content stringlengths 86 994k | meta stringlengths 288 619 |
|---|---|
Decimal/Binary Conversion of Integers in App Inventor - Exploring Binary
Decimal/Binary Conversion of Integers in App Inventor
To complete my exploration of numbers in App Inventor, I’ve written an app that converts integers between decimal and binary. It uses the standard algorithms, which I’ve just translated into blocks.
Overview of Base Conversion
For our purposes, we will be converting between strings in different bases. For example, if the user types in a string of characters representing a decimal number, the app spits out a string of
characters representing a binary number. To use built-in math to do the conversion, the input string is first converted to a numeric representation, which is then manipulated to create the output
(For more discussion on string-to-string, string-to-numeric, and numeric-to-string conversion, see my articles Base Conversion In PHP Using Built-In Functions, Base Conversion in PHP Using BCMath,
and Converting Floating-Point Numbers to Binary Strings in C.)
App Design
Here’s a screenshot listing my app’s components:
Components of the Conversion App
Here’s how it looks in the emulator:
Layout of the Conversion App (Shown In the Emulator)
I didn’t make the app commercial-grade; I just wanted to demonstrate decimal/binary conversion.
The Blocks
When I first wrote the blocks I made separate routines for decimal to binary conversion and binary to decimal conversion. However, for brevity, I decided to abstract them into one routine called
baseConvert(), with parameters “from base” and “to base”.
App Inventor Blocks To Convert Integers From One Base To Another (click thumbnail to enlarge)
The code only converts positive integers, and does not check whether the inputs are valid numerals. It can convert between numbers in any base from two to ten.
Code Notes
The “get fromDigit” block in the for each loop returns a character, but App Inventor converts it to a numeric value (as we want it to). In a language like C, we would do this manually by subtracting
the ASCII value for ‘0’ from it. Similarly, the “get toDigit” block in the while loop returns a number, but App Inventor converts it to a character (also as we want it). In C, we would manually add
the ASCII value for ‘0’ to it.
The Calls for Decimal to Binary and Binary to Decimal Conversion
Here’s the call to convert decimal to binary:
The Call to Convert Decimal to Binary
Here’s the call to convert binary to decimal:
The Call to Convert Binary to Decimal
Here is the app converting 25 to binary (11001) and then back to decimal:
Conversion of 25 to Binary and Then Back to Decimal
Here is the app converting a large integer (1234567890123456789012) to binary (10000101110110100010010001110110000101111011000001000000011101000010
100) and then back to decimal:
Conversion of a Large Integer to Binary and Then Back to Decimal
(The input in the binary box is too long to display — only the ending bits are shown.)
The app can convert arbitrarily large integers, taking advantage of App Inventor’s big integer implementation. (In fact, it was in writing this conversion app that I discovered its use of big
I tested the code only in the emulator, not on a real device.
Source Code
Here is the source code file, EB_d2b_b2d.aia, which you can import into App Inventor. (I did not make an “.apk” file — you can do that if you want after you import it.)
4 comments
1. How set the local variable??
2. @Victor ,
Sorry, I don’t understand your question.
3. Thank you for this tutorial!
Quick question: Would the “baseConvert” procedure also be able to convert Hexadecimal (base 16) numbers?
4. @Thales Ferreira,
You would have to add code that converts the letters a-f to the numbers 10-15 (and vice versa).
This site uses Akismet to reduce spam. Learn how your comment data is processed.
(Cookies must be enabled to leave a comment...it reduces spam.) | {"url":"https://www.exploringbinary.com/decimal-binary-conversion-of-integers-in-app-inventor/","timestamp":"2024-11-04T18:18:48Z","content_type":"text/html","content_length":"58001","record_id":"<urn:uuid:2ae938f7-1b26-4879-9bd5-bee2d86e7c5a>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00829.warc.gz"} |
How to determine the diameter of a circleHow to determine the diameter of a circle 🚩 determining the diameter 🚩 Math.
Diameter is the segment connecting two arbitrary points on the circle and passing through its center. Therefore, if the diameter you need to find by knowing the radius of the given circle, multiply
the numerical value of the radius by two, and measure the value found in the same units as the radius. Example: the Radius of a circle 4 centimeters. To find the diameter of this circle. Solution:
the Diameter is 4 cm*2=8 cm Answer: 8 inches.
If the diameter of the need to find through the length of the circle, then we must act using the first step. There is a formula to calculate the length of a circle: l=2PR, where l is the length of
the circumference, 2 is a constant, n is the number equal to 3.14; R is the radius of the circle. Knowing that the diameter is double the radius, the above formula can be written as: l=пD, where D is
the diameter.
To Express this formula diameter of circle: D=l/n, And substitute into it the known values, calculate a linear equation with one unknown. Example: to Find the diameter of a circleif its length is 3
meters. Solution: the diameter is equal to 3/3 = 1 m. Answer: the diameter equal to one meter.
Useful advice
In the mathematical tasks are often permitted to use the PI as just 3, and 3.14. | {"url":"https://eng.kakprosto.ru/how-33953-how-to-determine-the-diameter-of-a-circle","timestamp":"2024-11-12T15:42:18Z","content_type":"text/html","content_length":"29741","record_id":"<urn:uuid:0cc0f615-82b0-433c-9901-e775bf0533b5>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00106.warc.gz"} |
Time to Decimal Calculator with steps - Definition | Chart
Whether you want to know the equivalent of time expressed in hh:mm:ss in its decimal form, or do some math, such as adding or subtracting hours, minutes, or seconds – you will eagerly look for a tool
like our Time to Decimal Calculator. It is a very handy and easy-to-use tool that instantly converts any time into its decimal equivalent.
Would you like to explore more of our calculators related to time, times zones, drive times or times conversions? Take a look at the list below of some calculators we recommend you to check out:
Decimal Time
What does decimal time represent?
It represents the time of day using decimal units. It often refers to the system of time, which was most popular in France in 1792 during the French Revolution.
The system at that period worked by dividing the day into 10 decimal hours. In addition, each decimal hour consisted of 100 decimal minutes and each minute was made of 100 decimal seconds. Therefore,
one day was equivalent to 100000 decimal seconds.
In opposition, a standard UTC works by dividing the day into 24 hours. Each hour equals 60 minutes, and each minute is made of 60 seconds. Therefore, one day in UTC standard format is equal to 86400
Considering which time system is simpler and easier for conversion and calculations, the winner is a decimal format system. Why? Because it’s easier to interpret it and perform conversions if the
time is expressed in decimals.
For instance, let’s assume you want to interpret 1:25:30 (hh:mm:ss) as a time in decimal form. How would it look in a decimal format?
• 1.2530 – hours
• 125.30 – minutes
• 12530 – seconds
Or, for example, when you want to convert 3 hours in minutes and seconds. Again, it is straightforward: 3 hrs = 300 minutes or 30,000 seconds.
When did it all start, and at what period in history decimal time system was used?
In the 1790s in France, after the introduction of metric weights and measures, people divided a day into 10 hours, where each hour equalled 100 minutes and each minute was equivalent to 100 seconds.
However, it’s important to mention that the old time system failed to become popular and spread its usage worldwide.
In 1795 when a standardised system was introduced, which was way more useful and effective, these localised weights and measures, decimals system pretty much was already dead.
Today, standardised systems are used everywhere, including time. Therefore, there are zero chances that the decimal system may come into use and become a part of everyday life.
Decimal vs Standard 24-hour System
We learned about the decimal time system and its origins, but what about the 24-hour time system? What are its origins?
Interestingly, Egyptians were the first people who used a 24-hour system, and they divided a day into 24 equal parts. They used a sexagesimal system, with a base of 60, which was afterwards accepted
and used by Babylonians.
Additionally, Babylonians divided both the circle and the year into 360 segments. Our todays’ way of measuring and expressing time originates from the old Egyptian time.
What does 24-hour standard time look like?
In the 24-hour system, we express hours as 60 minutes and minute as 60 seconds. For instance, there are 1,440 minutes in a day, and 1 hour in seconds is equal to 3600. Whenever you see this format
“hh:mm” or extended “hh:mm:ss”, it indicates that the time is expressed in the standard 24-hour time system.
Example: 4:10:15 – we read as “four hours, ten minutes and fifteen seconds”. Additionally, it means that there are four periods of 60 minutes, one period of 10 minutes and 10 seconds.
What does a decimal time look like?
The decimal time system uses base 10 to express time. Therefore, each place in a decimal number ranges from 0 and 9.
Example: If we use the same example from above (4:10:15), how can we write it as a decimal time?
We express it as 4 hrs – 250 min – 15,015 sec in decimal format.
How to Convert Time to Decimal – Formula
There are cases when you need to express your time in decimals, converting time to decimal hours, minutes and seconds. Therefore, you can see all three formulas required for converting time to
decimals below.
Sometimes you need to express time in decimal hours. Therefore this is the formula you should utilize:
Hours = hh + (mm \div 60) + (ss \div 3600)
• hh – hours component of the time
• mm – minutes component of the time
• ss – seconds component of the time
For instance, converting 3:05:00 to decimal hours:
Hours = 3 + (10 \div 60) + (0 \div 3,600) Hours = 3 + 0.16 + 0 = 3.16
If you want to convert it to decimal minutes only, choose this formula instead:
Minutes = (hh \times 60) + mm + (ss \div 60)
For example, converting 3:05:00 to decimal minutes:
Minutes = (3 \times 60) + 5 + (0 \div 60) Minutes = 180 + 5 + 0 = 185
For converting it to decimal seconds, check out the formula below:
Seconds = (hh \times 3600) + (mm \times 60) + ss
For example, converting 3:05:00 to decimal seconds:
Seconds = (3 \times 3,600) + (5 \times 60) + 0 Seconds = 10,800 + 300 + 0 = 11,100
Time Conversion Chart
Here is the decimal chart (table) of the most common time expressed in decimal hours, minutes, and seconds. Instead of calculating it manually or using calculators, you can take the calculated values
from the chart below.
Time Hours Minutes Seconds
00:00:00 0 0 0
00:10:00 0.1667 10 600
00:30:00 0.5 30 1,800
00:40:00 0.6667 40 2,400
01:00:00 1 60 3,600
01:30:00 1.5 90 5,400
01:40:00 1.667 100 6,000
02:00:00 2 120 7,200
02:30:00 2.5 150 9,000
03:00:00 3 180 10,800
03:30:00 3.5 210 12,600
04:00:00 4 240 14,400
04:30:00 4.5 270 16,200
05:00:00 5 300 18,000
05:30:00 5.5 330 19,800
06:00:00 6 360 21,600
06:30:00 6.5 390 23,400
07:00:00 7 420 25,200
07:30:00 7.5 450 27,000
08:00:00 8 480 28,800
08:30:00 8.5 510 30,600
09:00:00 9 540 32,400
09:30:00 9.5 570 34,200
10:00:00 10 600 36,000
The most common times expressed in decimal hours, minutes, and seconds.
Time to Decimal Calculator – How to Use?
We’ve come to the most interesting part. If you are in a hurry or just lazy and not up to doing math calculations, we have developed a special calculator (converter). Our converter efficiently
listens to your input (in hh:mm:ss format) and instantly converts it to decimal hours, minutes and seconds.
Note: An extra feature you can enable in the calculator is the “step-by-step solution” mode which, besides returning the result, displays a step-by-step process of conversion using all three formulas
you saw in the previous sections.
How to utilize our calculator?
• Enter the parameters in the calculator (hours, minutes and seconds)
• The calculator calculate the total time and displays it in decimal hours, minutes and seconds
• Additionally, go through a step-by-step explanation of the conversion by setting “see step-by-step solutions” to “Yes”
Time to Decimal Calculator – Example
Okay, let’s take an imaginary time and use our calculator to turn it to a decimal. We will enable the “step-by-step solutions” option, as well.
Scenario: Convert 03:22:15 (hh:mm:ss) to decimal hours, minutes and seconds using the calculator.
1. Enter each value to a defined parameter field in the calculator (03 to hours, 22 to minutes and 15 to seconds)
2. Time to Decimal Calculator uses all three formulas to calculate and convert your time to a decimal. Therefore, in our case, it returns:
Hours: 3.371
Minutes: 202.25
Seconds: 12,135
What is 1 hour and 30 minutes as a decimal?
1 hour and 30 minutes written as a decimal equals 1.5
hours, 90 minutes and 5400 seconds.
How do you convert time into decimals?
You can convert time into decimals using one of the following formulas:
Hours = hh + (mm \div 60) + (ss \div 3600)
Minutes = (hh \times 60) + mm + (ss \div 60)
Seconds = (hh \times 3600) + (mm \times 60) + ss
How to convert 30 minutes to hours?
Converting 30 minutes to decimal hours equals 0.5 hrs.
How to write 1:20:50 as a decimal?
It is equivalent to:
– 1.3472 hours
– 80.83 minutes
– and 4,850 seconds
How to convert 45 minutes to decimal minutes?
You can convert 45 minutes to decimal minutes by using the formula below:
Minutes = (hh \times 60) + mm + (ss \div 60)
So, you need to:
– multiply the number of hours by 60
– add the number of minutes to the product of the previous calculation
– divide the number of seconds by 60 and add the product to the sum
How to convert 3:32:10 to decimal?
It is equivalent to:
– 3.536 hours
– 212.17 minutes
– and 12,730 seconds | {"url":"https://calconcalculator.com/everyday-life/time-to-decimal-calculator/","timestamp":"2024-11-09T19:23:04Z","content_type":"text/html","content_length":"102650","record_id":"<urn:uuid:7412847e-5325-4439-8840-8a30226b9254>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00222.warc.gz"} |
Transactions Online
Tzuen-Hsi HUANG, Yuan-Ru TSENG, Shang-Hsun WU, "A 1-V, 6.72-mW, 5.8-GHz CMOS Injection-Locked Quadrature Local Oscillator with Stacked Transformer Feedback VCO" in IEICE TRANSACTIONS on Electronics,
vol. E93-C, no. 4, pp. 505-513, April 2010, doi: 10.1587/transele.E93.C.505.
Abstract: This paper presents a real integration of a 5.8-GHz injection-locked quadrature local oscillator that includes two LC-tuned injection-locked frequency dividers (ILFDs) and a wide-tuning
stacked-transformer feedback voltage-controlled oscillator (VCO) operated in double frequency. A symmetric differential stacked-transformer with a high coupling factor and a high quality factor is
used as a feedback component for the wide-tuning VCO design. The wide tuning range, which is greater than three times the desired bandwidth, is achieved by selecting a greater tuning capacitance
ratio available from high-voltage N-type accumulation-mode MOS varactors and a smaller self-inductance stacked-transformer. Since the quality factors of the LC-resonator components can sustain at a
high enough level, the wide-tuning VCO does not suffer from the phase noise degradation too much. In addition, the tuning range of the local oscillator is extended simultaneously by utilizing
switched capacitor arrays (SCAs) in the ILFDs. The circuit is implemented by TSMC's 0.18-µm RF CMOS technology. At a 1-V power supply, the whole integrated circuit dissipates 6.72 mW (4.05 mW for the
VCO and 2.67 mW for the two ILFDs). The total tuning range frequency is about 500 MHz (from 5.54 GHz to 6.04 GHz) when the tuning voltage V[tune] ranges from 0 V to 1.8 V. At around the output
frequency of 5.77 GHz (at V[tune] = 0.5 V), the measured phase noise of this local oscillator is -119.4 dBc/Hz at a 1-MHz offset frequency. This work satisfies the specification requirement for IEEE
802.11a UNII-3 band application. The corresponding figure-of-merit (FOM) calculated is 186.3 dB.
URL: https://global.ieice.org/en_transactions/electronics/10.1587/transele.E93.C.505/_p
author={Tzuen-Hsi HUANG, Yuan-Ru TSENG, Shang-Hsun WU, },
journal={IEICE TRANSACTIONS on Electronics},
title={A 1-V, 6.72-mW, 5.8-GHz CMOS Injection-Locked Quadrature Local Oscillator with Stacked Transformer Feedback VCO},
abstract={This paper presents a real integration of a 5.8-GHz injection-locked quadrature local oscillator that includes two LC-tuned injection-locked frequency dividers (ILFDs) and a wide-tuning
stacked-transformer feedback voltage-controlled oscillator (VCO) operated in double frequency. A symmetric differential stacked-transformer with a high coupling factor and a high quality factor is
used as a feedback component for the wide-tuning VCO design. The wide tuning range, which is greater than three times the desired bandwidth, is achieved by selecting a greater tuning capacitance
ratio available from high-voltage N-type accumulation-mode MOS varactors and a smaller self-inductance stacked-transformer. Since the quality factors of the LC-resonator components can sustain at a
high enough level, the wide-tuning VCO does not suffer from the phase noise degradation too much. In addition, the tuning range of the local oscillator is extended simultaneously by utilizing
switched capacitor arrays (SCAs) in the ILFDs. The circuit is implemented by TSMC's 0.18-µm RF CMOS technology. At a 1-V power supply, the whole integrated circuit dissipates 6.72 mW (4.05 mW for the
VCO and 2.67 mW for the two ILFDs). The total tuning range frequency is about 500 MHz (from 5.54 GHz to 6.04 GHz) when the tuning voltage V[tune] ranges from 0 V to 1.8 V. At around the output
frequency of 5.77 GHz (at V[tune] = 0.5 V), the measured phase noise of this local oscillator is -119.4 dBc/Hz at a 1-MHz offset frequency. This work satisfies the specification requirement for IEEE
802.11a UNII-3 band application. The corresponding figure-of-merit (FOM) calculated is 186.3 dB.},
TY - JOUR
TI - A 1-V, 6.72-mW, 5.8-GHz CMOS Injection-Locked Quadrature Local Oscillator with Stacked Transformer Feedback VCO
T2 - IEICE TRANSACTIONS on Electronics
SP - 505
EP - 513
AU - Tzuen-Hsi HUANG
AU - Yuan-Ru TSENG
AU - Shang-Hsun WU
PY - 2010
DO - 10.1587/transele.E93.C.505
JO - IEICE TRANSACTIONS on Electronics
SN - 1745-1353
VL - E93-C
IS - 4
JA - IEICE TRANSACTIONS on Electronics
Y1 - April 2010
AB - This paper presents a real integration of a 5.8-GHz injection-locked quadrature local oscillator that includes two LC-tuned injection-locked frequency dividers (ILFDs) and a wide-tuning
stacked-transformer feedback voltage-controlled oscillator (VCO) operated in double frequency. A symmetric differential stacked-transformer with a high coupling factor and a high quality factor is
used as a feedback component for the wide-tuning VCO design. The wide tuning range, which is greater than three times the desired bandwidth, is achieved by selecting a greater tuning capacitance
ratio available from high-voltage N-type accumulation-mode MOS varactors and a smaller self-inductance stacked-transformer. Since the quality factors of the LC-resonator components can sustain at a
high enough level, the wide-tuning VCO does not suffer from the phase noise degradation too much. In addition, the tuning range of the local oscillator is extended simultaneously by utilizing
switched capacitor arrays (SCAs) in the ILFDs. The circuit is implemented by TSMC's 0.18-µm RF CMOS technology. At a 1-V power supply, the whole integrated circuit dissipates 6.72 mW (4.05 mW for the
VCO and 2.67 mW for the two ILFDs). The total tuning range frequency is about 500 MHz (from 5.54 GHz to 6.04 GHz) when the tuning voltage V[tune] ranges from 0 V to 1.8 V. At around the output
frequency of 5.77 GHz (at V[tune] = 0.5 V), the measured phase noise of this local oscillator is -119.4 dBc/Hz at a 1-MHz offset frequency. This work satisfies the specification requirement for IEEE
802.11a UNII-3 band application. The corresponding figure-of-merit (FOM) calculated is 186.3 dB.
ER - | {"url":"https://global.ieice.org/en_transactions/electronics/10.1587/transele.E93.C.505/_p","timestamp":"2024-11-07T02:39:51Z","content_type":"text/html","content_length":"66594","record_id":"<urn:uuid:60545984-e366-4c83-aea2-61a7eb923233>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00234.warc.gz"} |
Methodology | RDP 1999-09: Australian Banking Risk: The Stock Market's Assessment and the Relationship Between Capital and Asset Volatility
RDP 1999-09: Australian Banking Risk: The Stock Market's Assessment and the Relationship Between Capital and Asset Volatility 2. Methodology
In this section we present the contingent-claim model used to derive four measures of bank risk, namely: (1) the volatility of the return on economic assets; (2) the economic capital ratio; (3) the
probability of closure; and (4) the value of the potential public liability.^[1] Initially, we outline a contingent-claim model that can be applied to all leveraged firms. This model is then modified
to accommodate bank-specific factors. We then explain how the model is estimated and discuss how the model is used to infer the aforementioned measures of risk.
2.1 A Contingent-claim Model of a Bank
Consider a firm that has assets with an economic value of A[T] and liabilities with an economic value of B[T] due at date T. If assets exceed liabilities at maturity of the debt, the firm will
continue operating, with the value of equity being the difference between assets and liabilities.^[2] If, on the other hand, assets are less than liabilities, equity holders will relinquish control
of the firm and draw on their position of limited liability; as a result, the value of equity has a lower bound of zero. Thus, the value of equity at date T is:
Equity is described as a contingent claim since a positive payoff to equity is contingent upon the bank being solvent at date T. At any time prior to T, the total market value of a bank's equity can
be calculated using the same valuation techniques used to price other contingent claims, such as options. Equity holders are often viewed as having a long call option on the assets of the firm, where
the strike price is equal to the face value of liabilities, because the payoff from such a position at maturity of the option is characterised by Equation (1).
Following Levonian (1991a), this basic model is augmented to capture factors that relate specifically to banks. The first adjustment to the model is to incorporate a bank ‘licence value’. This
licence value captures the intangible asset value in a firm. Financial institutions that are granted a banking licence benefit from being called a ‘bank’. Specifically, the fact that banks are
perceived to have some form of public-sector backing generally enables them to pay a rate of interest on deposits that is approximately equal to the risk-free rate, since depositors anticipate little
risk of default.^[3] Furthermore, some customers are willing to pay an added premium to transact with banks since banks are viewed as being a superior source of both credit and debit services. It
should be noted that these benefits of a banking licence are partially offset by compliance costs, since banks must abide by portfolio-constraining directives issued by the Australian Prudential
Regulation Authority (APRA) (such as limits on the concentration of exposures to counterparties). Beyond these regulatory concerns, a large part of a bank's licence value will be due to its franchise
value (value of its brand name or goodwill). Since such branding distinguishes each bank from the others, competitive forces do not necessarily drive the franchise value down to zero. In our model,
the licence value is modelled as a constant fraction (φ) of liabilities and is received by equity holders at date T only if the bank remains open.^[4]
As in Cordell and King (1992), a second adjustment is made to account for the impact of dividends on the value of equity. The payment of dividends reduces assets and, hence, the value of the
contingent claim, but it also transfers value directly to shareholders. In our model, it is assumed that dividends, γA[T] are paid at date T.
The third modification allows for the fact that regulators have some discretion over whether a bank continues to operate – the so-called closure threshold need not be the point of actual insolvency,
as was the case in Equation (1). In our model, regulators are assumed to monitor banks at discrete intervals with a view to deciding whether to close the bank (the present date is taken to be t=0 and
the next monitoring date t=T).^[5] It is further assumed that regulators will close a bank if that bank's capital ratio, k[T], is less than c, where k[T] is defined as:^[6]
If the bank's capital ratio is above the closure threshold, equity holders receive the full value of the bank as a going concern. If a bank remains open at date T, shareholders receive the
dividend-adjusted difference between assets and liabilities ((1 − γ)A[T] − BT), a lump sum equal to the rents conferred by a banking licence (φB[T]) and a dividend payment (γA[T]).^[7] This is the
case regardless of whether the closure threshold is positive or negative.
If the bank is closed, it is assumed that equity holders manage to appropriate one final dividend payment from the firm. This assumption is made for algebraic convenience and, since dividend payments
are small relative to total assets, does not materially affect our results. The full payout for equity holders, however, depends upon whether the closure threshold is positive or negative. If
regulators only close banks with negative capital then, at closure the equity holders will only receive the final dividend (γA[T]). If, however, regulators apply a positive closure threshold, then at
time T, equity holders would receive the net tangible assets of the bank ((1 − γ)A[T] − B[T] + γA[T]) – once the bank is closed the licence value falls to zero.
A number of countries (including the US and Japan) have introduced prompt corrective action schemes whereby troubled banks must be closed before book-value equity falls below zero. Nevertheless, the
slippages that have been seen in the implementation of prompt corrective action schemes (see for example Benston and Kaufman (1998)), and the past experience of resolving troubled financial
institutions in Australia, suggest that it can be difficult for regulators to close a bank at positive economic capital ratios.^[8] As a result, we exclude the possibility of closure at positive
capital ratios from the model.
Reflecting adjustments for licence value, dividend payments and regulatory closure, the value of equity at the monitoring date, T, can be stated as:
where c is less than or equal to zero.
To obtain estimates of the value of equity prior to the monitoring date, it is necessary to make some assumptions regarding the stochastic processes followed by assets and liabilities. As in most
contingent-claim models, assets are assumed to follow Geometric Brownian Motion. The change in assets, d[A], can be expressed as:
where t is a time index, μ[A] is the expected instantaneous rate of return on assets per unit of time, σ[A] is the instantaneous standard deviation of the rate of return on assets and dz is the
differential of a Wiener process.^[9] Put simply, the first term on the right-hand side describes the drift of assets (or the average rate of return) over time while the second term can be regarded
as adding noise or variability to the path followed by assets. For simplicity, the market value of liabilities is assumed to be constant. Provided that liabilities are repriced frequently, any move
in interest rates will be offset by changes in future cash flows, leaving the present value of liabilities roughly constant. Any risk that does arise from liabilities (for example, from those with
fixed payments) will appear in our estimate of the volatility of the return on assets.
An important determinant of the value of equity is the probability that the bank will close. The relationship between the key determinants of closure is delineated in Figure 1. For illustrative
purposes, the figure is presented in the context of the basic model outlined in Equation (1).
Figure 1: Contingent-claim Model
Knowledge of the current market value of assets and the stochastic process followed by assets makes it possible to estimate the distribution of asset values at the monitoring date. As Figure 1 shows,
the probability that assets will be less than liabilities at the monitoring date depends on: (1) the expected value of assets at the monitoring date, E(A[T]) (i.e. the first moment of the
distribution); (2) the variability of assets (i.e. the second moment of the distribution); and (3) the level of liabilities. The shaded area of the distribution in Figure 1 represents this
The Black-Scholes option-pricing formula incorporates all the factors described above in estimating the value of a contingent claim. Using this formula, the value of equity at date t=0 is:
N(·) is the cumulative standard-normal distribution function and variables without subscripts denote present values.^[10] As discussed previously, equity holders can be viewed as having a long
European call option position in the value of the firm; equity holders have the option of either paying out the debt holders and acquiring the firm (receiving (1 − γ)A + φB + γA) or letting their
claims to the firm expire (still receiving γA).^[11] The probability that this option will be exercised (i.e. the probability that the bank will remain open) is represented in Equation (5) by
Given values for the market capitalisation of the firm (E), bank liabilities (B), the regulatory monitoring interval (T), the capital-ratio closure threshold (c), the licence value ratio (φ), and the
rate of dividend payments relative to assets (γ), the two remaining unknowns in Equation (5) are the value of assets (A), and the volatility of assets (σ[A]).^[12] Clearly, to compute the two
unknowns a second independent equation is needed.
Marcus and Shaked (1984) suggest applying Ito's Lemma to the expression for the value of equity, to yield a second equation involving the volatility of equity and the volatility of assets. They
follow Merton (1974) in deriving the relationship:
The basic idea of Equation (7) is that the volatility of a bank's equity is a magnified version of the volatility of a bank's assets, where the magnification factor depends on leverage and how
changes in assets are divided between liabilities and equity (that is, the elasticity of equity to assets).^[13]
Differentiating Equation (5) with respect to A yields:
where N′ (·) is the standard-normal density function and θ=1/(1−c)−(1−φ).^[14] Using Equation (8), the relationship defined by Equation (7) can be expressed as:
If σ[E] is observable, this equation also has A and σ[A] as the only unknowns; hence, Equation (5) and Equation (9) can be solved simultaneously for values of these two variables. Thus, under the
assumption that the value of bank equity is determined as in Equation (5), the market capitalisation of a bank can be used to infer the market value of assets and asset volatility.
2.2 Measures of Banking Risk
Given this theoretical framework, the central issue of this paper can be posed more explicitly. The key measure of the riskiness of a bank is the probability of closure. This overall risk measure can
then be broken down into two components – ‘financial risk’ and ‘operating risk’. (The term ‘operating risk’ is used in the literature. It should not be confused with operational risk, which is the
risk of earnings volatility not caused by market or credit factors.)
Financial risk is aligned with the bank's leverage. Regardless of the proclivity of banks to take risks, exogenous events can result in banks incurring large losses. A bank's ability to withstand
such large losses will depend on its level of capital. In terms of Figure 1, financial risk is inversely related to the difference between the economic assets and liabilities of the firm (that is,
the mean of the distribution relative to B[0]). The second component of overall risk, which is denoted operating risk, increases if the volatility of assets increases. If a bank takes on a portfolio
of assets characterised by a more uncertain income stream then, ceteris paribus, the chance of it incurring crippling losses increases. In Figure 1, this risk shows up in the shape of the
distribution; specifically, closure is more likely, the more volatile the assets.
In addition to the probability of closure, we present an alternative measure of overall risk, the expected losses borne by banks' creditors. We include this measure for purposes of comparison as this
measure is widely used in the literature. This measure originated in those countries where the repayment of deposits is guaranteed. For these countries (including France, Germany, Japan, United
Kingdom and the United States), the losses borne by creditors are transferred to the deposit insurer. The expected creditor losses can, therefore, be thought of as the value of the deposit guarantee,
or the expected amount that the guarantor will have to pay depositors at the monitoring date. In Australia, the system of depositor protection is quite different. While the Banking Act 1959 places a
duty on APRA to exercise its powers and functions to protect depositors, the repayment of deposits is not guaranteed.^[15] Despite the absence of explicit depositor protection, market participants
may hold the view that depositors are protected from financial loss, whether as a result of pre-emptive action by the supervisor, or due to compensation payments. Regardless of whether any liability
is borne by a government authority or the debt holders themselves, the size of this contingent liability can be estimated in a similar way to how we estimate the value of equity.
For simplicity, it is assumed that all creditors of a failed bank will be protected from financial loss if a government authority steps in whenever equity holders do not exercise their option. This
assumption enables us to treat the claims covered by any deposit guarantee as the total value of liabilities, B. If a bank fails, the payout under the guarantee may take the form of direct
restitution to depositors. In this situation, the guarantor will liquidate the assets of the bank and will pay depositors the amount they are owed. When liquidating the assets on behalf of
depositors, the guarantor is only able to sell the bank's tangible assets. Therefore, to compute the contingent liability under this scenario it is important to distinguish between the tangible
assets of the bank, A, and the intangible assets, φB. An alternative approach which the guarantor could follow is to locate a purchaser for the failed bank; the acquirer would receive all assets of
the bank – both tangible and intangible – and would assume all of the liabilities. If the assumed liabilities exceed the combined value of the tangible assets and the licence, the deposit guarantor
makes up the difference.
The size of the contingent liability depends on which action the regulator is likely to choose. Clearly, the liability will be lower under the second scenario since the licence value is being used to
reduce the payout to depositors following the bank's failure. As with the other risk measures, we are more interested in movements of the contingent liability rather than the level. Thus, given our
assumption that the licence value is constant through time, our conclusions will not be affected by what action we assume the guarantor will take following a bank's failure. Since it seems likely
that the guarantor would take the action that limits its liability wherever possible, we assume that the guarantor will find a purchaser for the failed bank. In this situation, the payout by the
guarantor at the monitoring date, V[T], is:
The guarantor has sold a put option to equity holders that has the same characteristics as the long call option held by equity holders (i.e. same strike price, etc).^[16] If the value of the firm
falls below liabilities, equity holders will exercise their put option and will force the guarantor to pay the debt holders the shortfall (of course, if the government does not step in, debt holders
will suffer the loss – they can be viewed as having a short put option position). Using the Black-Scholes formula once again, the value of the contingent payout in Equation (10) is:
Therefore, once values for A and σ[A] have been obtained it is straightforward to calculate the deposit-guarantee liability. Ceteris paribus, bank risk has increased if the size of this deposit
guarantee has increased.
The application of this technique to leveraged firms was pioneered by Merton (1974), the central framework of which has been employed in a number of subsequent papers (see, for example, Markus and
Shaked (1984) and Cordell and King (1992)). [1]
The value of a firm's economic liabilities (including equity) is equal to the value of a firm's economic assets, since the former represents a complete set of claims on the cash flows that accrue
from the assets. [2]
Of course, protected deposits are still risky in the sense that the real return on deposits can fluctuate unexpectedly. [3]
The main motivation for assuming that the licence value is proportional to liabilities is technical modelling convenience. It is likely that the value is positively related to bank size. As will be
discussed later, assets in the model are assumed to be stochastic; thus, if the licence value was related to assets in the model rather than liabilities, then its value would also be stochastic. This
would introduce an additional element of random fluctuation into the value of shares, thereby complicating the theoretical development of the model without substantially adding to the analysis. The
assumption is appropriate to the extent that the value of the licence reflects the opportunity to use deposits as a low cost source of funds. [4]
In this formulation, debt holders are assumed to play a passive role in the decision of whether the bank will be closed. One rationale for this assumption is that the operations of the bank are
sufficiently opaque to prevent debt holders from observing the market value of the bank's assets. In this environment, debt holders are reliant on the supervisor to act in their best interests. [5]
The capital-asset ratio is defined exclusive of the licence value. It, therefore, captures only the tangible assets of the bank. For this reason, the firm can still be solvent with a capital-asset
ratio less than zero. This is discussed further in Section 3.3. [6]
In a multi-period setting, φB and γA would reflect the discounted value of the future stream of rents and dividends respectively. [7]
Supporting this claim, the final annual report of the State Bank of Victoria gives a capital-asset ratio of about −0.06 when considering the State Bank group as a whole in the absence of government
assistance. [8]
This assumption implies that assets have a log-normal distribution and, hence, the returns on assets have a normal distribution. [9]
Note that, for simplicity, we have assumed that the expected periodic rate of return on assets, μ[A], is equal to zero. [10]
Using standard option terminology: assets are the underlying security; liabilities are the strike price of the option; and the value of equity is the price, or premium, of the option. [11]
The values of E, B, T, C, φ and γ used in the estimation are discussed in Section 3. [12]
A substantial change in the market value of assets will influence the bank's probability of default and, therefore, the market value of liabilities. [13]
The partial derivative [14]
The Act provides APRA with a range of powers designed to protect depositors. In particular, where a bank is likely to become unable to meet its obligations, the Act confers power on APRA to
investigate the bank's affairs and assume control of the bank for the benefit of its depositors. The Act also provides that the assets of the bank in Australia shall be available to meet its deposit
liabilities in Australia in priority to all other liabilities. For a more extensive discussion of APRA's powers and objectives see Goldsworthy, Lewis and Shuetrim (1999). [15]
In actual fact, the long call option position, discussed in Section 2.1, is equivalent to equity holders owning the assets of the firm and having a long put option in those assets. [16] | {"url":"https://www.rba.gov.au/publications/rdp/1999/1999-09/methodology.html","timestamp":"2024-11-02T05:23:52Z","content_type":"application/xhtml+xml","content_length":"58711","record_id":"<urn:uuid:805d7f5a-a129-4ff9-8223-e085f682de69>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00544.warc.gz"} |
The observed number counts in luminosity distance space
Next generation surveys will provide us with an unprecedented number of detections of supernovae Type Ia and gravitational wave merger events. Cross-correlations of such objects offer novel and
powerful insights into the large-scale distribution of matter in the universe. Both of these sources carry information on their luminosity distance, but remain uninformative about their redshifts;
hence their clustering analyses and cross-correlations need to be carried out in luminosity distance space, as opposed to redshift space. In this paper, we calculate the full expression for the
number count fluctuation in terms of a perturbation to the observed luminosity distance. We find the expression to differ significantly from the one commonly used in redshift space. Furthermore, we
present a comparison of the number count angular power spectra between luminosity distance and redshift spaces. We see a wide divergence between the two at large scales, and we note that lensing is
the main contribution to such differences. On such scales and at higher redshifts the difference between the angular power spectra in luminosity distance and redshift spaces can be roughly 50%. We
also investigate cross-correlating different redshift bins using different tracers, i.e. one in luminosity distance space and one in redshift, simulating the cross-correlation angular power spectrum
between background gravitational waves/supernovae and foreground galaxies. Finally, we show that in a cosmic variance limited survey, the relativistic corrections to the density-only term ought to be
Journal of Cosmology and Astroparticle Physics
Pub Date:
August 2023
□ galaxy clustering;
□ galaxy clusters;
□ gravitational waves / theory;
□ supernova type Ia - standard candles;
□ Astrophysics - Cosmology and Nongalactic Astrophysics
34 pages, 9 figures, 1 table. Agrees with published version | {"url":"https://ui.adsabs.harvard.edu/abs/2023JCAP...08..050F","timestamp":"2024-11-08T15:01:53Z","content_type":"text/html","content_length":"42116","record_id":"<urn:uuid:537f29e0-05fd-40a9-8252-4cd908969057>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00226.warc.gz"} |
Get Online Business Math Tutors
Learn Business Math Online with Best Business Math Tutors
Business Math is challenging, but it doesn’t have to be.
Our experienced business math tutors will work with you one-on-one to help you understand the concepts, solve problems, and ace your exams.
Sign up for our business math tutoring program starting at just $28/hour.
What sets Wiingy apart
Expert verified tutors
Free Trial Lesson
No subscriptions
Sign up with 1 lesson
Transparent refunds
No questions asked
Starting at $28/hr
Affordable 1-on-1 Learning
Top Business Math tutors available online
2040 Business Math tutors available
Responds in 33 min
Student Favourite
Business Math Tutor
5+ years experience
Business Mathematics expert with a burning desire to teach the many aspects of the subject. Possess multiple years of coaching experience in the field of study and willing to help with tests,
presentations and assignments
Responds in 36 min
Business Math Tutor
9+ years experience
I am a focused tutor who has mastered the skill of instructing youngsters in Basiness Math, and I can assist with homework and exam preparation. I hold a master's degree with 9 years of expertise and
assisted students.
Responds in 3 min
Star Tutor
Math Tutor
6+ years experience
An experienced math tutor, holding a Masters degree in Mathematics, also offers personalized lessons and provides assignment help on time.
Responds in 10 min
Star Tutor
Math Tutor
1+ years experience
Expert private online math tutor with 1 year of tutoring experience. Helps students learn a new concept, homework help, and test prep. Also provides test-taking strategies and boosts confidence.
Responds in 12 min
Student Favourite
Math Tutor
12+ years experience
Dive into complex math techniques with a tutor holding a Bachelor's degree and 12 years of experience. Master advanced topics and enhance your algebra skills with expert guidance.
Responds in 1 min
Star Tutor
Math Tutor
2+ years experience
Mathematics graduate and online Math tutor with 2+ years of tutoring experience. Provides customized lessons, test prep, and assignment help in Algebra, Calculus, Geometry, Statistics and more.
Responds in 6 min
Star Tutor
Math Tutor
2+ years experience
Expert Math Tutor dedicated to simplifying complex concepts and explaining every topics while solving the doubts. With 2+ years of tutoring experience for students of all levels. Holds a Master's
degree in Mathematics.
Responds in 8 min
Star Tutor
Math Tutor
2+ years experience
Expert in Math with Masters in Mathematics and 2+ years of experience of teaching math concepts to high school and college students in CA and UK.
Responds in 2 min
Star Tutor
Math Tutor
1+ years experience
Expert Math tutor for High School students. Provides detailed 1-on-1 sessions, homework help, and test-prep to US students.
Responds in 4 min
Star Tutor
Math Tutor
2+ years experience
Best Mathematics tutor have online teaching experince with US school students from 2+ years; Provides customised lessons, stem by stem problem solving strategies and test prep.
Responds in 14 min
Star Tutor
Math Tutor
7+ years experience
Talented Maths tutor with 7+ years of experience. Provides interactive concept clearing lessons test preparation and projects help to students. Holds a master's degree in Economics.
Responds in 10 min
Star Tutor
Math Tutor
2+ years experience
Math tutor with over 2 years of teaching experience, providing personalized classes and tailored lessons for high school and university students from various regions. Possesses a Bachelor's Degree in
Responds in 12 min
Star Tutor
Math Tutor
3+ years experience
Experienced Tutor for Math, 3+ Years of Tutoring, Well-Organized Sessions (Detailed & Provides assignment help)
Responds in 14 min
Student Favourite
Math Tutor
4+ years experience
Experienced Math tutor with a Bachelor's in Mathematics and 4 years of teaching experience. Provides personalized tutoring, homework help, and test preparation for students from elementary to high
Responds in 7 min
Star Tutor
Math Tutor
2+ years experience
Qualified Math, Chemistry and Coding Tutor ( Front-end Development and C Programming) with 2+ Years of Experience
Responds in 11 min
Student Favourite
Math Tutor
2+ years experience
Excellent Math Tutor having more than 2 years of online tutoring experience with school and college students. Provides assignment help and homework help.
Responds in 34 min
Student Favourite
Math Tutor
5+ years experience
Qualified Math tutor with 5+ years of teaching expertise, providing personalized online tutoring, practical exercises Holds a Master's degree in Mathematics Education.
Responds in 13 min
Star Tutor
Math Tutor
4+ years experience
A skilled and strategic Math tutor with a Bachelor's degree in Mathematics and over 4 years of experience. Provides interactive 1-on-1 concept clearing lessons, homework help, and test prep to school
and college students.
Responds in 5 min
Star Tutor
Math Tutor
4+ years experience
Top-tier Math with 4+ years of tutoring experience for high school to university students. hold a Master's Degree in Mathematics Education assist with test prep and clarify doubts.
Responds in 1 min
Star Tutor
Math Tutor
1+ years experience
Expert in Math. Bachelor's Degree from IIT, Madras, India, Personalized Sessions (Individual attention & Provides assignment help.
Responds in 8 min
Star Tutor
Math Tutor
3+ years experience
Learn and master mathematics. A highly skilled tutor who has a knack for breaking down complex topics and elaborating on them. A bachelor's degree tutor with 3 years of expertise in encouraging
Responds in 2 min
Star Tutor
Math Tutor
10+ years experience
Experience Math tutor with 10+ years of online tutoring experience with high school and college students. Subject expertise in Algebra, Calculus, Probability and, Statistics upto college level.
Responds in 7 min
Star Tutor
Math Tutor
13+ years experience
Achieve excellence in Math with expert support from a tutor holding a Master’s degree and 13 years of experience. Enhance your problem-solving abilities and mathematical understanding.
Responds in 12 min
Star Tutor
Math Tutor
5+ years experience
Seasoned Maths Tutor with 5 years of experience offers personalized instruction and comprehensive assistance. Proficient in calculus concepts, provides tailored guidance to students at both high
school and university levels.
Responds in 27 min
Student Favourite
Math Tutor
2+ years experience
Excellent Math tutor for kids and college students with 2+ years of experience. Provides assignment and homework help.
Responds in 14 min
Student Favourite
Math Tutor
6+ years experience
Experienced Math tutor of 6+ years of online teaching experience with IB curriculum. Best in Algebra, Calculus, Trigonometry, Geometry and Differential Equations. Helps with homework and assignments.
Responds in 4 min
Star Tutor
Math Tutor
10+ years experience
Excellent Maths tutor with 10+ years of experience delivering online instruction to students from grade 5 to university level. Holds a master's Degree and offers assessments and test practice.
Responds in 8 min
Student Favourite
Math Tutor
5+ years experience
Top-notch Math tutor with a Bachelor's in Mathematics, with an experience of 5 years, provides interactive concept-clearing lessons, homework help, and test prep to school and college students.
Responds in 35 min
Student Favourite
Math Tutor
12+ years experience
Accomplished Math, Algebra, Calculus and Geometry tutor with M.S. in Mathematics and 12+ years of 1-on-1 tutoring experience. Also gives test prep for AP Cal and SAT-Math.
Responds in 14 min
Student Favourite
Math Tutor
7+ years experience
Certified Math tutor online for high school students. BTech, with 7+ years of tutoring experience. Expert in providing one-to-one assisstance with difficult math concepts, assignments, and test prep.
Business Math topics you can learn
• Arithmetic and percentages
• Interest and investments
• Discounted cash flows
• Profit and loss calculations
• Break-even analysis
• Financial ratios and analysis
• Time value of money
• Depreciation and amortization
• Business statistics
• Inventory management
Try our affordable private lessons risk-free
• Our free trial lets you experience a real session with an expert tutor.
• We find the perfect tutor for you based on your learning needs.
• Sign up for as few or as many lessons as you want. No minimum commitment or subscriptions.
In case you are not satisfied with the tutor after your first session, let us know, and we will replace the tutor for free under our Perfect Match Guarantee program.
What is business math?
Business math is the application of math concepts and business problems. It is used in many business settings, including accounting, finance, marketing, and operations management.
It covers many topics, including basic arithmetic, algebra, geometry, statistics, and probability.
Uses of business math
Business math is used in solving a wide range of problems, such as:
1. Calculating profits and losses: Business math is used to calculate income and expenses and to determine the profits made and losses incurred.
2. Determining the best price for a product or service: Businesses use business math to set prices for their products and services, considering their costs, the competition, and the demand for them.
3. Sales forecasting: It is used to forecast future sales of businesses, which helps them plan their production and inventory levels.
4. Calculating taxes: Business math is also used to calculate taxes to be paid to the government.
5. Preparing financial statements: Businesses also use business math to prepare their financial statements, such as balance sheets and income statements, to show performance.
Why study business math?
Studying business math is important for anyone who wants to be successful in business. It can help develop the skills needed to make informed business decisions, better understand the financial
aspects of a business, and be more competitive.
Business math is also valuable for students aiming to take up business roles in their careers. Many employers look for employees who already have strong business math skills.
Business math is fundamental to study for career fields such as accounting, finance, marketing, and operations management.
Essential information about your Business
Math lessons
Average lesson cost: $28/hr
Free trial offered: Yes
Tutors available: 1,000+
Average tutor rating: 4.8/5
Lesson format: One-on-One Online | {"url":"https://wiingy.com/tutoring/subject/business-math-tutors/","timestamp":"2024-11-12T00:55:04Z","content_type":"text/html","content_length":"483739","record_id":"<urn:uuid:2242b520-9279-4651-a3e0-36926f77399d>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00219.warc.gz"} |
TFHE Deep Dive - Part III - Key switching and leveled multiplications
May 18, 2022
Ilaria Chillotti
> Part I: Ciphertext types
> Part II: Encodings and linear leveled operations
> Part III: Key switching and leveled multiplications
> Part IV: Programmable Bootstrapping
This blog post is part of a series of posts dedicated to the Fully Homomorphic Encryption scheme called TFHE (also known as CGGI, from the names of the authors Chillotti-Gama-Georgieva-Izabachène).
Each post will allow you to go deeper into the understanding of the scheme. The subject is challenging, we know! But don’t worry, we will dive into it little by little!
Disclaimer: If you have watched the video TFHE Deep Dive, you might find some minor differences in this series of blog posts. That’s because here there is more content and a larger focus on examples.
All the dots will connect in the end!
In the previous blog post we described how to perform homomorphic additions and homomorphic multiplications with small constants. We also described a few encodings used in TFHE.
In this post, we will continue describing some more leveled homomorphic operations and building blocks. The goal is to create a solid basis to be able to understand bootstrapping in the next blog
Homomorphic multiplication by a large constant
In the previous blog post we described the GLWE homomorphic multiplication by a small constant polynomial and we said that the noise grew proportionally with respect to the size of the coefficients
of the polynomial.
But what happens if we try to multiply by a large constant polynomial?
To even simplify the understanding, let's suppose that the polynomial is a constant (only the constant coefficient is different from 0).
By following the same notations as before, let's still consider a GLWE ciphertext encrypting a message $M \in \mathcal{R}_p$ under the secret key $\vec{S} = (S_0, \ldots, S_{k-1}) \in \mathcal{R}^k$:
C = (A_0, \ldots, A_{k-1}, B) \in GLWE_{\vec{S}, \sigma}(\Delta M) \subseteq \mathcal{R}_q^{k+1}.
Let's now consider a large constant $\gamma \in \mathbb{Z}_q$.
If we try to do the same as before, we multiply every component of the ciphertext by $\gamma$ (in $\mathcal{R}_q$). But now, as $\gamma$ is large and the noise grews proportionally with respect to
its size, what happens is that the noise compromises the result cause it grows too much.
To solve this noise problem, we need to use a very easy trick, combining decomposition and inner products. The idea consists in taking the large constant, and to decompose it into a small base $\
\gamma = \gamma_1 \frac{q}{\beta^1} + \gamma_2 \frac{q}{\beta^2} + \ldots + \gamma_\ell \frac{q}{\beta^\ell}
where the decomposed elements $\gamma_1, \ldots, \gamma_\ell$ are in $\mathbb{Z}_\beta$, and so they are small. We note $\mathsf{Decomp}^{\beta,\ell}(\gamma) = (\gamma_1, \ldots, \gamma_\ell)$. For
simplicity, we will suppose that both $q$ and $\beta$ are powers of two. In practice we often chose them as powers of two, if they are not, a rounding should be applied.
As the elements of the decomposition are small, now we should be able to perform the multiplication with the ciphertext and have just a small impact on the noise. But, in order to obtain as a result
the product of $\gamma$ and the message $M$ we need to be able to invert the decomposition, and so to recompose $\gamma$.
To do that, instead of multiplying the decomposed elements times a GLWE encryption of $M$, we multiply them times the GLev encryption of $M$, that, by definition, encrypts $M$ times different powers
of the decomposition base:
\overline{C} = (C_1, \ldots, C_\ell) \in \left( GLWE_{\vec{S}, \sigma}\left(\frac{q}{\beta^1} M\right) \times \ldots \times GLWE_{\vec{S}, \sigma}\left(\frac{q}{\beta^\ell} M\right) \right) = GLev^{\
beta, \ell}_{\vec{S}, \sigma}(M) \subseteq \mathcal{R}_q^{\ell \cdot (k+1)}.
In practice, we will perform an inner-product-like operation, so we will multiply every element of the decomposition times the corresponding element of the GLev and then add them all together
\langle \mathsf{Decomp}^{\beta,\ell}(\gamma), \overline{C} \rangle = \sum_{j=1}^\ell \gamma_j \cdot C_j \in GLWE_{\vec{S}, \sigma'}\left(\gamma \cdot M\right) \subseteq \mathcal{R}_q^{k+1}
where $\sigma'$ is the new standard deviation of the noise.
In this case, the noise grows way slowly, since the multiplication times the small constants have a smaller impact on the noise, and the following addition as well.
There is just one problem here: the GLWE in output has no more $\Delta$, so the new message potentially occupies the entire space $\mathbb{Z}_q$. We do not use this operation by itself in practice,
but we use it as a building block for more complex operations such as key switching and homomorphic multiplication between ciphertexts. We will talk about these operations in the rest of the post.
Approximate decomposition
In the figures that we just shown, we implicitly used an exact decomposition, or better we decomposed the constant with full precision (using $\beta^\ell = q$). But this is not always necessary: we
could decide to do an approximate decomposition and decompose up to a fixed precision (using $\beta^\ell < q$). In practice this means that we will do a rounding in the LSB part before decomposing:
if the decomposition parameters are chosen properly, this will not affect the correctness of the computations because in the LSB part there is always noise and the information we are interested into
keeping -- the message -- is in the MSB part. This is actually very convenient in some of the homomorphic operations that we will describe in the following sections.
Multiplication by a large polynomial
Now multiplication by a large polynomial follows the same footprint: you decompose the polynomial into small polynomials and then perform a polynomial inner product with the GLev. Assuming that the
polynomial we want to decompose is $\Lambda = \sum_{i=0}^{N-1} \Lambda_i \cdot X^i$, then the decomposition is equal to:
\mathsf{Decomp}^{\beta,\ell}(\Lambda) = (\Lambda^{(1)}, \ldots, \Lambda^{(\ell)})
where $\Lambda^{(j)} = \sum_{i=0}^{N-1} \Lambda_{i,j} \cdot X^i$, with $\Lambda_{i,j} \in \mathbb{Z}_\beta$, such that:
\Lambda = \Lambda^{(1)} \frac{q}{\beta^1} + \ldots + \Lambda^{(\ell)} \frac{q}{\beta^\ell}.
If the decomposition is approximate, the equality becomes an approximation.
This operation is going to be the main building block operation to describe the key switching and the homomorphic multiplication in the following sections. But first, let's fix the ideas with a toy
Toy example
The goal of this toy example is not to perform a multiplication between a large constant and a ciphertext, but to fix ideas about the decomposition. In particular, we show how to decompose a large
polynomial by using approximate signed decomposition, which is the one we mostly use in practice. For this toy example, we use our usual parameters $q=64$, $p=4$, $\Delta = q/p = 16$, $N=4$ and $k =
Now let's choose a random large polynomial in $\mathcal{R}_q$, so a polynomial of degree smaller than $N = 4$ and with coefficients in $\{ -32, -31, \ldots,$ $ -1, 0, 1, 2, \ldots, 30, 31 \}$:
\Lambda = \Lambda_0 + \Lambda_1 X + \Lambda_2 X^2 + \Lambda_3 X^3 = 28 - 5 X - 30 X^2 + 17 X^3
Let's choose a base for the decomposition $\beta = 4$ and $\ell = 2$, so $\beta^\ell = 16$. This means that we will decompose the $4$ MSB of each coefficient. But before we decompose, we need to
round all the coefficients.
We start by writing them in their binary decomposition first (MSB on the left and LSB on the right), and by performing the rounding of the $2$ LSB:
• $\Lambda_0 = 28 \longmapsto (0, 1, 1, 1 {\color{red} |} 0, 0)$ which after rounding becomes $\Lambda'_0 \longmapsto (0, 1, 1, 1);$
• $\Lambda_1 = -5 \longmapsto (1, 1, 1, 0 {\color{red} |} 1, 1)$ which after rounding becomes $\Lambda'_1 \longmapsto (1, 1, 1, 1);$
• $\Lambda_2 = -30 \longmapsto (1, 0, 0, 0 {\color{red} |} 1, 0)$ which after rounding becomes $\Lambda'_2 \longmapsto (1, 0, 0, 1);$
• $\Lambda_3 = 17 \longmapsto (0, 1, 0, 0 {\color{red} |} 0, 1)$ which after rounding becomes $\Lambda'_3 \longmapsto (0, 1, 0, 0).$
Next step is the decomposition. We start from the LSB and, since the base $\beta = 4$, we need to extract $2$ bits at every round. We want the decomposition to be signed, so we want coefficients in $
\{ -2, -1, 0, 1 \}$. So when in the binary decomposition we find $(0,0)$ -- corresponding to $0$ -- or $(0,1)$ -- corresponding to $1$ -- we simply read and record the value. When instead we find $
(1,0)$ -- corresponding to $2$ -- or $(1,1)$ -- corresponding to $3$ -- we subtract $4$ to the block, and we add $+4$ to the next block in the decomposition, like a carry. Every carry that goes
beyond the MSB, is thrown away. Let's do it in practice, it will be easier to understand.
In $\Lambda'_0 \longmapsto (0, 1, 1, 1)$:
• The two LSB are $(1,1)$, corresponding to the value $3$. We subtract $4$ and obtain $3-4 = -1$ as the first element of the decomposition.
• The next block is $(0,1)$, but since we subtracted $4$ before, we need to add it back now, which corresponds to add $1$ to $(0,1)$, that becomes $(1,0)$, corresponding to the value $2$. As before
we subtract $4$, obtaining $2-4 = -2$ as the second element of the decomposition. Since we reached the MSB, the $+4$ that should have been performed in the next block is simply thrown away.
In $\Lambda'_1 \longmapsto (1, 1, 1, 1)$:
• The two LSB are $(1,1)$, corresponding to the value $3$. We subtract $4$ and obtain $3-4 = -1$ as the first element of the decomposition.
• The next block is $(1,1)$, but since we subtracted $4$ before, we need to add it back now, which corresponds to add $1$ to $(1,1)$, that becomes $(0,0)$, corresponding to the value $0$, which
will be the second element of the decomposition.
In $\Lambda'_2 \longmapsto (1, 0, 0, 1)$:
• The two LSB are $(0,1)$, corresponding to the value $1$, which will be the first element of the decomposition.
• The next block is $(1,0)$, corresponding to the value $2$. As before we subtract $4$, obtaining $2-4 = -2$ as the second element of the decomposition.
In $\Lambda'_3 \longmapsto (0, 1, 0, 0)$:
• The two LSB are $(0,0)$, corresponding to the value $0$, which will be the first element of the decomposition.
• The next block is $(0,1)$, corresponding to the value $1$, which will be the second element of the decomposition.
Now that the decomposition of the coefficients is done, we can write explicitly the decomposed polynomials as:
\Lambda^{(1)} = -2 -2 X^2 + X^3 \\
\Lambda^{(2)} = -1 - X + X^2 \\
Observe that the coefficients of $\Lambda^{(2)}$ are the first elements of the decomposition, while the coefficients of $\Lambda^{(1)}$ are the second elements of the decomposition. These polynomials
can be used now in the inner products with the GLev ciphertexts.
To verify that the decomposition is correct, we can invert it by computing:
\Lambda^{(1)} \cdot \frac{q}{\beta^1} + \Lambda^{(2)} \cdot \frac{q}{\beta^2} = (-2 -2 X^2 + X^3) \cdot 16 + (-1 - X + X^2) \cdot 4 = 28 - 4 X - 28 X^2 + 16 X^3 \in \mathcal{R}_q
which is, as expected, an approximation of the original polynomial $\Lambda$.
Key switching
At this point you might wonder if multiplying for a large constant (as large as the ciphertext modulo $q$) would be really useful in practice. The answer is yes, if the large constant is/are another
ciphertext. Ciphertexts are in fact large vectors, polynomials, vectors of polynomials, and so on, composed by integers modulo $q$, that look uniformly random, so they might be quite large in
The trick combining decomposition and inner products with GLev ciphertexts, that we shown in previous paragraph, will now be very useful to start defining more complex operations, and in particular
multiplications between ciphertexts.
The first operation, looking like a multiplication, that we will focus on, is called key switching. This is an homomorphic operation largely used in all the (Ring)LWE-based schemes and, as the name
suggests, it is used to switch the secret key to a new one.
Let’s take a GLWE ciphertext encrypting a message $M \in \mathcal{R}_p$ under the secret key $\vec{S} = (S_0, \ldots, S_{k-1}) \in \mathcal{R}^k$:
C = (A_0, \ldots, A_{k-1}, B) \in GLWE_{\vec{S}, \sigma}(\Delta M) \subseteq \mathcal{R}_q^{k+1}
where the elements $A_i$ for $i\in [0..k-1]$ are sampled uniformly random from $\mathcal{R}_q$, and $B = \sum_{i=0}^{k-1} A_i \cdot S_i + \Delta M + E \in \mathcal{R}_q$, and $E \in \mathcal{R}_q$
has coefficients sampled from a Gaussian distribution $\chi_{\sigma}$, as we have already seen many times in previous blog posts.
Now, to switch the key we will try to cancel the secret key $\vec{S}$ and re-encrypt under a new secret key $\vec{S}'$, and we will try to do this homomorphically. The idea is to compute:
B - \sum_{i=0}^{k-1} A_i \cdot S_i = \Delta M + E \in \mathcal{R}_q
but instead of using the elements $S_i$ in clear, we will provide them encrypted under the new secret key $\vec{S}'$.
It is here that the multiplication between a ciphertext and a large constant comes to play: the ciphertext will be the GLev encryption of $S_i$ and the large constant will be $A_i$ that, by
construction, is a uniformly random polynomial in $\mathcal{R}_q$.
Let's call key switching key the list of GLev encryptions of the secret key elements $S_i$ under the new secret key $\vec{S}'$, and let's note them as:
\mathsf{KSK}_i \in \left( GLWE_{\vec{S}', \sigma_{\mathsf{KSK}}}\left(\frac{q}{\beta^1} S_i\right) \times \ldots \times GLWE_{\vec{S}', \sigma_{\mathsf{KSK}}}\left(\frac{q}{\beta^\ell} S_i\right) \
right) = GLev^{\beta, \ell}_{\vec{S}', \sigma_{\mathsf{KSK}}}(S_i) \subseteq \mathcal{R}_q^{\ell \cdot (k+1)}.
In practice, the key switching is performed as follows:
C' = \underbrace{\overbrace{(0, \ldots, 0, B)}^{\text{Trivial GLWE of } B} - \sum_{i=0}^{k-1} \overbrace{ \langle \mathsf{Decomp}^{\beta,\ell}(A_i), \mathsf{KSK}_i \rangle}^{\text{GLWE encryption of
} A_i S_i} }_{\text{GLWE encryption of } B - \sum_{i=0}^{k-1} A_i S_i = \Delta M + E}\in GLWE_{\vec{S}', \sigma'}(\Delta M) \subseteq \mathcal{R}_q^{k+1}.
The secret key has switched from $\vec{S}$ to $\vec{S}'$ but the message is the same.
Observe that this corresponds to the homomorphic evaluation of the first step of the GLWE decryption, but since we do not evaluate the second step -- which is the re-scaling by $\Delta$ and rounding
-- we do not reduce the noise. On the contrary, the noise in the result of the operation is larger than the one in the input ciphertext $C$, and we note it $\sigma'$.
Various types of key switching
There exists various types of key switching that we use in practice. Some examples (not an exhaustive list) are the:
• Key switching from one LWE to one LWE,
• Key switching from one RLWE to one RLWE,
• Key switching from one LWE to one RLWE, putting the message encrypted in the LWE into one of the coefficients of the RLWE ciphertext,
• Key switching from many LWE to one RLWE, packing the messages encrypted in the many LWE inputs into the RLWE ciphertext.
Other uses of the key switching
The key switching is not only used to switch the key, but can be also used to switch the parameters. In fact, the output key could be defined with $N$ and $k$ parameters that might be different from
the ones of the input key. This is actually something that happens very often in the key switching: a practical example (involved in the bootstrapping) will be shown in next blog post: stay tuned!
External product
Now that we know how to perform a key switching, the external product operation will be straight forward.
With the external product our goal would be to homomorphically multiply two ciphertexts such that the result is an encryption of the product of messages.
What we know, until here, is that to multiply a large constant to a ciphertext, we need to decompose the large constant and recompose by using a GLev as a ciphertext. We also know that a GLWE
ciphertext is a list of large constants. So how to combine these two ideas to do multiplication?
We will use a similar approach as the one used for key switching, so we will take one of the two ciphertexts as a GLWE (the ciphertext we will decompose), and the other ciphertext will be a list of
GLev ciphertexts. The difference, this time, is that with key switching we wanted only the mask of the first ciphertext to be multiplied by the GLev's encrypting the secret key, while this time we
want both mask and body.
In the first blog post we introduced a ciphertext, composed by GLev ciphertexts, that is just right for us: GGSW.
So, to summarize, the external product we will build is an operation that allows to multiply two ciphertexts -- a GLWE and a GGSW -- and that returns in output a new ciphertext -- a GLWE one --
encrypting the product of the two messages encrypted in the inputs. The two inputs are:
• a GLWE ciphertext encrypting a message $M_1 \in \mathcal{R}_p$ under the secret key $\vec{S} = (S_0, \ldots, S_{k-1}) \in \mathcal{R}^k$:
C = (A_0, \ldots, A_{k-1}, B) \in GLWE_{\vec{S}, \sigma}(\Delta M_1) \subseteq \mathcal{R}_q^{k+1}
where the elements $A_i$ for $i\in [0..k-1]$ are sampled uniformly random from $\mathcal{R}_q$, and $B = \sum_{i=0}^{k-1} A_i \cdot S_i + \Delta M + E \in \mathcal{R}_q$, and $E \in \mathcal{R}
_q$ has coefficients sampled from a Gaussian distribution $\chi_{\sigma}$, as we have already seen before.
• a GGSW ciphertext encrypting a message $M_2 \in \mathcal{R}_p$ under the same secret key $\vec{S} = (S_0, \ldots, S_{k-1}) \in \mathcal{R}^k$:
\overline{\overline{C}} = (\overline{C}_0, \ldots, \overline{C}_{k-1}, \overline{C}_k) \in GGSW^{\beta, \ell}_{\vec{S}, \sigma}(M_2) \subseteq \mathcal{R}_q^{(k+1) \times \ell (k+1)}
where $\overline{C}_i \in GLev^{\beta, \ell}_{\vec{S}, \sigma}(-S_i M_2)$ for $i \in [0..k-1]$ and $\overline{C}_k \in GLev^{\beta, \ell}_{\vec{S}, \sigma}(M_2)$.
Then the external product is noted with the symbol $\boxdot$ and is computed as:
C' &= \overline{\overline{C}} \boxdot C
= \langle \mathsf{Decomp}^{\beta,\ell}(C), \overline{\overline{C}} \rangle \\
&= \underbrace{
\overbrace{\langle \mathsf{Decomp}^{\beta,\ell}(B), \overline{C}_k \rangle}^{\text{GLWE encrypt. of } B M_2} + \sum_{i=0}^{k-1} \overbrace{\langle \mathsf{Decomp}^{\beta,\ell}(A_i), \overline{C}_i \
rangle}^{\text{GLWE encrypt. of } - A_i S_i M_2}
}_{\text{GLWE encrypt. of } B M_2 - \sum_{i=0}^{k-1} A_i S_i M_2 \approx \Delta M_1 M_2}
\in GLWE_{\vec{S}, \sigma^{\prime\prime}}(\Delta M_1 M_2) \subseteq \mathcal{R}_q^{k+1}
The noise in the result of the operation is larger than the one in the input ciphertext $C$, and we note it $\sigma^{\prime\prime}$.
External Product vs. Key Switching
A few more observations on differences and similarities between key switching and external product:
• The external product is like a key switching with an additional element to the key switching key (the $\overline{C}_k$ GLev ciphertext).
• The external product is like a key switching where we do not switch the key. In fact, in the GGSW ciphertext, the secret key used for encryption and the one used inside the GLev ciphertexts are
the same.
• An external product that takes as input a GGSW ciphertext, that uses a different secret key for encryption (say $\vec{S}'$) and uses the same secret key $\vec{S}$ as the GLWE ciphertext inside
the encryption, is called functional key switching. In fact it applies a function (multiplication by an encrypted constant) and switches the key at the same time.
Internal product
The external product is called external because is a product on GLWE that needs an external GGSW ciphertext to work (the external product notation is as instance used in mathematics for module over
rings, which are abelian groups with an external law). There is no internal product between GLWE ciphertexts, or at least not one that could be done in a straight way. In B/FV, as instance, a product
between GLWE ciphertexts is performed but it requires a key-switching-like operation.
There exists, however, an internal product between GGSW ciphertexts that can be defined from the external product. In fact, a GGSW ciphertext is a list of Glev ciphertexts, and each GLev is a list of
GLWE ciphertexts. So, in practice, a GGSW is a list of GLWE ciphertexts. Since the external product is a product between GLWE and GGSW ciphertexts, we can define the internal product as a list of
independent external products between one of the GGSW ciphertexts in input and all the GLWE ciphertexts composing the second GGSW input. The result of all these external products are going to be the
GLWE ciphertexts that will compose the GGSW output.
To be more explicit, the internal product takes in input:
• a GGSW ciphertext encrypting a message $M_1 \in \mathcal{R}_p$ under a secret key $\vec{S} = (S_0, \ldots, S_{k-1}) \in \mathcal{R}^k$:
\overline{\overline{C}}_1 = (\overline{C}_0, \ldots, \overline{C}_{k-1}, \overline{C}_k) \in GGSW^{\beta, \ell}_{\vec{S}, \sigma}(M_1) \subseteq \mathcal{R}_q^{(k+1) \times \ell (k+1)}.
where, for $i \in [0..k-1]$:
\overline{C}_i = (C_{i,1}, \ldots, C_{i,\ell}) \in GLev^{\beta, \ell}_{\vec{S}, \sigma}(-S_i M_1) \subseteq \mathcal{R}_q^{\ell \cdot (k+1)}
with $C_{i,j} \in GLWE_{\vec{S}, \sigma}\left(\frac{q}{\beta^j} (-S_i M_1) \right)$ for $j \in [1..\ell]$, and:
\overline{C}_k \in (C_{k,1}, \ldots, C_{k,\ell}) \in GLev^{\beta, \ell}_{\vec{S}, \sigma}(M_1) \subseteq \mathcal{R}_q^{\ell \cdot (k+1)}
with $C_{k,j} \in GLWE_{\vec{S}, \sigma}\left(\frac{q}{\beta^j} M_1 \right)$ for $j \in [1..\ell]$.
• a GGSW ciphertext encrypting a message $M_2 \in \mathcal{R}_p$ under the same secret key $\vec{S} = (S_0, \ldots, S_{k-1}) \in \mathcal{R}^k$:
\overline{\overline{C}}_2 \in GGSW^{\beta, \ell}_{\vec{S}, \sigma}(M_2) \subseteq \mathcal{R}_q^{(k+1) \times \ell (k+1)}.
Then the internal product is noted with the symbol $\boxtimes$ and is computed as:
\overline{\overline{C}}' = \overline{\overline{C}}_2 \boxtimes \overline{\overline{C}}_1
= (\overline{\overline{C}}_2 \boxdot C_{0,1},
\overline{\overline{C}}_2 \boxdot C_{0,\ell},
\overline{\overline{C}}_2 \boxdot C_{k,1},
\overline{\overline{C}}_2 \boxdot C_{k,\ell}).
Observe that:
\overline{\overline{C}}_2 \boxdot C_{i,j} &\in GLWE_{\vec{S}, \sigma^{\prime\prime}}\left( \frac{q}{\beta^j} (-S_i M_1 M_2) \right) &\text{for } i \in [0..k-1], j \in [1..\ell] \\
\overline{\overline{C}}_2 \boxdot C_{k,j} &\in GLWE_{\vec{S}, \sigma^{\prime\prime}}\left( \frac{q}{\beta^j} (M_1 M_2) \right) &\text{for } j \in [1..\ell]
\left( \overline{\overline{C}}_2 \boxdot C_{i,1}, \ldots, \overline{\overline{C}}_2 \boxdot C_{i,\ell} \right) &\in GLev^{\beta, \ell}_{\vec{S}, \sigma^{\prime\prime}}\left( -S_i M_1 M_2 \right) &\
text{for } i \in [0..k-1] \\
\left( \overline{\overline{C}}_2 \boxdot C_{k,1}, \ldots, \overline{\overline{C}}_2 \boxdot C_{k,\ell} \right) &\in GLev^{\beta, \ell}_{\vec{S}, \sigma^{\prime\prime}}\left( M_1 M_2 \right) &
So, putting everything together, we observe that:
\overline{\overline{C}}' = \overline{\overline{C}}_2 \boxtimes \overline{\overline{C}}_1
\in GGSW^{\beta, \ell}_{\vec{S}, \sigma^{\prime\prime}}(M_1 M_2) \subseteq \mathcal{R}_q^{(k+1) \times \ell (k+1)}.
The noise in the result of the operation is larger than the one in the input ciphertext $C$, and we note it $\sigma^{\prime\prime}$ (the same as the one in the external product).
Internal Product vs. External Product
The external product is way more efficient than the internal product, but it is not composable. This means that the result of an internal product can be used as input of another internal product, but
the result of an external product (GLWE ciphertext) can be used only in one of the two inputs of another external product (the GLWE one) but not on the other (the GGSW one). So, if we use external
products, the GGSW input should be a fresh one (just encrypted).
TFHE proposes a technique to build a GGSW ciphertext homomorphically from LWE, called circuit bootstrapping: we will not describe this technique in this series of blog posts, but if you are curious
to know more about it, you can check this paper.
In TFHE, we use mainly the external product and avoid as much as possible the internal one.
Before we finish, let us describe a last operation, maybe the one that is mainly used in TFHE, and which will be one of the main building blocks of bootstrapping: the CMux operation.
The CMux operation is the homomorphic version of a Mux gate, also known as multiplexer gate. A Mux gate is, in practice, an if condition: it takes three inputs, a selector (a bit $b$ in the figure
below) and two options ($d_0$ and $d_1$ in the figure below), and, depending on the value of the selector, it makes a choice between the two options.
It is evaluated in clear by computing:
b \cdot (d_1 - d_0) + d_0 = d_b
To evaluate it homomorphically, we encrypt $b$ as a GGSW ciphertext, and $d_0$ and $d_1$ as GLWE ciphertexts. Then the multiplication in the cleartext formula is evaluated as an external product,
while the other operations (addition and subtraction) are evaluated as homomorphic additions and subtractions. The result, encrypting $d_b$ is a GLWE ciphertext.
This blog post was more challenging than the ones before. If you made it until here, congratulations! Are you still curious to know how we use all these operations to build bootstrapping? Read part
IV to understand the bootstrapping of TFHE.
Additional links
• ⭐️ Like our work? Support us and star TFHE-rs on Github
• 👋 Questions? Join the Zama community.
• 💸 Join the Zama Bounty Program. Solve FHE problems, write tutorials and earn rewards, more than €500,000 in prizes.
Read more related posts
Ilaria Chillotti
May 4, 2022
TFHE Deep Dive - Part I - Ciphertext types
This blog post is part of a series of posts dedicated to the Fully Homomorphic Encryption scheme called TFHE by Ilaria Chillotti.
Read Article
Ilaria Chillotti
May 11, 2022
TFHE Deep Dive - Part II - Encodings and linear leveled operations
This second blog post of the series shows you how to perform operations on the ciphertexts used in TFHE.
Read Article
Ilaria Chillotti
June 2, 2022
TFHE Deep Dive - Part IV - Programmable Bootstrapping
This fourth and last part of the blog post of the TFHE series is dedicated to bootstrapping.
Read Article | {"url":"https://www.zama.ai/post/tfhe-deep-dive-part-3?utm_source=tfhe_deep_dive_part_3&utm_medium=substack&utm_campaign=blogpost","timestamp":"2024-11-13T18:10:16Z","content_type":"text/html","content_length":"82897","record_id":"<urn:uuid:574f9979-8970-4d89-bedf-1a1b03c064f2>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00671.warc.gz"} |
Abstract Scope Thermodynamic properties of the Si-B binary system were calculated using thermo-Calc software at a pressure of 1 bar. Gibbs free energy, enthalpy, entropy, and activity were calculated
and plotted against temperature and composition. Gibbs free energy curves for all phases were discussed and found to be consistent with phase diagram. It is found that Gibbs free energy becomes more
negative with temperature, and it shows minima at certain composition values for different phases. Enthalpy curves for all phases are plotted with composition, and they are consistent with Gibbs free
energy curves, then plotted against temperature and showed increase from negative values to positive values. Entropy and activity curves for all phases were plotted against composition and
temperature and thoroughly discussed. The largest value of boron activity is about 0.49, and the curves shown maxima then a monotonic decrease with temperature, where activity of boron becomes about | {"url":"https://www.programmaster.org/PM/PM.nsf/ApprovedAbstracts/A2674C63A1F3960485258437003C9A5B?OpenDocument","timestamp":"2024-11-06T21:04:43Z","content_type":"text/html","content_length":"11371","record_id":"<urn:uuid:d0376d57-93ce-4374-96f1-efe79449027d>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00499.warc.gz"} |
charles proteus steinmetz Archives - Eric P. Dollard - Official Homepage
Posted on — Leave a comment
versor algebra part 2 by eric dollard – paperback on amazon available now
NEW RELEASE – Versor Algebra Part 2 by Eric Dollard is here!
This is a continuation of Part 1 – Charles Proteus Steinmetz’s original math model is a natural outgrowth of Nikola Tesla’s polyphase power systems. Tesla was the discoverer, but Steinmetz was the
builder who first applied Versor Algebra to the analysis of alternating current power systems.
Get the new book here: Versor Algebra, Part 2
In my presentation and book Four Quadrant Representation of Electricity, my extension of Steinmetz’s work is presented in the most simple way possible using very simple analogies, pictures and
diagrams. It was a very difficult task as the goal was to facilitate an understanding for the layman.
That presentation was given at the 2013 Energy Science & Technology Conference and shortly thereafter, the book version was released, which went into more detail that was not covered in the
Tesla’s polyphase power system was originally four poles or four phases. Steinmetz is the one who adapted it into a three pole or three phase system, which is the prominent system of today. The
complication is that three phase systems cannot be explained by conventional mathematics. With three phase systems, there is no plus or minus and that is the reason why the conventional math doesn’t
work anymore.
That left a big gap in polyphase power systems until Dr. Fortescue came up with the system of Symmetrical Coordinates. This laid the groundwork for polyphase mathematics for any number of phases. And
ultimately, it can be extended into the Pythagorean understanding of numbers.
Get the new book here: Versor Algebra, Part 2
The “Fortescue Method” was never fully developed because of its complexity. The proper name for this is “Sequence Algebra” and the rudiments were presented in my presentation and book Four Quadrant
Representation of Electricity. Even though the system has become adopted for general engineering usage, Versor Algebra as Applied to Polyphase Power Systems and/or Versor Algebra Vol. II, Special
Theories of Sequence Operators as Applied to Power Engineering is the first theoretical basis that has ever been presented on the subject.
Versor Algebra as Applied to Polyphase Power Systems and/or Versor Algebra Vol. II, Special Theories of Sequence Operators as Applied to Power Engineering is the next logical step after Four
Quandrant Representation of Electricity as it takes the reader into the mathematical journey of the mathematical model and theory that is necessary to realize the unique electrical waves that exist
in polypahse power systems.
These waves are actually beyond the original understanding of Tesla and Steinmetz with regard to polyphase power systems. It is important to understand that this is all possible with simple 9th grade
algebra. Eric takes the reader through a step-by-step process from very basic algebra and log-rhythms into the more complex subject. The process involves very simple but numerous steps to guide the
reader into the understanding of polyphase mathematics.
Through Eric’s own journey in writing this Versor Algebra book, I have been able to unify the polyphonic music of Bach and his contemporaries as this music follows the logic of sequence algebra
perfectly. In fact, the book was written when listening to this music, which aided the process greatly.
Get the new book here: Versor Algebra, Part 2
For information on Versor Algebra Part 1 on Amazon – go here: https://ericpdollard.com/2019/07/25/new-paperback-versor-algebra-as-applied-to-polyphase-power-systems-part-1-by-eric-dollard/
Posted on — Leave a comment
Eric Dollard New Release March 22nd
• VERSOR ALGEBRA
• FUNDRAISER COMING
• VIDEO TRIBUTE TO ERIC DOLLARD
• ENERGY CONFERENCE
Here are a few important updates relating to Eric Dollard and his recent work…And thank you to the Eric Dollard Fan Club – your donations have helped to carry the lab’s operations through some tough
times – especially over the last 4-5 months! And thank you to all those who send donations directly to Eric and through PayPal. Without all of your generous support, EPD Laboratories could not
On Sunday, March 22nd, we’re releasing the long anticipated book by Eric Dollard called Versor Algebra as Applied to Polyphase Power Systems.
Nikola Tesla developed the polyphase power system but originally it was four phases. Charles Proteus Steinmetz was hired by General Electrical to break Tesla’s patent and to analyze it
mathematically, which he was the first to do with Versor Algebra. But Steinmetz adapted Tesla’s four phase system to a three phase system, which is still in use today. Conventional mathematics were
not suitable for analyzing this three phase system without Dr. Fortescue’s Symmetrical Coordinate method.
Although this made it possible to analyze any number of phases, there has never been a theoretical basis presented for this method until Eric Dollard created it – this book is actually Part 2 to this
book and lecture from the 2013 Energy Science & Technology Conference called Four Quadrant Representation of Electricity. He also simplified the method of Symmetrical Coordinates beyond anything that
has been done in engineering history while completely preserving it’s functionality.
This book shows the methods that not only apply to power systems, but can actually be used to analyze ANYTHING that uses multiphases such as the music of Bach, which has multiple phases overlayed on
top of each other. This math is also known as Sequence Algebra and we are only at the tip of the iceberg with what Eric made tangible for so many more engineers, those who study music and more.
There is an interview with Eric where he gives a great outline of what this book is about and why it is important and it is put into a YouTube video – this will be available on the homepage for the
new book. We’ll send a link to this on Sunday.
Please make sure to support Eric Dollard’s work by purchasing a copy this coming Sunday – 70% of the sales goes to EPD Laboratories Inc. to help make the building payment (no longer rent since
they’re the official buyers now!), utilities and other building related costs.
On Wednesday, April 1st, we’re launching a new fundraising campaign to help EPD Laboratories, Inc., which is a 501(c)(3) non-profit organization.
GOOD NEWS! What many people don’t know is that Eric Dollard recently paid almost $50,000 to David Wittekind to help pay off his debt for the building. This money was provided by a private donor.
Thank you J.!!!
EPD Laboratories, Inc. is officially the new buyer and for the organization to own the building outright, there is only a balance of $29,000 left! Paying off this debt will relieve a large monthly
financial obligation, which will allow more rapid progress since the money can go to parts and services necessary for the work at the lab, the Advanced Seismic Warning System, etc.
It is important to remember – the Advanced Seismic Warning System is only ONE application of this long lines antenna project – Eric will be able to do many Tesla related experiments with this setup.
The other funds that are needed are for shelving so that Eric can start disassembling a lot of the electronics and organizing them, 1 year of operating capital for the building (utilities, taxes and
insurance), a used “Lines Truck” for digging holes and planting telephone poles needed for the end of the seismic line (they’re getting this truck for a fraction of what it is worth), an inexpensive
water well drilling apparatus that will be used for for digging 30′ deep holes for the ground rods needed by each telephone pole for the seismic project (a few thousand dollars), enough parts for
Eric and John Polakowski to build all the analog networks that plugs into the lines on the poles, gas money so Steve Hilsz (“The Glommeister” and director of EPD Laboratories) can come to Spokane to
pick up a literal ton of Glom donated by Mark McKay (Spokane1 of Energetic Forum), etc.
We’ll be sending a link soon…please support EPD Laboratories by donating to this campaign once it is launched on April 1st and please pass it on to your friends.
Imaginarium Labs is making a series of short videos to help build awareness for Eric Dollard’s work. They’re fans of Eric and want to do what they can to support him. Please check out this video and
give it a thumbs up to show your support:
The conference schedule has been updated – you can see it here: http://energyscienceconference.com/energy-conference-schedule/2015-schedule/
Eric’s presentation is THE POWER OF THE AETHER AS RELATED TO MUSIC AND ELECTRICITY and is scheduled for 3 hours, but could run much longer.
Check this out – you can see that Eric Dollard has inspired a few young Electrical Engineering students. They won food, lodging and the conference fee made possible by a donation from the Eric
Dollard Fan Club! http://energyscienceconference.com/2015/03/13/energy-conference-winners-congratulations/
Out of 150 limited seats, there are only 60 something left. If you want to hang out with Eric for a weekend, pick his brain and meet other pioneers in this field, register now while you can. There
are 3.75 months left until the conference and the seats are going fast. http://energyscienceconference.com | {"url":"https://ericpdollard.com/tag/charles-proteus-steinmetz/","timestamp":"2024-11-02T18:34:54Z","content_type":"text/html","content_length":"192549","record_id":"<urn:uuid:bc6662dc-4e39-4bb9-b8b1-b4e630fe9bb1>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00760.warc.gz"} |
2009 Matthew Good (band) Elimination
As far as I know it's been awhile since this has been done, and since we now have two more albums to include I think it's time we do it again. Format is the same, song vs. song, with the opportunity
to bring back some songs later on that may get eliminated earlier than they deserve because of unfavorable matchups. Most matchups are random, the exception being tracks that are different versions
of the same song, i.e. "Rooms." There are 144 songs to choose from as I threw in a few pre LOTGA tracks to bring the number to 144.
