text: If the player uses mana items a lot, then meteor armor tends to be the best choice for apparel next to Jungle Armor. In his Shinobi uniform, he wears a black cloak with a pointed hood, decorated with a red diamond. Jungle Armor Jungle armor is an armor set which grants bonuses to magical equipment. Right now I have: Meteor makes you have 0 mana usage on space gun while jungle armor increase mana and something else. Each piece is … Clarify? Can you not get magic cuffs in a crimson world? Crystal Helmet Heavy Armor Protection +15 Health +5 Evasion -10 without Heavy armor proficiency Weight: 3.5kg When Metor … Frezarite Armor is crafted using Frezarite.The leggings and boots allows the wearer to walk on water by instantly freezing the water beneath them. Hey guys! I'm in pre hard mode and I've beaten the King Slime and Eye of Cthulu. 1.6.3.0: Item reworked into a vanity helmet. 1.2.6.4: Introduced. Todd of Toddstuff has just released what may be the ultimate test of late medieval plate harness vs. warbows. It can also be combined with your Celestial Magnet. 100 armor → × 0.5 incoming physical damage (50… Jungle armor with its effects. Example: 10% armor penetration causes a target to take physical damage as though they had 10% less armor; 10% armor penetration is equivalent to fighting a target with 10% less armor. In total you need 105 bars for the armor set and the gun. Lethality redirects here. 1.6.1.0: Buffed defense from 4 to 5. Then, it's up to you whether or not you want extra damage and free Space Gun or extra critical chance, mana, and reduced mana usage. I'm doing my first expert mode run this summer, and I had a question about my armor loadout. So wearing multiple pieces of Magnetized armor does not … Notes. When he takes it off, he reveals his shiny, neck-length black hair, which is parted down the middle. But, I just recently acquired jungle armor, which would give me 80 extra mana, 1 extra defense, and a big increase to magic critical strike chance. It's comforting to know I can spam the space gun as much as I want without worrying about running out of mana, because 90% of my deaths as a magic user are because I run out of mana in the middle of a battle. The set bonus of the meteor armor does not affect the mana cost of the Laser Rifle.Be warned! So, jungle or meteor? Else, Jungle might be a tiny bit better. All of the pieces require 30 Ranged and 30 Defence to wear and can be made through the Crafting skill from snakeskins. I've never used jungle armor before, and I'm hesitant to make the switch because I really love the space gun set bonus with meteor. Other than that, I personally like Meteor … Bone Armour shares a cooldown with other Guard skills. Meteorite Armor - It appears to be fortified armor made from unrecognizable meteorites. Armors are equippable items in Rumble Quest that can be obtained in multiple ways. The enchantment level does not stack with each piece of armor. If so, you can get a nice item called Magic Cuffs, which restore mana when you are damaged. Shadow armor is the melee armor at this level, and Meteor armor is for mage characters. Bulb Armor is a craftable armor set crafted from Petals and Jungle Spores.It takes a total of 30 Petals and 18 Jungle Spores to craft a full suit.. Check out the reviews, videos and comparisons for the Oakley vs Under Armour boot reviews and you make the best decision for your feet. This set used to be a rare drop from various Jungle mobs, but as of Patch 1.0.5, it is no longer a drop, and must be crafted. i.e. Yes, damage is generally better. It's the way infantry armor is supported and articulated (or mostly just attached to the mail and harness) that makes the difference. I’m not going to judge a pair of boots based on the laces but I did think that was odd. Jungle armor is a pre-Hardmode armor set consisting of the Jungle Shirt, the Jungle Hat and Jungle Pants.It is crafted from items found in the Underground Jungle.. A full set of Jungle armor grants the wearer 17 / 15 defense and boosts magic weapons, reducing mana usage by 16% and creating a grass particle effect when moving. I've had them before on my other Crimson character, I just don't use them because I've yet to find a band of starpower. For the 2013 Mastery, see Lethality. $\begingroup$ @Rekesoft Weight-wise, you're looking at about 20kg vs 25kg in total, mostly because cavalry could get away with more padding and under-layers. Each individual piece increases maximum mana by … Meteor Helm Heavy Armor Protection +18 Evasion -10 without Heavy armor proficiency Weight: 2.2kg This dark metal helmet seems to be shaped from a single block of meteor stone. Jungle Armor was previously known as Cobalt Armor, but Cobalt was replaced with Jungle Armor to fit the Jungle setting. I'm doing my first expert mode run this summer, and I had a question about my armor loadout. Note: This is ONLY to be used to report spam, advertising, and problematic (harassment, fighting, or rude) posts. Yah.. Better be quick if you want to use it. Meteor Armor vs. Less mana is Really great espially with juingle rose and mana potion combine. As for his clothing, Risho wears black pa… I'm primarily a magic user, I use the space gun and the water bolt, but I also use my boomstick and thorn chakram quite a bit. Meteor armor is good to have it but once you have Cobalt series, it is obsolete. Español - Latinoamérica (Spanish - Latin America). There's enough power to … History. At this point there is no specialized ranged armor. Jungle Armor. I'd go with jungle armor if you've found a really good magic weapon, like Flamelash. All trademarks are property of their respective owners in the US and other countries. I mean damage is fantastic cuz it makes every shot hit like a truck but I like the idea of critting alot (eventho its random) and the less mana useage means I can spam more spells. The Meteor Mask is a craftable Pre-Hardmode vanity helmet. Within the first month my boot laces broke. If you use the Space Gun, I'd say definitely go with Meteor. Armor pieces are either crafted, purchased from NPCs, or dropped from enemies. Yah I read what both do so im basicaly asking whats better, a flat inc in damage a la meteor or the crit and less mana cost form the jungle?? Try getting a diamond robe+wizard hat+meteor leggings, Go for potions then. The armor is crafted from Meteorite Ingots. They give you the buffs too. Armor can also be placed in vanity slots to change the player's appearance without affecting stats. This category contains pages and images related to Ranged armour. In order to find the potential gold cost, fill in the bottom 2 boxes. 25 armor → × 0.8 incoming physical damage (20% reduction, +25% effective health). It takes a total of 46 tanned snakeskins to make a full set. Press J to jump to the feed. Personally, I recommend using jungle armor over meteor armor as the increased mana and critical strike chance affect a lot of the weapons in a much better way than the meteor armor. New comments cannot be posted and votes cannot be cast. Nothing is impossible in this action-packed adventure game. Meteorite Armor is perhaps the easiest armor to craft from the mod. Hey guys! Meteor armor obviously gives you the free blasting with the space gun. The helmet and the chestplate will not. With this, you're far less likely to run out of mana since you get restoration stars from far away and you restore mana when hurt. Even though you get 21% damage from the meteorite set, the critical strikes deal much more damage, and with the increased mana you can keep firing good weapons such as the water bolt and others. Jungle armor gives you more crit, at the cost of having to manage your mana a little more. Jungle Armor's combined -16% Mana usage and +9% crit might not be *quite* as good as Meteor's +21% damage, but they're close, though. Jungle armor is constructed from jungle spores found in the jungle biome. If you want to use many different magic weapons, go for jungle. The Bulb Hood gives 4 defense, increased maximum life by 5, increased maximum mana by 10, and 3% increased magic damage; the Bulb Chestpiece gives 5 defense, increased maximum … Best Terraria build list: cool summoner, melee and ranged builds in … … He also has pronounced eyebags. [Top 10] Terraria Best Armor and How To Get Them | GAMERS … I'm primarily a magic user, I use the space gun and the water bolt, but I also use my boomstick and thorn chakram quite a bit. Do you have a world with corruption? Armor penetration is a champion statistic that reduces the effective armor of the target (vs. the champion with armor penetration). Meteorite Armor is a set of armor added by Falling Meteors. Under Armour Jungle Rat review: I purchased these boots 6 months ago. In order to find the potential power of a weapon, fill in all 3 boxes. The Mod I'm using: /Terraria_Forum/thread/Gameiki_Terraria_Mod-27 EVERY SINGLE 1.2.4 ITEM: /wiki/124 All rights reserved. Details all stats for each move and each Pokémon that can learn it If you want to use one magic weapon, the space gun, and have good defense, go with meteor. It consists of a Bulb Hood, Bulb Chestpiece and Bulb Leggings.. You can craft Meteor armor using the ore you mine, as well as some cool weapons. Snakeskin armour is a type of ranged armour. anyone who had any piece of the Cobalt Armor had it replaced with the Jungle Armor of the same type.. Each piece of armor comes pre-enchanted with Magnetization 1. Maybe you're in the wrong place. Meteor BEM (隕石ベム, Inseki Bemu) (40-41) 1 Character History 2 Personality 3 Arsenal 4 Behind the Scenes 4.1 Portrayal 5 Notes 6 References Created by Radiguet's Bio Dimensional Bug fusing in a meteor made of an anti-Birdronic metal, this monster weakens the Jetmen with his very presence and nearly kill them were it not for the sudden appearance of Neo-Jetmen. Good luck! Armor pieces are equipped by placing them in the armor slots of the player's inventory. Although his nose is relatively small, it is slightly pointy in the middle; in the manga his nose is longer, making him resemble a puppet. He is a rather thin and tall young man, with a well built posture. Incoming physical damage is multiplied by a factor based on the target's armor stat:$\pagecolor{Black}\color{White}{\rm Damage\ multiplier}=\begin{cases}{100 \over 100+{\it Armor}}, & {\rm if\ }{\it Armor} \geq 0\\2 - {100 \over 100 - {\it Armor}}, & {\rm otherwise}\end{cases}$Examples: 1. Press question mark to learn the rest of the keyboard shortcuts. And any other suggestions for armor, accessories or weapons are welcome. Also defeating Eater of World drops one piece of Shadow armor, I don't think you need to make them unless you want specific piece. I've been using my meteorite set, which gives me more magic damage overall, and makes the space gun free to use. … Meteor Beam Attack & Move listings for Pokémon Sword & Shield. Dig, fight, explore, build! Basically, warbow arrows sail through riveted mail and backing fiber armor. I'm not sure what you mean. Also, my accessories are spectre boots, arcane lava charm, guarding shield of Cthulu, band of regeneration, and celestial magnet. Mid Prehardmode (Jungle and Dungeon) Armor was designed for comfort first (even more so than today), specifically to allow for spending … © Valve Corporation. The effect takes the highest enchantment level available from any piece of armor. ; A feature worth noting is that the suit in its entirety resembles a Space Suit, likely due to the fact that it is made from a space-related material. To add an article, image, or category to this category, append [[Category:Ranged armour]] to the end of the page. Armor is a set of equipable defense items that reduce damage taken from enemies and most other damage sources. Since you say you are not using that much mana items it might not be for you (silver has 3 more armor then meteor) but once you do need plenty of mana it's 2nd best with the jungle armor in front. Thanks! This calculator calculates the potential power (Spell Power, Stamina, Physical Power) of Armor and the cost for it be upgraded. 2. Bone Armour is an active skill granted by Bone Barrier, a Notable Ascendancy passive skill. Each piece, when crafted, will also allow the player to right click with the armor piece in hand to create an enchant on that particular armor piece: The Oakley SI Assault boot is lightweight but rugged enough for tough terrain and military operations. Results in more total damage done, a more stable damage output, and yes, bigger crits when you do crit. The results are not too surprising based on what we already know. Risho is pale and has narrow, brown eyes. I work in an indoor environment walking on cement floors for 12+ hours a day, 4 days a week. Jungle armor increases Mana usage more so Jungle Armor is more useful. Additionally, it's got +60 Mana. The world is your canvas and the ground itself is your paint. Each piece is crafted from Meteor Chipsusing normal armor crafting patterns. Oakley SI Assault boot review by Lucas Ill keep these boots forever. I'm in pre hard mode and I've beaten the King Slime and Eye of Cthulu.


