G$ is odd.

## Proof

Let $\order x$ be odd.

Then:

$\exists n \in \Z_{> 0}: x^{2 n - 1} = e$

from the definition of the order of an element.

Conversely, suppose that

$\exists n \in \Z_{> 0}: x^{2 n - 1} = e$

Then $\order x$ is a divisor of $2 n - 1$ from Element to Power of Multiple of Order is Identity.

Hence $\order x$ is odd.

So $\order x$ is odd if and only if $\exists n \in \Z_{> 0}: x^{2 n - 1} = e$.

Then:

 $\displaystyle \order x$ $=$ $\displaystyle 2 n - 1$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle x^{2 n - 1}$ $=$ $\displaystyle e$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle x^{2 n}$ $=$ $\displaystyle x$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle \paren {x^n}^2$ $=$ $\displaystyle x$

Hence the result.

$\blacksquare$


text:
# Geosynchronous Orbit

## Homework Statement

A satellite is in a circular, geosynchronous orbit (makes one revolution around Earth in 24 hours). Calculate the distance in meters from the satellite to the earth's center. Mass of earth = 5.97 x 1024 kg

## Homework Equations

T=((2*Pi)*r^(3/2))/sqrt(GM)

## The Attempt at a Solution

I thought T equals the period 2Pi. So the the 2Pi's cancel and then you just plug in the variables. Unfortunately I'm not getting the right answer which is supposed to be: 4.2*10^7.

Any help would be greatly appreciated.

## Answers and Replies

gneill
Mentor
Check to see what units the formula will produce.

Would the units for T be radians?

gneill
Mentor
Would the units for T be radians?
You're guessing? Why don't you expand the units from the formula and actually see what it is?


text:
+0

+1
64
1
+457

Perhaps the most common error in ranking poker hands involves confusing the order of a straight and a flush.

A straight is 5 cards with consecutive rank, like 45678. The cards rank A,2,3,4,5,6,7,8,9,10,J,Q,K,A. Note that an ace ("A") can be used as high (as in 10JQKA) or as low (as in A2345), but it can't be used as both high and low in the same hand. (That is, you can't use an ace to loop around. KA234 does not count as a straight.)

A flush is 5 cards of the same suit.

If 5 cards are drawn from the deck at random (without replacement) to form a hand, which hand is more likely: a straight or a flush?

I know that to calculate flush its:

$$4 \cdot {13 \choose 5} =5148$$

But I'm not sure how to calculate the straights.

I know the answer is straight since I've played poker before (with my family), but I don't know how to calculate it..

thanks

:)

May 2, 2020
edited by lokiisnotdead  May 2, 2020

#1
+657
0

Is this the same question?

https://web2.0calc.com/questions/perhaps-the-most-common-error-in-ranking-poker-hands-involves-confusing-the-order-of-a-straight-and-a-flush-a-straight-is-5-cards-with-cons