Round 1 A
1) Tripoli vs. Tripoli (rooms)
- Tripoli lost to Tripoli (rooms) in round 1 in 2005
- Tripoli (rooms) finished 9th overall
- MY VOTE - Tripoli
2) On Nights Like Tonight vs. Look Happy, It's the End of the World
- On Nights Like Tonight did not exist in 2005
- Look Happy, It's the End of the World lost to Weapon in round 2 in 2005
- MY VOTE - On Nights Like Tonight
3) Weapon vs. Failing the Rorschach Test
- Weapon finished 7th in 2005
- Failing the Rorschach Test lost to The Rat Who Would Be King in round 1 in 2005
- MY VOTE - Weapon
4) Born Losers vs. Raygun
- Born Losers didn't exist in 2005
- Raygun lost to Lullaby For a New World Order in round 1 in 2005
- MY VOTE - Born Losers
5) Hopeless vs. The War is Over
- Rematch from 2005
- The War is Over was victorious, but lost in round 2 to Under The Influence
- MY VOTE - Hopeless
6) Girl Wedged Under the Front of a Firebird vs. I Am Not Safer Than a Bank
- both tracks did not exist in 2005
- MY VOTE - I Am Not Safer Than a Bank
7) Big City Life vs. Jenni's Song
- Big City Life lost to While We Were Hunting Rabbits in round 2 in 2005
- Jenni's Song lost to Sort of a Protest Song in round 3 in 2005
- MY VOTE - Jenni's Song
8) My Out of Style is Coming Back vs. Metal Airplanes
- My Out of Style is Coming Back lost to Truffle Pigs in round 2 in 2005
- Metal Airplanes did not exist in 2005
- MY VOTE - Out of Style
9) Comfortable Criminals vs. Flashdance II
- Comfortable Criminals lost to Everything is Automatic in round 2 in 2005
- Flashdance II lost to Strange Days in round 3 in 2005
- MY VOTE - Flashdance II
Voting is open until noon November 4th
Edited by rebellious_L
1. Tripoli (Rooms)
2. On Nights...
3. Weapon
4. Born Losers
5. War Is Over
6. I Am Not Safer...
7. Big City Life
8. Metal Airplanes
9. Flashdance II
Ha, the reason I signed up onto these boards were solely because of these competitions.
1) Tripoli
2) Look Happy, It's the End of the World
3) Weapon
4) Born Losers
5) The War is Over
6) Girl Wedged Under the Front of a Firebird
7) Jenni's Song
8) My Out of Style is Coming Back
9) Flashdance II
Edited by Shortcut To Moncton
Look Happy, It's the End of the World
Failing the Rorschach Test
Born Losers
The War is Over
I Am Not Safer Than a Bank
Big City Life
My Out of Style is Coming Back
Comfortable Criminals
1. Tripoli
2. On Nights Like Tonight
3. Failing The Rorschach Test
4. Raygun
5. The War Is Over
6. I Am Not Safer Than A Bank (both these "songs" are terrible)
7. Jenni's Song
8. My Out Of Style Is Coming Back
9. Flashdance II
Tripoli (rooms)
On Nights Like Tonight
Failing the Rorschach Test
Born Losers
I Am Not Safer Than a Bank
Big City Life
My Out of Style is Coming Back
Comfortable Criminals
1) Tripoli
2) On Nights Like Tonight
3) Weapon
4) Born Losers
5) Hopeless
6) I Am Not Safer Than a Bank
7) Jenni's Song
8) Metal Airplanes
9) Flashdance II
1) Tripoli
2) On Nights Like Tonight
3) Weapon
4) Raygun
5) Hopeless
6) I Am Not Safer Than a Bank
7) Jenni's Song
8) Metal Airplanes
9) Flashdance II
1) Tripoli
2) Look Happy It's The End Of The World
3) Weapon
4) Born Losers
5) Hopeless
6) I Am Not Safer Than A Bank
7) Big City Life
8) My Out Of Style Is Coming Back
9) Flashdance II
1. (Rooms)
2. On Nights Like Tonight
3. Weapon
4. Raygun
5. Hopeless (This one was incredibly hard for me)
6. Bank
7. Big City Life (not that fond of either)
8. Metal Airplanes
9. Flashdance II
1. Rooms
2. On Nights Like Tonight
3. Weapon
4. Born Losers
5. The War is Over
6. Firebird
7. Jenni's Song
8. Out of Style
9. Flashdance II
1. Rooms tripoli
2. Look happy it
1)Tripoli (rooms)
2) On Nights Like Tonight
3)Failing the Rorschach Test
4) Born Losers
5)The War is Over
6) Girl Wedged Under the Front of a Firebird
7) Big City Life
8) Metal Airplanes
9)Flashdance II
On Nights Like Tonight
Failing the Rorschach Test
I am not safer than a bank
On Nights Like Tonight
Failing the Rorschach Test
Born Losers
The War is Over
I Am Not Safer Than a Bank
Jenni's Song
My Out of Style is Coming Back
Flashdance II
Edited by Manchalivin
1. Tripoli (Rooms)
2. On Nights Like Tonight
3. Weapon
4. Born Losers
5. The War is Over
6. I am Not Safer Than a Bank
7. Big City Life
8. Metal Airplanes
9. Flashdance II
1) Tripoli
2) On Nights Like Tonight
3) Weapon
4) Born Losers
5) The War is Over
6) I Am Not Safer Than A Bank
7) Jenni's Song
8) Metal Airplanes
9) Flashdance II | {"url":"https://www.nearfantastica.com/bored/topic/13316-2009-matthew-good-band-elimination/","timestamp":"2024-11-05T22:37:56Z","content_type":"text/html","content_length":"240479","record_id":"<urn:uuid:2bda5a68-3678-4a62-b4a8-e608555dd007>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00706.warc.gz"} |
The Mapmakers: An Essay in Four Parts
The Mapmakers: An Essay in Four Parts
Mapmaking: 16th Century
Mapmaking: 17th Century
Mapmaking: 18th Century
Mapmaking: Map Production
16th Century
Maps (of land surfaces) and charts (of sea coasts) are scaled down representations of the earth's surface. For this reason they are ideal documents to prove that a discovery has taken place and
provide the means for the exploration to be repeated by others.
Maps are made up of three measurable elements: location, direction and distance. On some maps symbols and elevation are added. Symbols give more meaning to locations while elevation adds altitudinal
distance. The precision of these elements and their exact placement on maps relative to each other, is what separates accurate maps from poor ones. Accuracy is dependent on:
• the precision of the instruments available to make observations,
• the observer's knowledge of the earth's shape and size, and its relationship to various celestial bodies,
• the number of precise observations that form the basis of the map,
• advances in the nature of mathematics used to make observations and render these into maps, and
• the skill and training of the observer.
By the 16th century there was a general agreement that position be recorded by latitude and longitude. Due to the unvarying relationship between the earth's axis and the sun and stars, latitude (the
angle between a place, the centre of the earth and the equator) could be easily calculated. This was done either by measuring the height of the sun at noon above the horizon and correcting that
observation for the day of the year (sun's declination); or, by measuring the height of the North Star (Polaris) and compensating slightly for the difference between the position of Polaris and the
geographic pole, since the two do not exactly coincide. To do these tasks two instruments could be used; the astrolabe, mainly used for measurements on land, and the cross-staff (also called Jacob's
Staff) for observations at sea.
Sixteenth century measurements of latitude such as Jacques Cartier's were accurate to about one-quarter to one-half of a degree (one degree latitude equalling about 111 km). Longitude, the angle
between a place, the earth's axis and a prime meridian (today the prime meridian is the longitude of Greenwich, England), was impossible to calculate accurately until John Harrison invented the
marine chronometer, (a large pocket watch set on Greenwich mean time) in 1773. Since the ancient Greeks, geographers had known that longitude could best be determined by calculating the difference in
solar time between two places. Since the earth is 360° in circumference and rotates on its axis every 24 hours, one hour of time equals 15 degrees longitude. One degree therefore, equals four minutes
of time and about 111 km at the equator. Since time-pieces were not generally available until late in the 18th century, longitude had to be obtained by estimating east-west distances from a place of
departure to a destination. On land, distances were estimated by travel time -- for example the distance an average man could walk in an hour (one league or about five kilometres). The French called
this the "lieu d'une heure de chemin." Similarly, at sea, the estimated speed of a ship was converted into distance. This was called "dead reckoning." A navigator kept very careful note of all his
speeds, course changes, encounters with currents, etc. in a log book. At the end of the day, he would convert all his observations into distances and plot them on his chart according to his compass
By the 16th century, the mariner's compass was in general use. It was divided into 32 "points" or "winds", rather than degrees. Each point was equal to 11°15'. Compasses were not accurate enough to
sail by degrees. Since a compass points to the magnetic pole and maps are on the geographic pole (true north) compasses had to be corrected for this difference (magnetic declination). In the 16th
century, few mariners knew how to do that, or considered it to be unimportant. Nor did many know that magnetic declination varied across the earth's surface and that it changed over time (variation).
A result of all this confusion was that compass bearings on 16th century maps tended not to be very accurate. Most were in fact magnetic bearings, giving these maps a peculiar orientation to modern
Due to the twin problems of measuring direction and distance over the open sea, most 16th century navigators preferred to minimize guesswork through "parallel" (or "latitude") sailing. A captain
would sail along the coast of Europe until he reached the latitude of the place he wanted to go to. He would then depart the European coast and use the one instrument he trusted, his cross-staff, to
stay on that latitude until he got to the other side. On this journey he would then have to estimate his distance along a relatively straight course. This distance would then become the distance
between Europe and his destination on his map along the one line of latitude he had sailed.
By means of a table calculated by mathematicians for every line of latitude (parallels), the navigator could now mark off his lines of longitude (meridians). The more often he travelled over a route,
the better his observations got. Once he reached his destination, he would sail within sight of the coast, taking compass bearings of the coastline and of prominent features, estimating distances
and, weather permitting, calculating the latitudes of places. Bays, river mouths, hills, etc. were sketched on the chart as the ship sailed past them. These rough reconnaissance surveys formed the
bases of most 16th century maps. Another method for calculating distance sailed was the rule 'to raise or lay a degree of latitude'. This was an early form of 'plane sailing' (using right-angled
triangles) wherein a navigator would lay out a course with his compass. When he had crossed one degree of latitude by observation with his cross-staff (the adjacent side of his triangle) he could
look up the distance he had sailed (hypotenuse of his triangle), and longitudinal distance traversed (side opposite his course angle), in a set of tables calculated by mathematicians. The invention
of trigonometry made these tables redundant. It was not until the early 17th century, motivated by the search for harbours and locations for settlement, that more accurate maps were produced with
better instruments. | {"url":"https://webarchiveweb.wayback.bac-lac.canada.ca/web/20070515213157/http://www.collectionscanada.ca/explorers/h24-230-e.html","timestamp":"2024-11-06T10:47:37Z","content_type":"text/html","content_length":"41755","record_id":"<urn:uuid:ef3efc47-64ea-4710-b900-88aa6af762cd>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00402.warc.gz"} |
Find the \(k\) shortest paths between two vertices — k_shortest_paths
Find the \(k\) shortest paths between two vertices
Finds the \(k\) shortest paths between the given source and target vertex in order of increasing length. Currently this function uses Yen's algorithm.
weights = NULL,
mode = c("out", "in", "all", "total")
The input graph.
The source vertex of the shortest paths.
The target vertex of the shortest paths.
These dots are for future extensions and must be empty.
The number of paths to find. They will be returned in order of increasing length.
Possibly a numeric vector giving edge weights. If this is NULL and the graph has a weight edge attribute, then the attribute is used. If this is NA then no weights are used (even if the graph has
a weight attribute). In a weighted graph, the length of a path is the sum of the weights of its constituent edges.
Character constant, gives whether the shortest paths to or from the given vertices should be calculated for directed graphs. If out then the shortest paths from the vertex, if in then to it will
be considered. If all, the default, then the graph is treated as undirected, i.e. edge directions are not taken into account. This argument is ignored for undirected graphs.
A named list with two components is returned:
The list of \(k\) shortest paths in terms of vertices
The list of \(k\) shortest paths in terms of edges
Yen, Jin Y.: An algorithm for finding shortest routes from all source nodes to a given destination in general networks. Quarterly of Applied Mathematics. 27 (4): 526–530. (1970) doi:10.1090/qam/
See also
shortest_paths(), all_shortest_paths()
Other structural.properties: bfs(), component_distribution(), connect(), constraint(), coreness(), degree(), dfs(), distance_table(), edge_density(), feedback_arc_set(), girth(), is_acyclic(), is_dag
(), is_matching(), knn(), reciprocity(), subcomponent(), subgraph(), topo_sort(), transitivity(), unfold_tree(), which_multiple(), which_mutual()
Related documentation in the C library | {"url":"https://r.igraph.org/reference/k_shortest_paths.html","timestamp":"2024-11-01T23:52:48Z","content_type":"text/html","content_length":"13005","record_id":"<urn:uuid:6ba2bf5f-4e09-4e75-8231-504ea6097ed5>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00352.warc.gz"} |
Re: matrix multiplication inIML within a SAS macro
SAS uses the "*" for comments.
IML uses the "*" for matrix multiplication.
How can I use matrix multiplication in SAS/IML? It makes the text after the "*" purple indicating a comment???
-- Note: I am using Proc IML in a macro. Shouldn't matter though as the "*" is for comments in macros, just as in base SAS.
Please also email me directly your answers!! (steveseoul@yahoo.com)
07-28-2010 12:41 PM | {"url":"https://communities.sas.com/t5/SAS-IML-Software-and-Matrix/matrix-multiplication-inIML-within-a-SAS-macro/m-p/56682/highlight/true","timestamp":"2024-11-11T14:51:29Z","content_type":"text/html","content_length":"165264","record_id":"<urn:uuid:61153cc7-2f66-4cb3-b799-eea9688fba4a>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00537.warc.gz"} |
Millennium Prize: The Poincaré Conjecture - International Maths Challenge
Millennium Prize: The Poincaré Conjecture
The problem’s been solved … but the sweet treats were declined. Back to the Cutting Board
In 1904, French mathematician Henri Poincaré asked a key question about three-dimensional spaces (“manifolds”).
Imagine a piece of rope, so that firstly a knot is tied in the rope and then the ends are glued together. This is what mathematicians call a knot. A link is a collection of knots that are tangled
It has been observed that DNA, which is coiled up within cells, occurs in closed knotted form.
Complex molecules such as polymers are tangled in knotted forms. There are deep connections between knot theory and ideas in mathematical physics. The outsides of a knot or link in space give
important examples of three-dimensional spaces.
Torus. Fropuff
Back to Poincaré and his conjecture. He asked if the 3-sphere (which can be formed by either adding a point at infinity to ordinary three-dimensional Euclidean space or by gluing two solid
three-dimensional balls together along their boundary 2-spheres) was the only three-dimensional space in which every loop can be continuously shrunk to a point.
Poincaré had introduced important ideas in the structure and classification of surfaces and their higher dimensional analogues (“manifolds”), arising from his work on dynamical systems.
Donuts to go, please
A good way to visualise Poincaré’s conjecture is to examine the boundary of a ball (a two-dimensional sphere) and the boundary of a donut (called a torus). Any loop of string on a 2-sphere can be
shrunk to a point while keeping it on the sphere, whereas if a loop goes around the hole in the donut, it cannot be shrunk without leaving the surface of the donut.
Many attempts were made on the Poincaré conjecture, until in 2003 a wonderful solution was announced by a young Russian mathematician, Grigori “Grisha” Perelman.
This is a brief account of the ideas used by Perelman, which built on work of two other outstanding mathematicians, Bill Thurston and Richard Hamilton.
3D spaces
Thurston made enormous strides in our understanding of three-dimensional spaces in the late 1970s. In particular, he realised that essentially all the work that had been done since Poincaré fitted
into a single theme.
He observed that known three-dimensional spaces could be divided into pieces in a natural way, so that each piece had a uniform geometry, similar to the flat plane and the round sphere. (To see this
geometry on a torus, one must embed it into four-dimensional space!).
Thurston made a bold “geometrisation conjecture” that this should be true for all three-dimensional spaces. He had many brilliant students who further developed his theories, not least by producing
powerful computer programs that could test any given space to try to find its geometric structure.
Thurston made spectacular progress on the geometrisation conjecture, which includes the Poincaré conjecture as a special case. The geometrisation conjecture predicts that any three-dimensional space
in which every loop shrinks to a point should have a round metric – it would be a 3-sphere and Poincaré’s conjecture would follow.
In 1982, Richard Hamilton published a beautiful paper introducing a new technique in geometric analysis which he called Ricci flow. Hamilton had been looking for analogues of a flow of functions, so
that the energy of the function decreases until it reaches a minimum. This type of flow is closely related to the way heat spreads in a material.
Hamilton reasoned that there should be a similar flow for the geometric shape of a space, rather than a function between spaces. He used the Ricci tensor, a key feature of Einstein’s field equations
for general relativity, as the driving force for his flow.
He showed that, for three-dimensional spaces where the Ricci curvature is positive, the flow gradually changes the shape until the metric satisfies Thurston’s geometrisation conjecture.
Hamilton attracted many outstanding young mathematicians to work in this area. Ricci flow and other similar flows have become a huge area of research with applications in areas such as moving
interfaces, fluid mechanics and computer graphics.
Ricci flow. CBN
He outlined a marvellous program to use Ricci flow to attack Thurston’s geometrisation conjecture. The idea was to keep evolving the shape of a space under Ricci flow.
Hamilton and his collaborators found the space might form a singularity, where a narrow neck became thinner and thinner until the space splits into two smaller spaces.
Hamilton worked hard to try to fully understand this phenomenon and to allow the pieces to keep evolving under Ricci flow until the geometric structure predicted by Thurston could be found.
This is when Perelman burst on to the scene. He had produced some brilliant results at a very young age and was a researcher at the famous Steklov Institute in St Petersburg. Perelman got a Miller
fellowship to visit UC Berkeley for three years in the early 1990s.
I met him there around 1992. He then “disappeared” from the mathematical scene for nearly ten years and re-emerged to announce that he had completed Hamilton’s Ricci flow program, in a series of
papers he posted on the electronic repository called ArXiv.
His papers created enormous excitement and within several months a number of groups had started to work through Perelman’s strategy.
Eventually everyone was convinced that Perelman had indeed succeeded and both the geometrisation and Poincaré conjecture had been solved.
Perelman was awarded both a Fields medal (the mathematical equivalent of a Nobel prize) and also offered a million dollars for solving one of the Millenium prizes from the Clay Institute.
He turned down both these awards, preferring to live a quiet life in St Petersburg. Mathematicians are still finding new ways to use the solution to the geometrisation conjecture, which is one of the
outstanding mathematical results of this era.
For more such insights, log into www.international-maths-challenge.com.
*Credit for article given to Hyam Rubinstein* | {"url":"https://international-maths-challenge.com/millennium-prize-the-poincare-conjecture/","timestamp":"2024-11-09T09:37:05Z","content_type":"text/html","content_length":"150713","record_id":"<urn:uuid:2f517825-0e7a-4b6f-aa9a-f44f31e9e616>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00303.warc.gz"} |
History of logic
Jump to navigation Jump to search
The history of logic deals with the study of the development of the science of valid inference (logic). Formal logics developed in ancient times in India, China, and Greece. Greek methods,
particularly Aristotelian logic (or term logic) as found in the Organon, found wide application and acceptance in Western science and mathematics for millennia.^[1] The Stoics, especially Chrysippus,
began the development of predicate logic.
Christian and Islamic philosophers such as Boethius (died 524), Ibn Sina (Avicenna, died 1037) and William of Ockham (died 1347) further developed Aristotle's logic in the Middle Ages, reaching a
high point in the mid-fourteenth century, with Jean Buridan. The period between the fourteenth century and the beginning of the nineteenth century saw largely decline and neglect, and at least one
historian of logic regards this time as barren.^[2] Empirical methods ruled the day, as evidenced by Sir Francis Bacon's Novum Organon of 1620.
Logic revived in the mid-nineteenth century, at the beginning of a revolutionary period when the subject developed into a rigorous and formal discipline which took as its exemplar the exact method of
proof used in mathematics, a hearkening back to the Greek tradition.^[3] The development of the modern "symbolic" or "mathematical" logic during this period by the likes of Boole, Frege, Russell, and
Peano is the most significant in the two-thousand-year history of logic, and is arguably one of the most important and remarkable events in human intellectual history.^[4]
Progress in mathematical logic in the first few decades of the twentieth century, particularly arising from the work of Gödel and Tarski, had a significant impact on analytic philosophy and
philosophical logic, particularly from the 1950s onwards, in subjects such as modal logic, temporal logic, deontic logic, and relevance logic.
Logic in the East[edit]
Logic in India[edit]
Logic began independently in ancient India and continued to develop to early modern times without any known influence from Greek logic.^[5] Medhatithi Gautama (c. 6th century BC) founded the
anviksiki school of logic.^[6] The Mahabharata (12.173.45), around the 5th century BC, refers to the anviksiki and tarka schools of logic. Pāṇini (c. 5th century BC) developed a form of logic (to
which Boolean logic has some similarities) for his formulation of Sanskrit grammar. Logic is described by Chanakya (c. 350-283 BC) in his Arthashastra as an independent field of inquiry.^[7]
Two of the six Indian schools of thought deal with logic: Nyaya and Vaisheshika. The Nyaya Sutras of Aksapada Gautama (c. 2nd century AD) constitute the core texts of the Nyaya school, one of the six
orthodox schools of Hindu philosophy. This realist school developed a rigid five-member schema of inference involving an initial premise, a reason, an example, an application, and a conclusion.^[8]
The idealist Buddhist philosophy became the chief opponent to the Naiyayikas. Nagarjuna (c. 150-250 AD), the founder of the Madhyamika ("Middle Way") developed an analysis known as the catuṣkoṭi
(Sanskrit), a "four-cornered" system of argumentation that involves the systematic examination and rejection of each of the 4 possibilities of a proposition, P:
1. P; that is, being.
2. not P; that is, not being.
3. P and not P; that is, being and not being.
4. not (P or not P); that is, neither being nor not being.Under propositional logic, De Morgan's laws imply that this is equivalent to the third case (P and not P), and is therefore superfluous;
there are actually only 3 cases to consider.
However, Dignaga (c 480-540 AD) is sometimes said to have developed a formal syllogism,^[9] and it was through him and his successor, Dharmakirti, that Buddhist logic reached its height; it is
contested whether their analysis actually constitutes a formal syllogistic system. In particular, their analysis centered on the definition of an inference-warranting relation, "vyapti", also known
as invariable concomitance or pervasion.^[10] To this end, a doctrine known as "apoha" or differentiation was developed.^[11] This involved what might be called inclusion and exclusion of defining
Dignāga's famous "wheel of reason" (Hetucakra) is a method of indicating when one thing (such as smoke) can be taken as an invariable sign of another thing (like fire), but the inference is often
inductive and based on past observation. Matilal remarks that Dignāga's analysis is much like John Stuart Mill's Joint Method of Agreement and Difference, which is inductive.^[12]
In addition, the traditional five-member Indian syllogism, though deductively valid, has repetitions that are unnecessary to its logical validity. As a result, some commentators see the traditional
Indian syllogism as a rhetorical form that is entirely natural in many cultures of the world, and yet not as a logical form—not in the sense that all logically unnecessary elements have been omitted
for the sake of analysis.
Logic in China[edit]
In China, a contemporary of Confucius, Mozi, "Master Mo", is credited with founding the Mohist school, whose canons dealt with issues relating to valid inference and the conditions of correct
conclusions. In particular, one of the schools that grew out of Mohism, the Logicians, are credited by some scholars for their early investigation of formal logic. Due to the harsh rule of Legalism
in the subsequent Qin Dynasty, this line of investigation disappeared in China until the introduction of Indian philosophy by Buddhists.
Logic in the West[edit]
Prehistory of logic[edit]
Valid reasoning has been employed in all periods of human history. However, logic studies the principles of valid reasoning, inference and demonstration. It is probable that the idea of demonstrating
a conclusion first arose in connection with geometry, which originally meant the same as "land measurement".^[13] The ancient Egyptians discovered geometry, including the formula for the volume of a
truncated pyramid.^[14] Ancient Babylon was also skilled in mathematics. Esagil-kin-apli's medical Diagnostic Handbook in the 11th century BC was based on a logical set of axioms and assumptions,^
[15] while Babylonian astronomers in the 8th and 7th centuries BC employed an internal logic within their predictive planetary systems, an important contribution to the philosophy of science.^[16]
Ancient Greece before Aristotle[edit]
While the ancient Egyptians empirically discovered some truths of geometry, the great achievement of the ancient Greeks was to replace empirical methods by demonstrative proof. Both Thales and
Pythagoras of the Pre-Socratic philosophers seem aware of geometry's methods.
Fragments of early proofs are preserved in the works of Plato and Aristotle,^[17] and the idea of a deductive system was probably known in the Pythagorean school and the Platonic Academy.^[14] The
proofs of Euclid of Alexandria are a paradigm of Greek geometry. The three basic principles of geometry are as follows:
• Certain propositions must be accepted as true without demonstration; such a proposition is known as an axiom of geometry.
• Every proposition that is not an axiom of geometry must be demonstrated as following from the axioms of geometry; such a demonstration is known as a proof or a "derivation" of the proposition.
• The proof must be formal; that is, the derivation of the proposition must be independent of the particular subject matter in question.^[14]
Further evidence that early Greek thinkers were concerned with the principles of reasoning is found in the fragment called dissoi logoi, probably written at the beginning of the fourth century BC.
This is part of a protracted debate about truth and falsity.^[18] In the case of the classical Greek city-states, interest in argumentation was also stimulated by the activities of the Rhetoricians
or Orators and the Sophists, who used arguments to defend or attack a thesis, both in legal and political contexts.^[19]
It is said Thales, most widely regarded as the first philosopher in the Greek tradition,^[20]^[21] measured the height of the pyramids by their shadows at the moment when his own shadow was equal to
his height. Thales was said to have had a sacrifice in celebration of discovering Thales' theorem just as Pythagoras had the Pythagorean theorem.^[22]
Thales is the first known individual to use deductive reasoning applied to geometry, by deriving four corollaries to his theorem, and the first known individual to whom a mathematical discovery has
been attributed.^[23] Indian and Babylonian mathematicians knew his theorem for special cases before he proved it.^[24] It is believed that Thales learned that an angle inscribed in a semicircle is a
right angle during his travels to Babylon.^[25]
Before 520 BC, on one of his visits to Egypt or Greece, Pythagoras might have met the c. 54 years older Thales.^[26] The systematic study of proof seems to have begun with the school of Pythagoras
(i. e. the Pythagoreans) in the late sixth century BC.^[14] Indeed, the Pythagoreans, believing all was number, are the first philosophers to emphasize form rather than matter.^[27]
Heraclitus and Parmenides[edit]
The writing of Heraclitus (c. 535 – c. 475 BC) was the first place where the word logos was given special attention in ancient Greek philosophy,^[28] Heraclitus held that everything changes and all
was fire and conflicting opposites, seemingly unified only by this Logos. He is known for his obscure sayings.
This logos holds always but humans always prove unable to understand it, both before hearing it and when they have first heard it. For though all things come to be in accordance with this logos,
humans are like the inexperienced when they experience such words and deeds as I set out, distinguishing each in accordance with its nature and saying how it is. But other people fail to notice
what they do when awake, just as they forget what they do while asleep.
In contrast to Heraclitus, Parmenides held that all is one and nothing changes. He may have been a dissident Pythagorean, disagreeing that One (a number) produced the many.^[29] "X is not" must
always be false or meaningless. What exists can in no way not exist. Our sense perceptions with its noticing of generation and destruction are in grievous error. Instead of sense perception,
Parmenides advocated logos as the means to Truth. He has been called the discoverer of logic,^[30]^[31]
For this view, that That Which Is Not exists, can never predominate. You must debar your thought from this way of search, nor let ordinary experience in its variety force you along this way,
(namely, that of allowing) the eye, sightless as it is, and the ear, full of sound, and the tongue, to rule; but (you must) judge by means of the Reason (Logos) the much-contested proof which is
expounded by me. (B 7.1–8.2)
Zeno of Elea, a pupil of Parmenides, had the idea of a standard argument pattern found in the method of proof known as reductio ad absurdum. This is the technique of drawing an obviously false (that
is, "absurd") conclusion from an assumption, thus demonstrating that the assumption is false.^[32] Therefore, Zeno and his teacher are seen as the first to apply the art of logic.^[33] Plato's
dialogue Parmenides portrays Zeno as claiming to have written a book defending the monism of Parmenides by demonstrating the absurd consequence of assuming that there is plurality. Zeno famously used
this method to develop his paradoxes in his arguments against motion. Such dialectic reasoning later became popular. The members of this school were called "dialecticians" (from a Greek word meaning
"to discuss").
Let no one ignorant of geometry enter here.
—Inscribed over the entrance to Plato's Academy.
None of the surviving works of the great fourth-century philosopher Plato (428–347 BC) include any formal logic,^[34] but they include important contributions to the field of philosophical logic.
Plato raises three questions:
• What is it that can properly be called true or false?
• What is the nature of the connection between the assumptions of a valid argument and its conclusion?
• What is the nature of definition?
The first question arises in the dialogue Theaetetus, where Plato identifies thought or opinion with talk or discourse (logos).^[35] The second question is a result of Plato's theory of Forms. Forms
are not things in the ordinary sense, nor strictly ideas in the mind, but they correspond to what philosophers later called universals, namely an abstract entity common to each set of things that
have the same name. In both the Republic and the Sophist, Plato suggests that the necessary connection between the assumptions of a valid argument and its conclusion corresponds to a necessary
connection between "forms".^[36] The third question is about definition. Many of Plato's dialogues concern the search for a definition of some important concept (justice, truth, the Good), and it is
likely that Plato was impressed by the importance of definition in mathematics.^[37] What underlies every definition is a Platonic Form, the common nature present in different particular things.
Thus, a definition reflects the ultimate object of understanding, and is the foundation of all valid inference. This had a great influence on Plato's student Aristotle, in particular Aristotle's
notion of the essence of a thing.^[38]
The logic of Aristotle, and particularly his theory of the syllogism, has had an enormous influence in Western thought.^[39] Aristotle was the first logician to attempt a systematic analysis of
logical syntax, of noun (or term), and of verb. He was the first formal logician, in that he demonstrated the principles of reasoning by employing variables to show the underlying logical form of an
argument. He sought relations of dependence which characterize necessary inference, and distinguished the validity of these relations, from the truth of the premises. He was the first to deal with
the principles of contradiction and excluded middle in a systematic way.^[40]
The Organon[edit]
His logical works, called the Organon, are the earliest formal study of logic that have come down to modern times. Though it is difficult to determine the dates, the probable order of writing of
Aristotle's logical works is:
• The Categories, a study of the ten kinds of primitive term.
• The Topics (with an appendix called On Sophistical Refutations), a discussion of dialectics.
• On Interpretation, an analysis of simple categorical propositions into simple terms, negation, and signs of quantity.
• The Prior Analytics, a formal analysis of what makes a syllogism (a valid argument, according to Aristotle).
• The Posterior Analytics, a study of scientific demonstration, containing Aristotle's mature views on logic.
These works are of outstanding importance in the history of logic. In the Categories, he attempts to discern all the possible things to which a term can refer; this idea underpins his philosophical
work Metaphysics, which itself had a profound influence on Western thought.
He also developed a theory of non-formal logic (i.e., the theory of fallacies), which is presented in Topics and Sophistical Refutations.^[40]
On Interpretation contains a comprehensive treatment of the notions of opposition and conversion; chapter 7 is at the origin of the square of opposition (or logical square); chapter 9 contains the
beginning of modal logic.
The Prior Analytics contains his exposition of the "syllogism", where three important principles are applied for the first time in history: the use of variables, a purely formal treatment, and the
use of an axiomatic system.
The other great school of Greek logic is that of the Stoics.^[41] Stoic logic traces its roots back to the late 5th century BC philosopher Euclid of Megara, a pupil of Socrates and slightly older
contemporary of Plato, probably following in the tradition of Parmenides and Zeno. His pupils and successors were called "Megarians", or "Eristics", and later the "Dialecticians". The two most
important dialecticians of the Megarian school were Diodorus Cronus and Philo, who were active in the late 4th century BC.
The Stoics adopted the Megarian logic and systemized it. The most important member of the school was Chrysippus (c. 278–c. 206 BC), who was its third head, and who formalized much of Stoic doctrine.
He is supposed to have written over 700 works, including at least 300 on logic, almost none of which survive.^[42]^[43] Unlike with Aristotle, we have no complete works by the Megarians or the early
Stoics, and have to rely mostly on accounts (sometimes hostile) by later sources, including prominently Diogenes Laërtius, Sextus Empiricus, Galen, Aulus Gellius, Alexander of Aphrodisias, and Cicero
Three significant contributions of the Stoic school were (i) their account of modality, (ii) their theory of the Material conditional, and (iii) their account of meaning and truth.^[45]
• Modality. According to Aristotle, the Megarians of his day claimed there was no distinction between potentiality and actuality.^[46] Diodorus Cronus defined the possible as that which either is
or will be, the impossible as what will not be true, and the contingent as that which either is already, or will be false.^[47] Diodorus is also famous for what is known as his Master argument,
which states that each pair of the following 3 propositions contradicts the third proposition:
□ Everything that is past is true and necessary.
□ The impossible does not follow from the possible.
□ What neither is nor will be is possible.
Diodorus used the plausibility of the first two to prove that nothing is possible if it neither is nor will be true.^[48] Chrysippus, by contrast, denied the second premise and said that the
impossible could follow from the possible.^[49]
• Conditional statements. The first logicians to debate conditional statements were Diodorus and his pupil Philo of Megara. Sextus Empiricus refers three times to a debate between Diodorus and
Philo. Philo regarded a conditional as true unless it has both a true antecedent and a false consequent. Precisely, let T[0] and T[1] be true statements, and let F[0] and F[1] be false
statements; then, according to Philo, each of the following conditionals is a true statement, because it is not the case that the consequent is false while the antecedent is true (it is not the
case that a false statement is asserted to follow from a true statement):
□ If T[0], then T[1]
□ If F[0], then T[0]
□ If F[0], then F[1]
The following conditional does not meet this requirement, and is therefore a false statement according to Philo:
Indeed, Sextus says "According to [Philo], there are three ways in which a conditional may be true, and one in which it may be false."^[50] Philo's criterion of truth is what would now be called
a truth-functional definition of "if ... then"; it is the definition used in modern logic.
In contrast, Diodorus allowed the validity of conditionals only when the antecedent clause could never lead to an untrue conclusion.^[50]^[51]^[52] A century later, the Stoic philosopher
Chrysippus attacked the assumptions of both Philo and Diodorus.
• Meaning and truth. The most important and striking difference between Megarian-Stoic logic and Aristotelian logic is that Megarian-Stoic logic concerns propositions, not terms, and is thus closer
to modern propositional logic.^[53] The Stoics distinguished between utterance (phone), which may be noise, speech (lexis), which is articulate but which may be meaningless, and discourse (logos
), which is meaningful utterance. The most original part of their theory is the idea that what is expressed by a sentence, called a lekton, is something real; this corresponds to what is now
called a proposition. Sextus says that according to the Stoics, three things are linked together: that which signifies, that which is signified, and the object; for example, that which signifies
is the word Dion, and that which is signified is what Greeks understand but barbarians do not, and the object is Dion himself.^[54]
Medieval logic[edit]
Logic in the Middle East[edit]
The works of Al-Kindi, Al-Farabi, Avicenna, Al-Ghazali, Averroes and other Muslim logicians were based on Aristotelian logic and were important in communicating the ideas of the ancient world to the
medieval West.^[55] Al-Farabi (Alfarabi) (873–950) was an Aristotelian logician who discussed the topics of future contingents, the number and relation of the categories, the relation between logic
and grammar, and non-Aristotelian forms of inference.^[56] Al-Farabi also considered the theories of conditional syllogisms and analogical inference, which were part of the Stoic tradition of logic
rather than the Aristotelian.^[57]
Ibn Sina (Avicenna) (980–1037) was the founder of Avicennian logic, which replaced Aristotelian logic as the dominant system of logic in the Islamic world,^[58] and also had an important influence on
Western medieval writers such as Albertus Magnus.^[59] Avicenna wrote on the hypothetical syllogism^[60] and on the propositional calculus, which were both part of the Stoic logical tradition.^[61]
He developed an original "temporally modalized" syllogistic theory, involving temporal logic and modal logic.^[56] He also made use of inductive logic, such as the methods of agreement, difference,
and concomitant variation which are critical to the scientific method.^[60] One of Avicenna's ideas had a particularly important influence on Western logicians such as William of Ockham: Avicenna's
word for a meaning or notion (ma'na), was translated by the scholastic logicians as the Latin intentio; in medieval logic and epistemology, this is a sign in the mind that naturally represents a
thing.^[62] This was crucial to the development of Ockham's conceptualism: A universal term (e.g., "man") does not signify a thing existing in reality, but rather a sign in the mind (intentio in
intellectu) which represents many things in reality; Ockham cites Avicenna's commentary on Metaphysics V in support of this view.^[63]
Fakhr al-Din al-Razi (b. 1149) criticised Aristotle's "first figure" and formulated an early system of inductive logic, foreshadowing the system of inductive logic developed by John Stuart Mill
(1806–1873).^[64] Al-Razi's work was seen by later Islamic scholars as marking a new direction for Islamic logic, towards a Post-Avicennian logic. This was further elaborated by his student
Afdaladdîn al-Khûnajî (d. 1249), who developed a form of logic revolving around the subject matter of conceptions and assents. In response to this tradition, Nasir al-Din al-Tusi (1201–1274) began a
tradition of Neo-Avicennian logic which remained faithful to Avicenna's work and existed as an alternative to the more dominant Post-Avicennian school over the following centuries.^[65]
The Illuminationist school was founded by Shahab al-Din Suhrawardi (1155–1191), who developed the idea of "decisive necessity", which refers to the reduction of all modalities (necessity, possibility
, contingency and impossibility) to the single mode of necessity.^[66] Ibn al-Nafis (1213–1288) wrote a book on Avicennian logic, which was a commentary of Avicenna's Al-Isharat (The Signs) and
Al-Hidayah (The Guidance).^[67] Ibn Taymiyyah (1263–1328), wrote the Ar-Radd 'ala al-Mantiqiyyin, where he argued against the usefulness, though not the validity, of the syllogism^[68] and in favour
of inductive reasoning.^[64] Ibn Taymiyyah also argued against the certainty of syllogistic arguments and in favour of analogy; his argument is that concepts founded on induction are themselves not
certain but only probable, and thus a syllogism based on such concepts is no more certain than an argument based on analogy. He further claimed that induction itself is founded on a process of
analogy. His model of analogical reasoning was based on that of juridical arguments.^[69]^[70] This model of analogy has been used in the recent work of John F. Sowa.^[70]
The Sharh al-takmil fi'l-mantiq written by Muhammad ibn Fayd Allah ibn Muhammad Amin al-Sharwani in the 15th century is the last major Arabic work on logic that has been studied.^[71] However,
"thousands upon thousands of pages" on logic were written between the 14th and 19th centuries, though only a fraction of the texts written during this period have been studied by historians, hence
little is known about the original work on Islamic logic produced during this later period.^[65]
Logic in medieval Europe[edit]
"Medieval logic" (also known as "Scholastic logic") generally means the form of Aristotelian logic developed in medieval Europe throughout roughly the period 1200–1600.^[1] For centuries after Stoic
logic had been formulated, it was the dominant system of logic in the classical world. When the study of logic resumed after the Dark Ages, the main source was the work of the Christian philosopher
Boethius, who was familiar with some of Aristotle's logic, but almost none of the work of the Stoics.^[72] Until the twelfth century, the only works of Aristotle available in the West were the
Categories, On Interpretation, and Boethius's translation of the Isagoge of Porphyry (a commentary on the Categories). These works were known as the "Old Logic" (Logica Vetus or Ars Vetus). An
important work in this tradition was the Logica Ingredientibus of Peter Abelard (1079–1142). His direct influence was small,^[73] but his influence through pupils such as John of Salisbury was great,
and his method of applying rigorous logical analysis to theology shaped the way that theological criticism developed in the period that followed.^[74]
By the early thirteenth century, the remaining works of Aristotle's Organon (including the Prior Analytics, Posterior Analytics, and the Sophistical Refutations) had been recovered in the West.^[75]
Logical work until then was mostly paraphrasis or commentary on the work of Aristotle.^[76] The period from the middle of the thirteenth to the middle of the fourteenth century was one of significant
developments in logic, particularly in three areas which were original, with little foundation in the Aristotelian tradition that came before. These were:^[77]
• The theory of supposition. Supposition theory deals with the way that predicates (e.g., 'man') range over a domain of individuals (e.g., all men).^[78] In the proposition 'every man is an
animal', does the term 'man' range over or 'supposit for' men existing just in the present, or does the range include past and future men? Can a term supposit for a non-existing individual? Some
medievalists have argued that this idea is a precursor of modern first-order logic.^[79] "The theory of supposition with the associated theories of copulatio (sign-capacity of adjectival terms),
ampliatio (widening of referential domain), and distributio constitute one of the most original achievements of Western medieval logic".^[80]
• The theory of syncategoremata. Syncategoremata are terms which are necessary for logic, but which, unlike categorematic terms, do not signify on their own behalf, but 'co-signify' with other
words. Examples of syncategoremata are 'and', 'not', 'every', 'if', and so on.
• The theory of consequences. A consequence is a hypothetical, conditional proposition: two propositions joined by the terms 'if ... then'. For example, 'if a man runs, then God exists' (Si homo
currit, Deus est).^[81] A fully developed theory of consequences is given in Book III of William of Ockham's work Summa Logicae. There, Ockham distinguishes between 'material' and 'formal'
consequences, which are roughly equivalent to the modern material implication and logical implication respectively. Similar accounts are given by Jean Buridan and Albert of Saxony.
The last great works in this tradition are the Logic of John Poinsot (1589–1644, known as John of St Thomas), the Metaphysical Disputations of Francisco Suarez (1548–1617), and the Logica
Demonstrativa of Giovanni Girolamo Saccheri (1667–1733).
Traditional logic[edit]
The textbook tradition[edit]
Traditional logic generally means the textbook tradition that begins with Antoine Arnauld's and Pierre Nicole's Logic, or the Art of Thinking, better known as the Port-Royal Logic.^[82] Published in
1662, it was the most influential work on logic after Aristotle until the nineteenth century.^[83] The book presents a loosely Cartesian doctrine (that the proposition is a combining of ideas rather
than terms, for example) within a framework that is broadly derived from Aristotelian and medieval term logic. Between 1664 and 1700, there were eight editions, and the book had considerable
influence after that.^[83] The Port-Royal introduces the concepts of extension and intension. The account of propositions that Locke gives in the Essay is essentially that of the Port-Royal: "Verbal
propositions, which are words, [are] the signs of our ideas, put together or separated in affirmative or negative sentences. So that proposition consists in the putting together or separating these
signs, according as the things which they stand for agree or disagree."^[84]
Dudley Fenner helped popularize Ramist logic, a reaction against Aristotle. Another influential work was the Novum Organum by Francis Bacon, published in 1620. The title translates as "new
instrument". This is a reference to Aristotle's work known as the Organon. In this work, Bacon rejects the syllogistic method of Aristotle in favor of an alternative procedure "which by slow and
faithful toil gathers information from things and brings it into understanding".^[85] This method is known as inductive reasoning, a method which starts from empirical observation and proceeds to
lower axioms or propositions; from these lower axioms, more general ones can be induced. For example, in finding the cause of a phenomenal nature such as heat, 3 lists should be constructed:
• The presence list: a list of every situation where heat is found.
• The absence list: a list of every situation that is similar to at least one of those of the presence list, except for the lack of heat.
• The variability list: a list of every situation where heat can vary.
Then, the form nature (or cause) of heat may be defined as that which is common to every situation of the presence list, and which is lacking from every situation of the absence list, and which
varies by degree in every situation of the variability list.
Other works in the textbook tradition include Isaac Watts's Logick: Or, the Right Use of Reason (1725), Richard Whately's Logic (1826), and John Stuart Mill's A System of Logic (1843). Although the
latter was one of the last great works in the tradition, Mill's view that the foundations of logic lie in introspection^[86] influenced the view that logic is best understood as a branch of
psychology, a view which dominated the next fifty years of its development, especially in Germany.^[87]
Logic in Hegel's philosophy[edit]
G.W.F. Hegel indicated the importance of logic to his philosophical system when he condensed his extensive Science of Logic into a shorter work published in 1817 as the first volume of his
Encyclopaedia of the Philosophical Sciences. The "Shorter" or "Encyclopaedia" Logic, as it is often known, lays out a series of transitions which leads from the most empty and abstract of
categories—Hegel begins with "Pure Being" and "Pure Nothing"—to the "Absolute, the category which contains and resolves all the categories which preceded it. Despite the title, Hegel's Logic is not
really a contribution to the science of valid inference. Rather than deriving conclusions about concepts through valid inference from premises, Hegel seeks to show that thinking about one concept
compels thinking about another concept (one cannot, he argues, possess the concept of "Quality" without the concept of "Quantity"); this compulsion is, supposedly, not a matter of individual
psychology, because it arises almost organically from the content of the concepts themselves. His purpose is to show the rational structure of the "Absolute"—indeed of rationality itself. The method
by which thought is driven from one concept to its contrary, and then to further concepts, is known as the Hegelian dialectic.
Although Hegel's Logic has had little impact on mainstream logical studies, its influence can be seen elsewhere:
• Carl von Prantl's Geschichte der Logik in Abendland (1855–1867).^[88]
• The work of the British Idealists, such as F.H. Bradley's Principles of Logic (1883).
• The economic, political, and philosophical studies of Karl Marx, and in the various schools of Marxism.
Logic and psychology[edit]
Between the work of Mill and Frege stretched half a century during which logic was widely treated as a descriptive science, an empirical study of the structure of reasoning, and thus essentially as a
branch of psychology.^[89] The German psychologist Wilhelm Wundt, for example, discussed deriving "the logical from the psychological laws of thought", emphasizing that "psychological thinking is
always the more comprehensive form of thinking."^[90] This view was widespread among German philosophers of the period:
• Theodor Lipps described logic as "a specific discipline of psychology".^[91]
• Christoph von Sigwart understood logical necessity as grounded in the individual's compulsion to think in a certain way.^[92]
• Benno Erdmann argued that "logical laws only hold within the limits of our thinking".^[93]
Such was the dominant view of logic in the years following Mill's work.^[94] This psychological approach to logic was rejected by Gottlob Frege. It was also subjected to an extended and destructive
critique by Edmund Husserl in the first volume of his Logical Investigations (1900), an assault which has been described as "overwhelming".^[95] Husserl argued forcefully that grounding logic in
psychological observations implied that all logical truths remained unproven, and that skepticism and relativism were unavoidable consequences.
Such criticisms did not immediately extirpate what is called "psychologism". For example, the American philosopher Josiah Royce, while acknowledging the force of Husserl's critique, remained "unable
to doubt" that progress in psychology would be accompanied by progress in logic, and vice versa.^[96]
Rise of modern logic[edit]
The period between the fourteenth century and the beginning of the nineteenth century had been largely one of decline and neglect, and is generally regarded as barren by historians of logic.^[2] The
revival of logic occurred in the mid-nineteenth century, at the beginning of a revolutionary period where the subject developed into a rigorous and formalistic discipline whose exemplar was the exact
method of proof used in mathematics. The development of the modern "symbolic" or "mathematical" logic during this period is the most significant in the 2000-year history of logic, and is arguably one
of the most important and remarkable events in human intellectual history.^[4]
A number of features distinguish modern logic from the old Aristotelian or traditional logic, the most important of which are as follows:^[97] Modern logic is fundamentally a calculus whose rules of
operation are determined only by the shape and not by the meaning of the symbols it employs, as in mathematics. Many logicians were impressed by the "success" of mathematics, in that there had been
no prolonged dispute about any truly mathematical result. C.S. Peirce noted^[98] that even though a mistake in the evaluation of a definite integral by Laplace led to an error concerning the moon's
orbit that persisted for nearly 50 years, the mistake, once spotted, was corrected without any serious dispute. Peirce contrasted this with the disputation and uncertainty surrounding traditional
logic, and especially reasoning in metaphysics. He argued that a truly "exact" logic would depend upon mathematical, i.e., "diagrammatic" or "iconic" thought. "Those who follow such methods will ...
escape all error except such as will be speedily corrected after it is once suspected". Modern logic is also "constructive" rather than "abstractive"; i.e., rather than abstracting and formalising
theorems derived from ordinary language (or from psychological intuitions about validity), it constructs theorems by formal methods, then looks for an interpretation in ordinary language. It is
entirely symbolic, meaning that even the logical constants (which the medieval logicians called "syncategoremata") and the categoric terms are expressed in symbols.
Modern logic[edit]
The development of modern logic falls into roughly five periods:^[99]
• The embryonic period from Leibniz to 1847, when the notion of a logical calculus was discussed and developed, particularly by Leibniz, but no schools were formed, and isolated periodic attempts
were abandoned or went unnoticed.
• The algebraic period from Boole's Analysis to Schröder's Vorlesungen. In this period, there were more practitioners, and a greater continuity of development.
• The logicist period from the Begriffsschrift of Frege to the Principia Mathematica of Russell and Whitehead. The aim of the "logicist school" was to incorporate the logic of all mathematical and
scientific discourse in a single unified system which, taking as a fundamental principle that all mathematical truths are logical, did not accept any non-logical terminology. The major logicists
were Frege, Russell, and the early Wittgenstein.^[100] It culminates with the Principia, an important work which includes a thorough examination and attempted solution of the antinomies which had
been an obstacle to earlier progress.
• The metamathematical period from 1910 to the 1930s, which saw the development of metalogic, in the finitist system of Hilbert, and the non-finitist system of Löwenheim and Skolem, the combination
of logic and metalogic in the work of Gödel and Tarski. Gödel's incompleteness theorem of 1931 was one of the greatest achievements in the history of logic. Later in the 1930s, Gödel developed
the notion of set-theoretic constructibility.
• The period after World War II, when mathematical logic branched into four inter-related but separate areas of research: model theory, proof theory, computability theory, and set theory, and its
ideas and methods began to influence philosophy.
Embryonic period[edit]
The idea that inference could be represented by a purely mechanical process is found as early as Raymond Llull, who proposed a (somewhat eccentric) method of drawing conclusions by a system of
concentric rings. The work of logicians such as the Oxford Calculators^[101] led to a method of using letters instead of writing out logical calculations (calculationes) in words, a method used, for
instance, in the Logica magna by Paul of Venice. Three hundred years after Llull, the English philosopher and logician Thomas Hobbes suggested that all logic and reasoning could be reduced to the
mathematical operations of addition and subtraction.^[102] The same idea is found in the work of Leibniz, who had read both Llull and Hobbes, and who argued that logic can be represented through a
combinatorial process or calculus. But, like Llull and Hobbes, he failed to develop a detailed or comprehensive system, and his work on this topic was not published until long after his death.
Leibniz says that ordinary languages are subject to "countless ambiguities" and are unsuited for a calculus, whose task is to expose mistakes in inference arising from the forms and structures of
words;^[103] hence, he proposed to identify an alphabet of human thought comprising fundamental concepts which could be composed to express complex ideas,^[104] and create a calculus ratiocinator
that would make all arguments "as tangible as those of the Mathematicians, so that we can find our error at a glance, and when there are disputes among persons, we can simply say: Let us calculate."^
Gergonne (1816) said that reasoning does not have to be about objects about which one has perfectly clear ideas, because algebraic operations can be carried out without having any idea of the meaning
of the symbols involved.^[106] Bolzano anticipated a fundamental idea of modern proof theory when he defined logical consequence or "deducibility" in terms of variables:^[107]
Hence I say that propositions ${\displaystyle M}$, ${\displaystyle N}$, ${\displaystyle O}$,… are deducible from propositions ${\displaystyle A}$, ${\displaystyle B}$, ${\displaystyle C}$, ${\
displaystyle D}$,… with respect to variable parts ${\displaystyle i}$, ${\displaystyle j}$,…, if every class of ideas whose substitution for ${\displaystyle i}$, ${\displaystyle j}$,… makes all
of ${\displaystyle A}$, ${\displaystyle B}$, ${\displaystyle C}$, ${\displaystyle D}$,… true, also makes all of ${\displaystyle M}$, ${\displaystyle N}$, ${\displaystyle O}$,… true. Occasionally,
since it is customary, I shall say that propositions ${\displaystyle M}$, ${\displaystyle N}$, ${\displaystyle O}$,… follow, or can be inferred or derived, from ${\displaystyle A}$, ${\
displaystyle B}$, ${\displaystyle C}$, ${\displaystyle D}$,…. Propositions ${\displaystyle A}$, ${\displaystyle B}$, ${\displaystyle C}$, ${\displaystyle D}$,… I shall call the premises, ${\
displaystyle M}$, ${\displaystyle N}$, ${\displaystyle O}$,… the conclusions.
This is now known as semantic validity.
Algebraic period[edit]
Modern logic begins with what is known as the "algebraic school", originating with Boole and including Peirce, Jevons, Schröder, and Venn.^[108] Their objective was to develop a calculus to formalise
reasoning in the area of classes, propositions, and probabilities. The school begins with Boole's seminal work Mathematical Analysis of Logic which appeared in 1847, although De Morgan (1847) is its
immediate precursor.^[109] The fundamental idea of Boole's system is that algebraic formulae can be used to express logical relations. This idea occurred to Boole in his teenage years, working as an
usher in a private school in Lincoln, Lincolnshire.^[110] For example, let x and y stand for classes let the symbol = signify that the classes have the same members, xy stand for the class containing
all and only the members of x and y and so on. Boole calls these elective symbols, i.e. symbols which select certain objects for consideration.^[111] An expression in which elective symbols are used
is called an elective function, and an equation of which the members are elective functions, is an elective equation.^[112] The theory of elective functions and their "development" is essentially the
modern idea of truth-functions and their expression in disjunctive normal form.^[111]
Boole's system admits of two interpretations, in class logic, and propositional logic. Boole distinguished between "primary propositions" which are the subject of syllogistic theory, and "secondary
propositions", which are the subject of propositional logic, and showed how under different "interpretations" the same algebraic system could represent both. An example of a primary proposition is
"All inhabitants are either Europeans or Asiatics." An example of a secondary proposition is "Either all inhabitants are Europeans or they are all Asiatics."^[113] These are easily distinguished in
modern propositional calculus, where it is also possible to show that the first follows from the second, but it is a significant disadvantage that there is no way of representing this in the Boolean
In his Symbolic Logic (1881), John Venn used diagrams of overlapping areas to express Boolean relations between classes or truth-conditions of propositions. In 1869 Jevons realised that Boole's
methods could be mechanised, and constructed a "logical machine" which he showed to the Royal Society the following year.^[111] In 1885 Allan Marquand proposed an electrical version of the machine
that is still extant (picture at the Firestone Library).
The defects in Boole's system (such as the use of the letter v for existential propositions) were all remedied by his followers. Jevons published Pure Logic, or the Logic of Quality apart from
Quantity in 1864, where he suggested a symbol to signify exclusive or, which allowed Boole's system to be greatly simplified.^[115] This was usefully exploited by Schröder when he set out theorems in
parallel columns in his Vorlesungen (1890–1905). Peirce (1880) showed how all the Boolean elective functions could be expressed by the use of a single primitive binary operation, "neither ... nor ...
" and equally well "not both ... and ...",^[116] however, like many of Peirce's innovations, this remained unknown or unnoticed until Sheffer rediscovered it in 1913.^[117] Boole's early work also
lacks the idea of the logical sum which originates in Peirce (1867), Schröder (1877) and Jevons (1890),^[118] and the concept of inclusion, first suggested by Gergonne (1816) and clearly articulated
by Peirce (1870).
The success of Boole's algebraic system suggested that all logic must be capable of algebraic representation, and there were attempts to express a logic of relations in such form, of which the most
ambitious was Schröder's monumental Vorlesungen über die Algebra der Logik ("Lectures on the Algebra of Logic", vol iii 1895), although the original idea was again anticipated by Peirce.^[119]
Boole's unwavering acceptance of Aristotle's logic is emphasized by the historian of logic John Corcoran in an accessible introduction to Laws of Thought^[120] Corcoran also wrote a point-by-point
comparison of Prior Analytics and Laws of Thought.^[121] According to Corcoran, Boole fully accepted and endorsed Aristotle's logic. Boole's goals were "to go under, over, and beyond" Aristotle's
logic by 1) providing it with mathematical foundations involving equations, 2) extending the class of problems it could treat — from assessing validity to solving equations — and 3) expanding the
range of applications it could handle — e.g. from propositions having only two terms to those having arbitrarily many.
More specifically, Boole agreed with what Aristotle said; Boole's 'disagreements', if they might be called that, concern what Aristotle did not say. First, in the realm of foundations, Boole reduced
the four propositional forms of Aristotelian logic to formulas in the form of equations — by itself a revolutionary idea. Second, in the realm of logic's problems, Boole's addition of equation
solving to logic — another revolutionary idea — involved Boole's doctrine that Aristotle's rules of inference (the "perfect syllogisms") must be supplemented by rules for equation solving. Third, in
the realm of applications, Boole's system could handle multi-term propositions and arguments whereas Aristotle could handle only two-termed subject-predicate propositions and arguments. For example,
Aristotle's system could not deduce "No quadrangle that is a square is a rectangle that is a rhombus" from "No square that is a quadrangle is a rhombus that is a rectangle" or from "No rhombus that
is a rectangle is a square that is a quadrangle".
Logicist period[edit]
After Boole, the next great advances were made by the German mathematician Gottlob Frege. Frege's objective was the program of Logicism, i.e. demonstrating that arithmetic is identical with logic.^
[122] Frege went much further than any of his predecessors in his rigorous and formal approach to logic, and his calculus or Begriffsschrift is important.^[122] Frege also tried to show that the
concept of number can be defined by purely logical means, so that (if he was right) logic includes arithmetic and all branches of mathematics that are reducible to arithmetic. He was not the first
writer to suggest this. In his pioneering work Die Grundlagen der Arithmetik (The Foundations of Arithmetic), sections 15–17, he acknowledges the efforts of Leibniz, J.S. Mill as well as Jevons,
citing the latter's claim that "algebra is a highly developed logic, and number but logical discrimination."^[123]
Frege's first work, the Begriffsschrift ("concept script") is a rigorously axiomatised system of propositional logic, relying on just two connectives (negational and conditional), two rules of
inference (modus ponens and substitution), and six axioms. Frege referred to the "completeness" of this system, but was unable to prove this.^[124] The most significant innovation, however, was his
explanation of the quantifier in terms of mathematical functions. Traditional logic regards the sentence "Caesar is a man" as of fundamentally the same form as "all men are mortal." Sentences with a
proper name subject were regarded as universal in character, interpretable as "every Caesar is a man".^[125] At the outset Frege abandons the traditional "concepts subject and predicate", replacing
them with argument and function respectively, which he believes "will stand the test of time. It is easy to see how regarding a content as a function of an argument leads to the formation of
concepts. Furthermore, the demonstration of the connection between the meanings of the words if, and, not, or, there is, some, all, and so forth, deserves attention".^[126] Frege argued that the
quantifier expression "all men" does not have the same logical or semantic form as "all men", and that the universal proposition "every A is B" is a complex proposition involving two functions,
namely ' – is A' and ' – is B' such that whatever satisfies the first, also satisfies the second. In modern notation, this would be expressed as
${\displaystyle \forall \;x{\big (}A(x)\rightarrow B(x){\big )}}$
In English, "for all x, if Ax then Bx". Thus only singular propositions are of subject-predicate form, and they are irreducibly singular, i.e. not reducible to a general proposition. Universal and
particular propositions, by contrast, are not of simple subject-predicate form at all. If "all mammals" were the logical subject of the sentence "all mammals are land-dwellers", then to negate the
whole sentence we would have to negate the predicate to give "all mammals are not land-dwellers". But this is not the case.^[127] This functional analysis of ordinary-language sentences later had a
great impact on philosophy and linguistics.
This means that in Frege's calculus, Boole's "primary" propositions can be represented in a different way from "secondary" propositions. "All inhabitants are either men or women" is
${\displaystyle \forall \;x{\Big (}I(x)\rightarrow {\big (}M(x)\lor W(x){\big )}{\Big )}}$
whereas "All the inhabitants are men or all the inhabitants are women" is
${\displaystyle \forall \;x{\big (}I(x)\rightarrow M(x){\big )}\lor \forall \;x{\big (}I(x)\rightarrow W(x){\big )}}$
As Frege remarked in a critique of Boole's calculus:
"The real difference is that I avoid [the Boolean] division into two parts ... and give a homogeneous presentation of the lot. In Boole the two parts run alongside one another, so that one is
like the mirror image of the other, but for that very reason stands in no organic relation to it'^[128]
As well as providing a unified and comprehensive system of logic, Frege's calculus also resolved the ancient problem of multiple generality. The ambiguity of "every girl kissed a boy" is difficult to
express in traditional logic, but Frege's logic resolves this through the different scope of the quantifiers. Thus
${\displaystyle \forall \;x{\Big (}G(x)\rightarrow \exists \;y{\big (}B(y)\land K(x,y){\big )}{\Big )}}$
means that to every girl there corresponds some boy (any one will do) who the girl kissed. But
${\displaystyle \exists \;x{\Big (}B(x)\land \forall \;y{\big (}G(y)\rightarrow K(y,x){\big )}{\Big )}}$
means that there is some particular boy whom every girl kissed. Without this device, the project of logicism would have been doubtful or impossible. Using it, Frege provided a definition of the
ancestral relation, of the many-to-one relation, and of mathematical induction.^[129]
This period overlaps with the work of what is known as the "mathematical school", which included Dedekind, Pasch, Peano, Hilbert, Zermelo, Huntington, Veblen and Heyting. Their objective was the
axiomatisation of branches of mathematics like geometry, arithmetic, analysis and set theory. Most notable was Hilbert's Program, which sought to ground all of mathematics to a finite set of axioms,
proving its consistency by "finitistic" means and providing a procedure which would decide the truth or falsity of any mathematical statement. The standard axiomatization of the natural numbers is
named the Peano axioms in his honor. Peano maintained a clear distinction between mathematical and logical symbols. While unaware of Frege's work, he independently recreated his logical apparatus
based on the work of Boole and Schröder.^[130]
The logicist project received a near-fatal setback with the discovery of a paradox in 1901 by Bertrand Russell. This proved Frege's naive set theory led to a contradiction. Frege's theory contained
the axiom that for any formal criterion, there is a set of all objects that meet the criterion. Russell showed that a set containing exactly the sets that are not members of themselves would
contradict its own definition (if it is not a member of itself, it is a member of itself, and if it is a member of itself, it is not).^[131] This contradiction is now known as Russell's paradox. One
important method of resolving this paradox was proposed by Ernst Zermelo.^[132] Zermelo set theory was the first axiomatic set theory. It was developed into the now-canonical Zermelo–Fraenkel set
theory (ZF). Russell's paradox symbolically is as follows:
${\displaystyle {\text{Let }}R=\{x\mid xot \in x\}{\text{, then }}R\in R\iff Rot \in R}$
The monumental Principia Mathematica, a three-volume work on the foundations of mathematics, written by Russell and Alfred North Whitehead and published 1910–13 also included an attempt to resolve
the paradox, by means of an elaborate system of types: a set of elements is of a different type than is each of its elements (set is not the element; one element is not the set) and one cannot speak
of the "set of all sets". The Principia was an attempt to derive all mathematical truths from a well-defined set of axioms and inference rules in symbolic logic.
Metamathematical period[edit]
The names of Gödel and Tarski dominate the 1930s,^[133] a crucial period in the development of metamathematics – the study of mathematics using mathematical methods to produce metatheories, or
mathematical theories about other mathematical theories. Early investigations into metamathematics had been driven by Hilbert's program. Work on metamathematics culminated in the work of Gödel, who
in 1929 showed that a given first-order sentence is deducible if and only if it is logically valid – i.e. it is true in every structure for its language. This is known as Gödel's completeness theorem
. A year later, he proved two important theorems, which showed Hibert's program to be unattainable in its original form. The first is that no consistent system of axioms whose theorems can be listed
by an effective procedure such as an algorithm or computer program is capable of proving all facts about the natural numbers. For any such system, there will always be statements about the natural
numbers that are true, but that are unprovable within the system. The second is that if such a system is also capable of proving certain basic facts about the natural numbers, then the system cannot
prove the consistency of the system itself. These two results are known as Gödel's incompleteness theorems, or simply Gödel's Theorem. Later in the decade, Gödel developed the concept of
set-theoretic constructibility, as part of his proof that the axiom of choice and the continuum hypothesis are consistent with Zermelo–Fraenkel set theory. In proof theory, Gerhard Gentzen developed
natural deduction and the sequent calculus. The former attempts to model logical reasoning as it 'naturally' occurs in practice and is most easily applied to intuitionistic logic, while the latter
was devised to clarify the derivation of logical proofs in any formal system. Since Gentzen's work, natural deduction and sequent calculi have been widely applied in the fields of proof theory,
mathematical logic and computer science. Gentzen also proved normalization and cut-elimination theorems for intuitionistic and classical logic which could be used to reduce logical proofs to a normal
Alfred Tarski, a pupil of Łukasiewicz, is best known for his definition of truth and logical consequence, and the semantic concept of logical satisfaction. In 1933, he published (in Polish) The
concept of truth in formalized languages, in which he proposed his semantic theory of truth: a sentence such as "snow is white" is true if and only if snow is white. Tarski's theory separated the
metalanguage, which makes the statement about truth, from the object language, which contains the sentence whose truth is being asserted, and gave a correspondence (the T-schema) between phrases in
the object language and elements of an interpretation. Tarski's approach to the difficult idea of explaining truth has been enduringly influential in logic and philosophy, especially in the
development of model theory.^[136] Tarski also produced important work on the methodology of deductive systems, and on fundamental principles such as completeness, decidability, consistency and
definability. According to Anita Feferman, Tarski "changed the face of logic in the twentieth century".^[137]
Alonzo Church and Alan Turing proposed formal models of computability, giving independent negative solutions to Hilbert's Entscheidungsproblem in 1936 and 1937, respectively. The Entscheidungsproblem
asked for a procedure that, given any formal mathematical statement, would algorithmically determine whether the statement is true. Church and Turing proved there is no such procedure; Turing's paper
introduced the halting problem as a key example of a mathematical problem without an algorithmic solution.
Church's system for computation developed into the modern λ-calculus, while the Turing machine became a standard model for a general-purpose computing device. It was soon shown that many other
proposed models of computation were equivalent in power to those proposed by Church and Turing. These results led to the Church–Turing thesis that any deterministic algorithm that can be carried out
by a human can be carried out by a Turing machine. Church proved additional undecidability results, showing that both Peano arithmetic and first-order logic are undecidable. Later work by Emil Post
and Stephen Cole Kleene in the 1940s extended the scope of computability theory and introduced the concept of degrees of unsolvability.
The results of the first few decades of the twentieth century also had an impact upon analytic philosophy and philosophical logic, particularly from the 1950s onwards, in subjects such as modal logic
, temporal logic, deontic logic, and relevance logic.
Logic after WWII[edit]
After World War II, mathematical logic branched into four inter-related but separate areas of research: model theory, proof theory, computability theory, and set theory.^[138]
In set theory, the method of forcing revolutionized the field by providing a robust method for constructing models and obtaining independence results. Paul Cohen introduced this method in 1963 to
prove the independence of the continuum hypothesis and the axiom of choice from Zermelo–Fraenkel set theory.^[139] His technique, which was simplified and extended soon after its introduction, has
since been applied to many other problems in all areas of mathematical logic.
Computability theory had its roots in the work of Turing, Church, Kleene, and Post in the 1930s and 40s. It developed into a study of abstract computability, which became known as recursion theory.^
[140] The priority method, discovered independently by Albert Muchnik and Richard Friedberg in the 1950s, led to major advances in the understanding of the degrees of unsolvability and related
structures. Research into higher-order computability theory demonstrated its connections to set theory. The fields of constructive analysis and computable analysis were developed to study the
effective content of classical mathematical theorems; these in turn inspired the program of reverse mathematics. A separate branch of computability theory, computational complexity theory, was also
characterized in logical terms as a result of investigations into descriptive complexity.
Model theory applies the methods of mathematical logic to study models of particular mathematical theories. Alfred Tarski published much pioneering work in the field, which is named after a series of
papers he published under the title Contributions to the theory of models. In the 1960s, Abraham Robinson used model-theoretic techniques to develop calculus and analysis based on infinitesimals, a
problem that first had been proposed by Leibniz.
In proof theory, the relationship between classical mathematics and intuitionistic mathematics was clarified via tools such as the realizability method invented by Georg Kreisel and Gödel's
Dialectica interpretation. This work inspired the contemporary area of proof mining. The Curry-Howard correspondence emerged as a deep analogy between logic and computation, including a
correspondence between systems of natural deduction and typed lambda calculi used in computer science. As a result, research into this class of formal systems began to address both logical and
computational aspects; this area of research came to be known as modern type theory. Advances were also made in ordinal analysis and the study of independence results in arithmetic such as the
Paris–Harrington theorem.
This was also a period, particularly in the 1950s and afterwards, when the ideas of mathematical logic begin to influence philosophical thinking. For example, tense logic is a formalised system for
representing, and reasoning about, propositions qualified in terms of time. The philosopher Arthur Prior played a significant role in its development in the 1960s. Modal logics extend the scope of
formal logic to include the elements of modality (for example, possibility and necessity). The ideas of Saul Kripke, particularly about possible worlds, and the formal system now called Kripke
semantics have had a profound impact on analytic philosophy.^[141] His best known and most influential work is Naming and Necessity (1980).^[142] Deontic logics are closely related to modal logics:
they attempt to capture the logical features of obligation, permission and related concepts. Although some basic novelties syncretizing mathematical and philosophical logic were shown by Bolzano in
the early 1800s, it was Ernst Mally, a pupil of Alexius Meinong, who was to propose the first formal deontic system in his Grundgesetze des Sollens, based on the syntax of Whitehead's and Russell's
propositional calculus.
Another logical system founded after World War II was fuzzy logic by Azerbaijani mathematician Lotfi Asker Zadeh in 1965.
See also[edit]
1. ^ ^a ^b Boehner p. xiv
2. ^ ^a ^b Oxford Companion p. 498; Bochenski, Part I Introduction, passim
3. ^ Gottlob Frege. The Foundations of Arithmetic (PDF). p. 1.
4. ^ ^a ^b Oxford Companion p. 500
5. ^ Bochenski p. 446
6. ^ S. C. Vidyabhusana (1971). A History of Indian Logic: Ancient, Mediaeval, and Modern Schools, pp. 17–21.
7. ^ R. P. Kangle (1986). The Kautiliya Arthashastra (1.2.11). Motilal Banarsidass.
8. ^ Bochenski p. 417 and passim
9. ^ Bochenski pp. 431–7
10. ^ Matilal, Bimal Krishna (1998). The Character of Logic in India. Albany, NY: State University of New York Press. pp. 12, 18. ISBN 9780791437407.
11. ^ Bochenksi p. 441
12. ^ Matilal, 17
13. ^ Kneale, p. 2
14. ^ ^a ^b ^c ^d Kneale p. 3
15. ^ H. F. J. Horstmanshoff, Marten Stol, Cornelis Tilburg (2004), Magic and Rationality in Ancient Near Eastern and Graeco-Roman Medicine, p. 99, Brill Publishers, ISBN 90-04-13666-5.
16. ^ D. Brown (2000), Mesopotamian Planetary Astronomy-Astrology , Styx Publications, ISBN 90-5693-036-2.
17. ^ Heath, Mathematics in Aristotle, cited in Kneale, p. 5
18. ^ Kneale, p. 16
19. ^ "History of logic". britannica.com. Retrieved 2 April 2018.
20. ^ Aristotle, Metaphysics Alpha, 983b18.
21. ^ Smith, Sir William (1870). Dictionary of Greek and Roman biography and mythology. p. 1016.
22. ^ Prof.T.Patronis & D.Patsopoulos The Theorem of Thales: A Study of the naming of theorems in school Geometry textbooks. Patras University. Retrieved 2012-02-12.
23. ^ (Boyer 1991, "Ionia and the Pythagoreans" p. 43)
24. ^ de Laet, Siegfried J. (1996). History of Humanity: Scientific and Cultural Development. UNESCO, Volume 3, p. 14. ISBN 92-3-102812-X
25. ^ Boyer, Carl B. and Merzbach, Uta C. (2010). A History of Mathematics. John Wiley and Sons, Chapter IV. ISBN 0-470-63056-6
26. ^ C. B. Boyer (1968)
27. ^ Samuel Enoch Stumpf. Socrates to Sartre. p. 11.
28. ^ F.E. Peters, Greek Philosophical Terms, New York University Press, 1967.
29. ^ http://www.bard.edu/library/arendt/pdfs/Cornford-Parmenides.pdf
30. ^ R. J. Hollingdale (1974). Western Philosophy: an introduction. p. 73.
31. ^ http://www.wilbourhall.org/pdfs/From_religion_to_philosophy.pdf
32. ^ Kneale p. 15
33. ^ "The Numismatic Circular". 2 April 2018. Retrieved 2 April 2018 – via Google Books.
34. ^ Kneale p. 17
35. ^ "forming an opinion is talking, and opinion is speech that is held not with someone else or aloud but in silence with oneself" Theaetetus 189E–190A
36. ^ Kneale p. 20. For example, the proof given in the Meno that the square on the diagonal is double the area of the original square presumably involves the forms of the square and the triangle,
and the necessary relation between them
37. ^ Kneale p. 21
38. ^ Zalta, Edward N. "Aristotle's Logic". Stanford University, 18 March 2000. Retrieved 13 March 2010.
39. ^ See e.g. Aristotle's logic, Stanford Encyclopedia of Philosophy
40. ^ ^a ^b Bochenski p. 63
41. ^ "Throughout later antiquity two great schools of logic were distinguished, the Peripatetic which was derived from Aristotle, and the Stoic which was developed by Chrysippus from the teachings
of the Megarians" – Kneale p. 113
42. ^ Oxford Companion, article "Chrysippus", p. 134
43. ^ [1] Stanford Encyclopedia of Philosophy: Susanne Bobzien, Ancient Logic
44. ^ K. Huelser, Die Fragmente zur Dialektik der Stoiker, 4 vols, Stuttgart 1986-7
45. ^ Kneale 117–158
46. ^ Metaphysics Eta 3, 1046b 29
47. ^ Boethius, Commentary on the Perihermenias, Meiser p. 234
48. ^ Epictetus, Dissertationes ed. Schenkel ii. 19. I.
49. ^ Alexander p. 177
50. ^ ^a ^b Sextus Empiricus, Adv. Math. viii, Section 113
51. ^ Sextus Empiricus, Hypotyp. ii. 110, comp.
52. ^ Cicero, Academica, ii. 47, de Fato, 6.
53. ^ See e.g. Lukasiewicz p. 21
54. ^ Sextus Bk viii., Sections 11, 12
55. ^ See e.g. Routledge Encyclopedia of Philosophy Online Version 2.0 Archived 2015-05-03 at WebCite, article 'Islamic philosophy'
56. ^ Feldman, Seymour (1964-11-26). "Rescher on Arabic Logic". The Journal of Philosophy. Journal of Philosophy, Inc. 61 (22): 724–734. doi:10.2307/2023632. ISSN 0022-362X. JSTOR 2023632. [726].
Long, A. A.; D. N. Sedley (1987). The Hellenistic Philosophers. Vol 1: Translations of the principal sources with philosophical commentary. Cambridge: Cambridge University Press. ISBN
57. ^ Dag Nikolaus Hasse (September 19, 2008). "Influence of Arabic and Islamic Philosophy on the Latin West". Stanford Encyclopedia of Philosophy. Retrieved 2009-10-13.
58. ^ Richard F. Washell (1973), "Logic, Language, and Albert the Great", Journal of the History of Ideas 34 (3), pp. 445–450 [445].
59. ^ ^a ^b Goodman, Lenn Evan (2003), Islamic Humanism, p. 155, Oxford University Press, ISBN 0-19-513580-6.
60. ^ Goodman, Lenn Evan (1992); Avicenna, p. 188, Routledge, ISBN 0-415-01929-X.
61. ^ Kneale p. 229
62. ^ Kneale: p. 266; Ockham: Summa Logicae i. 14; Avicenna: Avicennae Opera Venice 1508 f87rb
63. ^ ^a ^b Muhammad Iqbal, The Reconstruction of Religious Thought in Islam, "The Spirit of Muslim Culture" (cf. [2] and [3])
64. ^ ^a ^b Tony Street (July 23, 2008). "Arabic and Islamic Philosophy of Language and Logic". Stanford Encyclopedia of Philosophy. Retrieved 2008-12-05.
65. ^ Dr. Lotfollah Nabavi, Sohrevardi's Theory of Decisive Necessity and kripke's QSS System Archived 2008-01-26 at the Wayback Machine, Journal of Faculty of Literature and Human Sciences.
66. ^ Dr. Abu Shadi Al-Roubi (1982), "Ibn Al-Nafis as a philosopher", Symposium on Ibn al-Nafis, Second International Conference on Islamic Medicine: Islamic Medical Organization, Kuwait (cf. Ibn
al-Nafis As a Philosopher Archived 2008-02-06 at the Wayback Machine, Encyclopedia of Islamic World).
67. ^ See pp. 253–254 of Street, Tony (2005). "Logic". In Peter Adamson and Richard C. Taylor (edd.). The Cambridge Companion to Arabic Philosophy. Cambridge University Press. pp. 247–265. ISBN
978-0-521-52069-0.CS1 maint: Uses editors parameter (link)
68. ^ Ruth Mas (1998). "Qiyas: A Study in Islamic Logic" (PDF). Folia Orientalia. 34: 113–128. ISSN 0015-5675.
69. ^ ^a ^b John F. Sowa; Arun K. Majumdar (2003). "Analogical reasoning". Conceptual Structures for Knowledge Creation and Communication, Proceedings of ICCS 2003. Berlin: Springer-Verlag., pp.
70. ^ Nicholas Rescher and Arnold vander Nat, "The Arabic Theory of Temporal Modal Syllogistic", in George Fadlo Hourani (1975), Essays on Islamic Philosophy and Science, pp. 189–221, State
University of New York Press, ISBN 0-87395-224-3.
71. ^ Kneale p. 198
72. ^ Stephen Dumont, article "Peter Abelard" in Gracia and Noone p. 492
73. ^ Kneale, pp. 202–3
74. ^ See e.g. Kneale p. 225
75. ^ Boehner p. 1
76. ^ Boehner pp. 19–76
77. ^ Boehner p. 29
78. ^ Boehner p. 30
79. ^ Ebbesen 1981
80. ^ Boehner pp. 54–5
81. ^ Oxford Companion p. 504, article "Traditional logic"
82. ^ ^a ^b Buroker xxiii
83. ^ (Locke, An Essay Concerning Human Understanding, IV. 5. 6)
84. ^ Farrington, 1964, 89
85. ^ N. Abbagnano, "Psychologism" in P. Edwards (ed) The Encyclopaedia of Philosophy, MacMillan, 1967
86. ^ Of the German literature in this period, Robert Adamson wrote "Logics swarm as bees in springtime..."; Robert Adamson, A Short History of Logic, Wm. Blackwood & Sons, 1911, page 242
87. ^ Carl von Prantl (1855-1867), Geschichte von Logik in Abendland, Leipsig: S. Hirzl, anastatically reprinted in 1997, Hildesheim: Georg Olds.
88. ^ See e.g. Psychologism, Stanford Encyclopedia of Philosophy
89. ^ Wilhelm Wundt, Logik (1880–1883); quoted in Edmund Husserl, Logical Investigations, translated J.N. Findlay, Routledge, 2008, Volume 1, pp. 115–116.
90. ^ Theodor Lipps, Grundzüge der Logik (1893); quoted in Edmund Husserl, Logical Investigations, translated J.N. Findlay, Routledge, 2008, Volume 1, p. 40
91. ^ Christoph von Sigwart, Logik (1873–78); quoted in Edmund Husserl, Logical Investigations, translated J.N. Findlay, Routledge, 2008, Volume 1, p. 51
92. ^ Benno Erdmann, Logik (1892); quoted in Edmund Husserl, Logical Investigations, translated J.N. Findlay, Routledge, 2008, Volume 1, p. 96
93. ^ Dermot Moran, "Introduction"; Edmund Husserl, Logical Investigations, translated J.N. Findlay, Routledge, 2008, Volume 1, p. xxi
94. ^ Michael Dummett, "Preface"; Edmund Husserl, Logical Investigations, translated J.N. Findlay, Routledge, 2008, Volume 1, p. xvii
95. ^ Josiah Royce, "Recent Logical Enquiries and their Psychological Bearings" (1902) in John J. McDermott (ed) The Basic Writings of Josiah Royce Volume 2, Fordham University Press, 2005, p. 661
96. ^ Bochenski, p. 266
97. ^ Peirce 1896
98. ^ See Bochenski p. 269
99. ^ Oxford Companion p. 499
100. ^ Edith Sylla (1999), "Oxford Calculators", in The Cambridge Dictionary of Philosophy, Cambridge, Cambridgeshire: Cambridge.
101. ^ El. philos. sect. I de corp 1.1.2.
102. ^ Bochenski p. 274
103. ^ Rutherford, Donald, 1995, "Philosophy and language" in Jolley, N., ed., The Cambridge Companion to Leibniz. Cambridge Univ. Press.
104. ^ Wiener, Philip, 1951. Leibniz: Selections. Scribner.
105. ^ Essai de dialectique rationelle, 211n, quoted in Bochenski p. 277.
106. ^ Bolzano, Bernard (1972). George, Rolf, ed. The Theory of Science: Die Wissenschaftslehre oder Versuch einer Neuen Darstellung der Logik. Translated by George Rolf. University of California
Press. p. 209. ISBN 9780520017870.
107. ^ See e.g. Bochenski p. 296 and passim
108. ^ Before publishing, he wrote to De Morgan, who was just finishing his work Formal Logic. De Morgan suggested they should publish first, and thus the two books appeared at the same time,
possibly even reaching the bookshops on the same day. cf. Kneale p. 404
109. ^ Kneale p. 404
110. ^ ^a ^b ^c Kneale p. 407
111. ^ Boole (1847) p. 16
112. ^ Boole 1847 pp. 58–9
113. ^ Beaney p. 11
114. ^ Kneale p. 422
115. ^ Peirce, "A Boolean Algebra with One Constant", 1880 MS, Collected Papers v. 4, paragraphs 12–20, reprinted Writings v. 4, pp. 218-21. Google Preview.
116. ^ Trans. Amer. Math. Soc., xiv (1913), pp. 481–8. This is now known as the Sheffer stroke
117. ^ Bochenski 296
118. ^ See CP III
119. ^ George Boole. 1854/2003. The Laws of Thought, facsimile of 1854 edition, with an introduction by J. Corcoran. Buffalo: Prometheus Books (2003). Reviewed by James van Evra in Philosophy in
Review.24 (2004) 167–169.
120. ^ JOHN CORCORAN, Aristotle's Prior Analytics and Boole's Laws of Thought, History and Philosophy of Logic, vol. 24 (2003), pp. 261–288.
121. ^ ^a ^b Kneale p. 435
122. ^ Jevons, The Principles of Science, London 1879, p. 156, quoted in Grundlagen 15
123. ^ Beaney p. 10 – the completeness of Frege's system was eventually proved by Jan Łukasiewicz in 1934
124. ^ See for example the argument by the medieval logician William of Ockham that singular propositions are universal, in Summa Logicae III. 8 (??)
125. ^ Frege 1879 in van Heijenoort 1967, p. 7
126. ^ "On concept and object" p. 198; Geach p. 48
127. ^ BLC p. 14, quoted in Beaney p. 12
128. ^ See e.g. The Internet Encyclopedia of Philosophy, article "Frege"
129. ^ Van Heijenoort 1967, p. 83
130. ^ See e.g. Potter 2004
131. ^ Zermelo 1908
132. ^ Feferman 1999 p. 1
133. ^ Girard, Jean-Yves; Paul Taylor; Yves Lafont (1990) [1989]. Proofs and Types. Cambridge University Press (Cambridge Tracts in Theoretical Computer Science, 7). ISBN 0-521-37181-3.
134. ^ Alex Sakharov. "Cut Elimination Theorem". MathWorld.
135. ^ Feferman and Feferman 2004, p. 122, discussing "The Impact of Tarski's Theory of Truth".
136. ^ Feferman 1999, p. 1
137. ^ See e.g. Barwise, Handbook of Mathematical Logic
138. ^ The Independence of the Continuum Hypothesis, II Paul J. Cohen Proceedings of the National Academy of Sciences of the United States of America, Vol. 51, No. 1. (Jan. 15, 1964), pp. 105-110.
139. ^ Many of the foundational papers are collected in The Undecidable (1965) edited by Martin Davis
140. ^ Jerry Fodor, "Water's water everywhere", London Review of Books, 21 October 2004
141. ^ See Philosophical Analysis in the Twentieth Century: Volume 2: The Age of Meaning, Scott Soames: "Naming and Necessity is among the most important works ever, ranking with the classical work
of Frege in the late nineteenth century, and of Russell, Tarski and Wittgenstein in the first half of the twentieth century". Cited in Byrne, Alex and Hall, Ned. 2004. 'Necessary Truths'. Boston
Review October/November 2004
Primary Sources
• Alexander of Aphrodisias, In Aristotelis An. Pr. Lib. I Commentarium, ed. Wallies, Berlin, C.I.A.G. vol. II/1, 1882.