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# Working with numpy-like arrays¶

Warning

This feature should be considered experimental. Please report any bug you may find on xarray’s github repository.

Numpy-like arrays (duck array) extend the numpy.ndarray with additional features, like propagating physical units or a different layout in memory.

DataArray and Dataset objects can wrap these duck arrays, as long as they satisfy certain conditions (see Integrating with duck arrays).

Note

For dask support see Parallel computing with Dask.

## Missing features¶

Most of the API does support duck array objects, but there are a few areas where the code will still cast to numpy arrays:

## Extensions using duck arrays¶

Here’s a list of libraries extending xarray to make working with wrapped duck arrays easier:

format: plain


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Does a multiclassed rogue/barbarian get sneak attack almost always? [duplicate]

The rogue gets a sneak attack on attacks where you have advantage.

Barbarians have advantage on attacks using strength with the reckless attack feature, which doesn't seem to have a limit on usage.

Does that mean I would be able to use the sneak attack on every attack using strength?


text:
A man is walking at a speed of 12 km/h. After every km, he takes rest for 3 minutes. How much time will he take to cover a distance of 6 km?

Free Practice With Testbook Mock Tests

Options:

1. 42 minutes

2. 40 minutes

3. 48 minutes

4. 45 minutes

Correct Answer: Option 4 (Solution Below)

This question was previously asked in

SSC CHSL Previous Paper 110 (Held On: 21 Oct 2020 Shift 3)

Solution:

Given:

Speed = 12 km/hr.

And after every km, A takes rest for 3 minutes.

Formula used:

Time = Distance/Speed

Concept used:

Time taken to cover the last 1 km distance will not include the rest time as the entire distance has been covered.

Calculation:

According to the question:

Time taken to cover 1 km distance = Distance/Speed

⇒ Time = 1/12 hr. = (1/12 × 60) = 5 min.

Including rest time, time taken to cover 1 km distance = (5 + 3)min = 8 min.

Time taken to cover 5 km distance = (5 × 8) min. = 40 min.

And time taken to cover 6 km distance = (40 +1 × 5) min.

⇒ 40 + 5 = 45 min.

Time A will take to cover a distance of 6 km is 45 minutes.


text:
American Institute of Mathematical Sciences

February & March  2004, 11(2&3): 615-619. doi: 10.3934/dcds.2004.11.615

A note on periodic orbits for singular-hyperbolic flows

 1 Instituto de Matemàtica, Universidade Federal do Rio de Janeiro, C. P. 68530, CEP 21945-970, Rio de Janeiro, Brazil

Received  May 2003 Revised  February 2004 Published  June 2004

A singular-hyperbolic set for flows is a partially hyperbolic set with singularities (hyperbolic ones) and volume expanding central direction [7]. Several properties of hyperbolic systems have been conjectured for the singular-hyperbolic sets [8, p. 335]. Related to these conjectures we shall prove the existence of transitive, isolated, singular-hyperbolic set without periodic orbits on any $3$-manifold. In particular, the periodic orbits are not necessarily dense in the limit set of a isolated singular-hyperbolic set.
Citation: Carlos Arnoldo Morales. A note on periodic orbits for singular-hyperbolic flows. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 615-619. doi: 10.3934/dcds.2004.11.615
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2018 Impact Factor: 1.143


text:
# Fractional Derivatives Of Sums

I have a question regarding the definition of a fractional derivative. I've searched, but I can't find a definition of fractional derivatives that explain the concept in terms of an operator on some function. I guess what I'm trying to ask is the following:

Let f(z) be an analytic function. Let $_0D_z^v(f(z))$ denote the $v^{th}$ fractional derivative of $f$ in the Riemann-Liouville sense. Let $D_z^v(f(z))$ denote the $v^{th}$ fractional derivative of $f$ in the Weyl sense. Assume $f(z)$ can be written as $f(z)=g(z)+h(z)$, where $g$ and $h$ are analytic (NOTE: one of $g$ or $h$ could be the zero function). Let $D^v$ be the operator defined by $D^v(f(z))= _0D_z^v(g(z)) + D_z^v(h(z))$, whenever this makes sense i.e. when each part is defined.

Could one say that $D^v(f(z))$ here is a fractional derivative in the sense that the operator satisfies:

1. This operator is linear
2. The operator preserves analytic properties, i.e. the result of applying the operator is again analytic.
3. When $v$ is a positive integer, the operator corresponds to the $n^{th}$ derivative of $f$ with respect to $z$. Furthermore, when $n$ is a negative integer the operator corresponds to the $n$ fold integral of $f$.
4. When $v=0$ applying $D^v$ leaves $f(z)$ unchanged.
5. $D^{v}D^r(f(z))= D^{v+r}(f(z))$.

