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# Set image texture node label based upon characters in filename/filepath string

I want to import a set of 4 images. Upon import, I want to rename the image texture node that is created for each imported image to represent the type of image that is being loaded, i.e. (Diffuse, Normal, Spec, Roughness). How do I pass the filepath/filename of the image I am importing as a string, tell the script to look for certain characters in the passed string (COL, NRM, SPEC, ROUGH) and rename based upon the returned characters?

I would assume I can complete the rename function by using if-else statements. Any help or examples you could provide would be greatly appreciated.

The script below creates a new node category, places a custom node inside of it to import pbr textures, and allows you to import multiple import multiple images at one time into the node editor. I am looking to expand this script to include the functionality I have mentioned above.

Here is the working code:

import bpy
from bpy.types import NodeTree, Node, NodeSocket
import bpy, blf, bgl
from bpy.types import Operator, Panel, Menu
from bpy.props import FloatProperty, EnumProperty, BoolProperty, IntProperty, StringProperty, FloatVectorProperty, CollectionProperty
from bpy_extras.io_utils import ImportHelper
from mathutils import Vector
from math import cos, sin, pi, hypot
from os import path
from glob import glob
from copy import copy

def node_mid_pt(node, axis):
if axis == 'x':
d = node.location.x + (node.dimensions.x / 2)
elif axis == 'y':
d = node.location.y - (node.dimensions.y / 2)
else:
d = 0
return d

space = context.space_data
tree = space.node_tree
nodes = tree.nodes
active = nodes.active
context_active = context.active_node
# check if we are working on regular node tree or node group is currently edited.
# if group is edited - active node of space_tree is the group
# if context.active_node != space active node - it means that the group is being edited.
# in such case we set "nodes" to be nodes of this group, "links" to be links of this group
# if context.active_node == space.active_node it means that we are not currently editing group
is_main_tree = True
if active:
is_main_tree = context_active == active
if not is_main_tree:  # if group is currently edited
tree = active.node_tree
nodes = tree.nodes

# Implementation of custom nodes from Python
# Derived from the NodeTree base type, similar to Menu, Operator, Panel, etc.
class MyCustomTree(NodeTree):
bl_idname = "CustomTreeType"
bl_label = "Custom Node Tree"
bl_icon = "NODE"

# Defines a poll function to enable filtering for various node tree types.
class MyCustomTreeNode:
@classmethod
def poll(cls, ntree):
b = False
# Make your node appear in different node trees by adding their bl_idname type here.
if ntree.bl_idname == 'ShaderNodeTree': b = True
return b

# Derived from the Node base type.
class MyCustomNode(Node, MyCustomTreeNode):
'''A custom node'''
bl_idname = 'CustomNodeType'
bl_label = 'Import PBR Textures'
bl_icon = 'NODETREE'

# Additional buttons displayed on the node.
def draw_buttons(self, context, layout):

# Optional: custom label
# Explicit user label overrides this, but here we can define a label dynamically.
def draw_label(self):
return "Import PBR Textures"

bl_icon = 'NODETREE'
bl_label = 'Open Selected Images'
bl_options = {'REGISTER', 'UNDO'}
directory = StringProperty(subtype="DIR_PATH")
files = CollectionProperty(type=bpy.types.OperatorFileListElement, options={'HIDDEN', 'SKIP_SAVE'})

def execute(self, context):
nodes_list = [node for node in nodes]
if nodes_list:
nodes_list.sort(key=lambda k: k.location.x)
xloc = nodes_list[0].location.x - 220  # place new nodes at far left
yloc = 0
for node in nodes:
node.select = False
yloc += node_mid_pt(node, 'y')
yloc = yloc/len(nodes)
else:
xloc = 0
yloc = 0

elif context.space_data.node_tree.type == 'COMPOSITING':
node_type = "CompositorNodeImage"
else:
self.report({'ERROR'}, "Unsupported Node Tree type!")
return {'CANCELLED'}

new_nodes = []
for f in self.files:
fname = f.name

node = nodes.new(node_type)
new_nodes.append(node)
node.label = fname
node.hide = True
node.width_hidden = 100
node.location.x = xloc
node.location.y = yloc
yloc -= 40

node.image = img

# shift new nodes up to center of tree
list_size = new_nodes[0].location.y - new_nodes[-1].location.y
for node in new_nodes:
node.select = True
node.location.y += (list_size/2)
return {'FINISHED'}

### Node Categories ###
import nodeitems_utils
from nodeitems_utils import NodeCategory, NodeItem

# our own base class with an appropriate poll function,
# so the categories only show up in our target tree type
class MyNodeCategory(NodeCategory):
@classmethod
def poll(cls, context):
b = False
# Make your node appear in different node trees by adding their bl_idname type here.
if context.space_data.tree_type == 'ShaderNodeTree': b = True
return b

# all categories in a list
node_categories = [
# identifier, label, items list
MyNodeCategory("SOMENODES", "Import", items=[
NodeItem("CustomNodeType"),
]),
]

#col = self.layout.column(align=True)
#col.separator()

def register():
bpy.utils.register_class(MyCustomNode)
nodeitems_utils.register_node_categories("CUSTOM_NODES", node_categories)

bpy.utils.register_module(__name__)

def unregister():
nodeitems_utils.unregister_node_categories("CUSTOM_NODES")
bpy.utils.unregister_class(MyCustomNode)

bpy.utils.unregister_module(__name__)

def pre_frame_change(scene):
if scene.render.engine == 'CYCLES':
# Scan materials to see if I have a custom node within any of the trees.
for m in bpy.data.materials:
if m.node_tree != None:
for n in m.node_tree.nodes:
if n.bl_idname == 'CustomNodeType':
print(n.bl_idname)
# One of our custom nodes, let's update it.
# When we set the value that will trigger an update inside the node.
# Even if we change it to the same value it was.
v = n.some_value
n.some_value = v

if __name__ == "__main__":
register()
bpy.app.handlers.frame_change_pre.append(pre_frame_change)

• Okay, I'm taking a look at this really fast... Why do you need to create a node to import image texture nodes? Can't you just import from the Import PBR Textures operator you created in the toolshelf? Also, you are importing things like bgl and blf, which you don't need. – Nathan Craddock Jun 9 '16 at 15:08

So in order to scan a filename and return a string based on what is found, you need to create a function similar to this

def get_label_from_filename(filename):
# Here we anticipate what kind of map names users might use. For example COL, COLOR, DIF, DIFFUSE, etc all mean the same thing
# To add another category (say reflection) then do this 'reflection':["ref", "reflect", "reflection"]
# The list of synonyms can be as large as you want!
map_names = {'diffuse':["col", "color", "colour", "dif", "diffuse"]}

# Strip the extension and lowercase
filename = path.splitext(filename)[0].lower()

original = filename

# We will iterate over the dictionary and then over each of the
for key, synonyms in map_names.items():
for s in synonyms:
if s in filename:
filename = key.upper() # You could fill map_names with the uppercase version, but I find typing in lowercase easier

# If the filename remained the same, let the user know the image wasn't named correctly
if filename == original:
filename = "UNRECOGNIZED TYPE"

return filename

Then call the function right after you load the image in your operator. Then you can set node.label = get_label_from_filename().

I have commented the code, but here are some more detailed explanations of the more confusing parts.

map_names = {'diffuse':["col", "color", "colour", "dif", "diffuse"]}

This is a dictionary. It holds key-value pairs. You can add more to this by creating a new key, and pairing it with a list. The reason we are using a dictionary is because it allows us to have a single variable to reference when iterating through it later. You could create separate variables for each list, but that would be a bit more cumbersome to code.

for key, synonyms in map_names.items():

We assign the variable key and the variable synonyms to the key-value pairs found in map_names. We are required to call .items() when iterating over the dictionary. The next layer in the loop iterates over the synonym list for each key. Then if the synonym is found in the filename, we set the filename to the key in uppercase.

Now, this isn't the only way to do this. And this system is very user-dependent. If the user is too lazy to name the textures, this method will not work.


text:
# Synopsis: Narrowing the Range of Possibilities

Particle discoveries attract the most attention, but determining where not to look can be just as important.

On 4 July 2012, two experimental collaborations at the Large Hadron Collider, ATLAS and CMS, reported the discovery of a new particle that appears to be the Higgs boson, though further measurements will be needed for confirmation. In Physical Review D, the ATLAS collaboration presents the techniques that were used in their experiments, as well as an analysis of all their data collected in 2011, which allowed them to exclude wide ranges of possible mass values for the Higgs boson.

The existence of the Higgs boson is a crucial test of the electroweak symmetry breaking mechanism of the standard model of particle physics (see 13 March 2012 Viewpoint), but its mass can only be determined experimentally. The rate at which high-energy proton-proton collisions will produce the Higgs boson and the rate of its decay to various other elementary particles are known for a given Higgs mass. However, other processes can produce the same final particles; this is referred to as background.