• Avicenna, Avicennae Opera Venice 1508.
• Boethius Commentary on the Perihermenias, Secunda Editio, ed. Meiser, Leipzig, Teubner, 1880.
• Bolzano, Bernard Wissenschaftslehre, (1837) 4 Bde, Neudr., hrsg. W. Schultz, Leipzig I-II 1929, III 1930, IV 1931 (Theory of Science, four volumes, translated by Rolf George and Paul Rusnock, New
York: Oxford University Press, 2014).
• Bolzano, Bernard Theory of Science (Edited, with an introduction, by Jan Berg. Translated from the German by Burnham Terrell – D. Reidel Publishing Company, Dordrecht and Boston 1973).
• Boole, George (1847) The Mathematical Analysis of Logic (Cambridge and London); repr. in Studies in Logic and Probability, ed. R. Rhees (London 1952).
• Boole, George (1854) The Laws of Thought (London and Cambridge); repr. as Collected Logical Works. Vol. 2, (Chicago and London: Open Court, 1940).
• Epictetus, Epicteti Dissertationes ab Arriano digestae, edited by Heinrich Schenkl, Leipzig, Teubner. 1894.
• Frege, G., Boole's Logical Calculus and the Concept Script, 1882, in Posthumous Writings transl. P. Long and R. White 1969, pp. 9–46.
• Gergonne, Joseph Diaz, (1816) Essai de dialectique rationelle, in Annales de mathématiques pures et appliquées 7, 1816/7, 189–228.
• Jevons, W.S. The Principles of Science, London 1879.
• Ockham's Theory of Terms: Part I of the Summa Logicae, translated and introduced by Michael J. Loux (Notre Dame, IN: University of Notre Dame Press 1974). Reprinted: South Bend, IN: St.
Augustine's Press, 1998.
• Ockham's Theory of Propositions: Part II of the Summa Logicae, translated by Alfred J. Freddoso and Henry Schuurman and introduced by Alfred J. Freddoso (Notre Dame, IN: University of Notre Dame
Press, 1980). Reprinted: South Bend, IN: St. Augustine's Press, 1998.
• Peirce, C.S., (1896), "The Regenerated Logic", The Monist, vol. VII, No. 1, p pp. 19-40, The Open Court Publishing Co., Chicago, IL, 1896, for the Hegeler Institute. Reprinted (CP 3.425–455).
Internet Archive The Monist 7.
• Sextus Empiricus, Against the Logicians. (Adversus Mathematicos VII and VIII). Richard Bett (trans.) Cambridge: Cambridge University Press, 2005. ISBN 0-521-53195-0.
• Zermelo, Ernst (1908). "Untersuchungen über die Grundlagen der Mengenlehre I". Mathematische Annalen. 65 (2): 261–281. doi:10.1007/BF01449999. English translation in Heijenoort, Jean van (1967).
"Investigations in the foundations of set theory". From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. Source Books in the History of the Sciences. Harvard Univ. Press. pp.
199–215. ISBN 978-0-674-32449-7..
Secondary Sources
• Barwise, Jon, (ed.), Handbook of Mathematical Logic, Studies in Logic and the Foundations of Mathematics, Amsterdam, North Holland, 1982 ISBN 978-0-444-86388-1 .
• Beaney, Michael, The Frege Reader, London: Blackwell 1997.
• Bochenski, I.M., A History of Formal Logic, Indiana, Notre Dame University Press, 1961.
• Boehner, Philotheus, Medieval Logic, Manchester 1950.
• Buroker, Jill Vance (transl. and introduction), A. Arnauld, P. Nicole Logic or the Art of Thinking, Cambridge University Press, 1996, ISBN 0-521-48249-6.
• Church, Alonzo, 1936-8. "A bibliography of symbolic logic". Journal of Symbolic Logic 1: 121–218; 3:178–212.
• de Jong, Everard (1989), Galileo Galilei's "Logical Treatises" and Giacomo Zabarella's "Opera Logica": A Comparison, PhD dissertation, Washington, DC: Catholic University of America.
• Ebbesen, Sten "Early supposition theory (12th–13th Century)" Histoire, Épistémologie, Langage 3/1: 35–48 (1981).
• Farrington, B., The Philosophy of Francis Bacon, Liverpool 1964.
• Feferman, Anita B. (1999). "Alfred Tarski". American National Biography. 21. Oxford University Press. pp. 330–332. ISBN 978-0-19-512800-0.
• Feferman, Anita B.; Feferman, Solomon (2004). Alfred Tarski: Life and Logic. Cambridge University Press. ISBN 978-0-521-80240-6. OCLC 54691904.
• Gabbay, Dov and John Woods, eds, Handbook of the History of Logic 2004. 1. Greek, Indian and Arabic logic; 2. Mediaeval and Renaissance logic; 3. The rise of modern logic: from Leibniz to Frege;
4. British logic in the Nineteenth century; 5. Logic from Russell to Church; 6. Sets and extensions in the Twentieth century; 7. Logic and the modalities in the Twentieth century; 8. The
many-valued and nonmonotonic turn in logic; 9. Computational Logic; 10. Inductive logic; 11. Logic: A history of its central concepts; Elsevier, ISBN 0-444-51611-5.
• Geach, P.T. Logic Matters, Blackwell 1972.
• Goodman, Lenn Evan (2003). Islamic Humanism. Oxford University Press, ISBN 0-19-513580-6.
• Goodman, Lenn Evan (1992). Avicenna. Routledge, ISBN 0-415-01929-X.
• Grattan-Guinness, Ivor, 2000. The Search for Mathematical Roots 1870–1940. Princeton University Press.
• Gracia, J.G. and Noone, T.B., A Companion to Philosophy in the Middle Ages, London 2003.
• Haaparanta, Leila (ed.) 2009. The Development of Modern Logic Oxford University Press.
• Heath, T.L., 1949. Mathematics in Aristotle, Oxford University Press.
• Heath, T.L., 1931, A Manual of Greek Mathematics, Oxford (Clarendon Press).
• Honderich, Ted (ed.). The Oxford Companion to Philosophy (New York: Oxford University Press, 1995) ISBN 0-19-866132-0.
• Kneale, William and Martha, 1962. The development of logic. Oxford University Press, ISBN 0-19-824773-7.
• Lukasiewicz, Aristotle's Syllogistic, Oxford University Press 1951.
• Potter, Michael (2004), Set Theory and its Philosophy, Oxford University Press.
External links[edit] | {"url":"https://static.hlt.bme.hu/semantics/external/pages/logikai_form%C3%A1t%C3%B3l/en.wikipedia.org/wiki/History_of_logic.html","timestamp":"2024-11-03T06:15:31Z","content_type":"text/html","content_length":"336163","record_id":"<urn:uuid:13ec8daf-cfb5-481b-938e-d1da60be69db>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00602.warc.gz"} |
Mathematical Induction
Download Mathematical Induction
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Document related concepts
Gödel's incompleteness theorems wikipedia , lookup
Foundations of mathematics wikipedia , lookup
Law of thought wikipedia , lookup
Mathematical logic wikipedia , lookup
Axiom wikipedia , lookup
Surreal number wikipedia , lookup
Novum Organum wikipedia , lookup
New riddle of induction wikipedia , lookup
List of first-order theories wikipedia , lookup
Axiom of reducibility wikipedia , lookup
Naive set theory wikipedia , lookup
Ordinal arithmetic wikipedia , lookup
Inquiry wikipedia , lookup
Mathematical proof wikipedia , lookup
Peano axioms wikipedia , lookup
Language, Proof and
Mathematical Induction
Chapter 16
Induction – a powerful method of proof
x (P(x)P(x+1))
------------------------------------x P(x)
• Person #1 knows the secret
• For all n, if person # n knows the secret, then so does person # n+1
• For all n, person # n knows the secret
Inductive definitions and inductive proofs
Example of an inductive definition: wff
1. Every atomic wff is a wff
2. If P is a wff, so is P
3. If P1,…,Pn are wffs, so are (P1…Pn) and (P1…Pn)
4. If P and Q are wffs, so are (PQ) and (PQ)
5. If P is a wff and x is a variable, xP and xP are wffs;
6. Nothing is a wff unless it is generated by repeated applications of 1-5.
An inductive definition of a set consists of:
• a base clause, which specifies the basic elements of the defined set,
• one or more inductive clauses, which tell us how to generate additional
elements, and
• a final clause, which tells us that all the elements are either basic or
generated by the inductive clauses.
Inductive definitions and inductive proofs
Example of an inductive definition: pal
1. Each letter in the alphabet (a,b,...,z) is a pal.
2. If a string is a pal, so is the result of putting any letter of the
alphabet both in front and in back of (e.g. aa, bb, etc.)
3. Nothing is a pal unless it is generated by repeated applications of 1-2.
An inductive definition of a set consists of:
• a base clause, which specifies the basic elements of the defined set,
• one or more inductive clauses, which tell us how to generate additional
elements, and
• a final clause, which tells us that all the elements are either basic or
generated by the inductive clauses.
Inductive definitions and inductive proofs
Example of an inductive definition: pal
1. Each letter in the alphabet (a,b,...,z) is a pal.
2. If a string is a pal, so is the result of putting any letter of the
alphabet both in front and in back of (e.g. aa, bb, etc.)
3. Nothing is a pal unless it is generated by repeated applications of 1-2.
Given an inductive definition of a set S, an inductive proof of the fact
that a certain property holds of all elements of S requires:
• a basis step, which shows that the property holds of the basic
elements, and
• an inductive step, which shows that if the property holds of some
elements, then it holds of any elements generated from them by
the inductive clauses.
The assumption that begins the inductive step is called the inductive
Inductive definitions and inductive proofs
Example of an inductive definition: pal
1. Each letter in the alphabet (a,b,...,z) is a pal.
2. If a string is a pal, so is the result of putting any letter of the
alphabet both in front and in back of (e.g. aa, bb, etc.)
3. Nothing is a pal unless it is generated by repeated applications of 1-2.
By induction, prove that:
a) Every pal has an odd length
b) Every pal is a palindrome
Inductive definitions in set theory
Making the final clause of an inductive definition more precise:
The set P of pals is the smallest set --- i.e. the intersection of all sets --such that:
1. Each letter in the alphabet (a,b,...,z) is in P.
2. If a string is in P, so is the result of putting any letter of the
alphabet both in front and in back of (e.g. aa, bb, etc.)
Induction on the natural numbers
The inductive definition of natural numbers:
1. 0 is a natural number.
2. If n is a natural number, then n+1 is a natural number.
3. Nothing is a natural number except in virtue of repeated
applications of (1) and (2).
The set N of natural numbers is the smallest set satisfying:
1. 0N
2. If nN, then n+1N.
To prove by induction that P(x) is true of all natural numbers:
1. Prove P(0) (basis step)
2. Prove x[P(x)P(x+1)] (inductive step)
Induction on the natural numbers
Proposition 4. For every natural number n, the sum of the first n
natural numbers is n(n-1)/2.
Basis: The sum of the first 0 natural numbers is indeed 0.
Inductive step: Assume the sum of the first k natural numbers is
k(k-1)/2 (inductive hypothesis). We want to show that then the same
is true for k+1 instead of k, that is, the sum of the first k+1 natural
numbers is (k+1)((k+1)-1)/2, i.e. it is k(k+1)/2.
But indeed, the sum of the first k+1 natural numbers is X+k, where
X is the sum of the fist k natural numbers. By the inductive hypothesis,
X= k(k-1)/2 . Thus, the sought X+k is
k(k-1)/2+k = (k(k-1)+2k)/2 = k(k-1+2)/2 = k(k+1)/2, as desired.
Axiomatizing the natural numbers
Peano Arithmetic PA
Language: =, 0, s, +, (s(a) means a+1)
1. x(s(x)0)
2. xy (s(x)=s(y) x=y)
Gödel’s Incompleteness Theorem:
These axioms are not sufficient to
prove every true arithmetical sentence.
Nor would any bigger set of axioms be
3. x (x+0 = x)
5. x (x0 = 0)
Axiom 7 is a scheme, for every
Q. If Q contains additional
variables z1,...,zn, then the whole
thing should be prefixed with
z1... zn
6. xy [xs(y) = (xy)+x]
This axiom is called the
induction scheme
4. xy [x+s(y) = s(x+y)]
7. [Q(0) x (Q(x) Q(s(x)))] xQ(x)
Informal proof of x(s(x)=s(0)+x)
What we need:
Axiom 3: x (x+0 = x)
Axiom 4: xy [x+s(y) = s(x+y)]
Axiom 7: [Q(0) x (Q(x) Q(s(x)))] xQ(x)
with s(x)=s(0)+x in the role of Q(x)
Basis: Q(0), i.e. s(0)=s(0)+0. Follows from Axiom 3.
Inductive step:
Assume (induction hypothesis) Q(n), i.e. s(n)=s(0)+n
We want to show that then
Q(s(n)), i.e. s(s(n))=s(0)+s(n)
But indeed:
s(s(n)) = s(s(0)+n) by induction hypothesis.
= s(0)+s(n) by Axiom 4.
Induction in Fitch
Peano Induction:
n P(n)
Where n does not occur outside
the subproof where it is introduced
Ordering the Natural Numbers
A binary relation R is said to be a total strict ordering iff it is:
1. Irreflexive:
2. Transitive:
3. Trichotomous: xy (xRy x=y yRx)
The ordinary relation < on natural numbers is a total strict ordering.
In PA, “x<y” can be treated as an abbreviation of “z(x+s(z)=y)”.
An alternative approach, taken in Fitch, is to treat < as a legitimate
symbol of the language, defined by the (additional Peano) axiom
xy[x<y z(x+s(z)=y)]
Strong Induction
Strong induction (other names: complete induction; course of values
induction) in Fitch takes the following form:
n x[x<n P(x)]
Where n does not occur outside
the subproof where it is introduced
This principle does not allow us to prove anything new because it is
equivalent to ordinary induction (either one follows from the other --see the textbook). However, it can often offer greater convenience.
Example: Using strong induction, prove the so called Fundamental
Theorem of Arithmetic, according to which every natural number
greater than 1 is either prime or the product of some primes. | {"url":"https://studyres.com/doc/8456273/mathematical-induction","timestamp":"2024-11-10T02:55:51Z","content_type":"text/html","content_length":"72490","record_id":"<urn:uuid:8c5dbbd6-1fb6-47fd-b178-6de3c7fec30f>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00666.warc.gz"} |
Rationale: Every logical argument must be defined in some language, and every language has limitations. Attempting to construct a logical argument while ignoring how the limitations of language might
affect that argument is a bizarre approach. The correct acknowledgment of the interactions of logic and language explains almost all of the paradoxes, and resolves almost all of the contradictions,
conundrums, and contentious issues in modern philosophy and mathematics.
Site Mission
• To promulgate the understanding that the validity of a logical argument is not necessarily independent of the way in which language is used by that argument.
• To rid the fields of philosophy and mathematics of arcane and irrational notions which have resulted in numerous contradictions.
• To ensure that future generations of young people will not be put off the study of mathematics and philosophy by the mystical and illogical notions that are currently widespread in those
Please see the menu for numerous articles of interest. Please leave a comment or send an email if you are interested in the material on this site.
Interested in supporting this site?
You can help by sharing the site with others. You can also donate at [] where there are full details.
David Hilbert
In 1900 the mathematician David Hilbert posed 23 major problems that were at that time all unanswered. Problem 10 was the question as to whether there can be a finite process which can definitively
tell whether there are natural number solutions to a certain type of equation known as a Diophantine equation.
In 1970 Yuri Matiyasevich claimed to have proved that the answer to Hilbert’s question was that it is impossible for there to be any such process. But is Matiyasevich’s proof rock-solid? See the
As site owner I reserve the right to keep my comments sections as I deem appropriate. I do not use that right to unfairly censor valid criticism. My reasons for deleting or editing comments do not
include deleting a comment because it disagrees with what is on my website. Reasons for exclusion include:
Frivolous, irrelevant comments.
Comments devoid of logical basis.
Derogatory comments.
Long-winded comments.
Comments with excessive number of different points.
Questions about matters that do not relate to the page they post on. Such posts are not comments.
Comments with a substantial amount of mathematical terms not properly formatted will not be published unless a file (such as doc, tex, pdf) is simultaneously emailed to me, and where the mathematical
terms are correctly formatted.
Reasons for deleting comments of certain users:
Bulk posting of comments in a short space of time, often on several different pages, and which are not simply part of an ongoing discussion. Multiple anonymous user names for one person.
Users, who, when shown their point is wrong, immediately claim that they just wrote it incorrectly and rewrite it again - still erroneously, or else attack something else on my site - erroneously.
After the first few instances, further posts are deleted.
Users who make persistent erroneous attacks in a scatter-gun attempt to try to find some error in what I write on this site. After the first few instances, further posts are deleted.
Difficulties in understanding the site content are usually best addressed by contacting me by e-mail.
Based on HashOver Comment System by Jacob Barkdull
The Lighter Side
A mathematician had tired of academia, so he decided to join the fire department. During his training, the chief asked him,
“If you were walking down an alley and came across a rubbish bin that was on fire, what would you do?”
“Why, I’d get a hose and extinguish it, of course!”
“Correct,” said the chief. “And what would you do if you were walking down an alley and came across a rubbish bin that was not on fire?”
“I suppose I’d set the bin on fire.”
“You’d do what?” shouts the chief.
“I’d set it on fire,” says the mathematician, “thereby reducing it to a previously solved problem.”
James R Meyer
Recently added pages
A new section on set theory
How to setup Dark mode for a web-site
I have set up this website to allow a user to switch to a dark mode, but which also allows the user to revert back to the browser/system setting. The details of how to implement this on a website are
given at How to setup Dark mode on a web-site.
Decreasing intervals, limits, infinity and Lebesgue measure
The page Understanding sets of decreasing intervals explains why certain definitions of sets of decreasing intervals are inherently contradictory unless limiting conditions are included, and the page
Understanding Limits and Infinity explains how the correct application of limiting conditions can eliminate such contradictions. The paper PDF On Smith-Volterra-Cantor sets and their measure has
additional material which gives a more formal version.
Easy Footnotes
How to set up a system for easy insertion or changing of footnotes in a webpage, see Easy Footnotes for Web Pages.
New section added to paper on Gödel’s flawed paper
After comments that my PDF paper on the flaw in Gödel’s incompleteness proof is too long, I have added a new section which gives a brief summary of the flaw, while the remainder of the paper details
the confusion of levels of language. The paper can be seen at The Fundamental Flaw in Gödel’s Proof of his Incompleteness Theorem.
Cantor’s Grundlagen and associated papers
To understand the philosophy of set theory as it is today requires a knowledge of the history of the subject. One of the most influential works in this respect was Georg Cantor’s set of six papers
published between 1879 and 1884 under the overall title of Über unendliche lineare Punktmannig-faltigkeiten, which were published between 1879 and 1884. I now have English translations of Part 1,
Part 2, Part 3 and the major part, Part 5 (Grundlagen). There is also a new English translation of Cantor’s “A Contribution to the Theory of Sets”.
A brief history of meta-mathematics
A look at how the field of meta-mathematics developed from its early days, and how certain illogical and untenable assumptions have been made that fly in the face of the mathematical requirement for
strict rigor.
For pages with a comment section, you can leave a comment.
Printer Friendly
The pages of this website are set up to give a good printed copy without extraneous material.
Easy Footnotes
How to set up a system for easy insertion or changing of footnotes in a webpage, see Easy Footnotes for Web Pages.
New section added to paper on Gödel’s flawed paper
After comments that my PDF paper on the flaw in Gödel’s incompleteness proof is too long, I have added a new section which gives a brief summary of the flaw, while the remainder of the paper details
the confusion of levels of language. The paper can be seen at The Fundamental Flaw in Gödel’s Proof of his Incompleteness Theorem.
Cantor’s Grundlagen and associated papers
To understand the philosophy of set theory as it is today requires a knowledge of the history of the subject. One of the most influential works in this respect was Georg Cantor’s set of six papers
published between 1879 and 1884 under the overall title of Über unendliche lineare Punktmannig-faltigkeiten, which were published between 1879 and 1884. I now have English translations of Part 1,
Part 2, Part 3 and the major part, Part 5 (Grundlagen). There is also a new English translation of Cantor’s “A Contribution to the Theory of Sets”.
A brief history of meta-mathematics
A look at how the field of meta-mathematics developed from its early days, and how certain illogical and untenable assumptions have been made that fly in the face of the mathematical requirement for
strict rigor.
For pages with a comment section, you can leave a comment.
Printer Friendly
The pages of this website are set up to give a good printed copy without extraneous material. | {"url":"https://www.jamesrmeyer.com/","timestamp":"2024-11-02T04:26:12Z","content_type":"text/html","content_length":"68734","record_id":"<urn:uuid:51f8e166-3683-4b48-88fd-a205a737748e>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00107.warc.gz"} |
Next: Radioactivity Up: Nuclear Physics Previous: Nuclear Physics
Atoms in nature generally are electrically neutral, as they have equal numbers of protons in the nucleus and orbiting electrons. However, within the nucleus there are other particles called neutrons,
which are electrically neutral but have about the same mass as protons. There are two numbers used to characterize a nucleus:
Z , the atomic number, which equals the number of protons;
A , the mass number, which equals the number of nucleons (protons and neutrons).
An element X is defined by the atomic number Z , while A denotes the particular isotope of that element. The usual notation for an element X is ^A[Z] X - for example, there are four common isotopes
of Carbon: ^11[6] C , ^12[6] C , ^13[6] C , and ^14[6] C , with ^12[6] C being the most abundant ( > 98% ).
It is convenient in some circumstances to measure masses in terms of the unified mass unit, u, which is defined so that ^12 C has a mass of 12 u exactly; in SI units,
1 u = 1.66 x 10^- 27 kg . (1)
There is also another convenient unit of mass which arises from Einstein's Special Theory of Relativity. We have not covered this in this course, so we simply quote the relevant (well-known)
relation: which associates an energy E to a mass m , with c = 3.0 x 10^8 m/s being the speed of light. Thus, dimensionally, E/c^ 2 is a unit of mass. It is customary to express this unit in terms of
MeV/c ^2, where 1 MeV=10 ^6 eV= 1.6 x 10^- 13 J - note that ``c'' here is considered part of the unit, and one does not substitute the numerical value of 3.0 x 10^8 m/s in it. This will be
illustrated later in some examples. Through this relation one can find the energy equivalent of 1 u:
E = (1.67 x 10^- 27 kg )(3.0 x 10^8 m/s )^2
= 1.50 x 10^- 10 J = 9.39 x 10^8 eV = 939 MeV , (3)
which is then written as
One important illustration of the equivalence of mass and energy of Eq. (29.2) has to do with what is called the binding energy of the nucleus. It is observed that the mass of any nucleus is always
less than the sum of the masses of the individual constituent nucleons which make it up. This ``loss'' of mass which then results when nucleons form a nucleus is attributed to a ``binding energy'',
and is a measure of the strength of the strong force holding the nucleons together.
Next: Radioactivity Up: Nuclear Physics Previous: Nuclear Physics | {"url":"http://theory.uwinnipeg.ca/physics/nucl/node2.html","timestamp":"2024-11-12T21:43:11Z","content_type":"text/html","content_length":"8284","record_id":"<urn:uuid:ee820aec-1d30-4a57-8eed-7dcc18939e56>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00353.warc.gz"} |
seminars - Invitation to crystal bases for quantum symmetric pairs
2023-02-15 (Wed) AM 10:00 ~ 12:00
2023-02-17 (Fri) AM 10:00 ~ 11:00
The theory of crystal bases for quantum symmetric pairs, i.e., $\imath$crystal bases, which is still in progress, is an $\imath$quantum group (also known as ``quantum symmetric pair coideal
subalgebra'') counterpart of the theory of crystal bases.A goal of the theory of $\imath$crystal bases is to provide a way to recover much information about the structures of representations of $\
imath$quantum groups from its crystal limit, just like the theory of crystal bases for quantum groups.In these three hours of lecture, we first review basic theory of canonical bases and crystal
bases for quantum groups, and $\imath$canonical bases for $\imath$quantum groups. Then, we introduce a recent progress on the theory of $\imath$crystal bases of quasi-split locally finite type. As
mentioned above, the theory of $\imath$crystal bases of arbitrary type is not completed yet. Toward a next step, we discuss how the already known theory of $\imath$crystal bases could be generalized
to locally finite types. It would be a great pleasure for the speaker if the audience would be interested in and develop this ongoing project.
*This seminar will be held on Zoom. | {"url":"https://www.math.snu.ac.kr/board/index.php?mid=seminars&sort_index=speaker&order_type=desc&page=81&l=ko&document_srl=1033000","timestamp":"2024-11-10T17:49:09Z","content_type":"text/html","content_length":"48938","record_id":"<urn:uuid:a6e4ae44-d6c5-4924-bf19-a7e70bd23fa2>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00601.warc.gz"} |
Older Versions
Older Versions#
Version 0.12.1#
October 8, 2012
The 0.12.1 release is a bug-fix release with no additional features, but is instead a set of bug fixes
Version 0.12#
September 4, 2012
• Various speed improvements of the decision trees module, by Gilles Louppe.
• GradientBoostingRegressor and GradientBoostingClassifier now support feature subsampling via the max_features argument, by Peter Prettenhofer.
• Added Huber and Quantile loss functions to GradientBoostingRegressor, by Peter Prettenhofer.
• Decision trees and forests of randomized trees now support multi-output classification and regression problems, by Gilles Louppe.
• Added LabelEncoder, a simple utility class to normalize labels or transform non-numerical labels, by Mathieu Blondel.
• Added the epsilon-insensitive loss and the ability to make probabilistic predictions with the modified huber loss in Stochastic Gradient Descent, by Mathieu Blondel.
• Added Multi-dimensional Scaling (MDS), by Nelle Varoquaux.
• SVMlight file format loader now detects compressed (gzip/bzip2) files and decompresses them on the fly, by Lars Buitinck.
• SVMlight file format serializer now preserves double precision floating point values, by Olivier Grisel.
• A common testing framework for all estimators was added, by Andreas Müller.
• Understandable error messages for estimators that do not accept sparse input by Gael Varoquaux
• Speedups in hierarchical clustering by Gael Varoquaux. In particular building the tree now supports early stopping. This is useful when the number of clusters is not small compared to the number
of samples.
• Add MultiTaskLasso and MultiTaskElasticNet for joint feature selection, by Alexandre Gramfort.
• Added metrics.auc_score and metrics.average_precision_score convenience functions by Andreas Müller.
• Improved sparse matrix support in the Feature selection module by Andreas Müller.
• New word boundaries-aware character n-gram analyzer for the Text feature extraction module by @kernc.
• Fixed bug in spectral clustering that led to single point clusters by Andreas Müller.
• In CountVectorizer, added an option to ignore infrequent words, min_df by Andreas Müller.
• Add support for multiple targets in some linear models (ElasticNet, Lasso and OrthogonalMatchingPursuit) by Vlad Niculae and Alexandre Gramfort.
• Fixes in decomposition.ProbabilisticPCA score function by Wei Li.
• Fixed feature importance computation in Gradient-boosted trees.
API changes summary#
• The old scikits.learn package has disappeared; all code should import from sklearn instead, which was introduced in 0.9.
• In metrics.roc_curve, the thresholds array is now returned with it’s order reversed, in order to keep it consistent with the order of the returned fpr and tpr.
• In hmm objects, like hmm.GaussianHMM, hmm.MultinomialHMM, etc., all parameters must be passed to the object when initialising it and not through fit. Now fit will only accept the data as an input
• For all SVM classes, a faulty behavior of gamma was fixed. Previously, the default gamma value was only computed the first time fit was called and then stored. It is now recalculated on every
call to fit.
• All Base classes are now abstract meta classes so that they can not be instantiated.
• cluster.ward_tree now also returns the parent array. This is necessary for early-stopping in which case the tree is not completely built.
• In CountVectorizer the parameters min_n and max_n were joined to the parameter n_gram_range to enable grid-searching both at once.
• In CountVectorizer, words that appear only in one document are now ignored by default. To reproduce the previous behavior, set min_df=1.
• Fixed API inconsistency: linear_model.SGDClassifier.predict_proba now returns 2d array when fit on two classes.
• Fixed API inconsistency: discriminant_analysis.QuadraticDiscriminantAnalysis.decision_function and discriminant_analysis.LinearDiscriminantAnalysis.decision_function now return 1d arrays when fit
on two classes.
• Grid of alphas used for fitting LassoCV and ElasticNetCV is now stored in the attribute alphas_ rather than overriding the init parameter alphas.
• Linear models when alpha is estimated by cross-validation store the estimated value in the alpha_ attribute rather than just alpha or best_alpha.
• GradientBoostingClassifier now supports staged_predict_proba, and staged_predict.
• svm.sparse.SVC and other sparse SVM classes are now deprecated. The all classes in the Support Vector Machines module now automatically select the sparse or dense representation base on the
• All clustering algorithms now interpret the array X given to fit as input data, in particular SpectralClustering and AffinityPropagation which previously expected affinity matrices.
• For clustering algorithms that take the desired number of clusters as a parameter, this parameter is now called n_clusters.
• 267 Andreas Müller
• 52 Vlad Niculae
• 44 Nelle Varoquaux
• 30 Alexis Mignon
• 30 Immanuel Bayer
• 16 Subhodeep Moitra
• 13 Yannick Schwartz
• 12 @kernc
• 9 Daniel Duckworth
• 8 John Benediktsson
• 7 Marko Burjek
• 4 Alexandre Abraham
• 3 Florian Hoenig
• 3 flyingimmidev
• 2 Francois Savard
• 2 Hannes Schulz
• 2 Peter Welinder
• 2 Wei Li
• 1 Alex Companioni
• 1 Brandyn A. White
• 1 Bussonnier Matthias
• 1 Charles-Pierre Astolfi
• 1 Dan O’Huiginn
• 1 David Cournapeau
• 1 Keith Goodman
• 1 Ludwig Schwardt
• 1 Olivier Hervieu
• 1 Sergio Medina
• 1 Shiqiao Du
• 1 Tim Sheerman-Chase
• 1 buguen
Version 0.11#
May 7, 2012
• Gradient boosted regression trees (Gradient-boosted trees) for classification and regression by Peter Prettenhofer and Scott White .
• Simple dict-based feature loader with support for categorical variables (DictVectorizer) by Lars Buitinck.
• Added Matthews correlation coefficient (metrics.matthews_corrcoef) and added macro and micro average options to precision_score, metrics.recall_score and f1_score by Satrajit Ghosh.
• Out of Bag Estimates of generalization error for Ensembles: Gradient boosting, random forests, bagging, voting, stacking by Andreas Müller.
• Randomized sparse linear models for feature selection, by Alexandre Gramfort and Gael Varoquaux
• Label Propagation for semi-supervised learning, by Clay Woolam. Note the semi-supervised API is still work in progress, and may change.
• Added BIC/AIC model selection to classical Gaussian mixture models and unified the API with the remainder of scikit-learn, by Bertrand Thirion
• Added sklearn.cross_validation.StratifiedShuffleSplit, which is a sklearn.cross_validation.ShuffleSplit with balanced splits, by Yannick Schwartz.
• NearestCentroid classifier added, along with a shrink_threshold parameter, which implements shrunken centroid classification, by Robert Layton.
Other changes#
• Merged dense and sparse implementations of Stochastic Gradient Descent module and exposed utility extension types for sequential datasets seq_dataset and weight vectors weight_vector by Peter
• Added partial_fit (support for online/minibatch learning) and warm_start to the Stochastic Gradient Descent module by Mathieu Blondel.
• Dense and sparse implementations of Support Vector Machines classes and LogisticRegression merged by Lars Buitinck.
• Regressors can now be used as base estimator in the Multiclass and multioutput algorithms module by Mathieu Blondel.
• Added n_jobs option to metrics.pairwise_distances and metrics.pairwise.pairwise_kernels for parallel computation, by Mathieu Blondel.
• K-means can now be run in parallel, using the n_jobs argument to either K-means or cluster.KMeans, by Robert Layton.
• Improved Cross-validation: evaluating estimator performance and Tuning the hyper-parameters of an estimator documentation and introduced the new cross_validation.train_test_split helper function
by Olivier Grisel
• SVC members coef_ and intercept_ changed sign for consistency with decision_function; for kernel==linear, coef_ was fixed in the one-vs-one case, by Andreas Müller.
• Performance improvements to efficient leave-one-out cross-validated Ridge regression, esp. for the n_samples > n_features case, in RidgeCV, by Reuben Fletcher-Costin.
• Refactoring and simplification of the Text feature extraction API and fixed a bug that caused possible negative IDF, by Olivier Grisel.
• Beam pruning option in _BaseHMM module has been removed since it is difficult to Cythonize. If you are interested in contributing a Cython version, you can use the python version in the git
history as a reference.
• Classes in Nearest Neighbors now support arbitrary Minkowski metric for nearest neighbors searches. The metric can be specified by argument p.
API changes summary#
• covariance.EllipticEnvelop is now deprecated. Please use EllipticEnvelope instead.
• NeighborsClassifier and NeighborsRegressor are gone in the module Nearest Neighbors. Use the classes KNeighborsClassifier, RadiusNeighborsClassifier, KNeighborsRegressor and/or
RadiusNeighborsRegressor instead.
• Sparse classes in the Stochastic Gradient Descent module are now deprecated.
• In mixture.GMM, mixture.DPGMM and mixture.VBGMM, parameters must be passed to an object when initialising it and not through fit. Now fit will only accept the data as an input parameter.
• methods rvs and decode in GMM module are now deprecated. sample and score or predict should be used instead.
• attribute _scores and _pvalues in univariate feature selection objects are now deprecated. scores_ or pvalues_ should be used instead.
• In LogisticRegression, LinearSVC, SVC and NuSVC, the class_weight parameter is now an initialization parameter, not a parameter to fit. This makes grid searches over this parameter possible.
• LFW data is now always shape (n_samples, n_features) to be consistent with the Olivetti faces dataset. Use images and pairs attribute to access the natural images shapes instead.
• In LinearSVC, the meaning of the multi_class parameter changed. Options now are 'ovr' and 'crammer_singer', with 'ovr' being the default. This does not change the default behavior but hopefully
is less confusing.
• Class feature_selection.text.Vectorizer is deprecated and replaced by feature_selection.text.TfidfVectorizer.
• The preprocessor / analyzer nested structure for text feature extraction has been removed. All those features are now directly passed as flat constructor arguments to
feature_selection.text.TfidfVectorizer and feature_selection.text.CountVectorizer, in particular the following parameters are now used:
• analyzer can be 'word' or 'char' to switch the default analysis scheme, or use a specific python callable (as previously).
• tokenizer and preprocessor have been introduced to make it still possible to customize those steps with the new API.
• input explicitly control how to interpret the sequence passed to fit and predict: filenames, file objects or direct (byte or Unicode) strings.
• charset decoding is explicit and strict by default.
• the vocabulary, fitted or not is now stored in the vocabulary_ attribute to be consistent with the project conventions.
• Class feature_selection.text.TfidfVectorizer now derives directly from feature_selection.text.CountVectorizer to make grid search trivial.
• methods rvs in _BaseHMM module are now deprecated. sample should be used instead.
• Beam pruning option in _BaseHMM module is removed since it is difficult to be Cythonized. If you are interested, you can look in the history codes by git.
• The SVMlight format loader now supports files with both zero-based and one-based column indices, since both occur “in the wild”.
• Arguments in class ShuffleSplit are now consistent with StratifiedShuffleSplit. Arguments test_fraction and train_fraction are deprecated and renamed to test_size and train_size and can accept
both float and int.
• Arguments in class Bootstrap are now consistent with StratifiedShuffleSplit. Arguments n_test and n_train are deprecated and renamed to test_size and train_size and can accept both float and int.
• Argument p added to classes in Nearest Neighbors to specify an arbitrary Minkowski metric for nearest neighbors searches.
• 282 Andreas Müller
• 198 Gael Varoquaux
• 129 Olivier Grisel
• 114 Mathieu Blondel
• 103 Clay Woolam
• 28 flyingimmidev
• 26 Shiqiao Du
• 17 David Marek
• 14 Vlad Niculae
• 11 Yannick Schwartz
• 9 fcostin
• 7 Nick Wilson
• 5 Adrien Gaidon
• 5 Nelle Varoquaux
• 5 Emmanuelle Gouillart
• 3 Joonas Sillanpää
• 3 Paolo Losi
• 2 Charles McCarthy
• 2 Roy Hyunjin Han
• 2 Scott White
• 2 ibayer
• 1 Brandyn White
• 1 Carlos Scheidegger
• 1 Claire Revillet
• 1 Conrad Lee
• 1 Jan Hendrik Metzen
• 1 Meng Xinfan
• 1 Shiqiao
• 1 Udi Weinsberg
• 1 Virgile Fritsch
• 1 Xinfan Meng
• 1 Yaroslav Halchenko
• 1 jansoe
• 1 Leon Palafox
Version 0.10#
January 11, 2012
API changes summary#
Here are the code migration instructions when upgrading from scikit-learn version 0.9:
• Some estimators that may overwrite their inputs to save memory previously had overwrite_ parameters; these have been replaced with copy_ parameters with exactly the opposite meaning.
This particularly affects some of the estimators in linear_model. The default behavior is still to copy everything passed in.
• The SVMlight dataset loader load_svmlight_file no longer supports loading two files at once; use load_svmlight_files instead. Also, the (unused) buffer_mb parameter is gone.
• Sparse estimators in the Stochastic Gradient Descent module use dense parameter vector coef_ instead of sparse_coef_. This significantly improves test time performance.
• The Covariance estimation module now has a robust estimator of covariance, the Minimum Covariance Determinant estimator.
• Cluster evaluation metrics in cluster have been refactored but the changes are backwards compatible. They have been moved to the metrics.cluster.supervised, along with
metrics.cluster.unsupervised which contains the Silhouette Coefficient.
• The permutation_test_score function now behaves the same way as cross_val_score (i.e. uses the mean score across the folds.)
• Cross Validation generators now use integer indices (indices=True) by default instead of boolean masks. This make it more intuitive to use with sparse matrix data.
• The functions used for sparse coding, sparse_encode and sparse_encode_parallel have been combined into sparse_encode, and the shapes of the arrays have been transposed for consistency with the
matrix factorization setting, as opposed to the regression setting.
• Fixed an off-by-one error in the SVMlight/LibSVM file format handling; files generated using dump_svmlight_file should be re-generated. (They should continue to work, but accidentally had one
extra column of zeros prepended.)
• BaseDictionaryLearning class replaced by SparseCodingMixin.
• sklearn.utils.extmath.fast_svd has been renamed randomized_svd and the default oversampling is now fixed to 10 additional random vectors instead of doubling the number of components to extract.
The new behavior follows the reference paper.
The following people contributed to scikit-learn since last release:
• 246 Andreas Müller
• 242 Olivier Grisel
• 220 Gilles Louppe
• 183 Brian Holt
• 166 Gael Varoquaux
• 144 Lars Buitinck
• 73 Vlad Niculae
• 60 Robert Layton
• 44 Noel Dawe
• 3 Jan Hendrik Metzen
• 3 Kenneth C. Arnold
• 3 Shiqiao Du
• 3 Tim Sheerman-Chase
• 2 Bala Subrahmanyam Varanasi
• 2 DraXus
• 2 Michael Eickenberg
• 1 Bogdan Trach
• 1 Félix-Antoine Fortin
• 1 Juan Manuel Caicedo Carvajal
• 1 Nelle Varoquaux
• 1 Tiziano Zito
• 1 Xinfan Meng
Version 0.9#
September 21, 2011
scikit-learn 0.9 was released on September 2011, three months after the 0.8 release and includes the new modules Manifold learning, The Dirichlet Process as well as several new algorithms and
documentation improvements.
This release also includes the dictionary-learning work developed by Vlad Niculae as part of the Google Summer of Code program.
API changes summary#
Here are the code migration instructions when upgrading from scikit-learn version 0.8:
• The scikits.learn package was renamed sklearn. There is still a scikits.learn package alias for backward compatibility.
Third-party projects with a dependency on scikit-learn 0.9+ should upgrade their codebase. For instance, under Linux / MacOSX just run (make a backup first!):
find -name "*.py" | xargs sed -i 's/\bscikits.learn\b/sklearn/g'
• Estimators no longer accept model parameters as fit arguments: instead all parameters must be only be passed as constructor arguments or using the now public set_params method inherited from
Some estimators can still accept keyword arguments on the fit but this is restricted to data-dependent values (e.g. a Gram matrix or an affinity matrix that are precomputed from the X data
• The cross_val package has been renamed to cross_validation although there is also a cross_val package alias in place for backward compatibility.
Third-party projects with a dependency on scikit-learn 0.9+ should upgrade their codebase. For instance, under Linux / MacOSX just run (make a backup first!):
find -name "*.py" | xargs sed -i 's/\bcross_val\b/cross_validation/g'
• The score_func argument of the sklearn.cross_validation.cross_val_score function is now expected to accept y_test and y_predicted as only arguments for classification and regression tasks or
X_test for unsupervised estimators.
• gamma parameter for support vector machine algorithms is set to 1 / n_features by default, instead of 1 / n_samples.
• The sklearn.hmm has been marked as orphaned: it will be removed from scikit-learn in version 0.11 unless someone steps up to contribute documentation, examples and fix lurking numerical stability
• sklearn.neighbors has been made into a submodule. The two previously available estimators, NeighborsClassifier and NeighborsRegressor have been marked as deprecated. Their functionality has been
divided among five new classes: NearestNeighbors for unsupervised neighbors searches, KNeighborsClassifier & RadiusNeighborsClassifier for supervised classification problems, and
KNeighborsRegressor & RadiusNeighborsRegressor for supervised regression problems.
• sklearn.ball_tree.BallTree has been moved to sklearn.neighbors.BallTree. Using the former will generate a warning.
• sklearn.linear_model.LARS() and related classes (LassoLARS, LassoLARSCV, etc.) have been renamed to sklearn.linear_model.Lars().
• All distance metrics and kernels in sklearn.metrics.pairwise now have a Y parameter, which by default is None. If not given, the result is the distance (or kernel similarity) between each sample
in Y. If given, the result is the pairwise distance (or kernel similarity) between samples in X to Y.
• sklearn.metrics.pairwise.l1_distance is now called manhattan_distance, and by default returns the pairwise distance. For the component wise distance, set the parameter sum_over_features to False.
Backward compatibility package aliases and other deprecated classes and functions will be removed in version 0.11.
Version 0.8#
May 11, 2011
scikit-learn 0.8 was released on May 2011, one month after the first “international” scikit-learn coding sprint and is marked by the inclusion of important modules: Hierarchical clustering, Cross
decomposition, Non-negative matrix factorization (NMF or NNMF), initial support for Python 3 and by important enhancements and bug fixes.
Several new modules where introduced during this release:
Some other modules benefited from significant improvements or cleanups.
• Initial support for Python 3: builds and imports cleanly, some modules are usable while others have failing tests by Fabian Pedregosa.
• PCA is now usable from the Pipeline object by Olivier Grisel.
• Guide How to optimize for speed by Olivier Grisel.
• Fixes for memory leaks in libsvm bindings, 64-bit safer BallTree by Lars Buitinck.
• bug and style fixing in K-means algorithm by Jan Schlüter.
• Add attribute converged to Gaussian Mixture Models by Vincent Schut.
• Implemented transform, predict_log_proba in LinearDiscriminantAnalysis By Mathieu Blondel.
• Refactoring in the Support Vector Machines module and bug fixes by Fabian Pedregosa, Gael Varoquaux and Amit Aides.
• Refactored SGD module (removed code duplication, better variable naming), added interface for sample weight by Peter Prettenhofer.
• Wrapped BallTree with Cython by Thouis (Ray) Jones.
• Added function svm.l1_min_c by Paolo Losi.
• Typos, doc style, etc. by Yaroslav Halchenko, Gael Varoquaux, Olivier Grisel, Yann Malet, Nicolas Pinto, Lars Buitinck and Fabian Pedregosa.
People that made this release possible preceded by number of commits:
• 159 Olivier Grisel
• 96 Vlad Niculae
• 32 Paolo Losi
• 7 Lars Buitinck
• 6 Vincent Michel
• 4 Thouis (Ray) Jones
• 4 Vincent Schut
• 3 Jan Schlüter
• 2 Julien Miotte
• 2 Yann Malet
• 1 Amit Aides
• 1 Feth Arezki
• 1 Meng Xinfan
Version 0.7#
March 2, 2011
scikit-learn 0.7 was released in March 2011, roughly three months after the 0.6 release. This release is marked by the speed improvements in existing algorithms like k-Nearest Neighbors and K-Means
algorithm and by the inclusion of an efficient algorithm for computing the Ridge Generalized Cross Validation solution. Unlike the preceding release, no new modules where added to this release.
• Performance improvements for Gaussian Mixture Model sampling [Jan Schlüter].
• Implementation of efficient leave-one-out cross-validated Ridge in RidgeCV [Mathieu Blondel]
• Better handling of collinearity and early stopping in linear_model.lars_path [Alexandre Gramfort and Fabian Pedregosa].
• Fixes for liblinear ordering of labels and sign of coefficients [Dan Yamins, Paolo Losi, Mathieu Blondel and Fabian Pedregosa].
• Performance improvements for Nearest Neighbors algorithm in high-dimensional spaces [Fabian Pedregosa].
• Performance improvements for KMeans [Gael Varoquaux and James Bergstra].
• Sanity checks for SVM-based classes [Mathieu Blondel].
• Refactoring of neighbors.NeighborsClassifier and neighbors.kneighbors_graph: added different algorithms for the k-Nearest Neighbor Search and implemented a more stable algorithm for finding
barycenter weights. Also added some developer documentation for this module, see notes_neighbors for more information [Fabian Pedregosa].
• Documentation improvements: Added pca.RandomizedPCA and LogisticRegression to the class reference. Also added references of matrices used for clustering and other fixes [Gael Varoquaux, Fabian
Pedregosa, Mathieu Blondel, Olivier Grisel, Virgile Fritsch , Emmanuelle Gouillart]
• Binded decision_function in classes that make use of liblinear, dense and sparse variants, like LinearSVC or LogisticRegression [Fabian Pedregosa].
• Performance and API improvements to metrics.pairwise.euclidean_distances and to pca.RandomizedPCA [James Bergstra].
• Fix compilation issues under NetBSD [Kamel Ibn Hassen Derouiche]
• Allow input sequences of different lengths in hmm.GaussianHMM [Ron Weiss].
• Fix bug in affinity propagation caused by incorrect indexing [Xinfan Meng]
People that made this release possible preceded by number of commits:
• 14 Dan Yamins
• 2 Satrajit Ghosh
• 2 Vincent Dubourg
• 1 Emmanuelle Gouillart
• 1 Kamel Ibn Hassen Derouiche
• 1 Paolo Losi
• 1 VirgileFritsch
• 1 Xinfan Meng
Version 0.6#
December 21, 2010
scikit-learn 0.6 was released on December 2010. It is marked by the inclusion of several new modules and a general renaming of old ones. It is also marked by the inclusion of new example, including
applications to real-world datasets.
• New stochastic gradient descent module by Peter Prettenhofer. The module comes with complete documentation and examples.
• Improved svm module: memory consumption has been reduced by 50%, heuristic to automatically set class weights, possibility to assign weights to samples (see SVM: Weighted samples for an example).
• New Gaussian Processes module by Vincent Dubourg. This module also has great documentation and some very neat examples. See example_gaussian_process_plot_gp_regression.py or
example_gaussian_process_plot_gp_probabilistic_classification_after_regression.py for a taste of what can be done.
• It is now possible to use liblinear’s Multi-class SVC (option multi_class in LinearSVC)
• New features and performance improvements of text feature extraction.
• Improved sparse matrix support, both in main classes (GridSearchCV) as in modules sklearn.svm.sparse and sklearn.linear_model.sparse.
• Lots of cool new examples and a new section that uses real-world datasets was created. These include: Faces recognition example using eigenfaces and SVMs, Species distribution modeling, Wikipedia
principal eigenvector and others.
• Faster Least Angle Regression algorithm. It is now 2x faster than the R version on worst case and up to 10x times faster on some cases.
• Faster coordinate descent algorithm. In particular, the full path version of lasso (linear_model.lasso_path) is more than 200x times faster than before.
• It is now possible to get probability estimates from a LogisticRegression model.
• module renaming: the glm module has been renamed to linear_model, the gmm module has been included into the more general mixture model and the sgd module has been included in linear_model.
• Lots of bug fixes and documentation improvements.
People that made this release possible preceded by number of commits:
Version 0.5#
October 11, 2010
New classes#
• Support for sparse matrices in some classifiers of modules svm and linear_model (see svm.sparse.SVC, svm.sparse.SVR, svm.sparse.LinearSVC, linear_model.sparse.Lasso,
• New Pipeline object to compose different estimators.
• Recursive Feature Elimination routines in module Feature selection.
• Addition of various classes capable of cross validation in the linear_model module (LassoCV, ElasticNetCV, etc.).
• New, more efficient LARS algorithm implementation. The Lasso variant of the algorithm is also implemented. See lars_path, Lars and LassoLars.
• New Hidden Markov Models module (see classes hmm.GaussianHMM, hmm.MultinomialHMM, hmm.GMMHMM)
• New module feature_extraction (see class reference)
• New FastICA algorithm in module sklearn.fastica
• API changes: adhere variable names to PEP-8, give more meaningful names.
• Fixes for svm module to run on a shared memory context (multiprocessing).
• It is again possible to generate latex (and thus PDF) from the sphinx docs.
External dependencies#
• Joblib is now a dependency of this package, although it is shipped with (sklearn.externals.joblib).
Removed modules#
• Module ann (Artificial Neural Networks) has been removed from the distribution. Users wanting this sort of algorithms should take a look into pybrain.
• New sphinx theme for the web page.
Version 0.4#
August 26, 2010
Major changes in this release include:
• Coordinate Descent algorithm (Lasso, ElasticNet) refactoring & speed improvements (roughly 100x times faster).
• Coordinate Descent Refactoring (and bug fixing) for consistency with R’s package GLMNET.
• New metrics module.
• New GMM module contributed by Ron Weiss.
• Implementation of the LARS algorithm (without Lasso variant for now).
• feature_selection module redesign.
• Migration to GIT as version control system.
• Removal of obsolete attrselect module.
• Rename of private compiled extensions (added underscore).
• Removal of legacy unmaintained code.
• Documentation improvements (both docstring and rst).
• Improvement of the build system to (optionally) link with MKL. Also, provide a lite BLAS implementation in case no system-wide BLAS is found.
• Lots of new examples.
• Many, many bug fixes …
The committer list for this release is the following (preceded by number of commits):
• 143 Fabian Pedregosa
• 35 Alexandre Gramfort
• 34 Olivier Grisel
• 11 Gael Varoquaux
• 5 Yaroslav Halchenko
• 2 Vincent Michel
• 1 Chris Filo Gorgolewski
Earlier versions#
Earlier versions included contributions by Fred Mailhot, David Cooke, David Huard, Dave Morrill, Ed Schofield, Travis Oliphant, Pearu Peterson. | {"url":"https://scikit-learn.org/dev/whats_new/older_versions.html","timestamp":"2024-11-04T12:05:34Z","content_type":"text/html","content_length":"148076","record_id":"<urn:uuid:a8b5e4c3-981e-4917-9bca-be7b15b6c8df>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00577.warc.gz"} |
d Re
Extended Renovation Theory and Limit Theorems for Stochastic Ordered Graphs,
2003, v.9, Issue 3, 413-468
We extend Borovkov's renovation theory to obtain criteria for coupling-convergence of stochastic processes that do not necessarily obey stochastic recursions. The results are applied to an ``infinite
bin model'', a particular system that is an abstraction of a stochastic ordered graph, i.e., a graph on the integers that has $(i,j)$, $i < j$, as an edge, with probability $p$, independently from
edge to edge. A question of interest is an estimate of the length $L_n$ of a longest path between two vertices at distance $n$. We give sharp bounds on $C=\lim_{n\to\infty} (L_n/n)$. This is done by
first constructing the unique stationary version of the infinite bin model, using extended renovation theory. We also prove a functional law of large numbers and a functional central limit theorem
for the infinite bin model. Finally, we discuss perfect simulation, in connection to extended renovation theory, and as a means for simulating the particular stochastic models considered in this
Keywords: stationary and ergodic processes,renovation theory,functional limit theorems,weak convergence,coupling,perfect simulation
Please log in or register to leave a comment
There are no comments yet | {"url":"https://math-mprf.org/journal/articles/id979/","timestamp":"2024-11-12T07:16:19Z","content_type":"text/html","content_length":"14410","record_id":"<urn:uuid:2c66af87-17f2-4ca9-a1e6-e8eeaa019980>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00576.warc.gz"} |
Correcting Huberman's Statistics
Probability 101
If you are active on the internet and interested in statistics, then the probability of seeing Andrew Huberman being wrong about statistics is 0.999.
In case you are an outlier, here is the summary:
In his podcast on Fertility Huberman said this:
If you have a 20% chance of pregnancy in any given month, the chance of being pregnant after 6 months is 120%.
That’s clearly wrong. Here is why:
The correct formula to calculate such examples is this:
The formula is the cumulative distribution function (CDF) of a geometric distribution.
But that’s too complicated, so let’s bring it down to grandma’s level.
From the statement above we know for a fact that the probability of pregnancy in a given month is 20%:
If we use the complement rule we can also calculate the probability of not being pregnant.
The complement of an event A is the event that A does not occur. The sum of the probabilities of an event and its complement is always 1.
But that is only for 1 month. In the original example, he mentioned 6 months. Using the multiplication rule for independent events, we can do the math.
Two events are independent if the occurrence of one event does not affect the occurrence of the other.
If events A and B are independent, then:
We have 6 independent events here, so we raise the probability of no pregnancy in a single trial to the power of 6:
If we combine the above we get that:
So, the probability of being pregnant at least once in 6 months is approximately 73.79%.
Now let’s simulate the situation using numpy:
Credit for this code goes to Craig Chirinda:
The output of this code is:
Estimated probability of at least one pregnancy in 6 months: 0.73713 Monthly cumulative probabilities: {1: 0.2, 2: 0.36, 3: 0.49, 4: 0.59, 5: 0.67, 6: 0.74}
The correction
After the reactions online, Huberman released a statement accepting his mistakes.
Please note that during this calculation we don’t consider any biological or other factors. It is purely based on statistics!
Thanks for reading Data Ground Up! Subscribe for free to receive new posts and support my work. | {"url":"https://datagroundup.com/p/correcting-hubermans-statistics?open=false#%C2%A7thats-clearly-wrong-here-is-why","timestamp":"2024-11-08T12:24:45Z","content_type":"text/html","content_length":"191308","record_id":"<urn:uuid:1957b698-66d7-45d7-bc09-d35420679de2>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00684.warc.gz"} |
Curve Stablecoin - Deep dive
March 18, 2024
Curve Stablecoin - Deep dive
Statemind recently finished an audit of Curve stablecoin and decided that a protocol this intricate and complex deserves another good article describing its inner workings. The article is based on
the original work by Paco with added insights from our auditing team.
Curve Stablecoin Interpretation
Curve Stablecoin is a new generation stablecoin protocol designed by the Curve team. Compared to traditional stablecoin protocols or lending protocols such as MakerDAO, Liquity, Compound, etc., the
main improvement of Curve Stablecoin lies in its built-in AMM (Automated Market Maker), which more efficiently resolves liquidation issues, thereby greatly reducing the risk of bad debt. At the same
time, this AMM can help smooth the selling of assets, thereby reducing the impact of market volatility.
Traditional lending/stablecoin protocols
In the illustration, user Alice collateralizes 10 ETH at an ETH price of $1200 and mints stablecoins. Let's assume Alice's liquidation line is at $900. When the price reaches $900, Bob, as a
liquidator, buys 5 ETH (liquidating half of the assets) from Alice's account for $810 (a 90% discount) and helps Alice repay a debt of $4050. After the liquidation, Alice has 5 ETH left in her
collateral, her debt is reduced by $4050, and her margin rate returns to a healthy level. If the price continues to fall to $700, Alice will face a second liquidation. In this liquidation, Bob buys
2.5 ETH from Alice's account for $630, helping Alice repay a debt of $1260. If the price continues to fall, Alice will continue to be liquidated until all her assets are liquidated.
This liquidation process has the following problems:
1. When the user's margin rate is insufficient, the protocol needs to liquidate a portion of the user's assets. If the user has too many assets, this means a large number of assets will need to be
liquidated at once.
2. Liquidators often use flash loans to improve capital utilization. After the liquidation, they need to immediately sell the liquidated assets to repay the flash loan, which can create significant
selling pressure in the spot market. When there's a large position of assets awaiting liquidation, the spot market prices will likely plummet.
3. If asset prices continue to fall and liquidation cannot be completed in time, the protocol faces the risk of bad debt.
4. To ensure liquidators are sufficiently incentivized, users' assets are sold at a low price, leading to significant losses for the users with each liquidation.
5. Once liquidation occurs, it results in irreversible losses for the user. Even if the price of the collateral assets increases after liquidation, the losses incurred cannot be recovered.
Regarding the protocol, bad debt risk mentioned above, there's a real-life example: How AAVE’s $1.6 Million Bad Debt Was Created.
A major player, Avraham Eisenberg (aka Avi, the attacker of the Mango protocol), collateralized a large number of assets in AAVE to borrow CRV and manipulated the price of CRV by selling the tokens.
However, due to market conditions, retail investors started actively buying CRV and driving up the price, resulting in the liquidation of Avi's position in AAVE. Avi's position was so large that the
entire liquidation process took several tens of minutes. Due to the insufficiently timely liquidation, AAVE eventually incurred a bad debt of $1.6 million.
In this example, AAVE, as a lending platform, failed to properly control risks, leading to the creation of bad debt. Furthermore, regardless of whether CRV's price rose or fell, and whether the
person being liquidated was Michael or Avi, given their large positions, it was difficult for AAVE to avoid the outcome of bad debt.
Curve Stablecoin
Curve Stablecoin was created to address the aforementioned drawbacks. In terms of liquidation, Curve Stablecoin has the following improvements:
• It comes with a specially designed AMM for liquidation, reducing reliance on external liquidity.
• It shifts from phase-based liquidation to gradual liquidation, making the liquidation process smoother and reducing the losses for users being liquidated.
• The liquidation process is reversible. When the price of a user's assets increases, the AMM will help the user repurchase the liquidated assets, further reducing the user's losses.
The entire protocol is composed of the following components (the following image is excerpted from the whitepaper):
• LLAMMA (Lending-Liquidating AMM Algorithm) is a specially designed AMM for liquidation. The name is a play on words for 'llama' (the mascot of the Curve community).
• Controller is responsible for user interaction logic and managing user liquidity in LLAMMA.
• Monetary Policy is a contract used to dynamically adjust the interest rate of the stablecoin.
• PegKeeper is a set of contracts designed to help stabilize the price of crvUSD.
• Stable Pool is the pool for crvUSD in Curve V1.
• Arbitrageurs help in the liquidation of user assets by arbitraging against LLAMMA.
Among the many components of Curve Stablecoin, the most core component is LLAMMA, which is an AMM specifically designed for liquidation.
Suppose crvUSD uses ETH as collateral; then, LLAMMA would be a crvUSD-ETH AMM.
LLAMMA utilizes a design similar to Uniswap V3. For example, like Uniswap V3, LLAMMA uses ranges (called bands in LLAMMA) to segment liquidity, allowing users to provide liquidity to a specific range
within the AMM.
However, a significant difference between LLAMMA and Uniswap V3 is the pattern of how the balances of the two tokens in the AMM pool change with price, as exemplified by ETH/USD:
In Uniswap V3 (as depicted in the upper half of the dashed line in the diagram), users choose to provide liquidity within a range from Lower to Upper Price. This range is referred to as a range in
Uniswap. When the ETH price is greater than or equal to the Upper Price, all of the user's liquidity will be in USD. As the price falls, the USD in the user's liquidity is converted into ETH. When
the ETH price is less than or equal to the Lower Price, all of the user's liquidity will be in ETH.
In LLAMMA (as shown in the lower half of the dashed line in the diagram), users collateralize ETH to borrow crvUSD. Their ETH is then added to a specific range (band) in LLAMMA to provide liquidity,
where the upper and lower bounds of this band are the start/end prices of liquidation. Initially, the price will be above the Liquidation begin price of the band, and all liquidity in the band will
be in ETH. As the price falls, ETH in the band starts to be converted into crvUSD. When the price falls to the lower bound price (external price) of the band, all of the user's assets will be
converted into crvUSD.
It can be observed that in LLAMMA, the change in the token balance within a band is exactly the opposite of Uniswap. This achieves the purpose of liquidating user assets: when the price is high, the
user's ETH assets remain untouched within the liquidity. As the price falls to the upper limit of the user's liquidity range, ETH begins to be converted into crvUSD. When the price falls to the lower
limit of the range, all of the user's ETH is converted into crvUSD. Assuming the conversion yields more crvUSD than the user's debt, the protocol will not face bad debt risk, and the entire process
does not require the involvement of liquidators from traditional lending protocols.
The advantages of this approach are:
• The AMM itself provides liquidity, reducing reliance on external liquidity (though not completely independent).
• Traditional lending protocol liquidators are not needed (but forced liquidation may still occur in extreme situations). The entire liquidation process is carried out with the rebalance of the
AMM's internal balance as prices fall.
• Liquidation is continuous, not phased. Thus, each time a user's assets are sold, they are sold at a level closer to the external market price, minimizing the loss for the user.
• If prices rebound, crvUSD will be converted back into ETH, and the user's assets will be repurchased. This avoids permanent losses for the user in a volatile market (although even if prices
return to their original levels, it's difficult to achieve completely lossless transactions, as LLAMMA always sells assets in the Pool at a discount).
LLAMMA Design
In the diagram, the horizontal axis ${p}_{o}p_o$ represents the external price of ETH, while the vertical axis represents the internal price within the AMM. The blue line is a straight line with a
slope of 1, indicating that the internal AMM price follows the changes in the external price. ${p}_{cu}p_\left\{cu\right\}$ and ${p}_{cd}p_\left\{cd\right\}$ represent the upper and lower bound
prices of a particular band within the AMM, while ${p}_{↑}p_↑$ and ${p}_{↓}p_↓$ respectively represent two selected reference boundary prices in the external price coordinates.
We can find that when the external price ${p}_{o}p_o$ increases, both ${p}_{cu}p_\left\{cu\right\}$ and ${p}_{cd}p_\left\{cd\right\}$ as well as the internal AMM price, will increase at a faster rate
(forming a convex curve). Conversely, when the external price ${p}_{o}p_o$ decreases, ${p}_{cu}p_\left\{cu\right\}$ and ${p}_{cd}p_\left\{cd\right\}$ along with the internal AMM price, will decrease
at a faster rate.
Furthermore, when ${p}_{o}={p}_{↑}p_o=p_↑$, ${p}_{cd}p_\left\{cd\right\}$ precisely satisfies ${p}_{cd}={p}_{o}={p}_{↑}p_\left\{cd\right\}=p_\left\{o\right\}=p_↑$, and when ${p}_{o}={p}_{↓}p_o=p_↓$，
${p}_{cu}p_\left\{cu\right\}$ precisely satisfies ${p}_{cu}={p}_{o}={p}_{↓}p_\left\{cu\right\}=p_o=p_↓$.
If we can design an AMM that dynamically changes its range's upper and lower bound prices with the external price, then in this AMM:
• When the external price ${p}_{o}>={p}_{↑}p_o >= p_↑$, the lower bound price of the band ${p}_{cd}>={p}_{o}p_\left\{cd\right\} >= p_o$. This corresponds to the scenario in Uniswap V3 where the
range is out of the money, and the external price is less than the range's lower bound. Just like in Uniswap V3, this range will be entirely ETH.
• When the external price ${p}_{o}<={p}_{↓}p_o <= p_↓$, the upper bound price of the band ${p}_{cu}<={p}_{o}p_\left\{cu\right\} <= p_o$. This corresponds to the scenario in Uniswap V3 where the
range is out of the money, and the external price is higher than the range's upper bound. Thus, just like in Uniswap V3, this range will be entirely crvUSD.
This way, the AMM's range can have an internal balance change that is exactly the opposite of Uniswap V3 when ${p}_{o}p_o$ fluctuates within the range of $\left[{p}_{↑}-{p}_{↓}\right]\left[p_↑ - p_↓\
right]$. Assuming the initial state is ${p}_{o}>{p}_{↑}p_o > p_↑$, as ${p}_{o}p_o$ decreases to ${p}_{↑}p_↑$, the ETH held by the user in the band begins to be converted into crvUSD, until ${p}_{o}=
{p}_{↓}p_o = p_↓$, at which point all of the user's assets will have been converted into crvUSD.
LLAMMA Implementation
As previously mentioned, LLAMMA needs to know the external asset price ${p}_{o}p_o$. In its actual implementation, LLAMMA uses the Oracle price from Curve V2 and applies EMA (exponential moving
average) processing to it. LLAMMA predefines a continuous series of prices on the external price coordinate, which forms a geometric sequence. Users can select any two prices as the upper and lower
reference prices for their liquidity (${p}_{↓}p_↓$ and ${p}_{↑}p_↑$) Therefore, we have:
$\frac{{p}_{↓}}{{p}_{↑}}=\frac{A-1}{A}\frac\left\{p_↓\right\}\left\{p_↑\right\} = \frac\left\{A-1\right\}\left\{A\right\}$
Where $AA$ is a constant greater than 1, set when the LLAMMA pool is created. The larger the value of $AA$, the smaller the ratio between the boundary prices on the external price coordinate,
resulting in a denser distribution. Currently, the value of $AA$ most of the markets is 100. This design is similar to the tick design in Uniswap, but it is important to note that here, ${p}_{↓}p_↓$
and ${p}_{↑}p_↑$ refer to the external prices, not the internal AMM prices. In its specific design, LLAMMA also satisfies an equation similar to Uniswap V3:
$I=\left(x+f\right)\left(y+g\right)I = \left(x + f\right)\left(y + g\right)$
Where $xx$ represents crvUSD and $yy$ represents ETH. As with Uniswap V3, the price in the AMM (ETH price) can be represented as:
$p=\frac{x+f}{y+g}p = \frac\left\{x + f\right\}\left\{y + g\right\}$
Unlike Uniswap, in the equation above, both $ff$ and $gg$ are dynamic variables and are related to the external price ${p}_{o}p_o$ . The functions for $ff$ and $gg$ are as follows:
$f=\frac{{p}_{o}^{2}}{{p}_{↑}}A{y}_{0}\text{}g=\frac{{p}_{↑}}{{p}_{o}}\left(A-1\right){y}_{0}f = \frac\left\{p_o^2\right\}\left\{p_↑\right\}Ay_0\ \ g=\frac\left\{p_↑\right\}\left\{p_o\right\}\left
In the formulas, ${y}_{0}y_0$ represents the amount of ETH in the interval when ${p}_{o}={p}_{↓}p_o = p_↓$. In this state, the AMM's token balance satisfies: $y={y}_{0}y = y_0$, $x=0x = 0$. Here we
can initially assume ${y}_{0}y_0$ is a constant (actually, ${y}_{0}y_0$ is not a constant, but we can ignore its change for now), and in the formula, ${p}_{↑}p_↑$ and $AA$ are also constants. Michael
did not provide the design process for these two functions, but we can see the purpose of such a design:
As the external price ${p}_{o}p_o$ increases, $ff$ grows exponentially and $gg$ decreases. Conversely, when ${p}_{o}p_o$ decreases, $ff$ decreases exponentially, and $gg$ increases.
Combining this with the AMM price formula $p=\frac{f+x}{g+y}p = \frac\left\{f+x\right\}\left\{g+y\right\}$, we can see that as the external price ${p}_{o}p_o$ rises, the internal price $pp$ within
LLAMMA also rises, and the rate of increase is much higher than that of the external price increase (the denominator decreases while the numerator increases exponentially). Conversely, when the
external price ${p}_{o}p_o$ falls, the internal price $pp$ within LLAMMA drops more rapidly.
Thus, LLAMMA achieves its first characteristic: even without any trades, the internal price in LLAMMA automatically changes with the external price, and it changes at a faster rate.
Through this method of automatically adjusting the AMM price, the token balance within LLAMMA exhibits the opposite behavior of Uniswap V3 in response to price changes. For example, when the external
price of ETH falls:
• In Uniswap, the internal price remains unchanged, and arbitrageurs will sell ETH in Uniswap at a relatively higher price, therefore the amount of ETH in the Uniswap Pool increases while USD
• In LLAMMA, the internal price will fall more than the external price, arbitrageurs will buy ETH in LLAMMA at a relatively lower price, therefore the amount of ETH in the LLAMMA Pool decreases
while USD increases.
It can be seen that when the price falls, LLAMMA attracts arbitrageurs by creating a price difference. The act of arbitrageurs buying ETH from LLAMMA is essentially liquidating the user's assets, and
the profit of the arbitrageurs is the buying price difference.
Similarly, when the price falls to the liquidation range and then begins to rebound, LLAMMA also raises the internal price through the price difference, allowing arbitrageurs to sell ETH back into
the LLAMMA pool, and helping users repurchase their assets.
However, LLAMMA may not necessarily help users buy back all of their original collateralized assets. This is due to the AMM Path Dependence problem caused by LLAMMA's proactive change in AMM price
(which can be referred to in this Tweet). Simply put, as the external price moves, LLAMMA is continuously giving profits to arbitrageurs (selling low and buying high), which will cause the loss of
user assets, even if prices return, the LLAMMA state is difficult to restore.
It is also worth noting that due to the exponential relationship between LLAMMA's internal price changes and external prices, if the intervention by arbitrageurs is not timely enough, or the external
price changes are too large, it will cause a significant price difference between LLAMMA and the external market, increasing the user's losses. To avoid large fluctuations in the external price,
Curve uses an EMA to process the external Oracle price before providing it to LLAMMA. To learn more about Curve oracle features see the Oracle section.
Swap Calculation
When ${p}_{o}={p}_{↑}p_o = p_↑$, the entire band consists of ETH, and the amount of ETH is ${y}_{0}y_0$, the token balance state within the AMM is $y={y}_{0}y = y_0$, $x=0x = 0$. Substituting $ff$
and $gg$ into the identity, we get:
$I={p}_{o}{A}^{2}{y}_{0}^{2}I = p_o A^2 y_0^2$
Extending this identity to any price ${p}_{o}p_o$ within the range of ${p}_{↑}\sim {p}_{↓}p_↑ \sim p_↓$
$\left(\frac{{p}_{o}^{2}}{{p}_{↑}}A{y}_{0}+x\right)\left(\frac{{p}_{↑}}{{p}_{o}}\left(A-1\right){y}_{0}+y\right)={p}_{o}{A}^{2}{y}_{0}^{2}\left\left(\frac\left\{p_o^2\right\}\left\{p_↑\right\} A y_0
+ x\right\right)\left\left(\frac\left\{p_↑\right\}\left\{p_o\right\} \left(A - 1\right)y_0 + y\right\right) = p_o A^2 y_0^2$
As previously mentioned, ${y}_{0}y_0$ is not a constant; in fact, it is also a function of ${p}_{o}p_o$.
The above formula can be written as a quadratic equation in terms of ${y}_{0}y_0$:
${p}_{o}A{y}_{0}^{2}-{y}_{0}\left(\frac{{p}_{↑}}{{p}_{o}}\left(A-1\right)x+\frac{{p}_{o}^{2}}{{p}_{↑}^{2}}Ay\right)-xy=0p_o A y_0^2 - y_0 \left\left(\frac\left\{p_↑\right\}\left\{p_o\right\} \left(A
- 1\right)x + \frac\left\{p_o^2\right\}\left\{p_↑^2\right\} Ay\right\right) - xy = 0$
At any moment, we know the balances of $x,yx, y$ in the AMM, the external price ${p}_{o}p_o$ and the reference upper limit price ${p}_{↑}p_↑$ of the band, then we can solve for the unknown ${y}_{0}
y_0$ in this equation.
In the Curve contract, the code for solving ${y}_{0}y_0$ corresponds to the AMM.get_y0() function:
After calculating ${y}_{0}y_0$, the aforementioned formula can be used to calculate the output amount for a trader swapping in the AMM. In the Curve contract, the code for calculating the swap output
amount corresponds to the AMM.calc_swap_out() function. This is a very lengthy function, so for brevity, the specific code implementation is not shown here.
A simple description of the calculation process for an $x\to yx \rightarrow y$ swap:
1. Find the band closest to the current price that has liquidity.
2. Read the values of $xx$ and $yy$ within the band, and calculate ${y}_{0}y_0$.
3. Calculate the needed $\mathrm{\Delta }x\Delta x$ when $y=0y = 0$.
4. If $\mathrm{\Delta }x\Delta x$ is less than or equal to the ${x}_{in}x_\left\{in\right\}$ - swap input amount, then the user swaps in this band and gets $yy$. Then, start from step 1 and enter
the next band for another swap.
5. If $\mathrm{\Delta }x\Delta x$ is greater than ${x}_{in}x_\left\{in\right\}$, the swap input amount, then set $x=x+{x}_{in}x = x + x_\left\{in\right\}$, calculate $\mathrm{\Delta }y\Delta y$, and
end the swap.
The amount of $yy$ obtained by the user is the sum of the results calculated in all the bands passed through during this swap. If the swap is $y\to xy \rightarrow x$, the calculation process is
largely the same.
Note that in this process, the swap calculations within a band treat ${y}_{0}y_0$ as a fixed value (using the state before the swap). This makes the engineering implementation easier and the error
will not be too large, provided the distance between bands is not too wide.
Band range price
For a given band, ${p}_{cu}p_\left\{cu\right\}$ and ${p}_{cd}p_\left\{cd\right\}$ are the upper and lower limit prices of the band within the AMM, respectively. When the internal AMM price reaches
the upper limit price ${p}_{cu}p_\left\{cu\right\}$, similar to Uniswap V3, the internal $y=0y = 0$ within the band. When the AMM price reaches the lower limit price ${p}_{cd}p_\left\{cd\right\}$,
the internal $x=0x = 0$ within the band. By substituting $x=0x = 0$ into LLAMMA's invariant, we can solve for $yy$, and at this point, the AMM's price is the lower limit price ${p}_{cd}p_\left\{cd\
right\}$. Similarly, the AMM price at $y=0y = 0$ represents the band's upper limit price ${p}_{cu}p_\left\{cu\right\}$. Simplifying these two price formulas, we get:
${p}_{cd}=\frac{{p}_{o}^{3}}{{p}_{↑}^{2}},\phantom{\rule{1em}{0ex}}{p}_{cu}=\frac{{p}_{o}^{3}}{{p}_{↓}^{2}}p_\left\{cd\right\} = \frac\left\{p_o^3\right\}\left\{p_↑^2\right\}, \quad p_\left\{cu\right
\} = \frac\left\{p_o^3\right\}\left\{p_↓^2\right\}$
For a band, ${p}_{↓}p_↓$ and ${p}_{↑}p_↑$ are fixed values, so the ${p}_{cd}p_\left\{cd\right\}$ and ${p}_{cu}p_\left\{cu\right\}$ within the band depend only on the external price ${p}_{o}p_o$.
We can identify the second characteristic of LLAMMA: the upper and lower limit prices of a band within LLAMMA also automatically change with the external price, and the rate of change is faster.
Let's illustrate these two characteristics of LLAMMA with a diagram (image captured from desmos):
In the diagram, the horizontal axis represents the external price, and the vertical axis represents the internal price of the AMM. The black solid line represents the reference external upper limit
price for a band (${p}_{↓}p_↓$ and ${p}_{↑}p_↑$), the green solid line represents the upper and lower limit prices of this band (${p}_{cu}p_\left\{cu\right\}$ and ${p}_{cd}p_\left\{cd\right\}$), the
blue dashed line is a dashed line with a slope of 1, indicating that the AMM price $pp$ is equal to the external price ${p}_{o}p_o$, and the red solid line represents the internal price of the AMM.
We can observe that as the external price increases, both the upper and lower limit prices of the band and the internal price of the AMM increase at a faster rate, and the internal price of the AMM
is always within the upper and lower limit prices of the band.
Relationship Between Bands
When creating an LLAMMA pool, a starting reference price, ${p}_{\text{base}}p_\left\{\text\left\{base\right\}\right\}$, needs to be set. Subsequently, all ${p}_{↑}p_↑$ and ${p}_{↓}p_↓$ can be
calculated, depending on the band number - $nn$:
${p}_{↑}\left(n\right)={\left(\frac{A-1}{A}\right)}^{n}{p}_{\text{base}},\phantom{\rule{1em}{0ex}}{p}_{↓}\left(n\right)={\left(\frac{A-1}{A}\right)}^{n+1}{p}_{\text{base}}p_↑\left(n\right) = \left\
left(\frac\left\{A-1\right\}\left\{A\right\}\right\right)^n p_\left\{\text\left\{base\right\}\right\}, \quad p_↓\left(n\right) = \left\left(\frac\left\{A-1\right\}\left\{A\right\}\right\right)^\left\
{n+1\right\} p_\left\{\text\left\{base\right\}\right\}$
Since ${p}_{↑}p_↑$ and ${p}_{↓}p_↓$ are fixed values, once the value of ${p}_{o}p_o$ is determined, the upper and lower limit prices ${p}_{cu}p_\left\{cu\right\}$ and ${p}_{cd}p_\left\{cd\right\}$ in
each band can be known.
This diagram shows the relationship between three bands (image captured from desmos):
In the diagram, the horizontal axis represents the external price, while the vertical axis represents the internal price. The blue dashed line is a straight line with a slope of 1. The diagram
includes four predefined reference external upper limit prices, which form three bands. In each band, the upper and lower limit prices within the AMM are represented by green, orange, and purple
curves, respectively. The solid lines indicate that the band is expected to be in an active state, meaning it can be used for swaps (liquidation), while the dashed lines suggest that the band is
likely to be out of the money, either not yet started for swaps or already completely swapped.
Upon closer examination, you can see that the blue dashed line intersects each band at the points where ${p}_{cu}={p}_{↓}p_\left\{cu\right\} = p_↓$ and ${p}_{cd}={p}_{↑}p_\left\{cd\right\} = p_↑$.
This ensures that internal price will always be within the respected price range (between ${p}_{cd}p_\left\{cd\right\}$ and ${p}_{cu}p_\left\{cu\right\}$ curves).
Moreover, ${p}_{cd}p_\left\{cd\right\}$ of the previous band will coincide with ${p}_{cu}p_\left\{cu\right\}$ of the next band, ensuring that there are no gaps between the bands and that the
liquidity within the bands is continuous across the internal price range of the AMM.
The following formulas can represent the aforementioned relationship:
$p\left(x=0,y>0,n\right)={p}_{cd}\left(n\right)={p}_{cu}\left(n-1\right)p\left(x = 0, y > 0, n\right) = p_\left\{cd\right\}\left(n\right) = p_\left\{cu\right\}\left(n - 1\right)$
$p\left(x>0,y=0,n\right)={p}_{cu}\left(n\right)={p}_{cd}\left(n+1\right)p\left(x > 0, y = 0, n\right) = p_\left\{cu\right\}\left(n\right) = p_\left\{cd\right\}\left(n + 1\right)$
In the diagram, it can also be observed that in the external price range, the ${p}_{cu}p_\left\{cu\right\}$ and ${p}_{cd}p_\left\{cd\right\}$ in the bands with lower external prices (on the left) are
actually higher than the ${p}_{cu}p_\left\{cu\right\}$ and ${p}_{cd}p_\left\{cd\right\}$ in the bands at higher external prices (on the right).
Liquidation/Redemption Outcome Estimation
In actual contracts, users' funds will be deposited into a group of continuous bands (at least 4). The ${p}_{↑}p_↑$ of largest band then becomes the start price for the liquidation of the user's
assets, and ${p}_{↓}p_↓$ of the smallest band is the end price for the liquidation. When this price is reached, all of the user's ETH will be converted into crvUSD. When users deposit ETH to mint
crvUSD, Curve needs to help users choose a suitable group of bands to deposit their ETH into. These bands should meet the requirement: when all the user's ETH is converted into crvUSD, the amount of
crvUSD obtained should be greater than the user's debt (in actual contracts, a coefficient is multiplied to leave some margin). This is to ensure that the protocol does not incur a bad debt.
Therefore, an estimation needs to be made when users deposit ETH into the bands: the amount of crvUSD obtained after all the ETH in the bands is traded into crvUSD as the external price of ETH
decreases, which we denote as ${x}_{↓}x_↓$. Similarly, we can estimate the amount of ETH obtained when the external price rebounds and all the crvUSD in the bands is traded back into ETH, denoted as
The estimation formula provided by Curve:
${y}_{↑}=y+\frac{x}{\sqrt{{p}_{↑}p}}y_↑ = y + \frac\left\{x\right\}\left\{\sqrt\left\{p_↑ p\right\}\right\}$
${x}_{↓}=x+y\sqrt{{p}_{↓}p}x_↓ = x + y \sqrt\left\{p_↓ p\right\}$
This set of formulas uses the most optimistic estimation method, that is, estimating the maximum possible outcome. Previously, we mentioned that changes in external prices lead to changes in LLAMMA
prices, creating a price difference with external prices. The entry of arbitrageurs causes losses to users. To estimate the maximum value of the results, we need to assume:
• The external price changes at an extremely slow pace, meaning that over a certain period, the price difference between the external price and the internal price in LLAMMA is minimal and can be
• Traders conduct transactions in LLAMMA, causing the internal price of LLAMMA to change synchronously with the external price. Since the price difference between the AMM internal price and the
external price can be ignored, users will not incur losses due to price differences during the trading process.
These two assumptions correspond to the meaning of 'adiabatically' as described in the Curve whitepaper. 'Adiabatically' in physics denotes a process occurring without heat transfer. In this context,
we can understand it as an ideal state that isolates arbitrage transactions due to price differences.
In such an ideal state, LLAMMA can directly use the Uniswap V3 formulas to calculate the swap results, which are the aforementioned two formulas. The derivation process for these formulas is:
${y}_{↑}=y+\mathrm{\Delta }yy_\uparrow = y + \Delta y$ $=y+\sqrt{I}\left(\frac{1}{\sqrt{{p}_{↑}}}-\frac{1}{\sqrt{p}}\right)= y + \sqrt\left\{I\right\} \left\left(\frac\left\{1\right\}\left\{\sqrt\
left\{p_\uparrow\right\}\right\} - \frac\left\{1\right\}\left\{\sqrt\left\{p\right\}\right\}\right\right)$ $=y+\frac{\sqrt{Ip}-\sqrt{I{p}_{↑}}}{{p}_{↑}p}= y + \frac\left\{\sqrt\left\{I p\right\} - \
sqrt\left\{I p_\uparrow\right\}\right\}\left\{p_\uparrow p\right\}$ $=y+\frac{\left(f+x\right)-f}{\sqrt{{p}_{↑}p}}= y + \frac\left\{\left(f + x\right) - f\right\}\left\{\sqrt\left\{p_\uparrow p\right
\}\right\}$ $=y+\frac{x}{\sqrt{{p}_{↑}p}}= y + \frac\left\{x\right\}\left\{\sqrt\left\{p_\uparrow p\right\}\right\}$ ${x}_{↓}=x+\sqrt{I}\left(\sqrt{{p}_{↓}}-\sqrt{p}\right)x_\downarrow = x + \sqrt\
left\{I\right\} \left\left(\sqrt\left\{p_\downarrow\right\} - \sqrt\left\{p\right\}\right\right)$ $=x+\sqrt{I}\left(\frac{1}{\sqrt{p}}-\frac{1}{\sqrt{{p}_{↓}}}\right)\sqrt{{p}_{↓}p}= x + \sqrt\left\
{I\right\} \left\left(\frac\left\{1\right\}\left\{\sqrt\left\{p\right\}\right\} - \frac\left\{1\right\}\left\{\sqrt\left\{p_\downarrow\right\}\right\}\right\right) \sqrt\left\{p_\downarrow p\right\}$
$=x+\left(\left(g+y\right)-g\right)\sqrt{{p}_{↓}p}= x + \left(\left(g + y\right) - g\right)\sqrt\left\{p_\downarrow p\right\}$ $=x+y\sqrt{{p}_{↓}p}= x + y \sqrt\left\{p_\downarrow p\right\}$
Through this estimation method, the Controller contract will help users add liquidity to the appropriate band based on the user's debt, collateral value, and the selected band width.
Furthermore, with the estimation of liquidation/redemption outcomes, it is possible to know at any given moment whether the user's assets, after being completely swapped, will be sufficient to repay
the debt. If at any point it is found that the user's assets, even if entirely swapped into crvUSD, are not enough to cover the debt, liquidators need to intervene for a forced liquidation. The
specific process is similar to the traditional lending liquidation process, which is not elaborated further in this text.