I believe at first glance (unless I'm making a mistake) that each property described above is satisfied by virtue that the Riemann-Liouville and Weyl fractional derivatives satisfy these properties as well.

I guess the above definition could give two different values (by switching the order of $g$ and $h$) but there are multiple ways of defining a fractional derivative as well. I don't think this is necessarily a problem. Could be wrong though.

I hope this question makes sense. Let me know if I need to be clearer about something.

Rick

P.S. I'm not sure that "operator" is the correct wording, but hopefully you understand what I mean by the above definition anyway.

Edit:

$D^v$ is not well-defined. Since I'm having an issue with a specific function, let $g(z)=1$ and $h(z) = f(z)-1$.

To be more specific, I'm trying to take the fractional derivative of:

$f(x) = 1 + n^{-x}$ where $n$ is an integer and $n\geq 2$.

I need the fractional derivative of $f(x)$ defined above when $0<v<1$.

In a perfect world, I'd really like for the derivative of $1$ to SLOWLY go to $0$ as $v$ gets closer to 1 and the derivative of the second piece to go to $\log(n)n^{-x}$ as $v$ gets close to 1. Just to suit my specific needs. But, I don't know if I can.

I really like the Riemann-Liouville fractional derivative of the constant term and the Weyl fractional derivative of the second piece, but it seems like I can't take the Weyl fractional derivative of $1$ and the Riemann-Liouville fractional derivative of $n^{-x}$ is "nasty" for my specific goal/desire.

• It looks like $D^v$ is not well-defined: given $f$, how do you determing $g$ and $h$? Feb 25 '14 at 10:10
• Ben, thanks for the response. See the edited question. Feb 26 '14 at 16:33


text:
## Oral 3: Learning from distribution shift

Moderators : Dustin Tran · Zsolt Kira

Wed 27 Apr 9 a.m. PDT — 10:30 a.m. PDT

Abstract:

Chat is not available.

Wed 27 April 9:00 - 9:15 PDT

(Oral)
##### Fine-Tuning can Distort Pretrained Features and Underperform Out-of-Distribution

Ananya Kumar · Aditi Raghunathan · Robbie Jones · Tengyu Ma · Percy Liang

When transferring a pretrained model to a downstream task, two popular methods are full fine-tuning (updating all the model parameters) and linear probing (updating only the last linear layer---the "head"). It is well known that fine-tuning leads to better accuracy in-distribution (ID). However, in this paper, we find that fine-tuning can achieve worse accuracy than linear probing out-of-distribution (OOD) when the pretrained features are good and the distribution shift is large. On 10 distribution shift datasets (BREEDS-Living17, BREEDS-Entity30, DomainNet, CIFAR $\to$ STL, CIFAR-10.1, FMoW, ImageNetV2, ImageNet-R, ImageNet-A, ImageNet-Sketch), fine-tuning obtains on average 2% higher accuracy ID but 7% lower accuracy OOD than linear probing. We show theoretically that this tradeoff between ID and OOD accuracy arises even in a simple setting: fine-tuning overparameterized two-layer linear networks. We prove that the OOD error of fine-tuning is high when we initialize with a fixed or random head---this is because while fine-tuning learns the head, the lower layers of the neural network change simultaneously and distort the pretrained features. Our analysis suggests that the easy two-step strategy of linear probing then full fine-tuning (LP-FT), sometimes used as a fine-tuning heuristic, combines the benefits of both fine-tuning and linear probing. Empirically, LP-FT outperforms both fine-tuning and linear probing on the above datasets (1% better ID, 10% better OOD than full fine-tuning).

Wed 27 April 9:15 - 9:30 PDT

(Oral)
##### Asymmetry Learning for Counterfactually-invariant Classification in OOD Tasks

S Chandra Mouli · Bruno Ribeiro

Generalizing from observed to new related environments (out-of-distribution) is central to the reliability of classifiers. However, most classifiers fail to predict label $Y$ from input $X$ when the change in environment is due a (stochastic) input transformation $T^\text{te} \circ X'$ not observed in training, as in training we observe $T^\text{tr} \circ X'$, where $X'$ is a hidden variable. This work argues that when the transformations in train $T^\text{tr}$ and test $T^\text{te}$ are (arbitrary) symmetry transformations induced by a collection of known $m$ equivalence relations, the task of finding a robust OOD classifier can be defined as finding the simplest causal model that defines a causal connection between the target labels and the symmetry transformations that are associated with label changes. We then propose a new learning paradigm, asymmetry learning, that identifies which symmetries the classifier must break in order to correctly predict $Y$ in both train and test. Asymmetry learning performs a causal model search that, under certain identifiability conditions, finds classifiers that perform equally well in-distribution and out-of-distribution. Finally, we show how to learn counterfactually-invariant representations with asymmetry learning in two physics tasks.

Wed 27 April 9:30 - 9:45 PDT

(Oral)
##### A Fine-Grained Analysis on Distribution Shift

Olivia Wiles · Sven Gowal · Florian Stimberg · Sylvestre-Alvise Rebuffi · Ira Ktena · Krishnamurthy Dvijotham · Ali Taylan Cemgil

Robustness to distribution shifts is critical for deploying machine learning models in the real world. Despite this necessity, there has been little work in defining the underlying mechanisms that cause these shifts and evaluating the robustness of algorithms across multiple, different distribution shifts. To this end, we introduce a framework that enables fine-grained analysis of various distribution shifts. We provide a holistic analysis of current state-of-the-art methods by evaluating 19 distinct methods grouped into five categories across both synthetic and real-world datasets. Overall, we train more than 85K models. Our experimental framework can be easily extended to include new methods, shifts, and datasets. We find, unlike previous work (Gulrajani & Lopez-Paz, 2021), that progress has been made over a standard ERM baseline; in particular, pretraining and augmentations (learned or heuristic) offer large gains in many cases. However, the best methods are not consistent over different datasets and shifts. We will open source our experimental framework, allowing future work to evaluate new methods over multiple shifts to obtain a more complete picture of a method's effectiveness. Code is available at github.com/deepmind/distributionshiftframework.

Wed 27 April 9:45 - 10:00 PDT

(Oral)
##### Sparse Communication via Mixed Distributions

António Farinhas · Wilker Aziz · Vlad Niculae · Andre Martins

Neural networks and other machine learning models compute continuous representations, while humans communicate mostly through discrete symbols. Reconciling these two forms of communication is desirable for generating human-readable interpretations or learning discrete latent variable models, while maintaining end-to-end differentiability. Some existing approaches (such as the Gumbel-Softmax transformation) build continuous relaxations that are discrete approximations in the zero-temperature limit, while others (such as sparsemax transformations and the Hard Concrete distribution) produce discrete/continuous hybrids. In this paper, we build rigorous theoretical foundations for these hybrids, which we call "mixed random variables.'' Our starting point is a new "direct sum'' base measure defined on the face lattice of the probability simplex. From this measure, we introduce new entropy and Kullback-Leibler divergence functions that subsume the discrete and differential cases and have interpretations in terms of code optimality. Our framework suggests two strategies for representing and sampling mixed random variables, an extrinsic ("sample-and-project'’) and an intrinsic one (based on face stratification). We experiment with both approaches on an emergent communication benchmark and on modeling MNIST and Fashion-MNIST data with variational auto-encoders with mixed latent variables.

