text:
I recently helped a colleague troubleshoot a problem where the same message was being consumed repeatedly when using Pulsar.

## Troubleshoot

I was skeptical when he told me about this phenomenon, based on my previous experience with Pulsar, which is explained in the official documentation and API.

Only when the ackTimeout for consumption is set and the consumption timeout is timed out will the message be recast, which is off by default and really not on by looking at the code.

Could that be a call to the negativeAcknowledge() method (which also triggers redelivery)? Because we use a third-party library https://github.com/majusko/pulsar-java-spring-boot-starter that only calls this method when an exception is thrown.

After checking the code there is no exception thrown anywhere, not even during the whole process; that’s a bit weird.

## Replication

In order to clarify the whole story, a detailed understanding of his use of the process.

In fact, there is a bug in the business, and he debugged and then debugged in a single step when the message was consumed, and when he finished debugging once, he received a generic message right away.

But the strange thing is that not every debug can be repeated after consumption, and we all say that if a bug can be 100% completely reproduced, it is basically a big part of the solution.

So the first step of our troubleshooting is to reproduce the problem completely.

Since the problem is generated when debug, it actually translates to the code which is sleep, so we plan to sleep directly in the consumption logic for a while to see if we can reproduce it.

After testing, sleep for a few seconds to a few tens of seconds could not be reproduced, and finally just sleep for a minute, something magical happened, every time successfully reproduced!

Since I can reproduce it successfully, it’s good to know that my own business code also uses Pulsar, so I’m going to reproduce it again in my own project to facilitate debugging.

The weird thing happened again, I couldn’t reproduce it again.

Although this is what is expected, it’s impossible to adjust it.

In the spirit of believing in modern science, the only difference between the two of us is that the project is different, for which I compared the code of both sides.

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18  @PulsarConsumer( topic = xx, clazz = Xx.class, subscriptionType = SubscriptionType.Shared ) public void consume(Data msg) { log.info("consume msg:{}", msg.getOrderId()); Lock lock = redisLockRegistry.obtain(msg.getOrderId()); if (lock.tryLock()) { try { orderService.do(msg.getOrderId()); } catch (Exception e) { log.error("consumer msg:{} err:", msg.toString(), e); } finally { lock.unlock(); } } }

As expected, my colleague’s code added a lock; it was a Redis-based distributed lock, so I slapped my head in the air and threw an exception due to a timeout while unlocking.

To verify this, I set a breakpoint at the Pulsar consumption on the basis that I could reproduce it.

Sure enough, the exception hint is already very clear: the locking has passed the timeout.

The negative message was directly entered after the exception, and the exception was eaten at the same time, so it was not found before.

After checking the source code of RedisLockRegistry, the default timeout is exactly one minute, so we were unable to reproduce the problem even after sleep for a few tens of seconds.

## Summary

Afterwards, I asked my colleague why we needed to add a lock here, because I didn’t see the need to add a lock at all; it turned out that he added it because he copied the code from someone else and didn’t think about it that much.

So there are some lessons to be learned from this.

• ctrl C/V is convenient, but you have to consider your business scenario.
• When using some third-party APIs, you need to fully understand their role, parameters.


