text:
Mathematics

# Classify the angles whose magnitudes are given below:$91^{o}$

##### SOLUTION
$91^{o}$ is an obtuse angle as it is more than $90^{o}$ and less than $180^{o}$

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Subjective Medium Published on 09th 09, 2020
Questions 120418
Subjects 10
Chapters 88
Enrolled Students 86

#### Realted Questions

Q1 Single Correct Medium
In which quadrilateral all sides and angles are equal?
• A. rectangle
• B. parallelogram
• C. kite
• D. square

1 Verified Answer | Published on 22nd 09, 2020

Q2 Single Correct Medium
ABC is an equilateral triangle of side 6 cm. If a circle of radius 1 cm is moving inside and along the sides of the triangle, then locus of the centre of the circle is an equilateral triangle of side_______.
• A. 5 cm
• B. 4 cm
• C. $\left( 3+\sqrt { 3 } \right) cm$
• D. $\left( 6-2\sqrt { 3 } \right) cm$

Asked in: Mathematics - The Triangle and Its Properties

1 Verified Answer | Published on 23rd 09, 2020

Q3 Subjective Hard
Two angles are complementary. The larger angle is $3^o$ less than twice the measure of the smaller angle. Find the measure of each angle.

Asked in: Mathematics - Lines and Angles

1 Verified Answer | Published on 09th 09, 2020

Q4 Single Correct Medium
The equation of a straight line passing through the point $(2,-7)$ and parallel to x-axis is
• A. $x=2$
• B. $x=-7$
• C. $y=2$
• D. $y=-7$

Asked in: Mathematics - Lines and Angles

1 Verified Answer | Published on 09th 09, 2020

Q5 Single Correct Medium
Which has a greater measure:
(i) an angle complementary to $15^{\circ}$
(ii) an angle supplementary to
$125^{\circ}$
• A. (ii)
• B. Both (i) and (ii) are equal.
• C. Both are Equal
• D. (i)

Asked in: Mathematics - Lines and Angles

1 Verified Answer | Published on 09th 09, 2020


text:
SMS scnews item created by Bill Unger at Mon 31 Oct 2011 1501
Type: Seminar
Distribution: World
Expiry: 3 Nov 2011
Calendar1: 3 Nov 2011 1500-1700
CalLoc1: Carslaw 535A
Auth: billu@daumier.maths.usyd.edu.au

# Computational Algebra Seminar: Mourao, Donnelly

Speaker: Michael Mourao (Warwick)
Title: The set of rational points of an algebraic curve
Time & Place: 3:05pm-3:55pm, Thursday 3 November, Carslaw 535

Abstract: In this talk we will introduce the methods of Chabauty and the
Mordell-Weil Sieve.  We will then show how a combination of these,
together with a "descent" argument, can be used to fully determine the set
of rational points of an algebraic curve (of genus greater than 1).

Speaker: Steve Donnelly (Sydney)
Title: Algorithms for Hilbert modular forms
Time & Place: 4:05pm-4:55pm, Thursday 3 November, Carslaw 535

Abstract: I will describe some of the algorithms used in Magma’s package
for Hilbert modular forms, and discuss data that I’ve collected for

format: plain


text:
# Modular Forms - Eichler quote

We have the following quote from Eichler: "There are five elementary arithmetical operations: addition, subtraction, multiplication, division, and… modular forms."

Why did Eichler consider modular forms "elementary arithmetical operations"? Not to mention that subtraction and division are just addition and multiplication with inverse elements. Can anybody shed some light on his train of thought?

-

The way I understand the quote is the following:

Modular forms are of course not "elementary", but they are a basic operation of arithmetic in the following sense: Modular forms can be used to generate sequences of integers that are most naturally defined and studied in the context of modular forms, and can also be used to prove arithmetic identities that would be significantly harder without modular forms. Furthermore, I think modular forms are unique in this respect of being a way of producing integers/rational numbers using means that cannot be reduced to the four operations of addition, subtraction, multiplication, and division.

The classic example of this is the Ramanujan tau function; tau(n) is most naturally defined as the coefficient of q^n in the unique weight 12 cusp form Delta for SL_2(Z) with leading term q. One can come up with a number of formulas for tau(n) in terms of things that can be defined using only the four standard operations of arithmetic (some of them can be found on the mathworld page above), but the easiest way to prove that these formulas all define the same function uses modular forms. Likewise, most of the other properties of tau listed on that Mathworld page are most easily understood in terms of propreties of the associated modular form Delta.

-

My interpretation of the quote is different. (Granted, whether Eichler really said such a thing is a different question). It should refer to solving the algebraic equations, to the legacy of Abel and Galois and to Kronecker's Jugendtraum! To the book of Taniyama and Shimura, whereby certain abelian equations are solved by adjoining the moduli of a relevant abelian variety. In this sense, from the point of view of exact solutions to algebraic equations, the modular functions such as $j$ are indeed a natural, inevitable complement to the basic arithmetic operations $+, \times, \sqrt[n]{\cdot}$. This falls into the rubric of explicit class field theory. For a connection to modular forms such as $\Delta(\tau)$, see Ribet's converse to Herbrandt's theorem; a great introduction to this circle of ideas is Mazur's article, "How can we construct abelian Galois extensions of basic number fields?" (Bull. AMS, vol. 48, no. 2, pp. 155-209).

-
This is also a very good answer! –  Alison Miller Feb 3 '13 at 17:22

One question is whether Eichler really did say it. One of the nice things about Google searches these days is that they let you trace the etymology of words, phrases, and aphorisms. I see a chain of references to this quote associated with Andrew Wiles and Fermat's Last Theorem. The chain seems to begin with a BBC interview in 1996 or 1997 in which Wiles says, "There's a saying attributed to Eichler that there are five fundamental operations of arithmetic: addition, subtraction, multiplication, division, and modular forms." On the other hand, this recent paper by Edixhoven, van der Geer, and Moonen calls the quote apocryphal.

I can't trace what happened before that interview. Since the documented history of this quip currently begins there, we can start with what Wiles himself meant. The interviewer wanted to know what a modular form is, but that's not easy to explain on television. Obviously it was a facetious (and witty) answer. It was also part of a general point in the interview that modular forms are widely studied and seem fundamental.

-

in p-adic Hodge theory, categories (semistable, crystalline, de-Rham, Hodge-Tate) of p-adic representations of Galois groups over local fields are proved to be equivalent to linear algebra categories (filtered modules with compatible actions). But for Galois representation of global fields, no such linear categories seem to exist, since the Galois group is more complicated, but then there are modularity results. Saying that these Galois representations correspond to modular forms (the most basic level). Of course, then according to Langlands Program, there are generalizations to automorphic forms. So, in some sense, modular forms are a "global patching" of all local linear data.

-
Can someone briefly explain why this was marked down? (N.B. I'm not disputing the reason one way or another, just curious) –  John R Ramsden Feb 3 '13 at 18:58
@John: I didn't vote down, but I can make a guess. The essay appears to be more of an exercise in free association than an honest attempt to answer the question. –  S. Carnahan Feb 4 '13 at 1:49


text:
Skip to main content

Easy hyperparameter optimization and automatic result saving across machine learning algorithms and libraries

# HyperparameterHunter

Automatically save and learn from Experiment results, leading to long-term, persistent optimization that remembers all your tests.

HyperparameterHunter provides a wrapper for machine learning algorithms that saves all the important data. Simplify the experimentation and hyperparameter tuning process by letting HyperparameterHunter do the hard work of recording, organizing, and learning from your tests — all while using the same libraries you already do. Don't let any of your experiments go to waste, and start doing hyperparameter optimization the way it was meant to be.

## Features

• Automatically record Experiment results
• Truly informed hyperparameter optimization that automatically uses past Experiments
• Eliminate boilerplate code for cross-validation loops, predicting, and scoring
• Stop worrying about keeping track of hyperparameters, scores, or re-running the same Experiments
• Use the libraries and utilities you already love

## How to Use HyperparameterHunter

Don’t think of HyperparameterHunter as another optimization library that you bring out only when its time to do hyperparameter optimization. Of course, it does optimization, but its better to view HyperparameterHunter as your own personal machine learning toolbox/assistant.

The idea is to start using HyperparameterHunter immediately. Run all of your benchmark/one-off experiments through it.

The more you use HyperparameterHunter, the better your results will be. If you just use it for optimization, sure, it’ll do what you want, but that’s missing the point of HyperparameterHunter.

If you’ve been using it for experimentation and optimization along the entire course of your project, then when you decide to do hyperparameter optimization, HyperparameterHunter is already aware of all that you’ve done, and that’s when HyperparameterHunter does something remarkable. It doesn’t start optimization from scratch like other libraries. It starts from all of the Experiments and previous optimization rounds you’ve already run through it.