Wed 27 April 10:00 - 10:15 PDT

(Oral)
##### Frame Averaging for Invariant and Equivariant Network Design

Omri Puny · Matan Atzmon · Edward Smith · Ishan Misra · Aditya Grover · Heli Ben-Hamu · Yaron Lipman

Many machine learning tasks involve learning functions that are known to be invariant or equivariant to certain symmetries of the input data. However, it is often challenging to design neural network architectures that respect these symmetries while being expressive and computationally efficient. For example, Euclidean motion invariant/equivariant graph or point cloud neural networks. We introduce Frame Averaging (FA), a highly general purpose and systematic framework for adapting known (backbone) architectures to become invariant or equivariant to new symmetry types. Our framework builds on the well known group averaging operator that guarantees invariance or equivariance but is intractable. In contrast, we observe that for many important classes of symmetries, this operator can be replaced with an averaging operator over a small subset of the group elements, called a frame. We show that averaging over a frame guarantees exact invariance or equivariance while often being much simpler to compute than averaging over the entire group. Furthermore, we prove that FA-based models have maximal expressive power in a broad setting and in general preserve the expressive power of their backbone architectures. Using frame averaging, we propose a new class of universal Graph Neural Networks (GNNs), universal Euclidean motion invariant point cloud networks, and Euclidean motion invariant Message Passing (MP) GNNs. We demonstrate the practical effectiveness of FA on several applications including point cloud normal estimation, beyond $2$-WL graph separation, and $n$-body dynamics prediction, achieving state-of-the-art results in all of these benchmarks.

Wed 27 April 10:15 - 10:30 PDT

(Oral)
##### F8Net: Fixed-Point 8-bit Only Multiplication for Network Quantization

Qing Jin · Jian Ren · Richard Zhuang · Sumant Hanumante · Zhengang Li · Zhiyu Chen · Yanzhi Wang · Kaiyuan Yang · Sergey Tulyakov

Neural network quantization is a promising compression technique to reduce memory footprint and save energy consumption, potentially leading to real-time inference. However, there is a performance gap between quantized and full-precision models. To reduce it, existing quantization approaches require high-precision INT32 or full-precision multiplication during inference for scaling or dequantization. This introduces a noticeable cost in terms of memory, speed, and required energy. To tackle these issues, we present F8Net, a novel quantization framework consisting in only ﬁxed-point 8-bit multiplication. To derive our method, we ﬁrst discuss the advantages of ﬁxed-point multiplication with different formats of ﬁxed-point numbers and study the statistical behavior of the associated ﬁxed-point numbers. Second, based on the statistical and algorithmic analysis, we apply different ﬁxed-point formats for weights and activations of different layers. We introduce a novel algorithm to automatically determine the right format for each layer during training. Third, we analyze a previous quantization algorithm—parameterized clipping activation (PACT)—and reformulate it using ﬁxed-point arithmetic. Finally, we unify the recently proposed method for quantization ﬁne-tuning and our ﬁxed-point approach to show the potential of our method. We verify F8Net on ImageNet for MobileNet V1/V2 and ResNet18/50. Our approach achieves comparable and better performance, when compared not only to existing quantization techniques with INT32 multiplication or ﬂoating point arithmetic, but also to the full-precision counterparts, achieving state-of-the-art performance.


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# Dualizing cartesian and cocartesian fibrations

## Clark Barwick, Saul Glasman and Denis Nardin

In this technical note, we proffer a very explicit construction of the dual cocartesian fibration of a cartesian fibration, and we show they are classified by the same functor to the $\infty$-category of $\infty$-categories.

Keywords: cocartesian fibrations, cartesian fibrations, quasicategories

2010 MSC: 18D30

Theory and Applications of Categories, Vol. 33, 2018, No. 4, pp 67-94.

Published 2018-01-08.

TAC Home


text:
# Difference between eigendecomposition and singular value decomposition for Hermitian matrices

Let consider the following Hermitian matrix

T =[   1.0000            -0.1000 - 0.0600i   0.3000 - 0.0300i
-0.1000 + 0.0600i   1.0000            -0.0400 + 0.0800i
0.3000 + 0.0300i  -0.0400 - 0.0800i   1.5000            ]

The SVD decomposition using matlab is [U,S,V] = svd(T)

U =

-0.4311 + 0.0000i   0.3253 - 0.0000i   0.8416 + 0.0000i
0.1302 - 0.1375i  -0.8697 + 0.1658i   0.4028 - 0.1345i
-0.8757 - 0.1074i  -0.2973 - 0.1484i  -0.3336 + 0.0024i

S =

1.0e+003 *

1.6662         0         0
0    1.0101         0
0         0    0.8237

V =

-0.4311             0.3253             0.8416
0.1302 - 0.1375i  -0.8697 + 0.1658i   0.4028 - 0.1345i
-0.8757 - 0.1074i  -0.2973 - 0.1484i  -0.3336 + 0.0024i

The eigendecomposition is [F,D] = eig(T_t)

F =

-0.8416 - 0.0059i  -0.2911 + 0.1453i   0.4279 - 0.0525i
-0.4037 + 0.1317i   0.7040 - 0.5368i  -0.1125 + 0.1524i
0.3336             0.3323             0.8822

D =

1.0e+003 *

0.8237         0         0
0    1.0101         0
0         0    1.6662

Theoretically, for Hermitian matrices both decompositions are equivalent. However, we observe that both decompositions give the same eigenvalues but the eigenvectors are different. Which decomposition is accurate?

• Welcome to SciComp! A tip: If you indent lines by 4 spaces, they'll be marked as a code sample. You can also highlight the code and click the "code" button (with "{}" on it). There's even syntax highlighting! – Christian Clason Jan 16 '15 at 11:52
• What is T_t in this context? Did you apply eig() to the right matrix? – Bill Barth Jan 16 '15 at 13:05
• "Theoretically, for Hermitian matrices both decompositions are equivalent". Actually, no. T may have negative eigenvalues, while its singular values must be nonnegative. So you need to allow for some sign flips at the very least. – Federico Poloni Jan 16 '15 at 15:55
• To add, one might consider the matrix, $\begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}$. – Nick Alger Jan 16 '15 at 16:52

If you entrywise divide the third column of $F$ by the first column of either $U$ or $V$, you will find that the two are just scaled versions of each other. Matlab returns eigenvectors to be unit normed, but they can be scaled arbitrarily by a complex number of modulus 1.


text:
Simple model of a bouncing ball

the nature of infinity

Thinking before calculating

The best units for a given problem

Trajectories in the Schwarzschild space-time

Using the Riemann package


text:
Announcements! ( See All )

# Material for the Midterm

### Prob140 Fall 2019

The midterm is during the lecture time on Thursday October 10. This is a summary of the material for the exam, grouped by main topic. The general techniques are in the sections Probability, Distribution, and Expectation. The next two sections consist of applications.

The labs relevant to the midterm are listed below by topic. Please also review all your homework, quizzes, and all the practice problems in the weekly Prep Guides.

Homework 6, due Tuesday October 8, consists of last semester’s midterm.

### Probability

• Chapter 1: Spaces, events, basic counting, exponential approximation
• Chapter 2: Addition and multiplication rules, conditioning and Bayes’ rule
• Chapter 5: Unions and intersections of several events
• Section 9.1: Probabilities by conditioning and recursion

### Distribution

• Chapter 3: Random variables, equality versus equality in distribution
• Chapter 4: Joint, marginals, conditionals, independence
• Section 5.4: Random permutations and symmetry

### Expectation

• Chapter 8: The main properties, including additivity, the method of indicators, and expectations of functions, and tail sum formula for the expectation of a non-negative integer valued variable
• Lab 3: Application of tail sum formula
• Sections 9.2, 9.3: Expectation by conditioning
• Lab 4: Waiting times till patterns: application of expectation by conditioning

### Random Counts

• Section 8.1: Bernoulli
• Section 8.1: Uniform on a, a+1, … , b
• Sections 6.1, 6.2, 6.4, 7.2: Binomial and multinomial
• Sections 5.4, 6.3, Lab 1: Hypergeometric
• Section 6.5, Chapter 7: Poisson
• Sections 8.1, 9.3: Geometric

### Markov Chains

• Sections 10.1, 10.2: Terminology and basics
• Sections 10.3, 10.4: The steady state distribution and its properties
• Section 11.1, 11.2: Detailed balance
• Sections 11.3, 11.4: Code Breaking and MCMC


text:
# On Relative Links and a Simplified URL structure

+ 0 like - 0 dislike
124 views

I realised a few days back, that while using the most simplified URL structure, /123/why-do-birds-sing, relative links don't work! The simplest URL structure for which it works is /?qa=123/why-do-birds-sing, Here is the exact description of the problem that I sent to polarkernel just today.