May 2, 2020


text:
## CryptoDB

### Paper: Perfectly Reliable and Secure Communication Tolerating Static and Mobile Mixed Adversary

Authors: Ashish Choudhary Arpita Patra AshwinKumar B.V Kannan Srinathan C. Pandu Rangan URL: http://eprint.iacr.org/2008/232 Search ePrint Search Google In the problem of perfectly reliable message transmission} (PRMT), a sender {\bf S} and a receiver {\bf R} are connected by $n$ bidirectional synchronous channels. A mixed adversary ${\mathcal A}_{(t_b,t_f,t_p)}$ with {\it infinite computing power} controls $t_b, t_f$ and $t_p$ channels in Byzantine, fail-stop and passive fashion respectively. Inspite of the presence of ${\mathcal A}_{(t_b,t_f,t_p)}$, {\bf S} wants to reliably send a message $m$ to {\bf R}, using some protocol, without sharing any key with {\bf R} beforehand. After interacting in phases\footnote{A phase is a send from {\bf S} to {\bf R} or vice-versa} as per the protocol, {\bf R} should output $m' = m$, without any error. In the problem of {\it perfectly secure message transmission} (PSMT), there is an additional constraint that ${\mathcal A}_{(t_b,t_f,t_p)}$ should not know {\it any} information about $m$ in {\it information theoretic} sense. The adversary can be either static\footnote{A static adversary corrupts the same set of channels in each phase of the protocol. The choice of the channel to corrupt is decided before the beginning of the protocol.} or mobile.\footnote{A mobile adversary can corrupt different set of channels in different phases of the protocol.} The {\it connectivity requirement}, {\it phase complexity} and {\it communication complexity} are three important parameters of any interactive PRMT/PSMT protocol and are well studied in the literature when the channels are controlled by a static/mobile Byzantine adversary. However, when the channels are controlled by mixed adversary ${\mathcal A}_{(t_b,t_f,t_p)}$ , we encounter several surprising consequences. In this paper, we study the problem of PRMT and PSMT tolerating "static/mobile mixed adversary". We prove that even though the connectivity requirement for PRMT is same against both static and mobile mixed adversary, the lower bound on communication complexity for PRMT tolerating mobile mixed adversary is more than its static mixed counterpart. This is interesting because against only "Byzantine adversary", the connectivity requirement and the lower bound on the communication complexity of PRMT protocols are same for both static and mobile case. Thus our result shows that for PRMT, mobile mixed adversary is more powerful than its static counterpart. As our second contribution, we design a four phase communication optimal PSMT protocol tolerating "static mixed adversary". Comparing this with the existing three phase communication optimal PSMT protocol against "static Byzantine adversary", we find that additional one phase is enough to design communication optimal protocol against static mixed adversary. Finally, we show that the connectivity requirement and lower bound on communication complexity of any PSMT protocol is same against both static and mobile mixed adversary, thus proving that mobility of the adversary has no effect in PSMT. To show that our bound is tight, we also present a worst case nine phase communication optimal PSMT protocol tolerating mobile mixed adversary which is first of it's kind. This also shows that the mobility of the adversary does not hinder to design constant phase communication optimal PSMT protocol. In our protocols, we have used new techniques which can be effectively used against both static and mobile mixed adversary and are of independent interest.
##### BibTeX
@misc{eprint-2008-17909,
title={Perfectly Reliable and Secure Communication Tolerating Static and Mobile Mixed Adversary},
booktitle={IACR Eprint archive},
keywords={foundations /},
url={http://eprint.iacr.org/2008/232},
note={An extended abstract of this paper is going to appear in ICITS 2008 arpitapatra_10@yahoo.co.in 14021 received 22 May 2008},
author={Ashish Choudhary and Arpita Patra and AshwinKumar B.V and Kannan Srinathan and C. Pandu Rangan},
year=2008
}


text:
Unlimited Plugins, WordPress themes, videos & courses! Unlimited asset downloads! From \$16.50/m

# Test Code Coverage: From Myth to Reality

Length:MediumLanguages:

There was a time when programmers were paid by the number of lines of code they wrote. They were treated as source code producing machines working in cubicles and in return they considered programming just a job that they do eight hours a day and then forget about it, for the rest of the day.

But times have changed. Most cubicle workplaces disappeared and programmers started loving their craft. With the advent of Agile techniques and the Software Craftsmanship movement, many new tools emerged to help the programmer and the process. TDD is slowly becoming the de facto way of writing code and the secrets of SCRUM or Kanban were revealed even to the programmers in the darkest corners of the cubicle world.

Automated testing and test driven development (TDD) are some of the essential techniques Agile provided to us programmers. And a tool that comes with those methodologies is used to produce test code coverage, which is the topic of this article.