## Getting Started

### 1) Environment:

Set up an Environment to organize Experiments and Optimization results.
Any Experiments or Optimization rounds we perform will use our active Environment.

from hyperparameter_hunter import Environment, CVExperiment
import pandas as pd
from sklearn.datasets import load_breast_cancer
from sklearn.model_selection import StratifiedKFold

data = load_breast_cancer()
df = pd.DataFrame(data=data.data, columns=data.feature_names)
df['target'] = data.target

env = Environment(
train_dataset=df,  # Add holdout/test dataframes, too
results_path='path/to/results/directory',  # Where your result files will go
metrics=['roc_auc_score'],  # Callables, or strings referring to sklearn.metrics
cv_type=StratifiedKFold,  # Class, or string in sklearn.model_selection
cv_params=dict(n_splits=5, shuffle=True, random_state=32)
)

### 2) Individual Experimentation:

Perform Experiments with your favorite libraries simply by providing model initializers and hyperparameters

Keras
# Same format used by keras.wrappers.scikit_learn. Nothing new to learn
def build_fn(input_shape):  # input_shape calculated for you
model = Sequential([
Dense(100, kernel_initializer='uniform', input_shape=input_shape, activation='relu'),
Dropout(0.5),
Dense(1, kernel_initializer='uniform', activation='sigmoid')
])  # All layer arguments saved (whether explicit or Keras default) for future use
model.compile(optimizer='adam', loss='binary_crossentropy', metrics=['accuracy'])
return model

experiment = CVExperiment(
model_initializer=KerasClassifier,
model_init_params=build_fn,  # We interpret your build_fn to save hyperparameters in a useful, readable format
model_extra_params=dict(
callbacks=[ReduceLROnPlateau(patience=5)],  # Use Keras callbacks
batch_size=32, epochs=10, verbose=0  # Fit/predict arguments
)
)

SKLearn
experiment = CVExperiment(
model_initializer=LinearSVC,  # (Or any of the dozens of other SK-Learn algorithms)
model_init_params=dict(penalty='l1', C=0.9)  # Default values used and recorded for kwargs not given
)

XGBoost
experiment = CVExperiment(
model_initializer=XGBClassifier,
model_init_params=dict(objective='reg:linear', max_depth=3, n_estimators=100, subsample=0.5)
)

LightGBM
experiment = CVExperiment(
model_initializer=LGBMClassifier,
model_init_params=dict(boosting_type='gbdt', num_leaves=31, max_depth=-1, min_child_samples=5, subsample=0.5)
)

CatBoost
experiment = CVExperiment(
model_initializer=CatboostClassifier,
model_init_params=dict(iterations=500, learning_rate=0.01, depth=7, allow_writing_files=False),
model_extra_params=dict(fit=dict(verbose=True))  # Send kwargs to fit and other extra methods
)

RGF
experiment = CVExperiment(
model_initializer=RGFClassifier,
model_init_params=dict(max_leaf=1000, algorithm='RGF', min_samples_leaf=10)
)

### 3) Hyperparameter Optimization:

Just like Experiments, but if you want to optimize a hyperparameter, use the classes imported below

from hyperparameter_hunter import Real, Integer, Categorical
from hyperparameter_hunter import optimization as opt

Keras
def build_fn(input_shape):
model = Sequential([
Dense(Integer(50, 150), input_shape=input_shape, activation='relu'),
Dropout(Real(0.2, 0.7)),
Dense(1, activation=Categorical(['sigmoid', 'softmax']))
])
model.compile(
optimizer=Categorical(['adam', 'rmsprop', 'sgd', 'adadelta']),
loss='binary_crossentropy', metrics=['accuracy']
)
return model

optimizer = opt.RandomForestOptPro(iterations=7)
optimizer.forge_experiment(
model_initializer=KerasClassifier,
model_init_params=build_fn,
model_extra_params=dict(
callbacks=[ReduceLROnPlateau(patience=Integer(5, 10))],
batch_size=Categorical([32, 64]),
epochs=10, verbose=0
)
)
optimizer.go()

SKLearn
optimizer = opt.DummyOptPro(iterations=42)
optimizer.forge_experiment(
model_initializer=AdaBoostClassifier,  # (Or any of the dozens of other SKLearn algorithms)
model_init_params=dict(
n_estimators=Integer(75, 150),
learning_rate=Real(0.8, 1.3),
algorithm='SAMME.R'
)
)
optimizer.go()

XGBoost
optimizer = opt.BayesianOptPro(iterations=10)
optimizer.forge_experiment(
model_initializer=XGBClassifier,
model_init_params=dict(
max_depth=Integer(low=2, high=20),
learning_rate=Real(0.0001, 0.5),
n_estimators=200,
subsample=0.5,
booster=Categorical(['gbtree', 'gblinear', 'dart']),
)
)
optimizer.go()

LightGBM
optimizer = opt.BayesianOptPro(iterations=100)
optimizer.forge_experiment(
model_initializer=LGBMClassifier,
model_init_params=dict(
boosting_type=Categorical(['gbdt', 'dart']),
num_leaves=Integer(5, 20),
max_depth=-1,
min_child_samples=5,
subsample=0.5
)
)
optimizer.go()

CatBoost
optimizer = opt.GradientBoostedRegressionTreeOptPro(iterations=32)
optimizer.forge_experiment(
model_initializer=CatBoostClassifier,
model_init_params=dict(
iterations=100,
eval_metric=Categorical(['Logloss', 'Accuracy', 'AUC']),
learning_rate=Real(low=0.0001, high=0.5),
depth=Integer(4, 7),
allow_writing_files=False
)
)
optimizer.go()

RGF
optimizer = opt.ExtraTreesOptPro(iterations=10)
optimizer.forge_experiment(
model_initializer=RGFClassifier,
model_init_params=dict(
max_leaf=1000,
algorithm=Categorical(['RGF', 'RGF_Opt', 'RGF_Sib']),
l2=Real(0.01, 0.3),
normalize=Categorical([True, False]),
learning_rate=Real(0.3, 0.7),
loss=Categorical(['LS', 'Expo', 'Log', 'Abs'])
)
)
optimizer.go()

## Output File Structure

This is a simple illustration of the file structure you can expect your Experiments to generate. For an in-depth description of the directory structure and the contents of the various files, see the File Structure Overview section in the documentation. However, the essentials are as follows:

1. An Experiment adds a file to each HyperparameterHunterAssets/Experiments subdirectory, named by experiment_id
2. Each Experiment also adds an entry to HyperparameterHunterAssets/Leaderboards/GlobalLeaderboard.csv
3. Customize which files are created via Environment's file_blacklist and do_full_save kwargs (documented here)
HyperparameterHunterAssets
|   Heartbeat.log
|
└───Experiments
|   |
|   └───Descriptions
|   |   |   <Files describing Experiment results, conditions, etc.>.json
|   |
|   └───Predictions<OOF/Holdout/Test>
|   |   |   <Files containing Experiment predictions for the indicated dataset>.csv
|   |
|   └───Heartbeats
|   |   |   <Files containing the log produced by the Experiment>.log
|   |
|   └───ScriptBackups
|       |   <Files containing a copy of the script that created the Experiment>.py
|
└───Leaderboards
|   |   GlobalLeaderboard.csv
|   |   <Other leaderboards>.csv
|
└───TestedKeys
|   |   <Files named by Environment key, containing hyperparameter keys>.json
|
└───KeyAttributeLookup
|   <Files linking complex objects used in Experiments to their hashes>

## Installation

pip install hyperparameter-hunter

If you like being on the cutting-edge, and you want all the latest developments, run:

pip install git+https://github.com/HunterMcGushion/hyperparameter_hunter.git

If you want to contribute to HyperparameterHunter, get started here.

## I Still Don't Get It

That's ok. Don't feel bad. It's a bit weird to wrap your head around. Here's an example that illustrates how everything is related:

from hyperparameter_hunter import Environment, CVExperiment, BayesianOptPro, Integer
from hyperparameter_hunter.utils.learning_utils import get_breast_cancer_data
from xgboost import XGBClassifier

# Start by creating an Environment - This is where you define how Experiments (and optimization) will be conducted
env = Environment(
train_dataset=get_breast_cancer_data(target='target'),
results_path='HyperparameterHunterAssets',
metrics=['roc_auc_score'],
cv_type='StratifiedKFold',
cv_params=dict(n_splits=10, shuffle=True, random_state=32),
)

# Now, conduct an Experiment
# This tells HyperparameterHunter to use the settings in the active Environment to train a model with these hyperparameters
experiment = CVExperiment(
model_initializer=XGBClassifier,
model_init_params=dict(
objective='reg:linear',
max_depth=3
)
)

# That's it. No annoying boilerplate code to fit models and record results
# Now, the Environment's results_path directory will contain new files describing the Experiment just conducted

# Time for the fun part. We'll set up some hyperparameter optimization by first defining the OptPro (Optimization Protocol) we want
optimizer = BayesianOptPro(verbose=1)

# Now we're going to say which hyperparameters we want to optimize.
# Notice how this looks just like our experiment above
optimizer.forge_experiment(
model_initializer=XGBClassifier,
model_init_params=dict(
objective='reg:linear',  # We're setting this as a constant guideline - Not one to optimize
max_depth=Integer(2, 10)  # Instead of using an int like the experiment above, we provide a space to search
)
)
# Notice that our range for max_depth includes the max_depth=3 value we used in our experiment earlier

optimizer.go()  # Now, we go

assert experiment.experiment_id in [_[2] for _ in optimizer.similar_experiments]
# Here we're verifying that the experiment we conducted first was found by optimizer and used as learning material
# You can also see via the console that we found experiment's saved files, and used it to start optimization

last_experiment_id = optimizer.current_experiment.experiment_id
# Let's save the id of the experiment that was just conducted by optimizer

optimizer.go()  # Now, we'll start up optimizer again...