Or if I click on a link to say, /4606/can-we-install-comment-rating, then I am instead directed to /4636/4606/can-we-install-comment-rating. [...]

One solution is that just before we go online, public, we make all the relative links absolute.

However, this would be very time consuming. Is there a way to probably edit the htaccess file itself, so that the relative links are still correct?

I have turned the simplest URL structure on for a while just so that you understand what I mean. Here are a few links:

4606

./4606

/4606

retagged Mar 16, 2014

+ 2 like - 0 dislike

This is a cautionary tale on the fact, that on a system which is that complex, changes should only be made if well understood and tested.

If we look at the source code of a page using this /123/why-do-birds-sing URL structure, we see that all relative links created by Q2A are now written as ../link, using a double-dot slash.

In order to rescue this situation, I have made a patch (fortunately possible within my OverSnow theme) that corrects all relative links in questions, answers and comments, so that the following types of links work correctly:

This should repair almost everything. Links made by hand by us administrators, like those in the upper side-panel, have to be absolute. We will have to think on that if we move to go online.

answered Mar 16, 2014 by (0 points)

Thanks! That greatly simplifies the work needed to do, and definitely saves the situation : )

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOverflo$\varnothing$Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.


text:
# Weak Algebraic Monge Arrays

1 ARCHI - Architecture
INRIA Rocquencourt
Abstract : An $n\times n$ matrix $C$ is called a {\em weak Monge\/} matrix iff $c_{ii}+c_{rs}\le c_{is}+c_{ri}$ for all $1\le i\le r,s\le n$. It is well known that the classical linear assignment problem is optimally solved by the identity permutation if the underlying cost-matrix fulfills the weak Monge property. In this paper we introduce higher dimensional weak Monge arrays and prove that higher dimensional axial assignment problems can be solved efficiently if the cost-structure is a higher dimensional weak Monge array. Additionally, the concept of weak Monge arrays is related to other Monge properties and extended in an algebraic framework. Finally, the problem of testing whether or not a given array can be permuted to become a weak Monge array is solved.
Keywords :
Type de document :
Rapport
[Research Report] RR-2501, INRIA. 1995
Domaine :

Littérature citée [2 références]

https://hal.inria.fr/inria-00074176
Contributeur : Rapport de Recherche Inria <>
Soumis le : mercredi 24 mai 2006 - 14:40:54
Dernière modification le : vendredi 16 septembre 2016 - 15:13:22
Document(s) archivé(s) le : mardi 12 avril 2011 - 15:41:41

### Identifiants

• HAL Id : inria-00074176, version 1

### Citation

Rüdiger Rudolf, Dominique Fortin. Weak Algebraic Monge Arrays. [Research Report] RR-2501, INRIA. 1995. 〈inria-00074176〉

### Métriques

Consultations de la notice

## 122

Téléchargements de fichiers


text:
In a certain sequence of 8 numbers, each number after the first is | Filo

Class 11

Math

Co-ordinate Geometry

Straight Lines

583
150

In a certain sequence of 8 numbers, each number after the first is 1 more than the previous number .If the first number is -5, how many of the numbers in the sequence are positive?

1. None
2. One
3. Two
4. Three
5. Four


text:
Contracting the Riemann tensor issues, p540 hobson

I am stuck trying to work through something on p540 in Hobson (General Relativity: An Introduction for Physicists), one is supposed to use the variation of the full Riemann tensor and then contract it to get $\delta R_{\mbox{ }\mu \nu }$: $$\delta R^{\sigma}_{\mbox{ }\mu \nu \rho} = \delta\partial_{\nu}\Gamma^{\sigma}_{\mbox{ }\mu\rho}-\delta\partial_{\rho}\Gamma^{\sigma}_{\mbox{ }\mu\nu} +\delta\Gamma^{\sigma}_{\mbox{ }\tau\nu}\Gamma^{\tau}_{\mbox{ }\rho\mu}-\delta\Gamma^{\sigma}_{\mbox{ }\tau\rho}\Gamma^{\tau}_{\mbox{ }\nu\mu}=\delta\partial_{\nu}\Gamma^{\sigma}_{\mbox{ }\mu\rho}-\delta\partial_{\rho}\Gamma^{\sigma}_{\mbox{ }\mu\nu}$$ and then contract it to show $$\delta R_{\mbox{ }\mu \nu }=[ \partial_{\nu}(\delta \Gamma^{\sigma}_{\mbox{ }\mu \sigma}) - \partial_{\sigma}(\delta \Gamma^{\sigma}_{\mbox{ }\mu \nu}) ]$$

MY ATTEMPT:

I thought to contract it by multiplying by $g_{\sigma \alpha}g^{\alpha \rho}$ $$g_{\sigma \alpha}g^{\alpha \rho}\delta R^{\sigma}_{\mbox{ }\mu \nu \rho}=\delta R_{\mbox{ }\mu \nu }$$ but then I get $$g_{\sigma \alpha}g^{\alpha \rho}(\delta\partial_{\nu}\Gamma^{\sigma}_{\mbox{ }\mu\rho}-\delta\partial_{\rho}\Gamma^{\sigma}_{\mbox{ }\mu\nu})=(\delta \partial_{\nu}\Gamma_{\mu\rho}-\delta \partial_{\rho}\Gamma^{\rho}_{\mbox{ }\mu\nu})$$ Can anyone help me get the correct answer (4th line), I may have made a mistake or could be completely wrong I am not sure.

• It seems like your penultimate Christoffel is missing an index, i.e. $\Gamma_{\mu \rho}$ needs to be $\Gamma^{\rho}{}_{\mu \rho}.$ In that case you're done, you have the correct answer! Apr 7 '13 at 22:59
• Looks like en.wikipedia.org/wiki/… is just what you want.
– user21433
Apr 8 '13 at 1:05
• If this is homework, please use the homework tag.
– user4552
May 8 '13 at 4:52
• it is not but i have done it now anyway
– user21119
May 9 '13 at 8:01

Apart from the missing index, which was already mentioned, you are fine. The partial derivative commutes with the variation, i.e. $$\partial_\rho(\delta{\Gamma^{\rho}}_{\mu\nu})=\delta\partial_\rho{\Gamma^{\rho}}_{\mu\nu}.$$
• surely though the $g^{\alpha \rho}$ takes away this index? If not thewn is $g_{\sigma\alpha}g^{\alpha rho} \Gamma^{\sigma}_{\mbox{ }\mu \rho}=\Gamma^{\rho}_{\mbox{ }\mu \rho}$