## Definition

"In computer science, code coverage is a measure used to describe the degree to which the source code of a program is tested by a particular test suite." ~ Wikipedia

The definition above, taken from Wikipedia, is one of the simplest ways to describe what code coverage means. Basically, in your project you have a bunch of production code as well as a bunch of test code. The test code exercises the production code and the test coverage tells you how much of your production code was exercised by the tests.

Information can be presented in various ways, from simple percentages to nice graphics or even real-time highlighting in your favorite IDE.

## Let's Check It in Action

We will use PHP as the language to exemplify our code. Additionally, we will need PHPUnit and XDebug to test our code and gather coverage data.

### The Source Code

Here is the source code we will use. You can also find it in the attached archive.

The above code contains a simple function that wraps text to a specified number of characters, per line.

### The Test Code

We wrote this code using Test Driven Development (TDD) and we have 100% code coverage for it. This means that by running our test, we exercise each and every line of the source code.

### Running the Tests in CLI With Text Only Coverage

One way to obtain coverage data is to run our tests in the CLI (command line interface) and analyze the output. For this example, we will assume a UNIX like operating system (Linux, MacOS, FreeBSD, etc). Windows users will need to slightly adapt the paths and executable names, but it should be fairly similar.

Let's open a console and change directories in to your test folder. Then run phpunit with an option to generate coverage data as plain text.

This should work out of the box on most systems if XDebug is installed, however in some cases, you may encounter an error related to time zones.

This can be easily fixed by specifying the suggested setting in your php.ini file. You can find the way to specify your timezone in this list. I am from Romania, so I will use the following setting:

Now, if you run the phpunit command again, you should see no error messages. Instead, the test results will be shown.

And the coverage data will be in the specified text file.

Let's analyze this a little bit.

• Classes: refers to how many classes were tested and how many of them were covered. WordWrap is our only class.
• Methods: same as with classes. We have only our wrap() method, nothing else.
• Lines: same as above, but for lines of code. Here we have a lot of lines because the summary contains all the lines from PHPUnit itself.
• Then we have a section for each class. In our case, that is only WordWrap. Each section has its own methods and line details.

Based on these observations, we can conclude that our code is 100% covered by tests. Exactly as we expected before analyzing the coverage data.

### Generating HTML Coverage Output

By just changing a simple parameter for PHPUnit, we can generate nice HTML output.

If you check your ./coverage directory, you will find a lot of files there. I won't paste the list here because it is quite extensive. Instead, I will show you how it looks in a web browser.

This is the equivalent of the summary section from the text version above. We can zoom in by following the proposed links and see more details.

### Coverage Inside Our IDE

The previous examples were interesting and they are quite useful, if your code is built on some remote server to which you have only SHH or web access to. But wouldn't it be nice to have all this info, live in your IDE?

If you use PHPStorm, everything is within the distance of a single click! Select to run your tests with coverage and all the info will just simply show up, magically.

The coverage information will be present in your IDE, in several ways and in several places:

1. Test coverage percentage will be shown near each directory and file.
2. In the editor, while editing code, on the left of the line numbers, a green or red rectangle will mark each line. Green represents tested lines, red represents untested ones. Lines without actual code (empty lines, only braces or parentheses, class or method declarations) will not have any marks.
3. On the right side there will be file browsers where you can quickly browse and sort files by coverage.
4. In the test output, you will see a line of text announcing to you that code coverage was generated.

## The Myths About Code Coverage

With such a powerful tool in the developer's hands and under the management's nose, it was inevitable for some myths to surface. After programmers refused to be payed by the number of lines of code they write, or managers realized how easy it is to game the system, some of them started paying programmers by the percentage of code coverage. Higher code coverage means the programmer was more careful, right? It's a myth. Code coverage is not a measure of how well you write code.

Sometimes programmers tend to think that code with 100% coverage has no bugs. Another myth. Code coverage merely tells you that you have tested each line of code. It is a measure of the number of lines exercised. It is not a measure of the number of lines correctly implemented. For example, half written algorithms with only half defined tests will still have 100% coverage. This does not mean the algorithm is finished or that it works correctly.