When $p={p}_{↑}={p}_{cd}p = p_↑ = p_\left\{cd\right\}$, $x=0x = 0$ within the AMM. Therefore, ${x}_{↓}=y\sqrt{{p}_{↓}{p}_{↑}}x_↓ = y \sqrt\left\{p_↓ p_↑\right\}$, this means that ideally, within a
band, the average selling price of user assets is the geometric mean of the reference upper and lower limit prices of this band: $\sqrt{{p}_{↓}{p}_{↑}}\sqrt\left\{p_↓ p_↑\right\}$. Similarly, the
price at which user assets are redeemed is also this geometric mean.
The functions in the contract related to the above description include Controller.get_y_effective(), AMM.get_x_down(), AMM.get_y_up(), Controller.liquidate(), etc. Due to space limitations, the
details of the code implementation are not explained here.
LLAMMA Summary
We summarize the characteristics of LLAMMA:
• The LLAMMA Pool contains predefined continuous bands. When adding liquidity, tokens can be added to a group of bands.
• Bands are determined by the external reference prices ${p}_{↑}p_↑$ and ${p}_{↓}p_↓$, which form a geometric sequence between them.
• Within the AMM, ${p}_{cu}p_\left\{cu\right\}$ and ${p}_{cd}p_\left\{cd\right\}$ represent the upper and lower limit prices of a band in the AMM.
• The LLAMMA AMM price, along with all the ${p}_{cu}p_\left\{cu\right\}$ and ${p}_{cd}p_\left\{cd\right\}$ of the bands, increases as the external price ${p}_{o}p_o$ increases, and the rate of
change is faster. The same is true in reverse.
• When users create debt, their assets are stored in a group of bands (these bands are always outside the AMM price, so only ETH is added).
• When the ETH price falls, reaching the upper reference price ${p}_{↑}p_↑$ of the largest band in user liquidity, theoretically (assuming enough arbitrageurs and traders), $p={p}_{cd}={p}_{↑}p =
p_\left\{cd\right\} = p_↑$ is satisfied, and the user's assets begin to be liquidated.
• If the ETH price continues to fall, reaching the lower reference price ${p}_{↓}p_↓$ of the smallest band in user liquidity, theoretically $p={p}_{cu}={p}_{↓}p = p_\left\{cu\right\} = p_↓$ is
satisfied, and all the user's assets are liquidated into crvUSD.
• If the ETH price begins to rebound, LLAMMA will help users buy back their collateralized assets, ETH.
• LLAMMA only exposes the swap interface to the outside and does not directly expose interfaces for adding/removing liquidity to users. Therefore, users cannot arbitrarily add liquidity to LLAMMA
without creating debt, and these related operations are managed by a dedicated Controller.
The Controller implements user interaction interfaces externally, while internally it interfaces with LLAMMA, responsible for depositing/withdrawing user assets into/from LLAMMA.
When users create debt by depositing, they need to specify these parameters:
• The amount of ETH collateral.
• The size of the crvUSD debt being created.
• The number of continuous bands in LLAMMA into which the ETH is deposited.
The third parameter, the number of bands, must be at least 4 and at most 50. The Controller calculates the bands in which the user needs to deposit based on these three parameters and deposits the
user's ETH into these bands (these bands are always out of the money, so liquidity is added as ETH only). Simultaneously, crvUSD is sent to the user, and the user's debt is equal to the amount of
crvUSD minted.
This process utilizes the liquidation outcome estimation formulas mentioned earlier. In the contract, the corresponding function is Controller.create_loan().
In addition, the Controller also has functions for adding collateral, repaying debt, withdrawing collateral, forced liquidation, maintaining loan interest, etc. These functions are not elaborated
upon in this text.
PegKeeper, also known as the stabilizer, is a set of contracts designed to help maintain the peg of crvUSD's price. These contracts primarily interact with the Curve V1 pool.
After the launch of crvUSD, corresponding pools will be created in Curve V1. For simplicity in this example, let's assume crvUSD creates individual Curve V1 pools with DAI and USDT. Suppose the price
of crvUSD is ${p}_{s}p_s$:
• When ${p}_{s}>1p_s > 1$ it indicates a shortage of crvUSD supply. PegKeeper will mint crvUSD and add it to the Curve V1 pool (adding crvUSD as a single asset), to increase the market supply of
• When ${p}_{s}<1p_s < 1$ it indicates an excess supply of crvUSD. PegKeeper will remove the previously added liquidity from the Curve V1 pool (removing crvUSD as a single asset), to reduce the
market supply of crvUSD.
The process is illustrated in the following diagram:
PegKeeper, in the process of carrying out these operations, functions similarly to 'buy low, sell high,' and can generate profits. For example:
When ${p}_{s}>1p_s > 1$ it indicates that there is a shortage of crvUSD in the Curve V1 pool. External accounts can call the PegKeeper contract to mint crvUSD and add it to the Curve V1 pool (adding
crvUSD as a single asset). It's important to note that the minted crvUSD, being uncollateralized and created out of thin air, can only be added to the Curve V1 pool by PegKeeper. The total amount
minted is recorded as a debt of the PegKeeper contract.
After the price ${p}_{s}p_s$ stabilizes or when ${p}_{s}>1p_s > 1$ external accounts can call the PegKeeper contract to withdraw the previously added liquidity from the Curve V1 pool. The withdrawal
only removes crvUSD as a single asset and repays the debt previously incurred by PegKeeper.
PegKeeper's actions of adding/removing crvUSD in Curve V1 are all done as single asset transactions, directly influencing the price of crvUSD in the Curve V1 pool to bring it closer to the pegged
Additionally, since crvUSD is added when its price is high and removed when it is low, PegKeeper will have surplus LP tokens after repaying its debts. These LP tokens represent the profits generated
by PegKeeper. A portion of these profits is distributed to external callers as gas compensation and incentives each time they call the PegKeeper contract. The remaining profits are retained as
protocol revenue.
However, external callers can not arbitrarily call PegKeeper. The contract has a series of checks to prevent malicious calls and ensure that the calls enable PegKeeper to generate profits.
It is important to note that PegKeeper's capacity to respond to market conditions is not entirely symmetrical in dealing with both scenarios:
• When ${p}_{s}>1p_s > 1$ PegKeeper can mint crvUSD out of thin air without collateral and add it to the Curve V1 pool for market regulation. Since PegKeeper doesn't require collateral, its market
regulation capacity is theoretically unlimited in this scenario.
• When ${p}_{s}<1p_s < 1$ PegKeeper can only remove the liquidity previously added to the Curve V1 pool for market regulation. At this time, PegKeeper's capacity is limited by the amount of crvUSD
minted when ${p}_{s}>1p_s > 1$. If PegKeeper had not minted any crvUSD previously, or if the price of crvUSD is still less than 1 after removing all liquidity, it leads to a situation as in
Chinese proverb: even the cleverest housewife can't cook rice without rice.
Therefore, to compensate for the insufficiency of PegKeeper's capacity when ${p}_{s}<1p_s < 1$ Curve Stablecoin also needs to adjust interest rates as a further means to help peg the price of crvUSD.
Monetary Policy
A dynamic component for adjusting lending rates, used to regulate interest rates based on market supply and demand, further helping to stabilize the price anchoring of crvUSD, primarily used in
scenarios where the crvUSD price ${p}_{s}<1p_s < 1$.
The logic for interest rates implemented in the open-source code of Curve Stablecoin is slightly different from that described in the whitepaper. Here, I will explain it according to the logic in the
actual code.
We denote the total debt generated by PegKeeper as ${d}_{pk}d_\left\{pk\right\}$, and the total debt generated by users through the Controller as ${d}_{t}d_t$, The ratio of these two debts is denoted
as ${r}_{d}r_d$. The price of crvUSD is denoted $pp$, and we define a base interest rate ${r}_{0}r_0$. The interest rate for crvUSD is:
$r={r}_{0}\cdot {e}^{\left(\frac{1-p}{\sigma }-\frac{{r}_{d}}{\alpha }\right)}r = r_0 \cdot e^\left\{\left\left(\frac\left\{1-p\right\}\left\{\sigma\right\} - \frac\left\{r_d\right\}\left\{\alpha\
In the above equation, $\sigma \sigma$ is a constant, and $\alpha \alpha$ is the target debt ratio.
After setting specific values for these parameters (currently, in Monetary Policy for ETH market $\sigma \sigma$ is set to 0.02 and target debt ratio equals 0.1), the curve of the interest rate
changes with the price is illustrated in the following diagram (screenshot from desmos):
We can observe:
• When the price of crvUSD is greater than 1, the interest rate is relatively low.
• When the price of crvUSD is less than 1, the interest rate sharply increases after a certain critical point (the rate of increase depends on $\sigma \sigma$)
The reason for this design:
• If the price of crvUSD is greater than 1, PegKeeper has sufficient capacity to regulate the market. In this case, maintaining a low-interest rate helps attract more users to mint crvUSD.
• If the price of crvUSD is less than 1, PegKeeper’s capacity is limited. At this point, to reduce the market supply of crvUSD, it is necessary to raise the interest rate to force users to repay
their debts. To avoid excessive price differences caused by detethering, the rate of interest increase will begin to rise rapidly after crvUSD falls below a certain critical point. The rate of
increase depends on the $\sigma \sigma$ parameter, and the price at which the interest rate starts to increase depends on the size of PegKeeper’s debt.
Additionally, we can observe the impact of the parameter and the PegKeeper debt ratio ${r}_{d}r_d$ on the actual interest rate changes:
It can be seen that the closer $\sigma \sigma$ is to 0, the steeper the curve becomes, and the faster the interest rate increases as the price decreases.
When the proportion of PegKeeper's debt is relatively large, it shifts the curve to the left, making the interest rate start to rise at a lower price.
The reason for this approach is that if PegKeeper's debt proportion is large, it indicates that PegKeeper holds a significant amount of liquidity in Curve V1. When the price of crvUSD is less than 1,
PegKeeper can first remove liquidity (crvUSD as a single asset) from Curve V1 to reduce its debt, simultaneously aiding the price of crvUSD to return to its peg. The larger the debt of PegKeeper, the
more the price of crvUSD needs to fall before the interest rate begins to rise rapidly. This approach helps PegKeeper control the size of its debt.
As mentioned earlier, Curve Stablecoin in LLAMMA needs to know the external price of the collateral ETH, and in PegKeeper and Monetary Policy, it needs to know the external price of crvUSD. For the
collateral price, Curve will fetch it from the Curve V2 pool, and the crvUSD price will be fetched from the corresponding Curve V1 pools (referring to prices in multiple pools).
Before using external prices, Curve Stablecoin first processes these prices with an Exponential Moving Average (EMA). As previously stated, if there are significant fluctuations in the external
Oracle prices, then a substantial price difference between LLAMMA prices and external prices can occur, resulting in losses for users. Using EMA prices can help reduce price volatility and also
increase the difficulty of manipulating Oracle prices.
Oracles functionality is updated based on changing market conditions and analysis of the influence of various parameters on the operation of the protocol. Thereby, the second version of oracles for
crvUSD takes into account TVLs of aggregated pools and adjusts weights based on that.
Collateral oracles fetch prices from pools and to prevent manipulations through flash loans, Curve employed a hybrid approach using TriCrypto, Chainlink, and Uniswap Twap Oracle prices, instead of
solely relying on TriCrypto prices. Oracle price is checked against anchor price (from Chainlink or Uni TWAP oracles). There is a threshold for price deviation, called safety limit, and if the price
difference hits the threshold, then the oracle uses the price from the anchor source. However, the Curve team did research showing that Chainlink (or, rather, market spot) prices cause more losses
than necessary in periods of high volatility. So, there was a proposal for disabling Chainlink limits.
Below are some common questions about Curve Stablecoin:
How can liquidity be provided to LLAMMA?
Users can only add liquidity to LLAMMA through the Controller, by pledging ETH to create crvUSD debt. The ETH will be added to LLAMMA by the Controller. Ordinary users don't have to consider market
making in LLAMMA, as the characteristics of LLAMMA mean that its Liquidity Providers (LPs) are very likely to incur losses in trading.
When minting crvUSD requires collateralizing ETH to provide liquidity to LLAMMA, where is the user's ETH placed?
Users need to specify the number of bands they wish to enter. Subsequently, the Controller, based on the user's debt size, automatically selects a group of bands that minimizes the user's risk. This
group consists of the bands with the lowest prices, while also ensuring that the protocol does not face bad debt risk.
How is the number of bands chosen?
When creating debt, users need to decide the number of bands into which their ETH will be placed.
When minting the same amount of crvUSD, the greater the number of bands, the more dispersed the collateral distribution, leading to a higher starting price for liquidation. Conversely, fewer bands
result in more concentrated collateral distribution, with a relatively lower starting price for liquidation.
If a higher loan-to-value ratio is desired, fewer bands should be chosen, but this also increases the risk of liquidation.
How does Curve specifically add a user's ETH to LLAMMA when they mint crvUSD?
• Since the user's collateral is only ETH, this definitely involves single-sided liquidity, meaning the band contains only ETH.
• Curve will try to add the user's ETH to bands with lower prices, but also needs to ensure that the protocol does not face bad debt risk.
Curve selects appropriate bands and adds liquidity based on the amount of ETH the user has, the width of the bands chosen by the user, and the amount of crvUSD minted. The specific logic is as
Suppose the user's collateralized ETH amount is $yy$. We can use Formula 10 from the whitepaper to calculate the amount of crvUSD obtained when the user's ETH in a certain band is traded into crvUSD
(for simplicity, we ignore loan_discount here):
${x}_{↓}=y\sqrt{{p}_{↑}{p}_{↓}}=y{p}_{↑}\sqrt{\frac{A-1}{A}}x_↓ = y \sqrt\left\{p_↑ p_↓\right\} = yp_↑ \sqrt\left\{\frac\left\{A-1\right\}\left\{A\right\}\right\}$
In LLAMMA, the user's ETH is evenly distributed across $NN$ bands, with each band containing an amount of ETH equal to $\frac{y}{N}\frac\left\{y\right\}\left\{N\right\}$. Assuming the band with the
highest price is numbered $n1n1$, after using the above formula to calculate for each band and summing up the results, we can obtain:
${x}_{↓}=\frac{y{p}_{↑}}{N}\cdot \sqrt{\frac{A-1}{A}}\cdot {\sum }_{k=0}^{N-1}{\left(\frac{A-1}{A}\right)}^{k}x_↓ = \frac\left\{yp_↑\right\}\left\{N\right\} \cdot \sqrt\left\{\frac\left\{A-1\right\}\
left\{A\right\}\right\} \cdot \sum_\left\{k=0\right\}^\left\{N-1\right\} \left\left(\frac\left\{A-1\right\}\left\{A\right\}\right\right)^k$
We define a variable ${y}_{\text{effective}}y_\left\{\text\left\{effective\right\}\right\}$ that is related only to the user's band width $NN$ and is independent of the band price:
${y}_{\text{effective}}=\frac{y}{N}\cdot \sqrt{\frac{A-1}{A}}\cdot {\sum }_{k=0}^{N-1}{\left(\frac{A-1}{A}\right)}^{k}y_\left\{\text\left\{effective\right\}\right\} = \frac\left\{y\right\}\left\{N\
right\} \cdot \sqrt\left\{\frac\left\{A-1\right\}\left\{A\right\}\right\} \cdot \sum_\left\{k=0\right\}^\left\{N-1\right\} \left\left(\frac\left\{A-1\right\}\left\{A\right\}\right\right)^k$
We also define ${x}_{↓}x_↓$ as ${x}_{\text{effective}}x_\left\{\text\left\{effective\right\}\right\}$. The above formula simplifies to:
${x}_{\text{effective}}={y}_{\text{effective}}\cdot {p}_{↑}x_\left\{\text\left\{effective\right\}\right\} = y_\left\{\text\left\{effective\right\}\right\} \cdot p_↑$
The ${p}_{↑}p_↑$ here refers to the highest ${p}_{↑}p_↑$ price in the bands where the user's ETH is added. In the code, the calculation of ${y}_{\text{effective}}y_\left\{\text\left\{effective\right
\}\right\}$ is implemented by the Controller.get_y_effective() function.
Next, we first assume that the user's ETH is placed in the band with the highest price (below the current AMM price). Suppose the number of this band is ${n}_{1}n_1$, then at this time ${x}_{\text
{effective}}x_\left\{\text\left\{effective\right\}\right\}$ is:
${x}_{\text{effective}}={y}_{\text{effective}}\cdot {p}_{↑\left({n}_{1}\right)}x_\left\{\text\left\{effective\right\}\right\} = y_\left\{\text\left\{effective\right\}\right\} \cdot p_\left\{↑\left
Assume the amount of crvUSD minted by the user is debt. To add the user's ETH to the band with the lowest price, we need to reduce ${x}_{\text{effective}}x_\left\{\text\left\{effective\right\}\right
\}$, but it must not be less than debt (otherwise, the protocol faces the risk of bad debt). That is:
$\frac{{y}_{\text{effective}}\cdot {p}_{↑\left({n}_{1}\right)}}{\text{debt}+1}\ge 1\frac\left\{y_\left\{\text\left\{effective\right\}\right\} \cdot p_\left\{↑\left(n_1\right)\right\}\right\}\left\{\
text\left\{debt\right\}+1\right\} \geq 1$
Next, our task is to find the largest value of ${n}_{1}n_1$ that satisfies the above equation. The larger ${n}_{1}n_1$ is, the lower the price of the band where the user's ETH is added. In the code,
logarithmic calculations are used to complete the above process:
$\mathrm{log}\left(\frac{{y}_{\text{effective}}\cdot {p}_{↑\left({n}_{1}\right)}}{\text{debt}+1}\right)\ge 0\log\left\left(\frac\left\{y_\left\{\text\left\{effective\right\}\right\} \cdot p_\left\{↑\
left(n_1\right)\right\}\right\}\left\{\text\left\{debt\right\}+1\right\}\right\right) \geq 0$
Suppose we need to increase the band number from ${n}_{1}n_1$ to ${n}_{1}+mn_1 + m$, then:
$\mathrm{log}\left(\frac{{y}_{\text{effective}}\cdot {p}_{↑\left({n}_{1}\right)}}{\text{debt}+1}\cdot {\left(\frac{A-1}{A}\right)}^{m}\right)\ge 0\log\left\left(\frac\left\{y_\left\{\text\left\
{effective\right\}\right\} \cdot p_\left\{↑\left(n_1\right)\right\}\right\}\left\{\text\left\{debt\right\}+1\right\} \cdot \left\left(\frac\left\{A-1\right\}\left\{A\right\}\right\right)^m\right\
right) \geq 0$
$\mathrm{log}\left(\frac{{y}_{\text{effective}}\cdot {p}_{↑\left({n}_{1}\right)}}{\text{debt}+1}\right)\ge m\cdot \mathrm{log}\left(\frac{A}{A-1}\right)\log\left\left(\frac\left\{y_\left\{\text\left\
{effective\right\}\right\} \cdot p_\left\{↑\left(n_1\right)\right\}\right\}\left\{\text\left\{debt\right\}+1\right\}\right\right) \geq m \cdot \log\left\left(\frac\left\{A\right\}\left\{A-1\right\}\
$m\le \frac{\mathrm{log}\left(\frac{{y}_{\text{effective}}\cdot {p}_{↑\left({n}_{1}\right)}}{\text{debt}+1}\right)}{\mathrm{log}\left(\frac{A}{A-1}\right)}m \leq \frac\left\{\log\left\left(\frac\left
\{y_\left\{\text\left\{effective\right\}\right\} \cdot p_\left\{↑\left(n_1\right)\right\}\right\}\left\{\text\left\{debt\right\}+1\right\}\right\right)\right\}\left\{\log\left\left(\frac\left\{A\
If $mm$ is an integer, then:
$m=⌊\frac{\mathrm{log}\left(\frac{{y}_{\text{effective}}\cdot {p}_{↑\left({n}_{1}\right)}}{\text{debt}+1}\right)}{\mathrm{log}\left(\frac{A}{A-1}\right)}⌋m = \left\lfloor\frac\left\{\log\left\left(\
frac\left\{y_\left\{\text\left\{effective\right\}\right\} \cdot p_\left\{↑\left(n_1\right)\right\}\right\}\left\{\text\left\{debt\right\}+1\right\}\right\right)\right\}\left\{\log\left\left(\frac\
The above $⌊⌋\lfloor \rfloor$ symbol represents the floor function, which rounds down to the nearest integer. Thus, we can calculate that the user's ETH is added to the bands in the range of $\left
[{n}_{1}+m,{n}_{1}+m+N\right]\left[n_1 + m, n_1 + m + N\right]$. In the code, this calculation is implemented in the Controller._calculate_debt_n1() function.
What is the maximum loan-to-value (LTV) ratio for Curve Stablecoin?
Depending on the Controller.loan_discount and the number of bands chosen by the user, the maximum amount of crvUSD that can be minted is highest when the number of bands selected is 4 (the minimum
As mentioned earlier, when a user creates debt, the ideal liquidation price estimated by Curve for ETH within a band is the geometric mean of ${p}_{↑}p_↑$ and ${p}_{↓}:\sqrt{{p}_{↑}\cdot {p}_{↓}}p_↓:
\sqrt\left\{p_↑ \cdot p_↓\right\}$
Assume the current ETH price = $pp$, the number of bands = $NN$, loan_discount = $rr$, and the amount of collateralized ETH = $yy$. To maximize the amount of crvUSD borrowed, we also assume $p\approx
{p}_{↑}p \approx p_↑$ (the highest ${p}_{↑}p_↑$ when placed into a band). Then the maximum amount of crvUSD that can be borrowed (considering the impact of loan_discount) is:
${x}_{\text{effective}}=\left(1-r\right)\cdot {y}_{\text{effective}}\cdot px_\left\{\text\left\{effective\right\}\right\} = \left(1 - r\right) \cdot y_\left\{\text\left\{effective\right\}\right\} \
cdot p$
Based on the earlier definition of ${y}_{\text{effective}}y_\left\{\text\left\{effective\right\}\right\}$, we obtain:
${x}_{\text{effective}}=\left(1-r\right)\cdot \frac{yp}{N}\cdot \sqrt{\frac{A-1}{A}}\cdot {\sum }_{k=0}^{N-1}{\left(\frac{A-1}{A}\right)}^{k}x_\left\{\text\left\{effective\right\}\right\} = \left(1 -
r\right) \cdot \frac\left\{yp\right\}\left\{N\right\} \cdot \sqrt\left\{\frac\left\{A-1\right\}\left\{A\right\}\right\} \cdot \sum_\left\{k=0\right\}^\left\{N-1\right\} \left\left(\frac\left\{A-1\
Then, the maximum loan-to-value ratio is:
$\frac{\left(1-r\right)}{N}\cdot \sqrt{\frac{A-1}{A}}\cdot {\sum }_{k=0}^{N-1}{\left(\frac{A-1}{A}\right)}^{k}\frac\left\{\left(1-r\right)\right\}\left\{N\right\} \cdot \sqrt\left\{\frac\left\{A-1\
right\}\left\{A\right\}\right\} \cdot \sum_\left\{k=0\right\}^\left\{N-1\right\} \left\left(\frac\left\{A-1\right\}\left\{A\right\}\right\right)^k$
Based on the settings in the test code, with $A=100A = 100$, $r=0.05r = 0.05$, and choosing the number of bands $N=4N = 4$, the highest loan-to-value ratio, approximately, is:
$\frac{1-0.05}{4}\cdot \sqrt{\frac{99}{100}}\cdot \left(\frac{1-{\left(\frac{99}{100}\right)}^{4}}{1-\frac{99}{100}}\right)\approx 93.12\mathrm{%}\frac\left\{1-0.05\right\}\left\{4\right\} \cdot \
sqrt\left\{\frac\left\{99\right\}\left\{100\right\}\right\} \cdot \left\left( \frac\left\{1-\left\left(\frac\left\{99\right\}\left\{100\right\}\right\right)^4\right\}\left\{1-\frac\left\{99\right\}\
left\{100\right\}\right\} \right\right) \approx 93.12\%$
At this point, the starting price for liquidation is $pp$, the ending price for liquidation is approximately $p\cdot 0.9{9}^{4}\approx 0.96pp \cdot 0.99^4 \approx 0.96p$, and the estimated average
liquidation price is about $p\cdot 0.9{9}^{2}\approx 0.98pp \cdot 0.99^2 \approx 0.98p$.
Note that these calculations are based on the ideal conditions described in the Curve whitepaper (adiabatic approximation). In actual scenarios, users might face forced liquidation earlier.
If the price of ETH falls, causing some of the user's ETH to be exchanged for crvUSD, and then the price of ETH rebounds above the liquidation line, will the user still incur losses?
It is highly likely, as the characteristics of LLAMMA result in the Pool engaging in 'buy high, sell low' transactions. Even if the price is restored, the assets in the Pool may still decrease.
What are the revenue sources at the protocol level for Curve Stablecoin?
The protocol has three sources of revenue: LLAMMA fees, crvUSD interest, and PegKeeper profits.
Does Curve Stablecoin completely avoid liquidation?
No, forced liquidation can still occur. When the estimated value of a user's assets after liquidation is less than their debt, liquidators can forcibly liquidate the user's assets. This means taking
the user's assets out of LLAMMA and repaying their debt in advance.
What arbitrage opportunities are created by Curve Stablecoin?
LLAMMA creates arbitrage opportunities due to price differences in DEX/CEX, and additionally, when crvUSD becomes unpegged, PegKeeper also presents arbitrage opportunities.
Which tokens can be used as collateral?
In theory, any token can be used as collateral. However, Curve Stablecoin is not completely independent of external liquidity. When the price of the collateral decreases and LLAMMA adjusts its price
downward, external DEX/CEX liquidity is still needed to work with LLAMMA for arbitrage opportunities to be profitable. Current launched markets are WBTC, wstETH, ETH, sfrxETH, tBTC markets, with
total debt value - 114 m$, and collateral value - 201 m$.
Will there be Liquidity Mining?
The LLAMMA contract is adapted to CurveDAO's Gauge interfaces, so it will likely support mining. The mining algorithm is such that the greater the value of the liquidated assets, the higher the
mining weight. This means you must borrow crvUSD and reach the liquidation line to generate mining profits. It seems like a game suited for 'degens' (degenerate gamblers in cryptocurrency slang).
Do you have any good hedging strategies?
The characteristics of LLAMMA, being an AMM opposite to Uniswap V3, suggest the following hedging strategy against liquidation risk:
• Suppose you borrow 1000 crvUSD, and your ETH is added to the LLAMMA bands in the [1300 ~ 1500] price range.
• At the same time, you could use stablecoins like USDC/DAI/USDT to add single-sided liquidity in Uniswap V3 within the [1300 ~ 1500] price range, with the amount of stablecoins also being 1000.
This way, when the ETH price falls to 1500, ETH will start to be sold in LLAMMA, but at the same time, ETH will start to be bought in Uniswap V3. This can maintain your ETH exposure roughly constant
at any price (although there will still be losses due to LLAMMA's wear and tear).
Additionally, thanks to @0xstan and @0xmc for their communication and discussion during the research process of the original article.
PS (@2023-05-10)
Additionally, some parameter settings differ from the assumptions made in this article. These are not exhaustively listed here; please refer to the official deployed contracts for accurate
The Statemind team extends gratitude to Paco for their foundational analysis of Curve Stablecoin.
The translation and adaptation of this article were undertaken by our dedicated team, with particular emphasis on the integration of sections related to oracles to enhance the understanding of this
complex topic. For further discussions and insights, follow Statemind and Paco on X. | {"url":"https://statemind.io/blog/curve-stablecoin-deep-dive","timestamp":"2024-11-11T13:06:58Z","content_type":"text/html","content_length":"1048937","record_id":"<urn:uuid:eb378d33-f56f-4513-860d-7a9aebc826d8>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00805.warc.gz"} |
Rafter Length Calculat
Rafter Length Calculator
Calculated Rafter Length:
7.21 m
Rafter Length Calculations
In the building and construction industry, precision is crucial, especially when it comes to roof framing. One of the most important calculations in this area is determining the length of a rafter. A
rafter is one of the sloping beams that support the roof structure and its covering material. Calculating the correct length of a rafter is essential for ensuring a roof's structural integrity,
proper load distribution, and aesthetics. This page explains how to use a Rafter Length Calculator and provides insight into its importance in construction projects. Whether you're a professional
carpenter or a DIY enthusiast, understanding how to calculate rafter length can significantly improve your ability to execute roofing tasks with confidence and accuracy.
The Basics: What is a Rafter?
Rafters are the inclined, structural components that extend from the ridge of the roof (the peak) to the wall plate or exterior walls. They are spaced evenly across the roof structure and support the
roof decking, insulation, and other materials. The length and size of the rafters affect the roof's ability to bear loads such as snow, wind, and the roof's materials.
Rafters are used in various roof designs, including gable roofs, hip roofs, and shed roofs. In each case, the precise calculation of the rafter length helps ensure that the roof is strong,
well-supported, and visually balanced. Incorrect rafter lengths can result in an improper fit, structural weaknesses, and other potential problems during the building process.
The Mathematics Behind Rafter Length Calculation
To calculate the length of a rafter, you need two key measurements:
1. The Rise: The vertical distance from the top of the wall plate (the base of the roof) to the peak of the roof (the ridge).
2. The Run: The horizontal distance from the wall plate to the point directly under the ridge. The run is usually half the width of the total building span.
The rafter length can be calculated using the Pythagorean Theorem, which is essential in geometry and trigonometry. The relationship is expressed as:
Rafter Length = √(Rise² + Run²)
This formula comes from the fact that the rise, run, and rafter length form a right-angled triangle, where:
• The rafter length is the hypotenuse (the longest side of the triangle),
• The rise is one leg (the vertical leg), and
• The run is the other leg (the horizontal leg).
Example Calculation
Let's take a practical example to better understand the process.
Suppose the rise of your roof is 8 feet, and the run is 10 feet. Using the Pythagorean Theorem:
Rafter Length = √(8² + 10²) = √(64 + 100) = √164 ≈ 12.81 feet
In this case, the length of the rafter would be approximately 12.81 feet. You would also need to add any overhang or eaves to this measurement if required.
Roof Pitch and Its Role in Rafter Length Calculation
An important concept in rafter calculations is the roof's pitch. Roof pitch is the slope or angle of the roof, often expressed as the rise over the run, such as 4/12 or 6/12. For example, a roof with
a 6/12 pitch means that for every 12 inches of horizontal run, the roof rises 6 inches vertically.
A higher pitch increases the rise and, consequently, the rafter length. In cases where the pitch of the roof is known, you can also use trigonometric functions such as sine, cosine, or tangent to
determine the rise, run, and rafter length.
Real-World Applications of Rafter Length Calculation
Rafter length calculations are used in many aspects of construction, particularly in roofing projects. Here are some common use cases:
1. Building a New Roof
Whether for residential or commercial structures, calculating the correct rafter length ensures that the roof is properly framed and supported. This is especially important for gable, shed, or hip
roofs where each rafter must be uniform to create a symmetrical, well-balanced structure.
2. Renovations and Extensions
When extending a house or making modifications to an existing roof, contractors need to calculate new rafter lengths to integrate the new structure with the old. Ensuring the correct length prevents
mismatches or misalignments in the roofing system.
3. Custom Roof Design
In modern architecture, custom roof designs are becoming more popular. Unique shapes, slopes, and angles require precise rafter length calculations to bring the design to life while maintaining
structural integrity. For example, a vaulted ceiling might require varying rafter lengths to achieve the desired effect.
4. Truss Design
Rafters are an integral part of trusses, which are pre-engineered, triangular roof structures. In this context, rafter length must be accurate to ensure the entire truss system works as intended.
5. Structural Load Bearing
When designing roofs in regions with heavy snow or wind loads, the rafter length and size must be calculated to withstand these additional forces. This often involves adjusting the spacing between
rafters or the type of wood or material used.
6. Solar Panel Installation
With the rise in sustainable construction, solar panel installation on roofs has become common. The length and slope of rafters can directly impact the optimal angle and positioning of solar panels,
making these calculations critical for energy efficiency.
Additional Considerations in Rafter Length Calculation
While the basic calculation using rise and run is straightforward, there are other factors that may need to be accounted for, including:
• Rafter Overhang: Many roofs have an overhang that extends beyond the walls, providing extra protection from the elements. This must be added to the total rafter length.
• Birdsmouth Cut: A birdsmouth cut is made at the point where the rafter rests on the wall plate. This cut affects the length of the rafter and needs to be considered when framing.
• Material Choice: The type of material (wood, metal, engineered lumber) can influence the strength and flexibility of the rafter. Different materials may require adjustments to rafter dimensions
or spacing.
• Building Codes: Always ensure that rafter lengths comply with local building codes and regulations. These codes may dictate minimum rafter sizes or maximum spans for specific types of buildings
and regions.
Using the Rafter Length Calculator
A Rafter Length Calculator simplifies the process by allowing you to input the rise and run, and it will automatically calculate the correct rafter length for you. This saves time and ensures
precision, particularly for complex or large-scale projects. By utilizing a tool like this, you minimize errors, improve efficiency, and focus on the other critical aspects of roof construction.
In conclusion, calculating the correct rafter length is essential for successful roof construction. With the help of tools like the Rafter Length Calculator, builders, architects, and DIY enthusiasts
can ensure accurate, efficient, and safe construction, resulting in durable and aesthetically pleasing roof structures. | {"url":"https://constructcalc.com/rafter.php","timestamp":"2024-11-09T04:10:53Z","content_type":"text/html","content_length":"19955","record_id":"<urn:uuid:15c57186-a042-4fe2-85ab-203a60f1ffaf>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00896.warc.gz"} |
Clusterjerk - Chris Blattman
Since yesterday’s pointy-headed statistics post proved unexpectedly viral, I assume you want more econometric rants. So here’s something that has been bothering me all week.
When I was in graduate school, economists discovered clustered standard errors. Or so I assume because it almost became a joke that the first question in any seminar was “did you cluster your
standard errors?”
Lately I’ve been getting the same question from referees on my field experiments, and to the best of my knowledge, this is wrong, wrong, wrong.
So, someone please tell me if I’m mistaken. And if I’m not, a plea to my colleagues: this is not something to write in your referee reports. Please stop.
[Read the follow up post here]
I guess I should explain what clustering means (though if you don’t know already there’s a good chance you don’t care and it’s not relevant to your life). Imagine people in a village who experience a
change in rainfall or the national price of the crop they grow. If you want to know how employment or violence or something responds to that shock, you have to account for the fact that people in the
same village are subject to the same unobserved forces of all varieties. If you don’t, your regression will tend to overstate the precision of any link between the rainfall change and employment. In
Stata, this is mindlessly and easily accomplished by putting “, cluster” at the end of your regression, and we all do it.
This makes sense if you have observational data (at least sometimes). But if you have randomized a program at the individual level, you do not need to cluster at the village level, or some other
higher unit of analysis. Because you randomized.
Or so I believe. But I don’t have a proof or citation to one. I have asked some of the very best experimentalists in the land this week, and all agree with me, but none have a citation or a proof. I
could run a simulation to prove it, but surely someone smarter and less lazy than me has attacked this problem?
While I’m on the subject, my related but nearly opposite pet peeves:
• Reviewing papers that randomize at the village or higher level and do not account for this through clustering or some other method. This too is wrong, wrong, wrong, and I see it happen all the
time, especially political science and public health.
• Maybe worse are the political scientists who persist in interviewing people on either side of a border and treating some historical change as a treatment, ignoring that they basically have a
sample of size of two. This is not a valid method of causal inference.
Update: The follow up post is here.
43 Responses
1. @jjmatta https://t.co/lahYBiJiFQ
https://t.co/nc0PqBQzPA https://t.co/9nSMnp3J5x
2. Peter, if I understand your question correctly, that sounds like a SUTVA violation, which is a serious issue but not solved by clustered standard errors (see also pp. 4-5 on peer effects in the
Weiss et al. paper that Stuart linked to).
3. Seems that you might still have to cluster even with individual-level random assignment. For instance, imagine the case where there is individual random assignment at the village level, and then
significant spillovers across treatment and control individuals within a village but not across villages. Seems to me this would require clustering across villages. Would love to hear if this is
wrong however and why.
4. Clusterjerk https://t.co/jWfgFUBXwO
5. Stuart, I think Mike, J.R., and Dan’s paper is very good, but I agree with Chris that “if you have randomized a program at the individual level, you do not need to cluster at the village level,
or some other higher unit of analysis”. (“You do not need” doesn’t mean “you absolutely shouldn’t”.)
As some other commenters have mentioned, if individuals were randomly assigned, then Neyman’s mode of randomization inference can justify the use of robust standard errors (not clustered at a
higher level) in large samples. This result is discussed in, among other places, Imbens & Rubin’s book (as Doug mentioned), and extended to regression-adjusted estimation of average treatment
effects in my paper “Agnostic Notes on Regression Adjustments to Experimental Data: Reexamining Freedman’s Critique”. Neyman’s mode of randomization inference tries to construct a confidence
interval for the average treatment effect on the experimental sample. It doesn’t try to generalize to a superpopulation or to what would happen if the same treatment were given by different
service providers, if the villages experienced different economic shocks, etc. Thus, at best, it answers a very narrow question and doesn’t capture all the uncertainty we should have about
broader policy-relevant questions. Nevertheless, I think this framework can be useful for lower bounds on our uncertainty, because it’s easier to agree on what the unit of randomization was than
to agree on what’s a reasonable model of the factors that affect outcomes.
When people discuss Chris’s question in a regression model framework instead of a randomization inference framework, they’re implicitly asking a different question. Model-based frequentist
inference considers hypothetical replications of the study in which each new replication brings not a new random assignment, but a new random draw of each person’s error term “epsilon”. Since
epsilon has to represent everything that determines the outcome besides treatment and the covariates in the regression model, we inevitably have some dependence between individuals’ epsilons,
e.g. because a group of patients share a service provider, because people in the same local labor market are subject to the same “random” economic shocks, or because students in the same
classroom experience the same “random” events such as a dog barking just outside the classroom on exam day (unless we want to hold service providers, economic shocks, and dog barks fixed across
our hypothetical replications of the study). One of the important contributions of Mike et al.’s paper is to show that addressing such dependence can be harder than people think and even
The desired inference depends on what we want to generalize to, and there’s no one right answer. I think it could often be a good idea to show more than one analysis. But sometimes the best we
can do on the broader questions is an informal discussion. E.g., suppose we randomly assign 40 schools in two school districts. Since we’re probably interested in the broader question of would
happen in other school districts, should we conclude that we have to cluster at the district level, where we have a sample size of 2? Or should we construct confidence intervals from SEs
clustered at the school level (perhaps using the method in Imbens & Kolesar’s paper “Robust Standard Errors in Small Samples: Some Practical Advice”), but acknowledge that the findings don’t
necessarily generalize to all districts?
This kind of issue also comes up in nonexperimental studies. Jeff Wooldridge (“Cluster-Sample Methods in Applied Econometrics”, American Econ Review, 2003) mentions that Donald & Lang criticized
Card & Krueger’s New Jersey – Pennsylvania minimum wage study for ignoring state-level clustering. Wooldridge points out that accounting for such clustering is impossible with only two states. He
writes, “The criticism in the G = 2 case is indistinguishable from a common criticism of difference-in-differences (DID) analyses: How can one be sure that any observed difference in means is due
entirely to the policy change?” Both here and in randomized experiments, my view is that formal statistical inference never captures all the uncertainty we care about. Confidence intervals and
tests can be useful for lower bounds on our uncertainty, but sources of uncertainty that they don’t capture should be acknowledged. Whether that should be done formally or informally may depend
on the situation.
See also Mosteller and Tukey’s 1977 book “Data Analysis and Regression” (sections on “Choosing an error term”, pp. 123-125, and “Supplementary uncertainty and its combination with internal
uncertainty”, pp. 129-131).
6. On clustering standard errors in experiments. Clusterjerk https://t.co/seZUoBtVLi
7. What do folks think of this MDRC paper?
Estimating the Standard Error of the Impact Estimator in Individually Randomized Trials with Clustering
| Michael J. Weiss, J. R. Lockwood, Daniel F. McCaffrey
In many experimental evaluations in the social and medical sciences, individuals are randomly assigned to a treatment arm or a control arm of the experiment. After treatment assignment is
determined, individuals within one or both experimental arms are frequently grouped together (e.g., within classrooms or schools, through shared case managers, in group therapy sessions, or
through shared doctors) to receive services. Consequently, there may be within-group correlations in outcomes resulting from (1) the process that sorts individuals into groups, (2) service
provider effects, and/or (3) peer effects. When estimating the standard error of the impact estimate, it may be necessary to account for within-group correlations in outcomes. This article
demonstrates that correlations in outcomes arising from nonrandom sorting of individuals into groups leads to bias in the estimated standard error of the impact estimator reported by common
estimation approaches.
8. I agree with Coady Wing and the paper by Cameron and Miller (now published in JHR) is helpful.
I have a little suspicion that Chris may have some confusions between conditions for coefficient consistency and standard error consistency. Randomization remove the correlation between error and
treatment variable, so it leads to consistency of OLS. But for standard errors, the standard formula for standard errors assume homoscedastic and uncorrelated errors. But if there is correlation
between errors, the assumed formula will assume different information content than we actually have, so the standard error formula is inconsistent to the true standard error. The more subtle
conditions for consistency are specified in the Camaron-Miller paper.
9. Some pegagogical simulations trying to make a similar point available here:
10. It’s been a very long time since I took an econometrics class and I seem to disagree with everyone else here so I’d appreciate somebody pointing out where I’m wrong.
My memory (plus some Googling) was that the default calculations of the standard errors of OLS coefficients assume spherical errors (as does the Gauss-Markov Theorem proving that OLS is BLUE).
Even if randomization occurred at the individual level, we’d still expect individuals within the same cluster to experience the same shocks. Wouldn’t this be likely to violate the spherical
errors assumption? In other words, I’d expect the variance of the error to vary by cluster (leading to homoscedasticity) and errors within clusters to be correlated.
I think this means that you need to correct your standard errors. Robust SEs are an option but SEs that specifically account for the clusters are more efficient.
Another issue (I think) occurs if you randomized at the individual level but selected people for participation in the study at the level of the cluster. It seems like this should further reduce
your effective sample size.
Tell me what I’m missing!
11. Let’s consider the extreme case in which all of the error is from the village-specific component. If you don’t cluster, you might claim to have 10,000 observations, but in practice, if there are
only 100 villages, there’s no way that you can have more than 100 independent observations in your sample. By failing to cluster, you are inflating your t-statistics by acting like you have
10,000 independent random draws.
12. Hi Dr. Blattman,
Others have already pointed out that looking at the moulton inflation factor shows that when assignment is at the individual level clustering isn’t necessary. If you’re like me, this probably
doesn’t help much with the intuition though. Here’s an alternate explanation: if you assume each unit has a defined outcome under treatment and control (i.e. SUTVA), then each treatment and
control unit in a randomized experiment is a random draw from one of two distributions — the distribution of potential treatment outcomes and the distribution of potential control outcomes. (This
is a slight simplification. For full details see page 87 of Imbens and Rubin, 2015.) Thus, the treatment and control means are averages of independent, identically distributed variables and the
usual estimate of the variance (without clustering) is justified. Note that this explanation does not make any assumptions whatsoever about the distribution of potential outcomes in the overall
population (other than the basic stuff necessary for the CLT to hold). Also note that this would not be the case if you randomized at a higher level.
On a related note, these same issues are present when testing for baseline balance in a randomized experiment. I have seen quite a number of papers where the authors randomize at a group level
and then to balance tests at the unit level and erroneously come to the conclusion that their randomization failed.
13. “Maybe worse are the political scientists who persist in interviewing people on either side of a border and treating some historical change as a treatment, ignoring that they basically have a
sample of size of two. This is not a valid method of causal inference.”
You’re stating this a little too strongly. You have to make some strong assumptions about what it means to be a treated unit, but you can do it. Melissa Dell mining mita paper does a decent job
outlining this.
14. (Not) Clustering in experiments, observational research, and something on natural experiments (borders) https://t.co/PWSJOy1GzM by @cblatts
15. All I gotta say is, ex post / ex ante.
16. Clusterjerk: Since yesterday’s pointy-headed statistics post proved unexpectedly viral, I assume you want more… https://t.co/kYASxjvlJd
17. “Maybe worse are the political scientists who persist in interviewing people on either side of a border and treating some historical change as a treatment, ignoring that they basically have a
sample of size of two. This is not a valid method of causal inference.”
Aren’t basically half of all the examples given by Robinson & Acemoglu in “Why Nations Fail” this right here?
18. RT @cblatts: Are you a clusterjerk or am I? (A bleg on whether I need to cluster std errors in a field experiment.) https://t.co/R21D5pkKpg
19. @cdsamii @FlorianFoos @BrendanNyhan @ClaytonNall @cblatts @cdsamii Yes! This paper may help: https://t.co/r6jnBQTdSl
20. I think you are asking the wrong question. The onus is on the other parties to show why some deviation from the simple model is needed, not on you to show why it is not needed. I can see why you
would prefer a handy proof or paper to show your case; just pointing out that there is something backward about having to prove some enhancement is not needed. Fwiw, it seems the underlying logic
for why s.errors are clustered would still apply. Obs within a group would still be within a group, and thus possibly correlated, regardless of how the sample was chosen. (Obviously, this depends
on the model.) But will follow the comments for arguments, proofs, to the contrary.
21. Forgot the link:
22. This review article by Cameron and Miller is helpful. On page 21, they write:
“First, given V[b] defined in (7) and (9), whenever there is reason to believe that both the regressors and the errors might be correlated within cluster, we should think about clustering defined
in a broad enough way to account for that clustering. Going the other way, if we think that either the regressors or the errors are likely to be uncorrelated within a potential
group, then there is no need to cluster within that group.”
I think the second sentence is what you care about.
You can see the idea more easily in the parameterized Moulton formula. It’s equation (6) in the paper. The equation shows that the “inflation” is a product of within cluster correlation in the
regressor (treatment) and within cluster correllation in the outcome. If either of those terms is equal to 0 then there is no variance inflation to worry about. In an individual level randomized
experiment then the within cluster correlation in the treatment will be zero and so there is no need to cluster.
The one regressor example that they give in section IIA applies well to experiments with person level random assignment and it does not use the parametric Moulton approach. It sets things up with
the sort of cluster standard errors that come out of the stata cluster option.
23. Clusterjerk – Chris Blattman https://t.co/6DpXD32Qd3
24. I had the same puzzle when using fixed effects at the level where you’d usually cluster. Seems overkill to do both.
25. RT @cblatts: Are you a clusterjerk or am I? (A bleg on whether I need to cluster std errors in a field experiment.) https://t.co/R21D5pkKpg
26. RT @cblatts: Are you a clusterjerk or am I? (A bleg on whether I need to cluster std errors in a field experiment.) https://t.co/R21D5pkKpg
27. @eduardo_leoni @BrendanNyhan @ClaytonNall @cblatts what I meant: this point doesn’t matter for consistency. It’s an efficiency issue.
28. @FlorianFoos @BrendanNyhan @ClaytonNall @cblatts they use the original Moulton factor, hides the key result (correlation in treatment).
29. @cdsamii @BrendanNyhan @ClaytonNall @cblatts Arceneaux and Nickerson 2009 PolAnalysis would be a good cite imo
30. @eduardo_leoni @BrendanNyhan @ClaytonNall @cblatts ? W/ design based methods, effects are always presumed to vary arbitrarily.
31. @BrendanNyhan @ClaytonNall @cblatts these are all implications of the generalized Moulton factor (cf Mostly Harmless).
32. @cdsamii @BrendanNyhan @ClaytonNall @cblatts if the effect of treatment varies across clusters you would still have to account for it.
33. @BrendanNyhan @ClaytonNall @cblatts this is because of the negative residual correlation of *treatment* vars within the group.
34. RT @cblatts: Are you a clusterjerk or am I? (A bleg on whether I need to cluster std errors in a field experiment.) https://t.co/R21D5pkKpg
35. @BrendanNyhan @ClaytonNall @cblatts if you assign *within* groups clustering can also be consistent and yield *smaller* s.e.
36. @BrendanNyhan @ClaytonNall @cblatts Chris is correct: clustering at the level of assignment is, typically, correct.
37. @ClaytonNall @cblatts @cdsamii per your other post, thinking about evaluating experiment w/RI instead suggests this doesn’t make sense
38. It’s not a proof, but just look at the Moulton formula for the design effect of clustering on standard errors. You only need to cluster if your outcome has a nonzero covariance across
observations within clusters, and the same is true of your main explanatory variable of interest. Randomizing at the individual level destroys the second of these within cluster covariances.
39. @ClaytonNall @cblatts i think we need @cdsamii on the case
40. RT @cblatts: Are you a clusterjerk or am I? (A bleg on whether I need to cluster std errors in a field experiment.) https://t.co/R21D5pkKpg
41. @BrendanNyhan @cblatts Who’s arguing this? (Note that HHs could be clusters, though, if you are not Kish sampling, etc.)
42. “Clusterjerk” https://t.co/EKFtVghF7B
43. @cblatts report back on the results! | {"url":"https://chrisblattman.com/blog/2015/12/08/clusterjerk/","timestamp":"2024-11-03T14:02:29Z","content_type":"text/html","content_length":"145483","record_id":"<urn:uuid:aaa26f53-0646-40f0-8f7d-19dfcd11834d>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00396.warc.gz"} |
Sherlock Holmes applies “meteorology” to geology
Sherlock Holmes and Dr. Watson are once again resting in their favorite location, in front of the fireplace. Holmes keeps staring at the mantelpiece, in silence. No drinks, no pipe.
– In our last after-dinner discussion, you brought up the word “subduction” – something that was unknown to me. Do you have more to say about this phenomenon, Dr. Watson asks.
Part 1: Sherlock Holmes and the Hippopotamus in the Basin
Holmes replies: Well – there is probably a lot to be said about this phenomenon. However, I will only touch on it briefly because today I have much bigger fish to fry. Maybe we might call what I will
tell you, “The Dinosaur in the Mantle”.
– Steady now Holmes, you might sound a bit more serious if you did not invent all these, well frankly, a bit silly expressions, Dr. Watson says.
Holmes, with a determined expression on his face: I am sorry to say, Watson, your lack of knowledge and imagination prevents you from seeing the necessity of such grand words. However, you will get
Holmes continues: Let’s finish off the subduction story and get on with the big stuff. Last time we spoke about basin formation due to uplift, erosion, and faulting caused by mantle upwelling. I also
told you that the reason for all of this is “subduction”.
– Exactly, Dr. Watson interrupts, and right now I was hoping for the continuation of your story.
Holmes explains: Well Watson, subduction is the process where old and dense oceanic crust, earlier formed in the deep oceans by upwelling magma, sinks in because of its density. You see, contrary to
ice that floats on water, the cold and dense oceanic crust does not “float” on the hot mantle below. Eventually, it will sink in. Because the oldest oceanic crust is found where the new oceanic crust
was first created – often near the coast of continents where rifting occurred – that is where it starts to sink in first. This is also the region where most of the sediments are brought in by rivers
and glaciers over the millennia. This aids in pushing the oceanic crust further down.
Holmes continues: But the thing is, since both the oceanic crust and the sediments were “born” in the sea, they contain a lot of seawater that subducts with it. This water, often bound to certain
minerals – that we will not take time to discuss – and its dissolved salts, are later liberated as the crust reaches certain depths. The fluids invade the mantle, make it less dense and less viscous,
and produces the mantle upwelling we spoke about and also produces volcanism. We will leave it there because as I said – we have bigger fish to fry.
– Ok then, I guess we might return to the subject on a later occasion if I can think of an intelligent question to ask, Dr Watson replies.
Holmes replies: Indeed, Watson. Have you ever heard about the French scientist Gaspard-Gustave Coriolis?
– No, Holmes, why should I?
Holmes: That is a question you will have the answer to in a minute.
He continues with the following story:
“A person on a moving train is pointing a rifle straight out the window. The trigger is pulled at the exact moment when the rifle points directly at a non-moving target situated by the railway. When
standing next to the target and seeing the bullet arrive – in slow motion of course, you might see the bullet deviating sideways and miss the target. If the difference in velocity between the target
and train/rifle is larger, or the bullet moves more slowly, the deviation will be greater.”
Holmes adds: This is an example of what is called the Coriolis Effect – it is not a force in itself, but forces may arise due to the relative movement of different objects contacting each other.
The face of Dr.Watson looks like a big question mark.
– What on earth is the connection between this rifle-man on a train, and geology, he exclaims.
Holmes replies: Well, Watson, good question, I am certain that many geologists would ask the same question. Many years ago, meteorologists observed that winds blowing towards a low-pressure system in
the atmosphere in the northern hemisphere made the air circulate counter-clockwise.
Holmes continues: And on the southern hemisphere, the air in a low-pressure system rotates clockwise. The reason is this: The Earth and the atmosphere may be regarded as a rigid body, rotating from
the west to the east. One revolution takes 24 hours, regardless of your location. Hence, the rotational speed is greatest along the Equator. Away from the Equator, the rotational speed is
proportional to the cosine of the latitudinal angle above/below the Equator, because this defines the radial distance between the axis of rotation and the surface of the Earth.
Holmes: Winds blowing northwards from the Equator are moving faster eastwards than the location of a low-pressure system further north. Therefore, just like the bullet on the train, the wind deviates
to the east of the low-pressure system. Winds blowing southwards towards the low-pressure start their journey moving at a slower pace eastward and deviating to the west of the low-pressure system. If
we regard the low-pressure system as some sort of wheel, the winds will combine to push it around in a counter-clockwise direction.
– I will have to make a drawing of that to understand what you are saying, Dr. Watson laments.
He makes the following drawing:
Holmes adds: Anyway, let it be no surprise that the air escaping from high-pressure systems, produces a rotation opposite to that of low-pressure systems. He continues: do you know what causes low
pressures and high pressures in the atmosphere, Watson?
– I am afraid not, Holmes, Dr.Watson replies.
Holmes explains: Low pressure systems are created by ascending light air that requires new air to flow in and replace it. High-pressure systems are created by descending air. Usually, these vertical
velocity components are not given much attention in weather forecasts. This might be because humans normally spend their time at the surface of the Earth. What goes on at five km elevation in the
atmosphere is not that important to common people.
Holmes continues: Anyway, in geology, vertical motion should not be overlooked. Rotational velocities are also a function of the radial distance from the Earth’s axis of rotation. In other words, the
Coriolis Effect is relevant for masses moving towards, or away from, the axis of rotation. This is true whether the masses are moving along the surface or moving up or down below the surface. Masses
moving away from the axis of rotation will deviate westwards because they are initially rotating slower towards the east than the regions further away from the axis. Masses moving towards the axis of
rotation will deviate eastwards relative to the region they are approaching. This is independent of what hemisphere we are observing.
Holmes: However, Watson, we are not finished yet. Do you remember the pressure systems in the air where rotation is created in the air flowing near the surface?
– Yes, Holmes, – If you don’t mind, I have a slightly better memory than a goldfish, Watson replies.
Holmes: Ok then Watson, let’s finish off these very rudimentary elaborations on the Coriolis Effect in the mantle.
He continues: As mantle masses approach a location where they are allowed to ascend, due to weakening of the mantle/crust above, they may be allowed to flow in from many directions just like the air
in the low-pressure system. Hence, in this situation, they will also rotate counter-clockwise on the northern hemisphere and clockwise on the southern hemisphere. On the Equator itself, rotation is
cancelled out but the westwards deviation will still be present. Now my friend, are you perhaps seeing some light at the end of the tunnel?
Holmes: This is the Dinosaur in the Mantle. Geologists seem to have overlooked the fact that subducting slabs, upwelling mantle, and continental drift are all subjected to the Coriolis Effect,
sometimes in three dimensions. They probably think that the effect is not visible since rocks are much more rigid than air. However, a fundamental principle in physics is not turned off just because
someone thinks it is. Fortunately, rocks tend to preserve signs that reveal their previous experiences. For this to occur, the forces arising from relative movements have to exceed the strength of
the rocks in question and produce long-lasting deformations. In other words, very old events may still be detectable in the structure of rocks if they were exposed to substantial Coriolis Effects.
– I am impressed, Dr. Watson states, have you checked if your theories actually might be valid?
Holmes: Of course, I have! Do you think that I am just rambling along to keep you amused? Where is that drink I ordered, by the way?
Holmes continues: Anyway, because of the Coriolis Effect and the slightly different behaviour on the northern and the southern hemispheres, the rocks will tell a story of a “Tectonic Equator”. Rigid
rocks will develop what is called strike-slip faults if they are subjected to large enough twisting forces. This will lead to so-called right-slip and left-slip faulting according to where the
twisting occurred, relative to the Equator. Mid-ocean, volcanic islands will show signs of rotation, just like any other low-pressure system, because they are located directly on the oceanic crust
with the deformable mantle below. Have a look at Fiji, for instance.
Holmes asks: Do you remember that I told you about the hiatus that will always be present in the vicinity, or within the big sedimentary basins?
– Yes, of course, is there more, Dr. Watson asks in anticipation.
Holmes says: Indeed, at the very bottom of intracratonic basins, where complete rifting has not destroyed the “evidence”, you may still observe a strike-slip fault in the basement to further confirm
the upwelling and twisting mantle millions of years ago. And, in addition, the type of strike-slip fault will tell you if the event occurred to the north or to the south of the Equator.
Holmes: There is so much more, and so many implications of what I have been telling you however, it is once again getting late. Now, do you believe that this is a Dinosaur in the Mantle?
Watson nods in silence.
– And you have just touched on “upwards” movements, without considering the velocity vectors of different kinds of movements on, or within the Earth. There is surely more to this story, he adds
Holmes exclaims: You impress me, Watson!
Holmes, getting slightly impatient and raising his voice: Where is that drink I ordered?
– Make it two, Watson shouts.
The butler seems to be in serious trouble now.
Inspired by Arthur Conan Doyle | {"url":"https://geo365.no/sherlock-holmes-applies-meteorology-to-geology/","timestamp":"2024-11-10T10:42:49Z","content_type":"text/html","content_length":"95840","record_id":"<urn:uuid:9bdd498f-0ee4-4857-90ab-99d19e44c77a>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00001.warc.gz"} |
Course Turkish Deniz Yapılarının Dinamiği
Name English Dynamics of Marine Structures
Course DTM Credit Lecture Recitation Laboratory
Code 502E (hour/week) (hour/week) (hour/week)
Semester - 3 3 - -
Course Language English
Course İsmail Hakkı Helvacıoğlu
Course Course Description Structural Dynamics; basic principles, single degree of freedom, undamped behavior, finding the natural frequency, types of loads, free oscillation, forced
Objectives vibration. Mathematical modeling of Real Elastic Structures; matrix theory of vibration, free vibration, matrix iteration. Determination of lowest natural frequency, proof of
convergence in iteration, orthogonality of the normal modes, uncoupled vibration, forced vibration. The energy or Rayleigh method of determining approximate frequencies; energy of
system, Rayleigh method of determining Approximate frequencies, approximate analysis of a general system, selection of the vibration shape, Improved Rayleigh method. Lagrange
equation of motion; Lagragian method, generalized force. Natural frequencies and mode shapes of marine risers; an approximate solution, strain energy due to bending, strain energy
due to tension, evaluating of integrals, total strain energy, kinetic energy. Rayleig-Ritz method for approximate frequencies. Lateral vibration of cables under tension; differential
equation of string under uniform tension, approximate frequencies of risers and pipelines, Newmark ?-method, numerical solution of differential equations, numerical solution of
differential equations. Natural Frequencies and mode shapes of uniform beams; boundary conditions, simple supported, clamped supported, free end. Natural frequencies of Buoy system.
Course Course Description Structural Dynamics; basic principles, single degree of freedom, undamped behavior, finding the natural frequency, types of loads, free oscillation, forced
Description vibration. Mathematical modeling of Real Elastic Structures; matrix theory of vibration, free vibration, matrix iteration. Determination of lowest natural frequency, proof of
convergence in iteration, orthogonality of the normal modes, uncoupled vibration, forced vibration. The energy or Rayleigh method of determining approximate frequencies; energy of
system, Rayleigh method of determining Approximate frequencies, approximate analysis of a general system, selection of the vibration shape, Improved Rayleigh method. Lagrange
equation of motion; Lagragian method, generalized force. Natural frequencies and mode shapes of marine risers; an approximate solution, strain energy due to bending, strain energy
due to tension, evaluating of integrals, total strain energy, kinetic energy. Rayleig-Ritz method for approximate frequencies. Lateral vibration of cables under tension; differential
equation of string under uniform tension, approximate frequencies of risers and pipelines, Newmark ?-method, numerical solution of differential equations, numerical solution of
differential equations. Natural Frequencies and mode shapes of uniform beams; boundary conditions, simple supported, clamped supported, free end. Natural frequencies of Buoy system.
Course Outcomes
Other References | {"url":"https://ninova.itu.edu.tr/en/courses/institute-of-science-and-technology/13404/dtm-502e/form","timestamp":"2024-11-02T12:20:07Z","content_type":"application/xhtml+xml","content_length":"12560","record_id":"<urn:uuid:d4634107-257d-48bd-8174-8491c89a56f9>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00191.warc.gz"} |
On overall measure of non-classicality of $N$-level quantum system and its universality in the large $N$ limit
Vahagn Abgaryan
In this report we are aiming at introducing a global measure of non-classicality of the state space of $N$-level quantum systems and estimating it in the limit of large $N$. For this purpose we
employ the Wigner function negativity as a non-classicality criteria. Thus, the specific volume of the support of negative values of Wigner function is treated as a measure of non-classicality of an
individual state. Assuming that the states of an $N$-level quantum system are distributed by Hilbert-Schmidt measure (Hilbert-Schmidt ensemble), we define the global measure as the average
non-classicality of the individual states over the Hilbet-Schmidt ensemble. We present the numerical estimate of this quantity as a result of random generation of states, and prove a preposition
claiming its exact value in the limit of $N\to \infty$ | {"url":"https://2020.dccn.ru/papers/134","timestamp":"2024-11-02T21:54:08Z","content_type":"text/html","content_length":"16602","record_id":"<urn:uuid:11728e7a-5d08-43e7-a65c-c0509aa8d1b9>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00112.warc.gz"} |
Writing Equations Of Lines Given A Graph Worksheet - Graphworksheets.com
Writing An Equation Given A Graph Worksheet – Graphing equations is an essential part of learning mathematics. It involves graphing lines and points, and evaluating their slopes. Graphing equations
of this type requires that you know the x and y-coordinates of each point. To determine a line’s slope, you need to know its y-intercept, which … Read more | {"url":"https://www.graphworksheets.com/tag/writing-equations-of-lines-given-a-graph-worksheet/","timestamp":"2024-11-02T21:33:26Z","content_type":"text/html","content_length":"47798","record_id":"<urn:uuid:91482d1b-ff24-408b-b911-5410f4bac21e>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00392.warc.gz"} |
Estimated Glomerular Filtration Rate
Estimated GFR Calculated Using the CKD-EPI Equation
See the prior memo, dated May 26, 2020.
On July 5, 2022, the laboratories at UWMC-ML, UWMC-NW, HMC, and SCCA began using the Chronic Kidney Disease Epidemiology Collaboration 2021 equation (2021 CKD-EPI) to calculate estimated glomerular
filtration rate (eGFR). This new equation is now recommended for estimating GFR from serum creatinine by the National Kidney Foundation (NKF) and the American Society of Nephrology (ASN). The
equation was modeled using the original CKD-EPI cohorts, but eliminated consideration of race as a variable. This has achieved a less biased approach to kidney function testing by recognizing that
race is a social construct and an ineffective variable in the biomedical environment.
Why is this change being made?
In 2021, the NKF-ASN Task Force on Reassessing the Inclusion of Race in Diagnosing Kidney Diseases concluded that race should not be a parameter in GFR estimating equations.
What this means
In 2020, UW Medicine shifted from using the MDRD equation to estimate GFR to using a form of the original 2009 CKD-EPI equation that lacked the race variable. During the development of the 2021
CKD‑EPI equation, race was excluded from the putative statistical models used to predict measured GFR. As a result, there will be slight shifts in the calculated eGFR for all patients. As shown in
the table, this could lead to the recategorization of many of our UW Medicine patients to a stage of CKD associated with better kidney function. A comparison of the performance of the 2009 equation
without race coefficient (rows) and the 2021 CKD-EPI equation (columns) in our patient population is shown here:
2021 CKD-EPI equation →
2009 CKD-EPI equation ↓ ≥90 (149,748) 60-89 (68,735) 45-59 (12,537) 30-44 (5,850) 15-29 (2,651) <15 (2,239)
calculated value (N)
≥90 (130,122) 130,122 (100%)
60-89 (82,986) 19,626 (23.6%) 63,360 (76.4%)
45-59 (15,786) 5,375 (34.0%) 10,411 (66.0%)
30-44 (7,349) 2,126 (28.9%) 5,223 (71.1%)
15-29 (3,095) 627 (20.3%) 2,468 (79.7%)
<15 (2,422) 183 (7.6%) 2,239 (92.4%)
No changes in ordering practices are indicated. However, a change in calculated eGFR before and on/after July 5, 2022 should be interpreted with this calculation change in mind.
For additional information, please contact Andy Hoofnagle, MD PhD, Director of UWMC Chemistry (ahoof@uw.edu), or Geoff Baird, MD PhD, Director of HMC Chemistry (gbaird@uw.edu).
Inker LA, et al. New Creatinine- and Cystatin C-Based Equations to Estimate GFR without Race. N Engl J Med 2021, 385:1737-1749. 34554658
Ghuman JK, et al. Impact of Removing Race Variable on CKD Classification Using the Creatinine-Based 2021 CKD-EPI Equation. Kidney Med 2022, 4:100471. 35756325
Associated Tests
Last updated 2022-11-14T16:22:17.423648+00:00 | {"url":"https://testguide.labmed.uw.edu/guideline/egfr?","timestamp":"2024-11-03T13:42:18Z","content_type":"text/html","content_length":"18731","record_id":"<urn:uuid:105b0618-c0c7-4f22-abc1-c6b901707675>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00448.warc.gz"} |
New prostate volume calculation formula to improve the specificity of PSA density.
Akira Kimura, Kiyoshi Hirasawa,Yuji Kurooka,and Kazuki Kawabe
Tokyo, Japan
[objective] PSA density is the quotient of PSA divided by transrectal ultrasound determined prostate volume.A common volume equation; prolate ellipsoid (height times width times length times π/6) has
a risk to increase false positives in BPH, because prolate ellipsoid has a tendency to underestimate the volume in BPH.We recently developed a new prostate volume calculation method using the full
information obtained from biplane transrectal ultrasonogram (Int J Urol 1997;4:152-156).The method; biplane planimetry calculates the volume accurately in BPH as well as in cancer.The usefulness of
prolate ellipsoid and biplane planimetry as denominator in PSA density was compared.
[method] In nineteen patients with prostatic cancer and twenty patients with BPH having PSA values of 4 to 10 ng/ml,prostatic volumes were calculated both by prolate ellipsoid (V[０]) and biplane
planimetry (V[１]).
[result] The averages of V[０] were 32ml in cancer and 49ml in BPH, while those of V[１] were 31ml and 52ml respectively. Accordingly, the averages of PSA/V[０] were0.27 in cancer and 0.13 in BPH,
while those of PSA/V[１] were 0.28 and 0.12 respectively. Using a PSA density cutoff of 0.15 as recommended in the literature, the sensitivity and specificity of PSA/V[０] were 74% and 65%, while
those of PSA/V[１] were 79% and 75%.
[conclusion] Because biplane planimetry does not underestimate the volume of BPH, the false positive rate of PSA density by biplane planimetry is smaller than that by prolate ellipsoid. | {"url":"https://www.akira-kimura.com/h/okada.html","timestamp":"2024-11-12T22:50:32Z","content_type":"text/html","content_length":"3181","record_id":"<urn:uuid:67ccdbe7-a833-4fa2-85c1-0520ca655d26>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00559.warc.gz"} |
Q1 If sin(A+B)=1 and tan(A+B)=31 find the value of
ii) tanA+c... | Filo
Question asked by Filo student
Q1 If and find the value of ii) (ii) Q3 If show that Q4 Evaluate Q5 is a right-angled triangle at . if find. (i) iii and (iii) (iv)
Not the question you're searching for?
+ Ask your question
Video solutions (1)
Learn from their 1-to-1 discussion with Filo tutors.
6 mins
Uploaded on: 2/1/2023
Was this solution helpful?
Found 7 tutors discussing this question
Discuss this question LIVE for FREE
6 mins ago
One destination to cover all your homework and assignment needs
Learn Practice Revision Succeed
Instant 1:1 help, 24x7
60, 000+ Expert tutors
Textbook solutions
Big idea maths, McGraw-Hill Education etc
Essay review
Get expert feedback on your essay
Schedule classes
High dosage tutoring from Dedicated 3 experts
Practice more questions on All topics
View more
Students who ask this question also asked
View more
Stuck on the question or explanation?
Connect with our Mathematics tutors online and get step by step solution of this question.
231 students are taking LIVE classes
Question Text Q1 If and find the value of ii) (ii) Q3 If show that Q4 Evaluate Q5 is a right-angled triangle at . if find. (i) iii and (iii) (iv)
Updated On Feb 1, 2023
Topic All topics
Subject Mathematics
Class Class 11
Answer Type Video solution: 1
Upvotes 73
Avg. Video Duration 6 min | {"url":"https://askfilo.com/user-question-answers-mathematics/q1-if-and-find-the-value-of-ii-ii-q3-if-show-that-q4-34303530303935","timestamp":"2024-11-11T00:22:40Z","content_type":"text/html","content_length":"441044","record_id":"<urn:uuid:1b9706c3-1f98-4d7a-9e7a-3d4b0661af59>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00143.warc.gz"} |
7.2 Round off Errors and the Derivative
Home | 18.013A | Chapter 7 Tools Glossary Index Up Previous Next
7.2 Round off Errors and the Derivative
How can d being too small cause problems?
Usually computations on a calculator or computer or by hand are not performed to perfect accuracy. There are very small errors. Normally, these very small errors (called round off errors) can be
ignored because the "noise" they represent in your evaluation is extremely small compared to the signal, which consists of the value of f itself. (A notable exception occurs when your answer is 0;
then the machine's answer will be only the error it has created.)
In general, if you take two very similar numbers, like f(x[0] + d) and f(x[0]) and take their difference, that difference will be very much smaller than either term and the information in the signal
represented by the difference will therefore be much smaller than the signal represented by either, while the noise level usually remains about the same for the terms and the difference.
Taking the result of the subtraction and dividing by a very small d (which is the same as multiplying by a huge
If you make d smaller than the accuracy of your machine's computation, your answer will typically be off by more than 1, or your program will accuse you of dividing by 0 when you divide by d.
The spreadsheet allows you to perform a very large number of calculations of this kind for a wide choice of d values with essentially no more work than is involved in one such calculation. This
usually gives you the power to look for yourself and see where round off error is causing significant error.
You will then be troubled by this effect only when the answer you are computing is too far from the correct answer for d values at which this effect becomes noticeable. In consequence we try to make
use of techniques that will allow us to get accurate estimates for as large d values as possible.
Set up a computation using one d value on one line of the spreadsheet, then on the next line set d = the old
If your estimate of the derivative were to home in on a value and stay there, that would probably be the derivative you seek. Alas this does not always happen. The estimates tend to home in then
start to move away again, as the effects of round off error make themselves felt.
(Fortunately modern computers keep greater accuracy in their computations than they display on the screen, so that you can tolerate some amount of loss of accuracy due to round-off error, without
even noticing it.)
However there is something much better that generally does home in on a value that is recognizable as the derivative you seek, and it takes no more work! Instead of computing | {"url":"https://ocw.mit.edu/ans7870/18/18.013a/textbook/HTML/chapter07/section02.html","timestamp":"2024-11-12T05:22:39Z","content_type":"text/html","content_length":"5596","record_id":"<urn:uuid:cce27117-2d7d-4346-b090-c0ad6578e0f3>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00732.warc.gz"} |
runCommand -package:structured-cli -package:github-release -package:process -package:io-streams is:exact -package:options -package:graphviz -package:xmonad-contrib -package:mongoDB -package:libmpd package:pandoc-plot
runCommand -package:structured-cli -package:github-release -package:process -package:io-streams is:exact -package:options -package:graphviz -package:xmonad-contrib -package:mongoDB -package:libmpd
Run a command within the PlotM monad. Stderr stream is read and decoded, while Stdout is ignored. Logging happens at the debug level if the command succeeds, or at the error level if it does not | {"url":"https://hoogle.haskell.org/?hoogle=runCommand%20-package%3Astructured-cli%20-package%3Agithub-release%20-package%3Aprocess%20-package%3Aio-streams%20is%3Aexact%20-package%3Aoptions%20-package%3Agraphviz%20-package%3Axmonad-contrib%20-package%3AmongoDB%20-package%3Alibmpd%20package%3Apandoc-plot","timestamp":"2024-11-04T04:52:44Z","content_type":"text/html","content_length":"204940","record_id":"<urn:uuid:c1b26d43-28ad-4f9f-889a-a542a6954b07>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00578.warc.gz"} |
ASVAB Arithmetic and Mathematics Preview
Are you planning on joining the US armed forces?
Then be prepared to write the ASVAB Arithmetic and Mathematics Test.
You might not have heard of it before or maybe you have but need more enlightenment.
Whichever it is, this article would teach you all you need to know about ASVAB Arithmetic and Mathematics.
The Absolute Best Book to Ace the ASVAB Math Test
Original price was: $24.99.Current price is: $14.99.
What is ASVAB?
ASVAB means Armed Services Vocational Aptitude Battery. It is a set of tests, set up by the United States Military Entrance Processing Command.
The ASVAB test can be written by any eligible candidate but the majority of people who take the test are U.S. high school students in grades 10th, 11th, and 12th.
The ASVAB test is very significant and used to decide the qualification for enlistment in the U.S. Army.
The test was introduced into the US Military System in 1968 but was embraced by all branches of the military in 1976.
The test is purely based on calculations which include Arithmetic and Mathematics.
ASVAB Test Requirements
Although the ASVAB test is brain-twisting, with the right preparation you will be able to pass it.
Each military branch has got its cut-off mark. For the Army, it is any score above 31 while for the Air Force, it is above 36. This implies that a standard score of 50 is an average score while a
score of 60 is above average.
If you are fully prepared, I can guarantee that you will score above the average score.
The most comprehensive workbook for the ASVAB Math test
Original price was: $21.99.Current price is: $16.99.
The Types and Duration of the ASVAB Test
The ASVAB Arithmetic and Mathematics preview is of two types. The first is the written format while the other one is the computerized format.
The testing procedure varies depending on the mode of administration.
The test currently contains 9 sections and takes three hours to complete. Although the duration of each test varies from 7 to 39 minutes.
The table below shows some of the tests you would write and the duration.
Computerized Format Test Written Format Test
Paragraph Comprehension(PC)-10 questions in 20 minutes Paragraph Comprehension (MC)-15 questions in 30 minutes
Arithmetic Reasoning (AR)-15 questions in 39 minutes Arithmetic Reasoning (AR)- 30 questions in 36 minutes
General Science (GS)-15 questions in 8 minutes General Science (GS)-45 questions in 20 minutes
Word Knowledge (WK)-15 questions in 8 minutes Word Knowledge (WK)-35 questions in 11 minutes
Mathematics Knowledge (MK)-15 questions in 20 minutes Mathematics Knowledge (MK)- 25 questions in 24 minutes
Assembling Object(AO)-15 questions in 40 minutes Assembling Object (AO)-25 questions in 15 minutes
The ASVAB Arithmetic and Mathematics test might seem difficult and hard to take nevertheless; you can solve all the questions within the given duration given you prepare well and adequately.
Study well and solve past questions if you can get your hands on any.
Even if you are not a lover of Arithmetic and Mathematics and you want to join the US Military, you have nothing to be scared of as long as you do all you need to in preparation.
Wishing you success if you take the test soon!
The Best Study Guide to Ace the ASVAB Math Test
Original price was: $19.99.Current price is: $14.99.
More from Effortless Math for ASVAB Test …
Are you planning on joining the US armed forces and need a complete and FREE Math course?
Take a look at our Ultimate ASVAB Math Course to help you review all ASVAB Math concepts.
Need to check out the most common ASVAB math formulas?
We have prepared a complete list of ASVAB Math formulas to help you practice and prepare for your test.
Also, have a look at our How to Prepare for the ASVAB Math Test and Top 10 Tips to ACE the ASVAB Math to prepare and succeed for the ASVAB test.
The Best Books to Ace the ASVAB Math Test
Original price was: $25.99.Current price is: $13.99.
Original price was: $20.99.Current price is: $15.99.
Original price was: $76.99.Current price is: $36.99.
Have any questions about the ASVAB Test?
Write your questions about the ASVAB or any other topics below and we’ll reply!
Related to This Article
What people say about "ASVAB Arithmetic and Mathematics Preview - Effortless Math: We Help Students Learn to LOVE Mathematics"?
No one replied yet. | {"url":"https://www.effortlessmath.com/blog/asvab-arithmetic-and-mathematics-preview/","timestamp":"2024-11-06T02:47:59Z","content_type":"text/html","content_length":"94444","record_id":"<urn:uuid:92e86fc2-685b-4310-91d2-ac6073287bc3>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00610.warc.gz"} |
Principal Component Analysis in R
What is Principal Component Analysis (PCA)?
Principal Component Analysis (PCA) is a method that helps simplify complex high dimensional data by mapping it onto a lower dimensional space. This process retains the information within the data.
PCA identifies directions (known as principal components) where the data shows the most variation making it possible to reduce dimensions without losing crucial details. It is commonly used for tasks
like visualizing data extracting features and recognizing patterns. PCA operates by computing the eigenvectors and eigenvalues of the datas covariance matrix then organizing them accordingly. The
eigenvectors associated with the eigenvalues represent the most important principal components facilitating the conversion of data, into a more streamlined format.
Why Use PCA in Data Analysis?
PCA is often used in data analysis because it offers a way to represent a dataset. It changes the variables into a new set of independent variables known as principal components reducing complexity
while preserving important information. This proves useful for managing datasets with many dimensions making analysis and visualization simpler.
Moreover PCA acts as a step by removing unnecessary or noisy features leading to better performance in tasks such as clustering or regression. Furthermore PCA aids in determining the required number
of components for modeling by assessing their impact, on the desired outcome.
Applications of PCA
PCA is widely used in fields, such as simplifying datasets by reducing dimensions while retaining key information. It aids in image and video processing tasks like image compression and face
recognition, supporting risk management and asset pricing in finance, analyzing gene expression data and DNA sequences in bioinformatics, examining survey data and facilitating clustering in social
sciences. Additionally PCA contributes to enhancing the precision of analysis through dimensionality reduction prior, to identifying discriminative features.
Understanding the Mathematics Behind PCA
Principal Component Analysis (PCA) relies on mathematical ideas like eigenvectors, eigenvalues, covariance matrices and singular value decomposition (SVD). Grasping these concepts is crucial, in
recognizing how PCA can convert complex data into a more straightforward format while retaining vital details.
Covariance Matrix
The covariance matrix is essential in PCA for understanding relationships between variables. It shows the covariance between pairs of variables, with diagonal elements representing variances and
off-diagonal elements representing covariances. By calculating the covariance matrix, patterns and relationships among variables can be identified, aiding in dimensionality reduction.
Singular Value Decomposition (SVD)
Singular Value Decomposition (SVD) is a technique used to decompose a matrix into three matrices extracting important information from the data. SVD finds applications in data analysis tasks such as
compressing images recognizing faces and creating recommendation systems. In Principal Component Analysis (PCA) SVD plays a role, in reducing the dimensions of datasets while preserving key
Proportion of Variance Explained
A scree plot is useful for showing how much of the variance is accounted for by each principal component, in PCA. It helps decide on the number of components to keep by displaying the percentage of
variance explained by each component ensuring that important variances are captured while filtering out noise.
Unit Variance
In PCA it's important to standardize data so that all variables have an impact on the analysis. This helps prevent any variable from overshadowing the results by ensuring that the dispersion of data
points, around the mean is consistent.
Largest Variance
In PCA it's crucial to pinpoint the variables that have the impact on dataset diversity. By working out eigenvalues and eigenvectors we can figure out the direction of variance which offers important
clues about the factors at play, in the dataset.
Implementing PCA in R Using the factoextra Package
To perform Principal Component Analysis (PCA) in R you can utilize the package. This package provides tools for extracting and displaying outcomes, from PCA well as other multivariate analyses.
Installing and Loading factoextra Package
To install and load factoextra, follow these steps:
This package provides tools for data visualization, clustering, and dimensionality reduction.
Loading Data for Analysis
When you're getting ready to analyze data make sure to bring in the package. For instance you can load up the USArrests dataset using this method --
Pre-processing Data for PCA
Prep work is crucial, for getting PCA outcomes. Key tasks to consider include —
• Dealing with Missing Data and Anomalies; Fix or substitute missing data points and spot anomalies to steer clear of outcomes.
• Standardizing Data; Make sure all data points have an average of zero and a variance of one to contribute equally.
• Checking for Multicollinearity; Tackle highly correlated data points to prevent skewed PCA outcomes.
By following these prep steps the data can be primed for PCA examination resulting in precise and dependable results. | {"url":"https://hyperskill.org/university/r-language/principal-component-analysis-in-r-670a1","timestamp":"2024-11-02T07:55:27Z","content_type":"text/html","content_length":"39635","record_id":"<urn:uuid:12026129-1aeb-45f4-ad9e-0d006772b057>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00639.warc.gz"} |
Binary magic card trick - Chalkdust
This post was part of the Chalkdust 2016 Advent Calendar.
Ah, the Christmas holidays. A time to be spent hiding from the cold, wearing pyjamas, eating too much food and solving integro-differential equations. (At least that’s how I think everyone spends
Christmas, right?) Come Christmas Day, after you’ve finally cracked Chalkdust Issue 4’s Crossnumber and stuffed your face with turkey, why don’t you stun your (slightly drunk) family and friends with
a very simple math-based magic trick?
The Trick
This trick follows a fairly standard magic setup: you are going to successfully guess (telepathically, of course) a number decided in secret by the unsuspecting watcher. Hand your target a set of
cards with numbers on (an example set of cards can be seen below):
Tell them to think of a number in between the smallest and largest number present on the cards (in our case, between 1 and 63). Then ask they hand you every single card with that number on it. Add up
the first number on each of the cards given to you, et voilà, you’ve ‘magically’ obtained their number.
Why does it work?
Afterwards, you’ll be asked how it works. Spend a good bit of time insisting you’re in touch with the Other Side, were recently struck by lightning and can now hear other peoples’ thoughts, or are
simply VERY good at guessing. But since, as mathematicians, we need to try and convince people maths is interesting and, more importantly, not a form of witchcraft, consider explaining to them how it
works (I’ve found talking over the television during the Christmas edition of Eastenders particularly effective).
Those of you with a keen eye will have clocked it already. Certainly, a big hint is that the first number on each card is a power of 2. The cards have been designed such that each combination of
cards uniquely represents a number in binary.
For those of you who are not sure what binary is, it is the base two numeral system. You will certainly be familiar with the base ten numeral system since that’s how we normally represent numbers. A
good explanation on base systems can be found here.
As an example, we can represent forty-seven in binary and base-ten as
\text{forty seven} &=& 4 \times 10^1 + 7 \times 10^0 \\
&=& 47 \,\,\,\, (\text{base ten}) \\
&=& 1 \times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 \\
&=& 101111 \,\,\,\, (\text{base two})
So, let us re-order and label the cards as follows
Card $N$ is characterised as containing all the numbers which contain a “$1 \times$” in front of the term $2^N$. Taking our example 47 (base ten), we see that 47 appears on cards [0,1,2,3,5]. That is
the say
47 \,\,\, \text{(base 10)} &=& 1 \times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 \\
&=& 101111 (\text{base two})
Ensuring that card $N$ has $2^N$ as the first number on the card (for your ease of reference), simply adding all the first digits on the cards handed back to you will give the correct answer.
We chose 63 as it is of the form $2^6 – 1$. It is best advised to choose a number of the form $2^M -1$ as your maximum number as it ensures each card has $M/2$ digits on it, raising the least | {"url":"https://chalkdustmagazine.com/advent-calendar/09-december/","timestamp":"2024-11-02T02:06:54Z","content_type":"text/html","content_length":"89230","record_id":"<urn:uuid:b1b46b84-be36-42e4-b7a0-8ebf2e497fac>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00873.warc.gz"} |
Divisibility Rule By 7: A Comprehensive Overview - LearnAboutMath
Divisibility Rule by 7: A Comprehensive Overview
Does 2315689 is divisible by 7?