# And we can see that this second optimization round learned from both our first experiment and our first optimization round
assert experiment.experiment_id in [_[2] for _ in optimizer.similar_experiments]
assert last_experiment_id in [_[2] for _ in optimizer.similar_experiments]
# It even did all this without us having to tell it what experiments to learn from

# Now think about how much better your hyperparameter optimization will be when it learns from:
# - All your past experiments, and
# - All your past optimization rounds
# And the best part: HyperparameterHunter figures out which experiments are compatible all on its own
# You don't have to worry about telling it that KFold=5 is different from KFold=10,
# Or that max_depth=12 is outside of max_depth=Integer(2, 10)

## Gotchas/FAQs

These are some things that might "getcha"

### General:

• Can't provide initial search points to OptPro?
• This is intentional. If you want your optimization rounds to start with specific search points (that you haven't recorded yet), simply perform a CVExperiment before initializing your OptPro
• Assuming the two have the same guideline hyperparameters and the Experiment fits within the search space defined by your OptPro, the optimizer will locate and read in the results of the Experiment
• Keep in mind, you'll probably want to remove the Experiment after you've done it once, as the results have been saved. Leaving it there will just execute the same Experiment over and over again
• After changing things in my "HyperparameterHunterAssets" directory, everything stopped working
• Yeah, don't do that. Especially not with "Descriptions", "Leaderboards", or "TestedKeys"
• HyperparameterHunter figures out what's going on by reading these files directly.
• Removing them, or changing their contents can break a lot of HyperparameterHunter's functionality

### Keras:

• Can't find similar Experiments with simple Dense/Activation neural networks?
• This is likely caused by switching between using a separate Activation layer, and providing a Dense layer with the activation kwarg
• Each layer is treated as its own little set of hyperparameters (as well as being a hyperparameter, itself), which means that as far as HyperparameterHunter is concerned, the following two examples are NOT equivalent:
• Dense(10, activation=‘sigmoid’)
• Dense(10); Activation(‘sigmoid’)
• We’re working on this, but for now, the workaround is just to be consistent with how you add activations to your models
• Either use separate Activation layers, or provide activation kwargs to other layers, and stick with it!
• Can't optimize the model.compile arguments: optimizer and optimizer_params at the same time?
• This happens because Keras’ optimizers expect different arguments
• For example, when optimizer=Categorical(['adam', 'rmsprop']), there are two different possible dicts of optimizer_params
• For now, you can only optimize optimizer, and optimizer_params separately
• A good way to do this might be to select a few optimizers you want to test, and don’t provide an optimizer_params value. That way, each optimizer will use its default parameters
• Then you can select which optimizer was the best, and set optimizer=<best optimizer>, then move on to tuning optimizer_params, with arguments specific to the optimizer you selected

### CatBoost:

• Can't find similar Experiments for CatBoost?
• This may be happening because the default values for the kwargs expected in CatBoost’s model __init__ methods are defined somewhere else, and given placeholder values of None in their signatures
• Because of this, HyperparameterHunter assumes that the default value for an argument really is None if you don’t explicitly provide a value for that argument
• This is obviously not the case, but I just can’t seem to figure out where the actual default values used by CatBoost are located, so if anyone knows how to remedy this situation, I would love your help!

## Download files

Download the file for your platform. If you're not sure which to choose, learn more about installing packages.

### Source Distribution

hyperparameter_hunter-3.0.0.tar.gz (208.9 kB view hashes)

Uploaded source

### Built Distribution

hyperparameter_hunter-3.0.0-py3-none-any.whl (233.3 kB view hashes)

Uploaded py3


text:
# How can I perform a chi-square test for independence on signal samples?

Let's say I have two signals $x$ and $y$, sampled $N$ times, i.e.

$$x = [ x_{1}, x_{2}, ..., x_{N} ]$$ $$y = [ y_{1}, y_{2}, ..., y_{N} ]$$

I would like to check whether $x$ and $y$ are statistically independent, with a certain level of probability.

I have been looking into the Chi-Square Test For Independence. However, since it can be applied to categorical data, I do not know how I can apply it to my signal samples.

As was suggested on this related question, once we compute the histogram for each signal, we do have categories on which a chi-squared test can be applied. But how do we use the histograms to generate the required contingency table?

For what it's worth, I am currently computing the histograms using this code:

n1 = hist(x);
n2 = hist(y);
n3 = hist3([x' y']);

Thank you for your help and suggestions.

EDIT

As an example, two signal samples would be the following:

xx = 0.2:0.2:34;
x = sin(xx);
y = randn(size(xx));

• It sounds as if the signals are measured on an interval or ratio level. If that's the case there's no benefit to converting to a categorical level in order to conduct a Chi-Square test. You can assess independence by creating a scatterplot and/or using Pearson correlation. – rolando2 Mar 10 '12 at 19:37
• @rolando2 But If you were to conduct a Chi-Square test, how would you do it? I was thinking of maybe using the 2D-histogram of their joint distribution as the contingency table, but didn't know whether it made sense.. With regards to using a scatter plot, how would that work? Also, with regards to using correlation, it can easily be done but is not what I want. Although independence implies correlation, the converse is not true. Two signals can easily be uncorrelated but dependent. – Rachel Mar 10 '12 at 20:49
• Fun fact: "Chi" squared looks like this: $X^2$. – user88 Mar 11 '12 at 10:01
• @mbq, shouldn't it look more like this: $\chi^2$? BTW, is this a subtle hint that you want me to switch my use of the words to mathjax? – gung - Reinstate Monica Mar 11 '12 at 13:37
• @gung Chi=$X$, chi=$\chi$. – user88 Mar 11 '12 at 14:03

I agree with @rolando2, if your data are continuous, a chi-square test is not really best. I would make a scatterplot and overlay a loess line on it. I don't really know MATLAB, but googling led me to these two pages. I can tell you that in R, the code would be plot(x,y); lines(lowess(y~x)). Although it's true that variables can be dependent while being uncorrelated, you should be able to assess this via your scatterplot. That is because correlation (specifically, Pearson's product-moment correlation) is a measure of linear dependence, so for example, a perfect parabola could be uncorrelated despite being dependent, or a sine wave that ran perfectly horizontally. My point is that you would be able to recognize that the correlation statistic was failing to capture the dependence by looking at your scatterplot. The smoothed fit would make this even easier to see.


text:
# Autodesk AutoCAD Full Version Product Key Full Free

18/06/2022

|

Auteur:

|

|

imprimer

## AutoCAD 20.0 Crack + [2022]

The program will prompt you to select the drive where you installed it. Select the folder where you keep the game. Press ‘Enter’. The game will load. Press ESC or F12. Press ‘Load’. The key is working. Q: In order to study a function on a normed vector space $X$, we usually choose a norm $||\cdot||$ such that $X$ is a Banach space and then study it on that space. However, we usually don’t check if $||\cdot||$ is a norm or not. Why can’t we study it in any other normed space? A: Let $X$ be a space equipped with a metric $d$. Let $\|\cdot\|$ be a norm. Then define $\varphi\colon X\to\mathbb{R}$ by $\varphi(x)=d(x,x_0)$. It is clear that $\varphi$ is a continuous function (since $d$ is a metric), but in general it is not a linear function. Suppose $X$ is a normed space with metric $d$, $x_0$ is an element of $X$, and $x_1$ is an element of $X$. Let $||\cdot||$ be a norm on $X$, and let $\varphi$ be as above. By the triangle inequality, $||x_1-x_0||\leq||x_1-x||+||x-x_0||$. But since $||\cdot||$ is a norm, $||x_1-x||\leq||x_1-x_0||+||x-x_0||$. Consequently, $\varphi(x_1)-\varphi(x_0)\leq d(x_1,x_0)$. Since $\varphi$ is non-negative, we have $\varphi(x_1)\leq\varphi(x_0)$. Therefore, $\varphi(x_1)$ is smaller than or equal to $\varphi(x_0)$. Consequently, $\varphi$ is non-decreasing. On the other hand, we have \$\varphi(x_0)\leq\


text:
Stacking angle-tunable photoluminescence from interlayer exciton states in twisted bilayer graphene

Abstract

Twisted bilayer graphene (tBLG) is a metallic material with two degenerate van Hove singularity transitions that can rehybridize to form interlayer exciton states. Here we report photoluminescence (PL) emission from tBLG after resonant 2-photon excitation, which tunes with the interlayer stacking angle, θ. We spatially image individual tBLG domains at room-temperature and show a five-fold resonant PL-enhancement over the background hot-electron emission. Prior theory predicts that interlayer orbitals mix to create 2-photon-accessible strongly-bound (~0.7 eV) exciton and continuum-edge states, which we observe as two spectral peaks in both PL excitation and excited-state absorption spectra. This peak splitting provides independent estimates of the exciton binding energy which scales from 0.5–0.7 eV with θ = 7.5° to 16.5°. A predicted vanishing exciton-continuum coupling strength helps explain both the weak resonant PL and the slower 1 ps−1 exciton relaxation rate observed. This hybrid metal-exciton behavior electron thermalization and PL emission are tunable with stacking angle for potential enhancements in optoelectronic and fast-photosensing graphene-based applications.

Introduction

The interlayer orbital overlap in twisted bilayer graphene (tBLG) produce sharp optical absorption resonances that tune with the layer-stacking angle, θ1,2,3,4,5. Currently, there are two prevailing models explaining resonant light absorption in tBLG: the free-electron van Hove singularity (vHs) model, and the strongly bound-interlayer exciton model1,6. Figure 1b sketches the vHs model as two optically-allowed, degenerate transitions X13, and X24 between anticrossing regions of band-structure. This band-flattened region of tBLG has recently been associated with many surprising many-body electronic effects, such as exciton effects, θ-tunable superconductivity, and metal-insulator phase transitions6,7,8. In this work, we use 2-photon resonant excitation to find resonant PL emission from similar flattened-band interlayer states in tBLG.

Near tBLG band-structure anticrossing regions, theory predicts strongly enhanced electron-hole interactions from interlayer orbital mixing6. The lower panel of Fig. 1b outlines this alternative bound-exciton model with the renormalized vHs labelled as unbound, degenerate exciton transitions, X13 and X24. Symmetric and antisymmetric mixing of these two transitions is predicted to yield hybridized interlayer exciton states; specifically, an optically bright state, XS and a strongly bound state, labeled XA. Using Bethe–Salpeter equation (BSE) simulations, Liang et al. predicted that the lower-lying exciton state, XA is strongly bound, optically dark with vanishing exciton to graphene continuum coupling strength, $$H_{X_A,k}$$ (see Fig. 1b)6. To search for such optically dark interlayer exciton states in tBLG, we loosely assume parity-based optical selection rules from the hydrogenic exciton model and detect photoluminescence (PL) and excited-state absorption (ESA) signals generated via resonant 2-photon excitation9,10,11.