Finally, gaming the system is very easy. Of course, if you use TDD, you are naturally having a high coverage value. On whole projects, 100% is impossible. But on small modules or classes, obtaining 100% coverage is very easy. Take for example our source code and imagine you have no tests at all. What would be the simplest test to exercise all the code?

That's it. Two assertions and full coverage. This is not what we want. This test is so far from descriptive and complete, that it is ridiculous.

## The Reality About Code Coverage

Code coverage is a status indicator, not a unit to measure performance or correctness.

Code coverage is for programmers, not for managers. It is a way to spot problems in our code. A way to find old, untested classes. A way to find paths not exercised by the tests that could lead to problems.

On real projects, code coverage will always be under 100%. Achieving perfect coverage is not possible, or if it is, it's rarely a must. However, to have 98% of coverage you must target 100%. Having anything else as your target is non-sense.

Here is the code coverage on Syneto's StorageOS configuration application.

The total is only about 35%, but the results need interpretation. Most of the modules are in the green, with more than 70% coverage. However there is a single folder, Vmware, which pulls down the average. It is a module with a lot of classes containing only definitions for the communication API. There is no reason to test those classes. They were automatically generated by trusted code. The programmers will know this and they will know how to interpret the results. A manager may insist on testing it because it is a red bar and it looks suspicious for someone not knowing the internal details of the project. Would it make any sense to test it? Not at all! It would be a useless test, that would take up precious tens of seconds of build time without any advantage.

## Final Thoughts

So here is where we are with code coverage: it's a great tool for programmers, a source of information to highlight possible problems, a misunderstood reality for most managers, and another tool to force and measure programmers' activities. As with any other tool, it is one that can be correctly used and misused easily.


text:
# What is the square root of 6 times the square root 2?

May 29, 2015

Rewriting the verbal description in symbols:
What is $\sqrt{6 \cdot \sqrt{2}}$ ?

$\sqrt{6 \sqrt{2}} = \sqrt[4]{72}$

...probably not what you were hoping for but that's pretty much as good as it gets.

You could evaluate this using a calculator (approximately $2.913$).


text:
This is a summary function for strings: regardless of the length of the input, it always returns a single string.

str_flatten(string, collapse = "", last = NULL)

## Arguments

string Character to flatten. String to insert between each piece. Optional string use in place of final separator.

## Value

A character vector of length 1

## Examples

str_flatten(letters)
#> [1] "abcdefghijklmnopqrstuvwxyz"str_flatten(letters, "-")
#> [1] "a-b-c-d-e-f-g-h-i-j-k-l-m-n-o-p-q-r-s-t-u-v-w-x-y-z"
str_flatten(letters[1:5], ", ", ", and ")
#> [1] "a, b, c, d, and e"str_flatten(letters[1:3], ", ", ", and ")
#> [1] "a, b, and c"str_flatten(letters[1:2], ", ", ", and ")
#> [1] "a, and b"str_flatten(letters[1], ", ", ", and ")
#> [1] "a"str_flatten(letters[0], ", ", ", and ")
#> [1] ""


text:
F v j

Parenthetically note that condition (4.88) is not automatically satisfied if iV vanishes at the surface, since this also implies that Eq. (4.87) has a singular point at r = R. In fact, Eq. (4.87) has a first integral that is quite convenient for our purposes. Setting the constant of integration equal to zero, one obtains the nonlinear equation dw

therefore ensuring that boundary condition (4.88) is satisfied provided the product pu vanishes at r = R.