text:
# Permutation Test Python

Here we use a permutation test to test the null hypothesis that two different groups come from the same distribution. Permutations, when considered as arrangements, are sometimes referred to as linearly ordered arrangements. Global options for elements of the permutation class can be set through the Permutations. You can specify where to start the slicing, and where to end. The second phase, either bubbles or oil droplets, can be introduced into the test section through a capillary island. The calculator is free and easy to use. your permutation as a parameter - as it seems to be the case in your poker example - simply use nested for-loops, 5 in this case. Multivariate distance matrix regression (MDMR) tests the significance of associations of response profile (dis)similarities and a set of predictors using permutation tests. java that take an integer command-line argument n and prints all n! permutations of the n letters starting at a (assume that n is no greater than 26). We define to be a permutation of the first natural numbers in the range. For sample the default for size is the number of items inferred from the first argument, so that sample(x) generates a random permutation of the elements of x (or 1:x). They are extracted from open source Python projects. Though note that this is not a good option unless the list is guaranteed to be very small. The permutation importance strategy does not require retraining the model after permuting each column; we just have to re-run the perturbed test samples through the already-trained model. Using a Python recipe? Installing ActivePython is the easiest way to run your project. raw download clone embed report print Python 0. Resampling and Monte Carlo Simulations¶ Broadly, any simulation that relies on random sampling to obtain results fall into the category of Monte Carlo methods. GeoGebra Math Apps Get our free online math tools for graphing, geometry, 3D, and more!. Followed by value f0r each test cases. I am feeding control points to a curve function, and want all permutations that result in unique curves. The methods of calculation for the perm order, parity, ranking and unranking are given. Permutations are printed in a lexicographic sorted order. If x is an integer, randomly permute np. is considered to be an absolute permutation if holds true for every. Permutations. So, for example, if the current iteration is on 'b', we’d want to find all the permutations of the string 'ac'. The Chi-Squared Test for Independence - Calculation with Numpy¶ In order to determine whether we accept or reject the null hypothesis. This sentence was stored by Python as a string. Though note that this is not a good option unless the list is guaranteed to be very small. In this post, we will take a look at the later. Permutations are emitted in lexicographic sort order. These are frequently contributed by fans, advocates, or the maintainers of frameworks. factorial function uses integers. Are there permutations of the population list that shuffle will never generate because the period of the random number generator is too small? It’s easy to check. This article provides a good general overview of permutation feature importance, its theoretical basis, and its applications in machine learning: Permutation feature importance. This post contains recipes for feature selection methods. permutation¶ numpy. One of the problems we have right now is that we're training on, for example, 90% of the data. Solving fifteen-puzzles is much more difficult: the puzzle in Figure 8 has a solution of 50 moves and required that 84702 vertices (different permutations of the puzzle) be visited and the maximum heap size was 72340. The lmPerm package provides permutation tests for linear models and is particularly easy to impliment. The inverse of a permutation is defined as the permutation that reverses this operation, i. the Permutation() method and the Permutation class. Update: For a more recent tutorial on feature selection in Python see the post: Feature Selection For Machine Learning in Python. A permutation is an arrangement or ordering of a number of distinct objects. A permutation is specified as each of several possible ways in which a set or number of things can be ordered or arranged. 13) CHEESE. org Read the doc Install it Contribute Get support. If no absolute permutation exists, print -1. Convert Decimal to Octal in Python. Parameter 'flags=re. It is an un-ordered collection of unique sizes. The Chi-Squared Test for Independence - Calculation with Numpy¶ In order to determine whether we accept or reject the null hypothesis. These are Python's bitwise operators. Python tries to re-use objects in memory that have the same value, which also makes comparing objects very fast in. Permutation First import itertools package to implement permutations method in python. DataCamp / 07-statistical-thinking-in-python-(part-2) / 3-introduction-to-hypothesis-testing / permutation-test-on-frog-data. If no such permutation exists, output a single integer -1 instead. For each test case, if a permutation satisfying the condition exists, output a single line containing N space-separated integers which denotes any such permutation. Like bootstrapping, a permutation test builds - rather than assumes - sampling distribution (called the “permutation distribution”) by resampling the observed data. y values are permuted among samples with the same group identifier. Thus, I can use a single set of labels for the permutation test, carrying out the relabelings and classifications in each person individually (since it's within-subjects cross-validation), but with the null distribution for each person built from the same relabelings. We plan to continue to provide bug-fix releases for 3. the Permutation() method and the Permutation class. Permutation test on frog data The average strike force of Frog A was 0. See Table 2 for the explicit list! Note that this is technically NOT considered a factorial (since we don't go all the way down to 1),. Python, 302 287 bytes. This Python Cheat Sheet will guide you to interactive plotting and statistical charts with Bokeh. 18 - a Python package on PyPI - Libraries. Resampling and Monte Carlo Simulations¶ Broadly, any simulation that relies on random sampling to obtain results fall into the category of Monte Carlo methods. I wish to determine a signficance threshold using permutation testing in order to correct for multiple testing. 7 is now released and is the latest feature release of Python 3. org interactive Python tutorial. Analysis of similarities (ANOSIM) provides a way to test statistically whether there is a significant difference between two or more groups of sampling units. Little point in permutation test for the mean: same result as t-test Permutation test is useful when we do not know how to compute the distribution of a test statistic. Join 575,000 other learners and get started learning Python for data science today! Welcome. A single equal sign (=) represents assignment for the Python. They are extracted from open source Python projects. By using exactly the same reasoning as before, there are 5 × 4 × 3 = 60 permutations. We define to be a permutation of the first natural numbers in the range. How many different cookie combinations can the baker create if each cookie has one type of frosting and one type of sprinkle?. So Python 3. This way the numbers 0 through n!-1 correspond to all possible permutations in lexicographic order. Python's itertools library is a gem - you can compose elegant solutions for a variety of problems with the functions it provides. When not specified, y values are permuted among all samples. The test is called Fisher's permutation test. Each test case will contain an integer N (1<= N <=10^9). My problem is to combine them. Implement the next permutation, which rearranges numbers into the numerically next greater permutation of numbers. The only line of each test case consists of two space-separated integers N, K respectively. Wrapping Up. two-three unit tests to test it. In this machine learning tutorial, we're going to cover shuffling our data for learning. Here computation is performed on MNE sample dataset between 40 and 60 ms. 3 permutations of previous ? 9. # # This recursive algorithm is usually referred to as Heap's # permutation, in honor of B. pl Abstract Python implementation of permutations is presented. If no absolute permutation exists, print -1. We have to compute p-value similar to the welch's t-test and ANOVA. The Python documentation has a lot of great examples that you can study to give you ideas of what can be done with this valuable library. You can use them to create your own iterators all by themselves or in combination with each other. Also generate the sign of the permutation which is +1 when the permutation is generated from an even number of swaps from the initial state, and -1 for odd. To perform tests for all experiments, where condition_1 is the reference and condition_2 is the test data, run follwoing command: $permtest my_data. The article in Wikipedia describes the algorithm invented by Narayana Pandit to changes the list in-place to generate the next permutation given a list or Array A. Hi all, I am performing permutation test for 500000 times. The conditional permutation test. I wish to determine a signficance threshold using permutation testing in order to correct for multiple testing. Write a Python program to print all permutations of a given string (including duplicates). A permutation is the arrangement of a set of items in different order. The second phase, either bubbles or oil droplets, can be introduced into the test section through a capillary island. By now, you probably know how to multiply and divide numbers in Python. Getting all the Permutations of String in Python Program: For permutations, we can use backtracking technique. (13 replies) Was able to find nice stuff on the python-list archives as to list permutations, list reversal, and removing duplicates from lists. Transposition Permutations. In our write-up on Python Iterables, we took a brief introduction on the Python itertools module. This free calculator can compute the number of possible permutations and combinations when selecting r elements from a set of n elements. Let's get started. So Python 3. This sentence was stored by Python as a string. Why Aptitude Permutation and Combination? In this section you can learn and practice Aptitude Questions based on "Permutation and Combination" and improve your skills in order to face the interview, competitive examination and various entrance test (CAT, GATE, GRE, MAT, Bank Exam, Railway Exam etc. For why permutation test can deal with dependency, I think that I need to study more about it. Vistocco DESPOTA 9/14 9/ 14 The algorithm retraces down-ward the tree, starting from the root of the dendrogram where all objects are classiﬁed in a unique cluster. and the permutation count is still high. Bonjour, Je dois trouver comment écrire la fonction qui à partir d'une permutation renvoie toutes les permutations possibles (au nombre de n!, où n est le nombre d'élément de la permutation). Members of the treatment group were given the treatment, and members of the control group were given a placebo. The general formula is which means "Find all the ways to pick k people from n, and divide by the k! variants". 7 and Python 3, the test package can be run as a script. Generate permutations of n elements (including repeating/duplicate elements) ab = aa, bb, ab, ba abc = aaa, aab, abc The number of combination should be n^k (n=number of elements, k=combination length). It offers off-the-shelf functions to implement many algorithms like linear regression, classifiers, SVMs, k-means, Neural Networks, etc. Like bootstrapping, a permutation test builds - rather than assumes - sampling distribution (called the "permutation distribution") by resampling the observed data. Number of ways of selecting 2 vowels from 4. Demonstrated proficiency in one or more of the following languages: Java, Python, SQL or JavaScript. The good news is that when n is small and the number of words is large, this approach runs very fast (because set membership testing is O(1)):. Permutation (nPr) and Combination (nCr) calculator uses total number of objects n and sample size r, r\leq n, and calculates permutations or combinations of a number of objects r, are taken from a given set n. IGNORECASE' can be used with all functions. permutations ( docs ). 3 permutations of previous ? 9. Permutation is an arrangement of objects in a specific order. It is to use itertools. Analysis of similarities (ANOSIM) provides a way to test statistically whether there is a significant difference between two or more groups of sampling units. If no absolute permutation exists, print -1. While it may not appear to be a permutation, python already ~$ cat test_assert_permutation. Given and , print the lexicographically smallest absolute permutation. hi, The above code will print all the permutations of the list [1,2,3,4]. In the context of non randomized studies, especially in rare diseases, where only selected patients undergo experimental therapies, matching is an approach to identify a proper set of controls for an unbiased comparison. The code I used was from itertools import permutations and then permutations([1,2,3]). is considered to be an absolute permutation if holds true for every. The function perm. Thus, I can use a single set of labels for the permutation test, carrying out the relabelings and classifications in each person individually (since it's within-subjects cross-validation), but with the null distribution for each person built from the same relabelings. The number of permutations on a set of n elements is given by n!. x though the end of 2018 and security fixes through 2021. The only line of each test case consists of two space-separated integers N, K respectively. Number of. A set can be of numbers or characters. The permutation 1, 4, 3, 2 for example is ambiguous, because its inverse permutation is the same. This feature is not available right now. 0 with limited support for ES6. Number of ways of selecting 3 consonants from 7. This post contains recipes for feature selection methods. 18 - a Python package on PyPI - Libraries. Now he wants to use the Bucket Sort algorithm to sort this huge set of permutations. It’s all free for learners and teachers. Join Richard Wells for an in-depth discussion in this video, Welcome, part of Unit Testing and Test Driven Development in Python. In this article, you are going to learn how to find permutation and combination using Python programming language. An ordered arrangement of sample data or sample points is called as a permutation. applying a permutation and then its inverse (or vice versa) does not modify the array. Codewars is where developers achieve code mastery through challenge. , then the total number of different permutations of N objects is $\frac{ N!. Python dictionary provides a member function update() i. We solve this problem in Python and analyze the time and space complexity of our approach. Next line contains $$N$$ space seperated integers representing the permutation $$P$$. So, for example, if the current iteration is on 'b', we’d want to find all the permutations of the string 'ac'. If you have made syntax mistakes, It will complain and don't give you the cookie ;). You can vote up the examples you like or vote down the ones you don't like. Train on kata in the dojo and reach your highest potential. For example, suppose we have a set of three letters: A, B, and C. applying a permutation and then its inverse (or vice versa) does not modify the array. The function perm. The notation and examples shown here are borrowed from Efron and Tibshirani's An Introduction to the Bootstrap [1]. Permutation test. To generate a set of feature scores requires that you have an already trained model, as well as a test dataset. negative side ? 