text: The Mean Time Before Failure (MTBF) number is calculated as follows: MTBF = 1,000,000,000 X 1/FR where FR = … FIT and MTTF / MTBF (4): What is FIT/Failure Rate? FIT and MTTF / MTBF (7): How does TDK calculate Failure Rate of MLCCs (Multilayer Ceramic Capacitors)? On the other hand, MTBF of repairable systems is the predicted value … These numbers are in no way a guarantee that the part will last for the stated amount of time. MTBF and the so called Failure Rate have a reciprocal relationship: MTBF = 1/Failure Rate, and Failure Rate = 1/ MTBF. FIT and MTTF / MTBF (1): What is MTBF (Mean Time Between Failures)? Mean time to failure sounds a lot like mean time between failure (MTBF), but they’re not the same. (E.g. Multilayer Ceramic Chip Capacitors. MTBF, and Annualized Failure Rate. These types of components often have a reliability metric quoted in terms of L10% life, the point in time where 10% of a population of items is expected to fail. Example. Failure In Time (FIT) is a Unit for expressing the expected failure rate of connectors, semiconductors and other electronic devices. FIT and MTTF / MTBF (2): What is the difference between MTBF and MTTF? Q. Reliability FIT-rate We are going to use a PIC16F873 in a design were we need a reliability prediction calculation. Both the FIT and MTBF (or MTTF) are shown. FIT and MTTF / MTBF (5): What are the failure periods? In other words, The Mean Time Between Failure (MTBF), or for components, the Mean Time To Failure (MTTF) is the distribution for a … FIT and MTTF / MTBF (3): What types of MTBF figures can be obtained from manufacturers? However, it is important to take into account that a capacitor does not function by itself. This is not solely my observation – the US Army has cancelled its use of part count method MTBF predictions (i.e. Is the a FIT value of the A/D-converter, the program memory and the IN/Outputs … Application Specific IC (ASIC) Development and Supply, 6,918 yrs x 8,760 hrs/yr = 60,601,680 hrs. To find the MTBF you would do the following calculations: MTBF = 1/(16.5 x 70) = 0.0008658008658 common method is to calculate the probability of failureor Rate of Failure (λ). One FIT equals one failure and is statistically projected from the results of accelerated test procedures. It is easy to convert between Mean Time Between Failures (MTBF), Failures In Time (FIT), and Parts Per Million (PPM) rates. The Failure In Time (FIT) number for the CY7C9689A device is 34 which is mentioned in the Qualification report. For our purposes, MTBF is typically expressed as a number of hours, years or some other measure of time. MTBF = 1 / Σcomponent FIT Rates For simplicity, the composite failure rates of subassemblies like power supplies are used rather than the sum of their individual component FIT rates. All of these capacitors have a FIT number of 16.5 (same as previous example). I find this task distasteful because I have never seen any indication that MTBF predictions are correlated in any way with field failure rates. When the layperson hears that a device has an MTBF of 10,000h, they often think that this means that To calculate the MTBF for the whole system, take the inverse of the sum of all the FIT figures. The calculators below convert between fail fraction and average fail rate FIT given an operation time. Q. Used for calculating MTBF and presenting the reliability curve for various distributions. Thus, the probability that a howitzer with a MTBF of 62.5 hours will fail an 18 hour mission is: $$F(18) = 1 – e^{\frac{-18}{62.5}} = .25$$ Conversely, the system has a 75% chance of completing the mission successfully. Both of these terms MTBF(Mean Time Between Failure) and MTTF … They are often misinterpreted and misused. The MTBF results from the inverse FIT or λ values. Failure rates are identified by means of life testing experiments and experience from the field. TI’s MTBF/FIT estimator search tool allows you to find generic data based on technology groupings to estimate these typical questions and shows conditions under which the rates were derived. To find the MTBF you would do the following calculations: For 70 capacitors in the circuit, the MTBF of all of them combine has dropped significantly. Q. MTTF and MTBF. Note the value in FIT that you wish to convert to MTBF. A MTBF of over 50 years can sound really impressive until you realize it means almost 20% failure in 10 years (MTBF is not the minimum time to failure). The formula for calculating the MTBF is = T/R. FIT and MTTF / MTBF (2): What is the difference between MTBF and MTTF? DPM in the traditional sense does not convert to MTBF… DPM is defects/defectives per million opportunities…. While MTBF seems to be more intuitive, it is quite difficult to handle in calculations, because MTBF is not an additive metric. To distinguish between the two, the concept of suspensions must first be … I just had another meeting where folks thought that specifications for Annualized Failure Rate (AFR), failure rate (λ), and Mean Time Between Failures (MTBF) were three different things – folks, they are mathematically equivalent. FIT and MTTF / MTBF (6): What factors affect the expected life and failure rates of MLCCs? FIT and MTTF / MTBF (5): What are the failure periods? Such control power supplies may have a FIT rate of 1,000x10-9 hours. These numbers are mainly used to calculate the probability a system will remain in working condition for a given time frame. Used for converting MTBF to FPMH, FIT, and L10. It is part of a circuit that performs some function for a larger system. FIT rate … The MTBF value (= Mean Time Between Failure) is defined as the time between two errors of an assembly or device. Q.FIT and MTTF / MTBF (8): How do users use MTBF and FIT values? Failures in Time, or ‘FIT’ rates are defined in ‘Failures per Billion Hours’ (FPBH). All rights reserved. Also, notice that the MTBF is now more reasonable, 865,800 hours is roughly 98 years. FIT and MTTF / MTBF (6): What factors affect the expected life and failure rates of MLCCs? EXAMPLE of MTTF calculator and MTBF calculator: INPUTS: Number of devices under test= 30, Duration of the test in Hours= 100 , Number of failures reported= 3 OUTPUTS: MTBF = 33.33 Hours/failure, MTTF= 3.33 hours/device MTBF Formula | MTTF formula. If the MTBF is known, one can calculate the failure rate as the inverse of the MTBF. The probability of the car not failing in 10 years would now be e-(10/593) = 0.983, or 98.3%. ÉÏZrÄó 0}(í}ã³óGýõoî=ý-À,4ª õÍ¯*3»/§Bëæ²ìïl32úé¿ÞUM¬RmUÚ\AÃ%Ú»VdC¿î)>$ÃçÛ4iæýÉçW/³ÝÊN±èe7D°ÕÖËPpWÙ®³ÝtåÚ±ðÆ¸+:öü§QXõ¨ÒQ®ø: ¡3N;*c°Q 4Åj\ÄLÂ \A2pYªqT ÝþÍÀñì§«XiÄ»#Æw8ZRÏí@:{7Qõ1 Æàk÷¼ìF0E2Cî Y(4f¾6Ø ³Á^wL³*îÂ8®Ú$áRâJÝ-|n£¸Ó»GÆ;Q¢ë»Æ+èl~ý .J4ÑÉ®¾ëèVs¶T#õ¥öG#tÁ0,ÄFFã®ãG æ²V¶§ä¾½@nxÓk½"|½ÒD¨pÎh£úóÛblæùâ¤dH¦È!½Ù&Ú»ê.ÔÝâ]6Ä 2\C'q. For example, let us suppose a TDK capacitor has a MTBF number of 6,918 years (16.5 FIT), this does not mean that the capacitor will last for that long (obviously no one will be able to test the part for that amount of time). Let’s look at an example to get a clearer understanding of what a FIT number means. Mean Time Between Failure (MTBF) There is a lot of confusion about the term MTBF. Failure Rate Calculation View PDF data sheet The steady-state FITs is calculated = λ G π Q π S π T (failures / billion hours) per Telcordia Technologies Special Report SR-332, Issue 1, May 2001. Ceramic Capacitors >  However, if it is used in a car, for example, then it would be much longer. In other words, MTBF is a maintenance metric, represented in hours, showing how long a piece of equipment operates without interruption. MTBF is then calculated as the inverse of this FIT rate. This tool can be used to make average failure rate estimates (or MTBF) for items that do not exhibit a constant failure rate, such as for mechanical components. >>Multilayer Ceramic Chip Capacitors Product site All terms used in the tool and definitions can be found on our … MTBF is not related to life expectancy of the operating life of the product and therefore has nothing to do with product lifetime. For any other product-related questions or inquiries, please contact us throughone of our sales representatives or this website. MTBF is also easily misinterpreted by the outside world. Q. Q. There are several problems with making this conversion. So for something that runs 24/7, the average fail time would be 9.88 years. The relationships between these different data formats are illustrated in the conversion examples below. Simply enter a part number, and the tool will present the data. Current reliability: MTBF (Hours): Failure rate (FPMH): A. Where MTTF uses non-repairable assets while MTBF deals with assets that are repairable—when they break down, they can be easily repaired without … FIT and MTTF / MTBF (1): What is MTBF (Mean Time Between Failures)? Reliability Parameters Conversion Calculator. Also consider that this is in terms of operating hours. Typical values lie between 300‘000 and 1‘200‘000 hours. FPMH = FPBH*10 -3 Failure rates are often expressed in engineering notation as failures per million, or 10 −6, especially for individual components, since their failure rates are often very low. Obviously, the higher the MTBF, the higher chance a part will perform for a longer period of time at the given conditions. The basic measure of reliability is mean time between failures (MTBF) for repairable equipment. The calculations is to be used in the IEC 61508 standard (Functional Safety). Use Calculator. As the sample size and observation time are limited for practical reasons, the number of actual failures may be low, so statistical tools such as chi-squared distribution are needed to be able to calculate the demonstrated MTBF figure with a reasonable amount of confidence (say, 95%). The failure rate that occurs during one billion device hours is called the Failure In Time (FIT). Enter the current reliability value and then select one or more of the conversion options (environment, temperature or part quality) by setting the "From" and "To" pull-down selections. FIT is number of failures per 10^9 device-hours.


text:
# Visualize Merge Sort

Merge sort is a sorting algorithm which works by splitting a given list in half, recursively sorting both smaller lists, and merging them back together to one sorted list. The base case of the recursion is arriving at a singleton list, which cannot be split further but is per definition already sorted.