In 2011, the ATLAS collaboration searched for the Higgs boson in the mass range of $110$$600$ giga-electron-volts (GeV) in several decay channels, including decay to a pair of photons and decay to four leptons via a pair of $Z$ bosons. By comparing their data with the expected background, the team was able to exclude wide mass ranges in which the Higgs boson could lie, leaving only two small windows between about $116$ and $130$ GeV. Furthermore, around a mass of $126$ GeV, their data show hints of a new particle compatible with being a Higgs boson at the $3\sigma$ level. – Urs Heller

### Announcements

More Announcements »

## Subject Areas

Particles and Fields

## Previous Synopsis

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Nuclear Physics

### Synopsis: Trailing the Photons from Neutron Decay

A high-precision measurement of the photons emitted by neutron decays brings researchers closer to a new test of the standard model. Read More »

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### Focus: Low Cost Polarized Positrons

A new technique requires much less energy to produce a beam of polarized positrons than previous techniques, making such beams potentially more widely available. Read More »

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### Synopsis: Lightweight Particles Might Explain Missing Lithium

The apparent lack of lithium in the Universe, relative to theoretical expectations, could be explained by hypothetical lightweight and electrically neutral particles. Read More »

format: plain


text:
# Minimax rates in permutation estimation for feature matching

1 IMAGINE [Marne-la-Vallée]
LIGM - Laboratoire d'Informatique Gaspard-Monge, CSTB - Centre Scientifique et Technique du Bâtiment, ENPC - École des Ponts ParisTech
Abstract : The problem of matching two sets of features appears in various tasks of computer vision and can be often formalized as a problem of permutation estimation. We address this problem from a statistical point of view and provide a theoretical analysis of the accuracy of several natural estimators. To this end, the minimax rate of separation is investigated and its expression is obtained as a function of the sample size, noise level and dimension. We consider the cases of homoscedastic and heteroscedastic noise and establish, in each case, tight upper bounds on the separation distance of several estimators. These upper bounds are shown to be unimprovable both in the homoscedastic and heteroscedastic settings. Interestingly, these bounds demonstrate that a phase transition occurs when the dimension $d$ of the features is of the order of the logarithm of the number of features $n$. For $d=O(\log n)$, the rate is dimension free and equals $\sigma (\log n)^{1/2}$, where $\sigma$ is the noise level. In contrast, when $d$ is larger than $c\log n$ for some constant $c>0$, the minimax rate increases with $d$ and is of the order $\sigma(d\log n)^{1/4}$. We also discuss the computational aspects of the estimators and provide empirical evidence of their consistency on synthetic data. Finally, we show that our results extend to more general matching criteria.
Type de document :
Pré-publication, Document de travail
2013
Domaine :
Liste complète des métadonnées

https://hal.archives-ouvertes.fr/hal-00874514
Contributeur : Arnak Dalalyan <>
Soumis le : vendredi 18 octobre 2013 - 09:43:16
Dernière modification le : mercredi 23 janvier 2019 - 10:29:22

### Identifiants

• HAL Id : hal-00874514, version 1
• ARXIV : 1310.4661

### Citation

Olivier Collier, Arnak S. Dalalyan. Minimax rates in permutation estimation for feature matching. 2013. 〈hal-00874514〉