Using 2-photon PL, we can differentiate between the prevailing vHs model and the bound-interlayer exciton model in Fig. 1b. The vHs model for tBLG predicts only non-resonant hot-electron and blackbody PL emission because interlayer optical excitations thermalize rapidly (~10–20 fs) by the efficient electron-electron scattering in graphene12. We further estimate exciton binding energy, Eb using both 2-photon PL excitation (PLE) spectra and excited-state absorption (ESA) transient spectra methods. These figures of merit directly impact recent optoelectronic applications of tBLG such as chiral-light sensitive photosensors and light-induced metal-insulator phase transitions13,14.

Results

Weak θ-tunable interlayer PL emission

Non-resonant hot-electron and blackbody PL from both graphene and tBLG have been recently reported15,16. Unexpectedly, we observe PL emitted from tBLG domains after resonant 2-photon excitation. This weak emission is θ-tunable and independent of substrates and preparation method. To clearly observe resonant PL emission, we employ 2-photon excitation to remove the non-resonant scattering background with shortpass or bandpass optical filter stacks. The basic experimental setup showing a modified scanning confocal microscope is outlined in Fig. 1a. PL spatial maps and transient dynamics of single-domain tBLG were collected over our 0.3–1.8 eV laser tuning range. Both PL maps were acquired by raster scanning a ~1 μm wavelength-tunable laser spot across tBLG supported on silicon nitride or fused silica. The filtered 2-photon PL is collected with a back-illuminated electron-multiplying CCD camera or a photomultiplier tube (PMT) and lock-in amplifier.

Figure 1c (circles) plots PLE spectra collected after 2-photon excitation and the associated θ-tunable linear absorption spectra (dashed lines). The PLE spectrum is plotted after subtracting the non-resonant background signals such as blackbody emission15,16. Gaussian fits of the 2-photon PLE spectrum (solid lines) are peaked at XA 2.61, 1.73, and 1.67 eV for the 16.3°, 11.5°, and 11.2° domains, respectively. Selective 10 nm wide bandpass filters were applied to confirm that the resonant PL emission was centered near ~2.7, 2.1, and 2.0 eV respectively. The domain angles were assigned by 1-photon linear absorption spectra which spectrally overlap with the corresponding 2-photon PLE peaks in Fig. 1c. Like PLE, the linear absorption spectra tune with stacking angle, θ but are blue shifted by an energy δ. In Fig. 1c, the 2-photon PLE and 1-photon absorption peak energies have a splitting, δ that is θ-tunable, ranging from δ = 160 to 380 meV (see Table 1). The absorption spectra after background-subtraction (σtBLG − 2σG) are also used to assign the θ-orientation of the tBLG domains with about ±0.2° precision (see Supplementary Figs. 13)4.

2-photon PL spatial maps of tBLG domains

The PL and absorption spectra plotted in Fig. 1c were compiled from full-frame spatial maps collected over spectrally tuned 1- and 2-photon excitation energies. For example, Fig. 2a shows a linear absorption contrast map with the 18.3° domains resonantly excited4. Figure 2b shows the corresponding tBLG spatial PL map collected at 1.26 eV excitation to search for allowed 2-photons transitions. Strikingly, two 17.5° domains show a ~5× enhancement of PL relative to the background hot-electron PL from the neighboring tBLG domains in Fig. 2b. Contributions below 2.65 eV were removed with an OD >6 filter stack. Selective bandpass optical filtering confirmed that emission energy matches the 1-photon absorption XS resonance of the 17.5° domain at ~2.8 eV. This process showing resonant emission is detailed in Fig. 2c on 20 μm wide θ ~ 14° domain. This suggests that the resonant emission relies on electronic thermalization with the spectrally overlapping dark XA state, as sketched in Fig. 3a. This bright–dark state splitting δ, was >150 meV, making tBLG emission inherently very weak.

Verification of PL origin

To verify the signals in Figs. 13 are θ-tunable PL emitted from interlayer excitons, we must exclude any contributions or leaks from other resonantly-enhanced 2-photon scattering processes such as second harmonic generation (SHG) or 2-photon resonant Raman. 1-photon resonant Raman is well-documented in tBLG for G-band splitting with an energy of ~0.2 eV4,17. The unexpected Gaussian-like lineshapes of these early Raman studies provided the early clues that excitons may be involved in tBLG photophysics4,18,19. For Raman, the G-band spectral Stokes and anti-Stokes energies need to be a θ-independent3,20,21. However, our peak splitting energies vary strongly with θ from δ = 0.16–0.38 eV, ruling out a 2-photon resonant Raman signal origin.

Previously observed hot-electron PL was also excluded as the signal origin15. Such thermally drivenc emission produces a broadband blackbody emission spectrum, whereas we show in Figs. 2b and 3b that there is a ~5× PL enhancement over the non-resonant background signal collected from bilayer graphene. In Fig. 2c, we show that the PL emission collected with resonant vs. non-resonant bandpass (BP) filters give starkly contrasting emission instensity for large θ = 14° domain after resonant excitation. We conclude the majority of the photons are emitted resonantly rather than hot-electron or blackbody emission. Further examples of PL maps are in the Supplementary Figs. 35.

Both resonant 2-photon PL and transient absorption (TA) signals show a counter-intuitive linear trend with laser fluence. This trend is expected based on our concurrently measured 1-photon spectra showing clear square root ($$\sqrt n$$) pump-fluence dependence on carrier density, n (see Supplementary Fig. 7 for detailed fluence-dependent measurements). This square root is functionally distinct from graphene’s sub-linear Fermi-filling dependence22. Instead, $$\sqrt n$$ results from Auger or exciton–exciton annihilation. The net result requires a linear 2-photon PL flux-dependence, $$I(n_{2ph}) \propto \sqrt {n_{2ph}^2}$$ where n2ph is proportional to the incident laser flux, which is typically 2 × 1014 photons/cm2 during 2-photon PL collection. As a control, we show in the supplement that our setup in Fig. 1a gives the typical 2-photon PL quadratic (I$$n_{2ph}^2$$) pump-power dependence when CdSe quantum dots were measured under identical experimental conditions. We further confirmed that the 1-photon TA signal in tBLG has a clear square root dependence, $$I(n_{1ph}) \propto \sqrt {n_{1ph}}$$ (see Supplementary Fig. 8)23,24. Very similar trends are reported for closely analogous materials such as carbon nanotubes (CNTs)24,25,26,27,28,29. This fluence dependence shows evidence that resonantly excited interlayer carriers undergo exciton–exciton annihilation.

Estimates of interlayer exciton binding energy, Eb

Using established 2-photon PL and intraband TA methods, we estimate Eb in resonantly excited tBLG10,11. Figure 3a shows how we measured Eb by expanding our energy range to include higher-lying 2-photon states predicted to cluster near the continuum edge for a hydrogenic exciton model. Upon 2-photon excitation at energy, Δ above the XS absorption resonance with Epump2, the carriers will relax to the lower-lying dark state XA. 2-photon excitation at either Epump1 or Epump2 results in PL emission from the bright XS state as the dark XA and XS state in Fig. 3c overlap spectrally. This bright–dark state splitting, δ quenches PL emission. A similar low-lying dark state is also thought to quench PL in CNTs30.

Figure 3b plots the 2-photon PL microscopy map of tBLG domains grown by chemical vapor deposition (CVD) and subsequently transferred to silicon nitride. Similar to the single-layer dry transfer tBLG in Fig. 2, we also show resonantly-enhanced PL (localized to the 16.3° domain). A partially overlapping absorption resonance results in the similarly stacked 15.2° domain emitting slightly-weaker PL. In Fig. 3c, the PLE spectral fits (magenta lines) for the 16.3° domain shows a bimodal distribution with two peaks spanning the 2.76 eV (green) absorption resonance at 2.61 and 3.10 eV. This spectral splitting energy is Δ ~ 0.34 eV, and this 2-photon PLE peak has almost twice the spectral width compared to the XS or XA peaks. This peak broadening may imply continuum edge mixing, suggesting we can estimate a lower bound for the exciton binding energy as Ebδ + Δ = 0.51 eV for the 16.3° domain shown. We summarize the θ–dependent peak splitting and Eb estimates in Table 1 below:

Verifying Eb using ultrafast intraband ESA

The hot-electron vHs model in Fig. 1b has sufficiently predicted scattering resonances in Raman and circular dichroism2,19,31,32. For example,scattering in tBLG Raman and ARPES experiments do not optically populate interlayer excitons. Resonant PL emission is not conventionally observed for metallic materials because of strong screening of electron-hole pair interactions, and fast thermalization with continuum states. For resonant optical absorption, strong electron-hole interactions are already well-accepted in graphene-based systems. For example in single-layer graphene, there is ~0.4 eV red-shift in the absorption resonance about the M saddle-point owning to band renormalization from strong exciton effects33,34. Likewise, Ju et al. recently showed that 0° Bernal stacked bilayer graphene enables a bandgap of strongly bound interlayer excitons under high-fields and low-temperatures8. 1D metallic CNTs also have bound excitons with Eb ≥ 50 meV35,36.

Pump-probe kinetic studies and lineshape analysis of absorption and Raman spectra offered the early experimental motivation to search for interlayer exciton states in tBLG6,17,23. BSE simulations by Liang et al. support a strongly bound-exciton model for tBLG6. Theory simulates that the bound-exciton state, XA has effectively zero electronic coupling with the lower metallic graphene continuum states6. Historically, such dark exciton states with $$H_{X_A,k}$$ = 0 (see Fig. 1b), have been termed ghost Fano resonances and observed in selected systems including some quantum dots6,23,37.

In Fig. 4, we plot the transient spectra and exciton relaxation dynamics of a 7.9° tBLG domain by raster scanning two collinear ~160 fs pump and probe pulses. By measuring the differential TA of the probe pulse, we plot the full transient spectra at t = 0.5 ps in Fig. 4b. The corresponding relaxation kinetics are shown in Fig. 4c. To spectrally resolve the intraband exciton transition (dashed blue arrows in Fig. 4a), the pump beam was optically resonant at $$E_{X_S}$$ = 1.54 eV and the near-IR probe energy was scanned from 0.4–1.2 eV. The dynamics intrinsic to the interlayer states are isolated by subtracting the graphene 0° bilayer TA contribution (ΔσB) from the tBLG TA response (ΔσT) for both Fig. 4b, c4.