A second relation between u and w can be obtained from Eqs. (4.3)-(4.6) and (4.50)-(4.51). In this case, Eq. (4.58) remains valid but Eqs. (4.59) and (4.60) must be replaced by

and where

1 GM p dw2

Again inserting these relations into Eq. (4.30), we obtain

Assuming electron-scattering opacity, one has

2Lr4

The function h is governed by the following inhomogeneous equation:

(Note that f = 0 in the convective core, where one assumes that Î2 = Œ0.) As explained in Section 4.2.1, when solving this equation one must always ensure the continuity of gravity across the core-envelope interface and across the free surface. One also has d2 f

df dr

Equations (4.89), (4.93), and (4.96) form a coupled system for the functions u, w, and h. Away from the boundaries, turbulent friction acting on the meridional flow is negligible so that one can replace Eq. (4.93) by u = uS + uf (4.98)

in the bulk of the radiative envelope. Near the core boundary, one can solve Eq. (4.93) along the lines presented in Section 4.3.1 (see Figure 4.1). Near the free surface, however, one readily sees from Eqs. (4.94)-(4.97) that the frictionless solution uS + uf behaves as 1/p. Following closely Eqs. (4.74)-(4.80), we shall thus let

S (pu f )r in the surface boundary layer. With this new definition for y, Eq. (4.93) becomes y d~y - x2"-2y = -x"-2. (4.100)

Note that Eq. (4.100) is very similar in structure to Eq. (4.76), with x"-2 merely replacing x2"-2 on the right-hand side. Conditions (4.77)-(4.79) remain unchanged but Eq. (4.80) must be replaced by y ^ -1, as x (4.101)

since the solution of Eq. (4.100) should match the frictionless solution at the bottom of the surface boundary layer. Figure 4.4 illustrates the solution of Eq. (4.100) that satisfies conditions (4.77)-(4.79) and (4.101). A uniformly valid solution of Eqs. (4.89), (4.93), and (4.96) can thus be obtained, all the way from the outer boundary to the core-envelope interface.

The above formulation corresponds to the case for which one has tH ^ tES so that we can let O. = Í2(r). As explained in Section 4.3.1, if one also assumes that tV ^ tES, the function O remains nearly equal to a constant. In that case, correct to order e, Eq. (4.86) can be rewritten in the form

After linearizing Eq. (4.89), we obtain

0.08

0.06

0.04

0.02

Fig. 4.4. Function y(x) in the surface boundary layer. The frictionless solutions, y = 1 /xn, are indicated by dashed curves. Source: Tassoul, J. L., and Tassoul, M., Astrophys. J. Suppl., 49,317, 1982.

0.06

0.04

0.02

Fig. 4.4. Function y(x) in the surface boundary layer. The frictionless solutions, y = 1 /xn, are indicated by dashed curves. Source: Tassoul, J. L., and Tassoul, M., Astrophys. J. Suppl., 49,317, 1982.

This equation must be solved with the condition ¡31 (Rc) = 0 so that £ = £0 at the core boundary r = Rc. (Condition [4.88] is automatically satisfied since one has pu = 0 at r = R.) This is a major simplification because it implies that f = uf = 0; the right-hand side of Eq. (4.96) is thus identically equal to zero. This, in turn, implies that Eq. (4.90) no longer depends on rotation. Hence, the function u can be calculated along the lines presented in Section 4.3.1. Thence, one can solve Eq. (4.103) to obtain the function jii. This is exactly the problem presented in Eq. (4.81), neglecting of course the d dependence of the function w1. Indeed, by making use of Eq. (4.29), one can easily show that the derivative of Eq. (4.103) is strictly equivalent to Eq. (4.82).

4.4 A consistent second-order solution

In Section 4.3.1 we have calculated the meridional velocity u and the angular velocity £ in a slowly rotating star, when the departures from solid-body rotation are uniformly small throughout the whole radiative zone. The circulation pattern consists of a single cell (or gyre) extending from the convective core boundary to the free surface, with interior upwelling at the poles that is compensated by interior downwelling at the equator (see Figure 4.3). Although turbulent friction acting on the circulation is negligible in the bulk of the radiative envelope, there exist thin layers in which turbulent friction prevents the formation of unwanted singularities near the inner and outer boundaries. Such boundary-layer solutions satisfy all the basic equations and all the boundary conditions, with the circulation velocities remaining uniformly small throughout the radiative envelope. Of course, in the boundary layers these velocities depend on the coefficient ¡¡V. Fortunately, because they depend, respectively, on (¡¡V)1/7 and (¡V)1/10 in the core and surface boundary layers, their dependence on this poorly known parameter is considerably reduced (see Eqs. [4.68] and [4.75]).