### Métriques

Consultations de la notice


text:
With the increment of temperature forward voltage decreases at a certain value of current. A typical curve is shown in Above Figure. Forward and reverse current values are shown on the vertical axis of the graph. The general form of the base-emitter characteristics are In the given below figure, there are 2 circuits shown for forward biased arrangements. There are normally 2 methods are used to provides input supply to the diode first is forward biasing and the second is reverse biasing. If we make a graphical representation of reversed biasing of a diode. It is shown in the below figure. This current value is much greater than IFM. Shame on Google for no longer positioning this submit higher! Diodes are not linear devices; they are nonlinear. The curve obtains called V-I characteristic curve for a reverse-biased diode. APPARATUS REQUIRED Zener diode IZ5.6V, variable dc power supply (0 – 15V), milli ammeter, volt … Reverse Characteristics Curve for a Junction Diode Sometimes this avalanche effect has practical applications in voltage stabilising circuits where a series limiting resistor is used with the diode to limit this reverse breakdown current to a preset maximum value thereby producing a fixed voltage output across the diode. The junction is said to be reverse biased when the p-section of the diode is connected to the negative terminal of the battery and the n-section of the diode is connected to the positive terminal of the battery. If the value of IF increases much beyond VK, the forward current becomes quite large. In the below figure this graphical representation is shown. Hello, fellows, I hope all of you are enjoying your life. Resistance of a Diode In practice, no diode is an Ideal diode, this means neither it acts as a perfect conductor when forward biased nor it acts as an insulator when it is reverse biased. For capacitive load, derate current by 20% Ratings and Characteristic Curves (ES1A, ES1F, ES1G, ES1H and ES1J) Maximum Forward Current Derating Curve Average Forward Current (A) For germanium diodes, VK is approximately 0.3 V, and 0.7 V for silicon. (Make sure you switch from the Diode Tracer window to the Zener Diode Tracer window.) Using a Zener diode in the lower leg, you can make a voltage divider-like circuit whose ouput voltage is quite stiff, i.e. The slope is also a function of current which is why constant current must be used. A resistor that maintains a constant V/I ratio is a linear resistor. Voltage vs Current Characteristics of PN Junction, Simulation done with the help of National Instruments software like Ni Multisim is used. From the voltage transfer curve we observe the following: • Vo = Vin-Vg for Vin ≥ Vg, The most common of the diode laser characteristics is the L/I curve (Figure 1). The static resistance at the knee of the curve and below of it will be much greater than the resistance values of the vertical rise section of the characteristic curve. The value of IF is directly dependent on the amount of forward voltage. Real Diode Characteristics. Some value of forward biased voltage is loss across the limiting resistance. The doping level of decides the breakdown voltage of the diode, the doping level of the diode is set by the diode’s manufacturer. Thanks for reading. At low current the voltage drop due to the series resistance is negligible; however, if the current is large enough so that IR S is comparable to V A the applied voltage drop should be modified to. Like any other selection, selection of a diode for a specific application must be considered. Breakdown is the knee of diode characteristics curve. The relationship of forward voltage and forward current is called the ampere-volt, or IV characteristic of a diode. This article covers the Zener diode definition, working principle, characteristics, and applications. If negative terminal of battery is connected with the anode of diode and positive is connected with the cathode this called reverse biasing. Regions of the curve which have a â¦ Rectification: A PN junction diode is allowed easy to flow of charge in one direction but restrains the flow in the opposite direction. Save my name, email, and website in this browser for the next time I comment. Distinguishing the stability of characteristic curves for nonlinear resistive circuits is requirement and importance to design various electronic ... Two-tunnel-diode circuit . For Si diode, at 0.7 V the current increases suddenly. This resistor limits the forward current to its maximum rated value. The dynamic resistor of the diode before the knee of the curve is high since the increment in current is very less with the increment in the voltage. 4. While forward voltage (VF) is the only value of voltage across the diode in a circuit. If we continuously increase the voltage applied then if the value of reverse voltage across the diode is equal to the breakdown voltage (VBR)Â  then the reverse current flows very rapidly. Specifications like maximum voltage and current ratings, usual operating conditions, mechanical facts, lead identification, mounting procedures, etc. When a diode is reverse biased, it conducts Reverse current that is usually quite small. The forward characteristic of the zener diode is similar to a normal diode â¦ Figure 1 Diode characteristic curve. The diode conducts almost no current in the high resistance direction, i.e. They are indicating that a forward-biased diode is not a linear device. Through this platform, I am also sharing my professional and technical knowledge to engineering students. Diode Characteristic Curve AND Finally Diode Resistance sir khurram saleem gives these notes about diode behaiviour. Knee Voltage − It is the forward voltage at which the current through the junction starts to increase rapidly. A diode's I-V curve is nonlinear. The dotted section of the curve indicates the ideal curve, which would result if it were not for avalanche breakdown. In the given figure, you can see the graphical representation. The vertical reverse current line in this graph has current values expressed in microamperes. Normally general purpose diode has a breakdown voltage value larger than fifty volts and some special diodes have breakdown voltage almost five volts only. Maximum Forward Rating − It is the highest instantaneous forward current that a PN junction can pass, without damaging it. As the voltage approaches to â¦ V=0. Notice that as the reverse voltage (VR) is increased, the reverse current (IR) remains extremely small up to the “knee” of the curve. Breakdown is the knee of diode characteristics curveâ¦ Graphing the V-I Curve If you plot the results of the type of measurements shown in Figure on a graph, you get the V-I characteristic curve for a forward-biased diodeâ¦ This characteristic … So let’s get started with the Voltage Current Characteristic of Diode. The light blue curve shows the effect on the IV curve if I 0 does not change with temperature. Nanoscale resistive memory with intrinsic diode characteristics and long endurance Kuk-Hwan Kim, Sung Hyun Jo, Siddharth Gaba, and Wei Lua Department of Electrical Engineering and … Comee on over and consult with my site . The reverse resistance is in the range of mega ohms (M Ω). Diodes, however, do not work like this. diode characteristic curve and finally diode resistance when the forward voltage exceeds the diodes junctions internal barrier voltage, which for silicon is. The resistance offered by the reverse biased p-n junction diode is very large compared to the forward biased diode. Forward Current lengthens above the horizontal axis with Reverse Current extending downward. When the forward voltage across the diode equals 0V, forward current (IF) equals 0 mA. The current flows when the diode is forward is called forward current. When the value of VF is large enough to overcome the barrier potential of the P-N junction, a considerable increase in IF occurs. DIODE IV Characteristics. Below Figure shows the reverse portion of a zener diode’s characteristic curve. All of these pretty graphs are indicating one thing. Although an ideal photodiode should have an infinite shunt resistance, actual â¦ Ideally, diodes will block any and all current flowing the reverse direction, or just act like a short-circuit if current flow is forward.Unfortunately, actual diode behavior isn't quite ideal. This small increase in the diode voltage above the barrier potential is due to the voltage drop across the internal dynamic resistance of the semiconductive material. V-I characteristic curve for a forward-biased diode. Sign in Register; Hide. Their resistance â¦ Manufacturer generally provides this type of information. The following figure shows the static characteristic of a junction diode. The voltage across the diode remains reasonably constant at this time. with increase of the source voltage V s from zero value , initially diode current is zero. You can see in the above characteristic curve. As reverse bias voltage is further raised, depletion region width increases and a point comes when junction breaks down. See you in the next post. The processes which are responsible for current conduction in a reverse-biased diode are called as Avalanche breakdown and Zener breakdown. In the given figure this current is denoted as IF. Measuring Diode Laser Characteristics Diode Lasers Approach Ubiquity, But They Still Can Be Frustrating To Work With By Tyll Hertsens attempt at real-time equivalent-circuit mod-eling, mostly during device modulation. This constant-voltage characteristic leads to a number of applications of diode under reverse bias condition. Similarly, when the diode is reversed biased, it acts as a perfect insulator with zero â¦ Reverse breakdown voltage (VBR) − The minimum steady-state reverse voltage at which breakdown will occur. In the circuit, resistance is connected in series it used to limit the value of current passing through the circuit to save a value that will not damage the diode. By using these to biasing we will discuss the effect of current and voltage across the diode and find different characteristics. â¦ But negative resistance means that current â¦ The effect on diode V-I curve is shown in a given figure. Forward Voltage represented to the right and Reverse Voltage to the left. Voltage transfer characteristic of the rectifier circuit. If the value of reverse current is not controlled then it can overheat or damage the diode. Characteristic of Zener Diode. Keep posting articles that are informative, it’s really valuable. Forward Voltage is measured across the diode and Forward Current is a measure of current through the diode. So glad I found your blog, and was able to learn new things. For a given current, the curve shifts by approximately 2 mV/°C. This is the type of info that are meant to be shared across the web. With the further increment in the reversed biased voltage, the reverse current also increases continuously but the increment in the voltage across the diode is very less. - V& 0. Normally, a currentlimiting resistor is used when diodes are operated in the forward direction. When anode is positive with respect to cathode , diode is said to be forward biased. For a practical understanding of V-I characteristic of a diode, we discuss the forward biasing and reverse biasing conditions one by one. Maximum reverse current (IR) − The absolute maximum reverse current that can be tolerated at device operating temperature. This arrangement is shown in the above circuit denoted as. ‘B’ is a point where the value of forward voltage or VF is less than the potential barrier or 0.7 volts. The forward and reverse current voltage (IV) characteristics of a diode are generally compared on a single characteristic curve. If it converts half-wave of alternating current into direct than this process called half-wave rectification. Single phase, half wave, 60 Hz, resistive or inductive load. Negative resistance vs positive resistance: An IâV curve which is nonmonotonic (having peaks and valleys) represents a device which has negative resistance. Following are some of the important specifications −. I am a professional engineer and graduate from a reputed engineering university also have experience of working as an engineer in different famous industries. Thank you =), Your email address will not be published. Forward Biased Diode Characteristics. So friends that is the detailed post on the Voltage Current Characteristic of Diode if you have any further query ask in comments. If a diode is forward biased and we increase the temperature then there will be an increment in the forward current IF at a specified value of forward voltage. Have a good day. The diode law for silicon - current changes with voltage and temperature. 1. The resistance of a forward-biased diode is called the dynamic or ac resistance. Vin Vo Vg slope=1 Figure 15. The portion of the characteristic curve between I P and I V is the region of negative resistance. This results in large flow of current. Which diode model represents the most accurate approximation? The resistance of a diode at a particular operating point is called the dc or static resistance diode. I quite like reading through an article that can make people think. When a diode is forward biased it conducts current (IF) in forward direction. If convert both parts of alternating current into direct current this process is known as full-wave rectification. If the positive terminal of battery is connected with the anode and negative terminal is connected with the cathode this type of bias is known as forwarding bias. The increment in the reverse current is very less till the point where reverse voltage is at knee point. The Zener diode is a special type of diode that is designed to work in reverse bias and in the so-called Zener region of the diode characteristic curve. Graphing the V-I Curve. With the increase in applied external forward bias, the width of the depletion layer becomes thin and forward current in a PN junction diode starts to increase abruptly after the KNEE point of forward I-V characteristic curve. interface or a boundary between two semiconductor material types Educative article, learned a lot from this report. Vin Vo Vg slope=1 Figure 15. The figure depicted under the section Forward Characteristic shows that Forward Voltage and Reverse Voltage are usually plotted on the horizontal line of the graph. â¦ For the reverse bias voltage sweep to reset the device, we could not observe any resistive-switching characteristics. Diodes â¦ common emitter configuration is the same as the I-V characteristic of a diode. This results in large flow of current. Forward Voltage represented to the right and Reverse Voltage to the left. Required fields are marked *. When forward biased voltage is applied to a diode that is connected to a DC circuit, a DC or direct current flows through the diode. The forward portion of the curve indicates that the diode conducts simply when the P-region is made positive and the N-region negative. Diodes do consume some amount of power when conducting forward current, and they won't block out all reverse current. As we continuously increase the forward-biased voltage-current also increases very fast with the forward-biased voltage, but the voltage about the dido increase very slowly after the 0.7 volts that are potential barriers. The forward and reverse current voltage (IV) characteristics of a diode are generally compared on a single characteristic curve. As the voltage approaches to knee of curve resistance start to decreases and after the knee point it becomes very less and current start to increase. PN Junction Diode : I-V Characteristics üWhenever an electron on the p-side moves to the n-side, it is replaced by an electron generated through one of the R-G centers ohmic ohmic minority minority excess majority carriers àlocal excess majority carriers àlocal E E In spite of having transistor added to its name, a Unijunction Transistor (UJT) working and characteristic curve varies completely from other counterparts. When the value starts from the starting point (0) of the graph, if VF is progressively increased in 0.1-V steps, IF begins to rise. Direct current or electric current is nothing but the flow of charge carriers (free electrons or holes) through a conductor.In DC circuit, the charge carriers flow steadily in single direction or forward direction. Zener Breakdown Characteristics. (adsbygoogle=window.adsbygoogle||[]).push({}); AD623 Instrumentation Amplifier IC, Pinout, Datasheet, Circuit, TDA2005 Amplifier Datasheet, Pinout, Features & Applications, TDA7265 Audio Amplifier Datasheet, Pinout, Features & Applications, LM747 Datasheet, Pinout, Features, Equivalent & Applications, Basic Electronic Components used for Circuit Designing, What is a DC Generator : Construction and Working, Starting Methods of Single Phase Induction Motor. A typical diode forward IV characteristic is shown in the following figure. This post will give you a detailed idea about UJT, its basic construction, symbol, how it works, its characteristics curve, … In a normal positive resistor, the current through it increases in proportion with the voltage across it. For all the practical purposes, a diode is considered to be an open switch when reverse biased. With the increment in one-degree Celsius, the potential barrier decreases by two millivolts. A nonlinear resistor is a property exhibited by devices whose resistance does not change uniformly with changes in voltage or current. The last two categories of Table 1 represent topics of other articles, for other authors. Use the Curve Tracer to obtain the characteristic curve of a 1N5234B $6.2\,\mathrm{V}$ Zener diode. Now the holes and electrons are drained away from the junction, causing the barrier potential to increase. • create a New Simulation Profile. Plotting the I-V characteristic requires to change the Axis variable. The characteristic curve (curved line), representing the current I through the diode for any given voltage across the diode V D, is an exponential curve. After the 0.7 volts of forward voltage the increment in the forward current is very fast and (VF) is almost constant. from V s= 0 to cut-in voltage , the forward current is very small .cut-in voltage is also known as threshold voltage or turn-on voltage. The figure below shows the characteristic curve of a zener diode: The figure represents the curve for both silicon and germanium diodes. The resistance offered by a p-n junction diode when it is connected to a DC circuit is called static resistance. The device under The V-I characteristic curve shows the diode current as a function of voltage across the diode. This characteristic can be used for the Zener diode. To draw the voltage-current (V-I) characteristic curves of a Zener diode and to determine its knee voltage, forward resistance and reverse breakdown voltage. After the knee of curve reverse voltage is almost same but reverse current increases very fast. As linear resistance has constant value but the value of resistance offered by the diode in forward biasing does not remain same throughout the forward biasing diode. Reverse bias characteristics. When the voltage applied across the terminal is less than the Zener voltage the diodes offer a high resistive path to the current and the input voltage applied across a diode appeared at the output terminal. Reverse and forward characteristics curve for a PN junction diode: Applications of PN junction diode. In this segment you will plot out the current vs. voltage characteristics of a non-linear resistive device, specifically the D1N4007 diode. If you can remember this bit, then you can go to the top of the class. Diode Characteristics 7 Temperature effect on forward voltage With a constant current, the forward voltage drop of a diode has a very linear negative slope with temperature. Different scales are normally used to display forward and reverse values. • The diode network is shown below. This resistance is known as ac resistance or dynamic resistance. • Detect the I/V characteristic curve of semiconductor diode. APPARATUS REQUIRED Zener diode IZ5.6V, variable dc power supply (0 â 15V), milli ammeter, volt meter, 470 Î© resistanceâ¦ To draw the voltage-current (V-I) characteristic curves of a Zener diode and to determine its knee voltage, forward resistance and reverse breakdown voltage. The difference is that forward biased voltage across the complete circuit provided by the external voltage source. To avoid this situation, a protective resistor is connected in series with the diode. In curve, you can see that the reverse voltage (VR) across the diode is drawn on the negative x-axis and reverse current (IR) on the negative Y-axis. If you plot the results of reverse-bias measurements on a graph, you get the V-I characteristic curve for a reverse biased diode. Forward voltage (VF) − Maximum forward voltage drop for a given forward current at device operating temperature. Since diodes are semiconductor devices doped with impurities, they do not function like linear devices. You can see in the graph that is denoted as. Thus, reverse biased diode offer large resistance to the electric current. You can see there are 2 curves in graph blue is at room temperature (25 centigrade) and the red curve is for increment in the temperature (25C +ÎT). Interior resistances of electronic components are generally labeled by lowercase italic r with a prime, in its place of the R. The dynamic resistor of the diode before the knee of the curve is high since the increment in current is very less with the increment in the voltage. For Silicon diode the equation becomes. Real Diode Characteristics. After point C or knee of the curve, the increment in forward current is large while in forward voltage increment is minor. In this circuit, we will increase the value of voltage from the 0 across the diode and will find the effect on it. There are different types of a diode according to their construction and working such as LED, OLED, Zener, etc. Diode Characteristics 6 The plot in Figure 5 includes the effect of leakage shunt resistance. The forward and reverse current voltage (IV) characteristics of a diode are generally compared on a single characteristic curve. The V-I characteristic of the Real diode is shown below:. It is because the value of reverse resistance is so high (R R > 100 MΩ) that is considered to be infinite for all practical purposes.. The resistance can be calculated by the formula,R=V/I. The following figure shows the static characteristic of a junction diode. In practical, the value or forward voltage is almost one volt according to the value of forward current. when the Pregion is made negative and the N-region is made positive. In reality, I 0 changes rapidly with temperature resulting in the dark blue curve… If we combine both forward biased and reverse biased curves of dido than we have a complete V-I characteristic curve. In other words, an actual diode offers a very small resistance (not zero) when forward biased and is called a forward â¦ This condition is indicated by the reverse current portion of the curve. This curve â¦ In general, this means that reverse current remains constant over a large part of reverse voltage. In the above figure, the circuit denoted as. Maximum Ratings and Electrical Characteristics Rating at 25°C ambient temperature unless otherwise specified. For example, a small diode may have a rating of 1 A; thus, it should not be used in a circuit with more than 1 A current (unless it is used in parallel with some other diodes). The ac resistance of a diode is the _____ of the characteristic curve at the Q-point of â¦ Static resistance or DC resistance of a PN junction diode defines the diode’s resistive nature when a DC source is connected to it. Peak Inverse Voltage − It is the maximum reverse voltage that can be applied to the PN junction, without damaging it. VI Characteristics of Zener Diode. It plots the drive current applied to the laser against the output light intensity. Forward and reverse current values are shown on the vertical axis of the graph. The diode, a nonlinear device, is in series with a linear circuit consisting of a resistor, R and a voltage source, V DD. In simple language, the current that flows through it is not proportional to the applied voltage. 1). Signal diodes, such as the 1N4148 only pass very small electrical currents as opposed to the high-current mains rectification diodes in which silicon diodes are usually used. The characteristic curve of an ideal diode is shown. Staticresistance is also defined as the ratio of DC voltage applied across diode â¦ Junction breakdown takes place due … The diode is also in the reverse working state, but the reverse voltage is large. If an external DC voltage is given to the circuit in which the semiconductor diode is a part of it, results in a Q-point or operating point on the PN junction diode characteristic curve that does not alter with time. Lets start. This result means that the device has an intrinsic rectifying characteristic, so we can ﬁt the curve with a Schottky diode equation. If show above circuit measurements on the graph this graphical representation is called the V-I characteristics curve for a forward-biased diode. The effect of a bypass diode on an IV curve can be determined by first finding the IV curve of a single solar cell with a bypass diode and then combining this curve with other solar cell IV curves. Power dissipation (PD) − The maximum power that the device can safely absorb on a continuous basis in free air at 25° C. Reverse recovery time (Trr) − The maximum time that it takes the device to switch from on to off stat. With incident sunlight, the I/V curve shifts up showing that there is external current flow from the solar cell to a resistive load as shown with the red curve. Rectifying Diode RR2LAM4STF ... 60Hz half sin wave, resistive load Symbol Limits Unit 400 Parameter Conditions Duty≦0.5 400 150 55 to 150 ... Electrical Characteristic Curves 1 10 100 0 5 10 15 20 25 30 f = 1MHz Ta = 25°C 1 10 100 1000 110 100 Tj = 25°C PEAK SURGE FORWARD CURRENT : I FSM (A) Minimum is the current passing through a diode maximum is the level of DC resistance. For capacitive load, derate current by 20%. Breakdown Voltage − It is the minimum reverse bias voltage at which PN junction breaks down with sudden rise in reverse current. Maximum forward current (IFM) − The absolute maximum repetitive forward current that can pass through a diode. DC or Static Resistance The application of a dc voltage to a circuit containing a semiconductor diode will result in an operating point on the characteristic curve that will not change with time. curve as we saw for the forward biased diode. The point at which this occurs is often called the knee voltage VK. As reverse bias voltage is further raised, depletion region width increases and a point comes when junction breaks down. The voltage transfer curve for this circuit is shown on Figure 15 and it is derived from the I-V characteristic of the diode model and Kirchhoff’s voltage law. Note: In this post, there are 2 types of voltage will be discussed the first one is froward biased voltage and the second is forward voltage (VF). V J =V A-IR S. The following Figure 5 shows how the series resistance influences the I-V curve of the pn junction diodes. The amount of minority current carriers that take part in conduction of reverse current is quite small. LIV Curves The fundamental test of a laser diode is a Light-Current-Voltage (LIV) curve, which simultaneously measures the electrical and optical output power characteristics of the device. At the breakdown voltage (VBR) point, current increases very rapidly. A diode is said to be an Ideal Diode when it is forward biased and acts as a perfect conductor, with zero voltage across it. In electronics, diode modelling refers to the mathematical models used to approximate the actual behaviour of real diodes to enable calculations and circuit analysis. The typical value of V BE for a silicon BJT is 0.7 V. Output characteristics are obtained between the output voltage V CE and output current I C at constant input current I B. The point of beginning or zero value is at the center of the graph. Characteristic â¦ The combined Forward Voltage and Forward Current values are located in the upper right part of the graph and Reverse Voltage and Reverse Current in the lower left corner. â¦ Which diode has negative resistance characteristics? forward bias diode : A diode is forward biased if the P-type pin is connected with the anode â¦ As the characteristic curve of this diode is not vertical similar to the ideal diode due to variation in Zener current (ÎIZ) its Zener voltage (ÎVZ) also increases. The increment in forward voltage is less due to internal resistance or dynamic resistance voltage drop and forward continuously increases. The curve shows the turn-on and the buildup of the forward bias current in the diode. i . Wonderful article! This article covers the Linear & Nonlinear Resistor Definition and Characteristic Curve. 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Directly dependent on the vertical reverse current values expressed in microamperes graph that is the region negative... Not work like this this platform, I hope all of you are enjoying your life diode a! Process is known as full-wave rectification over a large part of reverse current voltage ( )! The Real diode is used maximum reverse voltage ( IV ) characteristics of PN junction, without damaging.... Diode V-I curve is shown if ) equals 0 mA M Ω ) large... Diode in a normal positive resistor, the curve for a short of! Sharing my professional and technical knowledge to engineering students operation causes excessive heat to across! And I V is the expended form of V-I characteristic curve for both silicon and germanium diodes VK. Their construction and working such as ups and some special diodes have breakdown voltage it., they do not function like linear devices ; they are indicating one thing source voltage V s from value. 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# The Sink Password