Transient spectra in Fig. 4b provide a secondary, independent estimate of Eb. In analogous systems like CNTs, the exciton binding energy is estimated by an ESA spectrum of intraband exciton state transitions (see Fig. 4a, blue arrow)38. ESA transient absorption microscopy uses a NIR probe pulse to promote an exciton to the continuum band edge. The 2-photon selection rules are relaxed as this is an intraband transition within an exciton band of states11. In Fig. 4b, the TA spectrum (blue circles) is positively peaked at 0.7 eV, an ESA response. This suggests an intraband transition to the continuum edge-states illustrated in Fig. 4a, and that Eb ~ 0.7 eV for θ = 7.9° tBLG.

We verified the intraband ESA estimate of Eb by checking it with 2-photon TA microscopy (gray circles). Specifically, we scanned the 2-photon pump energy across the XA and XS + Δ state regions and probed the transient interband optical conductivity of the bright state XS. The resulting 2-photon absorption spectrum is shown in Fig. 4b (gray line). Two ground state bleach peaks are centered at 1.18 and 1.82 eV (gray line) and they straddle the 1-photon absorption peak at 1.54 eV (green line). Like the PLE in Fig. 3, the binding energy is estimated by the peak splitting, δ + Δ  0.69 meV. Both measurements methods shown in blue and gray in Fig. 4c gaive very similar estimates of Eb. Our θ-tunable exciton binding energies are summarized in Table 1 and they are similar to the theoretically simulated binding energy value of 0.7 eV for a 21° tBLG domain6. Such large values from Eb are comparable to analogous semiconducting 1D and 2D excitonic materials36,39,40,41,42.

Delayed thermalization of interlayer excitons

Lastly, we measure the interlayer carrier lifetimes in tBLG. In Fig. 4c, a least squares method is used to fit the decay of blue line to obtain an intraband lifetime of 1.05 ps that we associate with the exciton lifetime XA. The resonant interband (green line) tBLG carrier relaxation dynamics have been previously reported23. Comparing the two TA decay rates plotted in Fig. 4c (after subtraction of the much weaker, non-resonant 0° TA graphene response, ΔσB), we note the absence of the fast initial electronic thermalization component typical for graphene (gray line). This suggests that interlayer XA excitons may thermalize by exciton–exciton or exciton-phonon assisted processes, rather than by scattering from the electronic continuum22. Our observations of delayed thermalization of resonantly excited interlayer carriers for both the interband and the intraband kinetics supports the theoretical prediction that the bound-exciton has weak coupling to metallic continuum state ($$H_{X_A,k} \cong 0$$)6. Long lifetimes and predicted weak exciton-continuum coupling both suggests interlayer excitons may thermalize slower than they would in single-layer graphene.

Discussion

In summary, we observe resonant, θ-tunable emission of PL upon direct 2-photon excitation of tBLG interlayer electronic states. Assuming hydrogenic exciton selection rules loosely apply, two-photon excitation was employed to optically excite dark exciton states10. Figures 2 and 3 show a ~5x resonant PL enhancement that we justify using a strongly bound-interlayer exciton model. This observation of resonant PL conflicts with the vHs model, as here electrons will thermalize rapidly to lower metallic continuum states by electron-electron scattering. Both CVD bilayer growths and dry-transferred stacked tBLG show similar PL emission in Figs. 2 and 3 for both the silicon nitride and fused silica substrates used. The PL is not surface quenched, suggesting it is emitted from protected interlayer electronic states.

As a 2D metal, tBLG interlayer excitons differ from excitons in similar materials like CNTs. For 2D metallic systems, unbound excitonic effects near the graphene M-point are well-known31,32. For tBLG, we suggest excitation of bound excitons are possible by mixing the two degenerate interlayer Fano resonances illustrated in Fig. 1b. Our estimates of the tBLG exciton binding energies are summarized in Table 1. We obtained Eb, from the spectral peak splitting in both near-IR intraband TA and 2-photon PLE microscopy. The exciton binding energy scales with θ increasing from ~0.5 eV at 16.3° to 0.69 at 7.9°. This first measurement of Eb agrees with ~0.7 eV predictions from prior theory6. BSE theory predicts zero exciton-continuum coupling after the two interlayer degenerate transitions rehybridize. In accord, Fig. 4c (blue line) shows that interlayer excitons have much slower thermalization rate compared to 0° bilayer graphene (gray line). The ability to optically prepare interlayer excitons suggests tBLG is a 2D hybrid material with bound excitons coexisting alongside the metallic continuum states before decaying on a ~1.1 ps timescale. This unusual metal-exciton character of tBLG may enable stacking angle tunable materials applications from optically induced metal-insulator transitions to efficiency enhancements in graphene photosensing.

Methods

tBLG sample characterization

PL was observed from tBLG prepared both by CVD and the manual layer stacking by the dry transfer method43,44. The substrates were silicon nitride and fused silica and measurements were generally performed in ambient under a continuous nitrogen purge. 2-photon PL was observed for both the stacking methods and substrates used. The stacking angles of individual tBLG domains were assigned primarily with hyperspectral linear absorption microscopy after subtraction of the non-resonance graphene background (σG)4. θ–stacking assignments were further verified by transient absorption (TA) and dark-field TEM5,17,23.

2-photon PL measurements

For the 2-photon PL and interband ESA measurements, we use a Coherent Chameleon (80 MHz, 140 fs) pumping an APE Compact optical parametric oscillator (OPO) with wavelength range 680–4000 nm. For the 2-photon PL microscopy measurements, the pump beam was generated by the OPO. The beam is then raster scanned by the piezo-scanning mirror and the 1-photon back reflection off the sample on an InGaAs detector provided a way to map the 2-photon and 1-photon responses concurrently. Total beam flux was 2 to 4 × 1014 photons/cm2. The emitted photoluminescence was measured with a TE cooled, back-illuminated EMCCD camera (ProEm HS, Princeton Instruments, 95% QE) and a Hamamatsu Si PMT as a secondary detection confirmation (with lock-in amplifier). To exclude scattered laser light from Rayleigh, Raman and residual second harmonic generation (SHG), collection of photons with energy less than twice the 2-photon excitation energy were excluded, calibrated Chroma or Thorlabs shortpass and bandpass optical filters stacks were used in front of the camera for emission detection to maintain an OD > 6 blockage throughout. All the measurements were performed at 295 K unless specified, in a nitrogen-purged box to minimize the chance of sample damage by oxidation.

ESA and 2-photon transient absorption spectra

ESA TA experiments were used to probe intraband exciton transitions (Fig. 4, blue lines) using confocal transient absorption microscopy23. The pump beam was modulated at 1 MHz with an AO-modulator. Pump fluences were on average at ~1014 photons/cm2 with fluence dependence showing a square root exciton–exciton annihilation trend (see Supplementary Figs. 7 and 8).

Rigorous power normalization curves were taken in-situ for all spectrally resolved measurements. This includes the microscope objective transmission corrections, the spectral response of the detection system, and spectral characteristics of the optical filters were taken into account for each wavelength.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Acknowledgements

This research was supported by the Spectroscopy Society of Pittsburgh and AFOSR (FA 9550-10-1-0410) grant.

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The experiment was performed by H.P. Twisted bilayer graphene was grown by L.H., C.-J.K. and J.P. Data analysis was performed by H.P. and M.G. Manuscript was written by H.P. and M.G.

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Correspondence to Matt W. Graham.

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Patel, H., Huang, L., Kim, C. et al. Stacking angle-tunable photoluminescence from interlayer exciton states in twisted bilayer graphene. Nat Commun 10, 1445 (2019). https://doi.org/10.1038/s41467-019-09097-x

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# How to politely decline my salary due to feeling I don't currently deserve it? [closed]

I am a software engineer at a startup. I'm feeling burned out from my job and am not performing up to what I would consider par. Therefore, I do not believe it is ethical for me to accept my salary, at least the full amount. How can I politely ask my employer not to give it to me until I feel that I have improved?

I thought about removing my account number from our payroll system but not sure if that would work.

## closed as off-topic by gnat, WorkerWithoutACause, The Wandering Dev Manager, Rory Alsop, user9158 Nov 20 '16 at 20:21

This question appears to be off-topic. The users who voted to close gave these specific reasons:

• "Questions require a goal that we can address. Rather than explaining the difficulties of your situation, explain what you want to do to make it better. For more information, see this meta post." – The Wandering Dev Manager, Rory Alsop, Community
• "Questions seeking advice on company-specific regulations, agreements, or policies should be directed to your manager or HR department. Questions that address only a specific company or position are of limited use to future visitors. Questions seeking legal advice should be directed to legal professionals. For more information, click here." – gnat, WorkerWithoutACause
If this question can be reworded to fit the rules in the help center, please edit the question.

• Are you going to expect extra pay on days when you are on a roll, feeling great, and the code just flows? – Patricia Shanahan Nov 17 '16 at 20:00
• I'm feeling burned out from my job and am not performing up to what I would consider par. Therefore, I do not believe it is ethical for me to accept my salary Serious question - why do you jump to "don't pay me" instead of "I need a vacation" or "time to find a new job" or "I quit"? – HopelessN00b Nov 18 '16 at 3:48
• I added tags psychology, mental-health, burnout since in all sincerity the strong consensus here is those may be the actual underlying issue; the original question asked didn't have a legal answer. – smci Nov 18 '16 at 12:20
• Does this sound like you? It's very common amongst software engineers, especially the top performing ones. – Qix Nov 18 '16 at 19:28
• Is this a troll?! – Nicolas Barbulesco Nov 19 '16 at 14:45

In short, no, you should not remove an account number or refuse pay.