The execution of the algorithm on the list [1,7,6,3,3,2,5] can be visualized in the following way:

       [1,7,6,3,3,2,5]
/             \      split
[1,7,6,3]       [3,2,5]
/     \        /    \   split
[1,7]   [6,3]   [3,2]  [5]
/   \   /   \   /   \   |  split
[1] [7] [6] [3] [3] [2] [5]
\   /   \   /   \   /   |  merge
[1,7]   [3,6]   [2,3]  [5]
\     /         \   /   merge
[1,3,6,7]       [2,3,5]
\             /      merge
[1,2,3,3,5,6,7]

Write a program or function which takes a list of integers in any reasonable way as input and visualizes the different partitions of this list while being sorted by a merge sort algorithm. This means you don't have to output a graph like above, but just the lists are fine:

[1,7,6,3,3,2,5]
[1,7,6,3][3,2,5]
[1,7][6,3][3,2][5]
[1][7][6][3][3][2][5]
[1,7][3,6][2,3][5]
[1,3,6,7][2,3,5]
[1,2,3,3,5,6,7]

Furthermore, any reasonable list notation is fine, therefore the following would also be a valid output:

1 7 6 3 3 2 5
1 7 6 3|3 2 5
1 7|6 3|3 2|5
1|7|6|3|3|2|5
1 7|3 6|2 3|5
1 3 6 7|2 3 5
1 2 3 3 5 6 7

Finally, the way to split a list in two smaller lists is up to you as long as the length of both resulting lists differs at most by one. That means instead of splitting [3,2,4,3,7] into [3,2,4] and [3,7], you could also split by taking elements at even and odd indexes ([3,4,7] and [2,3]) or even randomize the split every time.

This is , so the shortest code in any language measured in bytes wins.

### Test cases

As noted above, the actual format and the way to split lists in half is up to you.

[10,2]
[10][2]
[2,10]

[4,17,1,32]
[4,17][1,32]
[4][17][1][32]
[4,17][1,32]
[1,4,17,32]

[6,5,4,3,2,1]
[6,5,4][3,2,1]
[6,5][4][3,2][1]
[6][5][4][3][2][1]
[5,6][4][2,3][1] <- Important: This step cannot be [5,6][3,4][1,2], because 3 and 4 are on different branches in the the tree
[4,5,6][1,2,3]
[1,2,3,4,5,6]

• @dylnan You can use another sorting algorithm or a built in sort function to do the sorting... Dec 14 '17 at 23:31
• Some golfing idea: the bottom half of the result can be generated by sorting each sublist in the first half and reversing the order. Dec 15 '17 at 0:46
• @Arnauld The [[1,2],[3],[4,5],[6]] stage is actually the correct solution, as merge sort is working recursively. That is if we start with [1,2,3,4,5,6] and split it into [1,2,3] and [4,5,6], then those lists are independently processed until they are merged in the final step. Dec 15 '17 at 11:10
• @l4m2 Ok, final try for an answer: 1. you need delimiters because also integers >9 should be supported. 2. This is not valid for the same reason as given in my comment above. If we split into [3] and [2,1], then those are on different branches, so we can't merge [3] and [2] after [2,1] is split into [2] and [1]. Dec 15 '17 at 11:34
• In fact the sentence after that answers my question exactly. Sorry for missing that. :P Dec 15 '17 at 19:05

(-2)+(-4)=(-6) bytes thanks to nimi! Changing it to collect all the steps in a list (again due to nimi) saves 12 more bytes.

Another 3 bytes due to Laikoni, with pattern guard binding and a clever use of list comprehension to encode the guard.

import Data.List
g x|y<-x>>=h=x:[z|x/=y,z<-g y++[sort<$>x]] h[a]=[[a]] h x=foldr(\x[b,a]->[x:a,b])[[],[]]x  Try it online! Splitting the lists into the odd and the even positioned elements is a shorter code than into the two sequential halves, because we're spared having to measure the length, then. Works by "printing" the lists, then recursing with the split lists (x >>= h) if there actually was any splitting done, and "printing" the sorted lists; starting with the one input list; assuming the non-empty input. And instead of actual printing, just collecting them in a list. The list produced by g[[6,5..1]], printed line-by-line, is: [[6,5,4,3,2,1]] [[6,4,2],[5,3,1]] [[6,2],[4],[5,1],[3]] [[6],[2],[4],[5],[1],[3]] [[2,6],[4],[1,5],[3]] [[2,4,6],[1,3,5]] [[1,2,3,4,5,6]]  • ... p=print and three times p. Try it online! – nimi Dec 15 '17 at 21:02 • @nimi great, again, and many thanks! now it really looks golfed. :) Dec 15 '17 at 21:03 • Instead of printing within function g you can collect all steps in a list and return it. Try it online! – nimi Dec 15 '17 at 21:31 • I don't think that we have a proper definition of "visualizing". More general the challenges asks for "outputting" the lists and per our defaults this can be done via a return value of a function. Other answers, e.g. 1, 2 do it this way, too. -- I don't think my suggestion is that much different, it just collects the intermediate lists instead of printing them. Feel free to edit it in. – nimi Dec 15 '17 at 22:01 • g x|y<-x>>=h=x:[z|x/=y,z<-g y++[sort<$>x]] saves some more bytes. Dec 15 '17 at 23:34

# Wolfram Language (Mathematica), 146127111 102 bytes

Join[u=Most[#/.a:{_,b=__Integer}:>TakeDrop[a,;;;;2]&~FixedPointList~#],Reverse@Most@u/.a:{b}:>Sort@a]&

Try it online!

Returns a List that contains the steps.

## Explanation

#/.a:{_,b=__Integer}:>TakeDrop[a,;;;;2]&

In an input, split all Lists containing 2 or more integers in half. The first sublist has the odd-indexed elements (1-indexed), and the second one has the even-indexed elements.

u=Most[... &~FixedPointList~#]

Repeat that until nothing changes (i.e. all sublists are length-1). Keep all intermediate results. Store this (the List of all steps) in u.

Reverse@Most@u

Drop the last element of u and reverse it.

... /.a:{b}:>Sort@a

From the result above, sort all occurrences of a list of integers.

# Clean, 228206168157140121 104 bytes

Builds the list of stages from the ends inwards, using the fact that the n-th element from the end is the sorted version of the n-th element from the beginning.

Idea from JungHwan Min's comment

import StdEnv
@u#(a,b)=splitAt(length u/2)u
=if(a>[])[[u]:[x++y\\y<- @b&x<- @a++ @a++ @a]][]++[[sort u]]

Try it online!

# Charcoal, 145133123 122 bytes

≔⟦⪪Ｓ ⟧θＷ⊖Ｌ§θ⁰«⟦⪫Ｅθ⪫κ ¦|⟧≔ＥＥ⊗Ｌθ§θ÷λ²✂κ﹪λ²Ｌκ²θ»⟦⪫ΦＥθ⪫ι ι|⟧Ｗ⊖Ｌθ«≔⮌θη≔⟦⟧θＷη«≔⊟ηε≔⟦⟧ζＦ⊟η«≔εδ≔⟦⟧εＦδ⊞⎇‹ＩμＩλζεμ⊞ζλ»⊞θ⁺ζε»⟦⪫Ｅθ⪫κ ¦|

Try it online! Link is to verbose version of code. Still having to work around Charcoal bugs... Edit: Saved 5 bytes by using a double Map as a poor man's array comprehension. Saved 4 bytes by using Pop to double iterate over an array. Saved 3 bytes by using concatenation instead of Push. Saved 10 bytes by coming up with a golfier while condition that also avoids a Charcoal bug. Saved 1 byte by discovering that Charcoal does indeed have a filter operator. Explanation:

≔⟦⪪Ｓ ⟧θ

Split the input on spaces, and then wrap the result in an outer array, saving that to q.

Ｗ⊖Ｌ§θ⁰«

Repeat while the first element of q has more than one element. (The first element of q always has the most elements because of the way the lists are divided into two.)