You are desperate to get past the guard and get into the party at The Sink (You're you, so you know why). To get in, you need to give the right password to the guard. You watch from the side at all of the guests coming in.

The first guest walks up to the guard. "Eleven," says the guard. "Four," responds the guest. He is allowed in.

The next guest comes up. "Twenty-nine." "Four." He walks in.

Next guest. "Nineteen." "Three," replies the guest. Made it in.

Next guest. "Fifty-six," says the guard. "Two," she replies. She is asked to leave. She must have been trying to figure it out too.

Next guest. "One hundred seventy-seven." Woah, big one. "Three," replies the guest. She is let in.

Another guest. "Four." "Zero," the guest replies. He is let in.

You start to recognize what may be going on. You get up and approach the guard. "Three hundred ninety-one thousand, eighteen," he says. Jerk.

What do you answer?

(Admittedly not an original premise, but a good one)

More Correct Responses:

Two million ~ Four
Eight thousand ~ Four
Thirty-seven ~ Three
One ~ Three
Five ~ One

Hint 1:

This place's passcode will likely be very different than their rival business The Source...

Hint 2:

It seems some of the guests were counting something on each hand...

• @Asterisk Numerical form is 391,018 – Nordii Jun 17 at 14:15
• Are the sample correct answers in the hint only valid in sequence, or are they standalone? – Mohirl Jun 18 at 10:53
• @Mohirl Standalone, not in sequence. – Nordii Jun 18 at 15:01

six

Jay basically answered this one.

Count the number of letters in the word when written out. then count the number of letters in that word when written out. Continue until you get to 4, which has 4 letters. The number of numbers you visit, other than 4, is the answer.
391018 -> 37 -> 11 -> 6 -> 3 -> 5 -> 4
has six steps.

Alrighty, so I actually had stumbled upon the right answer, as @Nordii mentioned after correcting the question.

6

This is because:

You apply the pattern "Count the number of letters in the number, then apply it".
For example: "eleven" would be "six" because "eleven" has 6 letters.

Then you continue applying that pattern until you reach the number "four" because that will result in an infinite loop ("four" has 4 letters). The answer is the number of times until you reach four (the "sink" as it's an infinite loop).
For example: "eleven" -> "six" -> "three" -> "five" -> "four". Given there were 4 "transitional" periods, this gives you the answer 4.

"Three hundred ninety-one thousand eighteen" -> "thirty-seven" -> "eleven" -> "six" -> "three" -> "five" -> "four"
As there were 6 transitions before arriving at "four", the answer is 6.

Below is my original answer:

Starting Thoughts

Given Hint 1 and the title of the question:

This problem could have to do with either a Computer-Science idea of "flow" (describes stuff "flowing" from a "source" to a "sink"), electronic flow, or thermal (heat) flow, or any idea that has to do with something flowing from a "source" to a "sink".

Given Hint 2, and Mohirl's observation that "All the responses are..."

All the responses are 0 to 4 (Thanks Mohirl!), which is something one can count to with their hands

This gave me a few ideas, which unfortunately didn't pay off but could be useful still as something to spark ideas:

- Number of syllables in the words, showing how the flow of the mouth goes. Immediately broken by "eleven".
- Number of times your tongue touches the roof of your mouth when you say the number. Also immediately broken by "eleven".
- Comparison of consonants vs vowels. Also broken immediately by "eleven".
- The classic "count the number of letters in the number, this gets you the next number". Then you see how many times you apply this until you get to 0. So "eleven" would be "six" because "eleven" has 6 letters. So it'll be "eleven" -> "six" -> "three" -> "five" -> "four". Given there were 4 "transitional" periods, this gives you the answer 4. This looked promising until I tried "twenty-nine", and got "four" instead of "three". ("twenty-nine" -> "ten" -> "three" -> "five" -> "four"). EDIT: This was actually a mistake in the question, the actual answer is 4.

Notable clues:

Hint 2 strongly suggests it can be simplified by doing something with your hands, so they must be counting something with those numbers.
"Four" -> "zero" is incredibly interesting. This is the only time it was answered with "zero", so what's so special about four? "one" -> "three" suggests it has less to do about the numerical value, unless they add/subtract something.

EDIT: Adjusted my answer thanks to Nordii's correction that "twenty-nine" -> "four".

• You caught an error in my question. Twenty-nine ~ Four is correct. Good find. – Nordii Jun 21 at 14:52
• @Nordii Oh, lol. I wasn't sure if I was on the right path. I'll try my answer to reflect that. – Jay Jun 21 at 21:01

Partial / Starting point thoughts

As a starting point, given hint 1

and the name The Sink, I'm guessing the response is subtracted from something related to the question, to give a target answer

I'd initially thought that the

He vs She in the story examples might be relevant, but since the "More correct responses" doesn't include that, it's clearly not

All the responses are

0-4, so something mod 5 might be involved?