Can you answer this question within seconds?
No, then you might need to understand the divisibility rule of 7. Divisibility rule by 7 helps us to determine whether number is divisible by 7 are not.
Divsbility rule by 7, tells us that if we find the difference doubling the number at the unit place and the remaining number, difference is divisible by 7, then given number is divisible by 7.
So, lets see how to use the divisibility rule to solve a long number, without doing long Divions. This article will focus on the divisibility rule by 7, one of the lesser-known rules. We will discuss
what it is, how to use it, and its real-world applications.
What are Divisibility Rules?
Can you tell me the number 23456 is divisible by 8.
How much time you need to find the solution.
Does there’s any shortcut to check whether number is divisible by 8 are not. Yes, there’s. You can determine whether a number is divisible by given number or not by a shortcut method. This shortcut
method is known as divisibility rules.
Divisibility rules tell you with in second given number is divisible by others number are not.
For example, the divisibility rule by 2 tells us that a number is divisible by 2 if its last digit is even. Similarly, the divisibility rule by 5 tells us that a number is divisible by 5 if its last
digit is 0 or 5.
What is the Divisibility Rule by 7?
Divisibility rule by 7 is the short cut method that tells you that given number is divisible by 7 or not.
A number is divisible by 7 if and only if the difference between twice the digit in the unit’s place and the remaining digits is divisible by 7.
For example, if we take a number like 532, we need to double the digit in the unit’s place, which is 2, and subtract it from the remaining digits, which gives us 53 – 2(2) = 49. If 49 is divisible by
7, then 532 is also divisible by 7.
Examples of Numbers Divisible by 7
Now check how divisibility rule by 7 works with some numbers.
Example 1
Check whether 315 is divisible by 7 or not?
Now use divisibility rule of 7 to identify whether number is divisible by 7 or not.
1. Double the digit in the units place, which is 5.
2. Now subtract it from the remaining digits, which gives us 31 – 2(5) = 21.
3. As 21 is divisible by 7, therefore 315 is also divisible by 7.
Example 2
Check whether 469 is divisible by 7 or not?
Now use divisibility rule of 7 to identify whether number is divisible by 7 or not.
1. Double the digit in the units place, which is 9.
2. Now subtract it from the remaining digits, which gives us 46 – 2(9) = 18.
3. As 28 is divisible by 7, therefore 469 is also divisible by 7.
How to Use the Divisibility Rule for 7
Now you are clear about what is divisibility rule of 7 and how does its Works. Move one to next step and check how we can use it.
Here are the three simple step though which you can apply divisibility rule.
Step :1 Identify the number at the unit place.
Step 2: Double the number and subtract form the reaming digits.
Step 3: Check whether the difference is divisible by 7 or not. If its is divisible by 7, then given number is also disvisble by 7.
Let’s take an example to illustrate this.
Check whether 728 is divisible by 7.
Step 1: Digit at unit place is 8.
Step 2: By doubling 8 we get 16.
Step 3: Subtracting 16 from the remaining digits, we get 72 – 16 = 56.
Since 56 is divisible by 7, we know that 728 is also divisible by 7.
Understanding the Math Behind the Rule
The divisibility rule by 7 may seem like magic, but it is based on sound mathematical principles. To understand why it works, we need to delve deeper into the properties of numbers.
Every number can be expressed as a sum of its digits multiplied by powers of 10.
For example,
We can write 532 can
5 x 100 + 3 x 10 + 2 x 1.
Now we can arrange the expression as
5 x (99 + 1) + 3 x (9 + 1) + 2 x 1,
which simplifies to
5 x 99 + 3 x 9 + 7.
Notice that the sum of the digits of the original number, 5 + 3 + 2 = 10, is the same as the remainder we get when we divide the expression by 9 (7 in this case).
Now, let’s apply this concept to the divisibility rule by 7.
If we take a number like 532, we can express it as
5 x 100 + 3 x 10 + 2 x 1.
Doubling the digit in the unit’s place and subtracting it from the remaining digits, we get
(5 x 10 – 2) x 10 + 3 x 1.
Rearranging this expression, we get (5 x 9 + 3) x 10 + 7.
Notice that the sum of the digits of the original number, 5 + 3 + 2 = 10, is the same as the remainder we get when we divide the expression by 7 . This is why the divisibility rule by 7 works.
Divisibility Rules for Other Numbers
The divisibility rule by 7 is just one of many divisibility rules. Some of the other popular rules are:
• Divisibility rule by 2: A number is divisible by 2 if its last digit is even.
• Divisibility rule by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
• Divisibility rule by 4: A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
• Divisibility rule by 5: A number is divisible by 5 if its last digit is 0 or 5.
• Divisibility rule by 6: A number is divisible by 6 if it is divisible by 2 and 3.
• Divisibility rule by 8: A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
• Divisibility rule by 9: A number is divisible by 9 if the sum of its digits is divisible by 9.
• Divisibility rule by 10: A number is divisible by 10 if its last digit is 0.
Practice Problems to Test Your Knowledge
Now that we have discussed the divisibility rule by 7 in detail, let’s test our knowledge with some practice problems. Try to solve these problems using the divisibility rule by 7.
• Is the number 357 divisible by 7?
• Is the number 924 divisible by 7?
• Is the number 8756 divisible by 7?
• Is the number 1234 divisible by 7?
• Is the number 7777 divisible by 7?
You can check your answers at the end of this article.
Tricks to Remember the Rule
Remembering the divisibility rule by 7 can be tricky, especially if you use it sparingly. Here are a few tricks that can help you remember the rule:
• Notice that twice the digit in the unit’s place is the same as subtracting 5 times the digit in the unit’s place from the number formed by the remaining digits.
□ For example, if we take the number 532, we can express it as 53 x 10 + 2 x 1. Notice that 2 x 2 = 4 is the same as subtracting 5 x 2 = 10 from 53.
• If you are testing a large number for divisibility by 7, you can break it down into smaller parts.
□ For example, if we take the number 1234567, we can break it down into 123 x 10000 + 4567. Applying the divisibility rule by 7 to this expression, we get (123 x 2 – 456) x 1000 + 7. Notice
that the expression in the parentheses is divisible by 7, so we only need to test whether 7 divides into 7, which it does.
Real-World Applications of the Divisibility Rule by 7
You may wonder why you would ever need to use the divisibility rule by 7 in real life. While it may not be a commonly used rule, it does have some practical applications. For example, it can be used
to check whether a credit card number is valid. Credit card numbers are typically 16 digits long and follow a specific pattern. The last digit is the check digit, calculated using a formula involving
the other digits in the number. One of the steps in this formula is to check whether a certain portion of the number is divisible by 7.
In conclusion, the divisibility rule by 7 is a useful shortcut that can save us a lot of time when performing calculations. By understanding the concept behind the rule and practicing with some
examples, you can become proficient in using it. Remember to use tricks like breaking down large numbers into smaller parts and noticing patterns to make the process easier. While the divisibility
rule by 7 may be used infrequently, it does have practical applications in fields like banking and finance. Keep practicing and honing your math skills; you’ll be amazed at what you can accomplish.
Practice Problems Answers
• Yes, 357 is divisible by 7.
• No, 924 is not divisible by 7.
• Yes, 8756 is divisible by 7.
• No, 1234 is not divisible by 7.
• Yes, 7777 is divisible by 7. | {"url":"https://learnaboutmath.com/divisibility-rule-by-7/","timestamp":"2024-11-03T19:28:23Z","content_type":"text/html","content_length":"341070","record_id":"<urn:uuid:0b7bc2c8-72ab-4524-9c10-262c1e345267>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00888.warc.gz"} |
Rasheed bought two kinds of candy bars, chocolate and toffee, that cam
Question Stats:
74% 26% (02:20) based on 1758 sessions
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of
variables and independent equations ensures a solution.
Rasheed bought two kinds of candy bars, chocolate and toffee, that came in packages of 2 bars each. He handed out 2/3 of the chocolate bars and 3/5 of the toffee bars. How many packages of chocolates
bar did Rasheed buy?
(1) Rasheed bought 1 fewer package of chocolate bars than toffee bars.
(2) Rasheed handed out the same number of each kind of candy bar.
Looking at the original condition, we can easily figure out that this is a “2 by 2” question, a common type of question in GMAT math. We can represent the information using a table as below:
GCDS enigma123 Rasheed bought(20151007).png [ 3.66 KiB | Viewed 59356 times ]
From above, you can see that there are 2 variables (C,T) and 2 equations from the 2 equations; the number of variables match that of the equations, so there is high chance that (C) is going to be our
Combining the 2 equations,
C=T-1, and 2C/3=3T/5 are sufficient to solve for the variables, so the answer becomes (C).
For cases where we need 2 more equation, such as original conditions with “2 variables”, or “3 variables and 1 equation”, or “4 variables and 2 equations”, we have 1 equation each in both 1) and 2).
Therefore, there is 70% chance that C is the answer, while E has 25% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5%
chance. Since C is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, D or E. | {"url":"https://gmatclub.com/forum/rasheed-bought-two-kinds-of-candy-bars-chocolate-and-toffee-that-cam-129470.html?kudos=1","timestamp":"2024-11-13T12:38:55Z","content_type":"application/xhtml+xml","content_length":"960992","record_id":"<urn:uuid:9f71cac2-a9e3-4b01-89af-02560ddf8450>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00494.warc.gz"} |
Bead Calculator: Determine the Number of Beads for Your Bracelet
Home » Simplify your calculations with ease. » Lifestyle Calculators »
Bead Calculator: Determine the Number of Beads for Your Bracelet
If you’re a fan of creating bracelets using beads, then you know how important it is to get the right number of beads for your project. Luckily, with the help of a bead calculator, you can easily
determine how many beads you need for a bracelet of a certain length and width. In this article, we’ll take a closer look at how this calculator works, its formula with examples, the benefits of
using the calculator, most common FAQs, who this calculator is for, how to use this calculator, and our conclusion.
How it Works
The bead calculator is a tool that uses a simple formula to calculate the total number of beads needed for a bracelet. The formula is as follows:
B = BRL * 10 / BW
Where B is the number of beads, BRL is the bracelet length in centimeters, and BW is the bead width in millimeters.
To use the calculator, you need to know the length and width of your bracelet. The length can be selected from a range of options, or you can input a custom value. Similarly, the width can be
selected from a range of options, or you can input a custom value. Once you have entered the length and width, the calculator will automatically calculate the number of beads needed for your
Formula with Examples
Let’s say you want to create a bracelet that is 7 inches long and 4mm wide. To calculate the number of beads needed, you need to convert the length from inches to centimeters and then apply the
First, we’ll convert the length from inches to centimeters:
7 inches * 2.54 cm/inch = 17.78 cm
Next, we’ll apply the formula:
B = 17.78 cm * 10 / 4 mm = 44.45 beads
So, for a bracelet that is 7 inches long and 4mm wide, you will need 44.45 beads.
Benefits of Using the Calculator
There are several benefits to using a bead calculator:
1. Saves time: Instead of manually calculating the number of beads needed, the calculator does the work for you.
2. Accuracy: The calculator uses a precise formula, which ensures that you get the exact number of beads needed for your bracelet.
3. Convenience: You can use the calculator from anywhere, at any time, as long as you have an internet connection.
4. Customization: You can input custom values for the length and width, which allows you to create bracelets of any size.
Most Common FAQs
Can I use the calculator for bracelets of any size?
Yes, you can input custom values for the length and width, which allows you to create bracelets of any size.
Do I need any special skills to use the calculator?
No, the calculator is designed to be user-friendly and easy to use. All you need is a basic understanding of the length and width of your bracelet.
Who is this Calculator For?
This calculator is for anyone who loves creating bracelets using beads. Whether you’re a beginner or an experienced crafter, the bead calculator can save you time and ensure that you get the right
number of beads for your project.
The bead calculator is an essential tool for anyone who loves creating bracelets using beads. It saves time, ensures accuracy, and is easy to use. With the ability to input custom values, you can
create bracelets of any size, and the option to select different units of measurement makes it convenient for users from around the world. So next time you’re planning a beaded bracelet project
Leave a Comment | {"url":"https://calculatorshub.net/lifestyle-calculators/bead-calculator/","timestamp":"2024-11-10T14:51:39Z","content_type":"text/html","content_length":"118272","record_id":"<urn:uuid:4d43aed3-7131-44d2-92e1-01fbb5f4af79>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00372.warc.gz"} |
Engaging with Loyal Customers
Problem E
Engaging with Loyal Customers
On the occasion of its $10$-year anniversary, an e-commerce company is running a campaign to engage with their loyal customers. They have prepared $m$ gifts, numbered from $1$ to $m$, to thank its
loyal customers where each customer will receive no more than one gift. The company has $n$ loyal customers, numbered from $1$ to $n$. In order to ensure that customers are satisfied with the gift
they receive, the company decided to conduct a customer survey. The customer survey result is recorded by the feedback cards, each of which can be described by a tuple of three positive integers $
(i,j,p)$ indicating that customer $i$ has a satisfaction level of $p$ if he or she receives the gift $j$. Each customer will give his or her level of satisfaction for every gift unless he/she has a
satisfaction level of $0$. At the end, the company receives $k$ feedback cards from their loyal customers.
Based on the result of the customer survey, your task is to determine how to send gifts to loyal customers to bring the greatest sum of satisfaction of all customers receiving the gifts.
• The first line contains three integers $m$, $n$, $k$ $(1 \leq m,n \leq 1\, 000, \; 1 \leq k \leq m \times n)$;
• The next $k$ lines describe the customer survey results, each of which contains three positive integers $i, j, p$ described above $(1 \leq i \leq n, \; 1 \leq j \leq m, \; 1 \leq p \leq 30\, 000)
$. It is guaranteed that no $2$ surveys have same $i$ and $j$.
• The first line contains an integer that is the greatest sum of satisfaction of all customers;
• The second line contains the integer $s$ – the number of gifts that must be sent to the customers;
• The next $s$ lines describe how the gifts are sent: each line contains two integers $x,y$ indicating that the customer $x$ receives the gift $y$.
If there are more than one solutions giving the greatest sum of satisfaction, you can print any of them.
Sample Input 1 Sample Output 1 | {"url":"https://hochiminh17.kattis.com/contests/hochiminh17/problems/engaging","timestamp":"2024-11-04T09:12:38Z","content_type":"text/html","content_length":"29341","record_id":"<urn:uuid:6e81dad9-890f-4071-a39e-80bd66795085>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00190.warc.gz"} |
Alejandro Erickson
These vividly colored icosahedrons look great when hung from the ceiling, or a wall, but the fun doesn't stop there. You'll notice that the colors seem to follow a pattern which is quite amazing.
I'll skip over the cool math that is involved, but try holding two opposite nodes still, and spin it by 1/5 of a rotation. What happens to all the yellows? All the greens? etc. Maybe I'll make a
video about it later!
My icosahedrons are made using colored skewers that are held together with wire. It is not glued, but the whole thing is dipped in acrylic medium to give it a nice shiny finish. They are very durable
and can be kicked around the house without breaking (don't try too hard though, ok?)
I will sell this by dutch auction, starting at $150, and reducing by $5 each week. You never know who's waiting for it to come down, so buy it quick!
This is a "Hexastix", shaped into a "Stellated Rhombic Dodecahedron". It casts a beautiful shadow, and belongs in a sunny window where the light hits something behind it. You'll need to remember that
if you buy it. :)
I make all of my sculptures in a small rented workshop in Victoria, BC, where I am a PhD candidate at the University of Victoria.
Beauty in mathematics, especially geometric beauty, is an ongoing passion. The asking price of this sculpture is very low, but it is worth selling, just to know someone else is in love with it.
Learn more about GeoBurst on http://geoburst.ca
I am pleased to announce the publication Japanese tatami mat tilings: No four tiles meet, in the Notes from the Margin, Volume II, 2011, from the Student Committee of the Canadian Mathematical
The title of the newsletter comes from the story behind Fermat's last theorem. He had written in the margins of his notebook:
it is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly
marvelous proof of this, which this margin is too narrow to contain. (wikipedia)
This article is a condensed version of a result in the conference proceedings: A. Erickson, M. Schurch, 2011: Enumerating tatami mat arrangements of square grids, in 22nd International Workshop on
Combinatorial Algorithms, University of Victoria, June 20-22, volume 7056 of Lecture Notes in Computer Science (LNCS), Springer Berlin / Heidelberg, 2011, p. 12 pages.
Tatami mats are the type of mats used in traditional Japanese rooms. Often, an arrangement in which four mats meet at a point is considered unlucky, perhaps because the word “four” sounds like the
word “death” in Japanese. So, a “lucky” layout has no “+” shapes formed by the lines where mats meet. Compare the following two arrangements.
Tatami tilings are accessible and fun, so go ahead and take a gander!
You'll no longer be sitting beside that sexy marine biologist dressed up as a sea anemone wishing that you could have a research themed costume too.
Finally, there is a combinatorics themed halloween costume! Sport this beautiful Arduino and LED based rendition of Cool-lex combinations (research by Aaron Williams and Frank Ruskey), and show off
your nerd times 2.
Here is the technical stuff on the combinations:
I used adafruit's LDP6803 LEDs. There are tutorials on how to use them on that page.
This week I had the pleasure of teaching a Hexastix workshop to a group of volunteer mathematics educators, at the invitation of the 2011 PIMS Education Prize winner, Veselin Jungic. As with my
previous experience teaching Hexastix at Math Camp, these beautiful mathematical objects will be the take-home project for students participating in Small Number, a program for First Nations math
Small Number is not precisely the name of the program, but of the protagonist in the associated animation, which is narrated in Blackfoot, Cree, and English on YouTube (embedded below). If you have
never heard Blackfoot or Cree, I strongly suggest you listen to those versions. The languages are moving and sounds fascinating. Here is a language map, just to give you an idea of how many First
Nations languages there are in British Columbia (I’m as amazed as you are!).
The animation and the school workshop program are products of several meetings at the Banff International Research Station beginning in November 2006. There, First Nations communities have partnered
with math educators to assess the performance of First Nations students in mathematics, and to find ways of providing them with more opportunities to learn it. The data presented at the workshop
shows, for example, that “Over the last seven years, only 5-7 percent of Aboriginal students have written and passed the Principles of Mathematics 12 provincial exam, compared to 25-27 percent of
non-Aboriginal students.”
My part is to provide the program with 15,000 coloured sticks, as well as other materials, and teach their volunteers how to make Hexastix. I used my blog post about it as a visual guide to my
workshop. The volunteers were challenged, and they enjoyed themselves. Here are some pictures of our workshop.
My first real math course - real analysis, in my case - was given by Dr. Jungic in 2004. His other students and I like to recall how he explained infinitesimals by saying “small small small small”,
and motioning with his hands, and then he would describe large numbers by saying “HUGE” and throwing up his hands.
Previous Page: 11 of 20 Next | {"url":"https://alejandroerickson.com/page11/","timestamp":"2024-11-05T07:21:51Z","content_type":"text/html","content_length":"22673","record_id":"<urn:uuid:964368e5-aec2-40c6-bcbf-b94f1492b635>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00898.warc.gz"} |
Testing Differences Between Proportions
Consider a study showing that 65% of 43 people aged 18 to 24 prefer Coca-Cola compared to 41% of 39 people aged 25 to 29. If we wish to test whether the difference between these proportions is
significant, we need to compute a p-Value (see Formal Hypothesis Testing for a general discussion of the logic of statistical testing).
The standard test of proportions
Introductory statistics courses and textbooks present a standard test of the difference between proportions.
Where and are the two proportions and and are the sample sizes:
The analysis of weighted data
The standard test makes a technical assumption known as i.i.d.. When data is weighted this assumption is violated.
The most straightforward modification of the test in this situation is to replace the sample size by the effective sample size and to compute using the weighted sample size. This approach is
adopted by most of the widely used commercial market research programs (e.g., SPSS IBM Data Collection Model programs, Uncle, WinCross, CfMC, Quantum), although sometimes with additional minor
variations (e.g., Yate's correction). These programs also commonly treat the test-statistic as a t-statistic, variously computing the number of degrees of freedom as the sum of the effective sample
sizes and minus one or minus two.
A more rigorous approach is to use variance estimation to calculate the standard error, as is done in Q, Displayr, SPSS Complex Samples, the R Survey Package, and the statistical software used by
government statistical agencies..
0 comments
Please sign in to leave a comment. | {"url":"https://the.datastory.guide/hc/en-us/articles/4611562791311-Testing-Differences-Between-Proportions","timestamp":"2024-11-06T09:14:31Z","content_type":"text/html","content_length":"41746","record_id":"<urn:uuid:b235a5d7-a0fb-4006-9685-d1d28c93d6cb>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00174.warc.gz"} |
Using A Calculator
Comment recorded on the 23 September 'Starter of the Day' page by Judy, Chatsmore CHS:
"This triangle starter is excellent. I have used it with all of my ks3 and ks4 classes and they are all totally focused when counting the triangles."
Comment recorded on the 5 April 'Starter of the Day' page by Mr Stoner, St George's College of Technology:
"This resource has made a great deal of difference to the standard of starters for all of our lessons. Thank you for being so creative and imaginative."
Comment recorded on the 28 September 'Starter of the Day' page by Malcolm P, Dorset:
"A set of real life savers!!
Keep it up and thank you!"
Comment recorded on the 9 October 'Starter of the Day' page by Mr Jones, Wales:
"I think that having a starter of the day helps improve maths in general. My pupils say they love them!!!"
Comment recorded on the 9 April 'Starter of the Day' page by Jan, South Canterbury:
"Thank you for sharing such a great resource. I was about to try and get together a bank of starters but time is always required elsewhere, so thank you."
Comment recorded on the 3 October 'Starter of the Day' page by S Mirza, Park High School, Colne:
"Very good starters, help pupils settle very well in maths classroom."
Comment recorded on the 14 September 'Starter of the Day' page by Trish Bailey, Kingstone School:
"This is a great memory aid which could be used for formulae or key facts etc - in any subject area. The PICTURE is such an aid to remembering where each number or group of numbers is - my pupils
love it!
Comment recorded on the 10 April 'Starter of the Day' page by Mike Sendrove, Salt Grammar School, UK.:
"A really useful set of resources - thanks. Is the collection available on CD? Are solutions available?"
Comment recorded on the 12 July 'Starter of the Day' page by Miss J Key, Farlingaye High School, Suffolk:
"Thanks very much for this one. We developed it into a whole lesson and I borrowed some hats from the drama department to add to the fun!"
Comment recorded on the 17 November 'Starter of the Day' page by Amy Thay, Coventry:
"Thank you so much for your wonderful site. I have so much material to use in class and inspire me to try something a little different more often. I am going to show my maths department your website
and encourage them to use it too. How lovely that you have compiled such a great resource to help teachers and pupils.
Thanks again"
Comment recorded on the 21 October 'Starter of the Day' page by Mr Trainor And His P7 Class(All Girls), Mercy Primary School, Belfast:
"My Primary 7 class in Mercy Primary school, Belfast, look forward to your mental maths starters every morning. The variety of material is interesting and exciting and always engages the teacher and
pupils. Keep them coming please."
Comment recorded on the 18 September 'Starter of the Day' page by Mrs. Peacock, Downe House School and Kennet School:
"My year 8's absolutely loved the "Separated Twins" starter. I set it as an optional piece of work for my year 11's over a weekend and one girl came up with 3 independant solutions."
Comment recorded on the 24 May 'Starter of the Day' page by Ruth Seward, Hagley Park Sports College:
"Find the starters wonderful; students enjoy them and often want to use the idea generated by the starter in other parts of the lesson. Keep up the good work"
Comment recorded on the 17 June 'Starter of the Day' page by Mr Hall, Light Hall School, Solihull:
"Dear Transum,
I love you website I use it every maths lesson I have with every year group! I don't know were I would turn to with out you!"
Comment recorded on the 19 June 'Starter of the Day' page by Nikki Jordan, Braunton School, Devon:
"Excellent. Thank you very much for a fabulous set of starters. I use the 'weekenders' if the daily ones are not quite what I want. Brilliant and much appreciated."
Comment recorded on the s /Indice 'Starter of the Day' page by Busolla, Australia:
"Thank you very much for providing these resources for free for teachers and students. It has been engaging for the students - all trying to reach their highest level and competing with their peers
while also learning. Thank you very much!"
Comment recorded on the 19 October 'Starter of the Day' page by E Pollard, Huddersfield:
"I used this with my bottom set in year 9. To engage them I used their name and favorite football team (or pop group) instead of the school name. For homework, I asked each student to find a
definition for the key words they had been given (once they had fun trying to guess the answer) and they presented their findings to the rest of the class the following day. They felt really special
because the key words came from their own personal information."
Comment recorded on the i asp?ID_Top 'Starter of the Day' page by Ros, Belize:
"A really awesome website! Teachers and students are learning in such a fun way! Keep it up..."
Comment recorded on the 1 May 'Starter of the Day' page by Phil Anthony, Head of Maths, Stourport High School:
"What a brilliant website. We have just started to use the 'starter-of-the-day' in our yr9 lessons to try them out before we change from a high school to a secondary school in September. This is one
of the best resources on-line we have found. The kids and staff love it. Well done an thank you very much for making my maths lessons more interesting and fun."
Comment recorded on the s /Coordinate 'Starter of the Day' page by Greg, Wales:
"Excellent resource, I use it all of the time! The only problem is that there is too much good stuff here!!"
Comment recorded on the 3 October 'Starter of the Day' page by Fiona Bray, Cams Hill School:
"This is an excellent website. We all often use the starters as the pupils come in the door and get settled as we take the register."
Comment recorded on the 8 May 'Starter of the Day' page by Mr Smith, West Sussex, UK:
"I am an NQT and have only just discovered this website. I nearly wet my pants with joy.
To the creator of this website and all of those teachers who have contributed to it, I would like to say a big THANK YOU!!! :)."
Comment recorded on the 10 September 'Starter of the Day' page by Carol, Sheffield PArk Academy:
"3 NQTs in the department, I'm new subject leader in this new academy - Starters R Great!! Lovely resource for stimulating learning and getting eveyone off to a good start. Thank you!!"
Comment recorded on the 14 October 'Starter of the Day' page by Inger Kisby, Herts and Essex High School:
"Just a quick note to say that we use a lot of your starters. It is lovely to have so many different ideas to start a lesson with. Thank you very much and keep up the good work."
Comment recorded on the 3 October 'Starter of the Day' page by Mrs Johnstone, 7Je:
"I think this is a brilliant website as all the students enjoy doing the puzzles and it is a brilliant way to start a lesson."
Comment recorded on the 2 May 'Starter of the Day' page by Angela Lowry, :
"I think these are great! So useful and handy, the children love them.
Could we have some on angles too please?"
Comment recorded on the 25 June 'Starter of the Day' page by Inger.kisby@herts and essex.herts.sch.uk, :
"We all love your starters. It is so good to have such a collection. We use them for all age groups and abilities. Have particularly enjoyed KIM's game, as we have not used that for Mathematics
before. Keep up the good work and thank you very much
Best wishes from Inger Kisby"
Comment recorded on the 11 January 'Starter of the Day' page by S Johnson, The King John School:
"We recently had an afternoon on accelerated learning.This linked really well and prompted a discussion about learning styles and short term memory."
Comment recorded on the 19 November 'Starter of the Day' page by Lesley Sewell, Ysgol Aberconwy, Wales:
"A Maths colleague introduced me to your web site and I love to use it. The questions are so varied I can use them with all of my classes, I even let year 13 have a go at some of them. I like being
able to access Starters for the whole month so I can use favourites with classes I see at different times of the week. Thanks."
Comment recorded on the 7 December 'Starter of the Day' page by Cathryn Aldridge, Pells Primary:
"I use Starter of the Day as a registration and warm-up activity for my Year 6 class. The range of questioning provided is excellent as are some of the images.
I rate this site as a 5!"
Karen Donnelly, Twitter
Thursday, June 29, 2017
Thursday, June 13, 2019
"I have just noticed something I hadn’t realised about the Windows calculator. When running in standard mode it operates LTR (2+3x5=25) while in scientific mode it obeys the normal order of
operations (2+3x5=17). Worth knowing when learning about BIDMAS or PEMDAS
Dave Grochocki, Twitter
Sunday, June 16, 2019
Tuesday, September 10, 2019
"And the sad thing is that I still remember some of these calculators that are now museum exhibits.
I took these photographs in the Whipple Museum of the History of Science in Cambridge in July 2019" | {"url":"https://transum.org/Software/sw/Starter_of_the_day/Similar.asp?ID_Topic=7","timestamp":"2024-11-15T01:26:22Z","content_type":"text/html","content_length":"58250","record_id":"<urn:uuid:999cdaf0-203e-4ec2-ba69-ea3529089046>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00486.warc.gz"} |
Calendar effect
Definition of Calendar effect
Calendar effect
The tendency of stocks to perform differently at different times, including such anomalies as
the January effect, month-of-the-year effect, day-of-the-week effect, and holiday effect.
Related Terms:
Result of a transaction that increases earnings per common share (e.g. by decreasing the
number of shares outstanding).
List of new issues scheduled to come to market shortly.
The grouping of investors who have a preference that the firm follow a particular financing
policy, such as the amount of leverage it uses.
Refers to the fact that the merger of two firms decreases the probability of default on
either firm's debt.
Result of a transaction that decreases earnings per common share.
An annual measure of the time value of money that fully reflects the effects of
Annualized interest rate on a security computed using compound interest techniques.
The strike price in an optional redemption provision plus the accrued interest to the
redemption date.
The convexity of a bond calculated with cash flows that change with yields.
In an interest rate swap, the date the swap begins accruing interest.
The duration calculated using the approximate duration formula for a bond with an
embedded option, reflecting the expected change in the cash flow caused by the option. Measures the
responsiveness of a bond's price taking into account the expected cash flows will change as interest rates
change due to the embedded option.
Used with SAT performance measures, the amount equaling the net earned spread, or
margin, of income on the assets in excess of financing costs for a given interest rate and prepayment rate
A measure of the time value of money that fully reflects the effects of compounding.
The gross underwriting spread adjusted for the impact of the announcement of the common
stock offering on the firm's share price.
A theory that nominal interest rates in two or more countries should be equal to the required real
rate of return to investors plus compensation for the expected amount of inflation in each country.
Information-content effect
The rise in the stock price following the dividend signal.
International Fisher effect
States that the interest rate differential between two countries should be an
unbiased predictor of the future change in the spot rate.
Low price-earnings ratio effect
The tendency of portfolios of stocks with a low price-earnings ratio to
outperform portfolios consisting of stocks with a high price-earnings ratio.
Neglected firm effect
The tendency of firms that are neglected by security analysts to outperform firms that
are the subject of considerable attention.
P/E effect
That portfolios with low P/E stocks have exhibited higher average risk-adjusted returns than high P/E stocks.
Small-firm effect
The tendency of small firms (in terms of total market capitalization) to outperform the
stock market (consisting of both large and small firms).
Synergistic effect
A violation of value-additivity whereby the value of the combination is greater than the
sum of the individual values.
Weekend effect
The common recurrent low or negative average return from Friday to Monday in the stock market.
Effective Annual Yield
Annualized rate of return on a security computed using compound
interest techniques
Effective Interest Rate
The rate of interest actually earned on an investment. It is
calculated as the ratio of the total amount of interest actually
earned for one year divided by the amount of the principal.
a measure of how well an organization’s goals
and objectives are achieved; compares actual output results
to desired results; determination of the successful accomplishment
of an objective
effective annual interest rate
Interest rate that is annualized using compound interest.
international Fisher effect
Theory that real interest rates in all countries should be equal, with differences in nominal rates reflecting differences in expected inflation.
Effective Exchange Rate
The weighted average of several exchange rates, where the weights are determined by the extent of our trade done with each country.
Policy-Ineffectiveness Proposition
Theory that anticipated policy has no effect on output.
Wealth Effect
The effect on spending of a change in wealth caused by a change in the overall price level.
Blue Ribbon Committee on Improving the Effectiveness of Corporate Audit Committees
A committee formed in response to SEC chairman Arthur Levitt's initiative to improve the financial
reporting environment in the United States. In a report dated February 1999, the committee
made recommendations for new rules for regulation of financial reporting in the United States that
either duplicated or carried forward the recommendations of the Treadway Commission.
Cumulative-Effect Adjustment
The cumulative, after-tax, prior-year effect of a change in accounting
principle. It is reported as a single line item on the income statement in the year of the
change in accounting principle. The cumulative-effect-type adjustment is the most common accounting
treatment afforded changes in accounting principle.
Cumulative Effect of Accounting Change
The change in earnings of previous years assuming
that the newly adopted accounting principle had previously been in use.
Cumulative Effect of a Change in Accounting Principle
The change in earnings of previous years
based on the assumption that a newly adopted accounting principle had previously been in use.
Effective Tax Rate
The total tax provision divided by pretax book income from continuing
Panel on Audit Effectiveness
A special committee of the Public Oversight Board that was created
to perform a comprehensive review and evaluation of the way independent audits of financial
statements of publicly traded companies are performed. The panel found generally that the
quality of audits is fundamentally sound. The panel did recommend the expansion of audit steps
designed to detect fraud.
Related to : financial, finance, business, accounting, payroll, inventory, investment, money, inventory control, stock trading, financial advisor, tax advisor, credit. | {"url":"http://www.finance-lib.com/financial-term-calendar-effect.html","timestamp":"2024-11-05T16:05:33Z","content_type":"text/html","content_length":"14442","record_id":"<urn:uuid:73b283df-5620-4558-b239-e9644d422bb3>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00832.warc.gz"} |
Decimal calculation
Search Engine users found us yesterday by typing in these algebra terms:
• ti calculator quadratic factoring
• download A level accounting books
• Uo p is hyperbolic filetype lecture :pdf
• t89 lettore pdf
• mixed fraction to decimals
• SAMPLE TESTS FRACTIONS
• algebra 2 problems
• introducing algebra
• maths year 8 exercises test
• add subtract multiply and divide percents
• How to solve Least Common Denominators in Algebra with letters
• geometers sketchpad hacked version
• math poem
• free pre algebra worksheets
• algebra II FINAL EXAM COMBINATION AND PERMUTATION
• algebrator simplify radicals
• real world problem simplifying expressions by combining like terms
• the basic rules of rearranging formulas gcse
• multiply square roots calculator
• Ch. 6 - 7 Cumulative Test
• can you put numbers in radical form on calc
• cost account book
• problem involving subtraction sample
• worksheet answers
• worksheets for solving formulas
• slope questions for algebra tests
• sample "math problems" density
• sample activity sheet for algebraic expression
• Solve a Simultaneous Set of Two Linear Equations using tensor
• Field Axioms
• workshets on dividing integers
• multiply radicals solver
• Learn algebra
• How to Write a Decimal as a Mixed Number
• adding,subtracting, multiplying & dividing negative and positive integers workshets
• worksheet for perfect square and perfect cubed
• free printable prime factorization worksheet
• theory on linear equation in two variable
• how do you do cube in calculator
• grade 8 algebra math worksheets
• convert 2nd order ode to first order
• exponents mix root mixed exponent multiply
• download ti 84 calculator emulator
• how to solve lcm
• linear algebra multiple variables equations matlab
• solving equations with fractions worksheet
• greatest common multiple, worksheets for 6th grade class
• 10 key calculator test
• simplify the expression fractions
• mixed numbers to decimals
• pictograph worksheets 4th grade
• answers to math homework
• 6TH GRADE MATH WORKSHEETS (DIVIDING AND MULTIPLYING FRACTIONS TO MAKE A MIXED NUMBER)
• maths worksheet ks3
• laws of exponent math SOLVER
• how to recognize linear equation
• dividing polynomials free worksheet
• advance algebra practice for sat
• how to solve prealgebra probability word problems
• aLGERBRA
• math worksheets slope
• 6th grade scale factor examples
• root formula
• algebra trivia questions
• sample algebra test print
• simple multiplication of linear equations
• solve for exponents
• worksheets square foot
• cost accounting download
• simple finding lcd gcf
• "Area" "Math" Worksheet "Grade 4" Kids
• 9th standard examination papers for practice
• scale factors in word problems
• online polar graphing caculator
• graphing linear equations worksheets
• math software paid
• simplifying exponents worksheet
• creative publications pre-algebra with pizzazz page 192
• glencoe.mcgraw-hill.com worksheets 6-3
• printable worksheets for 4th graders for division with remainders
• prentice hall algebra 1 workbook tennessee edition
• simplifying fractions square root
• modeling equations worksheets
• powerpoint presentations + differential equations
• simplify exponent square root
• How to solve alebra expressions
• program
• college algebra help
• TI-84 plus online calculator
• mathmatical number patterns
• Free Algebra Calculators
• logarithmic form with negative fractional exponents
• poems about math
• help solving rational expression equation
• evaluation of positive rational roots
• algibra
• 6th grade algebraic thinking ppt
• maths substitution year 8 worksheets
• practice worksheets to solve second degree equations
• free printable worksheets exponents
• variables and expressions worksheets
• free math problems for 6th graders
• blitzer introductory algebra 5th edition download
• Algerbra with Pizzazz pg 64 Riddle
• exponents properties worksheets
• conceptual physics hewitt answers practice page chapter 8
• scale factor worksheet
• multiplying and dividing integers test
• partial multiplication for 4th grade
• calculator for exponents
• mathamatic applications
• addison wesley algebra 2 worksheets
• D.A.V sample paper of std 8
• math tutor factoring
• how to find the least greatest fraction
• age problem on equations and inequalities
• factor cubed polynomials
• casio fx880 equation solver
• free polynomial factorer
• 7th grade Pre Algebra Textbook California prentice hall
• mcdougal littell algebra 2 even answers
• numerical solution simultaneous equations matlab
• printable problems and answers business maths
• algebra power
• singapore grade 8 algebra word question
• download free service accounting books
• subtraction properties- same
• mixed number to decimal conversion
• polynomial written as a product of factors
• "english workbook" fun free kids printable
• difference quotient solver
• how do you solve second order differentiation in matlab
• free worksheet on probability
• tests 9th grade california
• Search multiply divide numbers for kids
• math permutations for middleschooler
• what is the greatest common factor of 30 and 105?
• solve for the variable worksheet
• interactive college placement math tests
• free download of advanced accountancy author by pillai and bhagavathi in pdf document
• absolute value worksheet generator
• answers to algebra with pizzazz pg 146
• factoring quadratics on a ti-83 plus
• dividing polynomials free worksheets
• nonhomogeneous partial differential equations
• math investigatory problems
• Advanced Algebra - Quadratic Inequalities Domain and Range
• pre algebra test holt
• convert decimal number to fraction matlab
• ti 89 multiple statement solve
• sixth grade math programs
• Factor polynomials of higher degree using the GCF lesson plans
• substitutions into formulae quiz ks3
• order of Multiplyed Polynomials
• squaring numbers worksheets
• solving addition equations worksheet
• fraction to a power
• Pearson Prentice Hall pre-algebra textbook questions and answers to page 60
• algrebra problems
• mathcad solve substitution algebra
• accounting+free download+ebook+
• what is the importance of simplification in mathematics expressions?
• algebra pizzazz free worksheets
• algorithm division for texas ti 84
• how to calculate the area of an elipse
• free accounting books
• Solving Nonlinear Equations with Newton's Method using Matlab
• how to change decimals to fractions in MATLAB
• solve my math for free
• roots of nonlinear two-variable equation
• free mathematics exercise for 8 year old
• dividing factoring calculator
• first grade beginners worksheets
• decimal ordering worksheets
• algebra,pdf
• downloadable worksheets to multiply signed numbers
• factor tree worksheets
• balancing equations steps
• substitution method
• tutoring algebra software
• glencoe mathematics algebra 1
• hunter college high school entrance exam cheats
• use perfect squares to solve quadratic equations
• help with solving elimination problems in math
• help me factor this equation
• order least to greatest of fractions and decimals
• converting from intercept to vertex form
• determining multiplying or dividing fractions word problems'
• quadratic equation convert number to words basic form
• pratice hall
• download Jacob’s Math Training PPC
• math primary school sheets free
• prentice hall biology workbook answers
• Algebra GCSE WORKSHEETS
• multiplying and dividing fractions
• exponents and square root equations
• common denominator for all single digit numbers
• vertices of the quadratic formula
• how do you multiply uneven fractions
• Iowa Algebra Aptitude Test practice
• 8 grade math percent and proprotion work sheets
• softmath worksheets.com
• how do you do real number properties with roots
• simplifying square roots with radicals
• exercises for calculating squares, squares roots and cubes
• algebra theorem tutorials easy to learn
• mcdougal Littell Integrated mathematics algebra
• learn algebra fast
• examples of word problems involving integers
• advanced algebra answers
• multiplying rational expressions
• a cube +algebra formula squares formula
• wwwalgebra de baldor
• square root method
• factoring cubed polynomial
• holts math book pre algebra 6th grade with full pages from the book
• aptitude test papers with solution
• equasion
• finding roots of polynomials using a TI-83
• plus minus and subtraction in fraction
• Find a quadratic function in standard form for each set of points.
• solving coupled differential equations
• free math vocabulary
• adding dividing multiplying and dividing powers
• cost accounting standard free ebook download pdf
• lesson on square numbers for elementary kids
• cubic route excel
• casio calculators convert use
• math algebra practice book
• graphing linear equalities
• rational expressions online calculator
• solving nonlinear differential equation
• printable 8th grade pre algebra worksheets
• ti85 calculator rom
• example math trivias
• How to convert a mixed fraction and reduce it
• Artin algebra solutions
• adding radicals calculator
• variables as exponents
• printable prealgebra worksheets
• india mathematical area formula
• questions relating to boolean algebra
• online algebra trainer
• simultaneous equations for dummies
• free trig calculator
• radical math worksheet
• year 6 algebraic simplification
• complete the square grade 10
• algebra worksheets for kids
• discrete mathematics linear programming examples excel solver
• second order nonhomogeneous differential
• softmath.com+reviews
• Free Math Trivia
• a step by step way to solve radicals
• quadratic formula worksheet
• how to type log base 3 into TI-83 +
• factor loadings for variable
• Square Root (With Addition and Subtraction)
• free algebra worksheets graphing using intercepts
• pre-algebra test generator
• algebraic pyramids
• 3rd order polynomial
• least common denominator solver
• mathematics exams+Mcgraw Hill
• Learning Basic Algebra
• solving fraction problems involving money
• "lattice math" worksheet free
• beginner algebra games
• number patterns powerpoint for middle school
• Cost account book
• online factoring equations calculator
• McDougal Littell Inc answer key for algebra 2 chapter 3 cumulative review
• 11+ exams online test
• third order formula for polynomial equations
• n exponent equation javascript
• Search multiply divide numbers calculator
• polynomial division solver
• online math solver mathematics distance between two points and midpoint
• rational expressions in lowest terms
• software
• the slope formula worksheets
• prentice hall mathematics algebra 2 answers
• algebra made fun and simple
• DOWNLOAD ALGEBRATOR
• Introductory Algebra Problem Help
• measurement fraction worksheets
• evaluating variables and expressions substitution 6 grade
• applied math story problems free worksheet
• finding the LCM simple explanation
• cross multiple algebra problems free
• factoring with intergers worksheets
• free "music worksheets" 5th grade
• multiplying quantities cubed
• trig clep test free online
• free online games practicing symmetry
• sample activity sheets for math kumon
• printable percent circle
• 9th grade algebra 1 +quizs
• who invented the X-Y graphing technique
• printable 3rd grade poems
• finding ordered pairs solver
• prentice Hall PHysics answers
• simplifying fractions with exponents
• 2 equations solved by extracting square roots
• What is the nth term in each of the numerical patterns
• holt algebra 1
• accelerated math online help in pre-calc objective 82
• aptitude test questions and solutions
• matlab non linear square
• mathematics free exercises
• positive and negative numbers activities
• graphing linear equalities problems
• linear algebra solution to area
• formula to add fraction
• free printable tests for life's structure and function
• Free Math worksheets (Multiply and divide fractions
• solving 4 equations with 4 unknowns
• maths homework answers
• download Intro to Accounting: An Integrated Approach
• intermediate math problem solver
• simplify expressions on line
• convert linear metres into square metres
• free ks2 sats papers
• download engineer aptitude test
• prentice hall mathematics algebra 1 answer
• What Are Vertices in prealgebra
• T183 calculator
• subtracting polynomials printable worksheet
• Holt algebra 1 answer key
• free download fundamentals of indian accounting concepts
• how to solve a cubed polynomial
• GCSE worksheet download
• approximate square root on graphic calculator TI-84 plus
• the uses of trigonometry in daily life
• alg concepts teachers answer sheet
• simplified radical/rational coordinates
• online chatting and tutoring for maths from chennai
• convert mixed fraction to decimal
• mixed subtracting numbers calclator
• examples of decimal fractions in multiples of base 8
• College Help Software
• nineth grade study guides
• hrw algebra online textbook lesson tutorial chpt 5
• lowest common multiple of two equations
• simplifying expressions calculator
• graphing simple rational functions grapher
• roots and radical lesson plans
• graphing lesson plans for 8th grades
• circle theorem ks4 tangent worksheet
• is decimal fractions in muliples of base 8 examples
• Write in simplified radical form by rationalizing the denominator
• simplifying fractions with variables game
• Common factor analysis, SPSS
• TI-84 quad program download
• online factorer
• solving diff equation in matlab
• free online college algebra practice test
• adding and subtracting negative numbers worksheets
• how do i divide by radicals in calculus
• mcdougal littell world of chemistry answers
• software multiple choice exams algebra free
• example of Solving System of Linear Equation by substitution.presentation
• algebra 2 mcdougal littell workbook
• Factorization of polynomials - Lesson plan
• like terms calculator
• mathematica+simultaneous differential equations
• vertex form problem solver
• trig calculator
• Practice Grade Nine Probability Problems
• finding fourth roots
• online college algebra cheat
• Solving equations involving: higher-degree polynomials, by factoring
• lesson advanced combination permutation problems
• equation worksheets elementary
• Algebra For Beginners
• math word problems for dummies printables
• mathematics year 9 algebra factoring
• solving first order partial derivatives
• is the pattern for square numbers a geometric pattern
• how to convert fractions to simple form
• 11+ online exam
• a level maths quadratic graph inequalities
• free yr 7 maths worksheets getting ready for H.S basics skills test
• online factoring
• online calculator hexadecimal to radix
• scale factor lessons
• list of good aptitude books
• parabola calculator
• sample of problems-solution of speed and velocity
• algebra 2 answer
• solving for roots of a function in matlab
• solving with square root worksheets
• rational expressions calculator
• word problems involving linear equations + exercise + algebra 1
• get free math answers online
• free maths notes o-level (pdf)
• matlab solve differential equation
• aptitude exam papers of ias with solution
• solve for x games worksheet
• what is a scale in math
• ancova parallel intercept
• how to solve a problem with two equations but three variables
• real world second order differential equations
• math challenge books for 5th, 6th and middle school
• prentice hall teaching resources on chapter 9 chemical names and formulas answer key
• how to solve standard form of the equation of the circle
• sample algebra questions for grade 9
• accountancy printable worksheets
• hex do dec calc
• algebra-substitution
• solving second order nonlinear ode matlab
• problems on simplifying exponents
• "mathematical analysis" "problems" "online text"
• aptitude ques and ans
• steps in basic algebra
• calculating with scientific notation worksheet
• the equal value of quadratic equation square function visual basic form
• "high school algebra projects"
• prentice hall chemistry teaching resources on chapter 8 test answer key
• ontario cat test ged example
• algebrator using GCF
• matlab solving nonlinear simultaneous equations
• combining like terms puzzle sheet
• write function rules using one variable fourth grade
• free printable 9th grade worksheet
• aptitude problem with answer
• answers for algebra 1 glencoe mathematics
• Solve the following quadratic equations by completing the square.
• prentice hall integrated algebra new york state tutorial
• probability practice sheet
• algebra order of operations worksheet
• glencoe math taks practice grade 8 answer key
• NONHOMOGENEOUS FIRST ORDER ODE
• ti 89 applications laplace transforms
• how to dividing decimals by intergers steps
• square feet to linear feet online calculator
• a quotient of polynomial functions definition
• college physics investigatory project
• how do you CONVERT A MIXED FRACTION TO A DECIMAL
• accounting glencoe working papers answer key
• Adding and Subtracting Fractions Using Reading Strategies
• demo lesson for area and perimeter of irregular objects and polygons in 6th grade
• positive and negative integer worksheets
• past SAT qestions on pythagoras' theorem
• online polynomial solver
• solving quadratic complex formula
• hyperbola equation variables general equation
• decimal convert fractional formul
• excel slope formula
• algerbra poems
• simultaneous equations three unknowns
• solve y=-3/4+2
• free exam papers of grade 11 of physics
• stats5
• converting square root into a decimal point
• factoring equation calculator
• second order differential equation real world problems
• FOIL math worksheets
• FREE FACTORING TUTORIAL
• what is the term square root in +algabra
• ∑(summation)+java code
• aptitude questions with solutions in pdf
• Math Functions For Dummies
• practice fraction test adding and subtracting
• mcdougal littell algebra 2 answers for workbook
• free online stat graphing calculator
• examples of java programs on how to calculate the greatest commom divisor of two fractions
• fifth grade conversion worksheets
• aptitude questions and answers book downloadable
• how to solve trinomials
• Adding, Subtracting, Multiplying and Dividing Rationals
• ratio formula
• finite mat tutorials dvd
• solved aptitude test paper in pdf
• rules of square root
• writing a part of a fraction of a whole worksheet
• robotic simultaneous trigonometric equations
• Glencoe Algebra 2 workbook 6-2 Solving Quadratic Equations by Graphing
• orleans hanna algebra prognosis test questions
• "The C Answer Book" download free
• algebra worksheets practice exam paper
• ti 84 plus games downloaden
• ninth grade poem exam
• How to List Fractions from Least to Greatest
• ti 32 free online calculator
• ti 89 calculator downloads
• math trivias
• TI-83 plus permutation
• lesson plan for adding and subtracting positive square root
• mathematica tutorial video
• children free fraction problems download
• second order ordinary differential equations nonhomogeneous
• college algebra, software
• calculate a repeating decimal into a fraction on a ti83 calculater
• equation cubed
• 9th grade` algebra test practice
• solving graph problems
• algebra for dummies free online
• mcdougal littell algebra 2 answers
• find an expression for the funtion parabola
• prentice hall classics algebra 2 answers
• log calculations, TI-83
• polynomial factoring solver free
• Algebra 2 Answers
• convert a decimal amount to time
• multiplying functions calculator
• pdf on TI-89
• adding negative and positive numbers worksheet
• solving trinomials
• free worksheets add subtract integers
• algbrator
• algebra expression calculator
• add, subtract,and divide fractions
• square root of 30 in radical form
• HOW TO PASS MATRIC
• alebra for dummies
• worksheets picture adding
• conversting mixed numbersinto percents
• sample problems on linear programming with solutions
• how to enter a repeating decimal into a calculator
• worked solutions for rudin principles of mathematical analysis
• solving equations interactive games
• factorise quadratic calculator online
• college alegerbra software
• Cost Accounting EBook
• alebra 2 powerpoints
• example of a linear equation in real life
• printablemath work sheet for 9 year old
• mcdougal littell algebra 1 CHAPTER 7 ANSWERS
• online non verbal reasoning worksheet
• ti-89 +pdf converter
• free grade4 maths sample test in Singapore Syllabus
• slope if function in EXCEL
• graphing inequalities
• addition and subtraction of algebra for standard 6
• vertex graph worksheet
• aptitude questions and answer
• scott foresman math text errors 6th grade
• sample math problems adding and subtracting algebraic fractions adding
• free math word problems worksheets 8th grade
• algebra RATIONAL EXPRESIONS calculator
• solution algebra 2 math
• difference of two square
• how to calculate LCM
• factoring tree worksheets
• 9th grade online work
• radical form
• algebra 2/trig how to factor polynomials
• online algebra worksheets and solutions
• COST ACCOUNTING TUTOR
• algebra 2 HELP
• ppt presentation on maths problems in proportion
• solver game matric
• simplifying expressions with exponents worksheets
• latest math trivia with answers
• mcdougal littell inc. worksheet answers
• Free Simplifying trig calculator
• factor quadratics calculator
• intergers interactive worksheets
• discrete mathematics and its applications student solution manual download
• free online maths course in symmetry
• adding and subtracting fractions
• convert whole numbers to percentages
• percent formulas
• factor trinomials calculator
• fraleigh solution
• solving literal fractions
• where can I learn algebra online
• "fundamentals of algebra" practice book error
• advantages of using ode45 over ode23
• solved aptitude questions
• polynomial cubed
• math question solver
• ratios and algebra problems
• algebra worksheets with solutions
• any free ks3 english sats papers
• algebraic formula
• solving a linear nonhomogeneous partial differential equation
• java code 2 order equation
• nonlinear differential equation
• the sum of 20 divided by a number and that number divided by 20
• ordering fraction worksheets
• nc eoc intro to high school math practice test
• what is the formula for ratio
• balancing 180 and other equations examples
• 6th algebra sheets
• display fractions formula
• aptitude question and answers
• algebra 1 review worksheets, free
• holt, algebra 1
• simplify rational expressions calculator
• graphing inequalities on a coordinate plane
• algebra substitutions
• cheat on algebra
• free algebra II worksheet
• solve simultaneous equations online
• interactive scientific calculator cube root
• adding subtracting multiplying dividing integers worksheets
• log calculator download
• complex rational expressions with 2 variables
• algebra worksheets systems of equations substitution
• lcf and gcf in vin diagram
• online graphing calculator rational functions
• solutions rudin 7
• online rational equations calculators
• chemical engineering matlab programs database
• integers-adding and subtracting
• printable english lesson for first grade
• equation of ellipse when enter is not at origin
• rational expression calculator fractions
• strategies for problem solving workbook third edition answers
• adding and subtracting decimals free worksheets
• solve non-linear equations multiple variable online
• what is cubed in mathmattics
• third order quadratic equations
• year 9 probability worksheets
• Ninth Grade Algebra
• factoring cubes by grouping
• how do i put in x + y = 5 on a TI-83 calculator
• linear equations with fractions worksheets
• inverse operation worksheets ks2
• hungerford solution pdf
• Dividing Polynomials Calculator
• difference of two square composite
• mathematics transformation translation worksheet
• sideways parabolic equation graph
• high school entrance exam algebra worksheets
• factoring trinomial magic square
• convert decimal to mixed numbers
• all answers to algebra 1 book
• solve simultaneous equations with this software
• adding and subtracting mixed fractions worksheets
• brush up on your algebra skills for free
• free algebra 2 problem solver
• algebra 2 honors florida book
• quadratics non standard form
• mix fraction java
• adding and subtracting negative numbers- printable worksheets
• Reduce expression into lowest terms calculator
• Apptitude question Papers & answer
• glencoe chemistry concepts and applications answers to chapter assessments
• simplifying a sum of radical expressions
• solving trig functions on ti 83
• make an equation a perfect square
• printable excercise mathematics for 7-8 year old
• multiplying by power of ten printable worksheets ks2
• completing the square worksheets
• factorise quadratic calculator
• a word problem solving an equation with the variable on each side
• topics in algebra solutions manual
• accounting fomulas
• free download e-books on permutations combinations & probability
• "trigonometry exercises"
• TI-89 finding a quadratic equation through plotting
• LCM of 50 and 60
• how do i convert a decimal to a whole number
• arithmetic sequence trivia flash game
• free download papers of apptitude test
• least common denominator solver lcd
• who invented the tangent ratio
• grade 9 math texts online
• Free Algebra 2 Exam generator
• Ti 83 trace Y instead of X
• powerpoint on teaching how to solve simple equation
• ti 89 solve multiple linear equations
• free online iowa aptitude algebra test prep
• square root fraction calculator
• scientific calculator T183
• aptitude questions free download
• square root properties to solve radical equations
• how do you find the slope on a graphing calculator
• multi ply add and subtract fractions
• free ptintable work sheet for grade 2 india
• worksheet: word problems: scale
• printable algebra worksheets answer key
• rules for adding subtracting powers
• solving equations containing integers worksheets
• free algebra solver
• How can you write an equation as a function?
• mathcad-implicit equations
• holt math graphing linear equations
• solving functions calculator
• algebrator online
• elementary variables worksheets
• Free Online Math Problem Solvers
• worksheet download dividing mixed fractions
• solving general polynomials
• simplifying expressions math 5th
• algebra story problem finding month's rent
• Florida Glencoe Pre-Algebra Teacher's Edition
• mcgraw online graphing calculator
• worksheet and answer sheet for math translate each phrase
• games for solving equation for the classroom
• british factoring quadratics
• worlds hardest rearranging formula
• formula to convert decimal to fractions
• aptitude written test questions and answer dumps free downloads
• calculator Lowest common denominator
• grade 9 practice exam free
• introduction to slope math grade 9
• convert Quadratic function to a Quadratic equation
• addition and subtraction of fractions games
• multiplying decimals
• algebra and trigonometry book 2 answers for free
• free ebooks download for the preparation of aptitude test
• Aptitude questions with answer
• worksheet revision english math science
• substitution 6 grade math worksheets
• free math worksheets with answer exponents grade 6 algebraic expression
• convert decimal number to a fraction
• pre algebra work book free
• solve for y math answers
• problems on least common factors for grade 4
• area worksheets for third grade students for free online
• MS assessment practice for 3rd through 8th grade(printable)
• can't have radical in denominator
• math factor tree worksheet
• Softmath algebrator
• aptitude test paper with solution
• slope worksheet with answers
• math trivia with answers algebra
• dividing games
• using fractional coefficients in balancing equations
• algebra worksheets free printable
• finding the difference of two radical expressions
• simultaneous solutions and excel
• vertex of a quadratic equation
• free sample maths problem solving KS2
• multiply/subtract/add/divide/fractions worksheets
• solution of systems of simultaneous non linear equations + matlab
• find LCD of a number in java
• math answer finder
• four fundamental math concepts expression
• maths algebraic free trial
• simple and clear method for algebea 1
• how to write equations in vertex form
• decimals to percent conversion calculator
• free worksheets on transformations
• free eigenvector solver
• slope equation math 8th
• greatest common divisor java
• radical factoring calculators
• math trivia subtraction algebra
• free math worksheets on non-linear functions
• linear equations printable worksheets
• star test released questions grade 5 math
• distributive property fraction expressions
• learn algebra games two variables
• factoring polynomials
• algebra software
• program that factors equations
• the prime factorization of the denominator
• how to get a little number as an exponent on a TI-84 silver edition
• MATHMATICS FORMULA
• free+online+maths+aptitude+tests+high+school+students
• examples of java programs on how to calculate the greatest commom denominator of two fractions
• e books downloads free accounting
• balancing equation quiz game
• download calculator with trigonometry functions
• Free downloads on Aptitude test
• free math graph exercises for grade4
• polynomial third order solving
• Write java a program to calculate factorial of any given number
• linear equations worksheet
• easy addition
• School supply Stores, IOWA Algebra Aptitude Test
• free aptitude questions
• balancing equations online
• probability made easy
• "Operations with exponents"+"free worksheets"
• rational exponents solver
• BAsic Algebra Math pdf
• 6th grade VA SOL test prep
• holt physics study guide answer key
• math worksheets for fifth grade
• use java to get the coefficients a, b and c of a quadratic equation with the complex roots
• iowa algebra aptitude test
• Dividing polynomial Solver
• calculator square meters
• answers prentice hall algebra 1 workbook
• calculating greatest common factors
• radicals math worksheet
• algerbra
• uk free year 6 past exams paper year 6 free
• cost accounting basics
• algebra simplify equation
• factor a cubed number
• exponents and multiplication
• ordering fractions least to greatest
• math investigatory in geometry
• steps to dividing larger numbers
• algebra poems
• asset math for grade seven
• how to calculate sq root
• three equation solver
• printable homework help english for 9 yr olds
• how do you check your answer to simplifying rational expressions
• algebra sums\
• Factorization of quadratic expressions
• free fraction worksheets for 1st graders
• rules of exponents algebra printable worksheets free
• elementary algebra worksheets college
• how to solve for negative under square root
• percent discount worksheet
• download laplace t89
• evaluate versus simplify
• prime factor tree math free worksheets
• adding and subtracting customary units worksheet
• Algebra 2 holt california rinehart and winston homework teachers edition
• calculator to convert fractions to decimals
• mathematical equations for distance formulas
• permutation combination tricks
• Free Online Maths Test year 8
• ti 83 factoring
• recognize chemical equations being balanced and not balanced
• Combination Math
• mix numbers
• online practice test papers KS2
• converting decimal to fraction worksheet
• mathematical translation worksheet
• year 11 maths practice sheets
• multiplying and dividing integers games
• Difficulties of students in solving worded problems in mathematics
• solve quadratic equation TI-30x
• steps to balancing equations
• how to add, divide, multiply, subtract fractions
• added, subtracting, multiplying, and dividing fractions
• How to solve rational problem
• beginnier algebra math help
• variables on both sides Online calculator
• simplifying radicals in a fraction
• permutations for a 7th grader how to do them
• algebracator
• Math test for adding, subtracting, dividing, multiplying
• ti 84 plus emulator
• Fun Algebra Worksheets
• grade 11, math tutorial
• scale factor math projects
• third root of -1000
• free printable 9th grade math worksheets
• I need a free calculator to help with algebra
• solving equations with ti 83 plus
• ONLINE BOOKS FOR CLEP
• Holt algbra1 website
• arithmetic operations involving rational expressions
• modern CHemistry chapter 13 test for solutions answer key
• APTITUDE QUESTION
• y4 maths printable worksheets
• equation worksheets for 5th grade
• multiply divide fractions and decimals worksheet
• square root simplify calculator
• online summation calculator
• teaching integers worksheets
• Free Math Tutor Download
• Great common factor formula
• simultaneous equation solver 3 unknown
• simplified radicals calculator
• algebra 9th grade
• Mixed Number To Decimal
• aptitude test workbook (pdf)
• online equation solver involving square root
• McDougal Littell Algebra 1 Ebook docs
• comparing integers worksheets
• equivalent decimals
• free clep practice tests print outs
• common denominator calculator
• Linear differential equation ppt
• Aptitude Test Download
• system solver software
• Distributive law problem solver
• free 9th grade algebra worksheets
• factorials-TI 83
• substitution method - calculator
• ti-83 plus emulator
• standard form calculator
• hyperbola grapher
• When solving a rational equation, why it is OK to remove the denominator by multiplying both sides by the LCD
• free online polynomial equation solver
• exercise math elementary school+ppt
• complex quadratic equations
• percent printable worksheet
• dividing 6th radicals
• t1-83 plus emulator
• free beginner online albegra course
• kumon math answer key "word problems"
• how to put a square root symbol into a math equation calculator
• rules for adding and subtracting fractions
• prblem solving aptitude questions
• conceptual physics book answers online
• subtracting fractions with an unknown number
• convert decimals as a mixed number
• solve third order equation
• free elementary algebra worksheet
• simplifying 5th root
• sample of ninth grade algebra problems
• create math test slope
• C# solving polynomial system
• second order differential equation matlab
• free exam papers for class 9th
• solving permutation word problems
• printable worksheets on exponents
• square roots for dummies
• find vertex asymptote
• Solving Systems of Simultaneous Equations Involving Equations Of Degree 2
• integration by substitution calculator
• third grade one digit word problems free printable
• mathmatics fr 3rd grade
• McDougal littell Middle school math unit 2 chapter 5 test
• Nonhomogeneous Second Order Linear Equations solver
• help solving moles
• multi-step equation worksheets
• print out exel math for 6th graders
• examples of math trivia mathematics
• adding, subtracting, multiplying, and dividing fractions test
• simplified radical
• set theory calculator
• aptitude question with answer
• mcdougal littell science worksheets and answers
• Formula for Changing a Percent to Ratio
• equation calculator step by step
• algerbra 1free help
• Lattice Multiplication free lesson
• english aptitude papers of it for freshers with answers
• "third-root"
• free worksheets for fourth grade fractions
• ALEGEBRA EQUATIONS
• simple aptitude question
• systems of equations test
• algebra 2 online tutoring
• +polinomial function activity sheets
• 4th grade multiplication by points
• High School Algebra Worksheets Free
• algebra 1 online math solvers
• solve linear equations java code
• advanced math book for 5th and 6th grade
• Free Printable Consumer Math Worksheets
• factoring tutorial worksheets
• Free College Algebra Book
• algerbra solver
• ti calculator download
• online solve substitution methods calculator
• help with beginners algebra
• permutation combination pdf
• factors and multiples worksheet +maths
• simple expression worksheet
• algebraically removing an exponent from an equation
• Algebrator
• scale factor math cheat sheets
• advanced algebra help
• minimize quadratic equation
• elementary math text book GCD
• purpose of arithmetic sequence in daily life
• rationalizing the demonitor for simplified radical form
• glencoe mathmatics florida algebra 1 math book answers
• inequality worksheets
• online algebra calculator
• free printable ks2 sats papers
• 9th grade free math worksheets
• power engineering fourth class test answers
• 'practice multiplying and dividing fractions'
• how to solve y-intercept
• what is percentages? as a power point
• completing the square worksheet algebra I
• free worksheets multiply negatives
• negative subracting negative rule
• how to use fractions on graphing DIMENSIONS calculator
• formulaes
• least common denomination for 3, 5, 8
• arithematic
• texas instruments ti-84 math worksheets
• Plotting Linear Relations worksheet
• math trivia for kids
• easier way to do the different of mathematic
• 9th grade math worksheets
• where can i get free live help for algebra 1a
• download ti-83 plus emulator
• Trigonometry Word Problem Examples
• fifth grade solving equations
• Formula to Convert Decimal to Fraction
• algebreic calculator
• algebra verbal rule for writing expressions
• algebra de A. Baldor
• online statistic graph calculator
• Permutation Math Problems
• free downloadable PPT inequalities math GCSE
• college algebra cheat programs
• practice 5-3 adding and subtracting fractions mcgraw hill answers
• free worksheets on positive and negative numbers
• scale word problems
• a textbook should explain Greatest Common Multiple by
• rudin chapter 7 exercises solutions
• adding decimal worksheets
• TI 89 solve multi equation
• suare footage of a circumferance
• model aptitude paper with solution
• convert squared into cubed
• saxon math printable worksheets
• download Aptitude test for banks
• math trivia algebra
• ordering numbers from least to greatest c++
• free online math games 7th grade fractions
• fractions adding subtracting worksheets
• algebra and structure and method and mcdougal
• equation calculator with fractions algebra
• 26447
• ninth grade algebra
• worksheets slope graph
• addition and subtraction expression tables
• Least Common Denominator calculator
• Evaluating expressions worksheets
• java complex numbers converter program
• mixed number fraction to decimal conversion
• multiplying rational expressions calculator
• greatest common factor formula
• aptitute question with answer
• worksheets for grade 4 math multiplying and dividing
• solving a non-homogeneous differential equation
• math tutor "Operations with Functions"
• Multiplying and Dividing Rational Expressions calculator
• Combining like terms worksheet
• ged practice tes printouts
• rational numbers for dumbies
• maple fraction to decimal
• Coordinate Plane Worksheets
• converting quadratic functions to vertex form
• math circle worksheetmiddleschool
• www.mathe problems distance times rate freeware programs
• cost accounting tutorials
• how to remove algebrator
• free books on mathematical Induction
• loopmath
• www.algebrabasics.com
• complete the square calculator
• calculation on diffirential equation
• Glencoe algebra 2 cheat sheet.
• do my college algebra homework for me
• general aptitude questions
• mathmatical integer rules
• solving nonlinear differential equations
• worksheet on simple linear equations
• pre algebra how to combine like terms
• "Free worksheets" base "number systems"
• prentice hall mathmatics algebra1
• statistic for TI 89
• graphing help algebra
• activities on cube of numbers
• ged math practice sheets
Yahoo users found our website today by typing in these keyword phrases :
Math trivia, 9th grade properties of addition worksheets, converting a mixed fraction to a decimal, learning algebra free.
Matrix of system of linear equations(worded problems), glencoe algebra 1 glossary, simple aptitude question and answer, finding cubic roots on ti-83 radical.
Math variable worksheets, multiplying and dividing decimals, manipulation and +simplification of linear, quadratic, and exponential.
HOW TO Solve equations with fractions two variables, boolean algebra solver, books available free on internet on basics of financial accounting, FREE PRINTABLE MATH FOR 1ST GRADE, mathematics for
dummies, algebra simplify power is a fraction, free online cost accounting textbook.
Square root chart factor, Year 8 maths algebra revision, fraction test 4th grade, algebra 2 standard form and vertex form of quadratics, how to simply a number under a radical, javascript program for
finding the common divisor of two numbers.
Asset exam question paper class-8th, algebra online free, solving addition and subtraction with distribution equations worksheets, simplify the cube root of 1/5, java calculate median include math,
permutation and combination word problems.
Copyright © McDougal Littell Inc. Answer Key, free, two variable equation practice, precalculus trigonometry concepts applications paul a foerster 1st edition solutions manual, how to solve a second
order differential equations, definition of cauchy sequence ,in daily life where we use it.ppt.
Intermediate algebra computer drills, math trivias with answers, quadratic equation root calculator graph, latest math trivia, Math problem solver for 2nd Grade, Algebra 2 polynomial graphs, standard
Aptitude questions with answer, algebra word problems 5th grade, parabolas calculator(standard form), laplace für dummies, "algebra of programming" ebook, online free math sheets by saxon math,
sample abstract for tutorial grant.
Slope intercept form worksheet, linear equation slope math sheets for 6th grade, solving a nonhomogeneous differential equation with boundary conditions, ged cheating methods, northcarolina eog pre
test.com, MATHAMATICS, simplifying absolute values "a-b".
Math calculator radicals, factoring interactive games, math problem worksheets with answer keys printable 7th grade, programs for ti-84 plus, fraction multiplier calculator, math related poems.
Exams on subtracting real number, solving equations of third order, exponent rules worksheet, free download elemantry structural analysis by MC Graw Hill.
Sums with brackets ks2, how to solve an equation percentage, calculator simplifying sqrt, cheat sheets - grade 10 math, quadratic fractions, ex. of balancing chemical equation by redox method.
Free Algebra Answers, simultaneous nonlinear differential equations, ti-83 calculator download, answers for prentice hall algebra 1 california edition, Math Answers Free.
Parallel resistance addition algebra, solving limits of a function, Simplifying Rational Expressions on a calculator, ontario grade 10 math problems, KS3 Mathematics Homework pack D: level 6 answers.
Ks3 english free worksheets, glencoe math answers, Simplifying Rational Expressions on a ti-83, proportion worksheets for algebra, Java code to solve non-linear.
Multiplying integers games, nonlinear differential equations in matlab, ged math worksheets, year 5 maths area worksheet, 7th frade graphing functions worksheets.
Free simplify radicals calculator, how to put square roots on square roots ti 89 calculator, free printable TAKS math for 4th grade, converting metric units using decimals fifth grade worksheet,
fractional radical exponents calculator.
How to use a graphing calculator slop, Use the TI-84 graphing calculator to caculate inverse function, Worksheet that compares quantities of objects using the symbols =, <, >, Solving algebraic
exponential fractional equations, fourth grade algebraic expression variables, decimal equivalent of mixed number 6 3/5, partial volume of an ellipse calculator.
Second order homogeneous difference equation, math questions for 9th grade and answer sheet, free math problems for 6th graders printable.
Factoring with three variables, , powerpoint writing an equation problem solving strategy, 9th grade math exams for free.
Word problems with integers and exponents worksheets, online math book for 6th grade virginians, online factorising, grade 5 math work sheet.
How to do cubed roots on calculator, square number games, simplifying calculator algebra.
Finding formula quadratic sequence worksheet, algebra help parallelogram to a trinomial, creative publications answer sheet, passport to algebra and geometry answers.
Printable math coordinate plane, radical form, maths yr 11, adding pages worksheets, example of mathematics trivia, adding and subtracting integers worksheet, probability free worksheets.
Multiplying and dividing with scientific notation worksheet, download cost accounting textbook, algerbra for all, fraction calculator online, printable worksheets factoring sum of cubes.
Conceptual physics workbook answers, algebra, math cheats, calculator that does radical expressions, square roots with powers of seven, elementary worksheet equation, SLOPE PARABOLA CALCULATOR.
How to solve cubed roots, free online algebra expressions calculator, ti 89 quadratic equation, ti-89 pdf, completing the sqaure.
Cube root ti-83 plus, formula to transform decimal numbers in fractions, 5th grade algebra decimal factors and products, dividing polynomials by binomials free printable worksheets, write algebraic #
's on my PC, aptitude question and answers.
Free math solver online just type in your question, free worksheet subtracting integers, lesson plans for adding integers, 6th grade line graphs, graphing simple rational functions solver.
Dividing negative and positive in algebra, equation of circle, simplify rational expressions online calculator, worksheets repeating pattern using first next last.
Doing multiple equations in excel, how to solve fractions, linear equations and third grade lesson plans, grade 1 worksheets for problem solving by adding and subtracting, importance of difference of
two squares, 6th grade probability homework.
Graph of ellipse+mathcad+example, unusual quadratic equations solved by substitution, free online elementary algebra help, graphing intercepts worksheets free, prentice hall mathematics Algebra 1
answers, how to solve second order differential equations.
Do my algebra for me, CONVERT MIXED NUMBERS TO DECIMAL, basic, quadratic equation source code visual basic, algebra worksheets gcf, teaching least common multiple in elementary school.
Math help for 10th grade alebrac connections, adding and subtracting negative and positive numbers worksheet, distributive property of square roots.
Algebra witrh pizzazz awnsers, cost accounting for retail books, how to find zeros from vertex form, free lcm calculator, pre-algebra math question, accounting books +pdf +free download, down the
program cognitive tutor for algebra 1.
Elementary permutation worksheet, decimal to radical conversion, balancing equations ppt, Free ebook download of Aptitude, trinomial factoring calculator, math geometry trivia with answers, GCD
Glencoe mathmatics florida algebra 1 workbook answers, What is formula of square root to java, math tutor saint charles high.
Free site on how,to do 7th grade permutations, yr 8 maths games, how to solve radical expressions, factor polynomials three variables, program to solving a matrix.
Mcdougal littell algebra 1 answers, EXCEL FORMULA-SQUARE TWO, balancing equations grams moles solving, examples of problem the solution is subtraction, games to teach exponents, log de base 2 TI-83.
Simple equations to graph, find the slope in a coordinate graph for 6th grade math, free online graphing calculator, TI 83, free factor tree worksheets, dolciani modern school mathematics structure
and method 7, free printable symmetry worksheets.
Algebra ii answers, online cramers rule solver, mcdougal littell math answers.
Math 11 unit 9 linear equations answer key, summation of integer numbers java, how to square root exponents, equation worksheet, prentice hall algebra, multiply rational expressions calculator.
Multiply (4)(3)(7), program ti 84 simplify radical, solving algebraic equations gcse demand and supply, math program shows all steps.
Modern algebra tutorial, lessons to teach students to add integers, Math For Dummies + free, what are the functions are used for chemistry on a ti-89 calculator, ti graphing calculator emulator
Formula for adding percentages, simplify inequalitie calculator, equations, free printable "worlds hardest word search".
Vertex form and number of roots, slope online exercises, addition and subtraction of algebraic expression for standard 6, numerical analysis MCQs.
Solving quadratic equations game, how do you graph circles on a ti-84, pre-algebra worksheet 9 answer key, percentage formulas, Finding the Greatest common factor worksheets.
Graph ellipses, sove quadratic equations C code, "TI-84 plus App" download, adding subtracting multiplying with variables.
Algebra 1 glencoe study guide and assessment / answers skill and concepts, algebra associative rule free worksheets for 2nd graders, graphing curves, simplify expression multiplication, solve percent
word problems printables.
Slope formula algebra free worksheets, articles + accounting + pdf + free download, factorization of quadratic expression, nonlinear least squares equation solver, simplify algebra equaitons.
Math games scale factor, calculator- adding and subtracting rational expressions, adding and subtracting integers worksheets, otto bretscher linear algebra with applications free download, greatest
common divisor using for loop in java.
Common denominator with multiple variables, simplifying square root fractions, trivia on algebra, transformations logarithm exponential "fun activity", convert 10 digit time.
Show free simple linear function worksheets for real world, worksheets on dividing integers, Holt Chemistry Ch 9 assessment & answers, free download of accounting books, linear equation solver java.
Fraction problems word problem multiply divide, when addind and subtracting positive and negatives what is the outcome of the answer, algebra answer, common denominator algebra problems, free eog
test for third grade, square root functionalities + rules.
Perfect square roots using culator, online calculator that does Powers of Monomials, msb<<8 meaning calculate, implicit differentiation calculator.
Second order differential equation solver, solve my math problem by just typing it in, maryland assessment practice for 3rd through 8th grade(printable), online slope calculator, Q, free online
calculator that divides.
Second order differential solver, algebra 1 concepts and skills answers, functional algebra tests, free 9th grade algebra, free printable commutative property worksheets for fist grade.
Math homework answers, free homework for 7th graders, math for dummies, calculating fractional exponents, pre algebra practice midterm. Doc, algebra with pizzazz trigonometry ratios, sample math
investigatory problems.
Javascript multiply command, square roots with x ponents, steps for graphing on a ti - 84 plus calculator, coupled first order differential equations in matlab, 2nd order ode matlab.
Learning Algebra in a fun way for 6th standard, online factoring, Methods of Search for Solving Polynomial Equations, how to solve simple equations.
Write a polynomial with the given zeros + ti83+ basic, gauss online calc, ti-83 online, How to do sequencing on a TI-83 Graphing Calculator, download accounts books, quadratic equation vertex.
Hyperbola Graph, practice test - subtracting integers, ti-84 plus how do you plug in absolute values [], addidtion and subraction of fractions worksheets, math investigatory project, abstract algebra
help, mathematica algebra solver.
Gre formulas on stastics, math trivia with answers mathematics, define math algebra-scientific equation/solution, software for algorithms on ti 89, Algebra Printable Work, SOLVING SQUARE ROOTS.
How to change a mixed number into a decimal, what is the common denominator of a transaction?, trinomial solver, second order linear ode without first derivative.
Free math worksheets for the G.E.D, online check algebra equations, free help with college algebra, TEST QUESTIONS ON COMBINATION AND PERMUTATIONS, factoring polynomials with two variables.
How to use solver on graphing calculator, math tutor help in eugene, or, history of add,subtract,multiply,and divide complex numbers.
Mathematics trivia samples for grade 4, liner equation, math help algebra 1 solve by factoring, adding subtracting multiplying and dividing fractions worksheets for grade 8, decimal into radical,
maths tests for year 11.
Algebra depreciation formula, Solver linear equations excel, Download calculator texas TI-84 Plus, maths nth term gcse sequences changing difference, simplifying expressions for perimeters, solving
equations degree 3 1 variable, factoring cubed polynomials.
Simplify calculator Square Root online, math trivia about logarithm, determine lcm of two doubles in java, solved qurstion papers of general aptitude test, how to calculate the vertices of the
quadratic formula, log de base 2 TI83, free download aptitude test.
Square numbers activities, easy maths and english paper quiz, subtracting negative fraction, FREE PRINTABLE FUN MATH PAPERS.
Distributive property personal tutor, free intermediate accounting powerpoint, teachers guide for prentice hall's mathematics, multiplying exponents with a variable, student reference books for
Saxxon math tutor, algebra calculator for homework help, solved english sample paper for 6th class, convert sums to integral, fun math worksheets on foiling, download aptitude test for it
professional, scale and scale models printable worksheets.
Saxon advanced math "test form a", exponent linear parabolic cubic quadratic, DOWNLOAD FAS APTITUDE TEST.
Non linear differential equation matlab, solve this alegbra equation, greater than or less than fraction calculator, butane reaction chart.
Iowa algebra aptitude test sample, distance equals rate plus time worksheet, convert fraction to decimal to percent cheat sheet, definition of scale factor (math), taks study guide algebra for 3rd
graders, subtracting square roots calculator, Glencoe Algebra 1 Lesson 9.
Find slope and y intercept printable, gcd calculator, dividing decimals worksheet, math investigation in book of elementary algebra.
Mcdougal littell houghton mifflin algebra 2 worksheets, math +trivias, lesson plan for dividing polynomials, middle school freeprintable, order all the decimals from least to greatest, get delta
function on TI-89 Titanium.
Simplifying square roots calculator, free 9th grade worksheet, algebra 2 vertex form, algebra math cheat, algebra RATIONAL EXPRESSIONS calculator.
Factor calculator algebra, quadratic simultaneous equation solver, pearson textbooks 9th grade algebra 1, factor polynomials online calculator.
Algebra 2 Worksheets, solving operations involving rational expressions, ordering integer games, dividing polynomials by trinomials, 7th grade free worksheets on unit price and unit rate.
Count similar char exit how many time in string + java, using, t89 2 unknown variables, Solve for the roots of the equation, radical form calculator, least to greatest fractions (-8/9) (-7/8) (-22/
25), calculas.
Homework help scale factor, sats 2 maths papers downloads, algebra tutorials+ questions, simplifying complex rational expressions.
Poems about algebra, what are advantage and disadvantage of solving a system of equations by graphing., pre algebra with pizzazz answers, break even problems algebra, multiples grade nine printable
worksheets, INDIA 6TH STANDARD MATHS FORMULA.
Function and non function graph, 9th grade math games, fractions lessons number line least to greatest, basic opertions worksheet.
Fun liner equation worksheet, log de base 10 TI-83, TI 89 logs, free online factorer, solve using elimination method calculator, least common denominator worksheet, percentage conversion equation.
Example, algebraic expressions with square roots, square root properties, ti-89 generar pdf, equation with two variables worksheet, variable fraction calculator, scientific calculator for algebraic
expressions online.
Ratio Algebra problems, polar complex numbers ti 84, free linear algebra with business application for dummies, solving by elimination.
Square root algebra calculator, lesson plans for algebra Basics, online calculater algebra, lesson on elimination using addition and subtraction in algebra.
Teachers edition of addison-wesleychemistry second edition 1990, adding and subtracting integers games, 5th grade algebra activities.
Free elementary worksheets on similar figures in geometry, How do I learn algerbra?, Math Trivia for Kids, 5th grade level school sheet printouts, mcdougal littell science ch.19 test answers, solving
systems of linear equations using matrices worded problem.
Dividing decimal worksheet, algebrator free download, order numbers least to greatest, math tutor program high school.
Cauchy dirichlet neumann problem wave equation, download :aptitude questions, Converting vertex to root form "quadratic function", calculate common denominator, solving algebra problems, "Alien
Xperiment", compositions of functions with trinomials.
Tips to pass college math class 108, ONLINE TI 84 DOWNLOAD, how do you turn decimals into fractions worksheet, Algebra Calculator Program.
Freetype in Algebra Problem Get Answer, how to expand a trinomial cubed, explain year 6 algebra, Permutations and Combinations in GMAT, graph a ellipses calculator.
"Iowa Algebra Aptitude Test", FREE ONLINE ALGEBRA HELP, "abstract algebra" +dummit +foote +solutions, trig calculate.
Vertex to standard form, elementary algebra problem solver, free kumon j answer book, free line anglesprintable worksheets for elementary, convert decimel for fraction, printable coordinate plane,
calculator for adding radical expressions.
Algepra power is a fraction, exponent simplifying, sequences with changing difference, tips algebra tests, algebra percentages and averages, adding subracting multiplying dividing intergers, math
elimination calculator.
Word problems with ordering numbers from least to greatest, real time @merica 2B Workbook, Simplifying Radicals Calculator, solving equations worksheet simple, how to solve algebra equations.
Free usable calculator online, converting mixed numbers on a number line, worksheet slopes, java solve linear equations, how to do a square root of a product using product property of a radical on a
graphing calculator.
Download aptitude questions, what is 1/8 in decimal form?, dividing square roots with radicals.
2 digit multiplying worksheets, math sheets on radicals, uk past exams paper year 6 free, 8th grade california algebra 1 worksheets.
Download e-book for aptitude terst, Latest Trivia in math, 8th grade pre-algebra printable worksheets, mathematics trivia with answer, combining expressions algebra tiles.
Discount percentage exercises worksheet, free graphs on the internet for printing for kids, Download free ebooks for accounting.
"equation of linear function", worksheets with adding and subtracting fractions with like denominators, 7 class sample papers, mathematica simplifying algebraic expression, how to factor a cubed
polynomial, Mixed Numeral as a Decimal, square root -long division method.
Beginning fractions printable book grade two, mathtrivia algebra for 2nd year, coordinates worksheet ks4, aptitude question of java, gauss error function c#, add/subtract mixed numbers worksheet,
greatest common denominator With Variables.
How do you solve missing variable in equivalent fractions, "pre-algebra" tutor SOFTWARE, o'level maths past papers, least common denominator with variables.