Now, as was noted in Section 4.2.2, the claim has been made that there always exist two distinct cells separated by the level surface with density p = p* (say) given by

22 = 2n Gp *. Moreover, it has been shown that in a frictionless, differentially rotating star one always has ur a 1/p in the surface layers, thus leading to much larger surface velocities than Sweet's (1950). Both objections require that we retain the second-order terms in Eqs. (4.55) and (4.56). However, because differential rotation plays an essential role in the discussion, we shall also replace Eq. (4.55) by

where w0 is a function of the coordinates and time. Following Section 4.3.1, we shall consider the case for which one has tH ^ tV ^ tES, thus ensuring the convergence of expansion (4.104).

The general strategy is as follows. First, one solves to O(e1/2) the y component of the momentum equation for the large-scale motion. Neglecting the d dependence, we obtain p— = —— [iiyr4 — + kyr 3w0 , (4.105)

dt r4 dr V dr where we have retained the XV effect (see Eq. [3.133]). Thus, unless the parameter XV identically vanishes, the solution of Eq. (4.105) does not correspond to a solid-body rotation. For steady motions, we have

Since pV and XV are poorly known quantities, we shall merely prescribe that

where a is a constant. Second, one calculates the first-order velocity ui, which can be obtained from Eqs. (4.28) and (4.93), replacing w by w0 in definition (4.92). Third, once the problem has been solved to that order, one calculates the back reaction w1 (see Eq. [4.81]). Finally, collecting all the pieces together, one calculates the second-order velocity u2. To this order of approximation, however, one must retain the inertial terms u1 ■ grad u1 in the poloidal part of the momentum equation. Correct to O(e2), one has p = p0(r) + e [p1,0(r) + pu(r) Pz(p)]

+ e 2[p2,0(r) + P2,2(r) P2(p) + P2,4(r) P4P)] (4.108)

and similar expressions for p, T, and V. (Henceforth we shall omit the subscripts "0" from the function p0.) With the help of Eq. (4.4), we can also describe the meridional flow by means of a stream function. Thus, we let

pr2 dp pr2 dr (One also has u6 = — rup/sin d.) To the same order of approximation, one finds that dP2(p)

where we have defined the following functions:

It is to be noted that, to O(e2), the streamlines ^ = constant do depend on e, whereas they are independent of this small parameter in the first-order approximation. Correct to O(e2), the angular velocity can be brought to the form

where the y s are governed by a set of inhomogeneous equations.

In Table 4.2 we list the first-order functions u and r v (in cm s-1) for three values of a, in a Cowling point-source model with electron-scattering opacity (M = 3M0, R = 1.75R0, L = 93L0, and N = 6 in the boundary layers). Evidently, the case a = 0 corresponds to Sweet's problem, with uf = 0 and u = uS in the frictionless interior (see Eqs. [4.94] and [4.95]). In contrast, any model for which a = 0 has u = uS + uf in the frictionless interior, since uf = 0 when dw0/dr = 0. One can show that uf > 0 when a > 0 (i.e., when dw0/dr < 0); similarly, one finds that uf < 0 when a < 0 (i.e., when dw0/dr > 0). Since uS > 0, it follows at once that the sum uS + uf is always positive when a > 0 but may change its sign along the radius when a < 0. Therefore, to first order in e, the meridional flow consists of a single cell when a > 0, whereas it may consist of two cells when a < 0. This property is immediately apparent from the fourth column in Table 4.2.