7. Since, its first time for me to perform permutation test on PPI, Can someone guide me to a tool or a tutorial to perform permutation test on newly created PPI network. Permutations and combinations Before we discuss permutations we are going to have a look at what the words combination means and permutation. factorial function uses integers. Is it that a permutation won't allow for something like "000" or "111" or "333", etc all the way through to "999"?. Join 575,000 other learners and get started learning Python for data science today! Welcome. py -p [p-value-procedure] [item-file] [value-file] [significance-level] [k] > [output-file] The result is saved to the ‘output-file’. Basic String Operations. Each test case will contain an integer N (1<= N <=10^9). After completing Statistical Thinking in Python (Part 1), you have the probabilistic mindset and foundational hacker stats skills to dive into data sets and extract useful information from them. Tutorial: K Nearest Neighbors in Python In this post, we’ll be using the K-nearest neighbors algorithm to predict how many points NBA players scored in the 2013-2014 season. In this case, we just reverse the list to get the next permutation. This should be written in Python. Permutation test on frog data The average strike force of Frog A was 0. Naturally, I ended up getting more interested in the algorithm than in the original puzzle itself. PYTHON CODE. Test with permutations the significance of a classification score¶ In order to test if a classification score is significative a technique in repeating the classification procedure after randomizing, permuting, the labels. The permutation test will test the null hypothesis that the regression coefficient is 0 by assuming there is no relationship between X and Y, so pairs are as if at random. You can find the Combination and Permutation Calculator in Stat Trek's main menu under the Stat Tools tab. How to generate all Permutations - posted in C/C++ Tutorials: Problem: Given a string, print it’s all permutationsThis is a situation that might arise in a programmer’s every day project activities. One tests if the signal significantly deviates from 0 during a fixed time window of interest. Preamble: Twos-Complement Numbers. The following are code examples for showing how to use scipy. Thus, I can use a single set of labels for the permutation test, carrying out the relabelings and classifications in each person individually (since it's within-subjects cross-validation), but with the null distribution for each person built from the same relabelings. The test is called Fisher's permutation test. They can be defined as anything between quotes: astring = "Hello world!" astring2 = 'Hello world!' As you can see, the first thing you learned was printing a simple sentence. P = perms(v) returns a matrix containing all permutations of the elements of vector v in reverse lexicographic order. When data are tied and sample size is at most 100 permutation p-value is returned. The only difference is a small change in the latency window (cfg. Permutation is an arrangement of objects in a specific order. Permutations can denote in this weaker meaning a sequence of elements, where each element occurs just once, but without the requirement to contain all the elements of a given set. all sides = 0 ? 12. Like bootstrapping, a permutation test builds - rather than assumes - sampling distribution (called the "permutation distribution") by resampling the observed data. Below example will print only unique values. Python Statistics & Probability Tutorial (article) - DataCamp. For example, there are 2! = 2*1 = 2 permutations of {1, 2}, namely {1, 2} and {2, 1}, and 3!. In Python, strings are sequences of characters, which are effectively stored in memory as an object. This Python implementation takes two distance matrices (either redundant matrices or condensed vectors) and returns: the veridical correlation, the empirical p-value, and a standard score (z-score). Any suggestions? Thanks, D. yml file looks like for this project:. The basic idea of permutation importance is to permute the values of each feature and measure how much that permutation negatively impacts the scoring metric (which in our case is the Spearman's rank correlation). So, for example, if the current iteration is on 'b', we'd want to find all the permutations of the string 'ac'. Demonstrated proficiency in one or more of the following languages: Java, Python, SQL or JavaScript. In a permutation the order of occurence of the objects or the arrangement is important but in combination the order of occurence of the objects is not important. It’s all free for learners and teachers. Discover how to prepare data with pandas, fit and evaluate models with scikit-learn, and more in my new book, with 16 step-by-step tutorials, 3 projects, and full python code. The good news is that when n is small and the number of words is large, this approach runs very fast (because set membership testing is O(1)):. $$X^2 = \frac{(observed - expected)^2} {(expected)}$$. In the context of non randomized studies, especially in rare diseases, where only selected patients undergo experimental therapies, matching is an approach to identify a proper set of controls for an unbiased comparison. A generalization of the best bit counting method to integers of bit-widths upto 128 (parameterized by type T) is this: Charlie Gordon suggested a way to shave off one operation from the purely parallel version on December 14, 2005, and Don Clugston trimmed three more from it on December 30, 2005. Test exhaustiveness and test robustness is possible through property-based testing and Hypothesis, a Python testing library can be used for effective property tests. Test and deploy a Python application with GitLab CI/CD. Now he wants to use the Bucket Sort algorithm to sort this huge set of permutations. Number of ways of selecting 3 consonants from 7. one side larger than sum of others ? 10. Permutation definition is - often major or fundamental change (as in character or condition) based primarily on rearrangement of existent elements; also : a form or variety resulting from such change. Let’s see the demo. Why Aptitude Permutation and Combination? In this section you can learn and practice Aptitude Questions based on "Permutation and Combination" and improve your skills in order to face the interview, competitive examination and various entrance test (CAT, GATE, GRE, MAT, Bank Exam, Railway Exam etc. py -p [p-value-procedure] [item-file] [value-file] [significance-level] [k] > [output-file] The result is saved to the 'output-file'. Update pots (including getting rid of SVN) 2018-09-16 21:52 Regina Obe * [r16815] Prepping for 2. Using python itertools permutations example ( remove duplicates values) We can use python set for removing duplicate values from the output of itertools permutations function. Recall that the p -value is defined as the probability of getting data as extreme as the observed data when the null hypothesis is true. One interesting application is the rearrangement of characters in a word to create other words. Remember the Zen of Python, in particular: Explicit is better than implicit. Combinatorics. SAT Math Help » Arithmetic » Integers » Permutation / Combination Example Question #1 : How To Find The Greatest Or Least Number Of Combinations Mark has 5 pants and 7 shirts in his closet. This example will guide you how to run tests in your Python application and deploy it automatically as Heroku application. The First line contains an integer $$T$$ representing the number of test cases. Permutation and combination problems formula aptitude permutation and Each of these questions has four answers (A, B, C, or D). Example Question #1 : Permutation / Combination A chamber of commerce board has seven total members, drawn from a pool of twenty candidates. Python Program to Find Perfect Number between 1 to 1000. This means that, when we're testing, we're. Given two strings, write a function to determine if one is a permutation of the other. Some examples: Normal with mean 10 and standard deviation 4:. is considered to be an absolute permutation if holds true for every. Solving fifteen-puzzles is much more difficult: the puzzle in Figure 8 has a solution of 50 moves and required that 84702 vertices (different permutations of the puzzle) be visited and the maximum heap size was 72340. We can continue. Python dictionary provides a member function update() i. Permutation Tests. Then we show a very simple Information Retrieval system, and an example working on a tiny sample of Unix manual pages. Wrapping Up. Using a Python recipe? Installing ActivePython is the easiest way to run your project. Each of these arrangements is a permutation. The good news is that when n is small and the number of words is large, this approach runs very fast (because set membership testing is O(1)):. This article provides a good general overview of permutation feature importance, its theoretical basis, and its applications in machine learning: Permutation feature importance. The permutation feature importance depends on shuffling the feature, which adds randomness to the measurement. # Program 005: Combinations, Permutations, Repeated Combination. Using Hypothesis you can write parameterized tests derived from a source of examples. If no such index exists, the permutation is the last permutation. Permutations are emitted in lexicographic sort order. The authors recommend 50-100 permutations. This is what will be the point of focus today’s Python Itertools Tutorial. Berrett, Yi Wang, Rina Foygel Barber, Richard J. A handful of excellent resources exist for learning what functions are available in the. xchat-python Python bindings for xchat, now part of xchat 2. Substantially all of these recipes and many, many others can be installed from the more-itertools project found on the Python Package Index:. For every permutation previous permutation becomes the starting point and from there the steps are -. For example, if we have 3 variables (features), the ACC with all variables is 0. ∀ k apermutation testis designed to test the Null Hypothesisthat the two classesLk and Rk really belong to the same. Does anyone know if there's an easy way to generate these though? Thanks!. First import itertools package to implement permutations method in python. Still, python's is a superior algorithm because it can permutate arrays of elements of arbitrary size without swapping them around. The example I have showed here highlighted the problem of small datasets, but really the usage of permutations tests is not limited to this situation. If no absolute permutation exists, print -1. Let denote the value at position in permutation using -based indexing. The permutation test will test the null hypothesis that the regression coefficient is 0 by assuming there is no relationship between X and Y, so pairs are as if at random. To generate a set of feature scores requires that you have an already trained model, as well as a test dataset. Permutation Importance¶. Since we are calling permutation 0(n * n!) times (as an upper bound), and each one takes 0(n) time, the total runtime will not exceed O (n^2 * n!). Because permutations are discrete, "Fisher's permutation test" is actually more closely related to the Wilcoxon test. Train on kata in the dojo and reach your highest potential. Transposition Permutations. Number of ways of selecting 3 consonants from 7 and 2 vowels from 4. It shuffles the data and removes different input variables in order to see relative changes in calculating the training model. You can confidently say your friend is not being entirely forthcoming about how they chose the permutation if it has this form but they claim they chose it uniformly at random. Is it that a permutation won't allow for something like "000" or "111" or "333", etc all the way through to "999"?. Permutations and Derangements Nice algorithms for computing permutations, derangements, and variants. I wish to determine a signficance threshold using permutation testing in order to correct for multiple testing. This live programming mode of Python Tutor (code on GitHub) supports three languages: Python 2. This solution does not generate all of the permutations. For example, if we have 3 variables (features), the ACC with all variables is 0. py Find file Copy path wblakecannon massive cleanup 3d4fc49 Jan 9, 2018.$ python fastwy. Program to Find all the Permutations of a String on fibonacci, factorial, prime, armstrong, swap, reverse, search, sort, stack, queue, array, linkedlist, tree, graph etc. To make it easy for you to learn these questions, I have created this Python Interview Cheat Sheet with 14 interview questions from this article. DataCamp / 07-statistical-thinking-in-python-(part-2) / 3-introduction-to-hypothesis-testing / permutation-test-on-frog-data. Transposition Permutations. In keeping with that, I plan to explore methods for analyzing microarray data using Python rather than R and Bioconductor. 7 , hide frames of exited functions show frames of exited functions , inline primitives and nested objects render all objects on the heap , hide environment parent pointers show environment parent pointers , draw references using arrows use text labels for references , and. It can be used from within compiled languages such as C, C++, FORTRAN and Java, as well as interactively from interpreted languages such as Octave, Python, Perl and Tcl. Matrix P has the same data type as v, and it has n! rows and n columns. A permutation of the fifteen-puzzle. Output Format. It is interesting to note the similarities and differences between the bootstrap and the permutation test here. I can't figure out how to get permutations to return the actual permutation and not I tried a lot of different things to no avail. Statistics based on resampling in Python: bootstrapping and permutation tests Resampling techniques in statistics are appealing because they rely on few assumptions about the underlying distribution of one's data. Also generate the sign of the permutation which is +1 when the permutation is generated from an even number of swaps from the initial state, and -1 for odd. A t-test is one of the most frequently used procedures in statistics. It is an online math tool which determines the number of combinations and permutations that result when we choose r objects from a set of n objects. For any one SNP the z-statistic from a logistic. A permutation is an arrangement or ordering of a number of distinct objects. I have first seen this question in my college exam when we were asked to code the solution using C or C++ language. Visualize Execution Live Programming Mode. Hi all, I am performing permutation test for 500000 times. Labels to constrain permutation within groups, i. The basic idea of permutation importance is to permute the values of each feature and measure how much that permutation negatively impacts the scoring metric (which in our case is the Spearman's rank correlation). permutations (iterable [, r]) ¶ Return successive r length permutations of elements in the iterable. Using python itertools permutations example ( remove duplicates values) We can use python set for removing duplicate values from the output of itertools permutations function. Permutation importance works for many scikit-learn estimators. Python tries to re-use objects in memory that have the same value, which also makes comparing objects very fast in. Different permutations can be ordered according to how they compare lexicographicaly to each other; The first such-sorted possible permutation (the one that would compare lexicographically smaller to all other permutations) is the one which has all its elements sorted in ascending order, and the largest has all its elements sorted in descending. To get rid of such annoying sample test cases, you have to write a program which detects if a given permutation is ambiguous or not. Python Challenge The aim of this challenge is to write a procedure called listAllBinaryPermutations() that will take one parameter called numberOfBits and outputs all possible binary permutations depending on the number of bits. How permutation importance works. SAT Math Help » Arithmetic » Integers » Permutation / Combination Example Question #1 : How To Find The Greatest Or Least Number Of Combinations Mark has 5 pants and 7 shirts in his closet. The inverse of a permutation is defined as the permutation that reverses this operation, i. Next_permutation does the hard work, and there's a bit of fiddling when we have to increase the number of digits by adding a zero. This free calculator can compute the number of possible permutations and combinations when selecting r elements from a set of n elements. 3 permutations of previous ? 11. A more fair comparison would be to compare the python implementation of python's permutation algorithm: in my machine it took 0m8. In several cases, restricted permutations, in particular, suffered more than other procedures, in terms of loss of power, challenging the conventional wisdom. The methods of calculation for the perm order, parity, ranking and unranking are given. The basic idea of permutation importance is to permute the values of each feature and measure how much that permutation negatively impacts the scoring metric (which in our case is the Spearman's rank correlation). Each test case contains an integer $$N$$ representing the size of permutation $$P$$. The notation and examples shown here are borrowed from Efron and Tibshirani’s An Introduction to the Bootstrap [1]. Permutation tests work by resampling the observed data many times in order to determine a p-value for the test. If you have some background on ruby then surely, you have been used gem list to get the list of installed modules. To generate a set of feature scores requires that you have an already trained model, as well as a test dataset. In this article we will discuss how to add or append new key value pairs in a dictionary and also how to update value of existing keys. Python Sliding Window 90%. A permutation is an arrangement of all or part of a set of objects, with regard to the order of the arrangement. It mainly tests some one’s grip on recursion too.