You must be doing something right, because if you were really doing a bad job, you would be fired. It is understandable to feel like you aren't doing any good work or you are burned out, but the ultimate decision on how you are performing comes down to the company.

If you really need some form of validation, go ask your boss/colleagues/ceo of the startup for a quick status meeting and ask if your work lately is up to snuff and satisfying all requirements. If they say yes, then you should feel good. If you still feel bad, maybe that is indicative of something deeper and aligned with you not feeling challenged or engaged at work.

• There's a good chance the managers will take their cue from the OP in case the OP shows up with an even slightly unhappy face and asks more or less "Do you really think I'm working hard enough?". Their line of thinking may become "Oh, even khrit themselves think they're not doing well, there must be something to it!". Or even "Hey, here's an employee who's unsure about themselves and their performance, let's see what we can squeeze out of them before they fall over!" The employer is not an objective, unbiased evaluator of worth. Better to discuss this with an external counsellor first... – AllTheKingsHorses Nov 18 '16 at 8:59
• @SurprisedEuropean still, it's better than simply saying "don't pay me", right? But informal talk with colleagues at the same level or something more like "I feel you want from me more than I can give now, this job is burning me out" might be better, as it points to the real problem and does not put a blame on OP. – Mołot Nov 18 '16 at 9:40
• @Mołot Yes, way better. But still, the OP shouldn't give the managers an opportunity to justify the feeling of low (self-)worth without examining where the feeling comes from first. Better to use the "undeserved" money to get some counselling on depression, imposter syndrome or maybe just how to self-organise better so that they can improve their game. Which would likely benefit the employer as well because satisfied employees tend to deliver better quality. – AllTheKingsHorses Nov 18 '16 at 9:52
• +1 you should be having weekly catch-ups with your manager. There should be no surprises. If you're doing fine they can tell you. If you have to improve areas they can tell you. It's that simple, provided you listen, which is not always easy to do. – LoztInSpace Nov 18 '16 at 10:52
• @SurprisedEuropean Yes clearly the boss that offers you a raise that you didn't ask for is a Machiavellian genius. Paying people more than they expect is simply to trick them into admitting they could be working harder. Bwwwwahahaha! – JimmyJames Nov 18 '16 at 21:57

I sympathize greatly with you on this issue but I must agree with @dfundako that the ultimate responsibility of determining your worth to the company lies with the company.

Here's an 'outside of the box' suggestion: Consider donating whatever amount you are uncomfortable receiving to a worthy cause (Unemployment assistance, homeless shelter, etc.). That could ease your conscience without harming your future prospects.

How can I politely ask my employer not to give it to me until I feel that I have improved?

Chances are good that's illegal. I'm not a lawyer, or even an HR person, but there are state laws governing pay, including how often they're required to pay you. They can't just decide to not pay you.

Any decent company would not want their reputation tied to this kind of behavior (Did you hear they refuse to pay employees for performance reasons?) or want to enter into a payment agreement that would require all parties to keep it secret. There's absolutely nothing ethical about not paying someone working in what's supposed to be a paid position.

I would also think that this could cause problems with benefits (if they provide any) and employment taxes. Speaking of which:

I thought about removing my account number from our payroll system but not sure if that would work.

Do not do this.

If you wish to leave this company, then quit, but do not get yourself fired or worse by tampering with the payroll system.

• Have a discussion with your manager about your performance. Please take the at their word if they're happy with it. If not, discuss what you can change.
• Also talk to them about your workload/responsibilities. You may be able to ask to move into a different position or convince them to hire more help.
• Talk to someone who can help you manage your stress better. Sabotage is not a reasonable response to work stress; it worries me that you're thinking that way even if you're not entirely serious.
• Start job hunting. This job might just not be for you and that's OK.

Regarding the charity suggestions: Please don't take those as an OK to give away your entire paycheck.

• Tampering with accounts in payroll could very well land you in jail. That is fraud and is illegal in many countries. – HLGEM Nov 17 '16 at 22:18
• @HLGEM Even worse, it could look like the intent was to make the company make something illegal and them suing/extorting them. Additionally, a) no payroll software worth its name will fail to warn that it is supposed to pay someone with no account and b) Accounting should easily notice that there is more money in the bank than it was supposed to be there (that is one of the reasons of doing the whole accounting thing, although they usually try to prevent having less money than expected). – SJuan76 Nov 18 '16 at 9:30
• @HLGEM: That sounds a bit excessive. Our HR self-service system allows employees to change their account numbers directly. People change banks from time to time, and there's really no need to get HR staff involved in such trivia. (But you couldn't remove your number as it must be a valid IBAN number) – MSalters Nov 18 '16 at 11:20
• Something like, "I feel like I haven't been as effective as I could be recently. If you think so too, let's talk about what we might do about it." If he doesn't agree, then you can say that maybe you're imagining things and thank him for reassuring you. If he does agree, be ready with some ideas about changing your responsibilities or the like. – David Schwartz Nov 18 '16 at 11:54
• @MSaltersm changing something you are allowed to change through the user interface is acceptable, going into the backend and changing things without permission is not. I had made teh assumption that was what he was suggesting. Perhaps I was in error. But removing the bank account number woudl simply generate a paper check in any decently designed system as they legally have to pay you. Changing the backend so the system no longer tries to pay you would be illegal in most jurisdictions. – HLGEM Nov 18 '16 at 14:41

The market seems to be valuing you higher than you think you're worth. You should take the money.

First, it's odd to reject compensation. You will raise red flags with management. It's just strange behavior.

Second, it's not unethical to decline compensation but you will set a terrible precedent. They will remember this and may decide not to offer bonuses in the future...which may not be a problem unless you think you should get one.

If you find that you simply cannot bear to keep the money you can always give it to a charity; you don't need to keep it! The acpilot Retirement Charity and General Slush Fund is accepting donations and would be honored to relieve you of any money you believe you have not earned!! We prefer briefcases full of cash but will settle for a sack with a dollar sign on it.

Edit: At least one person took my charity pitch seriously so I guess this needs a warning label...

Caution: do not actually send money to anyone claiming to be a representative of the acpilot Retirement Charity and General Slush Fund. If anyone approaches you claiming to represent aRC&GSF, they are scamming you.

• Yes, if you feel uncomfortable with that extra money you think you don't deserve, there are tons of people that can better the world with it. Your therapyst for one. I bet your therapyst can better your world with that extra cash. – Mindwin Nov 17 '16 at 20:18
• Please let OP choose where to donate (if at all). Even if it's a non-rpofit organisation, your last paragraph basically is advertisment. – deviantfan Nov 17 '16 at 20:47
• However I am a Nigerian prince and have a huge fortune I would like to share with you... – IDrinkandIKnowThings Nov 17 '16 at 22:57
• @deviantfan It was a joke. If you notice "acpilot" is the poster's username plus the silly charity name, and the absurd payment delivery options are all signals this is a joke. – Erik Nov 17 '16 at 23:03

In addition to the existing answers regarding a reevaluation of the question whether this position is right for you and the possibility of donating a part of your salary, it is also possible that you have impostor syndrome.

It seems that you hold yourself to higher standards than your boss does. (Unless he should also be unhappy with your performance but hasn't told you just yet.) These standards are a positive thing as long as you realize that you're already on par and aim higher, they should not keep you from being satisfied with your performance in general.

Your asking this question raises major concerns about your health. I don't mean that as a criticism; mental health is incredibly important and the earlier you can intervene, the more problems you will save yourself.

I went through a health collapse a few years ago, and one of the effects was that I felt strongly about all sorts of ethical issues that aligned with my values. I presented arguments that seemed very convincing on paper, but in hindsight, they were completely lacking in perspective. Some were kind of crazy. I was a soul in desperate need of ways to be true to my beliefs.

I care deeply about many ethical issues; I always have. I think it is an honourable quality. But when my health deteriorated, I began to act on this in quite different ways to usual. The brain struggles with complex tasks when under a lot of stress; as a result I had a sort of "tunnel vision" and I could not see the big picture. I focussed on little details.

Let me give you a few thoughts that might help you see the big picture in your situation.

An ethical employer should see employees as more than just an expendable resource. The people in a business work together to create wealth, and the business should reward its employees with a share in that wealth. This goes beyond money; it can include rewarding work, a stimulating environment, social interaction, experience, training, opportunities, and various types of support, whilst employees contribute to a business in more ways than just completing tasks. In addition, employees should be seen as human beings and given a reasonable level of tolerance and leniency. A business that bleeds employees dry then kicks them out is clearly unethical. So, make sure you see yourself as more than an expendable resource.

Taking care of yourself is ethically vital. Over the long term, people who look after themselves can do much more for others. People who neglect themselves end up with nothing to give. It is certainly important for your employer that you restore (or at least stabilise) your health, otherwise you will eventually have to leave, which would mean your employer losing an asset.

Instead of asking "what am I worth?", ask "how can I become valuable again?" If your workload is too high, let your employer know. If you need to take some time off, let your employer know. (This makes a lot more sense to me than working full-time, inefficiently, for reduced pay.) If you are not sure what you need, talk to a healthcare professional, or someone you know; it is helpful to get an objective opinion.

Keep in mind that your burnout may be due to high workload and stress, but it also may be due to a deterioration of your personal life. Remember that a healthy diet, social interaction, sleep, exercise and attitude are important.

• "Remember that a healthy diet, social interaction, sleep, exercise and attitude are important." Everyone is different and I wouldn't be so sure about social interaction being important. Many software engineers are introverts which actually gain energy while alone and lose while in a group – kukis Nov 19 '16 at 15:24

Been there, done that :)

Excellent answers and comments. I'll add my bit to the pool. I do not have an answer to your question (since I was quite rude in rejecting the pay) but I'll try my best in helping you feel better.