From the solutions presented in Table 4.2, one readily sees that there is a definite intensification of the function u near the surface of models for which a = 0. Obviously, such an intensification does not occur in the limiting case a = 0. Close scrutiny of the second-order corrections indicates that there always exists a surface intensification of the radial component ur = eu\r + e2u2r, no matter whether one has w\ = 0 or w\ = 0. To be specific, in a frictionless model having w0 = 1, one has u1r a 1/p and u2r a 1/p in the surface layers. In contrast, letting w0 = 1 in a frictionless model, one finds that u1r a 1 and u2r a 1/p in these layers. Strictly speaking, then, the case w0 = 1 is mathematically singular since e2|u2r | may become larger than e |u1r | in the surface layers.* Accordingly, a consistent expansion method requires a small amount of differential rotation to O(e1/2), so that one has uir a 1/p in the frictionless solution near the surface. Of course, when turbulent friction is properly taken into account to all orders in the small parameter e, there are no singularities in the components of the

* The case w0 = 1 is the only one for which the function u 1r has no 1 /p singularity in the surface layers. This can happen only if there exists a centrifugal potential that is proportional to r 2[1 — P2(cos 0 )],that is to say, in the case of strict uniform rotation to O(el/2). Note that such a mathematical complication does not occur when the thermally driven currents are caused by disturbing forces other than the centrifugal force of rotation.

circulation velocity u. Figure 4.4 clearly illustrates how the frictional force acts to prevent the appearance of inordinately large radial velocities near the outer boundary. Because of the 1/p term in the frictionless solution that remains valid in the deep interior only, the function ur at first increases toward the surface and then drops rapidly to zero at the free boundary. To be specific, there is an intensification of the radial velocities below the surface, typically by two or three orders of magnitude (see Table 4.2). However, given the extreme smallness of the meridional currents in the bulk of a stellar radiative envelope, the maximum radial speed below the free surface does not exceed 1 cm s-1, which is a far remove from the various evaluations that can be found in the literature.

Figures 4.5 and 4.6 illustrate two second-order solutions for the meridional flow, respectively for a = + 10—3 and a = —10—3 (N = 6, ¡H/xV = 102, and e = 10—4). These curves are quite independent of the parameter e in the deep interior, where the second-order terms make a negligible contribution to the first-order solution. In contrast, it is immediately apparent that, even for a rather low value of e, the second-order terms make a sizeable contribution in the surface layers, where two or even three cells may occur. Note especially the cell in the equatorial belt, when the basic angular velocity decreases with depth (a < 0). Obviously, there is a definite interplay between the meridional flow and the spatial variations of the angular velocity in the surface layers of a stellar radiative envelope. This is quite unfortunate because the actual run of the angular velocity depends on the eddy viscosities, which are poorly known parameters.

4.4.1 Answer to the classical objections

Consider again a uniformly rotating, nonmagnetic barotrope. Neglecting viscosity and the inertial terms u ■ grad u in the momentum equation, one readily sees that the velocity u is present only in the equations expressing conservation of mass and energy (Eqs. [4.4] and [4.5]). In the case of a nonspherical star, then, Eqs. (4.3), (4.6), and (3.30) provide four scalar relations among the four functions p, p, T, and V. Indeed, letting p = p0 + epi, etc., and linearizing these equations, one can calculate unequivocally the four nonspherical corrections (i.e., p1, p1, T1, and V1) to a given spherical model. These four corrections are independent of the velocity u. Hence, the potential O is also completely determined, and it does not depend on the velocity u either. Because Eq. (4.47) is derived from Eq. (4.5), which is independent of the remaining equations, there is thus no reason to believe that the constraint (4.47) will be satisfied by the functions (g) and (g—1> that one has derived from the known potential O. Prima facie, this raises serious questions about the validity of Eq. (4.48).