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Black carbon and other light-absorbing impurities in snow in the Chilean Andes

Abstract

Vertical profiles of black carbon (BC) and other light-absorbing impurities were measured in seasonal snow and permanent snowfields in the Chilean Andes during Austral winters 2015 and 2016, at 22 sites between latitudes 18°S and 41°S. The samples were analyzed for spectrally-resolved visible light absorption. For surface snow, the average mass mixing ratio of BC was 15 ng/g in northern Chile (18–33°S), 28 ng/g near Santiago (a major city near latitude 33°S, where urban pollution plays a significant role), and 13 ng/g in southern Chile (33–41°S). The regional average vertically-integrated loading of BC was 207 µg/m2 in the north, 780 µg/m2 near Santiago, and 2500 µg/m2 in the south, where the snow season was longer and the snow was deeper. For samples collected at locations where there had been no new snowfall for a week or more, the BC concentration in surface snow was high (~10–100 ng/g) and the sub-surface snow was comparatively clean, indicating the dominance of dry deposition of BC. Mean albedo reductions due to light-absorbing impurities were 0.0150, 0.0160, and 0.0077 for snow grain radii of 100 µm for northern Chile, the region near Santiago, and southern Chile; respective mean radiative forcings for the winter months were 2.8, 1.4, and 0.6 W/m2. In northern Chile, our measurements indicate that light-absorption by impurities in snow was dominated by dust rather than BC.

Introduction

Chile is a long and narrow country spanning 38 degrees of latitude; it is therefore characterized by varied climate and varied response to climate drivers. The Andes act as a critical “water tower”, providing the main source of freshwater for Chile from snowpack and glacier runoff, wetlands and groundwater sources1,2,3,4,5 Climate warming along the Chilean Andes6,7 has resulted in the rapid loss of glaciers from the tropical Andes in the north8 to Patagonia and Tierra del Fuego in the South9. In central Chile, persistent drought in 2010–2015 reduced the snowpack, resulting in declines in river flow10. About half of the observed decline in precipitation since the 1970s in central Chile has been attributed to anthropogenic climate change11. Furthermore, general circulation models suggest that warming rates at higher elevations in the central Andes will outpace those for surrounding areas2,12,13 The past few decades have seen a regional enhancement of mid-tropospheric warming in the central Andes13 but the cause of the enhancement is unclear. Such warming increases the fraction of precipitation that falls as rain rather than snow, leading to earlier melting and peaks in river runoff1. For Chile, where the snowpack acts as a significant water reservoir, this can result in flooding in winter and spring, followed by water shortages in late spring and summer. Furthermore, model simulations predict a climatic trend of decreasing precipitation over the central and southern Andes, from 10°S to 45°S, even under strong reduction of greenhouse gas emissions14, with important implications for Chile’s water supply, particularly in central Chile (30–37°S) where 78% of the population is concentrated.

Because of the importance of snowpack as a source of freshwater for Chile, quantitative estimates of radiative forcing are key to predicting changes in runoff timing of snowpacks due to deposition of light-absorbing aerosol in snowpacks15. An important light-absorbing aerosol to quantify is black carbon (BC), an anthropogenic pollutant emitted during combustion, such as by coal-burning power plants, vehicles (particularly diesel), wood-burning stoves, agricultural fires, and forest fires16. Light-absorbing organic carbon (“brown carbon”) is often co-emitted with the BC as a component of carbonaceous aerosol. In the atmosphere, this carbonaceous particulate material contributes to respiratory problems and can interact with sunlight to affect climate16. When deposited on snow, light-absorbing carbonaceous particles reduce the albedo, leading to increased absorption of solar radiation. This triggers a positive feedback whereby the warmer snowpack ages more rapidly, further lowering albedo (through grain-size growth), ultimately accelerating surface snowmelt. Surface snowmelt then leads to the accumulation of particulate impurities at the snow surface17, further lowering albedo and accelerating melt. The darker underlying surface then becomes exposed earlier, leading to more absorption of sunlight and more melt18. Deposition of mineral dust similarly reduces snow albedo, with the same feedback processes, but scaled down by a factor of about 200 at visible wavelengths, by mass19. At high enough concentrations, the reduction of snow albedo by BC and mineral dust can significantly affect glacier mass balance and snow cover18,20. When dust thickness exceeds a critical depth of about 15 to 50 mm, it acts as an insulating layer and melting rates become lower than that of dust-free snow21.

It is expected that the main source of BC in northern Chile is emissions from diesel engines that power the mining industry and major astronomical observatories. Near Santiago (latitude 33.3°S), sources include emissions from transport (e.g. diesel), industrial pollution and residential heating. These emissions are trapped by the topography and the persistent wintertime temperature inversion22. In southern Chile (from 35°S to 47°S), the main source of heating is wood stoves, and thus the main source of particulate pollution is expected to be emissions from wood burning.

Surveys of BC in snow have been conducted in a variety of locations with various techniques, including thermo-optical methods and the SP2 instrument16, as well as filter-based methods, including the Light Absorption Heating Method23 and the Integrating Sphere Integrating SandWich (ISSW) spectrophotometer method24,25,26. The ISSW uses the wavelength dependence of measured light absorption to estimate the concentration of BC as well as the absorption Angstrom exponent (Åtot) of all light-absorbing insoluble particles. The latter is used to estimate the non-BC fraction of light absorption. Light-absorbing impurities in snow have been measured with the ISSW spectrophotometer in the Arctic24,26, northern China27,28 and central North America29. Median BC content was found to range from 3 ng/g in Greenland26 to 1220 ng/g in northeast China28.

Here, we present measurements of BC and other light-absorbing impurities made in the Chilean Andes using the ISSW spectrophotometer during two winters, one in northern Chile in 2015, and the other in central and southern Chile in 2016. As a first step toward quantifying the influence of BC in the Chilean Andes on snowmelt, we present albedo reductions and radiative forcings for a variety of snowpacks in the Chilean Andes.

The paper is organized as follows: Section 2 details the field campaigns and describes the methods of sample collection, filtration, and analysis. Sections 3–5 present results, discussion, and conclusions.

Methods

Sample sites

Snow was sampled at 22 locations between Putre (S 18°, W 69°) and Osorno (S 41°, W 72°). The sampling sites are assigned to one of three groups: northern Chile (18.1–32.9°S, Table 1 and Fig. 1a); the Santiago area (33.3–33.8°S, Table 2 and Fig. 2); and southern Chile (35.1–41.1°S, Table 3 and Fig. 1b). Topographic maps showing the sample sites are given in Figs S1S2 of the Supplemental. Sampling in the north occurred in one north-south transect from 4 to 27 July 2015. Sampling in the south occurred during two north-south transects in 2016: 13–21 July and 19 August – 1 September. Sampling near Santiago was done in both winters 2015 and 2016. The individual sample sites are described below; photographs of some of the sites are shown in Fig. 3. Date of sampling, latitude, longitude, elevation and total snow depth are given for each site in Tables 13. Sites are listed in latitude order; for each site where sampling occurred more than once, sequential samples are listed in order of date.

Sites in northern Chile

The Andes of northern Chile are bordered to the west by the Atacama Desert, and are characterized by being dry and dusty, with minimal vegetation. During our sampling in the far north of Chile the snow was thin and present only at higher altitude. Moving south, the snow cover and snow depth generally increased and the snow line generally dropped in altitude, allowing sampling to occur at lower altitudes, as indicated in Table 1. Sites are marked in the map in Fig. 1a. As indicated in Table 1, northern sites were visited sequentially from 4 July to 20 July 2015.

Sites near Santiago

Snow was sampled at five sites near Santiago, all within 1 degree of latitude of each other. Santiago is by far the largest city in Chile, with a large fraction of the country’s population (6.5 million people out of 17.9 million people in Chile as of 2017). The topography and persistent wintertime temperature inversion traps urban pollution and leads to smog, along with high levels of PM 2.5 and PM1022. During winter, the mixing layer height can drop to 100 m in early morning and has an average height of 200 m at local noon30. Within the daytime temperature inversion, the temperature increase is 3–6 K with an inversion top of 600 to 900 m31. Accumulations of soot particles are visible on balconies in the city. As shown in Table 2, sites near Santiago were visited in July 2015, and June, July and August 2016. Sites are marked in the map in Fig. 1b.

Sites in southern Chile

In contrast to northern and central Chile, southern Chile has frequent precipitation and abundant vegetation. Concentrations of mineral dust in snow are therefore expected to be low, relative to northern Chile. Southern Chile is considerably colder in winter (the average temperature in June is 17 °C in Arica and 7 °C in Osorno). Residential wood combustion is the main source of pollution for many cities in southern Chile32. Thus, a high incidence of black carbon is expected in snow in southern Chile, particularly relative to mineral dust. Sites are indicated on the map in Fig. 1.

Sample Collection and Filtration

Sample collection and filtration follows the methodology of Doherty et al.17,26,29. Samples were collected at vertical intervals of 5 to 8 cm depth from the surface (exposed to the air) down to the solid ground. Typically, two parallel vertical profiles were collected, separated by a horizontal distance of 1 to 2 m. To minimize pollution due to nearby sources of transportation, sites were chosen to be approximately 1 km away from any roads when possible.