There's no

clear pattern to the sequence of question numbers, in terms of increasing/decreasing, odd/even, number or sum of digits


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# Scale estimation in datasets for monocular odometry

Suppose I've developed a monocular SLAM or odometry algorithm. Then I want to test it on some dataset, for example KITTI, TUM etc. How should I deal with the absolute scale in this way? Thanks.

• Correct me if I understood you wrong. Suppose we want to initialize a scale (not to make its correction along the full trajectory). From dataset we got two images: first in $t_1$ and second in $t_2$ time moments. We can do 5-point algorithm and get a unit translation vector between these two frames. Also we have ground truth translation length between them. Now we can put the scale as the first length divided by the second one (or v. versa). Right? – Roman Dosaev Mar 9 '18 at 11:33


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Does doubling density (keeping average gas molecule speed the same), increase temperature recorded on a thermometer?

At the end of the day what the thermometer is measuring as temperature is energy of the air molecules (which could come in the form of kinetic energy). Now, imagine the following scenario :

1. Take a box with just one gas molecule (at speed x). It goes and hits the mercury of the thermometer. Thermometer will probably not record its proper temperature.
2. Now fill the box with a million molecules (at the same speed x), and the thermometer records right some temperature.
3. Now double the density (but keep the speed of individual air molecules the same). Fill the box with 2 million molecules, but keep the average speed of the molecules the same.

Will it record the same temperature?

My personal intuition is that the temperature recorded should increase, since more molecules are giving energy to that thermometer in the same amount of time.

• I have posted a new answer. Hope it helps Sep 9 '20 at 20:12
• A million (or two million) molecules isn’t really enough to give your box a measurable temperature. Unless it’s microscopically small, it will be a pretty good vacuum. Try using an Avogadro number of molecules (approx. 6E23). Sep 10 '20 at 7:32

Assuming an ideal gas, if you are keeping the average speed of the molecules the same then you are holding the temperature of the gas constant. Assuming the container keeps the same volume, by the ideal gas law it must be that the pressure of the gas increases as you add more molecules.

Yes there are more molecules hitting the thermometer, but that also means there are more molecules hitting the thermometer. What I mean by this is that energy can be transferred to the thermometer at a higher rate due to collisions, but collisions at the same increased rate will then transfer that energy from the thermometer back to the gas (and the same vice versa). This is how thermal equilibrium works.

So no, just because you have more gas at the same temperature doesn't mean you will record a higher temperature. More gas will just mean fewer fluctuations about the same temperature.

In addressing points made in the comments, technically the temperature of the thermometer will be somewhere between the starting temperature of the thermometer and the starting temperature of the gas, and the final temperature of the thermometer will approach the starting temperature of the gas as more gas is let in. However, there will come a point where additional gas will make the final temperature indistinguishable from the initial gas temperature, and ideally the thermometer shouldn't influence the temperature of the gas.

• I've added a comment to the answer by Bob D. I think the same comment will work for your explanation. Pressure, I concede will increase. But the equilibrium kinetic energy of all the molecules will increase if I'm adding more molecules that are on the higher energy side to the system. That should warrant a temperature increase. Sep 9 '20 at 16:01
• @Kshitij You said the added molecules have the same average speed, i.e. the same temperature Sep 9 '20 at 16:11
• You are talking macroscopically. I'm referring to whats actually going on between the air molecules and the mercury. Initially the air molecules are at some temperature T (lets say), and mercury at some temperature T ' lets say (without losing generality, assume T > T ' ). Thermal equilibrium between the molecules and the mercury will decide a new temperature Y that will be between T and T ' which will be the recorded temperature. Addition of more molecules which are at temperature T, will lead to a new recorded temperature Y ' which is more towards T than Y? Sep 9 '20 at 16:18
• @BioPhysicist What you mean in 2nd paragraph is that some molecules with lower energy take some heat away. This means their energies increase after collision , won't this require a coefficient of restitution greater than 1. Can you tell how energy is transferred back to gas? Sep 9 '20 at 16:48
• @Protein Even in elastic collisions energy can be transferred between objects. Sep 9 '20 at 17:42

Now double the density (but keep the speed of individual air molecules the same). Fill the box with 2 million molecules, but keep the average speed of the molecules the same. Will it record the same temperature?

The actual temperature of the gas will be the same prior to making a reading, but the final temperature of the gas and thermometer after making the reading will depend on the heat capacity of the gas before and after doubling the number of molecules as compared to the heat capacity of the thermometer (fluid plus glass), which is fixed.

If both before and after doubling the number of molecules the heat capacity of the gas is much greater than the thermometer, then the actual temperature of the gas should not change due to the act of measurement, and the thermometer reading should not change.

Let's say the heat capacity of the gas is $$C_H$$ and the heat capacity of the thermometer, is $$C_L$$. Before making the measurement the actual temperature of the gas is $$T_H$$ (higher temperature) and the temperature of the thermometer before making the measurement is $$T_L$$, (lower temperature). Then the final equilibrium temperature $$T$$ of each will be

$$C_{H}(T_{H}-T)=C_{L}(T-T_L)$$

Now let the heat capacity of each before doubling the molecules be equal, or $$C_{H}=C_{L}$$, then

$$T=\frac{(T_{H}+T_L)}{2}$$

Now we double the number of gas molecules but the pre-measurement temperature is the same $$T_H$$. But the heat capacity heat of the gas is now double that of the thermometer, or $$C_{H}=2C_L$$. The final equilibrium is now

$$T=2/3T_{H}+1/3T_L$$

Which is higher than before doubling the number of molecules and is closer to the actual pre-reading temperature of the gas, $$T_H$$.

The more molecules you add the closer the final equilibrium temperature of the thermometer is to $$T_H$$, until the heat capacity of the gas is so much higher than the thermometer that the addition of more gas at the same pre-measurement temperature has no effect on the final temperature of either one. If there are 1000 times more molecules than the original then $$T=\frac{1000T_H}{1001}+\frac{T_L}{1000}≈T_H$$

The above said, keep in mind that in order for a thermometer to measure the actual temperature of something (i.e., the temperature that something has before applying the device), the heat capacity of what is being measured must be much greater than the heat capacity of the liquid (and glass) of the thermometer so that the temperature of what is being measured does not change.

It is kind of a cardinal rule in making any measurement that the act of making the measurement should have a minimum effect on what is being measured. If doubling the amount of gas having the same pre-reading temperature gives you a different temperature reading, then perhaps you are using the wrong temperature measuring device.

Hope this helps.

Temperature of a gas is a measure of average speed of molecules. Since average speed of gas molecules is constant so is the temperature.

The increased density would increase thermal conductivity and not temperature (as avg speed of molecules is kept constant).The thermometer will attain equilibrium temperature faster and thus record temperature of gas in lesser time but the temperature recorded will be the same.

Your question seems to be about what the equilibrium temperature recorded by the thermometer will be after being placed with the gas.

Your intuition that 'the temperature recorded should increase, since more molecules are giving energy to that thermometer in the same amount of time' entirely depends on the thermometer starting at a colder temperature than the gas. In this case the thermometer will cool the gas with fewer molecules by a larger amount before reaching equilibrium.

If the thermometer starts and the same temperature of the gas there would be no net change. The thermometer will read the same in both instances.

If the thermometer started at a higher temperature than the gas then the eventual reading of the thermometer would be lower if there were more gas molecules, not higher. The thermometer would bring the gas with fewer molecules to a higher temperature before reaching equilibrium.

Doing some napkin math, your example is almost meaningless because there are so few gas molecules. The temperature changes from thermal radiation would outweigh any change due to the gas. If you want to talk about how the temperature reading would be affected, you need to specify what thermometer you are using. 1 million or 2 million gas molecules would have little impact on the thermometer. At this point the container obviously becomes important, as does the method of inserting the thermometer, the method of adding gas atoms to make sure they are all moving the same average speed, thermal radiation, etc.

Just the 0.61 g of mercury in a normal thermometer has about 1.83E21 molecules. With a specific heat of 0.14 J/gC it would take 0.0854 J to raise the temperature by 1 degree celsius. If your gas was a mixture of air, 1 million molecules would be about 3E-17 grams. At a specific heat of 1.02 J/gC that means for just the mercury in the thermometer to go up by 1 degree it would have to absorb 0.0854 J, which would decrease the temperature of the gas by 2.8E15 (2,800,000,000,000,000) degrees celsius.

• I understand your concern, but here we can arbitrarily decrease the size of the mercury thermometer to suit the experiment. Feel free to take 5 Avogadro number of gas molecules and then make them 10 Avogadro number of gas molecules likewise for the two cases. Also, citing your comment "your example is almost meaningless because there are so few gas molecules" implicitly agrees with the fact that temperature measurements change with the number of molecules (or likewise, regular experimental thermometers like mercury ones only work within a defined range of air density or pressure.) Sep 11 '20 at 9:08

The denser gas should bring the thermometer to the same temperature, but faster.

When there is only one molecule in the box, it doesn't have enough total energy to affect the thermometer. This is a thermometer failing. Our intention is to bring the thermometer to the equilibrium temperature without affecting the gas we're measuring. When there are a whole lot of gas molecules, that's true enough that the error doesn't matter. When there are only a few, the error gets big.

You seem to be talking about a traditional thermometer, such as a mercury thermometer (there are other devices that could be considered to be thermometers, such as infrared sensors). The idea behind a traditional thermometer is that it is a device whose temperature is easily seen. When such a device is allowed to reach thermal equilibrium with another substance, the thermometer's temperature is the same as the substance, and so we can determine the temperature of the substance by examining the thermometer.

Thermal equilibrium means that when molecules from the thermometer and the substance collide, there is on average no net transfer of energy. Sometimes energy will be transferred to the thermometer, and sometimes energy will be transferred from the thermometer, and overall they cancel out. Having more molecules means more collisions, but since each collision is on average not transferring energy, the increase in collisions will not affect the temperature.


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# Imaginary Action

1. May 1, 2007

### plxmny

Friends,

I am reading that the standard way to discuss tunneling is via imaginary action.
The reference cited is J.S. Langer, Ann. Phys. 1967.

(1) What is the full name of Ann. Phys. so I can look up this paper?
(2) What textbook can I acquire to learn this technique?