I made the mistake and it has several downsides (Don't consider the monetary downsides for now)

Background:-

It was an internship and I did loads of actual work. I worked for 8 - 9 hours a days. I had an amazing boss who would spend considerable time guiding me and teaching me stuff. The whole lot were very caring and genuine. They even tried convincing me to take a pay 7 to 8 times.

The work I do can be done by any programmer with a year to two of experience. I'm currently 20 years old and it's my 4 month here. As you probably know interns normally aren't given any importance but it isn't the case here. They try to involve me in every activity possible and ensure that I learn as much as possible.

They gave their best yet I could not produce anything tangible that would justify their investment for about 2 weeks

A month into the internship and I become super productive and contribute loads (the work that I do doesn't directly help them make money). At times they become very busy with client work and don't have much time to spare for me (I don't mean I'm entitled to their time). At times they've asked us to put in extra hours. And not to forget that I'm very respectful and punctual.

It's such a remarkable period. They understand my needs and I understand their. I regret being rude especially because I'm young.

Things tend to happen fast in internships. Jobs not so much. Since you've got the thought of refusing pay instead of quitting, it's kinda obvious that you respect the job and the people there. Periods where you feel like a ninja and periods where you feel like a snail tend to come and go. They might even stretch for relatively longer stretches of time.

### Firstly do not refuse pay.

• How would you feel about hiring a genuine person who puts in efforts and not paying him?
• Your boss can interpret it as being proud.
• When you feel that you deserve a pay, it'll become difficult to ask for a pay.
• If you consider switching jobs your future employers might value you less.
• They might go overboard in trying to help you out. Since you think so much, that might upset you as well.
• It does not guarantee that you'll become better and start contributing something worthwhile.
• Your boss might be able to do what you do in much lesser time. The time you put in is key and you have bills to pay.

### Things you should understand/do

• Ask for a different role that does not wear you out. You can switch between the two. I say this because you've mentioned that it's a startup and as far as I know you are given more freedom than well setup business houses.
• Take a mini vacation. Your productivity should shoot after that. Make sure you aren't being paid for the vacation :)

• Take sometime out and learn relevant stuff everyday so that you can get things done faster.

• A lot of jobs involve learning on the job. Hence the employees cannot contribute during that phase.

• We are all humans. We aren't brimming with energy all the time. It's okay to not be productive. Don't be harsh on yourself.
• Consistency is the key. You don't have to give 100% everytime. Work hard and try becoming better.
• The startup too will expect you to put in extra hours of work at times. Don't crib and support their decision.
• Companies invest in people. Investments aren't rosy all the time.
• Find out what others contribute. You might be having very high expectations from yourself.

When relationships are good, you can always make things work. You needn't take this extreme step.

Try solving the actual problem. Don't find a way to reduce guilt :)

It's not unethical.

What you are worth in terms of a salary is NOT determined by what YOU think.

What you are worth is determined by what you can get people to agree to pay you.

No different than a bar of soap being worth as much as a business can charge for it and still make enough sales to cover the cost of producing the soap.

For everyone's convenience and financial planning, salaries for long term jobs are based on your average performance, not your best days and not your worst days, and especially not the days you are off sick or on paid vacation. People have ups and downs, but continually adjusting salaries according to employee mood would be an administrative nightmare.

Much of the cost of carrying an employee is in payroll taxes, reserves for sick and vacation days, office space, furniture, computers, manager costs, HR costs, etc. Those things go on even if you are not being paid. Indeed, doing something unusual with your salary would actually increase your share of management, HR, payroll, and possibly legal advice costs. Your employer is more likely to want you working at your normal level and being paid normally than to want to save on your salary.

So, your objective should be to get back to normal productivity. There are several good strategies in prior answers, such as taking a vacation and checking with your doctor in case you are depressed. Adding financial worries due to reduced income is not one of them.

Pay is by salary. Your boss chose this pay structure. You pay a lawyer even if you lose a case. You pay a doctor even if you don't get better. You pay a pharmacist even if the medicine didn't work for you. Life isn't perfect.

I certainly have had my days when i got a lot done, and my days I didn't get hardly anything done.

Ethics does not come into the picture here. I assume you are a staff employee, and not a contract employee. In any case, the value of an employee is not just in the salary paid. You add value to the startup with your presence. Even if you didn't work productively for a few days, the cost of your salary would average out to normal.

• Even if the number of working hours are 8, the employees would do actual work only for around 6 hours.
• Some days, there will be a lot of work. Some days there won't. Some days employees will work to their full potential. Some days they won't. Some days they might need casual leaves because of some emergency.

Shocking fact: Actually, you and other employees should be paid much much more than what you are already earning. Google for how much a freelancer charges per hour for the same kind of work you do. Find out how much your company is charging your client per hour. What you get paid, is a small amount of your actual work, after the company subtracts operational expenses, costs of the two facts I've mentioned in points above and a whole lot of other expenses.

Another reason you are valuable even you aren't productive is:

• You have domain knowledge of the company product(s).
• You have familiarity with the technologies, co-workers and clients.

It is entirely worth it for a company to pay you just to keep you onboard even if you aren't doing any work for a while. The money you are getting is fully deserved. Keep it. Your manager/HR, no matter how kind hearted, will only see you as a simpleton if you make an offer to take no salary. You won't get ANY plus points in their eyes. Instead, it'll lower your standing because they will see you as a person who does not understand the corporate world. This is a corporate; not school or a religious institution.

If it's about ethics, then there is the Unethical Decision Making; Preventing Ethical Blindness online course will help you focus on what you really need to concentrate on, when considering ethics.

ps: We get taught a lot of "do good" things in school and religion. You need to understand that those teachings are only to give us a sense of security and right and wrong during childhood. The adult world is different. It's time to start learning how the corporate world works and use it to your advantage.

• What do you mean by "your client"? – khrit344 Nov 18 '16 at 20:52
• Who will eventually be buying and using the product you create? Some company or person? That company or person is your client. – Nav Nov 20 '16 at 22:07

You are a salaried employee. You get a fixed amout of money every month. At regular times, usually once an year, companies review their workers' salaries and handle raises, promotions, empty promises or pink slips, according to workers' results and the company policies.

Fixed salaries are great because they are stable. You get it if you are sick, if the company has no projects, if sales go down or even if you are stupid.

Owners on the other hand take the risk. They can get rich but they also risk losing it all. That's their bet, not yours.

You should not in any way feel bad for taking the salary you agreed to. You fully deserve it. For all the time you put in before, for all the good things you will do. For all the things you could have done have they not overworked you. It's part of the way business works.

As for burnout, that's absolutely the company's problem. They are now paying back in your reduced productivity their lack of proper management.

But they won't pay you for the stress they caused to you. Take care of that yourself. Make sure you don't work overtime. Use your holiday to have a really good time. Maybe see a counsellor if you feel depressive. Try to reconnect with a hobby you enjoy.

This is fully what you both need and deserve. It's not your mistake. You should not pay for it.

• More so, even if the company did misjudge and award the wrong salary, that's their problem, and they have to own it. If they realise their mistake later and decide to bring it down, they will. Pointedly trying to pre-empt them by asking for less money is unlikely to do anything to help the situation either way - and the idea of going into the payroll data and removing the account number is completely absurd. Rather, the things the OP should do are to ensure they're thinking clearly, & if they're still sceptical about their pay, then ask more generally for a review, maybe a training plan, etc – underscore_d Nov 19 '16 at 13:41

Khrit, first off, I do agree with the others that suggest you should very carefully seek input on your physical and mental position, as indeed unfortunately we often are prone to "make choices" that we really aren't making, when it comes down to it. How many times have we looked back later and thought "what in the world was I possibly thinking there?!?". It's easy to go astray when we hit a rough patch, and it's even easier when we're not OK physically. So please do seek out input from the right people regarding that.

That said, it's depressing that no one even seems to consider the remotest of possibilities that someone might legitimately make such a decision on moral/fairness/doing-the-right-thing grounds.

It could be driven underneath by strong religious/honor principles.
It could be driven by a passion to see the company do well. It could be because the person is rather quite "hippyish" on things like shunning worldly possessions.
There are lots of different reasons one could reach this mindset, and they're absolutely not all based on questions of sanity or intelligence. But it seems most here just want to convince you that you are wrong.

If someone's fundamental position is that "a man should get what he earns", shouldn't that truly work both ways? If someone is willing to request a raise when their boss doesn't offer one, shouldn't they in fairness also be willing to request a pay cut when they feel they aren't doing their fair share??

A pastor can turn down his salary. A humanitarian worker can venture out for nothing.
Heck, if a company we bought a service from comes back to us and says "gosh, we don't think we did a good job, it's on the house", no one would be in an uproar... so it seems troubling that people would instinctively dismiss your notion.

Maybe it really isn't all about what we can gain from life.
It won't be popular, but if you're a person reading this, and you think it just a great windfall when you score free benefits from others, but outlandish that one should ever do that himself, consider that mathematically there must always be an equal amount of giving to account for those taking.
Such thinking is with survival of the fittest.
Such thinking is also directly in line with the character of Scrooge.
I for one applaud that there may be people that do things thinking more about the bigger picture/others, and that passions for fairness can trump self gain.

It can be easy to make these choices for the wrong reasons, definitely. Look, Khrit, my advice is that you must you must first work towards strong internal reliability to make unbiased choices.
To be able to dig in, and seek out ulterior motives that may be clouding the picture.
You can't do this to feel better about yourself/pat yourself on the back.
Or to please someone (such as the boss).
Or because you secretly enjoy self-castigation.
Or because you want to feel owed by others.
Or feel at liberty to act as you please, no longer constrained by the weight that comes with salary.
Or you want to feel safer in your job position.
All of those belie there is indeed bias festering underneath. And indeed, none of these reasons will likely bring about your wish in the longterm.

And perhaps most applicably, don't do it because of stress, thinking that it will be the magical release and it will just make it all go away.