Samples were collected with a metal spatula into plastic food-handling bags, which were then packed in plastic Whirlpak bags. Food-handling bags were used for the inner bags because prior work indicated that BC adheres least to them, compared to other types of plastic26,28. To minimize the amount of particulates that adhered to the plastic bag with melting, the snow was stored in freezers or insulated containers, when possible, until immediately before filtration. The depth profile of snow density was also measured at each site.

Filtering was performed on a clean table or lab bench, pre-cleaned for dust and dirt. Gloves and lab coats were worn. Snow from Putre and San Pedro was filtered in temporary laboratories set up in local accommodations; snow from other locations was filtered in University laboratories. In southern Chile and near Santiago in 2016, bagged snow samples were stored in Styrofoam coolers to keep them at near-freezing temperatures during transportation to a freezer in a University; samples were thereafter kept frozen until they could be filtered in the University laboratory.

Because particulate matter adheres to glass less than it adheres to plastic, the snow was transferred to a glass beaker for melting, then the beaker was covered with a plastic bag to prevent external contamination. The snow was heated in a microwave oven until just melted, to avoid heating of the meltwater. A 0.4-µm Nuclepore filter was placed in a filter holder mounted on top of a side-arm flask. A stainless-steel funnel was clamped above the filter holder. The meltwater was then vacuum-filtered with a hand pump or electric pump through the filter. After filtering, filters were carefully placed in sterile petri dishes. The petri dishes were opened slightly to allow the filters to air dry, after which the petri dish lid was kept closed until optical analysis was performed. The mass of snow filtered was adjusted with the goal of producing filters with loadings within approximately 0.4 to 40 µg/cm2, to stay within the measurement-sensitivity range of the spectrophotometer (described in Section 2.3); filters determined to be outside the spectrophotometer measurement range were discarded. The meltwater volume was measured. The funnel, filter holder, and beaker were cleaned after each filtration with ultrapure water.

Optical Analysis

Optical analysis of filters is performed with the ISSW spectrophotometer described by Doherty et al.29, which is a modified version of the instrument developed by Grenfell et al.25. The ISSW configuration includes a sample filter sandwiched between two integrating spheres, positioned above and below the sample filter. Input light is fed via fiber optics through a side port into the lower sphere, and light exits through a side port in the upper sphere to the spectrometer via a fiber optic. Baffles prevent direct light from passing through the filter and prevent light from the filter from passing directly out through the side port; thus, only diffuse light is transmitted to the detector. The ISSW has a spectral range of 450–750 nm and has a measurement resolution of 1 nm which is then averaged to 10 nm.

Starting from the wavelength-dependent BC-equivalent loading, we calculate the quantities given in Table 4. These are described below in turn (after ref.29).

$${{\rm{C}}}_{{\rm{BC}}}^{{\rm{\max }}}$$ is the average BC-equivalent concentration, calculated by assuming all absorption in the 650–700 nm wavelength band is due to BC. To convert from the measured fullerene-equivalent loading (µg BC-equivalent per cm2) to concentration, the average 650–700 nm loading is multiplied by the exposed area on the filter (cm2) and divided by the mass of snow water filtered (g), yielding µg/g. This is converted to ng/g, giving the mixing ratio as ng BC per g of snow water. $${{\rm{C}}}_{{\rm{BC}}}^{{\rm{\max }}}$$ represents the maximum possible black carbon in the sample.

Also of interest is the amount of absorption due to non-BC particulate components. Here we use the absorption angstrom coefficient, Åtot, to discriminate between BC and non-BC components. Åtot is defined as:

$${{\rm{\AA }}}_{{\rm{tot}}}=\frac{-\mathrm{ln}\,[{{\rm{\tau }}}_{{\rm{abs}}}({{\rm{\lambda }}}_{1})/{{\rm{\tau }}}_{{\rm{abs}}}({{\rm{\lambda }}}_{2})]}{\mathrm{ln}({{\rm{\lambda }}}_{1}/{{\rm{\lambda }}}_{2})}$$
(1)

Herein, Åtot is calculated as a linear fit to absorption vs. wavelength in log-log space between the wavelengths λ1 = 450 nm and λ2 = 600 nm. The absorption optical depth τabs is the mass absorption coefficient of fullerene multiplied by the BC-equivalent loading. The mass absorption coefficient of fullerene is 8.9 m2/g at 450 nm and 6.5 m2/g at 600 nm. Pure BC has ÅBC = 1.1. For non-BC, we assume that ÅNBC = 5, based on prior work by Grenfell et al.25 and Doherty et al.26. The deviation of Åtot from 1.1 is therefore an indication of the amount of absorption by non-BC components in the snow.

$${{\rm{C}}}_{{\rm{BC}}}^{{\rm{est}}}$$ is calculated following the method of Grenfell et al.25:

$${{\rm{C}}}_{{\rm{BC}}}^{{\rm{est}}}={{\rm{C}}}_{{\rm{BC}}}^{{\rm{\max }}}(\frac{{\langle {{\rm{\tau }}}_{{\rm{BC}}}^{{\rm{est}}}\rangle }_{650-700}}{{\langle {{\rm{\tau }}}_{{\rm{BC}}}^{{\rm{MAX}}}\rangle }_{650-700}}),$$
(2)

where

$${{\rm{\tau }}}_{{\rm{BC}}}^{{\rm{est}}}({{\rm{\lambda }}}_{0})={{\rm{r}}}_{{\rm{BC}}}({{\rm{\lambda }}}_{0})\times {{\rm{\tau }}}_{{\rm{tot}}}({{\rm{\lambda }}}_{0}).$$
(3)

The value rBC0) is the fraction of absorption due to BC at a given wavelength λ0, and it is determined by assuming that

$${{\rm{\AA }}}_{{\rm{tot}}}={{\rm{r}}}_{{\rm{BC}}}{{\rm{\AA }}}_{{\rm{BC}}}+(1-{{\rm{r}}}_{{\rm{BC}}}){{\rm{\AA }}}_{{\rm{NBC}}},$$
(4)

where Åtot is the measured absorption Angstrom exponent of all particles on the filter and, as given above, ÅBC = 1.1 and ÅNBC = 5.0, with rBC defined over the same wavelength range as the values of Åtot. Rearranging Eq. (4) yields rBC. The value for τtot0) is L0 β0, where L0 is the BC-equivalent loading and β0 is the mass absorption efficiency of fullerene at the given wavelength. Furthermore, τ as a function of wavelength is given by

$${{\rm{\tau }}}_{{\rm{BC}}}^{{\rm{est}}}({\rm{\lambda }})={{\rm{\tau }}}_{{\rm{BC}}}^{{\rm{est}}}({{\rm{\lambda }}}_{0})\times {({\rm{\lambda }}/{{\rm{\lambda }}}_{0})}^{-{\dot{{\rm{A}}}}_{{\rm{BC}}}}.$$
(5)

To calculate $${{\rm{\tau }}}_{{\rm{BC}}}^{{\rm{\max }}}({\rm{\lambda }})$$ the first term on the right is replaced with τtot0).

$${{\rm{C}}}_{{\rm{BC}}}^{{\rm{equiv}}}$$ is the mass of BC that would be needed to account for the integrated 300–750 nm solar absorption by all particulates on the filter. It is calculated by weighting the spectral, all-component (total) absorption by the downwelling solar irradiance spectrum29. $${{\rm{C}}}_{{\rm{BC}}}^{{\rm{equiv}}}$$ differs from $${{\rm{C}}}_{{\rm{BC}}}^{{\rm{\max }}}$$ in that $${{\rm{C}}}_{{\rm{BC}}}^{{\rm{\max }}}$$ represents the maximum mass of BC possible, whereas $${{\rm{C}}}_{{\rm{BC}}}^{{\rm{equiv}}}$$ represents the amount of absorbing material present, whether BC or not, expressed as an equivalent BC mass. $${{\rm{C}}}_{{\rm{BC}}}^{{\rm{\max }}}$$ is quantified at 650–700 nm, whereas $${{\rm{C}}}_{{\rm{BC}}}^{{\rm{equiv}}}$$ is quantified for 300–750 nm; since non-BC components will absorb more strongly at shorter wavelengths, $${{\rm{C}}}_{{\rm{BC}}}^{{\rm{equiv}}}$$ is always greater than or equal to $${{\rm{C}}}_{{\rm{BC}}}^{{\rm{\max }}}$$.

The fraction of solar absorption integrated across 300–750 nm by non-BC particles, $${{\rm{f}}}_{{\rm{NBC}}}^{{\rm{est}}}$$, is calculated by “(1) extrapolating the absorption spectra linearly from 450 nm down to 300 nm25,26; (2) weighting the non-BC absorption spectra ($${{\rm{\tau }}}_{{\rm{BC}}}^{{\rm{tot}}}({\rm{\lambda }})$$$${{\rm{\tau }}}_{{\rm{BC}}}^{{\rm{est}}}({\rm{\lambda }})$$) and all-component (total) absorption spectra (τtot0)) by a downwelling solar irradiance spectrum appropriate for clear-sky, mid-latitude winter; (3) integrating these weighted spectra across the visible range; and (4) taking the ratio of the two integrals”29.

In addition to reporting concentrations in given snow layers, the vertically-integrated BC mass, “Int BC”, was calculated as follows: First, the snow water equivalent is calculated in each sample layer by multiplying the snow density (g/cm3) by the layer depth (cm). The BC amount within each sampled layer of snow is then calculated by multiplying $${{\rm{C}}}_{{\rm{BC}}}^{{\rm{est}}}$$ (ng/g) by the snow water equivalent (g/cm2), yielding a BC mass per unit area (ng/cm2). This is done for each sampled layer, and the BC mass from all layers summed together to give integrated BC for the snow column, which is reported in µg/m2. Because it is difficult to sample all the way to the ground without contamination by soil, the BC amount for the last few centimeters is typically estimated from $${{\rm{C}}}_{{\rm{BC}}}^{{\rm{est}}}$$ and the density of the layer sampled immediately above. For a given sample site, measurements within a few hundred meters, if made on the same day, are averaged. In such cases, a nearby measurement may be used to fill in missing values (e.g. in a case where filter loading from one sample was too high for an accurate measurement).