How to find slope on a ti-83 calculator, what are numbers called that will not simplify in radical form, AJmain, Free Accounting book, fluid mechanics fundamentals free downloads, formulas for ti-89.
"Operations with Functions" pattern domain, solving first-order, nonlinear differential equations, free algebra lessons for dummies, gcse math for grade 8, math formulas percentages, maths percentage
Algebra formulas answers, ti-89 symbolic manipulators, examples of trigonometric applications, "simultaneous nonlinear equation" + ti 89, how to do cube root with out calculator, Simplifying radical
Expressions Step by Step.
"Linear algebra" FRALEIGH ebook download, rules in multiplying dividing adding subtracting fractions, liner equation.
Free worksheets on adding integers, Importance of algebra, modeling adding fractions, radical fractions, radical expression quiz printable worksheet, simple ratio equation.
Middle school math with pizzazz! book E comparing & scaling, free printable slope worksheets, Edhelper factor tree test, square root and cube root activities.
Online symbolic equation system solver, logarithms ti, expression worksheet, hungerford solution, free trial Software for algebra with word problems.
Automatic LCM calculator, 9th Grade Pre-Algebra & Algebra Lessons & Worksheets, boolean equation simulator, methods of finding solutions of partial differential equations, adding and subtracting
fraction free worksheets, Learning Algebra 1 for free, proplem solvings.
Surds for dummies, factoring equations with fractions, free mathcad tutorials.
Area worksheets ks2, online alegbra formula solvers, free inequality worksheet, square root of 30 in fraction, EOC Test Workbook answers, polynomial with 3 unknowns.
Beginning algebera worksheets, fraction decimal worksheet, dowload question paper of IT aptitude test, algebra 1 percentage problems and answers, beginning algebra worksheets, how to calculate
equation for the vertex of a graph.
Printable practice simple algebra, rom image download, pearson prentice hall informal geometry exercises answers, maths sample question for standard 3, 7th math square root.
Prentice hall algebra tools for a changing world answers, After rewriting a general trinomial in descending powers of one variable what should be the next step as we try to factor the polynomial?,
algebra with pizzazz creative publications worksheets, math third grade combinations probability worksheet, aptitude questions and answer for maths, addition and subtraction of algebra for std 6,
solving quadratic sequences nth terms.
What is the greatest common factor of 30 and 105, binomial factor calculator, iowa test practice 6th grade, the easy way to find the coordinates in trivian, algebrator for college algebra, 5th grade
algebra word problems, pre algebra with pazzazz.
Simplify root of fractions, square root negative exponent, free exam papers in probability and statistics, solving 1-step equations worksheets "one side", free java aptitude questions and answer,
multiplying and dividing mixed numbers worksheet, free answers for algebra.
Prentice hall pre-algebra math workbook, LCM tutorials, solving systems of nonlinear nonhomogeneous differential equations in Matlab, Search how to turn decimals into fractions, graphing linear
equations, aptitude test download.
Explain why the restriction that a cannot be equal to 0 is given in the definition of quadratic equation?, aptitude test question and answer, answer key for modern chemistry chapter 7 mixed review
Factoring algebraic equations with variables and exponents, pre-algebra cheat sheet, algebra worksheets for year 7s, pythagoras theory calculator download software free, math worksheet 223 creative
publications, really hard equations solving worksheet, square root tables (no decimals).
Root square error method, number line for decimals and mixed numbers, online radical calculator, latest trivia in math, free on line college algebra practice quizz, dividing by subtracting
Mcdougal littell biology study guides, how to write a quadratic equation for graphs, "algebraic graphs".
Algebra With Pizzazz, work sheet for test of divisibility in maths, 6th grade math problem solving questions, 4th grade sol worksheets.
Understanding the language of fractions, log2 in ti calculator, how to do trigonometry grade 10 math prep, Introduction to cost Accounting book, Iowa Algebra Aptitude test sample questions, roots to
expressions using exponents, precalculus trigonomy word problems.
Maple solve, free english exam papers year 9, converting from base 8 to base 10, Legal Aptitude test sample paper, radical with fractions inside, cost accounting book free download.
When do you use graphing to solve a quadratic equation, free math worksheet on 1 step equations with answers, algebra trig electronic textbooks, Algabra 1, solving problems algebra 2.
Elements of Modern Algebra, 7th Edition study, free reasoning and aptitude books download, calculating what to make on final test if its 1/7 of semester grade, simplifying a radical expression with
addition, TI-83 plus cube root.
Fraction worksheets, subtracting decimals through thousands worksheet, FREE PRACTICE GED TEST PROGS, free algebra 2 answers, multiplying fractions worksheets for special ed.
Cool polar function pictures, coloring worksheet for finding slope, free download ebooks on permutations combinations & probability.
McDougal Littell Algebra 2 book answers, tricks to finding LCM, learning basic algebra, percent of a number equations, 6th grade circle graphs, algebra problems.
Calculator program solve cubes, factoring quadratic equations calculator, Ti 84 emulator.
Ancient tamil poem about mathematics numbers, nonlinear system of equations matlab, combinations and permutations worksheets, Algebra Grade 10, how do you input a Square Root symbol into powerpoint.
SOLVE MY ALGEBRA PROBLEM, algebra help for free applications, equation in standard form using integers calculator, factoring trinomial worksheet, algebra 2 book answers, grade10 maths tutor free
Prentice hall conceptual physics textbook answers, Decimal tests for grade 5, algebraic equations work sheet, simplify radicals algebrator, textbook +review +secondary +"algebra 2".
Free algebra practice tests order of operation, partial differntial equation, doenloadapptitute quetion for get, algebra online activities free, multiplying and dividing fractions worksheets, glencoe
physics review questions answers.
Cost accounting 7th edition answer key, algebrator download, java math algebra, Algebra 2 slope intercept form tip card, übertragungsfunktionen ti-89 pdf.
Change the radical to a decimal calculator, modern algebra tutorial pdf, ti 84 emulator, algebra worksheets GCSE, algerbra software, glencoe algebra.
Square numbers class activity, answers to algebra 1 homework, adding subtracting multiplying dividing fractions.
Minimize nonlinear equation, online parabola calculator, college algebra clep practice test free, taking roots longhand, 6th grade maths worksheets to print out, finding common denominators
Math formulas diagram sheet, glencoe math for ninth graders equations, answers forPassport to Mathematics book 2.
Linear equalities, simplifying exponents calculator, beginner intermediate algebra lial 4th, trig word problem solutions, subtracting negative numbers worksheets, solving equations worksheet.
Simplifying rational expressions worksheets, Fraction worksheets .doc, dividing polynomial applet, re balancing algebraic equations, agebra games, LCM ladder method 3 numbers, free online learning
math for 9th grade.
Fractions worksheets, substitution 6 grademath worksheets, nonlinear differential equations solution, fraction simplication worksheets, algebraic expresion calculator.
Simplifying radicals on a TI Calculator print out rules, square number activities, how to solve differential equation in matlab, freehighschoolhomework.com, solving non linear differential equations,
completing the square questions, adding subtracting multiplying and dividing powers.
Linear metre definition, free download solve nonlinear least squares with constraints, answer key to the real life math decimals and percent, free maths tests for grade 9, 12th grade glencoe
literature answer key, aptitude solve questions & answers, substitution calculator.
Algebra pdf, system of equations, 5th grade math practice, simplifying square roots with addition.
Math terms used in a poem, how to do radical expressions, how to solve a homogeneous differential equation, 5th grade STAR test papers, Evaluating square roots, Suare root, 6th-grade math pre-test.
Java calculator multiply 2 numbers, plotting points pictures, abstract algebra dummit foote solution, excell equation finder, aptiude paper of jeca, any equation solver non linear, graphing linear
equations ppt.
Solve your math software, fraction expressions and equations+6th grade, vertex of parabolas equation calculator, free math calculation and reasoning worksheets, study for exams for 9th grade pre
Download recent question paper for Management apptitute test, square root radical expressions, mathmatic equations practice hard, the answers to algebra 2 problem, maths puzzles to download ks3.
Addition and subtraction of algebraic terms, rom image ti 89 titanium, learning algebra the dummy way, probability homework video tutors permutations, algebra calcuator, dividing exponents
calculator, algebra solver.
Ti 83 identifying a function, quadratic to vertex form calculator, cost accounting Book, simple interest maths problems solved exercises india, postive and negative numbers activities, how algebra
sums are worked out.
Formula for 3 number digits in dividing, freesoftware for drawing hyperbola, freeware, software for solving equations.
The balance chemical equation of ammonia with phase, lineal metre definition, printable slope worksheets, solving addition and subtraction equations, FREE BASIC MATH PRACTICE SHEETS FOR COLLEGE, fun
linear equation worksheet, free solver for eigenvectors.
Finding slope made easy, sixth grade math curriculum + sanantonio, accounting notes free download, download aptitude test.
Ks3 maths worksheets, simplify expressions, similar figure free worksheet, inverse problem down for ti 84, math Poem Search.
Practice adding positive and negative intergers, ti 89 math made easy, Fractional equation worksheets.
11th grade math cheat sheet, 2 1/8 to decimals, adding and subtracting integer games for kids.
Algebra helper software, Free notes of cost Accounting, multiplying and dividing scientific notation.
Elimination method solve calculator, solving two step equations worksheet, online free "algebra review", year 11 maths exams free.
Prentice hall mathmatics algebra1 answers, removing fraction value in java, free online factoring polynomials calculator algebra, algebra square roots, answer to math homework.
Holt algebra one, www.mathsrevision.com level c homework 6 worksheet, radical, sum.
Excel solve equation, calculator casio use, mcdougal geometry book answers, calculate linear feet, algebra problems for bright children, integer add subtract worksheet, Free print math linear
equations work sheet.
Powerpoint slides linear equations in two variables, ebooks + download + General Ability + Aptitude, "lattice math" worksheet, 9th grade worksheets, fun maths questions worksheet.
Algebra for free, the answers to holt and rinehart 7th grade science test, study guide answers to mcdougal littell biology, INDIAN TEST PAPERS FOR 6TH GRADE.
Scale factor worksheets, synthetic division calculator program, algebra exponents calculator, permutation and combination in sas, calculate .871 to fraction.
Free Help With order fractions from least to greatest, free algebra calculators, 3rd class power engineering question bank, how can you make a line and and then see its equation on a graphing
calculator, how to solve differential equation of second order.
Online 7th grade help, poem in math algebra, cheat worksheet answers, "a poem on adding and subtracting integers".
Algebra 2 ( ordered triple) Need Help?, free math solver online where you just type in your question, algebra factoring rules, solving a differential equation with boundary conditions.
Equation 3 variables, worksheet for perfect cubed, 3rd order polynomial curve, write as a logarithmic equation calculator, free lecture note +download+basic equation+heat transfer, kumon level d
answer keys.
Simplifying algebraic equations, "second order differential" runge-kutta, adding multiple integer WORKSHEET, Free Cost Accounting Tutorials, balancing equations calculator.
How to find out the under root of square of any no., worlds hardest games, balancing equations animation, Glencoe Mathematics pre-algebra florida edition online version book, completing the square
work sheets, square root of a fraction.
Basic algebra practice worksheets, least common multiple lesson plan, ks3 coordinates worksheets, free formula sheet of trigonometry, program that will factor quadratic equations, convert fraction to
decimal to percent answer sheet, algebra with pizzazz! answers worksheet 160.
What does a fraction chart look like?, calculator for solving multiplying square roots unequal, How to convert the a square root into a decimal, solving nonhomogeneous differential equation,
WORKSHEET PRIMARY ALGEBRA RULES FOR NUMBER PATTERNS, Algebra Dummies Free, easy way to calculate prime numbers.
Adding and subtracting radical worksheets, best software for solve math problem, math triangle problem solver, Exponential Expressions, real life applications of linear equations, symmetry printable
quiz for fifth grade, free printable intro to algebra worksheets.
Teacher edition rudin, .055 equals what fraction?, ti-84 plus graphing calculator boolean algebra, PYTHAGORAS FOR IDIOTS, download Algebra 1: Concepts and Skills.
KS2 area worksheets, best math tutoring software, how to solve a complex trinomial, quadratic calc minus.
UCSMP Geometry problem, How to Solve Piecewise Functions, free algebra worksheets for 2 graders, free online algebra calculator, functions of cost accounting.
ALGEBRA SOLVE PROBLEM WITH NTH POWER, free algebra answers, adding like fractions worksheet, calculator cu radicali.
Math powerpoints-percentages, first order partial differential equations solution, online free teacher edition florida glencoe algebra 1, using like terms to solve equation.
Algorithm worksheets for 2nd grade, APTITUDE QUESTIONS OF ENGLISH, Approximate square roots on graphic calculator, chemical formula exponents, aptitude questions and answers, answers for algebra 2.
Maple; solve complex nonlinear equation, t1-83 calculator rule to 4th power, combinations permutations matlab, add,subtract,multiply, and divide fractions, where can i get free online help for
algebra 1a?.
Solving systems by graphing powerpoint, Symmetry Math Printable Worksheets Kids, the root of an exponent, indefinite integral ti 84 plus 83, igcse o level examination solved papers, linear equations
using t-charts worksheet.
Scale factor practice, MATHMATICS IN 8TH CLASS, solve cubed functions.
Free Math Problem Solvers Online, maths for dummies, Conjugate Cube-Rooting, explain slope usig y and x in math, simplifying complex numbers, ti 89 equation store.
TRYING TO START COLLEGE NEED HELP WITH MATH PRACTICE SHEETS, "Artin" + "solution to exercises", Dividing Variables Calculator.
Algebrator, LCM calculator, How to solve Least Common Denominators in Algebra 1, flowchart of lcm & gcd in one chart.
Lesson Plan of the first n terms of an Arithematic Sequences, worksheets adding positive and negative intergers, substitution method algebra.
Free rounding activities for 6th grade, convert lineal meters to square meters, Using set algebra simplify the following.
College algebra for dummies, trinomial calculator, finding the inverse of a cube root (x-6), I need an access code for student edition glencoe algebra I 2008, online fraction simplifying calculator.
Math permutations lesson plans, algebra balancing, substitution method to solve equation,worksheet, maths papers to print.
Elementary worksheets on coordinate grids, fraction formula, how to solve systems of equations using ti 83 plus, grade 12 solved mathematics exercise, adding subtracting multiplying dividing
integers, algebra 1 answers saxon.
Simplifying Radical expressions calculator, taking the square root of fractions, pdf to ti, solving a nonhomogeneous differential equation.
Simplifying fractions with square root, college+algebra+help, convert fraction to number in java, Exponents Calculator.
Completing the square basic, algebra worksheets, java convert decimal to any, lesson plans year 9 algebra, how to factor equations, multiplication using expanded method worksheets.
Solving nonlinear differential equations in matlab, Free Worksheet Intercepts of a Line, adding and multiplying powers, solving equations sheet, addition and subtractions equations, algebra
combinations and permutaion problems, integers worksheets 6th grade.
Fraction worksheets positive/negatives, find the quotient with variables and exponent, numerical integration calculator with steps, algebra structure and method book 1 answers, free online 4th grade
solving addition equations, convert mixed fractions to decimals.
Book cost accounting.pdf, SYSTEM OF EQUATIONS PROBLEMS AND ANSWERS TEST, solving linear combination, solving algebraic equation with exponents fraction, factoring trinomials calculator.
TI 84 graphic calculator emulator, solving linear systems by adding or subtracting, middle school Exponents basic formulas, algebraic equation for percentages.
Calculate Linear Feet, graphing calculator finding 3rd or more root, exponents, adding, subtracting, multiplying, and division, solve cubed roots, algebra with pizzazz answers worksheets.
Aptitude general test downloads, completing the square calculator, 3rd grade math review sheets, printable integer word problems.
Graphing calculator online clep, adding similar fractions worksheets, enter an algebra 2 problem and solve, ebook A. Baldor Algebra.
"geometric probability" worksheet, solve algebra problems, java ignore punctuation, pre-algebra revision worksheets, online Lattice math problems.
Fraction calculatorfinding slope of a line, solving radicals, advanced practice papers from algebra, add 3 two digit numbers worksheets, combination problems elementary school, aptitude questions to
solve with objective type.
Extracting square root, calculator mean standard deviation aggregated, logbase ti89 log2, algebra dividing square roots, factoring binomials calculator, approximate square roots using graphic
calculator, showing how to do alegebra the child way.
Trigonometric ratio worksheet, printable distributive property worksheet, how to solve for radical variable, example second order nonlinear differential equation matlab, algebrator radicals, i need
help on algebra 1a the book is prentice hall california edition.
Steps used in balancing equations, how to do algebra, solving quadratic equations with substitution, free maths excercise year 3.
Math problem solver, free apptitude question answers pdf download, common factor table, worksheet and flash cards on conditional type 2.
Algebra calculator free, (a+b ) square root formula, Adding and Subtracting Fractions Worksheet, partial products math sheet, sats maths question.
Solving simultaneous equations with quadratics, instant word problem solver, calculator converter decimals.
Printable math problems with pythagorean theorum, multiplying and dividing powers, adding and comparing freeworksheets, Help on radius of a circle for 5th graders.
How to use mathcad to solve differential equation, algebra 2 answers for the workbook, topic in alebra by herstein + problems and solutions.
RDcalc ti 89, Algebra Vertex Form, 3 unknown simultaneous equation solver, in algebraic standard form, does the variable always have to be x, unsolved sums of factorization, add subtract divide
multiply integers.
Worksheet + fractions, mcdougal littell geometry book chapter 5 test answers, system of equations ti-83, algebra with pizzazz creative publications, d'alembert for nonhomogeneous boundary.
NUMBER GAME, DEALING WITH SIMPLIFYING EQUATION, algorithm of division texas ti 84, converting decimal time, non homogeneous second order linear differential question, Maths Homework Assignments Book
7 Yr 8,9,10 free online, elementary math trivia, cube root on scientific calcualtor.
Getting rid of a radical in numerator, free college algebra calculators, "Prentice Hall 7th Grade Pre Algebra Textbook", samples of rational word problem+solutions, cost accounting books download,
probability combination algebra problems.
Free intermediate math problem solver, online graphing calculators that make tables, raising fractions to higher terms worksheets, algebraic aptitude questions, pre-algebra extra credit worksheet,
plotting eigenvalues in maple.
Review parabolas, how do add multiply divide and subtract fractions, Brain teaser permutation and combination classical probability, houghton mifflin graw hill 3rd grade IB.
Algebra games ks2, accounting pdf book free, easy adding math sheets, FRACTIONS FROM LEAST 2 GREATEST, when don't you need a common denominator to compare fractions, free factoring polynomials
calculator, printable math test.
Diflucan, Download material for aptitude, college algebra problems solver, discriminants for cubed equations, system of substitution worksheets to do online, chapter 4 linear programing, simplified
radical form.
Algebra worksheets with solutions free, vertex form algebra, FREE DOWNLOAD EASY way to Learn Math BOOK, solution of a third power equation.
How to solve a cube root equation, maths percentage formulae, Introductory Algebra second edition notes and answer.
Historical note of logarithms algebra and trigonometry, structure and method houghton, miffin, root square formula, How to get a Greatest Common Divisor using If and else Condition in java.
Math helo, Dividing Games, Ebooks aptitude free stuff.
Algebra trivia & answers, nets +math + 6th grade lesson plans, linear equations worksheets, ebooks on indian accounting free download, logarithm math help graphing calculator, sample of word problem
involving rational expressiions, rationalizing the denominator for simplified radical form.
Math factoring calculater, ebook for mathematics exercise for high school, what is a lineal metre?, Algebra 2 Answer Keys, chemistry addison-wesley answers, finding a number divisible by 3 in java.
Free english exercise books download, greatest common divisor calculator, what is a sideways parabola, printable exponent worksheets grade 5, Algebrator Online Special, How to calculate GCD of 2
numbers, CHEMICAL MIXING calculator vb6.
Download ti rom image, "calculas made easy", teach algebraic equations, solve quadratics expressions online.
Easier way how to do mathematic, solve nonlinear equations excel, visual math linear relationship worksheets.
Geometric formula sheet calculate chord area, fun square roots activities, rational expression solver, fun slope worksheets and activities, interactive example: reaction between two chemical
molecules, Introduction to cost Accounting book free, how to use the solver in ti-84.
Algebra by T. Hungerford, college algebra Barnett book help, pre-algebra percent group activity in oklahoma, decimal to square root converter.
Algebra for dummy, how do i recognize a linear equation, maths work sheet for 7th grade, maths in daily life-statistics, diene maths sheets.
Download aptitude pdf, hard math problems for modeling equations with quadratic functions, free college algebra clep quizzes, mathematical worksheet integers college.
Free clep practice college algebra, http 9th grade math exams for free, simplify radicals calculator, HOW CAN I GET ANSWER KEYS FOR ALGEBRA 2 PRACTICE WORKBOOK?, basic verbal aptitude question.
Ninth grade algebre worksheets, ti84 plus algegra quadratic, distributive property worksheets answer keys, algebra equation exponents.
Integers worksheet, LEARNING PRE ALGEBRA, differential equation calculator online.
Graphing equations powerpoint, when was the algebra invented, fractional powers equation, math work shhet site, solving nth order polynomials, prentice hall algebra 2 with trigonometry teacher
edition answers, Aptitude Questions With Answers.
Mathematical examples trivia, hyperbolas interactive high-school, find equation given roots, ks3 maths mental answers, fifth grade science question papers+download.
Prentice hall algebra 1 workbook, division rational expression, elimination method for solving equations calculator, Casio equation solver, Exponential Expressions and The Order of Operations, how
can I solve linear systems by addition or subtraction if the problem has a decimal.
Matlab algebraic solve, Adding and subtracting square root worksheets, math translation worksheets, factoring algebraic equations with variables, trigonometric calculator download, pdf notes SET exam
papers with solution in mathematics, free material for tricky mathmatics.
Worded problem in linear equation, algebra worksheet third grade, lnear equation divide fractions, convert decimal to square roots, java converting decimal to fraction, summation notation worksheets,
multiply and divide monomials worksheet.
Free college algebra ebook, +algebra equasion finding x, factoring with two variable, activities on square roots radicals, how to factor an equation with a calculator, coordinate plane worksheets.
Math calculator for radical expression, measurements in decimal converted fractions, "introduction to probability models" solution download, Integrated 3 McDougal Answers for Practice 38, principles
of mathematical analysis solutions manual.
Least common denominator tool, "Operations with Functions" rates of change, simplifying radicals algebrator, rational expression calculator, square root with variables.
Liner function algebra, solving complex equation in matlab, self paced algebra books, rotation work KS3, ORDER OF FRACTIONS.
Free worksheets, fundamentals of algerbra, revision maths form2, complete the square in 2 variables, free past revision science papers, equation of lines powerpoints.
Slope intercept worksheet, variance rules logarithm, sample GED test printable free, differentiate an equation through VBA, simultaneous equation solver, HOW DO YOU ADD FRACTIONS.
Elementary algebra worksheets, answers to glencoe algebra 2 worksheet, algabra, simplifying radicals geometry, "Hall mathematics course 3" Chapter 3 solutions.
Glencoe mathematics geometry chapter 4 practice test answers, online complex fraction calculator, linear equations and function worksheet, adding/subtracting restrictions, how to solve complex
algebraic equations, square root fractions, distance formula without square root.
Prentice hall mathematics algebra 1 workbook, what type of algebra is in heath algebra 1, Algebra 1 Lesson Presentation Transparencies Volume 1 Holt, Rinehart, and Winston, aptitude test paper free,
saxon advanced math solutions "test form a".
Show me a website where i can print off free pre-algebra worksheets, pre algebra problems for grade school students, excel solve right triangles, simplifying radicals puzzle, multiplying a number and
a variable with an exponent, spss.
Mcdougal littell world history texas edition, practice begginer algerbra, formula solver 3 unknowns, factor calculator equation, math translation worksheet, lesson\plan +Multiplication and Division
of Rational Expressions, simultaneous systems math.
How to solve an equation by finding the LCD, solving slopes/9th grade, math lesson plan squares and square roots, canada grade 11 math exam cheats, factorial gre story problems, decimal square,
worksheets for equations of motion.
Simple algebra questioms and answers, mcdougal littell inc/algebra chapter 9, algebra slope projects, find y values on graph of a calculator, multiply and divide decimals worksheets for 7th graders,
probability and stats worksheet + 5th grade.
Simplify square numbers, prentice hall algebra 1 solutions, Quadratic inequalities calculator.
Fluid mechanics solution manuals, pictures on graphing calculator, calculate one eighth of a penny to decimal, exaple of greatest common factor, aptitude Question and answer, algebra1/answers.com.
Modern Biology Active Reading worksheets, chapter 10, abstract algebra polynomial problems, MULTIPLY DECIMALS FOR GRADE 6 TEST, algebra vertex equation.
Market share algebra textbooks, free Factorization worksheets, math trivia with answers geometry, prentice hall mathematics algebra 1 workbook, hard algebra word problems online with answers.
Polynomial worksheets, free 7th grade pre-algebra revision worksheets, math properties worksheet, hardest algebra in the whole world, pre algebra project.
Rom "ti 83 plus" download, SQUARE ROOTS/RADICALS calculator, free sample algebra problems with answers or solutions.
Printable worksheet on writing function rule from tables, teacher stores, san antonio, mcdougal littell inc. world history worksheet answers, convert mixed numbers to decimals, third order
polynomial, solving third order system.
Introductory algebra games, mathtrivia, radical simplify online calculator.
Simplifying Radical Expressions with square roots, quadrating parabolas, mathematics trivias.
Dividing a whole number by a fraction worksheet, free download books site for cost accounting, difference of square roots, basic maths formula sheet, softmath, how to change a fraction or mixed
number as a decimal, free intermediate algebra lesson.
Foil method worksheets, Absolute Value, SAT prep workbook for 6th grade, converting mixed numbers into percents, converting mixed numbers to decimals.
Formula for nth term of function 1/x, solver simultaneous equations 3 unknown, adding positive negative numbers worksheet, formulae sheet of percentage in GMAT [PDF], vii th class sample question
papers, powers of radicals without calculators.
Reciprocal math practice 6th grade, free worksheets entrance exams, solve verbal problems by using two variables and systems of linear equations.
Coordinate system worksheet third grade, free prealgerbra help, converting decimals to mixed numbers, how to solve second order differential equation, heat in physics for mcqs and work sheet.
How do I convert whole numbers to decimal format?, Free downloadable test papers for equations for 11-12, algebra poems, online maths resources for yr 8 students, mathematical area formula ks2,
prentice hall mathematics california algebra pdf.
Steps in balancing chemical equations, find real numbers worksheets, iowa test for entering algebra, activities on square roots and cube roots.
Free online holt algebra 1 textbooks, solve nonlinear equations & c++ code, simplify cube root.
Decimal to radical, associative property worksheets, free glencoe textbook answers, ti- 89 solving a system of linear equations in three variables, Game Multiplying Adding Subtracting, logarithmic
equation slope, simplify calculator Square Root.
Free question paper for geometry for class 9th, grade 8 hard english and math test, formula for decimal to fractions.
Free 9th Grade Geometry Worksheets, algebraic word expression calculator, hard maths test online.
9th standard maths copy, arithmetic series sequence sum ppt, 9th grade pre algebra book.
Addition of fractions for dummies, mcdougall littell algebra 2 answers free practice workbook, maths trick for root, algebra equations fractions with variables, Break-Even Comparison using Matlab,
simplifying radicals solver.
Rudin solution manual, conceptual physics high school physics programme, how to simplify square roots in fraction, download mathtype 4 version for window free, how to solve imperfect square roots,
formula of square root in java, make a perfect square quadratic.
+math help finding square area in circl, aptitude question bank, square root worksheet for year 7.
Permutations and combinations basics, free math lesson plans, coordinate grids, equations to create pictures on calculator, worksheets on solving radicals, a math solving non linear simultaneous
Algebra with pizzazz answers, converting decimal values to decimal time, download application ti 84+cross product, properties of radicals calculator.
Convert quadratic function to vertex form, theory using equations of enthalpy change associated with neutralization, solving second order ODE imaginary roots, worksheet on adding and subtracting
scientific notation, java code for 2 order equation, math poems, school maths book in algebra.
Basic graph equations, solve the equation with variables as exponents, quad root calculate, linear equations and inequalities. 8th grade homework help, simplifying equations with multiple variables.
Algebra 1 formulas, users, smallest common denominator calculator, subtracting fractions with a number missing.
Multiply subtract negative, writing algebraic equations from word problems worksheets, math investigatory.com, multiple choice exponents practice, converting decimals to square roots on calculator,
real life example for application of exponential expression in mathematics, most common denominator calculator.
Online maths test year 7, learning alegebra, convert int biginteger java, simplifying variable expression with exponents, download aptitude sample papers, Download of Aptitude Test Papers .rar.
Cube root lessons, free printable 9th grade science worksheets, algebra practice problems for highschoolers.
Ti-84 factoring, ways to help students understand how to solve algebraic equations, algerba self help pamphlets, how to write 55% as a fraction, free complex fraction solver, maths trivia.
LOWEST COMMON DENOMINATOR CALCULATOR, free math trivia with answer, TI ROM IMAGE, glencoe algebra 1 resource masters chapter 7 book, 3rd grade algebra - multiply rule printables, sample act questions
Free worksheets positive and negative numbers, simplifying radicals on a graphing calculator, multiplying and dividing fractions centers, sample investigatory projects in math, ks3 maths quiz on
angles, finding real solutions of equations with fractional exponents.
Visual basic lcm, how to convert mixed numbers to decimals, special calculator for exponents, aptitude solved questions.
Rational expressions solver, roots of exponenets, ordering fractions from least to greatest, worksheets converting mixed numbers, decimal to square root.
Can i get an elementry education degree onlin at the university of phoenix if i live in chicago il, flow chart example for base exponent power, absolute value + worksheet + free, free fifth grade
geometry worksheets, fractions chart that show least to greatest.
How good are you at adding and subtracting, activities for simplifying radical expressions, algebra structure, formula for sqare, expansion and factorization of algebraic.
Decimal pictures, combining like terms worksheet, divide radicals multiple choice, finding coordinate points on a grid 6th grade worksheets, math study guides volume 6th grade.
How to cube fractions, radicals calculator, Conceptual Physics The High School Physics Program answers, algebra calculator.
'orleans hanna" test prep, algerbra, free workbooks high school, FINDING A COMMON DENOMINATOR.
Games with adding,subtracting,multiplying and dividing integers, free worksheets 4th grade, fraction equation practice, ALGEBRATOR, Scale Factor Charts, algebra expressions for beginners, kumon
Glencoe algebra 1 teachers edition, adding,subtracting, multiplying & dividing negative and positive integers worksheets, glencoe mcgraw hill worksheets.
How to Simplify square root of negative numbers, pre algebra substitution, simplify radical calculator, fator tree worksheets.
How to convert decimals to radicals, 8th grade math tests of slope and intercept forms, math cube square games, grade 10 algebra problems, turn decimals into fractions calculator, simplifying
radicals expression review answer, T184 CALCULATOR LITERATURE.
Polynominal, how to solve fractions that have subtraction and addition, ti-89 system solver, pre algebra worksheets, non homogeneous differential equation e^x.
Cost accounting books, write a mixed fraction as a decimal, subtracting integers free printable worksheets, boolean and calculation calculator.
Calculator simplifying conjugates, solving systems of non-linear equations worksheets, factoring algebra I worksheet, how to cube root on calculator, Calculator and Rational Expressions, square root
radical expressions multiple, system equations three variables software.
Free trinomial calculator, graph a system of equations, how to do Log on a TI 89, 2 step equations, sample question papers from m d university, cooperative learning worksheets.
Ti 83, system of equations, free test generator for algebra ii, kumon online answer book, ONLINE TI 84 CALCULATOR DOWNLOAD FOR COMPUTER, percent to fraction and decimal cheat, adding integers + fun
practice worksheets.
"gcd" +"10 to power", Graphing Quadratic Equations Online game, t1-83 plus linear equations, Glencoe textbook workbook answer, sample clep algebra.
Elementary algebra program, algebra probability problems, free high school algebra equations with variables worksheets, "word problem" "fraction" "garden", mcdougal littell geometry book answers,
simplify sum of products calculator.
How to put an equation in the form for a hyperbola, how to write equations with exponents, Lesson 6-3 Practice Dividing Polynomials Glencoe texas Algebra 2, decimals and mixed numbers.
Math formulas for percentages, aptitude question and answer, printable algebra worksheets, beginging algabra, algebra dividing fraction in equations, Fractions + Printable GED Test.
Factoring with cubes, rewriting division as multiplication, free math SOLVER for exponential growth.
Equation of a step graph, Printable Algebra Question & Answers for Grade-7, example of math trivia.
Linear programing free pdf book, p c combination permutation java app online, lesson plan multiplying integers, latest aptitude books free download, free maths worksheets ks4, algebra answers online
slope free, Algebra Poems.
Simplifying radicals calculator, First Grade Homework Worksheets, algebra for dummies, Solving addition problem worksheet, Ordering Fractions from Least to Greatest, converting square root into a
Integration problem of TI-89, factor worksheets 4th grade, clep calculator model, finding, simultaneous equations in two variables, solving simultaneous equations one linear and one non linear,
picture math sheets.
Cross product free worksheets, trig calculator activities, adding and subtracting fraction worksheets, times and divide fractions worksheet, how to generate equations from graph.
Algebra worksheet scale, ALGERBRA FOR BEGINNERS, math pre-algebra final exam, find square root of 2 numbers using java, multiplying and dividing decimals by 10 worksheets.
Elementary worksheets algebra, type in algebraic expressions and the computer give me answers, online pre algebra answers to my problems.
Maths worksheets translation, quadruple root calculator, Fractional LCD calculators, algebra simplify equation fraction power distribution, 8th grade math trivia, free kumon worksheets.
Synthetic division solver, algebra program teaches u algebra with 7 tutorials, practice, and tests, parabola rational function, calculator laplace, aptitude questions pdf, NYS 6th grade math test
samples, rational expressions caculator.
Answers to skills practice workbook, converting vertex to standard form, online rational calculator, Worksheets - Decimals - Add and Subtract - Grade 5, free apptitude book, free 6th grade iq test.
Square root with exponents, adding and subtracting fractions worksheets, math quiz/primary classes, elementary permutations and combinations, free special calculator for exponents, calculating
percentages on-line practice.
Biology mcdougal littell lesson plans, switching algebra simplifier, permutations and combination interactive tutorial, online trig graphing calculator.
Algebra properties of equality free worksheets, trigonomic equation calculator, free college algebra review.
3rd grade math finfing the range, substitution method solve calculator, prentice hall geometry answers free, algorithm solve symbolic equation.
Mathamatics, MATH TRICKS FOR ELEMENTARY GRADE 6 PRINTABLES, Vertex form, teacher supply stores in san antonio, Graphs of hyperbolas, sample cpm algebra 2 tests.
861032, free samples line graph interpretation practice, graphing linear equations :ppt, algebra problems 6th grade.
Mcdougal littell inc/algebra, adding subtracting multiplying and dividing fractions worksheets, poems with mathematical words, free college algebra quizzes, free download of aptitude and puzzles
ebooks, how do i enter cube root on a scientific calculator?.
Math lesson plans solving decimal equations, dividing decimals by intergers, add/subtract integer worksheets, prentice hall literature grade 7 workbook answers, evaluating expressions worksheets,
simplifying equation calculator, Prentice Hall Algebra 1 Chapter 10 answers.
Frobenius method+software, polynomial problem solver, factorization of monic quadratic pool expressions, calculation error basic algebra, dividing polynomials cheat, free polynomial calculator,
factoring cubed binomials.
Permutations for dummies, algebra 1 practice games, "algebra 1" free online textbook, the easiest way to learn advance algebra, solving complex quadratics, simplify square root of fractions,
interactive lessons for simple expressions.
School calculator download, free coordinates workshets+ks2, converting fractions to decimal worksheet, add subtract facts worksheets, McGraw Hill 6th grade math book.
Represent squares numbers and pattern with decimals, free worksheets on symmetry, 7th Grade Maths worksheets printouts.
Simultaneous equations calculator, worksheet+algebra factorization, free math printable work sheets with explanations, holt structure and method, multiplying integers worksheet, "Matrix Intermediate.
Teachers Book" -merlin -mareno, pretence hall answers.
Beginner calculus examples, finding slope matlab, fifth grade math worksheet, worksheet on completing the square, radicals math simple explanation.
Algebra program, printable math facts sheet partial sums 2nd grade, nonlinear ode matlab, math trivia and answers.
Mathematics investigatory project, advanced mathematics Precalculus + Richard G. Brown + free download, how to order numbers from least to greatest with multiples, Converting decimals into fractions
website that does it for you, Least common factor practice worksheets, advance algebra, mixed numbers to decimal calculator.
Free algebraic variable calculator, sequences and series power point presentation, worksheets on basic arithmetic adding and subtracting whole numbers, writing algebraic expressions free worksheets,
elementary font download.
Step by step free algebra solver, radical into decimal, cube root java, free elementary algebra tests, Math Trivia, mixed numbers to decimal.
IT aptitude question answers, free practice mathbook online, properties of addition worksheet, Example+Mathematics Year11, quadratic equation and expression.
Solving homogenous equation with matlab, square root solver, solving equations for a specific variable worksheet, how to solve for exponents, algebra 2 answers.
Yahoo visitors found us yesterday by using these algebra terms:
Download EBOOK FOR APTITUDE QUESTION, negative integer worksheet, pre algebra combining unlike terms, printable 3rd grade geometry, 11 year old exam papers, factored form expressions calculator.
Calculate a repeating decimal into a fraction on a ti83 calculator, Homework Worksheet For Kids, how to convert decimal fractions machine, maths worksheet fraction measurement.
Free math worksheet on coordinate plane, how to find square root in vb, Aptitude test papers with solution, ratio formular, MATHEMATICAL TRIVIA.
Permutation and combination basics, differential formula sheet, conceptual physics worksheets answers.
My daughters having problems with maths, free solution manual downloads ross elementary analysis, iMPERFECT square roots.
Free worksheet angle, how to do algebra 2, hard college algebra problems, free online trigonometry solver, free online graphing ellipses, square numbers activity, free adding and subtracting integers
Inverse variation in maths worksheet with answer, Elementary and intermediate algebra marvin bittinger, exponent square root, How to simplify algebra equations, algebra 1 fun activities to help with
inequalities, poems in mathematics, algebra solve for square root.
Numbers from least to greatest worksheets, ti-84 emulator, C Apptitude questions +pdf, Excel solver solution system of three equations.
Free worksheets converting improper fractions into mixed fractions, Free math symbols exams, algebra elimination/substitution calculator online free, answers to math book mathematics california
edition by houghton mifflin.
Excel basketball substitution spreadsheet, solving systems of equations addition subtraction, free matlab sheets, solving equations java.
Free online cognitive ability test practise for 3rd grade, Solving Systems of Simultaneous nonlinear Equations Degree 2, rudin math solutions "chapter 7", mathsprojectwork.com.
Choose form quadratic root, expanded notation maths free worksheets, solutions to integers, subtracting integers puzzle, printable math sheets for third graders, algebra + square root, worksheet and
answer key for math translate each phrase.
Egyptian math free worksheets, sample beginner alegra problems with worksheet, fraction equations finding lcd, convert fraction to decimal formula.
Direct variation worksheets, fraction exponents calculator, multiplying by 10, 20, 30 worksheets.
Free algebra textbook downloads, college algebra software, algebra the easy way download, fraction worksheets-4th grade, math sheets for 8 year olds, Free print linear equations work sheet, ti-89
Simple aptitude question and answers, d'alembert solution nonhomogeneous boundary, abstract algebra hungerford homework solutions.
Algebra Formulas Square Root, ti-89 simplify equation, how do you make a decimal into a radical, Mixed Number To Decimal Calculator, highest common factor of 28 and 32, synthetic division in
intermediate algebra.
Radical expression problems real world, one step inequality worksheets, iowa algebra aptitude sample test.
Lowest common denominator free calculator, converting decimal to mixed number, ti-84 calculator, how to solve trig ratios, online alegbra, multiplying scientific notation.
Download free games for a TI 84 plus, evaluating expressions activities, how do you convert a decimal to a mixed fraction, how to pass placement test for intermediate algebra, substracting mixed
number fraction worksheets.
Free online 9th grade math printout tests, commutative and associative rule free worksheets for 2nd graders, Formula for Scale Factors, how to teach probability to grade 7 help sheets, percents,
adding, subtracting, multiplying, and division, blank coordinate plane sheet with numbers, system of differential equations convert matrix maple.
Rational expression number games, The database file name cannot contain the following 3 characters: [ (open square brace), ] (close square brace) and ' (single quote, Numerical Aptitude Paper with
solutions, solve non linear equations of three variables, on-line Mathematical aptitude test, world problems in algebra step by step, Laplace determinante java-code.
Physics final test practice 9th grade, evaluate and simplify calculator, free cross product worksheets.
Free quizzes on greatest common factor and least common denominator, how do you do the substitution method in algebra, adding rational expression equation calculator, equations for liner
relationships examples in math, Iowa algebra aptitude test, three variable equation software.
6th grade equation story problems, boolean algebra questions, using prime factorization to reduce fractions worksheet, how to pass your algebra 1 final.
How to solve differential equation using ti 89 titanium, dividing polynomials glencoe algebra 2, online graphing calculator for statistics.
How do i solve a probability problem?, Three Value Least Common Multiple Calculator, gcse maths complete the square.
Free ks3 math work sheet expanding brackets with answerer, asset exam question paper class-8th 2009, free test maker 9 grade algebra, radical fractions with root divide, system of trigonometric
equations with maple, graphical calculator online, hyperbola equation.
Teach me how to use casio calculator silver 2 way power, How to Make a Repeating Decimal into a Fraction on ti83, fundamentals of physics answers 8th edition, negative integers and 5th grade, middle
school math with pizzazz book b pages, dividing polynomials on a ti 83 plus, Radical expression solver.
Year 7 maths questions ks3 algebra, ebooks indian basic accounting, usable TI-84 online, grade 10 algebra test, simplify square roots fractions, ti84 chemistry, order.
How to find square root pdf, games subtracting integers, aptitude questions pdf, year 10 maths ratio worksheet.
Simultaneous linear differential, aptitude arithmetic graph, work for 9th grade, free printable algebra math lessons and worksheets.
Simplifying fractions using radicals, solving equations with multiple variables, simple linear equation worksheets.
Faction to decimal conversion chart, +differentiated instruction lesson for function notation, pre algebra final exam papers, least common denominator finder.
Free online 8th grade my work, triganomotry cheat sheets, denominator worksheet.
How to get rid of perfect square, algebra formulas sales tax, maths worksheets for ks2.
Holt biology worksheets, pre algebra for 7th grade equations free print off sheet, how to you Determine The Prime Factorization Of A Given Number And Write In Exponential Form, clep college algebra
quiz, prentice hall algebra 1 test answers, Beginning and Intermediate algebra+answers+free, free algebra final practice tests.
3rd grade printables, calculator programs for algebra 2, represent square root, algebra calculator+factoring, algebra1 quik worksheets, similar triangles midpoint theorem proportion examples
questions and worksheets lessons, year 8 non-calculator maths worksheets.
Algebra simply and evaluate, how to calculate ellipses +Algebra, Algebra Fraction and Binomial products Calculator, free college algebra quiz.
6th grade algebra worksheets, everyday examples of polynomial equations usage, "Math test generator", math book 6 grade free.
Algebra 1 poems, copy of "orleans hanna test", algebra study sheet.
English aptitude questions, simplifying square root calculator, solve for the variable, where can i get a printable on a highschool algebra text book?.
Kinds of algebra books, factoring in the TI84 plus, quadratic equations 10th grade level.
Algebra cheating, How do you convert a mixed number to a decimal;, equation solver ti-89, highest common factor of 44 and 110, algebra tile games, algebraic factoring online, factor tree worksheet.
Locus maths rules, solve for interest rate. ti84, one step book on equations with integers, cost accounting download, Polynomials square root.
Rational expressions fraction calculators, solve differential equations for second solution ti-89, factoring/online graphing calculator, order of property math problems, grade 3 free worksheet, what
is a scale in math, visual basic programs for graphing calculator.
Factoring complex numbers, algebra help for free, college algebra trivia.
"grade 6""worksheet""math""printable", simplify polynomials factor, scott foresman 7th grade math book questions (2-step equations, pre algebra formula, algebric software.
Log base 2 on a ti 89, t-83 calculator program, "florida math connections", cardano cube fortran, cost accounting for dummies, Prentice Hall Mathematics.
Printable math test, log exponent solver, electrical "electrician aptitude test" questions, KS2 fractions worded problems.
9th grade math work sheets, hyperbola online graphing calculator, 8th grade math online free, what order trinomial is a quadratic equation, revise exponents for gmat, exam papers+grade 11, online
longhand calculator.
Excel solving homogeneous system of equation, ti-83 plus + cubed root, converting calculator bits into decimals, graphing inequalities problem solver.
9th grade printable worksheets for english, put decimal into radical calculator, 6th Grade math Scotts Foreman.
Converting mix numbers to decimals, Formula Greatest Common Divisor, how to subtract, multiply, and divide integers, mcgraw hill 6th grade textbook math, rationalizing polynomial denominator,
converting mixed numbers to decimals, ti-89 programs engineering.
Printable integer worksheets, Square roots of polynomials, "mean mode median" worksheet high school, rational expression PROBLEMS, Exponents on TI 83.
Graph a liner equation, algebra worksheets on adding brackets, online examination flash sample, "3d trigonometry" worksheet, parabola formula high school.
Online factoring, Radical expressions in real life, yr 8 math.
The university of chicago school mathematics project advanced algebra review, help with Elementary Algebra: Basic Operations with Polynomials, free downloadable aptitude test books, printable
worksheets yr 8, free answers to math algebra problems.
Practical uses algebra equations, Printable Math Handouts Ratio/Proportions, how to calculate least common denominator.
How to find the square root of a polynomial, ti calculator emulator rom, online yr 9 maths test, Biology Exam Paper for grade 8, download accounting bôk.
Spss warning only one component was extracted, pythagoras formula, java program to find if number is divisible by 3, online maths test free year 9, quadratic equation simplifier, houghton mifflin
math tests for teachers.
CliffsQuickReview Basic Math and Pre-Algebra free pdf, chemistry teacher facts power points, algebra calculator simplify, math trivia on algebra, Simplify dividing fraction exponents variables.
Maths tests for yr 7, 4x4 determinant tic tac toe, tutorial algebra grade 5, online T83 graphing calculator, casio graphics calculator emulator, factorization, quadratic formula, completing the
Freemathamatics downloads squarenumbers, 0.416666667 written as a fraction, how to solve "ellipses" with a calculator, simplifying expressions calculator, calcul radical, dividing polynomials
calculator, Percent Equations for Algebra.
Prime factored form, how to calculate RMS of linear equation, convert decimal to binary online mantissa, solving equation worksheet.
Lecture material mechanical measurements pdf ppt, graph relations equations solver, 3 times radical 2 plus 3 times radical 5 calculator, solving partial differential equations numerically in +Maple,
algebra vertex calculators, printable two step equation worksheets, poems about 8th grade math.
Online polynomial factoring calculator, formula sheets for CLEP algebra exams, Algebra for dummies free download, variable exponents, wwwmathcom.
Tips on teaching pre algebra, INVESTIGATORY project in mathematics, Students entering 8th grade pre-algebra worksheets, Quadratic Equation Calculator with square root, Free Algebra Readiness
Warm-Ups, c programming 3rd equation solution.
Downloading free ks3 study software, answers for algebra 2\, graphic calculator t184, ti-83, solve system of linear equations, Lesson Plans Elementary Algebra, quadratic equation programs for
calculator, online year 8 maths test.
Rules for adding negatives in algebraic equations, factor quadratic, solving third order polynomials, algebra square root of a, sample aptitude question for software company, PDF avec ti89, Solving
radical forms.
Download Banking Aptitude Questions, linear and nonlinear worksheets, chartered accountancy books free download, algebra worksheets combining terms, TAKS STUDY GUIDE 5TH GRADE MATHEMATICS, algebra
help software, graphing the slope of linesin excel.
Grade 10 math quadratic, importance of algebra, solving inequalities/game, solving an equation with multiple variables.
Free vocabulary printouts for middle schoolers, trigonometry fifth edition answers, college algebra quick learning clep, combination MATLAB.
Arithmetics problems worksheet, how to solve multiple monomials, where log in TI-89, Math exercices, instead of cramer's rule ti-83 plus.
Accounting for beginners glossary download, what is factor loading see table, answers for algebra 2, factoring trinomials diamond.
The order in which to solve algebra problems, year 2 freework sheets, integrate sqare root, inequalities worksheet 8th grade, solving partial fractions on ti89.
Used houghton mifflin text book 3rd grade,california, solve simultaneous differential equations using matlab, "additive property" worksheets, ti89 help solve simultaneous equation, practice on the
completing the square, free interest math problem solver.
Simultaneous linear differential solved by elimination, system of leaner equation, What is the difference between evaluation and simplification of an expression? Explain using an example, systems of
equations involving lines and circles, "grade 6 math""algebra test", graph solver, algebra worksheets for year 7.
Free worksheets for 8th grade, higher maths algebra calculator, Florida Glencoe class codes, Beginning and Intermediate algebra+answers, TI-86 Binary.
Subtracting like signs, how to subtract integers, taylor expansion multivariable, relevant of algebra in other subjects, how to calculate memory required for a cube, grade 9 math practise.
Adding, subtracting, multiplying, dividing exponents, "maths for year seven", website that can solve hard equations, exam papers on line statistics maths, how to find LCM kumon, worksheet for
fractional radicand.
How to solve quadratic equation using matrices, algebra: what are the rule in adding,subtracting,multiplying,dividing, advance algebra and trigonometry trivia, free downloadable aptitude ebooks, spss
factor analysis the solution cannot be rotated, holt algebra 1a final exam, how to pass college algebra.
Solving non-linear differential equations laplace, simplify algebraic expressions and answer, pre algabra help.
Math tutor, champaign, Illinois, cost accounting books, teachers copy of holt, rinehart and winston algebra 1 book answer key, free worksheets for integers for adding, subtracting and multiplying.
Common denominator calculator online, math trivia grade 4, aptitude question answer with explanation, college intermediate algebra help online, ged pre algebra.
Fifth grade algebra worksheets, clep algebra answers, holt physics book, graphing quadratic parabolas, how to find the square root in statistics, travel equation in Algebra, tutorial for solving for
square roots.
Solving first order partial integrals, free solve equations with rational expressions, simplest way to compute college algebra compound interest, Free 8th grade math worksheets, Free Math Problem
Solver, perimeter worksheets ks2, rationalize denominator polynomial.
Triangle formula in solving percentage, note for accounting-kids, multiplying negative numbers practice, nonlinear simultaneous equation.
Rational exponents ti-83 plus, simple algebra equations, algebraic translation worksheet, kumon math worksheets level 2, online maths paper, solve nonlinear differential equation matlab.
Grade 8 algebra questions, Baumgart History of Algebra, 'root word worksheets' for kids, casio fx-300w negative powers, circumference of a circle year 8 free maths worksheets.
Isolve mathcad, Free printable Testpapers on the topic Decimals for Primary 5, printable math problems for thied graders, What are the steps of the order of operations in alegra.
Solving 4 equations 4 unknowns in mathematica, log to the base 2 calculator, Finding Square roots by Prime Factorization + Power Point Presentations, learning algerbra, alegebra questions.
First grade math homework, prealgebra worksheets, trinomial factoring calculator, factoring a cubed equation, algebra problem solvers, solve standard form to vertex form.
Rules in subtracting,multiplying,and dividing integers, TI 89 cube route, sum of number in java, free primary year 3 sats example papers, aptitude test paper download for multimedia, math daily word
problem tutorial.
Grade 6 equations free online worksheets, 3rd grade mathmatic worksheets, integers lesson 5th grade.
Glencoe algebra 1 lesson 12-3, free math problem solver, exams ,test problems of mathematics quadratic algebra coordinate determinant, equation for factoring in final exam grade, Excel root
Grade ten trig questions, simplifying radical notation, javascript fraction calculation.
Prentice hall math book pages online algebra 1, source code minos linear relaxation, Florida Prentice Hall Mathematics Algebra 1.
Simplifying roots and exponents, 9th grade example division problems, how to solve tricky algebra equations.
Axioms in mathematics tutorial, easy middle school algebra equations, algebra 1 poem, maths for dummies, DOWNLOAD COMPUTER CALCULATER, merrill advanced mathematical concepts teacher download.
TI-89 Laplace Transforms, trigonometry problem solver, balancing equation root maths, exercises calculate fractions to decimal grade 5, finding the zeros using quadratics-grade 11.
Quadratic equation factorize exercise, lineal meters to square meters calculator, formula of decimal to fraction, calculate LCM.
Ti 89 titanium complex root, free 9th grade practice worksheets, rules in adding and subtracting integers, greatest common denominator equation.
Answer for english worksheets grade 9, multimedia aptitude question, sample math tests on line free percentage and fractions, free online ks2 math riddles.
Least common multiple solver, solving square root problems, finding slope and y-intercept with and equation worksheets, convert decimal to square foot, what is the relationship between solving a
function algebraiclly and graphically?, ks3 free english games online.
Math for dumies, mcdougal littell world history notes, 10 by 10 coordinate plane printables, program for permutations and combinations in C language, Free Printable geometry Worksheets for 9th grade.
Integers worksheet, solve simultaneous equations online, expanding trinomials, Excel equations.
Grade 8 algebra test alberta, glencoe algebra 2 answer key, special products and graphs relations functions, nys 6th grade math, EU test online ks3, aptitude question paper with answer, Elementary
Geometry for College Students solutions.
How to solve equations and inequalities with square roots, rational expressions and absolute values, worksheets for 8th grade, solve a complex expression by factoring, divide rationals, finding
exponents of variables, liner graph.
Algebra solving log calculator when base 3, circumference free math worksheets grade 7-10, adding radicals calculator.
Enter slope solve equation, convert hcf of fraction, online algebra games.
Maths worksheets adding subtracting negative numbers, trigonometry word problems with 2 unknowns, doing quadratic equations on ti-89.
Grade 8 algebra test, math poems on algebra, fractional exponents in demoninators.
College level tutor seattle linear programming, year 9 maths free work sheet, free word problem worksheets for fifth grade, basic algebra free online games.
TI-30x IIS instructions cubed, freeware calculator program, TI 83 Program basic trig programs, simplifying square root fractions variables, answers for key to algebra student workbook 9, McDougal
Littell Algebra 2 textbook-texas edition, Long Division Free Math Test.
Graph liner calculator, solve non linear equation system in MATLAB, complete the square calculator, online scientific calculator cubic root, mathematical advanced induction tutorials, free
downloadable kids maths workbooks, algebra half-life.
Thousands nonlinear equations, trivias about algebra 1, ti83 entering roots.
Free online pre-algebra worksheets, kumon+example+fraction+example, woksheets on permutations and conbinations, math induction equation.
Mixed number to decimal converter, like terms calculator, indirect lesson plan + algebra tiles, quadratics on the TI-89, fractions least terms using sci calculator, multiply rational expression
advanced, radical addition with roots.
How to subtract, multiply and divide integers math, multiply roots calculator, solving the quadratic equations by completing the square calculators, What would you recommend should be the steps for
factoring a polynomial?, trivia about mathematics.
Calculating slope radius from plan view radius, free download aptitude ebooks, non-compatible with algebra, quadratic equations +interactive activities, java, Finding number divisible by 5 and 6.
Simplyfying complicated fraction calculators, ti 84 calculator tricks for statistics, free advanced algebra.
Adding fractions with power signs, animation chemical formulae and equations, calculators that divide rational expressions, 6th grade eog samples, Study Sheet With Algebra Rules, "vocabulary for the
high school students"+pdf+free.
Online algebra expression calculator, algebra voice +tutorial, polynomials including quadratic equations, importance of algebra.
Trivia on algebra 1 in negative and positive sign, free 9th grade level math, ratio formula, simplifying radicals in Algebra 2.
8th grade pre-algebra test, lowest common denominator worksheets, 9th grade Estimation worksheets, Simplify square roots when would you use an approximate answer instead of an exact answer?, solve
quadratic equations with root.
Sample Maths Questions Ks3, free answers to complex rational expression, solving minimum and maximum problems using quadratic equations, discrete mathmatics outline.
Learn to add subtract multiply divide fractions, additional of monomial fractions, maths poems in Indian tradition, how to find regular price in 7th grade.
Incidence matrix matlab, Learn 3rd Grade math printable, free online math worksheets for kids going into eight grade, free download aptitude test papers, divide polynomials calculator, free rational
equation calculator.
Factoring in algebra, two variables second order differential equation: solution, office 2007 3th grade polynome, sat test for 6th grader, statics formulas ti89.
Free beginning algebra worksheets, Square Root Formula, What are the basic rules of graphing an equation or an inequality, online aptitude exam papers.
Elementary algebra measurements, multiply and simplify exponents, free factorisation worksheets, ti-86 graph in "terms of y".
Free printable math worksheet adding integers, rules on adding and subtracting integers, Online Calculator Square Root, download aptitude test.
Audio tutorials for factoring polynomials, solving differential equations by laplace transform with TI-89, online conic graphing calculator.
Excel aptitude test, function of investigatory project, how do you calculate a lowest common denominator, 9th grade algebra integers, matrix simultaneous equation solution tutorials, algebra factor
machine, algebra two problem solving sheets for free.
Whats the easiest way to turn a decimal to a fraction, free 9 th grade algebra sample test question, Free 8th Grade English Worksheets, printable pre algebra "show work", easy ways to factor
Free cost accounting ebooks, printable math worksheets grade 9-12, simple mathematicals trivias.
Sats test ks3 maths question breakdown, CONVERT SQUARE ROOTS, entering radicals in a t1-83, synthetic division calculator equation.
Thrid grade math test sheets, 6th grade honors algebra quiz, "Math Test Generator", logarithms tutorials for free, printable 8th grade work sheets, ti-89 boolean, parabolas. step by step.
Alegbra problem, partial sums addition, 11+ maths paper, working with signed numbers, GMAT answering cheats, rational expressions calculators.
Convert a number to a decimal in excel, free prealgebra worksheets, substitution calculator, NYS 7th grade math prep, math-foiling.
Agebraic Expressions, algebra with pizzazz answers, word problems with 2 unknowns worksheets, examples of trivia.
Learn 9th grade math online, square root multiply and simplify by factoring, PRE-ALGEBRA with pizzazz! worksheets answers, solving temperature two variable word problem, sample algerbra problems.
Multiple variable equations, multiplying and adding square roots, trinomial explanation, iowa pre algebra assessment, free online mathmatics tutorial 10th, equation to find square root of a number,
Maths problem solver.
Algrebra I free worksheets, grade 4 algebra- practice sheets, simplifying fractions with square roots, positive and negative coordinates worksheets, solve for equations for 3D data points,
subtracting algebraic expressions.
Prentice hall answer physics, REPRESENTATIONS CHEMICAL EQUATIONS BY DIAGRAMS, algabra solver, discrete mathmatics, free printable maths exercises primary, algebra tricks and trivia.
Ti-86+cubed root, ti84 factoring, Free TI 84 ROMs, Casio calculator reducing factoring Polynomials.
College algebra software, math for dummies free help, excel + addition and subtraction equations, free online calculator for solving intercepts, simplify (X squared+Y cubed)cubed, logic puzzles ks3
maths, maths-quadratics HELP.
Charts software free, Electrical Math formula sheets, online year 8 mental maths papers.
Solving equations using negative and positive numbers, free samples of pre-alegbra problems, yr 7 maths exam worksheets free printable, texas instruments t-86 directions.
Prealgebra & worksheet, VISUVAL BASIC BOOKS, simple algebra activity sheets.
Algebra I eoc word problems, english worksheets for tenth grade, importance of algebra, How to convert a Mixed number percentage to fractional notation, quadratic equation solver on ti83,
mathematical poems for 7th grade.
Free online math word problem solver, 8th grade math pre algebra, british & english multiplacation, rules for adding subtracting multiplying and dividing negative numbers, conic section exam problem
solved equation, maths for begginers.
Nonlinear simultaneous equations, prentice hall algebra 2 exercises online grade 10, importants of algebra, how to save formulas in TI 89, aptitude test paper for free, 9th grade science worksheets.
8th grade science taks question bank, answers to 2007 mcdougal littell course exam, steps in adding subtracting multiplying and dividing whole numbers, prealgebra problem solving, trinomial equation
solver, free online ks2 math riddles with answers, 1st grade printouts.
Easy solution in algebra, quadratic formula plug in, 8th grade math TAKS powerpoints, holt algebra 1 answers, numbers ordered from least to greatest, angles worksheet grade 5.
World's hardest math game, Solving Quadratic Equations by Finding Square Roots, aptitude test paper questions, Beginning Algebra Fourth Edition, how do we add subtract integers, free fifth grade math
Algebra ratio formula, convert linear metres into square meters, solve system of equations by elimination calculator, free online book for mcdougal littell algebra 1 teacher edition, Integrated
arithmetic and basic algebra 3rd edition online, 9th grade science taks test pretest, real life simultaneous equations.
Ks3 english test papers print outs, reflections maths worksheet free download, simplyfying fraction calculators, "equation of the line" solver.
Ti-83 plus using radicals, 9th grade algebra practice, college algebra help.
Help solving college algebra problems, how is vertex form of a parabola helpful, "pizzazz algebra", free cost accounting books, how can i help my daughter with grade 3 maths in SA.
Putting numbers least to greatest calculator, ti-83 plus programs summation notation, 9th grade algebra study sheets, free maths paper, general maths formula sheet.
Easy algebra, how to solve common denominator, solve wronskian.
Algebra for beginners free help, rational equations proportions paint, rules and steps to solve ratio, find the foci of the hyperbola defined by this equation calculator, ti 83 calculator programs
with steps, free algebra instructions and worksheets online.
Free pdf books on accounting, free physics gcse mulitple choice past papers, free online graphing calculator TI 83, second order linear differential nonhomogeneous, How to understand grade seven
math, algebra calculus beginners, worksheets on HYPERBOLA.
Mathematical relational symbols, solving equations for a variable online calculator, Ny 8th grade math sample test final test, help in intermate algrebra.
Maths worksheets of ks3 for free, college algebra glossary, downloadable free aptitude book, how do polynomial functions help you solve real world problems?, free math printouts, mathematics basic
statistics IMPORTANCE.
HARDest math problem, mcdougal algebra, matlab and nonlinear regression and finding equations, integers questions and answers for adding subtracting and multiplying, maths pratice, factoring
worksheets grade 10, parabala formula.
Trinomial calculator, math help ordering decimal least to greatest, steps to find nth term, how to chart a linear equation with a ti 83, math elipse, how to add,subtract,multiply,divide integers.
Elementary algebra worksheets, how to do cube roots on TI-83, sixth grade review worksheet, math test matric level, boolean simplifier, ti-89, lattice multiplication template.
Free year 5 number sequences worksheets, solve exponential equation matlab, define Writing linear equations, Easy Math Trivia.
Double conversion+precision+java, PRE-ALGEBRA terms of all the alphebet, easy way to find LCM, "teaching TAKS".
Extracting a common factor from an algebraic expression, installing the quadratic formula into TI-84, sample question paper of aplitude test in software company.
Algebra Mathematical Worksheet Extended Level Printable, free algebra worksheets for 9th grade for teachers, LCM calculator for exponents, online linear calculator.
Finding 4th root, soft math, pages of chapter 11, Geometry by McDougal Littell, how can you solve piecewise function using system of equation?, unit 3 resource book mcdougal littell biology.
Software to do algebra, worksheets on proportions, Expanding quadratic expressions advance, 9th grade subject worksheets, how to graph quadratic functions with Ti 83 plus.
Kumon type worksheets, Equivalent algebra equations, formulas to get to percentages.
Grade 7 algebra worded questions, 5th grade multiplying fractions and mixed numbers worksheet, java number fraction length, why is it when one number is a factor of another, the LCM is larger.
Help sheet maths y8 probability free, multiplying integers lesson plan, pocket pc algebra solver, percentage formulas, What are ellipse problems?.
Maths tests ks3, 9th grade printable test, 9th grade fractions review, taks 9th grade worksheets printable.
Free algebra graphing, fun integer math trivia, 5th grade worksheets, free, algebra solver download free, pass clep, power point pearson prentice hall algebra 1.
Algebra simplification rules for powers, scale factor-middle school math, Algebra with Pizzazz Answer Key, how can I find the quadratic equation if the points are fractions, am I ready for sixth
grade take a test to find out.
Alegebra 1, printable algebra games, solve my equation problem.
Algebra solve the trinomial, kumon math worksheets, practice sheets online for 7th grade math, Advanced Accounting midterm solutions, permutation combination, lining up decimal free worksheet.
Solve third order equations, secrt vii class square roots online test, simultaneous quadratic, TI-83 solve completing the square, Algebra Structure and Method Answers.
Mcdougal little powerpoints, adding and subtracting positive and negative numbers worksheets, Prime factorization of the denominator, help with solving college algebra graphing problems, notes on
permutation and combination for gre preparation, Permutations and combinations math tutorials pdf.
Subtract multiply and divide negative integers, free download + aptitude test, TI calculator accounting download.
Free maths paper online ks2, TI-84 plus factor, online algebra 1 math worksheets with answers, free printable english 6th grade.
TI-84 applications cheat SL math, solving linear diophantine equations with four unknowns, advanced 6th grade math concepts, root and exponent, fractions into decimals calculator.
Linear equality calculator, Algebra 1 Worksheets 9th Grade, mathematical formula to cracking a safe, putting quadratic formula on a ti-84.
Cubed equation, algebra trivia, 6th grade math sol practice ga, formulas in solving the square root of a number.
Quadratic equation quiz mcdougall, solving nonlinear equations in Excel, Algebra 2 McDougal Littell ch 9 help, 6th grade accelerated math final texas.
Need a calculator to solve trinomials math question, solving linear and non linear equations, "abstract algebra" powerpoint.
Worksheets on water grade 4, 9th grade math eog practice test, appitute questions, grade 8 scale factors, graph an ellipse free solve, pre algebra exam, easy way to learn algebra.
Relationship between Permutations and combinations math tutorials pdf, algebraic fractions calculator, graphs, how to simplify whole numbers with fraction exponents, linear algebra tutorial -
Easy way to solve radical expression, statistic in mathematics.ppt, Parabola maths homework practice doc, factoring cubed trinomials, second order differential equations non homogenous.
Percent worksheeets, balancing equations online, substitution math solver, free algerbra games for beginners, formula for ratio, pre allgebra learning.
Pythagorean theorem quadratic, Graph Linear Equations Worksheet, Percent formulas, math trivia questions, vocabulary flash cards chapter 1 conceptual physics, worksheet collecting like terms KS3,
rate of change formula.
Trinomial root solver, clep algebra, algebra 1 answers, Not an algebraic variable in expression TI 89, KS3 Mathematics homework pack c: level 5 answers.
Algebra1cheat sheets, excel combination calculation for variable, example of dividing integer.
Trig , free online help, math solver online, Things I need to know for a pre-allgebra exam, sample paper animation aptitude test, learn cost accouting free, maths sum difference between 120 and 160,
ti 83 plus calculator programs steps.
7th grade algebra practice - LCD, example problems quadratic absolute value inequalities, Simplifying square roots with exponents, Converting squared numbers and cubed nnumbers, Aptitude question and
Mathematical combinations, properties, samples of standard equation of ellipse, textbooks of english for 9th grade, real life examples of linear equations, parabolic graph calculator online.
Printable 5th grade algebra worksheets, learn basic algebra free, completing the square interactive, 4th grade fractions printables, calculator that factors trinomials, what are linear, quadratic,
and exponential functions and their formulas, year 8 exams papers.
Convert 6 divided by negative five, summation worksheets, Intermediate Algebra Trivia, Free Coordinate Grid Worksheet, 2 step equations, triginometry cheat chart.
How do i find the vertex on my calculator, math homework help nth power, free pre algebra finals test, parabola pictures and graphing calculators, Aptitude test qestion with answer, McDougal Littell
Geometry final.
Basic math tutor, convert whole numbers to decimal, applied algerbra 1, need answers for algebra american school, Evaluate Expressions quiz.
DIFFERENCE QUOTIENT SQUARE, college algebra CLEP, identity solver, free printable worksheets for fourth and fifth graders, McDougal Littell English Answer Key.
Ratio and proportion trivias for geometry, complete the`square calculator, free ti calculator download, differential function maple solve , Importance of Algebra, math tutor online free chAT.
Mixed review worksheet 8th grade, sample erb, solving simultaneous equations, c programming for binomial theorem, matlab combination.
Prealgebra online exam, scale factor activities, equa exam grade 3 practise, science worksheets for 8th grade, rational exponents and roots, "5 digit" square root problem worksheets -buy.
Algebra exercices, free download for recruitment aptitude tests, Algebra software for eighth graders, proportions worksheets.
Math formula sheet, 4th grade reducing fractions online worksheets, 5th grade math on mixed numbers,fractions, and decimals on the eog, permutation and combination basics, Monomial Solver, APTITUDE
Test + free template + answers, 5 digit square root problem worksheets.
Graphing linear equations worksheets, pearson algebra 1-2 exercises, free printable work for 4th graders.
What is the relevance and application of exponential functions in real life situations?, iowa algebra aptitude test, liner equation graph, Writing Decimals As Mixed Numbers, free downloads
mathamatics squear numbers downloads.
Online pre-algebra + free, algerbra games for beginners, accelerated math practice sheets for 4th grade, ti-83 sum, SQUARE NUMBERS AND CUBE ROOT GAMES, how to teach algebra, problems on ellipses.
Tenth grade mathmatics, how to multiplying fractions and integer, Kumon Material Matemática Download, mathematics for a dummie, vertex form to intercept form.
Algebra rational exponents and radicals solvers, positive and negative worksheets for kids, solving for a specified variable.
How to fing a printable grade sheet for class, algebra solver, chapter 11 algebra test final answers, 9th grade algebra tutorial, free simultaneous equations.
Algrabra dependent equations, 9th grade integrated algebra final, y intercept finder, basics exercices maths, how to do fractions on ti 83 plus, what do u download to get games on ti 84 plus.
Year 11 maths help, slope quadratic equations, brent method matlab, computer program multiply divide add subtract numbers, perfect numbers for dummies, maths + poems, yr 8 sample algebra questions.
"Algebra 1 pretest", free online study help for teens math/rates ratio, free math test printouts, Trigonometry printouts.
Polynomials under radicals, graph rules of graphing exponets, solve algebra, free examples of college algebra, sum of ten numbers in java, multiply and divide fractions of square roots, geometry
readiness worksheets.
Free solutions manual probability, what are the rules in adding,subtracting,multiplying,division, solve square root matrix in excel, 7th grade Algebra worksheet, grade 9 math questions, algebra
question trivia.
Example of the Gauss-Jordan Method for dummy, ti89 logs, Riemann sums solved examples, algebra finding the slope solver, factoring ti 84 plus, how to do algebra equations on TI-84.
FREE CONVERTION METRE, learn two step algebra, algebra equation of parabola passing through three points, very easy math problem, mathematics aptitude test formulas, x y intercept application in real
Nonlinear difference equation, algebra taks test for 9th grade, Ratio and Proportion trivias, relevance of algebra, algebra FOIL solvers, square root fractions.
Online multiple variable graphing calculator, equation solver for fractions, simplify radical expressions, quadratic word problems worksheet, What are the rules for adding, subtracting, multiplying
and dividing positive and negative integers?, simplifying ineqaulities, rational expression calculator graph.
Solving simultaneous nonlinear equations, the hardest inequalities in the world, Solving Factorials, parabola calculation, Holt Chemistry chapter review and answer, Solving 3rd order, maths dummies.
Properties of exponents lesson plans, free online TI-89 calculator, method square root, solving algebra equations, math hw anwsers, are there any free common entrance past papers.
Free quadratic factoring system, integers worksheets, grade 10 formula math.
Easy way to find the solutions to quadratic equations, nonlinear differential equations first order, square root of a binomial calculator, downloadable aptitude test, ks3 algebra worksheet, free
worksheet for 5th grade, how to pass the algebra 1 eoc.
Sum of rational expressions solver, Fractional and quadratic functions, gcse statistics free online practice papers, how to add subtract, multiply and divide integers, an introduction to programming
using visual basic 2005 6th edition answer key for the even problems, factoring by removing the greatest common factor.
Kids curves worksheets, easymathworksheets, slope-intercept equation.
Math tutor software, rules of adding,subtracting,multiplying,dividing algebraic expressions, signed numbers worksheet, steps for multiplying integers, holt rinehart and winston modern chemistry
chapter 13 quiz, in what way can we solve the addition,subtraction,dividing and multipliying.
Algebra 1 worksheet, permutation and combination expansions, simplifying fraction lesson plans, ks3 mathematics test, texas TI 84 plus (games).
Ti-89 on var, casio algebra log, english-test.ca, algebra tutoring, champaign, Illinois, factor problem.
Free 5th grade math fraction assignments teacher edition, hardest mathematical formula, 9th grade algebra equations finding X, "Easy ways to learn Algebra I", simplifying algebraic terms, easy steps
to do algebra, mixed numbers to decimals converter .
Practice test for algebra 1 final, online free worksheets of kumon, "fluid mechanics""pdf""free download".
How to calculate number of combinations, mathmatic, solving natural log equations, First Grade Printable Step-by-Step Worksheets, math halp, 8 grade free worksheets, free algebra ll problem
How to calculate a hyperbola, worksheet for fourth grade maths fourth grade, solving second degree polynomial inequality, pre algebra practice work book, parabola equation converter, algebra 2
practice workbook solution manual, adding ,multiplying ,division and subtraction worksheets.
Simplifying square roots worksheets, help with solving graphing algebra equations, neooffice how to mix fractions and numbers, calculating time value money solver on TI 84 PLUS SILVER, free algebra
books, free online linear equations test and answers, Math Trivia with Answers.
Algebra calculator solving for x, factorization calculator, How is the Distributive Property useful in calculating store savings?, how to factor an equation, free printables math for grade 2, how to
store formula ti 89.
Free algebra problems (grade 9), how to teach kids algebra, adding square roots general form, simple mathemathics trivias, glencoe precalculus answers, Are there any online calculators that divide
rational expressions?, problems and exercises+algebra+math.
Slopes to solve an application problem, math+cheats, algebra 2 key concepts, learn algebraic fractions on line, FREE PRINTOUTS FOR THIRD GRADERS, 10 grade algebra worksheet.
Rational expression calculator calculator, how to convert mixed numbers to decimals, pre algebra practice 9th grade math, express .12 as a fraction.
Calculation solver, hard equations, online factorise, polynomial Ti-83 program, focus of a circle, Expanding quadratic expressions advanced.
Trigonometry values, fractions for dummies, easy steps on understanding hyperbolas, 20 week mathematic semester lesson plan, year seven maths.
Examples of Quadratic Equation, year 8 maths past test papers - free, practice problems for the physics test for 7th grade, rational expression online calculator.
Scale factor, basic math for dummies, convert 2/3.
Free Grade 10 Maths Questions, multiplying roots calculator, 9th one algera free online learn, hard math equations, adding square root polynomials, 3rd grade math compare decimal sheets.
Synthetic division worksheet with answer, two and three step equations - fractions, Mathamatics.
Covert fractions to decimals calculator, homogeneous first order system of partial differential equations, System of Equations Addition calculator, thermometer question for integer, factoring
polynomials applet.
Simplify logarithm exponent, trigonometric expression as an algebraic expression, exponent exercises grade 6.
Download polysmlt ti 83+, accelerated math for 4th grade practice pages, solving 2nd order different equation, simplifying radicals calculator, 8th grade algebra printables, examples of integers,
permutations ti-83.
Math algebra poems, free trig practice problems, square method for factoring polynomials.
Free trial pre algebra worksheets, evaluating and solving with radicals, ti-84 greatest integer, test papers yr 8, online textbook grade 10, multiplying by 8 worksheets, maths test online for year 8.
Algebra de 8th grade graphics, free online exams +tests for primary section, find intercepts of quadratic with fractions, "variables in algebra powerpoints", importance of algebra college, printable
year 8 maths exam, math code words intermediate level examples of.
Math Area Sheets, easy lcm, I GRADE KIDS MATH PRINTOUTS, pdf algebra, matlab convert decimal to fraction, algebra year 8 test.
Hard algebra question, any question of college algebra, free mathematical advanced induction tutorials, mixed decimal, algerbra questions, real life polynomial division situations.
Algebrator download, practice college algebra clep test online, fractions, adding subtracting, multiply, dividing, subtracting rational fractions calculator.
Calculator for trinomials questions, logarithm advance tutorial, cheats for maths homework.
"algebraic long division" lesson plan, formula of ratio, xth maths-application of derivation, online TI-84.
Finding the greatest common factor of 3 numbers, math solver, college algebra by mark dugopolski, trig caculator.
Logs base on ti 83, free downloadable uk year 9 english wworksheets, cubed button ti-89.
Finding the common denominator, problem solving worksheets 5th grade, linear and absolute value inequalities union, discriminant math worksheet, solving rational equations printable worksheet,
simplify radicals calculator.
Second radical of real number, gui matlab calculator, Ti 89 Hyperbolas, beginning algebra parabolas.
Multiplying and dividing fractions with square roots, free algebra solver downloads, solving.
Practice Math Sheets for High Schoolers, factoring equation in the TI84, Rational expression calc subtraction, the square root with fractions, free accounting books.
Cubed root on a calculator, mathmatical answers, what is the difference between greatest common divisor and greatest common denominators, free download addison wesley algebra, prentice hall biology
homework worksheets, frre on line english test, calculation permutation using matlab.
Multi-step algebra, grade nine math worksheets, excel slope formula, 4TH GRADE ALGEBRA GAMES.
Problem for investigatory project, contemporary abstract algebra, free college algebra simplifying online, calculus made easy code, problem solving for dividing fractions, free download aptitude
Algerba online, solve quadrinomial, quadratic equation on ti-83 plus, printable math logs.
5th grade questions, percentage equations, Permutation(Solved Problems), help with grade 8 maths-algebra problems, free printable worksheets for 8th grade math.
Steps to solving real life parabola questions, free accounting BOOK, rearranging equations - yr 8, logic and proof of discrete mathematics+permutations and combinations+PPT.
Put in algebra 2 problem and get the answer free, Logarithm Solver Online, solving algebraic equations practice test, resistance of 1KM calculator, free algebra worksheets for 9th grade, +"solving
rational inequalities"ppt, Radicals+maths+grade 8.
Simplified radical form., simplifying sum of radical solver, prentice hall physics answers.
Gcse maths free test past papers, algebra 1b formulas learned california, intermediate algebra-software.
Making algebra easy, pie value, maths factorise calculator, free online 6th grade math book problems, algebra formulas for grade 9, algebra-standard form.
Easy way to use ti-84, c program to calculate sum of n numbers, addition and subtraction integers worksheet, basic logarithm quiz, free worksheets for sixth graders and every subject, divinding
fractions with meaning.
Free Online Algebra Tutor, maths sums for free online, free mathematice exercises and solution, simple interest calculator using java code, solve 3rd order polynomials with matrix.
Algebra test, solving pre algebra equations from college, math algebra simplification, solve quadratic equations with radical.
Algebra square root, importance of algebra, free online algebra solver, Holt Introductory algebra 1 tests online, rom ti, kumon answers, factoring with a graphing calculator.
Work sheets age 5 maths, free activity sheets on 7th grade math, multiplication and division rational expressions, partial method of balancing chemical equations, printable 9th grade math worksheet,
Adding Integer Fractions, excel formula square root.
Calculator that can find the square root of any number, how to find number which is evenly divisble by any prime number for RSA in java program, math trivia question about algebra, algebra multistep
equations online quiz, bm worksheets and answers(year 5), mental maths works sheets.
Maths formulae for tenth class, Trig answers, math solution software for algebra.
Year 11 math questions, surds solver, quadratic exponent equation solution, free 9th grade algebra worksheets, slope of line given two vertices.
C Aptitude Questions, Mcdougal littell work out solution key Algebra, help me with algebra, FLuid Mechanics Solutions Manual.
SOLVE MY HIGH DEGREE POLYNOMIAL, free algebra refresher, ONLINE LESSONS SCALE FACTOR, Solving Simultaneous Nonlinear Equations in TI-89, how to teach 1st graders about graphs'.
Glencoe maths, ONLINE MATHS PAPERS FOR 5TH GRADE, free reading worksheets grade 6, bisect polynomial root java, online graphing cal.
Eog math 6th, mastering physics answers, printabel work out sheets for 3rd grade, "online matrice calculator", rationalize square root calculator.
Science ks2 worksheet, texas calculator distance bearing programs, objective function of linear equation, grade 9 trig practice questions.
Teach yourself maths, who invented the distributive property, free mathematics test papers.
Math formulas+ percentages, grade 11 examination papers, measurement printables + third grade, second order differential equation solver, third grade worksheets to print, proportions worksheet, holt
online sample word.
Pythagoras solver, ti-84 algebra cheat apps, school work math printouts, help to resolve logarithms problems, hard equations involving variables for 7th graders.
Solving 2x2 linear equations calculator, ti 84 emulator, 8 grade math final practice exam, college algebra made simple, free 5th grade math fraction teacher edition.
Dsolve second order nonhomogeneous, dividing fractions in 4th gradr, 8th grade adding integers worksheets, how do you solve fractions on the TI-83, math handbook formula.pdf.
What is one basic principle that can be used to simplify a polynomial?, quadratic equation and real numbers, ks3 math worksheets algebra, rational exponents in rectangle, Lial Miller Trigonometry
Trig answers, sample math prayers, geometry worksheets for third grade, solving binomial equations, boolean equation solver.
Intermidate algebra fouth edition study guide, texas calculator distance bearing programs TI-84 plus, solving fifth order polynomial.
9th grade work for your child practice sheets math, Free Maths Practice exercise for beginners, second and first order in one equation differential equation in Matlab.
TI-84 interactive calculator applet, simplifying square root fraction difference, holt online sample word, "the c answer book" download, aptitude tests worksheets india.
6th & 7th Grade Free Printable Worksheets, algebra1 work, calculator radical.
Clep tutorial online, cost accounting test questions, solve my synthetic division problem, The Symbolic Method For solving an equation, online maths quiz for year 8, what are some examples from real
life in which you might use polynomial division, free year 8 maths exam papers. | {"url":"https://softmath.com/math-com-calculator/function-range/decimal-calculation.html","timestamp":"2024-11-10T06:21:18Z","content_type":"text/html","content_length":"214262","record_id":"<urn:uuid:8d6f6cb6-5943-4868-9d85-2b3e9a8dd80d>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00401.warc.gz"} |
Victor Han^1, Jianshu Chi^1, and Chunlei Liu^1,2
^1Electrical Engineering and Computer Sciences, University of California, Berkeley, Berkeley, CA, United States, ^2Helen Wills Neuroscience Institute, University of California, Berkeley, Berkeley,
CA, United States
When multiple RF fields are applied at different frequencies, multiphoton excitation can occur when the sums or differences of integer multiples of these frequencies equal the Larmor frequency. No RF
at the Larmor frequency is required. In this work, we describe the general principles of multiphoton pulsed selective excitation, providing a formalized treatment with design examples and
implementations on a 3T scanner with an additional homemade z-direction (B[z]) coil. With the additional B[z] coil, we demonstrate additional flexibility, where the same excitation can be
accomplished in several different ways with the same pulse duration.
NMR excitation can occur not only at the Larmor frequency, but also at subharmonics of the Larmor frequency, a phenomenon called multiphoton excitation^1–3. Although subharmonic excitation at high
$$$B_0$$$ fields is very inefficient, the efficiency can be greatly increased by using multiple frequencies to generate excitation instead of a single subharmonic. When multiple RF fields ($$$B_1$$$
fields) are applied at different frequencies, multiphoton excitation can occur when the sums or differences of integer multiples of these frequencies equal the Larmor frequency^4,5. With a proper
choice of the RF frequencies, multiphoton excitation becomes practical in high-field MRI where SAR may be a concern.
In this work, we describe the general principles of multiphoton pulsed selective excitation, providing a formalized treatment with design examples and implementations on a 3T scanner with an
additional homemade z-direction ($$$B_z$$$) coil. With the additional $$$B_z$$$ coil, we demonstrate additional flexibility, where the same excitation can be accomplished in several different ways
with the same pulse duration.