⟦⪫Ｅθ⪫κ ¦|⟧

Print the elements of q joined with spaces and vertical lines. (The array causes the result to print on its own line. There are other ways of achieving this for the same byte count.)

≔ＥＥ⊗Ｌθ§θ÷λ²✂κ﹪λ²Ｌκ²θ»

Create a list by duplicating each element of q, then map over that list and take half of each list (using the alternate element approach), saving the result back in q.

⟦⪫ΦＥθ⪫ι ι|⟧

Print the elements of q joined with spaces and vertical lines. Actually, at this point the elements of q are either empty or single-element lists, so joining them just converts them to empty strings or their elements. The empty strings would add unnecessary trailing lines so they are filtered out. A flatten operator would have been golfier though (something like ⟦⪫Σθ|⟧).

Ｗ⊖Ｌθ«

Repeat while q has more than one element. (The following code requires an even number of elements.)

≔⮌θη≔⟦⟧θ

Copy q to h, but reversed (see below), and reset q to an empty list.

Ｗη«

Repeat until h is empty.

≔⊟ηε

Extract the next element of h into e. (Pop extracts from the end, which is why I have to reverse q.)

≔⟦⟧ζ

Initialise z to an empty list.

Ｆ⊟η«

Loop over the elements of the next element of h.

≔εδ≔⟦⟧ε

Copy e to d and reset e to an empty list.

Ｆδ

Loop over the elements of d.

⊞⎇‹ＩμＩλζεμ

Push them to z or e depending on whether they are smaller than the current element of the next element of h.

⊞ζλ»

Push the current element of the next element of h to z.

⊞θ⁺ζε»

Concatenate z with any elements remaining in e and push that to q. This completes the merge of two elements of h.

⟦⪫Ｅθ⪫κ ¦|

Print the elements of q joined with spaces and vertical lines.

• Hang on. There's another bug? :/ Feb 28 '18 at 13:12
• @ASCII-only No, this was the while (...Map(...)...) bug I already told you about.
– Neil
Feb 28 '18 at 21:39

# Python 2, 145 144 bytes

Idea from JungHwan Min's comment
-1 byte thanks to Jonathan Frech

p=input()
l=[[p]]
i=0
while 2**i<len(p):l[i+1:i+1]=[sum([[x[1::2],x[::2]][len(x)<2:]for x in l[i]],[]),map(sorted,l[i])];i+=1
for s in l:print s

Try it online!

# Husk, 14 bytes

S+ȯ†O↔hUmfL¡ṁ½

Takes an array containing a single array. Try it online!

## Explanation

S+ȯ†O↔hUmfL¡ṁ½  Implicit input, say A = [[4,17,32,1]].
¡    Iterate this function on A:
ṁ½   Split each array in two, concatenate results: [[4,17],[32,1]]
Result is [[[4,17,32,1]],
[[4,17],[32,1]],
[[4],[17],[32],[1]],
[[4],[],[17],[],[32],[],[1],[]],
...
mfL     Map filter by length, removing empty arrays.
Result is [[[4,17,32,1]],
[[4,17],[32,1]],
[[4],[17],[32],[1]],
[[4],[17],[32],[1]],
...
U        Longest prefix of unique elements:
P = [[[4,17,32,1]],[[4,17],[32,1]],[[4],[17],[32],[1]]]
h         Remove last element: [[[4,17,32,1]],[[4,17],[32,1]]]
↔          Reverse: [[[4,17],[32,1]],[[4,17,32,1]]]
†O           Sort each inner array: [[[4,17],[1,32]],[[1,4,17,32]]]
S+ȯ             Concatenate to P:
[[[4,17,32,1]],
[[4,17],[32,1]],
[[4],[17],[32],[1]],
[[4,17],[1,32]],
[[1,4,17,32]]]
Implicitly print.

The built-in ½ takes an array and splits it at the middle. If its length is odd, the first part is longer by one element. A singleton array [a] results in [[a],[]] and an empty array [] gives [[],[]], so it's necessary to remove the empty arrays before applying U.

## JavaScript (ES6), 145 bytes

f=a=>a.join|+(a[0][1]?
${f([].concat(...a.map(b=>b[1]?[b.slice(0,c=-b.length/2),b.slice(c)]:[b])))} +a.map(b=>b.sort((x,y)=>x-y)).join|:)  Takes input as an array within an array, i.e. f([[6,5,4,3,2,1]]). Works by generating the first and last lines of the output, then splitting and calling itself again, until every sub-array has length 1. Here's a basic demonstration of how it works: f([[6,5,4,3,2,1]]): 6,5,4,3,2,1 f([[6,5,4],[3,2,1]]): 6,5,4|3,2,1 f([[6,5],[4],[3,2],[1]]): 6,5|4|3,2|1 f([[6],[5],[4],[3],[2],[1]]): 6|5|4|3|2|1 end f 5,6|4|2,3|1 end f 4,5,6|1,2,3 end f 1,2,3,4,5,6 end f  • So, was there a point where there were three answers tied on 145 bytes? – Neil Dec 16 '17 at 0:26 # Stax, 116(÷3>)38 29 bytesCP437 -9 bytes per comment by @recursive. Now the input is given as a singleton whose only element is an array of the numbers to be sorted. ƒ3s}óºE/ßB╢↕êb∩áαπüµrL╞¶è,te+  Try it online! Unpacked version with 35 bytes: {c{Jm'|*Pc{2Mm:f{fc{Dm$wW{{eoJm'|*P

## Explanation

The code can be splitted to two parts. The first part visualizes splitting and the second visualizes merging.

Code for visualizing splitting:

{                      w    Do the following while the block generates a true value
c                          Copy current nested array for printing
{Jm                       Use spaces to join elements in each part
'|*                    And join different parts with vertical bar
P                   Pop and print

c                  Copy current nested array for splitting
{2Mm              Separate each element of the array to two smaller parts with almost the same size
That is, if the number of elements is even, partition it evenly.
Otherwise, the first part will have one more element than the second.
:f            Flatten the array once
{f          Remove elements that are empty arrays

c         Copy the result for checking
{Dm\$     Is the array solely composed of singletons?
If yes, ends the loop.

The code for visualizing merging:

W              Execute the rest of the program until the stack is empty
{{eoJm        For each part, sort by numeric value, then join with space
'|*     Join the parts with vertical bar
P    Pop and print the result

## Old version, actually building the nested list structure.

{{cc0+=!{x!}Mm',*:}}Xd;%v:2^^{;s~{c^;<~sc%v,*{2M{s^y!svsm}M}YZ!x!Q,dmU@e;%v:2^{;%v:2^-N~0{c;={scc0+=Cc%v!C:f{o}{scc0+=C{s^y!svsm}?}Y!cx!P,dcm

cc0+= is used thrice in the code to check whether the top of stack is a scalar or an array.

{{cc0+=!{x!}Mm',*:}}X builds a block that recursively calls itself to output a nested array of numbers properly. (Default output in Stax vectorizes a nested array before printing).

{                  }X    Store the code block in X
{           m           Map each element in the list with block
cc                     Make two copies of the element
0+                   + 0. If the element is a scalar, nothing will change
If the element is an array, the element 0 will be appended
=!{  }M            If the element has changed after the 0+ operation
Which means it is an array
x!              Recursively execute the whole code block on the element

',*        Join the mapped elements with comma
:}      Bracerizes the final result

There are two other blocks that performs the splitting and the merging, respectively. They are too verbose and I don't care to explain (there are a little bit more information in the historical version of this post but don't expect too much).

• Very nice improvement. I don't totally understand it yet, but I think cH! can be used in place of cH%!. Feb 25 '18 at 17:19
• {Nd}M can be replaced by T also. Feb 25 '18 at 18:53
• I made some more adaptations. staxlang.xyz/… The Husk answer takes input as an array in an array, so I guess that's legal. Feb 25 '18 at 23:31
• I found a solution that should be 2 ascii characters shorter, but I discovered a bug in array transpose. Specifically, it mutates the array rows. I'll add it to the backlog for 1.0.4 Feb 25 '18 at 23:39
• OK. I look forward to the update. Feb 25 '18 at 23:44