Estimating albedo reduction and radiative forcing

The albedo reduction and radiative forcing due to light-absorbing impurities in snow were calculated using the method of Dang et al.33, and making use of the parameterization of Dang, Brandt, and Warren19, as follows. The all-sky broadband albedo reduction was calculated according to

$$\delta \alpha =CF\times \delta {\alpha }_{cloudy}+(1-CF)\delta {\alpha }_{clear},$$
(6)

where CF is the cloud fraction. Monthly mean cloud fraction was determined from the CERES Terra + Aqua Edition 4A34. Monthly means for each of the three winter months (June, July, August) were averaged from 2000 to 2017 (Fig. 4a). The albedo reduction was determined from the parameterization of Dang et al.19, which gives the albedo reduction for clear and overcast skies for a range of BC mass mixing ratios and snow grain radii. To get the albedo reduction due to BC, the albedo reduction was interpolated to the BC mass mixing ratio for a given value of $${{\rm{C}}}_{{\rm{BC}}}^{{\rm{est}}}$$. This was done for cloudy skies (yielding δαcloudy) and for clear skies (yielding δαclear), after which Eq. (6) was applied. These steps were repeated for $${{\rm{C}}}_{{\rm{BC}}}^{{\rm{equiv}}}$$ to calculate the albedo reduction for all light-absorbing impurities. Albedo reductions due to BC and due to all light-absorbing impurities were calculated for the months of June-July and for snow grain radii of 100 and 1000 µm.

The radiative forcing is calculated as the albedo reduction times the all-sky shortwave flux, where the latter is also taken from CERES (see above and Fig. 4b) and averaged over the years 2000 to 2017.

Albedo reductions and radiative forcings were not calculated for every site given in Tables 13, but rather for median values of $${{\rm{C}}}_{{\rm{BC}}}^{{\rm{est}}}$$ and $${{\rm{C}}}_{{\rm{BC}}}^{{\rm{equiv}}}$$ for each location (e.g. the median for all the Putre sites, etc.), except when the locations were separated by a large altitude difference or when the site was revisited on different days.

Results

Latitude dependence of light-absorbing impurities

$${{\rm{C}}}_{{\rm{BC}}}^{{\rm{est}}}$$, Å, $${{\rm{f}}}_{{\rm{NBC}}}^{{\rm{est}}}$$, and Integrated BC (Int BC) are shown as functions of latitude in Fig. 5. Sampling sites are color-coded and listed in the legend, from north to south; the color coding is the same as for the site labels in the maps shown in Figs 1 and 2. One point is plotted for each layer sampled; typically these are averages of two samples made for the same layer, although sometimes more or fewer samples were taken of a given layer. At some sites, samples were taken at different locations within the site area, or the site was revisited on a different date, as indicated in Tables 13.

For Fig. 5d (lower right), which shows the estimate of snow-column-integrated black carbon mass, values for sites within a few hundred meters and at a similar altitude were averaged. Multiple points are due to sample collections separated by a considerable distance, altitude, or for different conditions at the same site (e.g. La Ola, where one set of samples was made in a snow drift) or due to revisiting the site at a later date (e.g. Valle Nevado). Points are missing where snow density was not measured (e.g. Putre) or where there was no surface measurement.

Surface snow

Table 5 gives values for Åtot, $${{\rm{C}}}_{{\rm{BC}}}^{{\rm{est}}}$$, $${{\rm{C}}}_{{\rm{BC}}}^{{\rm{\max }}}$$, and $${{\rm{C}}}_{{\rm{BC}}}^{{\rm{equiv}}}$$ for surface samples at each location. The layer thickness of the surface sample is also shown in the table. Medians were calculated across all surface samples from a given location, except when the locations were separated by a large altitude difference or when the site was revisited on different days. (These values are given for each layer in each sampling site in Tables S1S3 of the Supplemental).

Regional results

Regional averages and standard deviations for measurements in northern Chile, the Santiago region, and southern Chile are given in Table 6. In northern and southern Chile, where there were sometimes large distances between sites, the average and standard deviation for each site were geographically weighted by the increment of latitude that the site was judged to represent. Values shown in the table include surface measurements of Åtot, $${{\rm{C}}}_{{\rm{BC}}}^{{\rm{est}}}$$, $${{\rm{C}}}_{{\rm{BC}}}^{{\rm{\max }}}$$, and $${{\rm{C}}}_{{\rm{BC}}}^{{\rm{equiv}}}$$(as well as the subsurface values for $${{\rm{C}}}_{{\rm{BC}}}^{{\rm{est}}})$$, the total snow depth, snow water equivalent, integrated black carbon, and average black carbon. For subsurface and snowpack measurements, averages in the north exclude Putre, where snow density was not measured. Also shown in Table 6 are the albedo reductions due to light-absorbing impurities in snow for BC alone and for all light-absorbing impurities and the resulting radiative forcing; these are discussed below.

Figure 4a,b show the cloud fraction and shortwave flux for the Austral winter months June, July, and August. While cloud fraction does not change much with month, it increases substantially from north to south. Shortwave flux changes considerably with both latitude and month.

Figure 4c shows the albedo reduction due to all light-absorbing impurities in snow for each site shown in Table 5 (for the values of $${{\rm{C}}}_{{\rm{BC}}}^{{\rm{equiv}}}$$ given in the table) assuming snow grain radii r = 100 µm (small symbols) and r = 1000 µm (large symbols). The choice of these representative grain sizes is explained by Wiscombe and Warren35 and Warren36. Because new snow is typically fine-grained, the albedo reduction is plotted for r = 100 µm for the beginning of winter (June in the north; July in the south). By contrast, melting snow is typically coarse-grained; thus the albedo reduction is plotted for r = 1000 µm for the melt season (July in the north, August in the south). The albedo reduction and radiative forcing are greatest in the far north and in the Santiago region.

Albedo reductions and radiative forcings for regional averages are given in Table 6 for 100 and 1000 µm snow grains and for BC-only and all light-absorbing impurities.

Discussion

Latitude dependence of non-BC light-absorbing impurities

The contribution of non-black carbon constituents to particulate absorption shows a trend with latitude, decreasing southward. The fraction of absorption from non-BC decreases from around 0.9 near San Pedro to 0.1 at the southern-most site of Osorno (Fig. 5c). The far north is a desert where large amounts of dust may be lofted into the air by natural and human causes (see, e.g. Fig. 6). In contrast, the southern region is characterized by frequent precipitation and the atmosphere is therefore less dusty.

How much the Angstrom exponent, Åtot, deviates from 1.1 is an indication of the relative contribution of non-BC particles to the light-absorption. Starting at the north end of Chile and moving south, Åtot first increases from about 2.5 at Putre (Site 1), to a maximum of ~5 at La Ola (Site 3; Fig. 5b), then decreases to a minimum of 1.1–1.9 at Osorno (Site 22), indicating minor contribution from non-BC at Osorno. This geographic trend in Åtot holds for samples from all layers (Fig. 5b). For surface samples, the trend is also the same, except that Åtot peaks at Site 2 above San Pedro (Åtot = 4.5; Table 5).

Particles with Åtot near 1.0 are essentially black, indicating their light-absorbing component is nearly all black carbon (Bond et al.16), indicating a combustion source for particulate matter. The low values of Åtot at our southern sites are consistent with a combustion source for the snow particulate matter. Added to the knowledge that wood-burning is the main source of heating in this region and the lack of dust sources we infer that light-absorbing particles in snow are dominated by carbonaceous aerosols. To the north, the high values of Åtot of 3 to >5 indicate a much larger contribution from non-BC sources. Previous measurements with the ISSW coupled with chemical analyses (refs26,28,29) indicate that these high values of Åtot could be due to very high brown carbon to black carbon ratios in combustion-sourced aerosol or, more likely for the highest values seen here (>3), due to dust. Dust as the dominant source of light-absorbing particles to snow in northern Chile is consistent with the low population density, the proximity of dust sources and the low snow-cover fraction.

Precise attribution of absorption to sources using Åtot is not possible. We have assumed that ÅNBC is 5.0. If the particulate absorption at Site 3 is in fact essentially all due to non-BC (e.g. dust), this supports the validity of using ÅNBC = 5.0 for this region. If, in fact, there is a significant contribution to absorption by BC even at Site 3, ÅNBC should be higher; in this case, our reported values of $${{\rm{C}}}_{{\rm{BC}}}^{{\rm{est}}}$$ will be too low and, correspondingly, our attribution of absorption to brown carbon and dust will be too high. Regardless, trends in Åtot are strongly indicative of compositional trends in light-absorbing particulate matter in snow.

Latitude dependence of black carbon

As shown in Fig. 5a, $${{\rm{C}}}_{{\rm{BC}}}^{{\rm{est}}}$$ decreased southward to a minimum above Elqui (Site 4), then increased continuing south to the Santiago area (Sites 7–11), and finally decreased gradually southward to Osorno (Site 22). In the north, this trend is consistent with the trend for non-BC, suggesting that the snow becomes overall cleaner moving south from the far north of Chile (Putre) to Elqui.

The trend for $${{\rm{C}}}_{{\rm{BC}}}^{{\rm{est}}}$$ described above is in keeping with the regional averages, for which $${{\rm{C}}}_{{\rm{BC}}}^{{\rm{est}}}$$ is highest near Santiago. Near Santiago there is a lot of variation in $${{\rm{C}}}_{{\rm{BC}}}^{{\rm{est}}}$$ from site to site, which may be linked to the influence of dry deposition, as will be discussed below.