With a small-tip angle approximation and ignoring relaxation, the Bloch equations can be written as
Denote the transverse magnetization and magnetic field as
For pulsed B fields over the time period from 0 to T, the solution to Eq. [1] is given by
$$m_{xy}(\textbf{r},T)=i\gamma M_0\int_0^TB_{xy}(\textbf{r},t)e^{-i\gamma\int_t^TB_z(\textbf{r},\tau)d\tau}dt.\:\:\:[4]$$
In the Larmor frequency rotating frame, let us define
These are the typical fields in MRI, except with the addition of a uniform RF field in the z-direction for multiphoton excitation. When the frequency of the xy-RF is far off from the Larmor
frequency, but satisfies the multiphoton resonance condition $$$\omega_{xy}-\omega_0=n\omega_z$$$^6, Eq. [4] can be rewritten as
$$m_{xy}(\textbf{r},T)=i\gamma M_0\int_0^TB_{1,xy}(t)e^{-i(n\omega_zt+\theta(t))}e^{-i\gamma\int_t^TB_{1,z}(\tau)\cos{(\omega_z\tau+\phi)}d\tau}e^{i\textbf{k}(t)\cdot\textbf{r}}dt,\:\:\:[7]$$
where $$$-\gamma\int_t^T\textbf{G}(\tau)d\tau=\textbf{k}(t)$$$ as in excitation k-space^7.
If $$$B_{1,z}(\tau)$$$ is slowly varying compared to $$$\cos{(\omega_{1,z}\tau+\phi)}$$$, then the integral of their product is approximately the product of $$$B_{1,z}(\tau)$$$ and the integral of
$$$\cos{(\omega_{1,z}\tau+\phi)}$$$. With this assumption, and using the Jacobi-Anger expansion shown below, where $$$J_m(-)$$$ is the Bessel function of the first kind of order m,
$$e^{i\frac{\gamma B_{1,z}}{\omega_z}\sin{(\omega_zt+\phi)}}=\Sigma_{m=-\infty}^\infty J_m\left(\frac{\gamma B_{1,z}}{\omega_z}\right)e^{im(\omega_zt+\phi)},\:\:\:[8]$$
Eq. [7] can be rewritten as
$$m_{xy}(\textbf{r},T)\approx i\gamma M_0\int_0^TB_{1,xy}(t)e^{-i(n\omega_zt+\theta(t))}\left(\Sigma_{m=-\infty}^\infty J_m\left(\frac{\gamma B_{1,z}(t)}{\omega_z}\right)e^{im(\omega_zt+\phi)}\right)
e^{-i\frac{\gamma B_{1,z}(t)}{\omega_z}\sin{(\omega_zT+\phi)}}e^{i\textbf{k}(t)\cdot\textbf{r}}dt.\:\:\:[9]$$
Only the term with $$$m=n$$$ contributes significantly to the integral, giving
$$m_{xy}(\textbf{r},T)\approx i\gamma M_0\int_0^TB_{1,xy}(t)e^{-i\theta(t)}J_n\left(\frac{\gamma B_{1,z}(t)}{\omega_z}\right)e^{-i(\frac{\gamma B_{1,z}(t)}{\omega_z}\sin{(\omega_zT+\phi)}-n\phi)}e^{i
Eq. [10] shows that $$$B_{xy}$$$ and $$$B_z$$$ contribute to the excitation profile in a similar way with $$$B_{1,xy}(t)$$$ and $$$J_n\left(\frac{\gamma B_{1,z}(t)}{\omega_z}\right)$$$ available for
amplitude modulation and $$$e^{-i\theta(t)}$$$ and $$$ e^{-i(\frac{\gamma B_{1,z}(t)}{\omega_z}\sin{(\omega_zT+\phi)}-n\phi)}$$$ available for phase modulation. This contrasts with the standard
one-photon case where we would just have $$$m_{xy}(\textbf{r},T)\approx i\gamma M_0\int_0^TB_{1,xy}(t)e^{-i\theta(t)} e^{i\textbf{k}(t)\cdot\textbf{r}}dt$$$.
To demonstrate the principles described in the theory, we simulated and implemented three sets of related pulses. To generate each pulse, the following procedure was followed.
1. Generate a prototype pulse using a conventional method like the SLR algorithm
2. If designing a standard one-photon pulse, directly set $$$B_{xy}$$$ to the prototype pulse and finish.
3. Else if designing a multiphoton pulse, choose $$$\omega_z$$$.
• Based on Eq. [10], choose values such that $$$B_{1,xy}(t)J_n\left(\frac{\gamma B_{1,z}(t)}{\omega_z}\right)$$$ equals the amplitude modulation of the prototype pulse, and $$$e^{-i\theta(t)}e^{-i
(\frac{\gamma B_{1,z}(t)}{\omega_z}\sin{(\omega_zT+\phi)}-n\phi)}$$$ equals the frequency modulation of the prototype pulse. For multiphoton excitation, we have more variables to choose which can
together achieve the same effects as in the one-photon case.
• Shift the center frequency of the $$$B_{xy}$$$ pulse by $$$n\omega_z$$$.
Base SLR prototype pulses were generated using SigPy.RF
. See https://github.com/LiuCLab/multiphoton-selective-excitation for complete details on pulse generation. $$$\omega_z/(2 \pi)=25$$$ kHz for all experiments.
Fig. 1 shows the setup using an additional $$$B_z$$$ coil. Fig. 2 shows the simulations and experimental results of one-photon, two-photon, and frequency-modulated one-photon pulses producing the
same slice selective excitation when designed to be equivalent. Frequency-modulated one-photon pulses are pulses which use $$$e^{-i\theta(t)}$$$ to imitate the effects of a $$$B_z$$$ pulse. Fig. 3
shows the two-photon pulse from Fig. 2, except shifted in position by three different methods. In Fig. 4, two-photon SLR pulses are demonstrated where the first pulse has amplitude modulation fully
in the $$$B_{xy}$$$ pulse, the second pulse has amplitude modulation fully in the $$$B_z$$$ pulse, and the third pulse has amplitude modulation in both the $$$B_{xy}$$$ and $$$B_z$$$ pulses.
Using the same one-photon and two-photon pulses as in Fig. 2, Fig. 5 shows the in-plane results of the one-photon and two-photon pulses in vivo under our institution's IRB approval. No significant
differences between the images are observed for this set of parameters.
Discussion and Conclusions
When $$$\omega_z$$$ is large enough, the distinction between the xy- and z-direction RF becomes smaller, and the ability to modulate $$$B_z$$$ instead of $$$B_{xy}$$$ gives extra flexibility to the
multiphoton RF designer. We demonstrated how the same slice profiles can be achieved in many ways. Although not explored here, the principles of using a $$$B_z$$$ field for selective excitation could
be extended to the use of arrays of z-direction coils for improving excitation homogeneity or tailoring excitation in general. Especially for low-field scanners where SAR is less of a concern, more
eclectic applications can also be envisioned. For example, the traditional xy-RF transmit chain could be simplified in favor of a lower frequency z-RF. Alternatively, since the multiphoton pulses do
not have any RF at the Larmor frequency, a pulsed version of simultaneous transmit and receive like in^10,11 could be implemented.
The authors thank Ekin Karasan and Miki Lustig for an introduction to HeartVista, Anita Flynn for improving lab spaces, and Karthik Gopalan for help and advice with mechanical engineering. This work
was supported in part by NIH grant R21EB030157.
1. Eles PT, Michal CA. Two-photon excitation in nuclear magnetic and quadrupole resonance. Progress in Nuclear Magnetic Resonance Spectroscopy. 2010;56(3):232-246. doi:10.1016/j.pnmrs.2009.12.002
2. Michal CA. Nuclear magnetic resonance noise spectroscopy using two-photon excitation. The Journal of Chemical Physics. 2003;118(8):3451-3454. doi:10.1063/1.1553758
3. Abragam A. The Principles of Nuclear Magnetism. Clarendon Press; 1961.
4. Eles PT, Michal CA. Two-photon two-color nuclear magnetic resonance. The Journal of Chemical Physics. 2004;121(20):10167-10173. doi:10.1063/1.1808697
5. Zur Y, Levitt MH, Vega S. Multiphoton NMR spectroscopy on a spin system with I =1/2. The Journal of Chemical Physics. 1983;78(9):5293-5310. doi:10.1063/1.445483
6. Han V, Liu C. Multiphoton magnetic resonance in imaging: A classical description and implementation. Magnetic Resonance in Medicine. 2020;84(3):1184-119
7. doi:https://doi.org/10.1002/mrm.281867. Pauly J, Nishimura D, Macovski A. A k-space analysis of small-tip-angle excitation. Journal of Magnetic Resonance (1969). 1989;81(1):43-56. doi:10.1016/
8. Pauly J, Roux PL, Nishimura D, Macovski A. Parameter relations for the Shinnar-Le Roux selective excitation pulse design algorithm (NMR imaging). IEEE Transactions on Medical Imaging. 1991;10
(1):53-65. doi:10.1109/42.75611
9. Martin J, Ong F, Ma J, Tamir J, Lustig M, Grissom W. SigPy.RF: Comprehensive Open-Source RF Pulse Design Tools for Reproducible Research. Proceedings of the ISMRM Annual Meeting and Exhibition.
Published online 2020:1045.
10. Brunner DO, Pavan M, Dietrich B, Rothmund D, Heller A, Pruessmann K. Sideband Excitation for Concurrent RF Transmission and Reception. Proceedings of the ISMRM Annual Meeting and Exhibition.
Published online 2011:625.
11. Brunner DO, Dietrich BE, Pavan M, Prüssmann KP. MRI with Sideband Excitation: Application to Continuous SWIFT. Proceedings of the ISMRM Annual Meeting and Exhibition. Published online 2012:150. | {"url":"https://cds.ismrm.org/protected/22MProceedings/PDFfiles/0451.html","timestamp":"2024-11-12T08:57:17Z","content_type":"application/xhtml+xml","content_length":"21897","record_id":"<urn:uuid:793efc54-7e25-4bd9-80ec-16f29b957ce3>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00644.warc.gz"} |
Index Match Function Help
Hi Smartsheet Gurus,
I am attempt to pull information from one sheet to another given a set criterion that is present in both. I believe that either a VLOOKUP or INDEX function should suffice but cannot seem to get
things to work.
A little background:
We would like to be able to have employees enter a form with data about the orders that they will work on. This form has a Created column that captures date and time which is considered the 'start
time'. Ideally, another form connected to a second sheet would be used to collect an end time. These two values would then be subtracted from each other to get total time between (I already have
functions for this set up).
A form submission on the first sheet looks something like this:
The second sheet that captures the end time looks like this:
Employees utilize the form on the first sheet to document up to six order numbers. I would like for the second sheet's form to just ask for the employee's name and to document any of the potential
six order numbers to be used as a unique identifier between both sheets. The unique identifier would be queried across the potential six columns of order numbers on the first sheet and used to pull
in an end time to the first sheet.
I have been able to get an INDEX(range, (MATCH(... to work on the second sheet. This obviously works as I am index the range of order numbers across the six columns against the unique identifier. I
am wondering if there is a way to pull back the end time into the first sheet as that's where most of the data is captured.
Thank you!
Best Answer
• The way you have the first sheet set up, it looks like you're going to need to put an End Time column after every Order # column. So after "Order #1" column you'll need a column called something
like "Order #1 End Time". Then the same for each order # on that sheet. So six of them. You'll need a formula in the "Order #1 End Time" column that's something like this:
=INDEX({Range 1}, MATCH([Order #1]@row, {Range 2}, 0))
The {Range 1} range should point to the entire "End Time HH:MM" column on sheet 2. The {Range 2} range should point to the entire "Order Number Identifier" column on sheet 2.
Then on Sheet 1 for the new column "Order #2 End Time" formula it would have this:
=INDEX({Range 1}, MATCH([Order #2]@row, {Range 2}, 0))
And so on for each of the 6 new columns for the End Times.
• The way you have the first sheet set up, it looks like you're going to need to put an End Time column after every Order # column. So after "Order #1" column you'll need a column called something
like "Order #1 End Time". Then the same for each order # on that sheet. So six of them. You'll need a formula in the "Order #1 End Time" column that's something like this:
=INDEX({Range 1}, MATCH([Order #1]@row, {Range 2}, 0))
The {Range 1} range should point to the entire "End Time HH:MM" column on sheet 2. The {Range 2} range should point to the entire "Order Number Identifier" column on sheet 2.
Then on Sheet 1 for the new column "Order #2 End Time" formula it would have this:
=INDEX({Range 1}, MATCH([Order #2]@row, {Range 2}, 0))
And so on for each of the 6 new columns for the End Times.
• This did the trick! I then utilize an INDEX COLLECT combo to pull out the nonblank cell to display the end time. Here's what I use in case it might help anyone:
=IFERROR(INDEX(COLLECT([Order #1]@row:[Order #6]@row, [Order #1]@row:[Order #6]@row, @cell <> ""), 1, 1), "In Progress")
Help Article Resources | {"url":"https://community.smartsheet.com/discussion/98876/index-match-function-help","timestamp":"2024-11-08T17:44:51Z","content_type":"text/html","content_length":"405789","record_id":"<urn:uuid:eeabedaf-2039-4591-90dd-b6c181c0c7d6>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00742.warc.gz"} |
Functions in C - Part 3 of 5
Recursion, as the name suggests, requires a function to be capable of calling itself. C supports implementation of recursive functions by allowing a function to call itself repeatedly. Let us look at
a couple of examples.
Example 1
A recursive function used to compute factorial of a number.
/* Compute factorial of n */
long int fact(int n)
if(n <=1)
return(n * fact (n -1));
Example 2
A recursive function used to generate numbers which are in Fibonacci sequence e.g. 1, 1, 2, 3, 5, 8, 13,...
The sequence of Fibonacci numbers f[n], satisfy the recurrence relationf[n] = f[n]-1 + f[n]-2, for n > 2 and f[1] = f[2] = 1.
#include <stdio.h>
int n;
printf("Give n: ");
scanf("%d", &n);
printf("\nfib(%d) = %d\n", n, fib(n));
/* Function to find first m numbers which are in Fibonacci sequence */
int fib(int m)
if(m == 0) return(1);
else if(m == 1) return(1);
else return(fib(m - 1) + fib(m - 2));
The only difference between a recursive function call and an ordinary function call is that a recursive call creates a second activation of the subprogram during the lifetime of its first activation.
If the second activation leads to another recursive call, then three activations may exist simultaneously, and so on. The only new element introduced by recursion is the multiple activations of the
same function that can exist simultaneously at some point during its execution. Thus, due to existence of several activation records, recursive functions can be quite memory intensive.
Although the recursive call feature enhances the power of the language and is appealing from the programmer's point of view, many problems, however, can be solved with the help of repetitive
statements, without taking recourse to recursive function call. For example, the following code illustrates the implementation of factorial computation without recursive call.
Implementation of a non recursive method to find factorial of a number:
#include <stdio.h>
int n, i, factorial;
printf("Enter a number: ");
scanf("%d", &n); fflush(stdin);
printf("The number input is: %5d\n", n);
if(n == 0)
printf("\nThe factorial of 0 is 1\n");
factorial = 1;
for(i=1; i <= n; i++)
factorial = factorial * i;
printf("The factorial of %5d is %5d\n", n, factorial);
It may be mentioned that from computational efficiency point of view it would be desirable to replace recursive function call by such repetitive statements, whenever possible. This is due to the
reason that implementation of the recursive function call will lead to creation of activation records and jumping to the function code and returning from it at the end of computation, as discussed
above. Such activities will involve additional computational overhead, which will not occur when the problem is solved using a repetitive block of statements. However there are many recursive cases
which cannot be implemented by mere repetitive statements.
To effectively substitute the recurrence relation, complex data structures like a user stack are used. To illustrate this, consider the following problem called Tower of Hanoi which requires a set of
disks of increasing diameter to be moved from first peg to second taking the help of a third peg, so that at every stage a disk of smaller diameter can be placed over another having larger diameter.
At no stage of the disk movement this ordering should be violated.
A program to solve the Tower of Hanoi problem:
#include <stdio.h>
int n;
printf("Give n: ");
scanf("%d", &n);
towers(n, 'A', 'B', C);
void towers(int m, char from, char to, char via)
if(m == 1)
printf("Move disk from peg %c to peg %c \n", from, to);
towers(m-1, from, via, to);
printf("Move disk %d from peg %c to peg %c\n", m, from, to);
towers(m-1, via, to, from);
Run the above code and observe the results for different values of n. Also, try to simulate the above program with pen and paper, by drawing the static code segment and activation records for a
smaller case where n=3 and figure out how the instructions will execute in the computer's memory, how multiple activation records will be created for the function towers, the sequence of execution
and the sequence in which activation records will be destroyed.
Is it possible to implement this problem without using recursive function call? If so, try it.
The following recursive function called Ackerman function (for arbitrary values of m and n), is difficult to implement without the help of recursive function call.
Ackerman's function A(m, n) is defined as:
A(m, n) = n + 1 - if m = 0
A(m-1, 1) - if n = 0
A(m1, A(m, n-1)) - otherwise
/* An implementation of Ackerman's function in C */
int a(int m, int n)
if(m == 0)
return n + 1;
if(n == 0)
return a(m-1, 1);
return a(m-1, a(m, n-1));
Can you express each of the following algebric formulae in a recursive form? If, yes, write C program to implement the same.
(a) y = x1 + x2 + ... + xn
(b) y = 1 + 2x + 4x2 + 8x3 + ... + 2nxn
(c) Y = (1 + x)n
Lab work
1. Write a C program to compute x^n where x is a floating point variable and n is an integer.
2. Write a C program to find the greatest common divisor of two integer numbers.
3. Write a C program to find the Length of a string of characters.
4. Write a C program to compress a string of character by eliminating blank spaces.
5. Write a C program to compute e^x, where x is an integer number using the formula:
e^x = 1 + x + x^2/2! + x^3/3! + ...
Compute to a given degree of accuracy ?, where ? = 0.0001
6. The combination comb(n, m) where n and m are integer variables and n ? m, can be computed using the following recurrence relation:
For n, m ? 1,
Comb(n, m) = comb(n-1, m) + comb(n-1, m-1)
Comb(n, m) = 1, if(n == 1) or (m == 0) or (n == m)
Write a recursive C function to compute comb using recursive relation. Can you compute comb without using recursive function call? | {"url":"https://www.how2lab.com/programming/c/functions-3","timestamp":"2024-11-10T00:12:35Z","content_type":"text/html","content_length":"29094","record_id":"<urn:uuid:e9259a40-12cd-45b2-9e24-a816ad3144f4>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00563.warc.gz"} |
I Traded The Volatility Surface For an Entire Week — Strongly Recommend.
I like my vol surface how I like my women — curvy, kinky, and hard-to-interpret.
Some time ago, we set out to become volatility kings.
By building out our own internal volatility surface, we were able to pretty accurately predict the realized volatility of nearly every stock with listed options. While this is a fantastic ability
largely attributable to the great minds before us, it raises a difficult dilemma:
You get a call telling you there’s a high chance that Boeing’s stock will move by 10% tomorrow. However, the call ends before you have a chance to hear the direction. How do you profit from this
This led us to search tirelessly for back-testable strategies based on surface changes; things like buying a straddle whenever the surface starts to expect significantly more short-term volatility.
However, because our investable universe is so wide (every optionable stock), a purely quantitative approach completely ignores the idiosyncratic (stock-specific) factors that determine what happens
next. This idiosyncratic factor is also important for deciding whether the volatility is priced too richly or cheaply — a key driver in the profitability of any options position.
So, instead of searching for a fixed, rule-based approach, what if we just went out there and traded? We’d use the volatility surface to find the dislocations and kinks, then we’d try our best to
find out why the market is anticipating more/less vol for the specific stock, then finally, we’d put on a trade that supports our view.
And that’s exactly what we did.
Vol Is Kinda Thicc Sometimes
Before diving deeper, let’s first get a general idea of what we’re looking for in the surface and how we’ll trade those observations.
First off, here’s what our surface will look like on a typical trading day:
Our main focus is on the first 4 columns:
• The underlying percent change: How the stock has performed since the prior day’s close.
• Vol Change: The change in implied volatility from the prior day. If yesterday the market implied that the stock would move 5% by expiration, but today it implies that the stock will move by 7%,
this is a vol change of 2%.
• Slope: The IV of next week minus the IV of this week. A slope of -10% means that this week’s IV is 10% higher than the IV for the expiration of next week.
• Slope diff: How much the slope has changed. If yesterday, the slope was -10% but today it’s -50%, this is a slope diff of -40.
As we’ll see later, idiosyncratic situations will lead to different kinds of trades — but in general, we will try to collect income when volatility expectations become lower, then we’ll try to long
volatility when expectations become affordably higher.
Short vol scenario: Yesterday, stock ABC reported earnings and fell 10% after-hours. Upon open, implied volatility decreases and the slope normalizes to a positive value. After our review of the
earnings and investor perspectives, we determine that the earnings are adequately priced-in and there isn’t likely to be any abnormal future volatility. So, we sell an iron condor right outside of
the implied move (if market expects 5% vol, we sell 6% OTM iron condor).
Long vol scenario: Yesterday, Stock A, a stock in the same industry and strongly correlated to Stock B, jumped 15% after reporting earnings. Stock B plans to report earnings next week. At open, the
market raises its volatility expectation for Stock B from 5% to 7% and the slope becomes negative by 10%. We take the view that Stock B’s earnings will be just as volatile and as the earnings date
approaches, volatility expectations will continue to increase. So, we buy a straddle and plan to sell it before the actual earnings event.
Our volatility surface will bring these potential trades straight to us, then it’s up to us to evaluate anything out of the ordinary.
So, now that you have a general idea of how we’ll be using the surface, let’s go over some trades:
Day One - Friday
At around the close of Friday, 02/02, here’s what our curve looked like:
This post is for paid subscribers | {"url":"https://www.quant-galore.com/p/i-traded-the-volatility-surface-for","timestamp":"2024-11-03T00:08:59Z","content_type":"text/html","content_length":"133809","record_id":"<urn:uuid:946e8aca-c698-4644-8477-8f1bad18cf78>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00400.warc.gz"} |
Turing Complete - Basic Logic Manual - SteamAH
Turing Complete – Basic Logic Manual
For Turing Complete players, this guide is a basic logic manual to help you to have a better understanding of the game.
Truth Table
A truth table is a table that assigns each combination of truth values of all variables in an expression with logical operator(s) to a truth value. The following are the truth tables of three common
logical operators: NOT, AND and OR.
Negation (NOT): the opposite of the truth value of a variable (denoted by
Conjunction (AND): true if both two variables are true (denoted by AB)
Disjunction (OR): true if one of two variables is true (denoted by A+B)
Notes: If no parenthesis is used, the priority of operation always follows the order: negation, conjunction, disjunction and from left to right if in the same order. For example, A[B]C+[A] equals ((A
Boolean Algebra
Boolean algebra is a binary operation structure that consists of logical variables, constants 0 and 1, and logical operations AND, OR and NOT satisfying the following properties:
A1=A, A0=0
A+0=A, A+1=1
De Morgan’s laws
In the game, we can optimize our design by using the properties above to minimize the number of logic gates and delay. For example, we can use the distributive property to reduce the number of logic
gates we use from three to two.
De Morgan’s laws allow us to convert AND, OR, NAND, or NOR gates into another. Circuits in the same color are logically equivalent (have the same truth value). Output is inverted when we convert
gates between left and right. Input is inverted when we convert gates between top and bottom.
Analysis of Logic Circuits
Control of conditions
For every logic circuit, we can use AND or OR gates to accurately control the circuit’s conditions, which means we can freely assign which combinations of truth values to 1. Due to the associative
and commutative property, the order of operations and inputs has no effect on the result of a circuit when any number of AND gates (or OR gates) are used individually. So,
Use AND gates when all conditions are required.
Use OR gates when one of conditions is required.
Karnaugh map and Boolean sum of products
A Karnaugh map is the truth table used for the simple analysis of a logic circuit with two up to four inputs. The head of rows or columns of a Karnaugh map represents every possible combination of
truth values of input (usually pairs of input appears in the head of either rows or columns) respectively. Each intersection of a row and a column represents the truth value to which the
corresponding combination is assigned. For example, in a four-variable Karnaugh map,
The truth value in the third row and the fourth column shows the output is 1 when the input A,B,C,D are 1,0,1,1 respectively. So we can write down the conjunction of each condition of input:
Similarly we have the rest of all conditions that make the output be 1:
• [A]B[C]D
• [A]BC[D]
• [A]BCD
• AB[C]D
Finally we get the Boolean sum of products of this truth table by writing down the disjunction of all the conjunctions listed above:
• [A]B[C]D+[A]BC[D]+[A]BCD+A[B]CD+AB[C]D
Simplification of a Boolean expression
Let X,Y be Boolean expressions. Based on the operation laws, we have
Therefore we can simplify the expression of any form XY+X[Y] to X. For example, in the sum of products mentioned above, we have
• [A]BC[D]+[A]BCD=[A]BC (X=[A]BC, Y=D)
• [A]B[C]D+AB[C]D=B[C]D (X=B[C]D, Y=A)
Replace the original terms with these, we have
Based on the distributive property, we can simplify the terms containing either B or D as needed. If we want to simplify the terms containing B, we have
This logic circuit requires three NOT gates, six AND gates and two OR gates. Readers can check the result.
Although Boolean sums of products cannot guarantee the optimal design, it helps us figure out designs that satisfy the condition of complex logic circuits.
That’s all we are sharing today in Turing Complete – Basic Logic Manual, if you have anything to add, please feel free to leave a comment below, you can also read the original article here, all the
credits goes to the original author qwerty | {"url":"https://steamah.com/turing-complete-basic-logic-manual/","timestamp":"2024-11-02T14:51:17Z","content_type":"text/html","content_length":"57132","record_id":"<urn:uuid:2b1c0ca9-0162-48b4-b965-af41687b6ebb>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00702.warc.gz"} |
Statistics - Probability of Combined Events - David The Maths TutorStatistics – Probability of Combined Events
Statistics – Probability of Combined Events
I ended my last post showing the probability of picking a type of card from a standard deck of 52 cards. For example, if the event of interest, A, is picking a Jack, then the probability of picking a
Jack from a shuffled deck of cards is
because there are 4 ways to pick a Jack out of 52 cards. Now let’s consider probabilities of events like “picking a Jack or a Heart” or “a face card and a Heart”.
If we let events A be picking a Jack, B be picking a Heart, and C be picking a face card (Jack, Queen, or King), then the maths notation for these statements are
\[P\left(A\cup B\right)=\mathrm{probability\ of\ picking\ a\ Jack\ or\ a\ Heart}\] \[P\left(B\cap C\right)=\mathrm{probability\ of\ picking\ a\ face\ card\ and\ a\ Heart}\]
The symbol “∪” stands for the union of two events, but in English, you can use the word “or”: A ∪ B = “A union B” or “A or B“. The symbol “∩” stands for the intersection of two events, but in
English, you can use the word “and”: B ∩ C = “B intersection C” or “B and C“. These concepts are easily seen in a Venn diagram:
Circle A is the set of all Jacks and circle B is the set of all Hearts. Now the probability of picking a card from set A is 4/52. The probability of picking a card from set B is 13/52. You may be
tempted so say that the probability of A or B is the sum of the two individual probabilities. But both of these probabilities include the Jack of Hearts so it is used twice. We have to subtract out
this intersection of the two probabilities, so in maths notation:
\[P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)\]
This equation can be rearranged to show that the probability of the intersection of the two events is equal to the sum of the individual probabilities minus the probability of the union:
\[P\left(A\cap B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cup B\right)\]
These two equations are different forms of what is called the addition rule of probability.
So P(A ∪ B) = 4/52 +13/52 – 1/52 = 16/52, because P(A ∩ B) is the probability of a Jack and a Heart. Only one card satisfies this, the Jack of Hearts, so the probability of that is 1/52.
Now let’s define event D as picking a Diamond and consider the probability of picking a Heart and a Diamond, P(B ∩ D). This is clearly 0 as a card cannot be both suits. The associated Venn diagram
looks like:
Events like this are called mutually exclusive, that is, you can pick one or the other, the picked card cannot be both. For mutually exclusive events:
\[P\left(B\cup D\right)=P\left(B\right)+P\left(D\right)\ \mathrm{and}\ P\left(B\cap D\right) =0\]
In my next post, I will discuss what is called conditional probabilities and explore the probability of picking a Jack given that the card is a Heart. | {"url":"https://davidthemathstutor.com.au/2022/09/05/statistics-probability-of-combined-events/","timestamp":"2024-11-03T12:59:39Z","content_type":"text/html","content_length":"47552","record_id":"<urn:uuid:0263009a-cfab-4223-bc67-e3e9365d037e>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00635.warc.gz"} |
Matrix inversion - (Numerical Analysis II) - Vocab, Definition, Explanations | Fiveable
Matrix inversion
from class:
Numerical Analysis II
Matrix inversion is the process of finding a matrix that, when multiplied by a given square matrix, results in the identity matrix. This is crucial in solving linear equations, as the inverse of a
matrix can be used to isolate variables and find solutions efficiently. Understanding matrix inversion also links to methods for solving systems of linear equations and assessing the stability of
numerical algorithms.
congrats on reading the definition of matrix inversion. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Not all matrices are invertible; a matrix must have a non-zero determinant to have an inverse.
2. The inverse of a matrix A is denoted as A^(-1) and satisfies the equation A * A^(-1) = I, where I is the identity matrix.
3. Computing the inverse can be done using various methods such as Gauss-Jordan elimination or through LU decomposition.
4. In numerical analysis, it's important to consider the conditioning of a matrix, as poorly conditioned matrices can lead to significant errors when inverting.
5. Matrix inversion is essential for solving systems of linear equations represented in matrix form, particularly in scenarios where direct solutions may be impractical.
Review Questions
• How does the concept of matrix inversion relate to solving systems of linear equations?
□ Matrix inversion plays a vital role in solving systems of linear equations by allowing us to express the solution in terms of matrix operations. If we have a system represented as Ax = b,
where A is the coefficient matrix, x is the vector of variables, and b is the output vector, we can find x by calculating x = A^(-1)b. Thus, finding the inverse of A enables us to isolate x
and solve for it efficiently.
• What conditions must be met for a matrix to have an inverse, and why are these conditions significant in numerical analysis?
□ For a matrix to have an inverse, it must be square and have a non-zero determinant. These conditions are significant because they determine whether solutions to linear equations can be
uniquely found using matrix methods. In numerical analysis, if a matrix is nearly singular (having a determinant close to zero), it may lead to large errors in computed solutions due to
instabilities in numerical algorithms.
• Evaluate the impact of using LU decomposition on the efficiency of calculating matrix inverses in computational applications.
□ Using LU decomposition enhances the efficiency of calculating matrix inverses in computational applications by breaking down the original matrix into simpler components: a lower triangular
matrix L and an upper triangular matrix U. This method not only simplifies the computation but also reduces the overall number of operations required compared to direct inversion methods.
Consequently, for large matrices or when multiple solutions need to be computed with varying right-hand sides, LU decomposition significantly speeds up calculations while maintaining
numerical stability.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website. | {"url":"https://library.fiveable.me/key-terms/numerical-analysis-ii/matrix-inversion","timestamp":"2024-11-12T16:44:06Z","content_type":"text/html","content_length":"150334","record_id":"<urn:uuid:1976a0e3-885e-41d9-bc1c-59930b1eba52>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00284.warc.gz"} |
Permutations with Repeats
<p><a target="_blank" href="https://www.prepswift.com/quizzes/quiz/prepswift-permutations-with-repeats">Permutations with Repeats Exercise</a></p><p>If, for example, we want to find the number of
ways of arranging the letters of a word where there are multiple characters of the same letter, we need to divide by the factorial of the number of repeats (of that character). </p> <p>For
example, consider the word MISSISSIPPI. That word has</p> <ul> <li>four Ss</li> <li>four Is</li> <li>two Ps</li> </ul> <p>Hence, the number of ways of arranging that word would be </p> <p>$$\
frac{11!}{4! \times 4! \times 2!}$$</p>Sorry, you need to log in to see this. Click here to log in. | {"url":"https://www.prepswift.com/content/permutations-with-repeats","timestamp":"2024-11-11T16:42:15Z","content_type":"text/html","content_length":"318189","record_id":"<urn:uuid:c2c8f808-37c2-4243-89aa-fcacdcb853d2>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00798.warc.gz"} |
How to Generate A Random Number In Haskell?
Generating a random number in Haskell involves using the random package, which provides functions for generating random values.
To generate a random number, you need to import the System.Random module. You can do this by adding the following line at the top of your Haskell file:
Once you have imported the System.Random module, you can use the randomR function to generate a random number within a specified range. The randomR function takes a range and a random number
generator as arguments and returns a random number within that range.
Here's an example that generates a random number between 1 and 10:
1 main = do
2 gen <- getStdGen
3 let (randomNumber, _) = randomR (1, 10) gen :: (Int, StdGen)
4 putStrLn $ "Random number: " ++ show randomNumber
In the above code, the getStdGen function is used to create a random number generator. The randomR (1, 10) gen expression generates a random number between 1 and 10 using the gen generator. The
result is a tuple containing the random number and a new generator. We extract only the random number using pattern matching and then print it using the putStrLn function.
Remember to compile and run your Haskell file using GHC or any other Haskell compiler. Each time you run the program, it will generate a new random number.
Are there any common pitfalls to be aware of when generating random numbers in Haskell?
Yes, there are a few common pitfalls to be aware of when generating random numbers in Haskell. Some of these pitfalls include:
1. Using the default random number generator: By default, Haskell uses the System.Random module, which provides a simple generator called StdGen. However, the StdGen generator has limited randomness
properties and its seed space is small (only 231 bits), which can lead to predictable sequences of random numbers. To overcome this, it is recommended to use a more advanced random number
generator like System.Random.MWC or System.Random.Mersenne, which provide better quality randomness.
2. Forgetting to supply a new seed: Haskell's random function relies on an initial seed to generate random numbers. If you forget to supply a new seed or reuse the same seed, you will get the same
sequence of random numbers. To avoid this, make sure to use a fresh seed, such as a new random number generator or a randomly generated seed value.
3. Not taking thread safety into account: If you intend to use random number generation in a concurrent or parallel setting, it is crucial to use a random number generator that is thread-safe. The
System.Random module's StdGen is not thread-safe, and using it in a concurrent or parallel program can lead to unexpected behavior or incorrect results. Consider using generators like
System.Random.MWC or System.Random.Mersenne, which are designed to be thread-safe.
4. Assuming uniform distributions: Haskell's random number generators provide functions for generating random numbers from various distributions, such as uniform, normal, or exponential
distributions. However, it is important to remember that these generators produce pseudo-random numbers, which means that the resulting distribution may not be perfectly uniform or follow the
desired behavior. It is a good practice to test and verify the distribution properties of the generated random numbers.
5. Not properly managing state: When working with random number generators, it is important to manage and propagate the generator's state correctly. If you accidentally use the same generator or
modify its state in multiple places, you may introduce unexpected correlations or patterns in the generated random numbers. Make sure to pass around or use separate instances of the generator as
By being aware of these pitfalls, you can ensure better quality and more predictable random number generation in Haskell.
Can we generate random integers in a specific range in Haskell?
Yes, we can generate random integers in a specific range in Haskell using the randomRIO function from the System.Random module.
Here's an example of how you can generate random integers in a specific range:
1 import System.Random
3 main :: IO ()
4 main = do
5 randNum <- randomRIO (1, 10) :: IO Int
6 putStrLn $ "Random number between 1 and 10: " ++ show randNum
In this example, randomRIO (1, 10) generates a random integer between 1 and 10 (inclusive). The :: IO Int syntax is used to specify the type of the generated random number. The random number is then
printed using putStrLn.
How does Haskell handle random number generation?
Haskell has a built-in module called System.Random that is used for random number generation. This module provides a pure interface for generating random values, making it deterministic and
The key concept in Haskell's random number generation is the StdGen type, which represents a random number generator state. It can be generated using the newStdGen function, which typically uses the
system's random number generator to create a new StdGen value.
One common way to generate random numbers is to use the random function from System.Random module. This function takes an StdGen as input and returns a random value along with a new state of the
generator. The type of random value depends on how it is requested. For example, random :: Random a => StdGen -> (a, StdGen) would generate a random value of type a.
Additionally, Haskell provides functions like randomR and randomRs, which generate random numbers within a specified range. For instance, randomR :: Random a => (a, a) -> StdGen -> (a, StdGen)
generates a random value within the given range.
It is worth mentioning that the StdGen state can be manually threaded through the program to maintain deterministic behavior, or it can be automatically threaded using the getStdGen and setStdGen
Overall, Haskell's approach to random number generation ensures purity and referential transparency by separating the generation of random numbers from their usage.
Can we generate random numbers with a specific distribution in Haskell?
Yes, we can generate random numbers with a specific distribution in Haskell using the random package. The package provides various functions for generating random numbers and distributions.
To ensure a specific distribution, we can make use of the randomR or randomRs functions, which generate random numbers within a given range. We can define a specific distribution by providing a
custom range and mapping function.
Here is an example of generating random numbers with a normal distribution:
1 import System.Random
3 -- Normal distribution parameters
4 mean :: Float
5 mean = 0.0
7 stdDev :: Float
8 stdDev = 1.0
10 -- Generates random numbers with a normal distribution
11 randomNormal :: IO Float
12 randomNormal = do
13 gen <- getStdGen
14 let (value, newGen) = randomR (-6.0, 6.0) gen -- Choose range based on desired distribution
15 setStdGen newGen
16 return $ stdDev * value + mean
18 main :: IO ()
19 main = do
20 values <- replicateM 10 randomNormal
21 putStrLn $ "Random Numbers with Normal Distribution: " ++ show values
This code uses the randomR function to generate random numbers within the range (-6.0, 6.0). The resulting values are then mapped to the desired normal distribution by multiplying with the standard
deviation and adding the mean.
Note that we use the getStdGen function to retrieve the current random number generator and setStdGen to update it with the new generator after generating each value.
By running the main function, we can generate 10 random numbers with a normal distribution. | {"url":"https://ubuntuask.com/blog/how-to-generate-a-random-number-in-haskell","timestamp":"2024-11-09T07:44:04Z","content_type":"text/html","content_length":"340669","record_id":"<urn:uuid:71c49f8b-7113-40c8-a83f-69a9d4776c9f>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00264.warc.gz"} |
This page is not finished yet
Most of the things in Warzone are easy to calculate, but some of them can be pretty confusing. Here are some of the formulas we use to calculate things.
Nearly all calculations in formulas are done with integers. This means that numbers are nearly always rounded down to the nearest integer. This includes damage bonuses, weight multipliers, damage
multipliers, experience bonuses, accuracy bonuses (down to the nearest percent).
However, note that time is usually rounded down to the nearest 0.1 second, and distance is usually rounded down to the nearest 1/128 tile.
Ever wonder how damage is calculated in Warzone? If you just want to know if armor is subtractive or multiplicative, it's subtractive:
DAMAGE = BASE DAMAGE − ARMOR
or DAMAGE = ^1/[3] × BASE DAMAGE
whichever is higher.
There are 2 kinds of armor: kinetic and thermal.
Kinetic armor protects tanks from "physical" damage (ie from machineguns, cannons, rockets, etc.). Thermal armor is used to protect the tank body from "heat" damage type weapons such as flamers,
lasers, thermite bombs, etc.
But to know exactly how much damage you're going to do, you need to know how to calculate base damage.
The formula
BASE DAMAGE = Weapon damage × Weapon damage upgrade modifier × Propulsion/Structure damage modifier
ARMOR = Target armor × Target armor upgrade modifier
Damage is BASE DAMAGE − ARMOR, or ^1/[3] of BASE DAMAGE, whichever is higher, rounded down (The exception is when rounding down would make damage 0, in which case damage is rounded up to 1).
• Weapon damage is the base damage of the weapon—what you see in the "Damage" column of the turret table.
• Weapon damage upgrade modifier is the best damage upgrade you've researched for the weapon's subclass.
• Propulsion/Structure damage modifier is the multiplier in the above damage tables.
• Target armor is the target's thermal armor for thermal weapons, and kinetic armor for kinetic weapons. In tanks and VTOLs, thermal armor depends on nothing but the vehicle body.
• Target armor upgrade modifier is the best kinetic/thermal armor upgrade your target has researched for it.
An example
Green's Tank Killer Scorpion Half-Tracks shoots Yellow's Heavy Cannon Python Tracks. Green has HESH Rocket Warhead, and Yellow has Dense Composite Alloys.
Tank Killer (an anti-tank weapon) has 180 damage. HESH Rocket Warhead gives 275%, and anti-tank weapons do 120% damage against tracks.
BASE DAMAGE = 180 × 275% × 120% = 524.
Tank Killer is a kinetic weapon, and Python has 20 Kinetic Armor. Dense Composite Alloys upgrades 220%.
ARMOR = 20 × 220% = 44
524 − 44 (480) is greater than 524 × ^1/[3] (174), so the tank killer does 480 damage each time it hits (Notice that it fires two salvos; if they both hit, the tank killer does 960 damage, a
significant proportion of the 2260 HP after upgrades that the heavy cannon tank has).
Hit points(HP) of Unit are calculated as summ of hit points of components of Unit
Each Unit have 3 components:
Note: Multiturret bodies (Dragon) can have 2 turrets so in that case unit has 4 components (body, propulsion, turret1, turret2).
Unit HP = (Body HP + Body HP x Propulsion HP modifier + Turret HP) x HP upgrade modifier
• Body HP base hit points of body
• Propulsion HP modifier hit points modifier for propulsion
• Turret HP hit points of turret (turrets). Some turrets have 0 HP whilc other turrets can have 500 HP
• HP upgrade modifier modifier from armor upgrades. Each researchof composite alloys increases this modifier.
Speed = Base Speed × Speed Penalty × Unit Experience Bonus
Speed = Propulsion Max Speed × Unit Experience Bonus
(Whichever is lower.)
Base Speed = Engine Power After Upgrades × Propulsion Speed × Propulsion Terrain Multiplier / Total Weight
Speed Penalty is 3/4 if using a medium body on VTOL, 1/4 if using a heavy body on VTOL, and 1 in all other cases.
This speed is in units of world-coordinates per second. There are 128 world-coordinates in a tile, so, for instance, a unit with a speed of 128 would go at a speed of one tile per second.
That "Terrain Multiplier" is pretty important. You'd expect it to be 1× for all propulsions flat terrain, but it's actually 2.5× for VTOLs.
Production Time (Build Points)
Production time (also known as Build points) is time required to produce unit in factory.
Note: Factory modules decrease production time of units
Production time (build points) of Unit are calculated as summ of build points of components of Unit
Each Unit have 3 components:
Note: Multiturret bodies (Dragon) can have 2 turrets so in that case unit has 4 components (body, propulsion, turret1, turret2).
Unit Build Points = (Body Build Points + Body Build Points x Propulsion Build Points modifier + Turret Build Points) x Production upgrade modifier
• Body Build Points base Build Points of body
• Propulsion Build Points modifier Build Points modifier for propulsion
• Turret Build Points Build Points of turret (turrets). Some turrets are produced very slow.
• Production upgrade modifier modifier from production upgrades (upgrades like Automatic Manufacturing, Robotic manufacturing)
Construction Time
Construction Time is time required to build a building by trucks/cyborg engineers
Note: more trucks = faster build
Construction time (seconds) = Structure build points / (Truck construct points x Engineering upgrade modifier x Count of trucks)
• Structure build points Build Points of structure. More build points means longer construction time.
• Truck construct points Construct points of truck/cyborg engineer. More construct points means faster build
• Engineering upgrade modifier modifier from engineering upgrades (upgrades like Improved Engineering, Advanced Engineering)
• Count of trucks how much trucks/cyborg engineers are used to build up structure | {"url":"https://warzone2100.pro/wz2100-database-project/Formulas.html","timestamp":"2024-11-07T19:07:49Z","content_type":"application/xhtml+xml","content_length":"16228","record_id":"<urn:uuid:4672899a-12f7-416f-b54d-4e453f9b87b0>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00154.warc.gz"} |
Pythagoras in the Forge - Wikibooks, open books for an open world
This article sheds light on the physical and music-theoretical background of the legend of Pythagoras in the Forge and proves that this legend could have a realistic basis. It is based on a previous
publication from 2012,^[1] and the appropriate German speaking Wikibook, which was widely translated with www.DeepL.com/Translator (free version).
The copper engraving Duynkirchen by Eberhard Kieser from the Thesaurus philopoliticus by Daniel Meisner (* 1585; † 1625) published in 1626.
On the left Pythagoras with an angle measure and on the right three blacksmiths with hammers at an anvil with five staves, on which the lettering "Guido" is shown.
The Latin and German texts of the images have the same content and read as follows:
Triplicibus percussa sonat varie ictibus incus.
Musica Pythagoras struit hinc fundamina princ(eps).
Der Amboß von drey Hämmern klingt, darauß dreyerley thon entspringt.
Pythagoras hie die Music findt, das hett kein Eselskopff gekönt.
In English:
The anvil rings with three hammers, and from it springs three sounds.
Pythagoras finds the music here, that no donkey's head can do.
The connections between sounds and numbers were not only studied in antiquity. In the Middle Ages, music, together with arithmetic and geometry, belonged to the four liberal arts of the quadrivium.
These subjects still offer a rewarding field for music-theoretical considerations and investigations, and this concerns various vocal temperaments still in use today as well as, for example,
music-aesthetic aspects or tonal theory. The author hopes that these remarks on the ancient legend can contribute to awakening or consolidating interest in the subject.
The invention of music
Pythagoras of Samos (* around 570; † after 510 B.C.) is said to have invented music, according to legend, through his visit to a forge. This does not mean that there had been no music before, but
that he is said to have been the first to give music a theoretical basis by assigning the ratios of the natural numbers six, eight, nine and twelve to the pure musical intervals prime, fourth, fifth
and octave.
The four integers 6, 8, 9 and 12 and their relations to the pure musical intervals prime, fourth, fifth and octave.
The following table shows the frequency ratios of such four tones with the exemplary frequencies 1200, 1600, 1800 and 2400 hertz:
Intervall Prime Quarte Quinte Oktave
${\displaystyle f}$ ${\displaystyle f_{6}}$ ${\displaystyle f_{8}}$ ${\displaystyle f_{9}}$ ${\displaystyle f_{12}}$
${\displaystyle f}$ 1200 Hz 1600 Hz 1800 Hz 2400 Hz
${\displaystyle {\frac {f}{f_{6}}}}$ ${\displaystyle {\frac {1}{1}}}$ ${\displaystyle {\frac {4}{3}}}$ ${\displaystyle {\frac {3}{2}}}$ ${\displaystyle {\frac {2}{1}}}$
${\displaystyle {\frac {f}{f_{8}}}}$ ${\displaystyle {\frac {3}{4}}}$ ${\displaystyle {\frac {1}{1}}}$ ${\displaystyle {\frac {9}{8}}}$ ${\displaystyle {\frac {3}{2}}}$
${\displaystyle {\frac {f}{f_{9}}}}$ ${\displaystyle {\frac {2}{3}}}$ ${\displaystyle {\frac {8}{9}}}$ ${\displaystyle {\frac {1}{1}}}$ ${\displaystyle {\frac {4}{3}}}$
${\displaystyle {\frac {f}{f_{12}}}}$ ${\displaystyle {\frac {1}{2}}}$ ${\displaystyle {\frac {3}{4}}}$ ${\displaystyle {\frac {2}{3}}}$ ${\displaystyle {\frac {1}{1}}}$
Example with the four Pythagorean tones c', f', g' and c" in notation with treble clef.
• Sound examples with the four Pythagorean tones c', f', g' and c" in pure Pythagorean tuning and all 15 combinations
• Variant A.
• Variant B.
In pairs, four Pythagorean tones can produce a total of four different lower-frequency combination tones, which result from the difference in the frequencies of the two respective tones under
consideration. With respect to each of the four Pythagorean tones, the combination tones each have an integral multiple of one-half, one-third, one-fourth or one-sixth of their frequency. With the
exemplary frequencies given in the table above, the four combination tones with the frequencies 200, 400, 600 and 800 hertz thus result. Because of the quite rational ratios, all combination tones
also sound in harmonic unison with the four Pythagorean tones.
The following table shows the four Pythagorean tones c', f', g' and c" with the vibration numbers of their tone frequencies, which correspond to the concert pitch A with 440 hertz:
Tone name Tone frequency ${\displaystyle f}$ in hertz Frequency ratio
to first Tone c
c' 261,6 ${\displaystyle {\frac {1}{1}}}$
f' 349,2 ${\displaystyle {\frac {4}{3}}}$
g' 392,0 ${\displaystyle {\frac {3}{2}}}$
c" 523,3 ${\displaystyle {\frac {2}{1}}}$
The Typus arithmeticae from the Margarita Philosophica of 1503 by the philosopher Gregor Reisch (* 1467; † 1525) with Boethius (left) and Pythagoras of Samos (right)
Guido of Arezzo (* c. 992; † 1050) instructs Bishop Theobald of Strasbourg († 1082) on the monochord. Vienna, Austrian National Library, 12th century.
Tradition in antiquity
Unfortunately, no writings by Pythagoras exist (he may not have left any at all), and the oldest sources date from many centuries after his death. Nicomachus of Gerasa recorded Pythagoras'
discoveries at least 600 years after his death.
But even these records have not survived, so that we have to resort to the late antique Latin writing De institutione musica ('Introduction to Music) by Boethius, which was written only about 1000
years after Pythagoras and presumably also refers to Nicomachus, among others. In any case, in the tenth chapter of De institutione musica it is described "how Pythagoras investigated the
relationships of the harmonic sounds."^[2]
According to the legend of Pythagoras in the forge, he passed by a workshop "by a divine hint" and noticed the harmony of the individual tones caused by five different hammer blows. Because he
suspected that the individual tones were caused by the type and force of the hammer blows, he induced the craftsmen to change the tools. He noticed that the individual tones were not connected with
the craftsmen, but with the tools and that the tools, which resonated together, were in certain whole-number weight relationships to each other.
According to the eleventh chapter of De institutione musica, he would have subsequently investigated these relationships when varying the tension weights of strings and finally also with the
monochord, and also researched different lengths and thicknesses of the strings.^[3]
Tradition in the Middle Ages
Another 500 years later, i.e. 1500 years after Pythagoras' work, the medieval music theorist and Benedictine Guido of Arezzo (* around 992; † 1050) in his Micrologus, also in Latin, again refers to
Boethius. In the twentieth chapter, Guido mentions "how music was invented from the sound of hammers".^[4]
This tradition of the legend of Pythagoras mentions that he passed a forge where forging was said to have been done with five hammers on an anvil. In the older tradition of Boethius, however, there
is no mention of the smiths leading the hammers or of an anvil.
A physical analysis of the facts that have been handed down reveals a number of contradictions.
Rod with length ${\displaystyle l}$ and cross-sectional area ${\displaystyle A}$ .
For this purpose, we consider an idealised hammer head in the form of a cuboid rod with the greatest length ${\displaystyle l}$ . Its volume ${\displaystyle V}$ together with its cross-sectional area
${\displaystyle A}$ results in:
${\displaystyle V=l\cdot A}$
The mass ${\displaystyle m}$ is at a density ${\displaystyle \rho }$ :
${\displaystyle m=V\cdot \rho =l\cdot A\cdot \rho }$
The weight force ${\displaystyle F}$ of the hammer head can be directly calculated from the mass ${\displaystyle m}$ by the proportional constant of the acceleration due to gravity ${\displaystyle g=
9.8{\frac {\text{m}}{{\text{s}}^{2}}}}$ can be calculated:
${\displaystyle F=m\cdot g=l\cdot A\cdot \rho \cdot g}$
The natural frequency or pitch ${\displaystyle f}$ of hammerheads made of the same material is usually not inversely proportional to their weight ${\displaystyle F}$ , but depends essentially on
their exact geometry. The longer the geometric extension in a body, the lower the natural frequency in this direction or of the associated longitudinal vibration mode. The lowest audible frequency is
therefore correlated with the greatest length ${\displaystyle l}$ of the hammer head.
The natural frequency ${\displaystyle f}$ of hammerheads, however, is practically not audible at all because it lies in a frequency range that is too high. The speed of sound ${\displaystyle v}$ in
steel is about 5000 metres per second, and with a typical forging hammer head length ${\displaystyle l}$ of 10 to 16 centimetres, ${\displaystyle f={\frac {v}{2\cdot l}}}$ results in natural
frequencies between 15 and 25 kilohertz, which cannot be perceived in connection with a pitch.
Vibrating string with length l and tension force F.
Finally, it should be noted that the tensile weight ${\displaystyle F}$ of a string of length ${\displaystyle l}$ is neither proportional nor inversely proportional to the frequency ${\displaystyle
f}$ of the string vibrations or to the pitch. Rather, it is proportional to the square root of the tension weight ${\displaystyle F}$ . Furthermore, the pitch is inversely proportional to the length
${\displaystyle l}$ and the thickness ${\displaystyle D}$ of the string:
${\displaystyle f\propto {\frac {\sqrt {F}}{l\cdot D}}}$
Attempted explanation
These contradictions can be eliminated if the following facts are considered or taken into account:
• Pythagoras may have witnessed or even accompanied the complicated and elaborate construction of the Tunnel of Eupalinos, over 1000 metres long, on his native island of Samos.
• During Pythagoras' lifetime, the monumental Heraion of Samos was built of limestone and marble.
• The Latin word faber does not have to be translated as blacksmith, but can also be translated as craftsman.
• There were certainly more workshops and craftsmen for stone working than for metal working at that time.
• The Latin word fabrica means workshop and not smithy.
• Workshops in which at least four craftsmen forged at the same time with hammers of different sizes were likely to have been rare.
• With chisels the pitch ${\displaystyle f}$ is in the well audible range.
• In chisels the pitch ${\displaystyle f}$ of the longitudinal vibrations is inversely proportional to their length ${\displaystyle l}$ .
• For chisels of equal cross-sectional area ${\displaystyle A}$ , the pitch ${\displaystyle f}$ is therefore also inversely proportional to their length ${\displaystyle l}$ , to their volume ${\
displaystyle V}$ , to their mass ${\displaystyle m}$ and to their weight ${\displaystyle F}$ .
• The pitch ${\displaystyle f}$ of a vibrating string is inversely proportional to its length ${\displaystyle l}$ .
• The pitch ${\displaystyle f}$ of a vibrating string is inversely proportional to its thickness ${\displaystyle D}$ .
The following sound examples illustrate the pitches of five metal rods or chisels of different lengths when mechanically excited along the longitudinal axis with one blow, for example by a hammer.
The metal bars all have the same cross-sectional area, and the lengths as well as the natural frequencies and the pitches are in a ratio of 12 to 9 to ${\displaystyle {\sqrt {72}}}$ to 8 to 6 length
Four Pythagorean chisels with the in harmonic length and mass and natural frequency ratios 12:9:8:6.
• Pitches of five metal bars of different lengths
• Basic tone with 12 length units.
• Fourth with 9 length units.
• Fifth with 8 length units.
• Octave with 6 length units.
• Tritone (exactly half an octave) with ${\displaystyle {\sqrt {72}}\approx }$ 8.485 units of length. The corresponding chisel is not shown in the picture above and has a length between the two
chisels with 8 and 9 length units.
The metal bars with the integer length units produce harmonious sounds in all combinations, whereas the metal bar with a non-rational length ratio of ${\displaystyle {\sqrt {72}}}$ sounds dissonant
to all others.
With some corresponding and plausible assumptions, a scenario emerges that could have happened in Pythagoras' time, without any contradictions with physical laws:
If the events of Boethius' tradition, which mentions neither forges nor anvils, took place in a workshop for stonemasons and was inaccurate in the point of naming the tools to the effect that not
only the hammers but ensembles of chisels of the same cross-section but of different lengths and hammers were meant, the sounds and pitches would have been clearly audible and caused by hammer blows
but attributable to the chisels. Under this assumption, the integer ratios of the pitches would have been identical to those of the lengths or weights of the chisels and completely independent of the
craftsmen and the hammers used.
Two parallel monochords on a common resonance box in Deutsches Museum in Munich.
When experimenting with a monochord and constant string tension and texture, Pythagoras would have found exactly the same relationship between string length and pitch for a certain string thickness
and exactly the same relationship between string thickness and pitch for a certain string length as between chisel length or chisel weight and pitch. A string twice as long with the same thickness or
a string twice as thick with the same length will sound exactly one octave lower than the string with the same thickness or length.
The ratios observed here with the products of the two natural numbers two and three correspond to the consonant intervals octave, fifth, fourth and prime. In relation to any fundamental, the
corresponding four Pythagorean tones result in a so-called tetrachord.
Further investigation of these ratios finally yielded the diatonic scale of the seven tones A - B - C - D - E - F - G. This heptatonic scale forms the basis for the ancient Systema Téleion of the
Greeks, which developed in the centuries after Pythagoras, as well as for the four main church keys Protus, Deuterus, Tritus and Tetrardus, which developed in the centuries after Boethius.
The ancient investigations with the tension weights of strings may have been carried out, but they are neither sufficient nor necessary for these findings. If the tension of the string is doubled,
the result is a frequency increased by a factor of the square root of two (≈ 1.4142), which corresponds to a tritone interval commonly perceived as dissonant. Nevertheless, this irrational number was
also known to both the Pythagoreans and, long before, the Babylonians.
In harmony theory, the four Pythagorean tones are of great importance, as they form the framework of one of the most frequently used final cadences consisting of tonic, subdominant, dominant and
tonic. The folowing example shows the cadence C major - F major - G major - C major with the respective root of the four chords in the bass voice.
The four Pythagorean notes c - f - g - c' are depicted 'blue.
Hearing example of the final cadence in C-Major.
These four Pythagorean notes are, for example, a central motif of the Impromptus (opus number 5) by Robert Schumann (* 1810; † 1856) for piano of 1833 on a theme by Clara Wieck.
Illumination "Pythagoras in the Forge" to the work "De musica cum tonario" by Johannes Cotto (around 1100) from a collected manuscript of the Cistercian Abbey of Aldersbach. The third chapter
describes the invention of music by Pythagoras. Among other things, the work also contains extensive instructions for the composition of monophonic Gregorian chant and the organum.
The first transcriptions of the melodies of Gregorian chant are made with adiastematic neumes, in which the direction of the pitch for the following tone could be recorded upwards, to the same pitch
or downwards, as well as the approximate duration of the tones. It was not until Guido of Arezzo introduced line notation with diastematic neumes in the 11th century that the intervals of diatonic
melodies could also be precisely notated. In the various traditions from the Middle Ages, there are slightly different melodic progressions for the liturgical Latin texts. C and F clefs were already
used, but a tuning pitch was not yet available, so that the absolute pitches are not fixed despite the naming of the seven tones on the diatonic scale.
The Gregorian antiphon Ad te levavi is sung as the Introit on the first Sunday in Advent. The melody in the VIIIth tone (tetrardus plagalis) with the finalis (final tone) G begins on the text "Ad te
levavi animam meam". The Latin text from the Nova Vulgata with the first three verses of the 25th Psalm and the corresponding Hebrew letters Aleph, Beth and Ghimel reads as follows:
Psalm 25 (24),1-3A^[5]
1 Aleph. Ad te, Domine, levavi animam meam,
2A Beth. Deus meus, in te confido; non erubescam.
2B Neque exsultent super me inimici mei,
3A Ghimel. etenim universi, qui sustinent te, non confundentur.
The text of the first verse appears again in Psalm 143 (Nova Vulgata):
Psalm 143 (142),8^[6]
Auditam fac mihi mane misericordiam tuam, quia in te speravi.
Notam fac mihi viam, in qua ambulem, quia ad te levavi animam meam.
The melody restored after the Graduale Novum consists of twenty notes in the first verse, fourteen of which belong to the Pythagorean tetrachord c - f - g - c' and the remaining six can be considered
as ornaments or passing notes. The melodic section ends on the note F, the repercussa (the sustaining note or tenor) is the C.
The beginning of the antiphon Ad te levavi from the first Sunday in Advent after the Graduale Novum notated in square notation with a C clef. The four Pythagorean tones c - f - g - c' are represented
in blue.
Audio sample of the beginning of the Introit Ad te levavi from the first Sunday of Advent after the Graduale Novum.
The following table gives the frequency of the Pythagorean tones for the four sections of Psalm 25 of the Introit according to the version of the Graduale Novum:
Verse Final tone Number Number Number Number Sum of Number Sum Part of
c f g c' Pythagorean others all Pythagorean
1 f 1 3 8 2 14 6 20 70,0%
2A g 0 4 8 10 22 10 32 68,8%
2B f 0 2 4 11 17 12 29 58,6%
3A g 0 2 13 3 18 17 35 51,4%
In all four sections, the four Pythagorean tones predominate, even clearly in the first two sections. This coincidence is quite striking, and it seems as if the anonymous medieval composer wanted to
point us to the Pythagorean origin of music theory and the systems of ancient and Gregorian modes with this first piece of Gregorian repertoire in the Christian church year.
Another example from the Gregorian repertoire is the Communion of Pentecost Sunday Factus est repente de caelo sonus (Made is suddenly from heaven a sounding)^[7] in the VIIth tone (Tetrardus
authenticus) with the tenor D and the finalis G. The text describes the Pentecost event in which the Holy Spirit descended on the Christian community with tongues of fire and brought about the
speaking in tongues. Except for the first three notes of the strongly accented group neume (porrectus flexus) with the two top notes f' and the passing note e' a semitone below, the elemental melody
of the first verse consists only of the Pythagorean notes g – c' – d.
The beginning of the antiphon Factus est repente from Pentecost Sunday according to the Graduale Romanum notated in square notation with a C clef. The Pythagorean tones g - c' - d' are represented in
Audio sample of the beginning of the Communion Factus est repente from Pentecost Sunday after the Graduale Romanum.
Frontispiece of the first volume of the "Musurgia Universalis, sive Ars Magna Consoni et Dissoni" published in Rome by the jesuit Athanasius Kircher (* 1602; † 1680) in 1650. In the lower left corner
there ist Pythagoras. Next to him on the right there are three smithies, who hit an anvil with their hammers.
There are two works from the end of the 17th century with explicit reference to the legend, both composed by pupils of the Italian composer Jean-Baptiste Lully (* 1632; † 1687).
The sounds caused by hammer blows were introduced in 1690 by the French-German organist and composer Georg Muffat (* 1653; † 1704) with the organ composition Nova Cyclopeias Harmonica in C major and
set to tones in 3/4 time. This composition is framed by an aria consisting of two parts of 16 bars each. Otherwise, it comprises eight variations of 21 bars each on the theme Ad Malleorum Ictus
Allusio (To allude to the blows of the hammers) and ends with the chant Summo Deo Gloria. The individual pieces are all built on the fundamental notes c-f-g-c', which are variedly played around and
harmonised. In the last four bars of the Aria and in the last seven bars of the Variations, at least one of these Pythagorean tones can be heard on every beat:
Score of the last seven bars of the first variation Ad Malleorum Ictus Allusio of the composition Nova Cyclopeias Harmonica of Georg Muffat. The Pythagorean tones C, F and G are marked blue.
Audio sample of the last seven bars of the first variation Ad Malleorum Ictus Allusio of the composition Nova Cyclopeias Harmonica by Georg Muffat.
Title page of the seven orchestral suites "Pythagorische Schmids=Fuencklein" by Rupert Ignaz Mayr from 1692 with an illustration by the German painter Johann Andreas Wolff (* 1652; † 1716).
Canon with blue Markings at the four Pythagorean tones g'-c"-d"-g" on the title page of the seven orchestral suites "Pythagorische Schmids=Fuencklein" by Rupert Ignaz Mayr from 1692..
In 1692, the German violinist, composer and court conductor Rupert Ignaz Mayr (* 1646; † 1712) published the seven orchestral suites dedicated to the Elector of Bavaria Maximilian II Emanuel:
Pythagorische Schmids=Fuencklein
Bestehend in unterschiedlichen Arien / Sonatinen / Ouverturen / Allemanden / Couranten / Gavotten / Sarabanden / Giquen / Menueten / &c.
Mit 4.Instrumenten und beygefügten General-Baß, Bey Tafel=Musicken / Comœdien / Serenaden / und zu anderen fröhlichen Zusammenkunfften zu gebrauchen.
The main keys of the seven suites for solo violin are F major, D major, G major, D minor, F major, D major and B flat major.
Canon in G major on the title page of the seven orchestral suites "Pythagorische Schmids=Fuencklein" by Rupert Ignaz Mayr from 1692.
Four-part canon in G major on the title page of the seven orchestral suites "Pythagorische Schmids=Fuencklein" by Rupert Ignaz Mayr from 1692.
The German composer Johann Sebastian Bach (* 1685; † 1750) created a magnificent work in 1741 with the Goldberg Variations, which, like the Nova Cyclopeoas Harmonica by Georg Muffat written over
fifty years earlier, consists of a two-part aria with variations. The two Ariae reveal a number of similarities.
As a young man, Johann Sebastian Bach had already composed an equally outstanding work for the organ, namely the Passacaglia and Fugue in C minor (Bach-Werke-Verzeichnis 582). The eight-bar main
theme of the Passacaglia consists of fifteen notes, ten of which correspond to the Pythagorean tones.
The theme of the Passacaglia in C minor (BWV 582) by Johann Sebastian Bach with the blue Pythagorean tones depicted..
Theme of the Passacaglia in C minor (BWV 582) by Johann Sebastian Bach.
Pythagoras in a forge with four stonemasons
The legend of Pythagoras in the forge may be based on an actual incident. Regardless of the question of which of the regularities described here were actually investigated and found in antiquity,
inaccuracies have obviously occurred in the medieval and modern traditions.
Furthermore, unhistorical additions have been made to the legend that has been handed down, but these need not be considered further for the interpretation of Boethius' tradition. Nevertheless,
inaccuracies in the traditions and the additions and changes that were not in line with actual practice have certainly contributed to the fact that even the oldest reports on Pythagoras'
investigations have been relegated to the realm of legends by many authors - but perhaps quite unjustly according to the above explanations.
The German theoretical physicist Werner Heisenberg (* 1901; † 1976) wrote in his 1937 essay Thoughts of Ancient Natural Philosophy in Modern Physics:
The abstractness of the modern concept of the atom and of the mathematical forms which serve today's atomistics as an image for the multiplicity of phenomena already leads over to the second
basic idea which the exact natural science of our time has taken over from antiquity: the thought of the sense-giving power of mathematical structures.
The harmonies of the Pythagoreans, which Kepler still believed to find in the orbits of the stars, have been sought by natural science since Newton in the mathematical structure of the dynamic
law, in the equation formulating this law.
The successes of this view of nature, which has in part led to a real mastery of the forces of nature and thus decisively intervened in the development of mankind, have proved the belief of the
Pythagoreans right to an unforeseeable degree.
This turn means a consistent implementation of the programme of the Pythagoreans insofar as the infinite multiplicity of natural events finds its faithful mathematical image in the infinite
number of solutions of an equation, such as Newton's differential equation of mechanics.
The main author thanks his teacher Lorenz Weinrich (*1929). He introduced him to medieval church music with his profound knowledge of the Middle Ages and Gregorian chant.
Summary of the project
• Target audience: Musicians, historians, natural scientists
• Learning objectives: Integer-rational relationships based on an ancient legend.
• Book sponsorship/contact person: User:Bautsch
• Are co-authors currently desired? Yes, very much so. Corrections of obvious errors directly in the text; content please via discussion.
• Guidelines for co-authors: Wikimedia-like. | {"url":"https://en.m.wikibooks.org/wiki/Pythagoras_in_the_Forge","timestamp":"2024-11-04T10:30:49Z","content_type":"text/html","content_length":"187983","record_id":"<urn:uuid:dcc576cd-537c-47db-acd3-63ddfce647e1>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00324.warc.gz"} |
The Stacks project
Definition 34.3.4. Compare Schemes, Definition 26.5.2. Let $T$ be an affine scheme. A standard Zariski covering of $T$ is a Zariski covering $\{ U_ j \to T\} _{j = 1, \ldots , m}$ with each $U_ j \to
T$ inducing an isomorphism with a standard affine open of $T$.
Comments (0)
There are also:
• 2 comment(s) on Section 34.3: The Zariski topology
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder,
this is tag 020R. Beware of the difference between the letter 'O' and the digit '0'.
The tag you filled in for the captcha is wrong. You need to write 020R, in case you are confused. | {"url":"https://stacks.math.columbia.edu/tag/020R","timestamp":"2024-11-11T16:21:57Z","content_type":"text/html","content_length":"14163","record_id":"<urn:uuid:0cf927e8-cd08-4ca7-b3e8-6b3db8eba2e9>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00143.warc.gz"} |
EDUC 260B: Advanced Statistical Methods for Observational Studies (CHPR 290, STATS 266)
Design principles and statistical methods for observational studies. Topics include: matching methods, sensitivity analysis, instrumental variables, graphical models, marginal structural models. 3
unit registration requires a small project and presentation. Computing is in R. Pre-requisites:
HRP 261
and 262 or STAT 209 (
HRP 239
), or equivalent. See
Terms: Spr | Units: 2-3 | Grading: Medical Option (Med-Ltr-CR/NC)
EDUC 261: Sociocultural Theories of Learning & Development: Vygotksy & Bakhtin
Grounded in theories of Vygotsky and Bakhtin, this course will review commonly used, but often misunderstood, concepts about how context enters theories of learning and development. Topics will
include: distinctions between development and learning; the place of culture in developing higher mental functions; the zone of proximal development, conceptions and misconceptions; contributions of
activity theory; importance of heterogeneity and multivocality; and role of language in ¿ideological becoming¿ or idea development. Focus will be on using theory to guide research.
Terms: not given this year | Units: 3 | Grading: Letter (ABCD/NP)
EDUC 262A: Curriculum and Instruction in English
Approaches to teaching English in the secondary school, including goals for instruction, teaching techniques, and methods of evaluation. (STEP)
Terms: Sum | Units: 2 | Grading: Letter (ABCD/NP)
EDUC 262B: Curriculum and Instruction in English
Approaches to teaching English in the secondary school, including goals for instruction, teaching techniques, and methods of evaluation. STEP secondary only.
Terms: Aut | Units: 3 | Grading: Letter (ABCD/NP)
EDUC 262C: Curriculum and Instruction in English
Approaches to teaching English in the secondary school, including goals for instruction, teaching techniques, and methods of evaluation. (STEP)
Terms: Win | Units: 3 | Grading: Letter (ABCD/NP)
EDUC 262D: Curriculum & Instruction Elective in English
Methodology of science instruction: teaching for English and language arts; linking the goals of teaching English with interdisciplinary curricula; opportunities to develop teaching materials.
Terms: Spr | Units: 3 | Grading: Letter or Credit/No Credit
EDUC 263A: Curriculum and Instruction in Mathematics
The purposes and programs of mathematics in the secondary curriculum; teaching materials, methods. Prerequisite: STEP student or consent of instructor. (STEP) 263A. Sum, 263B. Aut, 263C. Win
Terms: Sum | Units: 2 | Grading: Letter (ABCD/NP)
EDUC 263B: Curriculum and Instruction in Mathematics
The purposes and programs of mathematics in the secondary curriculum; teaching materials, methods. Prerequisite: STEP student or consent of instructor. (STEP) 263A. Sum, 263B. Aut, 263C. Win
Terms: Aut | Units: 3 | Grading: Letter (ABCD/NP)
EDUC 263C: Curriculum and Instruction in Mathematics
The purposes and programs of mathematics in the secondary curriculum; teaching materials, methods. Prerequisite: STEP student or consent of instructor. (STEP) 263A. Sum, 263B. Aut, 263C. Win
Terms: Win | Units: 3 | Grading: Letter (ABCD/NP)
EDUC 263D: Curriculum & Instruction Elective in Math
Methodology of math instruction: teaching for mathematical thinking and reasoning; linking the goals of teaching math with literacy and interdisciplinary curricula; opportunities to develop teaching
Terms: Spr | Units: 3 | Grading: Letter or Credit/No Credit | {"url":"https://swap.stanford.edu/was/20160311035101mp_/http:/explorecourses.stanford.edu/search?q=EDUC&view=catalog&academicYear=20152016&page=17&filter-departmentcode-EDUC=on&filter-coursestatus-Active=on","timestamp":"2024-11-04T06:28:22Z","content_type":"application/xhtml+xml","content_length":"175284","record_id":"<urn:uuid:103b78b0-6843-4f82-9507-65b3104ad204>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00614.warc.gz"} |
NCERT Solutions For Class 12 Physics Chapter 3 CBSE
NCERT Solutions For Class 12 Physics Chapter 3 Current Electricity
NCERT Solutions For Class 12 Physics Chapter 3 gives us an introduction to the motion of electrons, and current electricity, solving the problems at the end of the chapter. Class 12 Physics NCERT
Solutions Chapter 3 is provided in a detailed manner, so the students can easily understand and practice the solutions. This chapter introduces the concept of motion of electrons and the effects that
arise due to it, current electricity. This chapter is a continuation of the topics and concepts that are introduced in the previous chapter. They can read NCERT Class 12 Physics Solutions for more
solutions regarding other chapters. They can also check this NCERT Solutions For Class 12 Physics Chapter 3 post for reference.
Class 12 Physics NCERT Solutions Chapter 3 Current Electricity:
Topic-wise overview of NCERT Solutions For Class 12 Physics Chapter 3 is given in the following table.
Section Name Topic Name
3 Current Electricity
3.1 Introduction
3.2 Electric Current
3.3 Electric Currents in Conductors
3.4 Ohm’s law
3.5 Drift of Electrons and the Origin of Resistivity
3.6 Limitations of Ohm’s Law
3.7 Resistivity of Various Materials
3.8 Temperature Dependence of Resistivity
3.9 Electrical Energy, Power
3.10 Combination of Resistors — Series and Parallel
3.11 Cells, EMF, Internal Resistance
3.12 Cells in Series and in Parallel
3.13 Kirchhoff’s Rules
3.14 Wheatstone Bridge
3.15 Meter Bridge
3.16 Potentiometer
Class 12 Physics NCERT Solutions Chapter 3 For Problems Given at the End of the Lesson:
Question 3.1:
The storage battery of a car has an EMF of 12 V. If the internal resistance of the battery is 0.4Ω, what is the maximum current that can be drawn from the battery?
EMF of the battery, E = 12 V
The internal resistance of the battery, r = 0.4 Ω
Maximum current drawn from the battery = I
According to Ohm’s law,
V = IR
⇒ I = V/R
= 12/0.4 = 30 A
The maximum current drawn from the given battery is 30 A.
Question 3.2:
A battery of EMF 10 V and internal resistance 3 Ω is connected to a resistor. If the current in the circuit is 0.5 A, what is the resistance of the resistor? What is the terminal voltage of the
battery when the circuit is closed?
EMF of the battery, E = 10 V
The internal resistance of the battery, r = 3 Ω
Current in the circuit, I = 0.5 A
Resistance of the resistor = R
The relation for current using Ohm’s law is,
The terminal voltage of the resistor = V
According to Ohm’s law,
V = IR
= 0.5 × 17
= 8.5 V
Therefore, the resistance of the resistor is 17 Ω and the terminal voltage is
8.5 V.
Question 3.3:
(a) Three resistors 1 Ω, 2 Ω, and 3 Ω are combined in series. What is the total resistance of the combination?
(b) If the combination is connected to a battery of EMF 12 V and negligible internal resistance, obtain the potential drop across each resistor.
a) Three resistors of resistances 1 Ω, 2 Ω, and 3 Ω are combined in series. The total resistance of the combination is given by the algebraic sum of individual resistances.
Total resistance = 1 + 2 + 3 = 6 Ω
(b) Current flowing through the circuit = I
EMF of the battery, E = 12 V
The total resistance of the circuit, R = 6 Ω
The relation for current using Ohm’s law is,
I = V/R
⇒ 12/6 = 2A
The potential drop across 1 Ω resistor = V[1]
From Ohm’s law, the value of V[1 ]can be obtained as
V[1] = 2 × 1= 2 V … (i)
Potential drop across 2 Ω resistor = V[2]
Again, from Ohm’s law, the value of V[2 ]can be obtained as
V[2] = 2 × 2= 4 V … (ii)
Potential drop across 3 Ω resistor = V[3]
Again, from Ohm’s law, the value of V[3 ]can be obtained as
V[3] = 2 × 3= 6 V … (iii)
Therefore, the potential drop across 1 Ω, 2 Ω, and 3 Ω resistors are 2 V, 4 V, and 6 V respectively.
Question 3.4:
(a) Three resistors 2 Ω, 4 Ω and 5 Ω are combined in parallel. What is the total resistance of the combination?
(b) If the combination is connected to a battery of EMF 20 V and negligible internal resistance, determine the current through each resistor, and the total current drawn from the battery.
(a) There are three resistors of resistances,
R[1] = 2 Ω, R[2] = 4 Ω, and R[3] = 5 Ω
They are connected in parallel. Hence, the total resistance (R) of the combination is given by,
Therefore, the total resistance of the combination is 20/19 ohms.
(b) EMF of the battery, V = 20 V
Current (I[1]) flowing through resistor R[1] is given by,
Current (I[2]) flowing through resistor R[2] is given by,
Current (I[3]) flowing through resistor R[3] is given by,
Total current, I = I[1] + I[2] + I[3] = 10 + 5 + 4 = 19 A
Therefore, the current through each resister is 10 A, 5 A, and 4 A respectively and the total current is 19 A.
Question 3.5:
At room temperature (27.0 °C) the resistance of a heating element is 100 Ω. What is the temperature of the element if the resistance is found to be 117 Ω, given that the temperature coefficient of
the material of the resistor is 1.70 x 10^-4 ^∘C^-1
Room temperature, T = 27°C
Resistance of the heating element at T, R = 100 Ω
Let T[1] is the increased temperature of the filament.
Resistance of the heating element at T[1], R[1] = 117 Ω
Temperature co-efficient of the material of the filament,
Therefore, at 1027°C, the resistance of the element is 117Ω.
Question 3.6:
A negligibly small current is passed through a wire of length 15 m and uniform cross-section 6.0 × 10^−7 m^2, and its resistance is measured to be 5.0 Ω. What is the resistivity of the material at
the temperature of the experiment?
Length of the wire, l =15 m
Area of the cross-section of the wire, a = 6.0 × 10^−7 m^2
Resistance of the material of the wire, R = 5.0 Ω
The resistivity of the material of the wire = ρ
Resistance is related to the resistivity as
Therefore, the resistivity of the material is 2 × 10^−7 Ω m.
Question 3.7:
A silver wire has a resistance of 2.1 Ω at 27.5 °C, and a resistance of 2.7 Ω at 100 °C. Determine the temperature coefficient of resistivity of silver.
Temperature, T[1] = 27.5°C
Resistance of the silver wire at T[1], R[1] = 2.1 Ω
Temperature, T[2] = 100°C
Resistance of the silver wire at T[2], R[2] = 2.7 Ω
Temperature coefficient of silver = α
It is related with temperature and resistance as
Therefore, the temperature coefficient of silver is 0.0039°C^−1.
Question 3.8:
A heating element using nichrome connected to a 230 V supply draws an initial current of 3.2 A which settles after a few seconds to a steady value of 2.8 A. What is the steady temperature of the
heating element if the room temperature is 27.0 °C? The temperature coefficient of resistance of nichrome averaged over the temperature range involved is 1.70 × 10^−4 °C ^−1.
Supply voltage, V = 230 V
The initial current drawn, I[1] = 3.2 A
Initial resistance = R[1], which is given by the relation,
Steady-state value of the current, I[2] = 2.8 A
Resistance at the steady-state = R[2], which is given as
Temperature co-efficient of nichrome, α = 1.70 × 10^−4 °C ^−1
The initial temperature of nichrome, T[1]= 27.0°C
Study state temperature reached by nichrome = T[2]
T[2 ]can be obtained by the relation for α,
Therefore, the steady temperature of the heating element is 867.5°C
Question 3.9:
Determine the current in each branch of the network shown in fig 3.30:
Current flowing through various branches of the circuit is represented in the given figure.
I[1] = Current flowing through the outer circuit
I[2] = Current flowing through branch AB
I[3] = Current flowing through branch AD
I[2] − I[4] = Current flowing through branch BC
I[3] + I[4] = Current flowing through branch CD
I[4] = Current flowing through branch BD
For the closed circuit ABDA, potential is zero i.e.,
10I[2] + 5I[4] − 5I[3] = 0
2I[2] + I[4] −I[3] = 0
I[3] = 2I[2] + I[4] … (1)
For the closed circuit BCDB, potential is zero i.e.,
5(I[2] − I[4]) − 10(I[3] + I[4]) − 5I[4 ]= 0
5I[2] + 5I[4] − 10I[3] − 10I[4 ]− 5I[4 ]= 0
5I[2] − 10I[3] − 20I[4 ]= 0
I[2] = 2I[3] + 4I[4 ]… (2)
For the closed circuit ABCFEA, potential is zero i.e.,
−10 + 10 (I[1]) + 10(I[2]) + 5(I[2] − I[4]) = 0
10 = 15I[2] + 10I[1 ]− 5I[4]
3I[2] + 2I[1 ]− I[4] = 2 … (3)
From equations (1) and (2), we obtain
I[3] = 2(2I[3 ]+ 4I[4]) + I[4]
I[3] = 4I[3 ]+ 8I[4] + I[4]
− 3I[3] = 9I[4]
− 3I[4] = + I[3 ]… (4)
Putting equation (4) in equation (1), we obtain
I[3] = 2I[2 ]+ I[4]
− 4I[4] = 2I[2]
I[2] = − 2I[4] … (5)
It is evident from the given figure that,
I[1] = I[3 ]+ I[2] … (6)
Putting equation (6) in equation (1), we obtain
3I[2] +2(I[3 ]+ I[2]) − I[4] = 2
5I[2] + 2I[3 ]− I[4] = 2 … (7)
Putting equations (4) and (5) in equation (7), we obtain
5(−2 I[4]) + 2(− 3 I[4])[ −] I[4] = 2
− 10I[4] − 6I[4 ]− I[4] = 2
17I[4] = − 2
I[4] = -2∕17 A
Equation (4) reduces to
I[3] = − 3(I[4])
Therefore, current in branch AB = 4/17 A
In branch BC = 6/17 A
In Branch CD = -4/17 A
In Branch AD = (6/17) A
In Branch BD = (-2/17) A
Total Current =
Question 3.10:
(a) In a meter bridge [Fig. 3.27], the balance point is found to be at 39.5 cm from the end A, when the resistor Y is of 12.5 Ω. Determine the resistance of X. Why are the connections between
resistors in a Wheatstone or meter bridge made of thick copper strips?
(b) Determine the balance point of the bridge above if X and Y are interchanged.
(c) What happens if the galvanometer and cell are interchanged at the balance point of the bridge? Would the galvanometer show any current?
A meter bridge with resistors X and Y is represented in the given figure.
(a) Balance point from end A, l[1] = 39.5 cm
Resistance of the resistor Y = 12.5 Ω
The condition for the balance is given as,
Therefore, the resistance of resistor X is 8.2 Ω.
The connection between resistors in a Wheatstone or metre bridge is made of thick copper strips to minimize the resistance, which is not taken into consideration in the bridge formula.
(b) If X and Y are interchanged, then l[1] and 100−l[1] get interchanged.
The balance point of the bridge will be 100−l[1] from A.
100−l[1 ]= 100 − 39.5 = 60.5 cm
Therefore, the balance point is 60.5 cm from A.
(c) When the galvanometer and cell are interchanged at the balance point of the bridge, the galvanometer will show no deflection. Hence, no current would flow through the galvanometer.
Question 3.11:
A storage battery of EMF 8.0 V and internal resistance 0.5 Ω is being charged by a 120 V dc supply using a series resistor of 15.5 Ω. What is the terminal voltage of the battery during charging? What
is the purpose of having a series resistor in the charging circuit?
EMF of the storage battery, E = 8.0 V
Internal resistance of the battery, r = 0.5 Ω
DC supply voltage, V = 120 V
Resistance of the resistor, R = 15.5 Ω
Effective voltage in the circuit = V^1
R is connected to the storage battery in series. Hence, it can be written as
V^1 = V − E
V^1 = 120 − 8 = 112 V
Current flowing in the circuit = I, which is given by the relation,
Voltage across resistor R given by the product, IR = 7 × 15.5 = 108.5 V
DC supply voltage = Terminal voltage of battery + Voltage drop across R
Terminal voltage of battery = 120 − 108.5 = 11.5 V
A series resistor in a charging circuit limits the current drawn from the external source. The current will be extremely high in its absence. This is very dangerous.
Question 3.12:
In a potentiometer arrangement, a cell of EMF 1.25 V gives a balance point at 35.0 cm length of the wire. If the cell is replaced by another cell and the balance point shifts to 63.0 cm, what is the
EMF of the second cell?
EMF of the cell, E[1] = 1.25 V
Balance point of the potentiometer, l[1]= 35 cm
The cell is replaced by another cell of EMF E[2].
New balance point of the potentiometer, l[2] = 63 cm
The balance condition is given by the relation,
Therefore, EMF of the second cell is 2.25V.
Question 3.13:
The number density of free electrons in a copper conductor estimated in Example 3.1 is 8.5 × 10^28 m^−3. How long does an electron take to drift from one end of a wire 3.0 m long to its other end?