Try to really be very challenging with yourself, looking for what deeper reasons are in the WHY you're drawn towards making this choice.

Perhaps it would also be wise to seek some intimate counsel from someone who knows you closely. That's not always an option... and it can also be very tough in circumstances like these where many people are likely to have the same immediate reflex shown here to decry the idea on principle without considering the circumstances more deeply. But if you can find someone who is open and wise, they can offer perspective and help you dig in to see whether any other issues are at play in wanting to do this.

As to the details of it... there are certainly labor laws which indeed could make it challenging.

Ideally, shouldn't it be that a person could somehow legitimately request such a move like this? We're living in a world where the growing sentiment seems to be "you should be free to do whatever you feel like." You can cheat on your wife, slash your wrists, or gamble all your money away without any restriction or mandated repercussions, but cannot turn down a salary based on ethical positions??? Unfortunately, I do also see the other side... it'd likely be quickly taken advantage of by some unscrupulous companies, and be pretty tough to patrol.

That said, maybe there are still some feasible options to work without getting paid. Maybe you could shift into a volunteer position or intern. It would depend both on what the law would allow, and on how much the company is interested in respecting your position.

Additionally, since you did mention you are in a startup, perhaps a lot of your concern is rather in a wish to see the company survive the period and grow into its potential? If that is where it's at, you could see if there are legal ways you can get paid in stock rather than salary. You wouldn't be the first to do that.

Perhaps you should sit down for a quiet talk with the boss, and try to explain your feelings. Face the problem straight on. Trying to get around it by fiddling with the payroll system probably (almost certainly) just winds up making things more complicated. If you really feel this way, you have to do it right.
Your boss will probably struggle with it a bit, show a bit of shock, and may well propose that you're doing this for wrong reasons. But perhaps if you make your request earnestly and soberly, and you continue to show you're acting rationally and thoughtfully in your everyday interactions, maybe you could convince them to work something out after a little while.

Now it may well also not be taken well. The company may flatly reject your request without any real consideration. Perhaps that leads to things getting worse, maybe even driving you towards questions of whether your employers themselves make smart/caring/fair/committed decisions, whether this is the right spot to be. Or they may lose confidence in you, start to see you as a loose cannon. Or it could even give them the motivation to inspect your work more, and realize you really aren't the employee you want!

Also realize, if you do wish to keep such an act quiet (perhaps wishing not to be a distraction/spectacle, or to make others feel bad)... people at your job may well still get wind of your choice (perhaps because others need to be brought on board to facilitate it, perhaps because of gossip, perhaps because the boss doesn't feel comfortable keeping such secrets, etc). And once it's known, quite a few people may start to see you as some mix between being self-righteous and being legitimately nuts... and it may be much colder at the workplace if that happens... That said, if you're convinced you're doing it for the right reasons, society is always telling us not to make choices based upon what people will think.
And generally it will blow over with time, if people get to know you for who you really are.

Honestly, if we live in a world where a range of people from Donald Trump to Michael Bloomberg to Elon Musk and Mark Zuckerberg can all turn down salaries, for a variety of reasons, is it really so preposterous that we might carefully choose such? Even moreso if we are indeed convinced we have done bad work?

khrit, I really hope your work turns around, and that you can soon oncemore feel confident that you are doing very productive work. It may not be popular, but there are people out there in their right mind who have made this choice and it has worked out ok. You have my support in this challenging time sir! Best wishes with it all!

• Pretty sure those billionaires are in pretty unique circumstances. The normal rules don't apply when your monthly investment dividend is going to buy office blocks. – Gusdor Nov 18 '16 at 10:44
• This didn't really answer the question in a way that is suitable for a Q/A forum. You have some decent points but you have gone way too much into detail and tangents. Looking answers aren't nervously bad, but they should do so only when a long technical explanation is required. – user30031 Nov 19 '16 at 15:01

You should strive for better but you shouldn't worry, and definitely not deprecate yourself, as that would probably make it worse.

If you were really underperforming, you would been told so. Until then, don't worry, even the very best programmers have unproductive days/weeks, as Joel Spolsky, CEO of Stack overflow confesses in his blog

(...) Sure, I come into the office, putter around, check my email every ten seconds, read the web, even do a few brainless tasks like paying the American Express bill. But getting back into the flow of writing code just doesn't happen.

These bouts of unproductiveness usually last for a day or two. But there have been times in my career as a developer when I went for weeks at a time without being able to get anything done. As they say, I'm not in flow. I'm not in the zone. I'm not anywhere.

Software engineering (or programming) is a weird job. It is said that the average programmer produces 10 productive lines of code a day. Be that true of not, when you self evaluate if you are up to par, do it on a year or six month average, not based on the last weeks/month

(probably this is more a comment than a proper answer. Unfortunately I don't currently have the rights to comment)


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Pygame Particle Emitter Problems

so I've been trying to create a basic particle system in pygame which I can later expand upon but I've run into a spot of bother creating the particle emitter. Obviously what's supposed to happen is it's supposed to create a new particle every frame (or a set of particles) but instead it just alters the particle that's created first time round. I understand why this is happening, as far as the game is concerned, every frame I add the same object to the particle list so when I call the update function it calculates the positional and velocity sums multiple times, what I'm having trouble with is fixing it. How would I go about making a new particle every time?

My existing classes for this are as follows:

class Particle():
def __init__(self, pos, vel, accel, image):
self.pos = pos
self.vel = vel
self.accel = accel
self.life = 255
self.lifecount = 1
self.image = image

def update(self):
self.life -= self.lifecount

if self.life <= 0:
return True
else:
return False

class Emitter():
def __init__(self, pos):
self.pos = PVector(pos[0], pos[1])
self.particles = []
self.part_list = [Particle(self.pos, PVector(0,2), PVector(0,0), pygame.image.load("resources/particles/fireball.png"))]

def update(self):
self.particles.extend(self.part_list)
for p in self.particles[:]:
p.update()
self.particles.remove(p)

def draw(self, surface):
for p in self.particles:
x = p.pos.x
y = p.pos.y
temp = pygame.Surface((p.image.get_width(), p.image.get_height())).convert()
temp.blit(surface, (-x, -y))
temp.blit(p.image, (0, 0))
temp.set_alpha(p.life)
surface.blit(temp, (x, y))

The positions, velocities and accelerations are all part of a vector class with vector math functions.

Just create the particles in the update method:

# Load the fireball once

class Emitter():
def __init__(self, pos):
self.pos = PVector(pos[0], pos[1])
self.particles = []
self.part_list = [] # Create an empty list of particles

def update(self):
# Add a new particle
p_pos = PVector(self.pos.x, self.pos.y)
p = Particle(p_pos, PVector(0,2), PVector(0,0), fireball)
self.particles.append(p)

# Update existing particles
for p in self.particles[:]:
p.update()


text:
# A property regarding complete/perfect squares.

When a set of natural numbers is under consideration, if we add first consecutive 'n' odd natural numbers(i.e. from 1 ) we get a complete square whose root is 'n' itself.

e.g. first 5 consecutive odd natural numbers are,
1,3,5,7,9 so,
1+3+5+7+9 = 25.
We get √25 = 5 .
or
e.g. first 6 consecutive odd natural numbers are,
1,3,5,7,9,11 so,
1+3+5+7+9+11 = 36
We get √36 = 6 .

I observed this while working on an algorithm to identify perfect squares. Has this been observed by any mathematician before?

• Boy, is this well known. However, your discovering it by yourself is good. Don't worry whether some you discovered is new - it probably is not. That does not matter. – marty cohen Aug 24 '15 at 18:52
• Duplicate of this and this. – Lucian Aug 25 '15 at 0:03

Yes, it's kind of amazing isn't it! To explain we have: Odd number $2i-1$
\begin{align} \sum_{i=1}^{n}(2i-1) &=2\sum_{i=1}^{n}i+\sum_{i=1}^{n}1 \\ &=\frac{2(n)(n-1)}{2}+n \\ &=n^{2}-n+n=n^2 \end{align}


text:
#### Volume 21, issue 2 (2021)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 Author Index To Appear Other MSP Journals
Moduli spaces of Hecke modifications for rational and elliptic curves

### David Boozer

Algebraic & Geometric Topology 21 (2021) 543–600
##### Abstract

We propose definitions of complex manifolds ${\mathsc{𝒫}}_{M}\left(X,m,n\right)$ that could potentially be used to construct the symplectic Khovanov homology of $n$–stranded links in lens spaces. The manifolds ${\mathsc{𝒫}}_{M}\left(X,m,n\right)$ are defined as moduli spaces of Hecke modifications of rank $2$ parabolic bundles over an elliptic curve $X\phantom{\rule{-0.17em}{0ex}}$. To characterize these spaces, we describe all possible Hecke modifications of all possible rank $2$ vector bundles over $X\phantom{\rule{-0.17em}{0ex}}$, and we use these results to define a canonical open embedding of ${\mathsc{𝒫}}_{M}\left(X,m,n\right)$ into ${M}^{s}\left(X,m+n\right)$, the moduli space of stable rank $2$ parabolic bundles over $X$ with trivial determinant bundle and $m+n$ marked points. We explicitly compute ${\mathsc{𝒫}}_{M}\left(X,1,n\right)$ for $n=0,1,2$. For comparison, we present analogous results for the case of rational curves, for which a corresponding complex manifold ${\mathsc{𝒫}}_{M}\left({ℂℙ}^{1},3,n\right)$ is isomorphic for $n$ even to a space $\mathsc{𝒴}\left({S}^{2},n\right)$ defined by Seidel and Smith that can be used to compute the symplectic Khovanov homology of $n$–stranded links in ${S}^{3}$.

##### Keywords
Hecke modifications, rational curves, elliptic curves, vector bundles, parabolic bundles, Khovanov homology
##### Mathematical Subject Classification 2010
Primary: 14H52, 14H99