As was pointed out in Section 4.2.2, the Gratton-Mestel proof of the double-cell pattern rests on the fact that their frictionless, nonmagnetic body remains strictly barotropic in spite of the inexorable meridional flow. We may therefore ask the following question: Is it actually possible to obtain such a flow in a uniformly rotating body that has all the properties of a barotrope? Specifically, given the approximations made, Eq. (4.49) implies that one has u ■ grad (r2 sin2 d) = 0, that is, the streamlines of the meridional flow must coincide with the straight lines r sin d = constant (i.e., lines parallel to the rotation axis). This is an impossible requirement, since Eq. (4.38) implies that the streamlines must be closed curves. Moreover, because the meridional velocities in a frictionless system have unwanted singularities at the upper and lower boundaries, there is no reason

Equatorial Plane

Fig. 4.5. Second-order solution for the meridional flow in a Cowling point-source model, with electron-scattering opacity, M = 3M0, N = 6, /^v = 102, e = 10-4, and a = + 10-3. In the inner cell, interior upwelling along the rotation axis is compensated by interior downwelling in the equatorial belt. The sense of circulation is reversed in the outer cell that is adjacent to the rotation axis. Note that there are two cells in the outer layers: One of them is adjacent to the rotation axis, and the other is located in the equatorial belt. Source: Tassoul, M., and Tassoul, J. L., Astrophys. J., 440, 789, 1995.

Equatorial Plane

Fig. 4.6. Same as Figure 4.5, but for a = -10 3. Source: Tassoul, M., and Tassoul, J. L., Astrophys. J.., 440, 789, 1995.

to believe that one can apply the condition n ■ u dS = 0 (4.115)

on each level surface S, since this integral relation implicitly assumes that the velocity u is everywhere finite. We therefore conclude that the Gratton-Mestel result is the consequence of an excessively large number of conflicting assumptions that cannot be met in a realistic model.

Making use of Eq. (4.109), one also has, correct to O(e2),

Ur = eu(r)^(f) + e2 [uo(r) + U2(r)P2P + u4(r)P4(f)1, (4.116)

where u0 = — (1/5)(p12/p)u. A mere comparison with Eq. (4.44) shows that Opik's formula does not provide a reliable solution for the meridional flow in arotating barotrope. It must therefore be disregarded.

Yet, because one has uf a 1/p as r ^ R in Eq. (4.95), it is immediately apparent from Eq. (4.98) that a small amount of differential rotation can have a large effect in the surface layers. Does it imply that meridional velocities of the order of kilometers per second are the rule in the outer layers of an early-type star? The answer to this question is flatly no, because any formula that has a 1 /p singularity cannot possibly satisfy all the basic equations and all the boundary conditions. As a matter of fact, we have shown in this chapter that turbulent friction acting on the meridional flow always prevents huge surface velocities, having |ur | < 1 cm s—1 and \ue | < 102 cm s—1 in the surface layers of a 3M0 star in almost uniform rotation. Obviously, these speeds are much slower than those predicted on the basis of the formulae ur a e/p or ur a e2/p, which are completely inadequate in the outermost surface layers of a rotating star.

4.5 Meridional circulation in a cooling white dwarf

Consider a single, nonmagnetic white dwarf that produces its luminosity by cooling of its almost isothermal, degenerate interior. Following closely the analysis given in Sections 4.2.1 and 4.3.1, we shall consider a configuration in slow, almost uniform rotation. Hence, we shall expand about hydrostatic equilibrium in powers of the ratio of centrifugal force to gravity at the equator (see Eq. [4.9]). In spherical polar coordinates (r, l = cos d, the meridional velocity u is

u = e u(r) P2(ß)1r + e v(r)(1 - ß2)—^ 1ß, (4.117)

where, by virtue of Eq. (4.4), v is related to u by the relation

6 pr2 dr

The meridional flow is thus characterized entirely by the radial function u. (Recall that u0 = — ru^f sin d.) We also have

ß ( ) dP1(ß) , ß ( ) dP3(ß) ß1(r )—;-+ ß3(r )


text:
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