The integrated black carbon is a metric for total BC deposition over the period of snowpack accumulation. For regional averages, both integrated black carbon and snow depth increased southward. $${{\rm{C}}}_{{\rm{BC}}}^{{\rm{est}}}$$ for surface samples in the Santiago region is higher than the value for southern Chile; however Santiago has considerably less integrated BC. The values for average BC are the same. The explanation is that there can be high BC concentrations but low column-integrated BC mass if the snowpack is thin.

Wet vs. dry deposition

The above observations alone do not permit unambiguous identification of the dominant mechanism of BC deposition (dry deposition, in which BC settles onto snow from the atmosphere, or wet deposition, in which BC is transferred to snow via precipitation), because of a number of complicating factors. One complication is that the sets of numbers used to compute $${{\rm{C}}}_{{\rm{BC}}}^{{\rm{est}}}$$, the integrated BC, and the subsurface BC vary slightly due to missing data and variations in snow depth (e.g. there was not always a subsurface layer to sample). This is particularly important for the Santiago area (as well as in the north), where snow depth was typically thinner and where there were large variations in $${{\rm{C}}}_{{\rm{BC}}}^{{\rm{est}}}$$ (110%) and integrated BC (66%). In southern Chile the variations were smaller, 70% and 52%, respectively, indicating a more uniform snowpack. It is furthermore worth noting that if snow cover is intermittent at a given location, then the column-integrated BC mass will not reflect total BC deposition for the winter, whereas if there is no loss of snow particulate matter to melting, the column-integrated BC will reflect the total BC deposition since snowfall started. There are, however, situational factors that help constrain this distinction; an exploration of wet vs. dry deposition using a few case studies follows.

Meteorological data and observations suggest a lack of snowfall in northern and central Chile from 1 to 6 July 2015, with the exception of a light snowfall on 4 July at Site 1, Putre (observed by the sampling group). On 5 and 6 July a cold front may have resulted in some snowfall in the mountainous region east of Santiago. Another front brought snowfall to the Santiago area on 11 and 12 July and to Elqui on 13 July. Blizzard conditions with snow and wind were experienced at La Ola on 13 July, the day before sampling was performed. On 16 July a less intense front brought some snowfall to the mountains east of Site 4, Elqui, but no precipitation was measured at the station near the Elqui sampling site. There was no evidence of additional snowfall for the remainder of the 2015 field season. Further details are given in the Supplemental.

Based on the above as well as reports from locals near the field sites, Table 7 gives estimates of days since last snowfall, $${{\rm{C}}}_{{\rm{BC}}}^{{\rm{est}}}$$ for both the surface and sub-surface snow, and integrated BC for many of the sites for the 2015 field season.

In northern Chile and the Santiago area, the date of the most recent snowfall could be estimated from observations and weather reports. As shown in Table 7 and in Fig. 7, the contamination of the surface layer and the integrated black carbon mass both generally increased with time. Given that the subsurface samples are relatively clean everywhere, proximity to Santiago seems unlikely to be a major factor. Instead, the relative dirtiness of the surface snow suggests that dry deposition plays a major role. In addition, consolidation of BC in the surface layer with snow melt17 may be playing a role, as well as sublimation of the surface snow at high elevations in the dry Andes37.

Data from meteorological stations suggest that most of the precipitation from 27.2 to 33.4°S was due to the passage of a front on 12 July, with small amounts of precipitation in this latitude range the following day for stations from 33.3 to 33.4°S. However, precipitation was measured at Santo Domingo at 33.39°S on 5 July. Thus it is possible some of the subsurface snow sampled at Valle Nevado and Valle Maipo was seven days older, consistent with the dirtier subsurface snow observed there.

Near Santiago, measurements made on sequential days more clearly indicate the importance of dry deposition. At Yerba Loca, near Santiago, the full column of the snowpack was sampled two days in a row (25 and 26 July 2016) at nearby locations, allowing for determination of the dry-deposition rate. The snow was melting and the depth decreased from 10 to 5 cm, while the snow density increased from 0.19 to 0.44 g/cm3. The snow-column-integrated BC increased from 31 to 200 µg/m2, indicating that the new snowfall in the Santiago area was relatively clean, and light-absorbing impurities arose mainly from dry deposition.

Aerosol deposition rate near Santiago

The freshly fallen snow of 25 July 2016 at Yerba Loca was quite clean both in terms of the BC concentration and in terms of the non-BC contribution to absorption, which was negligible. On 26 July particulate absorption included a significant contribution by non-BC components, as Åtot increased from ~1.1 to 2.0 and $${{\rm{f}}}_{{\rm{NBC}}}^{{\rm{est}}}$$ increased from a negligible value to 0.3. Since no snow fell between these two days, this contribution must have been via dry deposition.

The deposition rate is estimated as the difference in integrated BC mass divided by the time difference, or 17.3 ng/cm2-day. This is equivalent to 5.2 mg/m2-month, which is roughly comparable to the estimated inter-monsoon dry-deposition rate of BC in the Himalayas (see Fig. 3a in Menegoz et al.38).

Trend in BC with distance from a major highway

To determine whether significantly more BC is deposited close to major highways, measurements were made at varying distances from the Trans-Andean highway to the southeast of Juncal, in the Juncal National Park. Measurements were made 3.64 km, 1.89 km, 1.05 km, and 0.561 km from the highway (Sites 8a-d). Moving along this transect toward the highway, the surface black-carbon estimate was 15, 15, 14, and 18 ng/g. Thus black carbon from vehicles does not seem to be preferentially deposited near the road.

Albedo reduction and radiative forcing due to impurities in snow in the Chilean Andes are found to be greatest in the far north and in the Santiago region. For example, under all-sky conditions, mean albedo reductions due to light-absorbing impurities in snow in the Chilean Andes are estimated to be 0.0150, 0.0160, and 0.0077 for northern Chile (in June), the region near Santiago (in June), and southern Chile (in July), respectively, for snow grain radii of 100 µm (representative of new snow). Likewise, mean radiative forcings are estimated to be 2.8, 1.4, and 0.6 W/m2, respectively.

In northern Chile, BC plays a smaller role in albedo reduction than non-BC. For example, albedo reduction for 100 µm snow grains due to BC alone is only about 43% of that for all light-absorbing impurities. By comparison, these albedo reductions are 53% and 82% near Santiago and in southern Chile, where a greater share of light absorption is due to BC.

Comparison to other measurements

The regional means and standard deviations of $${{\rm{C}}}_{{\rm{BC}}}^{{\rm{equiv}}}$$ for northern Chile measured in this work (refer back to Tables 5 and 6) are higher than values for measurements made in the Cordillera Blanca in Peru23. At an altitude of ~4900 m, Schmitt et al.23 measured mean effective black carbon values of 10 ± 7 ng/g and 17 ± 12 ng/g. By contrast, for the three sample sites at around 5000 m for this work, the mean value of $${{\rm{C}}}_{{\rm{BC}}}^{{\rm{equiv}}}$$ was 112 ± 29 ng/g. The mean for northern Chile, corresponding to altitudes of 1800–5300 m, was 72 ± 17 ng/g.

Regional averages of $${{\rm{C}}}_{{\rm{BC}}}^{{\rm{est}}}$$ in the top 5 cm of snow in the Arctic and North America were 27 ng/g and 29 ng/g (ref.33). This is comparable to the regional average near Santiago (28 ng/g; refer to Table 6), but is about twice as large as the regional averages for northern and southern Chile (~14 ng/g). The subsurface snow was clean in the Andes (7 to 15 ng/g) compared to snow in the Arctic and North America (21 and 30 ng/g; ref.33).

In northern Chile, a large fraction of the computed albedo reduction is due to non-BC. When only black carbon is considered, the albedo reduction due to BC is about 43% of that for all light-absorbing impurities (Table 6). This value and the value for snow near Santiago (53%) are roughly comparable to values for the Arctic, western North America, and China (53%, 49 to 78%, and 41%). By contrast, in southern Chile, black carbon accounts for a greater share of the albedo reduction (82%).

Conclusions

Based on snow-sampling measurements made in 2015 and 2016, the black-carbon mass-mixing ratio in snow in the Chilean Andes decreased southward from Putre (18.1°S) to the Elqui Valley above La Serena (29.9°S), after which it increased continuing southward to the Santiago area (33.3°S), then again decreased from the Santiago area south to Osorno (41.1°S).

The contribution of non-BC light-absorbing components to particulate absorption in snow decreased dramatically from north to south, as expected given the transition from the dusty, vegetation-sparse Atacama Desert to the wetter, vegetation-rich south of Chile.

Regional results for Chile are as follows: for surface snow, the average mass mixing ratio of BC was 15 ± 4 ng/g in the north, 28 ± 35 ng/g near Santiago, and 13 ± 3 ng/g in the south. The regional average vertically-integrated loading of BC was 207 ± 46 µg/m2 in the north, 780 ± 510 µg/m2 near Santiago, and 2500 ± 440 µg/m2 in the south, where the snow season was longer and the snow was deeper.

The estimated black carbon mass mixing ratio in surface snow generally increased with time since the last snowfall for a transect from northern to central Chile from 26.5 to 33.5°S, indicating that dry deposition plays a larger role than wet deposition. Sampling on two consecutive days near Santiago indicated a dry deposition rate of 5.2 mg/m2-month, which is roughly comparable to the inter-monsoon dry deposition rate in the Himalayas.