2. May 1, 2007

### nrqed

Annals of Physics

3. May 1, 2007

### plxmny

Thank you - I checked the local university library but they don't have back to 1967 without jumping through hoops. I'll try another tomorrow.

In the meantime I hope someone can shed some light on this imaginary action technique for me.

I tried posting the same query to crackpot-land (google) so we'll see.

4. May 2, 2007

### TriTertButoxy

The imaginary action technique might be a fancy term to describe Feynman's path integral formalism, whose central equation is

$$\langle x_b|e^{-i \hat{H} t/\hbar}|x_a\rangle=\int\mathcal{D}x(t)\,e^{i S[x(t)]/\hbar}\,,$$

where the LHS is the amplitude for a particle to travel (or tunnel) from position $x_a$ to $x_b$. The imaginary action might be the quantity in the exponent in the RHS.

5. May 2, 2007

### plxmny

I think that you are correct. I also heard that landau lifschitz nrqm has an exposition of the technique. i am trying to get a copy now to check.

Also I just learned that the cited reference was re-published in 2000 so I might have some luck there. I will let you all know.

kisses,
plxmny


text:
## 27.12 Quick review questions

A study compared the nutritional intake of $$n=50$$ anaemic infants in Lahore (Pakistan) with the amounts recommended.

The mean daily protein intake in the sample was 14g, with a standard deviation of 3g. The recommended protein intake was 13g.

The researchers wanted to see if the mean intake met the recommendation, or not.

1. The standard error of the mean (to four decimal places) is
2. The null hypothesis is:
3. The test statistic (to two decimal places) is
4. The two-tailed $$P$$-value is
5. True or false? We accept the null hypothesis.
6. There is to support the alternative hypothesis (that the mean daily protein intake is not 13g).

### References

Imtiaz S, Ali Z, Chaudhry KA, Abbas F, Anjum N. Assessment of nutritional status of anemic children. Pakistan Journal of Medical & Health Sciences. 2017;11(1):131–5.


text:
# Evaluate |-100|-98

The absolute value is the distance between a number and zero. The distance between and is .
Subtract from .
Evaluate |-100|-98


text:
Article | Open | Published:

# PinAPL-Py: A comprehensive web-application for the analysis of CRISPR/Cas9 screens

## Abstract

Large-scale genetic screens using CRISPR/Cas9 technology have emerged as a major tool for functional genomics. With its increased popularity, experimental biologists frequently acquire large sequencing datasets for which they often do not have an easy analysis option. While a few bioinformatic tools have been developed for this purpose, their utility is still hindered either due to limited functionality or the requirement of bioinformatic expertise. To make sequencing data analysis of CRISPR/Cas9 screens more accessible to a wide range of scientists, we developed a Platform-independent Analysis of Pooled Screens using Python (PinAPL-Py), which is operated as an intuitive web-service. PinAPL-Py implements state-of-the-art tools and statistical models, assembled in a comprehensive workflow covering sequence quality control, automated sgRNA sequence extraction, alignment, sgRNA enrichment/depletion analysis and gene ranking. The workflow is set up to use a variety of popular sgRNA libraries as well as custom libraries that can be easily uploaded. Various analysis options are offered, suitable to analyze a large variety of CRISPR/Cas9 screening experiments. Analysis output includes ranked lists of sgRNAs and genes, and publication-ready plots. PinAPL-Py helps to advance genome-wide screening efforts by combining comprehensive functionality with user-friendly implementation. PinAPL-Py is freely accessible at http://pinapl-py.ucsd.edu with instructions and test datasets.

## Introduction

Genetic screens using pooled CRISPR/Cas9 libraries are functional genomics tools that are becoming increasingly popular throughout the life sciences. Using these screens, researchers are able to find novel molecular mechanisms, better understand complex cellular systems, or find new drug targets1,2,3. Analysis of the raw sequencing output of these screens is a non-trivial task, given the size and diversity of these datasets. MAGeCK4 was the first bioinformatic workflow specifically aimed at analysis of CRISPR/Cas9 screens and has been used in numerous genome-wide screening studies since. However, while being a standard solution in the field, MAGeCK has to be run from a command line, and requires manual definition of the position of the 20 bp single guide RNA (sgRNA) sequence for proper read identification. It, thus, requires familiarity with working from a command-line, basic handling of raw fastq files, as well as knowledge of the read sequence composition, all of which might constitute obstacles to many biologists. BAGEL5 is an alternative command-line workflow, powerful in analyzing gene essentiality screens, but inapplicable to other screening experiments, such as resistance screens or reporter-based screens. caRpools6 is a R-package offering multiple analysis options, but installation and execution require proficiency with the R platform. Finally, the recently developed CRISPRcloud7 is an analysis workflow that runs as a web-service, thus offering superior ease of use. However, CRISPRcloud requires manual definition of a fixed 20 bp window (idential for each sample) to extract the sgRNA sequence from each read file. This makes the workflow incompatible with cases where the position of the sgRNA sequence varies between reads, for example if read staggering is used. Read staggering is a common technique and recommend by many sequencing facilities to increase sequencing yield as it mitigates the low base diversity problems in the initial sequencing cycles of PCR amplicons libraries like the ones generated in CRISPR/Cas9 screens8,9,10 (Fig. 4A). As a consequence, sequencing analysis of CRISPR/Cas9 screens still remains challenging for most laboratories since prior solutions are either hard to access for non-bioinformatic experts, or might not provide sufficient support for the datasets generated. The web-service we present, PinAPL-Py, addresses these issues by providing a comprehensive analysis workflow running through an intuitive web-interface that supports a broad class of CRISPR/Cas9 screening experiments as well as staggered reads. This facilitates standardized, reproducible data analysis that can be carried out directly by the scientists conducting the experiments.

## Methods

### Workflow description

PinAPL-Py is designed to analyze a generic layout of CRISPR/Cas9 screening experiments where a control group is compared to one or more conditions which can, for example, refer to cells exposed to a chemical compound, samples taken at different time points, or cells expressing a fluorescent reporter and sorted by FACS (Fig. 1A). PinAPL-Py is written in Python 2.7 and implements well-established methods to provide a fully automated workflow, taking the user from the raw sequence files to a list of candidate sgRNAs and genes while requiring only minimal manual input. In particular, the workflow comprises the following steps (Fig. 1B):

#### Sequence quality check

Sequence quality assessment is carried out using fastqc (http://www.bioinformatics.babraham.ac.uk/projects/fastqc/), which analyzes sequence composition, sequencing quality, and read depth.

Read alignment is carried out using Bowtie2 12 in the “local” mode. Bowties alignment parameters (seed length (-L), seed number (-N), and seed interval function (-i)) are set to values producing optimal results for 20 bp sgRNA sequences, but can be changed on the configuration page. A technical description of these parameters is available in the Bowtie2 manual (http://computing.bio.cam.ac.uk/local/doc/bowtie2.html#what-is-bowtie-2). After completion, alignments are processed based on the best (=primary) and second-best (=secondary) alignment scores achieved for each read (referred to as ‘AS’ or ‘XS’, respectively, in the Bowtie2 SAM output), and categorized as either “unique”, “tolerated”, “ambiguous”, or “failed”. An alignment is considered failed if either no primary score is reported by Bowtie2, or if the primary score reported is below the critical matching threshold (set to perfect matching in the default setting, but adjustable on the configuration page). An alignment is considered ambiguous if the difference between primary and secondary alignment is below a critical ambiguity threshold (adjustable on the configuration page). If this difference is above the threshold, the read is mapped to the primary match and classified as “tolerated”. If a secondary score is not reported (and the primary score satisfies the matching threshold), the alignment is considered unique. Only unique and tolerated reads are kept for further processing, while failed and ambiguous reads are being discarded.

#### Analysis of read count distributions

For each sample, descriptive statistics of normalized read counts are computed and reported on the results page as boxplots, histograms and various measures, such as median, standard deviation, maximum, minimum, sgRNA/gene representation, and the Gini coefficient.

In order to generate a candidate list of individual sgRNAs showing either the most significant enrichment or depletion (compared to the control samples), PinAPL-Py models normalized read counts as following a negative binomial distribution14, similarly to other read count analysis workflows4,15,16. For this, the average (normalized) read count μi and sample variance σi 2 for each sgRNA i is computed across all control replicates. The dispersion parameter D of the negative binomial distribution is then estimated via linear regression since σ2 = μ + D μ2 holds in a negative binomial model. With this estimated dispersion, a model variance $σ ˆ i 2$ is computed for each sgRNA via $σ ˆ i 2$ = μi + D μi 2. Thus, the read count of each sgRNA i, under the control condition, is modeled to follow a negative binomial distribution with parameters NB(μi, $σ ˆ i 2$).

#### sgRNA enrichment/depletion analysis

sgRNA counts from each treatment sample are evaluated on the basis of the control read count distribution, estimated in the previous step. In particular, for each normalized read count ci from a treatment sample, the fold-change to the control mean is taken, and a p-value is computed from NB(μi, $σ ˆ i 2$) to test either for ci > μi (in an enrichment screen) or ci < μi (in a depletion screen). Finally, these p-values are adjusted for multiple hypothesis tests using the Benjamini-Hochberg FDR procedure. Note that for the computation of p-values, PinAPL-Py requires the user to specify at least two control samples (only fold-change is reported otherwise). The results are returned to the user as sortable tables on the results page, and as spreadsheets after download. In addition, PinAPL-Py produces scatterplots of normalized counts for each treatment sample versus the control averages on which sgRNAs of individual genes can be interactively highlighted on the results page. Results of the sgRNA enrichment/depletion analysis are reported for each treatment sample separately, and correlation analysis is carried out for samples representing replicates of the same treatment.

#### Sample clustering

Unsupervised clustering analysis of all samples is done based on either the most variable or most abundant/depleted (depending on screen type) sgRNAs throughout the dataset. For most variable sgRNAs, the n sgRNAs with the highest variance across all samples are extracted (n = 25 by default, adjustable on the configuration page). For highest abundance/depletion, the n sgRNAs with the highest/lowest read counts are extracted for each sample, and the union set of all these sgRNAs is used for the clustering. Clustering is done through the gplots package in R.

#### Gene ranking

From the sgRNA enrichment/depletion analysis described above, a gene ranking is derived by combining the fold-change data from all sgRNAs targeting a single gene through one of two possible methods:

• Adjusted Robust Rank Aggregation (αRRA): This option follows the αRRA procedure described previously4. Individual sgRNAs are required to show enrichment/depletion at significance level of P 0  = 0.05−1 (by default, adjustable on the configuration page) in order to be taken into account for computation of the gene ranking score. Genes having no significant sgRNAs are given a score of 1.0; p-values for the gene ranking score are computed based on randomly assigning sgRNAs to target genes and running permutations (Np = 1,000 by default) to estimate the empirical distribution function.

• STARS: This option calls the STARS function to compute gene ranking scores and p-values based on a binomial model (http://portals.broadinstitute.org/gpp/public/software/index) Technical details are provided in the original publication17.

### Benchmarking & Quality control

#### Sequencing data

Datasets for comparisons with MAGeCK and CRISPRcloud were taken from CRISPR/Cas9 screening experiments in our laboratory (unpublished). In these, an A375 cell population was transduced with the full human GeCKOv2 library at ~100x coverage and split into treatment and control samples. Treatment samples were incubated with a toxin at ~90% lethal dose. Control samples were incubated with PBS. After 4 days, cells were harvested, sgRNA sequences were extracted with nested PCR10, and sequenced at a minimum depth of 4 million reads per sample.

#### Sequencing data analysis

PinAPL-Py was run with gene ranking set to “αRRA” with 1,000 permutations. Matching threshold was set to 40 (perfect match) with ambiguity threshold set to 2 (Fig. 2). Read count normalization was set to “total”. All other parameters were left at default. To generate the ranked list of top 25 genes (Fig. 3E), the gene list was sorted on two levels: First by αRRA FDR (low to high), and second by −log αRRA score (high to low).

MAGeCK v0.5.6 was run following the instructions on sourceforge (https://sourceforge.net/p/mageck/wiki/Home/#usage). Normalization was set to “total”. To generate the ranked list of top 25 genes (Fig. 3E), the list was sorted on two levels: first by FDR (“pos|fdr”, low to high), and second by −log αRRA score (“pos|score”, high to low).

CRISPRcloud (http://crispr.nrihub.org/) was run in the “Survival and Drop-out” mode, with analysis algorithm set to “DESeq2” (since this option is most comparable to the negative binomial model implemented in PinAPL-Py and MAGeCK). To generate the ranked list of top 25 genes (Fig. 3E), the gene list was sorted on three levels: 1: FDR (low to high), 2: hit ratio (HR, high to low), and 3: average sgRNA log10 fold-change (AVGFC, high to low). Venn diagrams were generated at http://bioinformatics.psb.ugent.be/webtools/Venn/. For the comparisons between tools in Fig. 3, a dataset without staggers in the reads was used.

### Availability of data and materials

The raw data read files are available in the NCBI Sequence Read Archive (SRA: SRP123360; BioProject: PRJNA416669). The source code is available at https://github.com/LewisLabUCSD/PinAPL-Py.

## Results and Discussion

### Comparison to other tools

We compared PinAPL-Py to MAGeCK and CRISPRcloud. A dataset from an unpublished CRISPR/Cas9 toxin resistance screen was used. Knock-out of the heparan-binding epidermal growth factor (HBEGF) is expected to confer resistance to the toxin used in our dataset, making this gene a positive control for the comparison. A comparison to caRpools was not possible since the tool could not be run successfully and support was no longer available.

#### Comparison to MAGeCK

As laid out above, PinAPL-Py requires less know-how from the user and offers more flexibility for read identification. Apart from differences caused by PinAPL-Py’s filtering of ambiguous reads, counts obtained with either PinAPL-Py or MAGeCK are nearly identical (Fig. 3A, left panel; Fig. 3B, left and middle panel). sgRNA fold-changes, as reported in either PinAPL-Py’s or MAGeCK’s enrichment analysis, show a high correlation and a large overlap of significant sgRNAs (Fig. 3C, left panel). HBEGF was retrieved with high significance (FDR < 0.005) as the top ranking gene by both PinAPL-Py and MAGeCK (Fig. 3D, left panel), and the ranked gene lists show good overlap (Fig. 3E). Finally, as mentioned above, MAGeCK has served as a highly appreciated analysis workflow in the field, but its usage is constrained by the user’s computational skill and computer platform. We conclude that PinAPL-Py constitutes an improvement over MAGeCK as its alignment protocol is more rigorous and flexible, and its web-interface makes sequencing analysis more easily accessible, while its functionality maintains the previously established standards.

#### Comparison to CRISPRcloud

CRISPRcloud also runs as a web-service, similar to PinAPL-Py. As with MAGeCK, read counts and sgRNA fold-changes, obtained with PinAPL-Py or CRISPRcloud, are almost identical, apart from differences caused by PinAPL-Py’s filtering procedure (Fig. 3A–C, right panels). CRISPRcloud and PinAPL-py differ in the way sgRNA results are combined into a ranking of genes. In particular, CRISPRcloud introduces novel gene scores (“directionality”, “hit ratio” and “conflict”) providing useful information about efficacy and consistency of the set of sgRNAs targeting each gene. However, these scores are fractional notions with number of sgRNAs per gene (which is typically no larger than 10) in the denominator, so they do not provide sufficient resolution to effectively sort a totality of 20,000 genes. As a consequence, HBEGF could not be retrieved as an outstanding positive hit by means of these scores alone. Taking the average log fold-change score into account as an additional criterion (see “Benchmarking & Quality Control”) yielded comparable results to PinAPL-Py’s gene ranking score (Fig. 3D,E). However, a gene ranking score like αRRA is more robust to single sgRNA outliers which is why CRISPRcloud’s top 25 gene list is slightly biased towards genes with fewer sgRNAs gaining significant enrichment (Fig. 3E, red fractions), and with fewer sgRNAs overall (Fig. 3E, grey arrow). As a significant advantage, PinAPL-Py allows processing of data involving staggered reads. Since PinAPL-Py’s adapter removal is based on sequence alignment, the position of the 20 bp sgRNA sequence can be flexible, and staggered reads can be correctly processed. In contrast, staggering is incompatible with CRISPRcloud as its workflow requires all reads to have the 20 bp sgRNA sequence at a fixed position across all samples (Fig. 4). Finally, PinAPL-Py makes no restrictions to the number of experimental conditions to be analyzed in the same run, whereas CRISPRcloud is limited to five groups at a time. We conclude that PinAPL-Py constitutes an improvement over CRISPRcloud because of its higher flexibility and increased functionality in read alignment and gene ranking while providing an equally user-friendly web-interface.

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## References

1. 1.

Wucherpfennig, K. W. & Cartwright, A. N. R. Genetic screens to study the immune system in cancer. Curr. Opin. Immunol. 41, 55–61 (2016).

2. 2.

Xue, H.-Y. et al. CRISPR-Cas9 for medical genetic screens: applications and future perspectives. J. Med. Genet. 53, 91–97 (2016).

3. 3.

Hartenian, E. & Doench, J. G. Genetic screens and functional genomics using CRISPR/Cas9 technology. FEBS J. 282, 1383–1393 (2015).

4. 4.

Li, W. et al. MAGeCK enables robust identification of essential genes from genome-scale CRISPR/Cas9 knockout screens. Genome Biol. 1–12 (2014).

5. 5.

Hart, T. & Moffat, J. BAGEL: A computational framework for identifying essential genes from pooled library screens. BMC Bioinformatics 17, 33068 (2016).

6. 6.

Winter, J. et al. CaRpools: An R package for exploratory data analysis and documentation of pooled CRISPR/Cas9 screens. Bioinformatics 32, 632–634 (2015).

7. 7.

Jeong, H.-H., Kim, S. Y., Rousseaux, M., Yoghbi, H. & Liu, Z. CRISPRcloud: A secure cloud-based pipeline for CRISPR pooled screen deconvolution. Bioinformatics 0–0 [Epub ahead of print] (2017).

8. 8.

Lundberg, D. S., Yourstone, S., Mieczkowski, P., Jones, C. D. & Dangl, J. L. Practical innovations for high-throughput amplicon sequencing. Nat. Methods 10, 999–1002 (2013).

9. 9.

Fadrosh, D. W. et al. An improved dual-indexing approach for multiplexed 16S rRNA gene sequencing on the Illumina MiSeq platform. Microbiome 2, 6 (2014).

10. 10.

Shalem, O. et al. Genome-Scale CRISPR-Cas9 Knockout Screening in Human Cells. Science 343, 84–87 (2014).

11. 11.

12. 12.

Langmead, B. & Salzberg, S. L. Fast gapped-read alignment with Bowtie 2. Nat. Methods 9, 357–359 (2012).

13. 13.

Love, M., Anders, S. & Huber, W. Differential analysis of count data - the DESeq2 package. Genome Biology 15 (2014).

14. 14.

Cameron, A. & Trivedi, K. Regression Analysis of Count Data (Cambridge University Press, 2013).

15. 15.

Love, M., Huber, W. & Anders, S. Moderated estimation of fold change and dispersion for RNA-seq data with DESeq2. Genome Biol. 15, 550 (2014).

16. 16.

Robinson, M. D., McCarthy, D. J. & Smyth, G. K. edgeR: A Bioconductor package for differential expression analysis of digital gene expression data. Bioinformatics 26, 139–140 (2009).

17. 17.

Doench, J. G. et al. Optimized sgRNA design to maximize activity and minimize off-target effects of CRISPR-Cas9. Nat. Biotechnol. 34, 184–191 (2016).

18. 18.

Kim, J. et al. MAGI: A Node.js web service for fast microRNA-Seq analysis in a GPU infrastructure. Bioinformatics 30, 2826–2827 (2014).

19. 19.

Sanjana, N. E., Shalem, O. & Zhang, F. Improved vectors and genome-wide libraries for CRISPR screening. Nat. Methods 11, 783–784 (2014).