Mean albedo reductions due to light-absorbing impurities in snow in the Chilean Andes are estimated to be 0.0150, 0.0160, and 0.0077 for northern Chile (in June), the region near Santiago (in June), and southern Chile (in July), respectively, for snow grain radii of 100 µm (representative of new snow). Corresponding mean radiative forcings are 2.8, 1.4, and 0.6 W/m2, respectively. In the north, BC plays a smaller role than non-BC; for example, the albedo reduction for 100 µm snow grains due to BC alone is only about 43% of that for all light-absorbing impurities. By comparison, these albedo reductions are 53% and 82% near Santiago and in southern Chile, where a greater share of light absorption is due to BC.

Data Availability

The datasets generated and analyzed during the current study are available from the corresponding author on reasonable request. Correspondence and requests for data should be addressed to P.M.R.

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Acknowledgements

S.N. was supported by NSF grant CHE-1306366, by a Lantz Senior Sabbatical Fellowship from the University of Puget Sound, and by the Fulbright Scholar program. E.S. acknowledges support from the University of Puget Sound. The support of the Consejo Nacional de Ciencias y Tecnología (CONICYT, Preis ANILLO ACT1410, 1161460, and 1151034), and the Universidad de Santiago de Chile (USACH, Preis USA1555) is gratefully acknowledged. We are grateful to Edgardo Sepúlveda, Francisca Quiroz, Juan Rayas, Camilo Guzman, Christian Brahm, Catalina Pino, Pedro Marconi, Nicole Torres, José Jorquera, Marta Caballero, and Andrea Sepúlveda, who participated in snow sampling or filtering, and to Dr. Delia Rodríguez for help with filtering and for the generous use of her lab. Maps of the cold fronts were generously provided by the weather service of Chile (Direccion Meteorologica de Chile).

Author information

Authors

Contributions

S.N. and R.R.C. conceived of the experiment. P.R., S.N., R.R.C., S.G.W., A.P. planned the field experiment in northern and central Chile in 2015. P.R., S.N., R.R.C., G.C., J.C., J.P., S.M., F.L., R.R., N.H. and F.F. participated in planning and coordinating the field experiment in central and southern Chile in 2016. P.R., R.R.C., S.G.W., E.S., A.P, G.C., M.S., F.F., S.M., F.L. and S.N. participated in snow sampling campaigns. P.R., R.R.C., S.G.W., E.S., A.P., M.S., J.P., N.H., S.M. and S.N. participated in snow filtering. A.D. analyzed satellite images to help identify field sites. S.J.D. analyzed filters to determine estimates of light-absorbing impurities. M.S., R.R. and J.C. provided weather information. P.R. wrote the manuscript with input from all authors.

Corresponding author

Correspondence to Penny M. Rowe.

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Rowe, P.M., Cordero, R.R., Warren, S.G. et al. Black carbon and other light-absorbing impurities in snow in the Chilean Andes. Sci Rep 9, 4008 (2019). https://doi.org/10.1038/s41598-019-39312-0

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text:
# Power of a r-cycle

Given $$n \ge 3$$ and $$2 \le r \le n$$, let's define the permutation $$\rho \in S_n$$ (the symmetric group of degree $$n$$) by:

• $$\rho(k)=k+1$$, for $$k \in I_<:=\lbrace 1,...,r-1 \rbrace$$
• $$\rho(r)=1$$
• $$\rho(k)=k$$, for $$k\in I_>:=\lbrace r+1,...,n \rbrace$$

I've thought to prove that $$\rho^r=\iota_{S_n}$$ -the identical permutation- in the following manner (reductio ad absurdum). Could you double check if the proof is correct, please?

Proof. Let's suppose that $$\rho^r \ne \iota_{S_n}$$; then, $$\exists \tilde k \in I:=\lbrace 1,...,n \rbrace$$, such that $$\rho^r(\tilde k)=\rho(\rho^{r-1}(\tilde k)) \ne \tilde k$$. By the very same definition of $$\rho$$, it follows that $$\rho^{r-1}(\tilde k) \in I_<^*:=\lbrace 1,...,r-1 \rbrace \cup \lbrace r \rbrace$$. Now, let's assume that $$\rho^j(\tilde k) \in I_<^*$$ for some $$0 \le j \le r-2$$ (inductive hypothesis); then $$\rho^{j-1}(\tilde k)=\rho^{-1}(\rho^j(\tilde k)) \in I_<^*$$, because the restriction $$\rho_{|I_<^*}$$ is a bijection of $$I_<^*$$ into itself; so, finally, $$\rho^j(\tilde k) \in I_<^*$$ $$\forall j=0,\dots,r-1$$. This means that $$\tilde k+j \in I_<^*$$, $$\forall j=0,\dots,r-1$$, which in its turn implies $$\tilde k=1$$: contradiction, because by definition of $$\rho$$, it is $$\rho^r(1)=1$$.

• "This means that $\tilde k+j \in I_<^*$, $\forall j=0,\dots,r-1$," Why? If you have intermediate claims, it makes sense to delimit them (and their proofs) clearly, particularly when you are trying to formalize the proof. – darij grinberg Nov 20 '18 at 4:04
• @darij grinberg - You're right, that conclusion is undue, and frankly I'm stuck there. Any chance to conclude on that way? – Luca Nov 20 '18 at 14:55
• A quick way is by proving (by induction) that $\rho^i \left(k\right) \in \left\{1,2,\ldots,r\right\}$ and $\rho^i \left(k\right) \equiv k+i \mod r$ for each $k \in \left\{1,2,\ldots,r\right\}$ – darij grinberg Nov 20 '18 at 16:05


text:
## Maximum distance separable codes in the $\rho$ metric over arbitrary alphabets

##### J. Algebr. Comb. 16(1), 71-81 (2002)
DOI: 10.1023/A:1020834531372

### Summary

Summary: We give a bound for codes over an arbitrary alphabet in a non-Hamming metric and define MDS codes as codes meeting this bound. We show that MDS codes are precisely those codes that are uniformly distributed and show that their weight enumerators based on this metric are uniquely determined.

### Keywords/Phrases

MDS codes, uniform distributions


text:
# unwanted white box over the corner of a figure [closed]

* UPDATE *

I tried a bunch of photos and found out that the issue only happens when the image is not in the shape of a square (i.e if the image is rectangular like my case)

I am also using a double column kind of layout for the paper

How do I solve this issue?

* UPDATE *

This is the portion of the code for the figure which I usually used for all figures. However, I have an issue with only one of the figures.

What I get is a white box covering the top right corner through the middle of the box.

I tried changing the place of the figure but with no luck.

I used texstudio, in the preview everything looks good, however when I access the saved PDF file I find that issue.

More details: the image is in PNG. It shouldn't be an issue because I never had this issue with other PNGs (and also with the same image in old compilation).

What could be the issue?

\begin{figure}[ht]
\centering
\includegraphics [scale=0.5]{figurefilename}
\caption{Figure Caption.}
\label{fig:label}
\end{figure}

This is only a portion of the image, I did not screenshot the whole image, just the problematic region.

## closed as unclear what you're asking by Henri Menke, Stefan Pinnow, TeXnician, Maarten Dhondt, JesseMay 17 '17 at 7:31

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• With this fragment of code and don't having the relevant .png file at hand there is not much to say about this. – user31729 May 16 '17 at 21:18
• that's distinctly odd, are you sure the image looks OK if you look at it in an image viewer before including it into latex? without a reproducible test file I think that will be impossible to debug. – David Carlisle May 16 '17 at 21:19
• @DavidCarlisle yes the image is fine. Plus I have used it before and ran and compiled the paper perfectly. Now I got back to the document after a few months and did some modification to the text as per reviewer comments. But I got shocked by this issue. – HappyBee May 16 '17 at 21:23
• @ChristianHupfer I'll try changing the image to see if the issue persists – HappyBee May 16 '17 at 21:24
• I just added an update to the question. It seems that the issue only happens for rectangular images. How do I solve this? – HappyBee May 16 '17 at 21:41


text:
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# Summation of Divergent Series

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In physics one often needs to sum divergent series. Here we give a simple example of how one can estimate the limit of x to infinity of a function f(x) given by its Taylor expansion around x = 0 when its radius of convergence is only 1.

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#### Solution Preview

An unknown function for which we only know the first few terms of a series expansion can often be computed quite accurately for values where the expansion diverges. The approximation can be improved if we have some information about the asymptotic behavior of the function. We'll illustrate this using the following example. A function f(x) has a series expansion around zero given by:

f(x) = x - x^3/3 + x^5/5 - x^7/7 + ... (1)

Of course, given this form we can guess that the nth term is (-1)^(n+1)x^(2n-1)/(2n-1) and that f(x) = arctan(x). In what follows we'll pretend that we don't know anything about f(x) beyond the first 4 terms of the series expansion given in (1), and that the function tends algebraically to a limit at infinity. We want to estimate this limit. This seems an impossible task as the radius of convergence of the series is only 1, for even small x the partial sums start to oscillate wildly. E.g. for x = 3 the partial sums are 3, -6, 42.6, -269.83,..., so it seems doubtful that one could say something sensible about the limit of x to infinity. However, it is possible to improve the convergence using conformal mappings. A practical method to estimate the behavior at infinity works as follows. We apply the conformal transform:

x = p z/(1-z^2) (2)

Where p is a parameter that has to be chosen in some optimal way. The functional form is chosen such that z=0 corresponds to x = 0, allowing us to expand the function f in powers of z. Then z = 1 corresponds to the limit of x to infinity, if this lies inside the radius of convergence, then we can get an accurate estimate. This need not be the case, it is enough if the terms become smaller, the radius of convergence can e.g. be zero for the resummed series to be useful.

Often one takes the denominator to be 1-z, but here we choose this to be 1-z^2 as that is more appropriate for an odd function. It turns out that the optimal way to choose the parameter p ...

#### Solution Summary

Over 1300 words of explained calculations, with a reference, to show how to estimate the limit of a function given by its Taylor expansion with a convergence radius of 1.

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