## Acknowledgements

We would like to acknowledge the valuable feedback from the participants of the CRISPR screening workshop organized by the San Diego Center for Systems Biology in October 2016 as well as the members of the UC San Diego CRISPR screening discussion group. In particular, we would like to thank Kristen Jepsen and her team at the UCSD Institute for Genomic Medicine. In addition, we would like to thank Julie la Cour Karrotki from the Center for Biosustainability at DTU for helpful feedback. This work was supported by generous funding from the Novo Nordisk Foundation provided to the Center for Biosustainability (NNF16CC0021858 and NNF10CC1016517), and grants from NIGMS (R35 GM119850 & P50 GM085764), NCI (R21 CA199292 & R21 CA177519), DOD (OC140179) and NHLBI (U54 HL108460 and U24 HL126127).

## Author information

### Affiliations

1. #### School of Medicine, Department of Pediatrics, University of California San Diego, La Jolla, CA, USA

• Philipp N. Spahn
•  & Nathan E. Lewis
2. #### Novo Nordisk Foundation Center for Biosustainability at UCSD, University of California San Diego, La Jolla, CA, USA

• Philipp N. Spahn
•  & Nathan E. Lewis
3. #### Glycobiology Research and Training Center, Department of Cellular and Molecular Medicine, University of California San Diego, La Jolla, CA, USA

• Ryan J. Weiss
•  & Jeffrey D. Esko
4. #### Division of Biomedical Informatics, University of California San Diego, La Jolla, CA, USA

• Tyler Bath
• , Jihoon Kim
•  & Olivier Harismendy
5. #### Moores Cancer Center, Department of Medicine, University of California San Diego, La Jolla, CA, USA

• Olivier Harismendy

### Contributions

Workflow development: P.S. Web-service implementation: T.B./J.K. Experimental data: R.W./P.S. Supervision (Experimental data): J.E. Supervision (Workflow development): N.L./O.H. Manuscript: P.S./N.L./O.H. All authors read and approved the final manuscript.

### Competing Interests

The authors declare that they have no competing interests.

### Corresponding authors

Correspondence to Nathan E. Lewis or Olivier Harismendy.


text:
## The Simple Linear Regression Model

The model involves a variable called the response and another called a predictor variable or feature. The assumptions are that the response is a linear function of the predictor variable, called a signal, plus random error, called noise:

$$\text{response} ~ = ~ \text{signal} ~ + ~ \text{noise}$$

Our job is to extract the signal. The data are observations on the predictor variable and the response of $n$ individuals. We assume that we can measure the predictor variable perfectly accurately, and that it is therefore not random. The only random component of the response is the noise.

Formally, for individuals $i = 1, 2, 3, \ldots, n$, the response is assumed to be

$$Y_i ~ = ~ \beta_0 + \beta_1x_i + \epsilon_i$$

where:

• $\beta_0$ and $\beta_1$ are unobservable constant parameters.
• $x_i$ is the value of the predictor variable for individual $i$ and is assumed to be constant (that is, not random).
• The errors $\epsilon_1, \epsilon_2, \ldots, \epsilon_n$ are i.i.d. normal $(0, \sigma^2)$ random variables.
• The error variance $\sigma^2$ is an unobservable constant parameter, and is assumed to be the same for all individuals $i$.

For each individual $i$, we would like to get as close as we can to the signal $\beta_0 + \beta_1x_i$. We can't see the true line $y = \beta_0 + \beta_1x$, but we can estimate it by the regression line. Thus our estimator of the $i$th signal $\beta_0 + \beta_1x_i$ is

$$\hat{Y}_i ~ = ~ \hat{\beta}_0 + \hat{\beta}_1x_i$$

where $\hat{\beta}_0$ and $\hat{\beta}_1$ are the interecept and slope of the regression line.

To estimate the true slope $\beta_1$, it is a good idea to examine the properties of the estimated slope $\hat{\beta}_1$. Before that, let's examine the response.

It is important to keep in mind that all the subsequent calculations in this chapter will be performed under the regression model. The model does not pretend to be the truth: is only a set of assumptions about reality. If the assumptions are not valid for your data, then the calculations of this chapter will not be valid either.

### Individual Responses

Fix $i$ in the range $1$ through $n$. The response $Y_i$ of individual $i$ is the sum of two pieces: the $i$th signal $\beta_0 + \beta_1x_i$ and the noise $\epsilon_i$.

The signal is a constant, and $\epsilon_i$ has the normal $(0, \sigma^2)$ distribution.

Therefore the response $Y_i$ of individual $i$ has the normal distribution with mean $\beta_0 + \beta_1x_i$ and variance $\sigma^2$.

The randomness in $Y_i$ comes only from $\epsilon_i$ as the $i$th signal is a constant. Since $\epsilon_1, \epsilon_2, \ldots, \epsilon_n$ are independent, the individual responses $Y_1, Y_2, \ldots, Y_n$ are independent of each other.

### Average Response

Let $\bar{Y} = \frac{1}{n}\sum_{i=1}^n Y_i$ denote the average response of the $n$ individuals.

Then $\bar{Y}$ is a linear combination of independent normally distributed random variables. So the distribution of $\bar{Y}$ is normal.

The mean of the distribution is

\begin{align*} E(\bar{Y}) ~ &= ~ \frac{1}{n}\sum_{i=1}^nE(Y_i) \\ &= ~ \frac{1}{n}\sum_{i=1}^n (\beta_0 + \beta_1 x_i) \\ &= ~ \frac{1}{n}\sum_{i=1}^n \beta_0 ~ + ~ \frac{1}{n}\sum_{i=1}^n \beta_1x_i \\ &= ~ \frac{1}{n} \cdot n\beta_0 ~ + ~ \beta_1\frac{1}{n}\sum_{i=1}^n x_i \\ &= ~ \beta_0 + \beta_1 \bar{x} \end{align*}

Thus the expected average response is the signal at the average value of the predictor variable.

Since the individual responses are independent of each other, we have

$$Var(\bar{Y}) ~ = ~ \frac{1}{n^2}\sum_{i=1}^n Var(Y_i) ~ = ~ \frac{1}{n^2}\cdot n\sigma^2 ~ = ~ \frac{\sigma^2}{n}$$

To summarize, the distribution of the average response $\bar{Y}$ is normal with mean $\beta_0 + \beta_1\bar{x}$ and variance $\frac{\sigma^2}{n}$.


text:
# What is the value of $\sum_{n=1}^\infty \frac{1}{n^{1+\varepsilon}}$?

I know that

$\displaystyle \sum_{n=1}^\infty x^n = \frac{1}{1-x}$ (Geometric series)

and that the harmonic series is divergent:

$\displaystyle \sum_{n=1}^\infty \frac{1}{n} \rightarrow \infty$

And I see quite often series of this form:

$s_\varepsilon =\displaystyle \sum_{n=1}^\infty \frac{1}{n^{1+\varepsilon}}$ with $\varepsilon > 0$

I know that $s_{\varepsilon > 0}$ converges due to the root test. But what is the value of those series?

So here is my question:

Let $\varepsilon > 0$. What is the value of $s_\varepsilon := \displaystyle \sum_{n=1}^\infty \frac{1}{n^{1+\varepsilon}}$?

-
Side note: The root test doesn't help here ($\limsup_{n\to\infty}\sqrt[n]{|a_n|} = 1$). In fact, this is mentioned in the Wikipedia article you linked to. –  Ayman Hourieh Sep 21 '12 at 16:02
$s_\epsilon \rightarrow \frac1\epsilon$ –  karakfa Sep 21 '12 at 16:06
@karakfa: What do you mean? –  Did Sep 21 '12 at 16:11
@karakfa: Does your answer mean anything else than "$s_\varepsilon$ diverges for $\varepsilon \rightarrow 0$"? –  moose Sep 21 '12 at 16:12
Well, but many systems will have numerical algorithms for the zeta function ... –  kjetil b halvorsen Sep 21 '12 at 17:17

Your answer is of course correct that you are looking at the $\zeta$ function. You don't mention $\epsilon$ being small, but you may also be interested in an expansion in small $\epsilon$, $$\sum_{n=1}^\infty\frac{1}{n^{1+\epsilon}}=\frac{1}{\epsilon}+\gamma_0-\gamma_1\epsilon+O\left(\epsilon^2\right)$$ with the $\gamma_i$ called Stieltjes constants.

-

As I was looking for an answer, I've noticed:

According to Wolfram|Alpha:

• $s_1 = \frac{\pi^2}{6}$ (Source) $= \zeta(2)$ (Source)
• $s_\frac{5}{4} = \zeta(\frac{5}{4})$ (Source)
• $s_{1+0.25} = 4.59511$ (Source)

This series has a name; it is the Riemann zeta function, which is exactly defined like this:

$\displaystyle\zeta(s) :=\sum_{n=1}^\infty\frac{1}{n^s}$

An expicit form for the value of $\zeta(s)$ seems to be unknowen.

-
Well, the Riemann zeta function is exactly what you are asking about. It doesn't answer your question. You can now rephrase your question, if you wanted, to ask what is the value of $\zeta(1 + \epsilon)$ for $\epsilon > 0$. The answer is that there are some values of $\epsilon$ where there is a known answer, for instance $\zeta(2n)$ is known for all positive integers $n$. But, $\zeta(2n+1)$ is not known in any nice way for any $n \geq 1$. You simply have a name for what you are asking, so it's easier to look things up, but you still don't know the answer for almost all $\epsilon$. –  Graphth Sep 21 '12 at 15:58
Knowing that it is not easy to find an answer is still much more than I knew before I've asked the question. And knowing the name is very important useful if you want to find an answer by yourself. –  moose Sep 21 '12 at 16:09


text:
# Problem: The specific heat of octane, C8H18 (l), is 2.22 J/g K.How many J of heat are needed to raise the temperature of 82.0 g of octane from 11.2 oC to 25.2 oC?

###### FREE Expert Solution

The amount of heat Q is given by:

$\overline{){\mathbf{Q}}{\mathbf{=}}{\mathbf{mc}}{\mathbf{∆}}{\mathbf{T}}}$

where m = mass (in grams), c = specific heat, ΔT = change in temperature = final T – initial T. We’re given the following values:

m = 82 g

c = 2.22 J/g • ˚C

ΔT = 25.2 ˚C – 11.2 ˚C = 14 ˚C

86% (402 ratings)
###### Problem Details

The specific heat of octane, C8H18 (l), is 2.22 J/g K.

How many J of heat are needed to raise the temperature of 82.0 g of octane from 11.2 oC to 25.2 oC?