The area of cross-section of the wire is 2.0 × 10^−6 m^2 and it is carrying a current of 3.0 A.
Number density of free electrons in a copper conductor, n = 8.5 × 10^28 m^−3 Length of the copper wire, l = 3.0 m
Area of cross-section of the wire, A = 2.0 × 10^−6 m^2
Current carried by the wire, I = 3.0 A, which is given by the relation,
I = nAeV[d]
e = Electric charge = 1.6 × 10^−19 C
V[d] = Drift velocity = Length of the wire (l) / Time taken to cover l (t)
Therefore, the time taken by an electron to drift from one end of the wire to the other is 2.7 × 10^4 s.
Question 3.14:
The earth’s surface has a negative surface charge density of 10^−9 C m^−2. The potential difference of 400 kV between the top of the atmosphere and the surface results (due to the low conductivity of
the lower atmosphere) in a current of only 1800 A over the entire globe. If there were no mechanism of sustaining atmospheric electric field, how much time (roughly) would be required to neutralize
the earth’s surface? (This never happens in practice because there is a mechanism to replenish electric charges, namely the continual thunderstorms and lightning in different parts of the globe).
(Radius of earth = 6.37 × 10^6 m.)
Surface charge density of the earth, σ = 10^−9 C m^−2
Current over the entire globe, I = 1800 A
Radius of the earth, r = 6.37 × 10^6 m
Surface area of the earth,
A = 4πr^2
= 4π × (6.37 × 10^6)^2
= 5.09 × 10^14 m^2
Charge on the earth surface,
q = σ × A
= 10^−9 × 5.09 × 10^14
= 5.09 × 10^5 C
Time taken to neutralize the earth’s surface = t
Current, I = q/t
⇒ t = q / i
Therefore, the time taken to neutralize the earth’s surface is 282.77 s.
Question 3.15:
(a) Six lead-acid types secondary cells each of emf 2.0 V and internal resistance 0.015 Ω are joined in series to provide a supply to a resistance of 8.5 Ω. What is the current drawn from the supply
and its terminal voltage?
(b) A secondary cell after long use has an EMF of 1.9 V and a large internal resistance of 380 Ω. What maximum current can be drawn from the cell? Could the cell drive the starting motor of a car?
(a) Number of secondary cells, n = 6
EMF of each secondary cell, E = 2.0 V
Internal resistance of each cell, r = 0.015 Ω
series resistor is connected to the combination of cells.
Resistance of the resistor, R = 8.5 Ω
Current drawn from the supply = I, which is given by the relation,
Terminal voltage, V = IR = 1.39 × 8.5 = 11.87 A
Therefore, the current drawn from the supply is 1.39 A and terminal voltage is
11.87 A.
(b) After a long use, emf of the secondary cell, E = 1.9 V
Internal resistance of the cell, r = 380 Ω
Hence, maximum current
Therefore, the maximum current drawn from the cell is 0.005 A. Since a large current is required to start the motor of a car, the cell cannot be used to start a motor.
Question 3.16:
Two wires of equal length, one of aluminum and the other of copper have the same resistance. Which of the two wires is lighter? Hence explain why aluminum wires are preferred for overhead power
cables. (ρ[Al] = 2.63 × 10^−8 Ω m, ρ[Cu] = 1.72 × 10^−8 Ω m, Relative density of Al = 2.7, of Cu = 8.9.)
Resistivity of aluminum, ρ[Al] = 2.63 × 10^−8 Ω m
Relative density of aluminum, d[1] = 2.7
Let l[1] be the length of aluminum wire and m[1] be its mass.
Resistance of the aluminum wire = R[1]
Area of cross-section of the aluminum wire = A[1]
Resistivity of copper, ρ[Cu] = 1.72 × 10^−8 Ω m
Relative density of copper, d[2] = 8.9
Let l[2] be the length of copper wire and m[2] be its mass.
Resistance of the copper wire = R[2]
Area of cross-section of the copper wire = A[2]
The two relations can be written as
It is given that,
l[1] = l[2]
Mass of the aluminum wire,
m[1] = Volume × Density
= A[1]l[1] × d[1 ]= A[1] l[1]d[1] … (3)
Mass of the copper wire,
m[2] = Volume × Density
= A[2]l[2] × d[2 ]= A[2] l[2]d[2] … (4)
Dividing equation (3) by equation (4), we obtain
It can be inferred from this ratio that m[1] is less than m[2]. Hence, aluminum is lighter than copper.
Since aluminum is lighter, it is preferred for overhead power cables over copper.
Question 3.17:
What conclusion can you draw from the following observations on a resistor made of alloy Manganin?
Current (A) Voltage (V) Current (A) Voltage (V)
0.2 3.94 3.0 59.2
0.4 7.87 4.0 78.8
0.6 11.8 5.0 98.6
0.8 15.7 6.0 118.5
1.0 19.7 7.0 138.2
2.0 39.4 8.0 158.0
It can be inferred from the given table that the ratio of voltage with current is a constant, which is equal to 19.7. Hence, Manganin is an ohmic conductor i.e., the alloy obeys Ohm’s law. According
to Ohm’s law, the ratio of voltage with current is the resistance of the conductor. Hence, the resistance of Manganin is 19.7 Ω.
Question 3.18:
(a) A steady current flows in a metallic conductor of non-uniform cross-section. Which of these quantities is constant along the conductor: current, current density, electric field, drift speed?
(b) Is Ohm’s law universally applicable for all conducting elements?
If not, give examples of elements which do not obey Ohm’s law.
(c) A low voltage supply from which one needs high currents must have very low internal resistance. Why?
(d) A high tension (HT) supply of, say, 6 kV must have a very large internal resistance. Why?
(a) When a steady current flows in a metallic conductor of a non-uniform cross-section, the current flowing through the conductor is constant. Current density, electric field, and drift speed are
inversely proportional to the area of the cross-section. Therefore, they are not constant.
(b) No, Ohm’s law is not universally applicable for all conducting elements. Vacuum diode semi-conductor is a non-ohmic conductor. Ohm’s law is not valid for it.
(c) According to Ohm’s law, the relation for the potential is V = IR
Voltage (V) is directly proportional to current (I).
R is the internal resistance of the source.
I = V/R
If V is low, then R must be very low, so that high current can be drawn from the source.
(d) In order to prohibit the current from exceeding the safety limit, a high tension supply must have a very large internal resistance. If the internal resistance is not large, then the current drawn
can exceed the safety limits in case of a short circuit.
Question 3.19:
Choose the correct alternative:
(a) Alloys of metals usually have (greater/less) resistivity than that of their constituent metals.
(b) Alloys usually have much (lower/higher) temperature coefficients of resistance than pure metals.
(c) The resistivity of the alloy Manganin is nearly independent of/increases rapidly with increase of temperature.
(d) The resistivity of a typical insulator (e.g., amber) is greater than that of a metal by a factor of the order of (10^22/10^3).
(a) Alloys of metals usually have greater resistivity than that of their constituent metals.
(b) Alloys usually have lower temperature coefficients of resistance than pure metals.
(c) The resistivity of the alloy, Manganin, is nearly independent of increase of temperature.
(d) The resistivity of a typical insulator is greater than that of a metal by a factor of the order of 10^22.
Question 3.20:
(a) Given n resistors each of resistance R, how will you combine them to get the (i) maximum (ii) minimum effective resistance? What is the ratio of the maximum to minimum resistance?
(b) Given the resistances of 1 Ω, 2 Ω, 3 Ω, how will be combine them to get an equivalent resistance of (i) (11/3) Ω (ii) (11/5) Ω, (iii) 6 Ω, (iv) (6/11) Ω?
(c) Determine the equivalent resistance of networks shown in Fig. 3.31.
(a) Total number of resistors = n
Resistance of each resistor = R
(i) When n resistors are connected in series, effective resistance R[1]is the maximum, given by the product nR.
Hence, maximum resistance of the combination, R[1 ]= nR
(ii) When n resistors are connected in parallel, the effective resistance (R[2]) is the minimum, given by the ratio R/n
Hence, minimum resistance of the combination, R[2] = R/n
(iii) The ratio of the maximum to the minimum resistance is,
(b) The resistance of the given resistors is,
R[1] = 1 Ω, R[2] = 2 Ω, R[3] = 3 Ω2
i. Equivalent resistance, R′ = 11/3 ohms
Consider the following combination of the resistors.
Equivalent resistance of the circuit is given by,
2. Equivalent resistance, R’ = 11/5 ohms
Consider the following combination of the resistors.
Equivalent resistance of the circuit is given by,
(iii) Equivalent resistance, R^’ = 6 Ω
Consider the series combination of the resistors, as shown in the given circuit.
Equivalent resistance of the circuit is given by the sum,
R^’ = 1 + 2 + 3 = 6 Ω
(iv) Equivalent resistance, R’ = 6/11 Ω
Consider the series combination of the resistors, as shown in the given circuit.
Equivalent resistance of the circuit is given by,
(c) (a) It can be observed from the given circuit that in the first small loop, two resistors of resistance 1 Ω each are connected in series.
Hence, their equivalent resistance = (1+1) = 2 Ω
It can also be observed that two resistors of resistance 2 Ω each are connected in series.
Hence, their equivalent resistance = (2 + 2) = 4 Ω.
Therefore, the circuit can be redrawn as
It can be observed that 2 Ω and 4 Ω resistors are connected in parallel in all the four loops. Hence, equivalent resistance (R^’) of each loop is given by,
The circuit reduces to,
All the four resistors are connected in series.
Hence, equivalent resistance of the given circuit is
(b) It can be observed from the given circuit that five resistors of resistance R each are connected in series.
Hence, equivalent resistance of the circuit = R + R + R + R + R
= 5 R
Question 3.21:
Determine the current drawn from a 12 V supply with internal resistance 0.5 Ω by the infinite network shown in Fig. 3.32. Each resistor has 1 Ω resistance.
The resistance of each resistor connected in the given circuit, R = 1 Ω
Equivalent resistance of the given circuit = R^’
The network is infinite. Hence, equivalent resistance is given by the relation,
Negative value of R^’ cannot be accepted. Hence, equivalent resistance,
R’ = (1+√3) = 1+1.73 = 2.73 Ω
Internal resistance of the circuit, r = 0.5 Ω
Hence, total resistance of the given circuit = 2.73 + 0.5 = 3.23 Ω
Supply voltage, V = 12 V
According to Ohm’s Law, current drawn from the source is given by the ratio, (12/3.23) = 3.72 A
Question 3.22:
Figure 3.33 shows a potentiometer with a cell of 2.0 V and internal resistance 0.40 Ω maintaining a potential drop across the resistor wire AB. A standard cell which maintains a constant EMF of 1.02
V (for very moderate currents up to a few mA) gives a balance point at 67.3 cm length of the wire. To ensure very low currents drawn from the standard cell, a very high resistance of 600 kΩ is put in
series with it, which is shorted close to the balance point. The standard cell is then replaced by a cell of unknown EMF ε and the balance point found similarly, turns out to be at 82.3 cm length of
the wire.
(a) What is the value ε ?
(b) What purpose does the high resistance of 600 kΩ have?
(c) Is the balance point affected by this high resistance?
(d) Is the balance point affected by the internal resistance of the driver cell?
(e) Would the method work in the above situation if the driver cell of the potentiometer had an EMF of 1.0 V instead of 2.0 V?
(f ) Would the circuit work well for determining an extremely small EMF, say of the order of a few mV (such as the typical EMF of a thermo-couple)? If not, how will you modify the circuit?
(a) Constant EMF of the given standard cell, E[1] = 1.02 V
Balance point on the wire, l[1 ]= 67.3 cm
A cell of unknown EMF, ε,replaced the standard cell. Therefore, new balance point on the wire, l = 82.3 cm
The relation connecting EMF and balance point is,
the value of unknown EMF is 1.247 V
(b) The purpose of using the high resistance of 600 kΩ is to reduce the current through the galvanometer when the movable contact is far from the balance point.
(c) The balance point is not affected by the presence of high resistance.
(d) The point is not affected by the internal resistance of the driver cell.
(e) The method would not work if the driver cell of the potentiometer had an EMF of 1.0 V instead of 2.0 V. This is because if the EMF of the driver cell of the potentiometer is less than the EMF of
the other cell, then there would be no balance point on the wire.
(f) The circuit would not work well for determining an extremely small EMF. As the circuit would be unstable, the balance point would be close to end A. Hence, there would be a large percentage of
The given circuit can be modified if a series resistance is connected with the wire AB. The potential drop across AB is slightly greater than the EMF is measured. The percentage error would be small.
Question 3.23:
Figure 3.34 shows a potentiometer circuit for comparison of two resistances. The balance point with a standard resistor R = 10.0 Ω is found to be 58.3 cm, while that with the unknown resistance X is
68.5 cm. Determine the value of X. What might you do if you failed to find a balance point with the given cell of EMF ε?
Resistance of the standard resistor, R = 10.0 Ω
Balance point for this resistance, l[1] = 58.3 cm
Current in the potentiometer wire = i
Hence, potential drop across R, E[1] = iR
Resistance of the unknown resistor = X
Balance point for this resistor, l[2] = 68.5 cm
Hence, potential drop across X, E[2] = iX
The relation connecting emf and balance point is,
= (68.5/58.3) x 10 = 11.749 Ω
Therefore, the value of the unknown resistance, X, is 11.75 Ω.
If we fail to find a balance point with the given cell of emf, ε, then the potential drop across R and X must be reduced by putting a resistance in series with it. Only if the potential drop across R
or X is smaller than the potential drop across the potentiometer wire AB, a balance point is obtained.
Question 3.24:
Figure 3.35 shows a 2.0 V potentiometer used for the determination of internal resistance of a 1.5 V cell. The balance point of the cell in open circuit is 76.3 cm. When a resistor of 9.5 Ω is used
in the external circuit of the cell, the balance point shifts to 64.8 cm length of the potentiometer wire. Determine the internal resistance of the cell.
Internal resistance of the cell = r
Balance point of the cell in open circuit, l[1] = 76.3 cm
An external resistance (R) is connected to the circuit with R = 9.5 Ω
New balance point of the circuit, l[2 ]= 64.8 cm
Current flowing through the circuit = I
The relation connecting resistance and emf is,
Therefore, the internal resistance of the cell is 1.68Ω. | {"url":"https://jobschat.in/ncert-solutions-for-class-12-physics-chapter-3-current-electricity/","timestamp":"2024-11-01T20:12:51Z","content_type":"text/html","content_length":"253464","record_id":"<urn:uuid:9d530d49-b8f2-49e4-84e9-57cb52901f0b>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00213.warc.gz"} |
MCMCorner: Hello World!
“You can never be absolutely certain that the MCMC is reliable, you can only identify when something has gone wrong.” Andrew Gelman
Model-based inference is, after all, based on the model. Careful research means being vigilant both regarding the choice of model and rigorously assessing our ability to estimate under the chosen
model. These two concerns pertain both to model-based inference of phylogeny—using programs such as RaXML or MrBayes—and to inferences based on phylogeny—such as the study of character evolution,
lineage diversification—and indeed to all model-based inference.
The first issue—model specification, which entails three closely related issues—is critically important for the simple reason that unbiased estimates can only be obtained under a model that provides
a reasonable description of the process that gave rise to our data. Model selection entails assessing the relative fit of our dataset to a pool of candidate models. Rankings are based on
model-selection methods that compare the relative fit of candidate modes based either on their maximum-likelihood estimates (which measures the fit of the data to the model at a single point in
parameter space), or on the marginal likelihood of the candidate models (which measures the average fit of the candidate models to the data). Model adequacy—an equally important but relatively
neglected issue—assesses the absolute fit of the data to the model. Model uncertainty is related to the common (and commonly ignored) scenario in which multiple candidate models provide a similar fit
to the data: in this scenario, conditioning on any single model (even the best) will lead to biased estimates, and so model averaging is required to accommodate uncertainty in the choice of model.
Much less concern is given to the second aspect of model-based inference: the ability to obtain reliable estimates under the chosen model(s). Our field is currently experiencing a “pioneering
era”—increasingly complex (and presumably realistic) phylogenetic models are being proposed and implemented at an unprecedented rate. Our frontier era, however, more closely resembles the ‘wild west’
of the 1760’s than the ‘space race’ of the 1960’s: the statistical behavior of many new models remains unchartered territory, and might aptly carry the warning label ‘Here be
dragons’. Nevertheless, most users implicitly assume, it seems, that if a model has been implemented correctly, and if that implementation has been “successfully” used to obtain an estimate from a
given dataset (i.e., the input file has been read into the program, the program has been run and an output file has been generated), then we must have performed valid inference under the model. This
would be perfectly sound reasoning if inferences were based on analytical methods.
Owing to the complexity of most phylogenetic models, however, it is not possible to estimate parameters analytically. Instead, parameter estimates are based on numerical methods. In the case of
maximum-likelihood estimation, these are typically hill-climbing algorithms that search the profile likelihood to identify the vector of point estimates for all phylogenetic model parameters that
jointly maximize the likelihood of observing the data under the model. The reliability of these algorithms can (and should) be assessed by comparing estimates obtained from repeated analyses that are
initiated from random points in parameter space. Because there is only one maximum likelihood estimate, the terminal parameter values of replicate searches should be identical (within the precision
of computer memory).
In the Bayesian statistical framework, inferences focus on the joint posterior probability density of phylogenetic model parameters, which is approximated by Markov chain Monte Carlo (MCMC)
algorithms. It may be comforting to know that, in theory, an appropriately constructed and adequately run MCMC simulation is guaranteed to provide an arbitrarily precise description of the joint
posterior probability density. In practice, however, even a given MCMC algorithm that provides reliable estimates in most cases will nevertheless fail in some cases and is not guaranteed to work for
any given dataset. This raises an obvious question: “When do we know that an MCMC algorithm provides reliable estimates for a given empirical analysis“. The answer is simple: Never.
Fortunately, this problem is not unique to Bayesian inference of phylogeny. Much of Bayesian inference outside our field also relies on MCMC algorithms to approximate the joint posterior probability
density of parameters: similar concerns regarding the reliability of those numerical approximations have motivated the development of a suite of diagnostic tools to assess MCMC performance. The trick
is learning how to use these tools effectively and rigorously, especially for analyses that entail complex phylogenetic models and/or large datasets. As a field, we have failed both to emphasize the
importance of assessing MCMC performance, and also to provide opportunities to develop the skills to do so.
I intend to use this blog as a venue to discuss issues related to the diagnosis of MCMC algorithms used for Bayesian inference of phylogeny. The goal of this thread is to raise general awareness on
this issue by stimulating discussion about specific diagnostics, successful strategies, common pathologies, best practices, etc., associated with assessing the performance of phylogenetic MCMC
methods. If you have questions or comments, please leave a message below or drop me an email.
2 thoughts on “MCMCorner: Hello World!”
1. BobThomson
Looking forward to this series of posts Brian. Let’s see some of those gnarly traces from your folder of MCMC nightmares! It could be fun to get readers to submit ‘problematic’ traces and try to
get the community to work out solutions together.
2. Rich Glor
Great idea Brian, I’m looking forward to these posts. Way to kick things off too; I bet there aren’t many other lessons on Bayesian MCMC that also manage to mention the wild west, the space race,
and dragons all in one sentence. | {"url":"https://treethinkers.org/mcmcorner/","timestamp":"2024-11-01T21:58:37Z","content_type":"text/html","content_length":"40278","record_id":"<urn:uuid:9085817d-1b87-4cb2-ae2e-77231657807b>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00398.warc.gz"} |
Professor Peter Schreier
Head of Group
Signal and System Theory Group
I received a Master of Science from the University of Notre Dame, Indiana, USA, in 1999, and a Ph.D. from the University of Colorado at Boulder, USA, in 2003, both in electrical engineering. In the
Fall semester of 1998, I was a visiting research student with the Coding Group at the University of Hawaii at Manoa, USA.
From 2004 until 2011, I was with the School of Electrical Engineering and Computer Science at The University of Newcastle, Australia, first as Lecturer, then Senior Lecturer, and finally Associate
Professor. In the Spring semester of 2008, I was a Visiting Associate Professor with the Department of Electrical and Computer Engineering at Colorado State University, Ft. Collins, USA.
Since 2011, I have been Professor and Head of the Signal and System Theory Group, and from 2019 until 2023, I served as Dean of the Faculty of Electrical Engineering, Computer Science & Mathematics
at the University of Paderborn, Germany. In 2018, I co-founded metamorphosis, which is a spin-off startup developing AI-based technologies for computer-assisted musculoskeletal surgery, and I have
since served as its Chief Executive Officer.
I have received fellowships from the State of Bavaria, the Studienstiftung des deutschen Volkes (German National Academic Foundation), and the Deutsche Forschungsgemeinschaft (German Research
From 2008 until 2012, I was an Associate Editor of the IEEE Transactions on Signal Processing, from 2010 until 2014 a Senior Area Editor for the IEEE Transactions on Signal Processing, and from 2015
to 2018 an Associate Editor for the IEEE Signal Processing Letters. I was the General Chair of the 2018 IEEE Statistical Signal Processing Workshop in Freiburg, Germany. From 2009 until 2014, I was a
member of the IEEE Technical Committee on Machine Learning for Signal Processing, and from 2016 until 2021, a member of the IEEE Technical Committee on Signal Processing Theory and Methods. I am a
Past Chair (2019-20) and Vice Chair (2021-22) of the Steering Committee of the IEEE Signal Processing Society’s (SPS) Data Science Initiative. | {"url":"https://sst-group.org/team/peter-schreier/","timestamp":"2024-11-01T23:51:59Z","content_type":"text/html","content_length":"148702","record_id":"<urn:uuid:0d211e70-1d2f-406a-815b-7ea7bc8cfcf1>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00188.warc.gz"} |
Illustrative Mathematics
Exponent Experimentation 1
Alignments to Content Standards: 6.EE.A.1
Decide whether each equation is true or false, and explain how you know.
1. $2^4=2\cdot 4$
2. $3+3+3+3+3=3^5$
3. $5^3=5\cdot5\cdot5$
4. $2^3=3^2$
5. $16^1=8^2$
6. $(1+3)^2=1^2+3^2$
7. $2\cdot2\cdot2\cdot3\cdot3\cdot3=6^3$
IM Commentary
The purpose of this task is to give students experience working with exponential expressions and to promote making use of structure (MP7) to compare exponential expressions. To this end, encourage
students to rewrite expressions in a different form rather than evaluate them to a single number. This may be best accomplished with a demonstration before students begin the task, like
Is $4^2=2^3$ true? Well, let's see. I can rewrite each side like this: $$4\cdot4=2\cdot2\cdot2$$ Then I can replace one of those $2\cdot2$'s with a $4$, like this: $$4\cdot4=4\cdot2.$$ Now I can
tell that this equation is not true.
For students who are accustomed to viewing the = sign as a directive that means "perform an operation," tasks like these are essential to shifting their understanding of the meaning of the = sign to
one that supports work in algebra, namely, "The expressions on either side have the same value."
1. $2^4=2\cdot 4$ is false because it says $2\cdot2\cdot2\cdot2=2\cdot4$ or $16=8$.
2. $3+3+3+3+3=3^5$ is false because it says $3+3+3+3+3=3\cdot3\cdot3\cdot3\cdot3$ or $15=243$.
3. $5^3=5\cdot5\cdot5$ is true because it says $5\cdot5\cdot5=5\cdot5\cdot5$ or $125=125$.
4. $2^3=3^2$ is false because it says $2\cdot2\cdot2=3\cdot3$ or $8=9$.
5. $16^1=8^2$ is false because it says $16=8\cdot8$ or $16=64$.
6. $(1+3)^2=1^2+3^2$ is false because it says $4^2=1+9$ or $16=10$.
7. $2\cdot2\cdot2\cdot3\cdot3\cdot3=6^3$ is true. We can use the meaning of exponents and the commutative and associative properties of multiplication to show this.
2\cdot2\cdot2\cdot3\cdot3\cdot3 &= 6^3\\
2\cdot3\cdot2\cdot3\cdot2\cdot3 &= 6^3\\
(2\cdot3)\cdot(2\cdot3)\cdot(2\cdot3) &= 6^3\\
6\cdot6\cdot6 &= 6^3\\
6^3 &= 6^3
Exponent Experimentation 1
Decide whether each equation is true or false, and explain how you know.
1. $2^4=2\cdot 4$
2. $3+3+3+3+3=3^5$
3. $5^3=5\cdot5\cdot5$
4. $2^3=3^2$
5. $16^1=8^2$
6. $(1+3)^2=1^2+3^2$
7. $2\cdot2\cdot2\cdot3\cdot3\cdot3=6^3$ | {"url":"https://tasks.illustrativemathematics.org/content-standards/6/EE/A/1/tasks/2225","timestamp":"2024-11-08T05:14:53Z","content_type":"text/html","content_length":"25937","record_id":"<urn:uuid:ee3fc87d-564a-4dcb-92b9-5789a6ef1038>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00654.warc.gz"} |
Hols yet again
Hi Everyone, i shall be off on leave as from tomorrow, leaving you all hopefully with maybe some decent weather , for those of you in the Suffolk area, especially around the ipswich area , its tough
news ,cus thats where i,m heading , and where ever i go , so does the rain ehehehheheheh .
Image of the week will be carried out by Phil this weekend.
So until a week on Sunday , i wish you Farewell , see you all on my return .
Hope you have a good time rog
come back to us all refreshed and full of vigour
Have a good one Rog :sunny:
Take your wellies mate, you might need 'em.
Have a nice time as well.
Kaptain Klevtsov
Enjoy yourself Rog
No clouds down here please, I need to work out this autoguiding lark!
Have a good time, Rog!
I've never known anyone have so many holidays!!!!
Have ANOTHER good one
Where you staying? Drop in for a cuppa if you want??
Hope you enjoy your leave?
Hi Bill
Orwell meadows camp site , meadow 1, 4th static on the left,
do u have sugar ehhehe
may see ya then
I've never known anyone have so many holidays!!!!
dont know, politicians dont do too bad.
This topic is now archived and is closed to further replies. | {"url":"https://stargazerslounge.com/topic/14395-hols-yet-again/","timestamp":"2024-11-13T22:42:59Z","content_type":"text/html","content_length":"164530","record_id":"<urn:uuid:3233880e-cb80-447b-b757-13c91de8d064>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00616.warc.gz"} |
25 interesting facts about Isaac Newton แ ๐ก MillionFacts ๐ต
25 interesting facts about Isaac Newton
Isaac Newton is widely regarded as one of the greatest scientists of all time. He is known for his contributions to the fields of physics and mathematics, among others. In this article, we’ll explore
some interesting facts about the life and work of this legendary figure.
1. Newton was born prematurely on Christmas Day, 1642, in Woolsthorpe, England.
2. As a child, Newton was a poor student and had little interest in formal education. However, he was fascinated by machines and gadgets, and spent much of his time tinkering with them.
3. When Newton was 17, his mother pulled him out of school to help run the family farm. However, he showed little interest in farming, and instead spent his time reading and conducting scientific
4. Newton enrolled at Trinity College, Cambridge in 1661. He initially intended to study law, but his interests quickly turned to mathematics and physics.
5. In 1665, the bubonic plague swept through Cambridge, and the university was closed. During this time, Newton returned to his family’s farm, where he spent nearly two years developing his ideas on
calculus, optics, and gravity.
6. In 1687, Newton published his most famous work, Philosophiรฆ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), which laid the foundations for classical mechanics
and revolutionized the field of physics.
7. Newton’s three laws of motion โ the law of inertia, the relationship between force and acceleration, and the principle of action and reaction โ are still taught in schools today.
8. Newton’s work on optics also revolutionized the field, and he is credited with discovering that white light is made up of a spectrum of colors.
9. Newton was appointed warden of the Royal Mint in 1696, and later served as Master of the Mint. During his time at the Mint, he introduced a number of reforms that helped to combat counterfeiting.
10. Newton was knighted by Queen Anne in 1705, and was the first scientist to be given this honour.
11. Newton was known for his prickly personality, and had a number of feuds with other scientists of the day. One of his most famous disputes was with German mathematician Gottfried Leibniz over who
had developed calculus first.
12. Newton suffered from depression and several nervous breakdowns throughout his life, and was known to have engaged in some unusual practices, such as self-experimentation with mercury.
13. Newton died on March 31, 1727, at the age of 84. He was buried in Westminster Abbey, and his tomb bears the Latin inscription “Here lies Isaac Newton, knight, who by a strength of mind almost
divine, and mathematical principles peculiarly his own, explored the course and figures of the planets, the paths of comets, the tides of the sea, and the dissimilarities in rays of light.”
14. Newton’s legacy continues to inspire scientists and thinkers around the world, and his contributions to the fields of physics and mathematics are still studied and celebrated today.
Isaac Newton was a remarkable figure in history who made many contributions to the fields of mathematics, physics, and astronomy. From his development of the laws of motion and universal gravitation
to his invention of the reflecting telescope and calculus, Newton’s impact on science and technology cannot be overstated. Beyond his intellectual accomplishments, Newton was also a complex
individual with a wide range of interests, including alchemy, theology, and even the occult. Although he lived over three centuries ago, his ideas and discoveries continue to shape our understanding
of the universe today.
Isaac Newton is widely regarded as one of the greatest scientists of all time. He is known for his contributions to the fields of physics and mathematics, among others. In this article, we’ll
explore some interesting facts about the life and work of this legendary figure. Isaac Newton was a… | {"url":"https://millionfacts.co.uk/post/25-interesting-facts-about-isaac-newton","timestamp":"2024-11-11T04:13:07Z","content_type":"text/html","content_length":"53492","record_id":"<urn:uuid:857eef48-f566-46bb-a3b6-62acf0114e94>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00879.warc.gz"} |
89 research outputs found
The maximality property was introduced in [9] in orthomodular posets as a common generalization of orthomodular lattices and orthocomplete orthomodular posets. We show that various conditions used in
the theory of e ect algebras are stronger than the maximality property, clear up the connections between them and show some consequences of these conditions. In particular, we prove that a Jauch
{Piron e ect algebra with a countable unital set of states is an orthomodular lattice and that a unital set of Jauch{Piron states on an e ect algebra with the maximality property is strongly order
In this thesis we study certain mathematical aspects of evolution. The two primary forces that drive an evolutionary process are mutation and selection. Mutation generates new variants in a
population. Selection chooses among the variants depending on the reproductive rates of individuals. Evolutionary processes are intrinsically random – a new mutation that is initially present in the
population at low frequency can go extinct, even if it confers a reproductive advantage. The overall rate of evolution is largely determined by two quantities: the probability that an invading
advantageous mutation spreads through the population (called fixation probability) and the time until it does so (called fixation time). Both those quantities crucially depend not only on the
strength of the invading mutation but also on the population structure. In this thesis, we aim to understand how the underlying population structure affects the overall rate of evolution.
Specifically, we study population structures that increase the fixation probability of advantageous mutants (called amplifiers of selection). Broadly speaking, our results are of three different
types: We present various strong amplifiers, we identify regimes under which only limited amplification is feasible, and we propose population structures that provide different tradeoffs between high
fixation probability and short fixation time
summary:We present three results stating when a concrete (=set-representable) quantum logic with covering properties (generalization of compatibility) has to be a Boolean algebra. These results
complete and generalize some previous results [3, 5] and answer partiallz a question posed in [2]
Balanced knockout tournaments are ubiquitous in sports competitions and are also used in decision-making and elections. The traditional computational question, that asks to compute a draw (optimal
draw) that maximizes the winning probability for a distinguished player, has received a lot of attention. Previous works consider the problem where the pairwise winning probabilities are known
precisely, while we study how robust is the winning probability with respect to small errors in the pairwise winning probabilities. First, we present several illuminating examples to establish: (a)
~there exist deterministic tournaments (where the pairwise winning probabilities are~0 or~1) where one optimal draw is much more robust than the other; and (b)~in general, there exist tournaments
with slightly suboptimal draws that are more robust than all the optimal draws. The above examples motivate the study of the computational problem of robust draws that guarantee a specified winning
probability. Second, we present a polynomial-time algorithm for approximating the robustness of a draw for sufficiently small errors in pairwise winning probabilities, and obtain that the stated
computational problem is NP-complete. We also show that two natural cases of deterministic tournaments where the optimal draw could be computed in polynomial time also admit polynomial-time
algorithms to compute robust optimal draws
We characterize atomistic effect algebras, prove that a weakly orthocomplete Archimedean atomic effect algebra is orthoatomistic and present an example of an orthoatomistic orthomodular poset that is
not weakly orthocomplete.Comment: 6 page
We consider the modified Moran process on graphs to study the spread of genetic and cultural mutations on structured populations. An initial mutant arises either spontaneously (aka \emph{uniform
initialization}), or during reproduction (aka \emph{temperature initialization}) in a population of $n$ individuals, and has a fixed fitness advantage $r>1$ over the residents of the population. The
fixation probability is the probability that the mutant takes over the entire population. Graphs that ensure fixation probability of~1 in the limit of infinite populations are called \emph{strong
amplifiers}. Previously, only a few examples of strong amplifiers were known for uniform initialization, whereas no strong amplifiers were known for temperature initialization. In this work, we study
necessary and sufficient conditions for strong amplification, and prove negative and positive results. We show that for temperature initialization, graphs that are unweighted and/or self-loop-free
have fixation probability upper-bounded by $1-1/f(r)$, where $f(r)$ is a function linear in $r$. Similarly, we show that for uniform initialization, bounded-degree graphs that are unweighted and/or
self-loop-free have fixation probability upper-bounded by $1-1/g(r,c)$, where $c$ is the degree bound and $g(r,c)$ a function linear in $r$. Our main positive result complements these negative
results, and is as follows: every family of undirected graphs with (i)~self loops and (ii)~diameter bounded by $n^{1-\epsilon}$, for some fixed $\epsilon>0$, can be assigned weights that makes it a
strong amplifier, both for uniform and temperature initialization | {"url":"https://core.ac.uk/search/?q=author%3A(Tkadlec%2C%20Josef)","timestamp":"2024-11-03T03:44:47Z","content_type":"text/html","content_length":"118808","record_id":"<urn:uuid:ce9682fa-b3e6-4e3d-8851-74eda8d1b724>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00407.warc.gz"} |
Solved: WPC 300 Quiz 5 Score for this attempt: 20 out of | Answerswave.com
Answered Questions
WPC 300
/ wpc-300-quiz-5-score-for-this-attempt-20-out-of-20-submitted-apr-15-at-10-10am-q-904
(Solved): WPC 300 Quiz 5 Score for this attempt: 20 out of 20 Submitted Apr 15 at 10:10am...
WPC 300 Quiz 5
Score for this attempt: 20 out of 20
Submitted Apr 15 at 10:10am
This attempt took 11 minutes.
Question 1
2 / 2 pts
A market analyst is developing a regression model to predict monthly household expenditures on groceries as a function of family size, household income, and household neighborhood (urban, suburban,
and rural). The "neighborhood" variable in this model is ________.
• A continuous variable
• A dependent variable
• A qualitative variable
• An independent variable
Question 2
2 / 2 pts
The unexplained variance in the regression analysis is also known as:
• Predicted variance
• Residual variance
• Regression variance
• Total variance
Question 3
2 / 2 pts
Which of the following statement is true based on the following regression equation?
IQ = 4.0 + Reading Label * 5.6
• A unit point change in IQ will result in 5.6 point increase in reading label
• A unit point change in reading label will increase IQ by 5.6 point
• A unit point change in IQ will result in 9.6 point increase in reading label
• Reading label is not a good predictor of IQ
Question 4
2 / 2 pts
The value of R-Squared always falls between ___________ and ___________ inclusive.
• -infinity to + infinity
• -1 and +1
• 0 and -1
• 0 and 1
Question 5
2 / 2 pts
A correlation coefficient between "college entrance exam" grades and scholastic achievement was found to be -1.08. On the basis of this, you would tell the university that:
• They should hire a new statistician.
• The exam is a poor predictor of success.
• Students who do best on this exam will make the worst students.
• The entrance exam is a good predictor of success.
Feedback: -1< r<1
Question 6
2 / 2 pts
You need to find out which customer is likely to buy your product. A sample data is available from your current customer base. Which of the following analysis method will be appropriate for this
• Linear regression
• Multiple linear regression
• Logistic regression
• Clustering
Question 7
2 / 2 pts
Which of the following is true about multi-collinearity?
• The effect of a dependent variable on another becomes difficult to isolate.
• P-value reduces significantly leading to rejection of the null hypothesis.
• Regression coefficients become clearer and are easier to interpret
• Is measured using the statistical variance inflation factor (VIF)
Question 8
2 / 2 pts
The correlation coefficient between the age of an auto and the money spent to repair is 0.9. Which of the following statement is true?
• 81% of the variation in the money spent on repairs is explained by the age of the auto
• 81% of money spent on repairs is explained by the age of an auto
• 90% of the money spent on repair is explained by the age of an auto
• 90% of the repair cost will be explained by the age of an auto
Feedback: R=0.9, R-squared = 0.81. Remember the definition of R-squared.
Question 9
2 / 2 pts
A manager wishes to predict the annual cost (y) of an automobile based on the number of miles (x) driven. The following model was developed: y = $1500 + 0.36x. If a car is driven 15000 miles, the
predicted cost of the car is:
Question 10
2 / 2 pts
Which of the following assumptions is not true for multiple linear regression?
• Residuals are normally distributed
• Presence of multi-collinearity effect
• Relationship between dependent and independent variables should be linear
• No correlations between the independent variables
We have an Answer from Expert View Expert Answer | {"url":"https://www.answerswave.com/ExpertAnswers/wpc-300-quiz-5-score-for-this-attempt-20-out-of-20-submitted-apr-15-at-10-10am-q-904","timestamp":"2024-11-10T02:20:08Z","content_type":"text/html","content_length":"33539","record_id":"<urn:uuid:36ab1db9-ea05-4627-b145-f1247d70805a>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00667.warc.gz"} |
A General Design Procedure for Bandpass Filters Derived from Low Pass Prototype Elements: Part I
K.V. Puglia
M/A-COM Inc.
Lowell, MA
Bandpass filters serve a variety of functions in communication, radar and instrumentation subsystems. Of the available techniques for the design of bandpass filters, those techniques based upon the
low pass elements of a prototype filter have yielded successful results in a wide range of applications. The low pass prototype elements are the normalized values of the circuit components of a
filter that have been synthesized for a unique passband response, and in some cases, a unique out-of-band response. The low pass prototype elements are available to the designer in a number of
tabulated sources ^1,2,3 and are generally given in a normalized format, that is, mathematically related to a parameter of the filter prototype.
This article presents a general design procedure for bandpass filters derived from low pass prototype filters, which have been synthesized for a unique filter parameter. A number of illustrated
examples are offered to validate the design procedure.
Low pass prototype filters are lumped element networks that have been synthesized to provide a desired filter transfer function. The element values have been normalized with respect to one or more
filter design parameters (cutoff frequency, for example) to offer the greatest flexibility, ease of use and tabulation. The elements of the low pass prototype filter are the capacitors and inductors
of the ladder networks of the synthesized filter networks as shown in Figure 1. This diagram also depicts the two possible implementations of the low pass prototype filter topologies. In both cases,
the network transfer function is
T(s) = e[2](s) / e[1](s)
s = s + jw , the Laplace complex frequency variable
Clearly, the transfer function, T(s), is a polynomial of order n, where n is the number of elements of the low pass filter prototype.
The illustrated circuit topologies represent a filter prototype containing an odd number of circuit elements. To represent an even number of elements of the prototype filter, simply remove the last
capacitor or inductor of the ladder network.
For purposes of illustration, an example representing a Chebyshev filter is offered. The power transfer function of the Chebyshev filter may be represented by
T(f') = 10log{1 + e cos^2 [ncos^-1 (f')]}
for f' > 1.0
T(f') = 10log{1 + e cosh^2 [ncosh^-1 (f')]}
for f' > 1.0
ε = log^-1(r[dB ]/ 10) – 1
r[dB ] = inband ripple factor in decibels
These equations represent the power transfer function of the Chebyshev low pass prototype filter with normalized filter cutoff frequency f' of 1.0 Hz. A graphical representation of the power transfer
function of the Chebyshev low pass prototype filter is shown in Figure 2.
The low pass prototype filter parameters for the low pass Chebyshev filter example are
n = 5
R'[0 ] = 1.0 W
r[dB ] = 0.5
A schematic representation of the prototype Chebyshev filter is shown in Figure 3. The prototype elements are from Matthaei, Young and Jones^1 where the normalized cutoff frequency is given in the
radian format w '[1 ] = 1.0 = 2p f'[1] .
If this five-section prototype filter were constructed from available tables of elements and a circuit simulation performed, the transfer function would be exactly as represented in the schematic. To
construct the filter at another frequency (1.0 GHz, for example) and circuit impedance level (R[0 ] = 50 W ), the element values must be adjusted (de-normalized) in accordance with
In addition to the tabulated data of low pass prototype filter elements, the values may be computed via execution of the equations found in Matthaei, et al.^1 , and repeated in Appendix A for
The schematic of the filter, which was derived from Chebyshev, the low pass prototype elements and the associated frequency response are shown in Figure 4. Note that the 0.5 dB in-band ripple results
in a return loss of -10 dB as expected.
A low pass filter may be converted to a bandpass filter by employing a suitable mapping function. A mapping function is simply a mathematical change of variables such that a transfer function may be
shifted in frequency. The mapping function may be intuitively or mathematically derived. A known low pass to bandpass mapping function may be illustrated mathematically as
f[0] = √f[1] f[2 ]
D f = f[2] – f[1]
and f[0] , f[1] and f[2] represent the center, lower cutoff and higher cutoff frequencies of the corresponding bandpass filter, respectively. If the substitution of variables is made within the
Chebyshev power transfer function, the power transfer function of the corresponding bandpass filter may be determined as shown in Figure 5.
The schematic diagram of the bandpass filter, which was derived from the low pass prototype filter via the introduction of complementary elements and producing shunt and series resonators, is shown
in Figure 6. This is a basic low pass to bandpass transformation, and unfortunately sometimes leads to component values, which are not readily available or have excessive loss. As described later,
the mapping function need not be considered as part of the bandpass filter design procedure. It is presented here as a supplement to the filter theory.
It bears repeating that the low pass prototype filter elements, that is, the g-values, are the result of network synthesis techniques to produce a desired characteristic of the prototype filter
transfer function. These desired characteristics might include a flat amplitude response, maximum out-of-band rejection, linear phase response, Gaussian or other amplitude response, minimum time
sidelobes and matched signal filters.
RF and microwave resonators are lumped element networks or distributed circuit structures that exhibit minimum or maximum real impedance at a single frequency or at multiple frequencies. The resonant
frequency f[0] is the frequency at which the input impedance or admittance is real. The resonant frequency may be further defined in terms of a series or shunt mode of resonance; the series mode is
associated with small values of input resistance at the resonant frequency, while the shunt mode is associated with large values of resistance at the resonant frequency. Some typical lumped and
distributed resonators are shown in Figure 7.
Resonators may be characterized by their unloaded quality factor Q[u] , which is the ratio of the energy stored to the energy dissipated per cycle of the resonant frequency. Resonators are also
characterized with respect to their reactance a or susceptance b slope parameters, which are defined, respectively, as
These are important resonator parameters because they influence Q[u] and the coupling factor between resonators in multiple resonator filters. Table 1 provides the reactance and susceptance slope
parameters of some common lumped element and distributed resonators.
Q[u] may also be defined in terms of the reactance or susceptance slope parameter as
Q[u ]= α / R[se] = ßR[sh]
R[se] = resonator series resistance
R[sh] = resonator shunt resistance
Together these resistances represent the resonator loss. The bandpass filter design examples will illustrate the utility of the slope parameters.
In many bandpass filter applications, particularly those applications where the filter is deployed at the front end of a receiver, it is important to know the Q[u] for the resonators in order to
accurately estimate the insertion loss of the filter. The insertion loss of a single transmission resonator can be mathematically represented as
At f = f[0] , this equation may further reduce to
Solving for Q[u] , yields
Therefore, a measurement of the single resonator insertion loss at the resonant frequency L(f[0] ) and the -3 dB bandwidth D f is sufficient to accurately determine Q[u] of any resonant structure.
The loaded quality factor Q[l] may be determined from a measurement of the resonant frequency and the -3 dB bandwidth from the equation
Q[1] = f[0] / Df
D f = -3 dB bandwidth
This measurement technique may also be employed to compare the quality factors (Q[u] ) of different types of resonant structures or as a method of comparing various plating or manufacturing
techniques for the filter if insertion loss is a critical parameter.
Consider an example. A PCS1900 transmit filter was required with minimum insertion loss and minimum size as critical design parameters. Rectangular, coaxial l /8 resonators were determined to offer
the minimum volume in an eight-resonator filter consistent with the maximum insertion loss requirement of less than 1.0 dB at the center frequency of 1960 MHz. A single resonator was constructed of
the type anticipated to be used within the filter. The single resonator is shown in Figure 8.
A single resonator structure was fabricated and plated with silver in order to obtain the maximum Q[u] . The resonant structure was tuned to the desired center frequency and the insertion loss L(f[0]
) and -3 dB bandwidth were measured. The measurement data is shown in Figure 9 where
L(f[0 ] = 1.950 GHz) = 0.533 dB
D f = 14.50 MHz
Q[u] 2250.
In executing the measurement, three notes of caution are required in the interest of accuracy: The coupling probes to the cavity should be equal and minimized to avoid load and source resistance
across the resonator; the source and load SWRs should be kept low; and the input SWR at f[0] should be minimized to avoid mismatch loss.
The equivalent circuit of the single resonator is shown in Figure 10. Note that the circuit element, which represents the resonator loss R[sh] , has been included. The value of R[sh] may be
determined with the aid of the susceptance slope parameter b from
Q[u ] = b R[sh]
ß =
^Y[0]⁄[2 ][(cot(θ[0]) + θ[0]csc2(θ[0])] = 0.01836
for θ[0 ]= ^p⁄[4] and Y[0 ]= ^1⁄[70] Mhos
R[sh ] = 122.5 kW
This measurement technique for estimating the value of Q[u] is completely general and applies to lumped element and distributed resonators.
Resonator coupling represents one of the most significant factors affecting filter performance. There are several methods to couple resonators. For ease of manufacturing and tuning, a common
resonator type and coupling method is generally preferable. Matthaei^1 proposes what have been termed J (admittance) and K (impedance) inverters both to permit a common type of resonator and to serve
as coupling elements for the resonators.
The J inverters may be represented as the admittance of the element or the value of the characteristic admittance of a quarter-wavelength line in the equivalent circuit that couples the resonators.
Similarly, the K inverters may be represented as the impedance of the element or the value of the characteristic impedance of a quarter-wavelength line that couples the resonators. This permits the
general expression of the coupling between resonators to be mathematically written as
for shunt-type and series-type resonators, respectively, where the coupling between the i^th and the i^th+1 resonators is represented by k[i,i+1] .
A similar approach to the general design of bandpass filters employing common types of resonators proposes specific coupling elements in the case of lumped resonator bandpass filters or specific
proximity methods of coupling in the case of distributed resonator bandpass filters. To illustrate, consider the coupled p -type of L-C resonators and the coupled l /8 transmission line distributed
resonators shown in Figure 11.
In the case of the coupled l -type L-C resonators, a series capacitor is inserted between the resonators to perform the coupling function, in which case the coupling coefficient is
In the case of the coupled l /8 transmission line resonators, the equivalent circuit shown in Figure 12 is useful in determining the coupling coefficient. The coupling coefficient may be determined
from the capacitive matrix parameters associated with the coupled lines, that is, the capacitance per unit length to ground C[g] and the mutual capacitance per unit length between the conductors C[m]
, where V is the velocity in the dielectric medium. The coupling coefficient may be written as
For the special case where θ[0 ] = p /4, the coupling may be written as
Another very popular type of resonator, which is frequently used in microwave bandpass filters, is the quarter-wavelength resonator. Figure 13 illustrates the coupling of symmetrical l /4 resonators
and the equivalent circuit. Note that l /4 resonators must be grounded at opposite ends to prevent the null coupling condition caused by cancellation of the electric and magnetic field modes.
The coupling coefficient for this configuration may be written as
For a given coupled line geometry, the l /4 lines offer closer coupling than the comb-line configuration.
The input, output and adjacent resonator coupling in a multi-element bandpass filter are the parameters that determine the amplitude, phase and SWR over the passband of the filter. This statement
understates the importance of resonator coupling to the bandpass filter parameters.
Recall that the elements of the low pass prototype filter, from which the bandpass filter is derived, determine completely the characteristics of the resulting filter. This fact will become evident
when the coupling between resonators is disclosed to be a function only of the fractional bandwidth and the low pass filter prototype elements. Fortunately, a measurement technique is available to
verify the coupling values of symmetrical resonators, and may also be utilized in multi-resonator filters. That measurement technique will now be explored.
The amplitude response of any pair of symmetrical resonators may be represented by^1
In the equation, k is the coupling coefficient between the symmetrical resonators, Q[u] is the unloaded quality factor of each resonator and Q[e] is the external quality factor. The external quality
factor is defined to differentiate the source and load coupling and loss from the losses associated with the individual resonators (Q[u] ).
If the overcoupled condition is satisfied such that
it is possible to determine the resonator coupling coefficient from
where f[0] , f[a] and f[b] are subsequently defined.
The utility of the equations will be demonstrated by two examples. Consider the symmetrical, lumped element resonators in the schematic shown in Figure 14 where two, p -type, L-C resonators are
coupled by the capacitor C[p ] , and the external source and load are coupled to the respective resonators by capacitor C[e] . Note also the Q[u] of each resonator is
The following variables have been assigned
With the assigned variables, the amplitude response is shown in Figure 15. The two peaks in the amplitude response correspond to the frequencies f[a] and f[b] . Using the assigned variables, the
calculated coupling coefficient is k[c ] = 0.05033.
A second illustrative example using coupled l /8 resonators is shown in Figure 16. An equivalent circuit representation of the coupled lines, external coupling and resonator loss is required in order
to quantify the coupling factor.
The variables have been assigned as
With the assigned variables, the amplitude response is shown in Figure 17. The two peaks in the amplitude response correspond to the frequencies f[a] and f[b] . Using the assigned variables, the
calculated coupling coefficient is k[c ] = 0.01999. Part II explores how to use the preceding principles and data to design bandpass filters using a variety of lumped and distributed element
The principal reference for the content of this article is the work of Matthaei, Young and Jones.^1 Many of the concepts within this reference have been investigated and interpreted in order to
provide a greater intuitive understanding of the bandpass filter design process. This text is strongly recommended for those having little familiarity with this work.
1. G.L. Matthaei, L. Young and E.M.T. Jones, Microwave Filters, Impedance-matching Networks and Coupling Structures, McGraw-Hill, New York, 1964.
2. E.G. Cristal, "Coupled Circular Cylindrical Rods Between Parallel Ground Planes," IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-12, July 1964, pp. 428-439.
3. W.J. Getsinger, "Coupled Rectangular Bars Between Parallel Plates," IRE Transactions on Microwave Theory and Techniques, Vol. MTT-10, January 1962, pp. 65?72.
4. A. Zverev, Handbook of Filter Synthesis, John Wiley and Sons, New York, 1967.
5. G.L. Matthaei, "Interdigital Band Pass Filters," IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-10, No. 6, November 1962.
6. G.L. Matthaei, "Comb-line Bandpass Filters of Narrow or Moderate Bandwidth," Microwave Journal, Vol. 6, No. 8, August 1963.
7. S.B. Cohn, "Parallel-coupled Transmission-line Resonator Filters," IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-6, No. 2, April 1958.
8. R.M. Kurzrok, "Design of Comb-line Bandpass Filters," IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-14, July 1966, pp. 351?353.
9. M. Dishal, "A Simple Design Procedure for Small Percentage Round Rod Interdigital Filters," IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-13, September 1965, pp. 696?698.
10. S.B. Cohn, "Dissipation Loss in Multiple-coupled Resonator Filters," Proceedings of the IRE, Vol. 47, No. 8, August 1959, pp. 1342?1348.
Kenneth V. Puglia holds the title of Distinguished Fellow of Technology at the M/A-COM division of Tyco Electronics. He received the degrees of BSEE (1965) and MSEE (1971) from the University of
Massachusetts and Northeastern University, respectively. He has worked in the field of microwave and millimeter-wave technology for 35 years, and has authored or co-authored over 30 technical papers
in the field of microwave and millimeter-wave subsystems. Since joining M/A-COM in 1971, he has designed several microwave components and subsystems for a variety of signal generation and processing
applications in the field of radar and communications systems. As part of a European assignment, he developed a high resolution radar sensor for a number of industrial and commercial applications.
This sensor features the ability to determine object range, bearing and normal velocity in a multi-object, multi-sensor environment using very low transmit power. Puglia has been a member of the
IEEE, Professional Group on Microwave Theory and Techniques since 1965. | {"url":"https://www.microwavejournal.com/articles/3102-a-general-design-procedure-for-bandpass-filters-derived-from-low-pass-prototype-elements-part-i","timestamp":"2024-11-06T14:28:38Z","content_type":"text/html","content_length":"87990","record_id":"<urn:uuid:db26b1e1-639a-462a-9f7a-04f9f636cffc>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00827.warc.gz"} |
Kurtosis - Types of Kurtosis | Business Statistics
Kurtosis – Types of Kurtosis | Business Statistics
➦ Even if we know about the measures of central tendency, dispersion, and skewness, we cannot fully comprehend a distribution.
➦ For a complete understanding of the shape of the distribution, we should also know another measure called Kurtosis.
➦ It is called the “convexity of a curve” by Prof. Karl Pearson. It measures the flatness of distributions.
➦ Kurtosis is another measure of the shape of a frequency curve. It is a Greek word, which means bulginess.
➦ While skewness signifies the extent of asymmetry, kurtosis measures the degree of peakedness of a frequency distribution.
➦ Kurtosis is a statistical measure of how much a distribution’s tails differ from the tails of a normal distribution.
➦ As such, kurtosis identifies whether a distribution features extreme values in the tails. Along with skewness, kurtosis is also used as a descriptive statistic to describe data distribution.
➦ However, the two concepts should not be confused with each other. Skewness is a measure of distribution symmetry, while kurtosis is a measure of tail heaviness.
➦ Financial risk is measured by kurtosis in finance. When the kurtosis is large, there is a high probability of extremely large and extremely small returns for an investment.
➦ A small kurtosis, on the other hand, indicates a low risk level because the probability of extreme returns is relatively low.
Excess Kurtosis
➦ An excess kurtosis metric compares a distribution’s kurtosis with the normal kurtosis. A normal distribution has a kurtosis of 3.
➦ So, it is easy to check whether there is excessive kurtosis by using the following formula:
Excess Kurtosis = Kurtosis – 3
➦ Excess kurtosis is a statistical measure that quantifies the degree to which a probability distribution deviates from the normal distribution in terms of its peakedness and tail behavior.
➦ In business statistics, excess kurtosis plays a crucial role in understanding the shape and characteristics of data distributions, particularly in financial analysis, risk management, and market
Types of Kurtosis
➦ The types of kurtosis are determined by the excess kurtosis of a particular distribution. The excess kurtosis can take positive or negative values, as well as values close to zero.
1) Mesokurtic
• [normal in shape]
• When the kurtosis = 0
➦ A Mesokurtic distribution will have a kurtosis of zero or close to zero. Therefore, if the data has a normal distribution, it also has a Mesokurtic distribution.
2) Leptokurtic
• [high and thin]
• When the kurtosis > 0, there are high frequencies in only a small part of the curve (i.e, the curve is more peaked)
➦ A positive excess kurtosis is indicated by leptokurtic. It is evident that there are large outliers on either side of the leptokurtic distribution.
➦ Leptokurtic distributions are prone to extreme values on either side of an investment return. A risky investment is one whose returns follow a leptokurtic distribution.
3) Platykurtic
• [flat and spread out]
• When the kurtosis < 0, the frequencies throughout the curve are closer to be equal (i.e., the curve is more flat and wide)
➦ There is a negative excess kurtosis with a Platykurtic distribution. According to the kurtosis, the distribution is flat.
➦ A flat tail indicates that an outlier has been found in the distribution. Investment returns are more likely to have a Platykurtic distribution in the financial context, since there is a small
chance that the investment will experience extreme returns.
• Kenton, W. (2023, October 1). Kurtosis Definition, types, and importance. Investopedia. https://www.investopedia.com/terms/k/kurtosis.asp#:~:text=
• Kurtosis: Definition, leptokurtic, platykurtic – Statistics How to. (2024, January 19). Statistics How To. https://www.statisticshowto.com/probability-and-statistics/statistics-definitions/
Related Posts
1 thought on “Kurtosis – Types of Kurtosis | Business Statistics”
1. It has been shown conclusively that kurtosis measures tailweight only, and nothing about the peak. For example, the beta(.5,1) distribution is infinitely peaked but has very low kurtosis. And the
.0001Cauchy + .9999U(0,1) distribution appears perfectly flat over 99.99% of the observable data, but has infinite kurtosis. Have a look at more current references.
Leave a Comment | {"url":"https://www.managementnote.com/kurtosis-types-of-kurtosis-business-statistics/","timestamp":"2024-11-05T03:31:33Z","content_type":"text/html","content_length":"210270","record_id":"<urn:uuid:5449461a-e2b0-41c7-913f-0cc99fb91607>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00667.warc.gz"} |
Lesson 2
Representations of Equal Groups of Fractions
Warm-up: Number Talk: Three, Six, Nine, Twelve (10 minutes)
This Number Talk encourages students to use their knowledge of multiplication facts, properties of operations, and the structure of the given expressions to mentally solve problems. The reasoning
elicited here will be helpful in upcoming lessons as students find products of whole numbers and non-unit fractions (such as \(3 \times \frac{6}{10}\) or \(6 \times \frac{9}{4}\)).
• Display one expression.
• “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time
• Record answers and strategy.
• Keep expressions and work displayed.
• Repeat with each expression.
Student Facing
Find the value of each expression mentally.
• \(3 \times 6\)
• \(3 \times 9\)
• \(6 \times 9\)
• \(12 \times 9\)
Activity Synthesis
• “What did you notice about the factors in all of the expressions?” (They are all multiples of 3.)
• “Did noticing that all the factors are multiples of 3 help you find the values?” (Sample responses:
□ Yes, I was able to think of “3 more groups of something.”
□ Yes, it helped me see how the factors were related, which helped me reason about the products.
□ No, it didn't, but it made me think that the values would be multiples of 3 as well.)
Activity 1: Card Sort: Expressions and Diagrams (25 minutes)
In this activity, students interpret multiplication expressions and diagrams as the number of groups and amount in each group and match representations of the same quantity. They then use their
insight from the matching activity to generate diagrams for expressions without a match and to find their values (MP2).
MLR8 Discussion Supports. Students should take turns finding a match and explaining their reasoning to their partner. Display the following sentence frames for all to see: “I noticed _____ , so I
matched . . .” Encourage students to challenge each other when they disagree.
Advances: Speaking, Conversing
Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. Give students a subset of the cards to start with and introduce the remaining cards once students have
completed their initial set of matches.
Supports accessibility for: Organization, Conceptual Processing
Required Preparation
• Create a set of cards from the blackline master for each group of 2.
• Groups of 2
• Give each group a set of cards from the blackline master.
• “Work with your partner to match each expression to a diagram that represents the same equal-group situation and the same amount.”
• “Be prepared to explain how you know the two representations belong together.”
• 5 minutes: partner work time
• Monitor for students who reason about the number of groups and amount in each group as they match.
• Pause for a discussion. Select students to share their matches and reasoning.
• Highlight reasoning that clearly connects one factor in the expression to the number of groups and the other factor to the size of each group.
• “Now you will complete an unfinished diagram for \(7 \times \frac{1}{8}\) and then draw a new diagram for an expression without a match.”
• 5 minutes: independent work time
Student Facing
Your teacher will give you a set of cards with expressions and diagrams.
1. Match each expression with a diagram that represents the same quantity.
2. Record each expression without a match.
3. Han started drawing a diagram to represent \(7 \times \frac{1}{8}\) and did not finish. Complete his diagram. Be prepared to explain your reasoning.
4. Choose one expression that you recorded earlier that didn't have a match.
Draw a diagram that can be represented by the expression. What value do the shaded parts of your diagram represent?
Advancing Student Thinking
If students are not yet matching expressions to appropriate diagrams, consider asking them to compare the diagrams for \(5 \times 3\) and \(5 \times \frac{1}{3}\) and reason about the number of
groups and the size of each group. Consider asking: “How are these alike? How are they different?”
Activity Synthesis
• “What was missing from Han’s diagram? How do you know?” (4 more groups of \(\frac{1}{8}\) were missing, because \(7 \times \frac{1}{8}\) means 7 groups of \(\frac{1}{8}\)and there are only 3 in
Han’s diagram.)
• “If the expression was for 7 groups of \(\frac{1}{3}\) instead of \(\frac{1}{8}\) how would Han’s diagram change?” (Each rectangle representing 1 would have 3 equal parts with 1 shaded.)
• Select students to share the diagrams they drew for the expressions without a match. Ask them to point out the number of groups and size of each group in each diagram.
Activity 2: Different Representations (10 minutes)
This activity prompts students to use their earlier observations to generate a diagram or expression that represents equal groups of unit fractions when one or the other is given. In one of the
problems, only the total quantity (\(\frac{7}{2}\)) is given, so students need to reason in about the number of groups and the size of each group that could lead to this value. Finally, they analyze
two different ways of representing \(4 \times\frac{1}{3}\) with a diagram, which further illustrates that the value of the expression is \(\frac{4}{3}\).
• Groups of 2
• “Turn to a partner and explain what needs to be done to complete the first problem.”
• “Complete the first problem independently. Afterwards, pause for a class discussion.”
• 5 minutes: independent work time
• Pause to discuss the fraction \(\frac{7}{2}\) in the first problem.
• “How did you know what diagram and expression would have the value \(\frac{7}{2}\)?” (Sample response:
□ For the diagram, the numerator, 7, is the number of groups, and the denominator, 2, shows how many parts are in 1 whole.
□ For the expression, I multiplied a whole number and a fraction. The whole number was the same as the number in the numerator of \(\frac{7}{2}\) and the fraction has the same number as the
denominator of \(\frac{7}{2}\).)
• “Work on the last problem with your partner.”
• 5 minutes: partner work time
Student Facing
1. Write a multiplication expression that represents the shaded parts of the diagram. Then, find the value of the expression.
2. Draw a diagram that the expression \(6 \times \frac{1}{3}\) could represent. Then, find the value of the expression.
Expression: \(6 \times \frac{1}{3}\)
3. Draw a diagram and write an expression that gives the value \(\frac{7}{2}\).
Value: \(\frac{7}{2}\)
2. To represent \(4 \times \frac{1}{3}\), Diego drew this diagram:
Elena drew this diagram:
Are they representing the same expression and value? Explain or show how you know.
Advancing Student Thinking
Students may be unsure about how to begin writing expressions for fractions. Remind students that the fraction will be written as a whole number times a unit fraction. Consider asking: “How might
this help to write the expression?”
Lesson Synthesis
“Today we analyzed expressions and diagrams that represent equal groups and created some of these representations.”
Display or sketch these diagrams:
“How do we know which diagram represents \(3 \times \frac{1}{5}\)? Where do we see each number in the diagram?” (B represents \(3 \times \frac{1}{5}\) because it shows 3 groups of \(\frac{1}{5}\).)
“What expression does the other diagram represent?” (A represents \(5 \times \frac{1}{3}\), because it shows 5 groups with \(\frac{1}{3}\) in each group.)
“What is the value of \(3 \times \frac{1}{5}\)? How do we know?” (\(\frac{3}{5}\). We can count the number of shaded fifths and see that there are 3.)
“What is the value of \(5 \times \frac{1}{3}\)?” (\(\frac{5}{3}\))
Cool-down: Equal Groups of Fractions (5 minutes) | {"url":"https://curriculum.illustrativemathematics.org/k5/teachers/grade-4/unit-3/lesson-2/lesson.html","timestamp":"2024-11-02T22:19:45Z","content_type":"text/html","content_length":"101523","record_id":"<urn:uuid:babe9cbc-e746-49db-b18f-176bc24e8c26>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00180.warc.gz"} |
Thermal bistability through coupled photonic resonances
We present a scheme for achieving thermal bistability based on the selective coupling of three optical resonances. This approach requires one of the resonant frequencies to be temperature dependent,
which can occur in materials exhibiting strong thermo-optic effects. For illustration, we explore thermal bistability in two different passive systems, involving either a periodic array of Si ring
resonators or parallel GaAs thin films separated by vacuum and exchanging heat in the near field. Such a scheme could prove to be useful for thermal devices operating with transition times on the
order of hundreds of milliseconds.
Rapid progress in the synthesis and processing of materials at small lengthscales has created demand for understanding thermal phenomena in nanoscale systems.^1,2 Recent interest in harnessing excess
heat that is readily available at the nanoscale has culminated in several proposed thermal devices^3 including both photonic and phononic realizations of various functionalities such as thermal
rectifiers,^4,5 thermal memory,^6–9 thermal transistors,^10,11 phononic logic gates,^12,13 and phonon waveguides.^14,15 In this paper, we propose a scheme to achieve thermal bistability based on the
radiative (photonic) coupling between three or more optical resonances. Our approach complements and builds on recently proposed ideas^6,8,9,16 in several ways, described further below.
A thermal bistable system can be used as a memory device that stores thermal information by maintaining the temperature of the system in one of the two or more possible states. Realizing such
temperature bistability requires a nonequilibrium thermal circuit supporting multiple steady states. Such a circuit was first proposed several years ago based on the concept of negative differential
thermal resistance (NDTR), which relies on the ability to achieve heat flux rates between objects that decrease with increasing temperature differences. While first proposed in a model system
consisting of a lattice of one-dimensional nonlinear mechanical oscillators,^6 recent implementations of NDTR have instead sought to exploit radiative energy transfer between slabs separated by
nanometer gaps and heated to very high ≈1500K temperatures.^16,17 Here, we propose a simple and experimentally feasible all-optical scheme based on a system of three optical resonances, which builds
and expands on a recently proposed and related scheme that achieves thermal bistability at a temperature ∼340K but requires materials supporting metal-insulator phase-transitions.^8,9 Our approach
exploits common materials exhibiting strong thermo-optic effects and relies instead on thermal bistability induced by a resonant mechanism involving three optical resonances—microring cavities
supporting travelling-wave resonances or polar–dielectric slabs supporting surface–propagating polaritonic resonances. This work extends previous studies of thermal rectification^4 and NDTR through
vacuum^16 and also parallels recent ideas based on exotic non-volatile memory systems,^17–19 which have recently been proposed as viable alternatives to traditional electrostatic memory.^20,21
We begin by briefly describing the main mechanism behind the proposed thermal bistability scheme, leaving quantitative predictions for later. Consider a system of three thermal bodies, shown
schematically in Fig. 1(a), two of which are maintained at constant temperatures T[h] and T[c], with T[h]>T[c], while the remaining body is thermally isolated from its surroundings and has variable
temperature T[0]. The hot and cold bodies exchange heat with the isolated body through flux rates J[h] and J[c], respectively, leading to a net heat influx J[t]=J[h] − J[c] and a steady-state
temperature satisfying (neglecting losses due to thermal conduction) $ρcpV∂T0∂t=Jt=0$, where ρ, c[p], and V are the density, specific heat capacity, and volume of the body, respectively. For typical
heat transfer mechanisms such as conduction^22 or radiation,^23 the heat flux between any two bodies increases with the increasing temperature difference, leading to monotonic J[t](T[0]) and thereby
giving rise to a single steady state, i.e., J[t](T[0])=0, as illustrated in Fig. 1(a). As recently illustrated in Ref. 16, NDTR can be realized in the context of radiative heat transfer between
bodies exhibiting significant thermo-optic effects: namely, by exploiting the monotonic increase in the frequency of planar resonances with increasing temperature. Here, we extend this idea by
considering a system of three bodies that support narrow and slightly detuned resonances of frequencies ω[j], with j ∈ {h, 0, c}. Consider a situation under which T[0]=T[c] and ω[0]<ω[c]<ω[h].
As the temperature T[0] is increased from T[c] → T[h], ω[0] sweeps over the frequencies of both the hot and cold resonators, whose temperatures and frequencies are held fixed. As ω[0] → ω[c], the two
resonators exchange heat more effectively and hence experience larger overall heat loss, causing J[t] to decrease considerably. As ω[0] moves past ω[c] and approaches ω[h], J[t] increases again due
to increasing coupling with the hot resonance, decreasing with increasing ω[0] as it moves past ω[h]. Thus, if properly engineered, such a system can lead to three steady states, consistent with zero
net heat exchange (J[t]=0). Such a situation is illustrated on the right half of Fig. 1(b), wherein the intermediate state (red dot) is unstable, while the remaining two (blue dots) are stable,
i.e., $∂Jt∂T0<0$. If, on the other hand, the initial configuration is such that ω[0]<ω[h]<ω[c] when T[0]=T[c], similar arguments imply the existence of a single steady state, as illustrated in
Fig. 1(c). While this NDTR scheme can be generalized to any system of resonances, below we consider and quantify the feasibility of observing thermal bistability using this scheme in realizations
based on Si photonic resonators and GaAs thin films exchanging heat in the near field. Note that nonlinear thermo-optic effects in driven photonic resonators have been shown to lead to optical
bistability,^24 but their use as ultrafast optical memory devalues their potential as a slow thermal memory. In this work, we focus on passive systems in line with previous implementations of thermal
memory, in which case no optical driving mechanisms are employed.
We first focus on a system of Si ring resonators,^25,26 which exhibit both large thermo-optic coefficients and long-lived resonances at mid infrared wavelengths ∼ peak thermal wavelength, λ[T]∼10μ
m. In particular, we consider three one dimensional arrays of ring resonators shown in the bottom schematic of Fig. 2(a) with period Λ, two of which are maintained at fixed T[h]=800K (left) and T
[c]=300K (right), while the middle one is suspended on insulating posts and has variable temperature T[0]. We ignore the negligible interactions between adjacent rings along the array ensured by a
sufficiently large Λ and obtain the flux rates by considering heat exchange for three coupled resonators shown schematically in the top inset of Fig. 2(a). Such a simplified system can be described
via the temporal coupled-mode theory framework,^27–29 which provides accurate predictions while circumventing the need for numerically intensive calculations.^30 In this framework, the resonances are
described by mode amplitudes a[j], normalized such that $|aj|2$ are mode energies,^30 and have frequencies ω[j] and decay/loss rates γ[j], where j={h, c, 0}. They are subject to thermal noise
sources ξ[j] described by delta-correlated, Gaussian noise terms satisfying $〈ξj*(ω)ξj(ω′)〉=δ(ω−ω′)Θ(ωj,Tj)$, where $〈⋯〉$ denotes the statistical ensemble average and $Θ(ω,T)=ℏω/(eℏωkBT−1)$ is
the Planck function. The three resonators are coupled to one another via coupling coefficients κ[h] and κ[c], allowing heat to flow from the hot to the cold resonator as described by the following
coupled-mode equations:
Here, ω[0] depends on the local resonator temperature through the thermo-optic effect,^25 with $ω0(T0)≈ω0(Tc)−ω0n∂n∂T0(T0−Tc)$, where n is the effective refractive index and $∂n∂T0$ is the
thermo-optic coefficient of the resonator. It follows that the spectral flux densities associated with the coupled modes, $Φh/c=2Im{κh/c}〈ah/c*a0〉$, are given by
where $Θjk(ω)=Θ(ω,Tj)−Θ(ω,Tk)$ and $Dj(ω)=i(ω−ωj)+γj$, for j, k ∈ {h, 0, c}, and
The net flux rates per unit length, $Jh/c=1Λ∫Φh/c(ω)dω2π$, are obtained by integrating overall frequencies.
As an illustration, we consider rings^25 of radii $Rj=4cnωj$ for j ∈ {h, 0, c} and equal width and height of 200nm, designed to support resonances at ω[c]=3.62×10^14rad/s (5.2μm), ω[h]=ω[c]
− 7γ[c], and ω[0](T[c])=ω[c]+5γ[c], with equal decay rates γ[j]=ω[c]/Q, where Q=500 is the lifetime. Material properties are thermo-optic coefficient $∂n∂T=2×10−4K−1$ and effective refractive
index n=3.42. For ring radii R[j] ∼1μm, a lattice period of Λ = 3μm is chosen to ignore the interactions between neighboring rings along the array and the arrays are placed such that the
coupling rates κ[h]=0.9γ[c] and κ[c]=2γ[c]. Figure 2(a) shows the flux rates J[h] and J[c] per unit length as a function of the temperature T[0] of the middle resonator. The net flux entering/
leaving the ring J[t] shown in Fig. 2(b) leads to two stable steady states at T[s][1]=437K and T[s][2]=713K (blue dots), along with an unstable state at T[u]=600K (red dot). Here, we ignore
the radiative decay into the surroundings as well as conductive losses into the mechanically supporting structures. These extraneous channels of heat transfer can be suppressed by suspending the
middle rings on thermally insulating posts to reduce conductive losses while also operating under vacuum to eliminate conductive/convective heat transfer through air, as discussed in Ref. 31. Apart
from stability against temperature perturbations, guaranteed here by large temperature gaps between steady states, robustness against flux perturbations will generally depend on the flux barrier and
hence net magnitude of the flux rates ∼Θ(ω, T)/Q, guaranteed here by operating with large wavelengths and relatively small Q. Figure 2(c) illustrates the relaxation of T[0] from $Th,Tc,Tu+, and Tu−$
to the nearest stable steady states T[s][1] and T[s][2], for V ∼ 0.25 μm^3, where V is the volume of the middle ring and the temperature-dependent values of c[p] and ρ are given in Ref. 32. While the
relaxation time can be increased arbitrarily by setting the initial condition close to T[u], we estimate the characteristic “thermal memory” timescale as the maximum time it takes for the middle ring
to reach the stable steady states when its starting temperature is taken to be that of either the hot or cold resonators, which are 0.1 s and 1s, respectively.
Compared to previous implementations based on phase-transition materials,^8,9 the transition times achieved here are of the same order of magnitude, while the range of operating temperatures is wider
by an order of magnitude. While the relaxation process can in principle be hastened by exploiting large thermo-optic coefficients and/or larger lifetimes Q, thus decreasing the operating temperature
range, the former are constrained by material choices, while the latter lead to decreased flux rates. Aside from careful engineering of the coupling rates and resonator frequencies needed to achieve
bistability, a thermal memory based on this setup requires good thermal insulation and a suitable choice of materials exhibiting large thermo-optic coefficients for speed and improved stability
One possible way to increase the speed of such a thermal memory device is to exploit planar polaritonic materials, which offer enhanced heat flux rates owing to the large number of surface localized
resonances they can support. In what follows, we consider one such example, shown schematically in Fig. 3(a), consisting of three GaAs thin films exchanging heat radiatively in the near field, where
the hot and cold films are again held at fixed temperatures T[h] and T[c], while the intermediate film is thermally insulated from its surroundings and hence described by a variable temperature T[0].
Such a three-body planar configuration has been studied previously using scattering formulations, with the various flux rates obtained through the straightforward calculation of the reflection/
transmission matrices in this geometry,^33 described in more detail in Ref. 34. We exploit this approach to consider a full calculation of the flux rates which includes thermo-optic effects in GaAs,
obtained from Ref. 35, assuming operating temperatures of T[h]=1100K and T[c]=300K, 200nm films, and vacuum gaps of d[h]=48nm and d[c]=45nm. Note that realization of thermal bistability
requires a suitable choice of operating parameters such as temperatures of hot and cold bodies and distances of separation for both configurations in addition to the NDTR effect, which is a necessary
Figure 3(a) shows the computed flux rates per unit area, J[h] and J[c], as a function of T[0], while the net flux entering/leaving the middle film J[t] is shown in Fig. 3(b). As before, the
thermo-optic induced NDTR results in two stable steady states at T[s][1]=430K and T[s][2]=900K (blue dots) along with an unstable steady state at T[u]=710K (red dot). Figure 3(c) illustrates
the relaxation of T[0] from $Th,Tc,Tu+,Tu−$ to the nearest stable steady states, which is substantially decreased compared to ring resonators due to the significantly larger flux rates $≳104W/m2$
attained in this setup. Moreover, the characteristic timescale associated with such a relaxation, i.e., the maximum time it takes for the middle film to reach a steady state when starting from T[h]
or T[c], is≈0.1s and can be further decreased by going to smaller separations or smaller film thicknesses.
Note that a related NDTR-based mechanism was suggested in Ref. 16 in a system of two planar SiC plates. In that work, the heat flux between the two plates was shown to vary nonmonotonically at very
high temperatures T ∼1500K and small separations d ∼15nm, in which case the application of a constant (temperature independent) external flux leads to thermal bistability. Another recent work
proposed a nanothermomechanical memory where NDTR is achieved at very high temperatures T ∼ 1100K by exploiting the nonmonotonic dependence of near field heat transfer on the separation between two
planar slabs, as actuated by the thermal expansion of a mechanical support. The three-body system explored in this work relaxes some of these stringent operating conditions, allowing for a wide range
of operating temperatures (steady state temperatures $≲1000K$) and flux rates. Furthermore, our proposed scheme also offers flexibility with respect to material choices in that it does not rely on
phase-change materials^8 and could be realized with a wide range of materials exhibiting strong thermo-optic effects, such as chalcogenide glasses,^36 silica,^37 and silicon carbide,^37 among many
others. While thermal memory devices based on phase-transition materials offer a smaller operating temperature range (close to room temperature in the case of vanadium dioxide^9) and have also been
shown to lead to multistability,^38 our three-body scheme leads to wider temperature differences between the steady states, thereby guaranteeing stability against temperature and flux perturbations.
We demonstrated a simple scheme to realize temperature bistability in all-passive systems comprising multiple coupled resonant modes. We provided concrete predictions of expected operating conditions
(including transition times of several hundred milliseconds) in realistic designs involving either suspended Si ring resonators or GaAs thin films. Since the underlying mechanism is very general and
not restricted to the proposed implementations, one possible direction forward could be to explore other geometries such as nanobeam resonators,^39 multilayered thin films,^40 nanostructured
materials,^41,42 and different choices of materials,^36,37 where one could potentially observe larger heat exchange. With rapidly advancing nanotechnology, the understanding of this and related
thermal phenomena could be important for nanoscale heat management.
We would like to thank Riccardo Messina and Weiliang Jin for useful comments. This work was partially supported by the National Science Foundation under Grant No. DMR-1454836 and by the Princeton
Center for Complex Materials, a MRSEC supported by NSF Grant No. DMR 1420541.
D. G.
W. K.
K. E.
G. D.
H. J.
, and
S. R.
, “
Nanoscale thermal transport
J. Appl. Phys.
, and
, “
Colloquium: Phononics: Manipulating heat flow with electronic analogs and beyond
Rev. Mod. Phys.
, and
, “
Near-field radiative thermal transport: From theory to experiment
AIP Adv.
C. R.
W. T.
, and
, “
Thermal rectification through vacuum
Phys. Rev. Lett.
, and
, “
Thermal diode: Rectification of heat flux
Phys. Rev. Lett.
, “
Thermal memory: A storage of phononic information
Phys. Rev. Lett.
C. T.
C. H.
, and
J. T. L.
, “
An electrically tuned solid-state thermal memory based on metal–insulator transition of single-crystalline VO[2] nanobeams
Adv. Funct. Mater.
, and
, “
Radiative bistability and thermal memory
Phys. Rev. Lett.
S. A.
, and
, “
Near field thermal memory based on radiative phase bistability of VO[2]
J. Phys. D: Appl. Phys.
, “
Near-field thermal transistor
Phys. Rev. Lett.
, and
, “
Negative differential thermal resistance and thermal transistor
Appl. Phys. Lett.
S. R.
, “
Splash, pop, sizzle: Information processing with phononic computing
AIP Adv.
, “
Towards boolean operations with thermal photons
Phys. Rev. B
, and
, “
Nanotube phonon waveguide
Phys. Rev. Lett.
, and
, “
Hyperbolic waveguide for long-distance transport of near-field heat flux
Phys. Rev. B
C. R.
, and
, “
Negative differential thermal conductance through vacuum
Appl. Phys. Lett.
, “
Near-field nanothermomechanical memory
Appl. Phys. Lett.
M. J.
, “
Thermally actuated buckling beam memory: A non-volatile memory configuration for extreme space exploration environments
Microsyst. Technol.
G. Y.
, and
C. M.
, “
Carbon nanotube-based nonvolatile random access memory for molecular computing
D. N.
S. M.
G. M.
, and
A. H.
, “
Radiation effects on advanced flash memories
IEEE Trans. Nucl. Sci.
, and
, “
Space and terrestrial radiation effects in flash memories
Semicond. Sci. Technol.
S. M.
Y. A.
, and
V. I.
Principles Statistical Radiophysics
), Vol. 2.
, and
, “
Multibistability and self-pulsation in nonlinear high-q silicon microring resonators considering thermo-optical effect
Phys. Rev. A
De Heyn
Van Vaerenbergh
De Vos
S. K.
Van Thourhout
, and
, “
Silicon microring resonators
Laser Photonics Rev.
, and
, “
Suspended si ring resonator for mid-ir application
Opt. Lett.
, and
, “
Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities
IEEE J. Quantum Electron.
, and
A. W.
, “
Thermal radiation from optically driven Kerr (χ(3)) photonic cavities
Appl. Phys. Lett.
M. B.
, and
T. S.
, “
Temporal coupled mode theory for thermal emission from a single thermal emitter supporting either a single mode or an orthogonal set of modes
Appl. Phys. Lett.
M. J.
M. R.
, and
G. N.
, “
Fabrication techniques for creating a thermally isolated tm-fpa (thermal microphotonic focal plane array)
MOEMS-MEMS 2008 Micro and Nanofabrication
International Society for Optics and Photonics
), p.
, and
, “
Three-body amplification of photon heat tunneling
Phys. Rev. Lett.
Pinar Mengüç
, and
, “
Solution of near-field thermal radiation in one-dimensional layered media using dyadic green's functions and the scattering matrix method
J. Quant. Spectrosc. Radiat. Transfer
J. S.
, “
Semiconducting and other major properties of gallium arsenide
J. Appl. Phys.
A. B.
, “
Chalcogenide glasses: A review of their preparation, properties and applications
J. Non-Cryst. Solids
E. D.
Handbook of Optical Constants of Solids
Academic Press
), Vol. 3.
, and
, “
Multilevel radiative thermal memory realized by the hysteretic metal-insulator transition of vanadium dioxide
Appl. Phys. Lett.
, “
Deterministic design of wavelength scale, ultra-high q photonic crystal nanobeam cavities
Opt. Express
S. V.
J. K.
, and
, “
Enhancement and tunability of near-field radiative heat transfer mediated by surface plasmon polaritons in thin plasmonic films
,” in
Multidisciplinary Digital Publishing Institute
), Vol. 2, pp.
, “
Nanoscale heat transfer–from computation to experiment
Phys. Chem. Chem. Phys.
A. K.
D. G.
, and
T. E.
, “
Temperature dependence of surface phonon polaritons from a quartz grating
J. Appl. Phys.
© 2017 Author(s). | {"url":"https://pubs.aip.org/aip/apl/article/111/8/083104/35104/Thermal-bistability-through-coupled-photonic","timestamp":"2024-11-03T09:57:55Z","content_type":"text/html","content_length":"263410","record_id":"<urn:uuid:4802b7da-19ff-4eed-8823-fa5bdfb68e13>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00245.warc.gz"} |
The Odd Couple: ZK and Optimistic Rollups on a Scalability Date
In this post, we present a joint partnership between AltLayer and RISC Zero that aims to bring “on-demand” ZK Fraud Proofs to optimistic rollups by merging ZK rollups and fault proofs together.
Bit of Background
There are two popular flavours of rollups today: Optimistic and ZK. Examples of optimistic rollups include Arbitrum and Optimism, while, ZKSync, Polygon zkEVM among others, have implemented the ZK
ZK rollups rely on cryptographic proofs of correct execution (aka validity proofs), that is to say, every new state proposed by a ZK rollup comes with a proof of correct execution of the underlying
state transition function on a given set of transactions. On the other hand, optimistic rollups assume that a state proposed by the rollup is valid until it’s not.
Unlike ZK rollups that are secured by cryptographic proofs, optimistic rollups are secure under the assumption that there exists at least one honest party that can detect an incorrect state. This
party, often called a challenger upon noticing an invalid state, engages with the rollup operator via what is called a bisection protocol – an off-chain interactive protocol between the challenger
and the rollup operator with the goal to prove the fault on-chain (via fault proofs).
Fault Proof vs. Validity Proofs
Fault proofs and validity proofs are the core mechanisms that make a rollup secure. As mentioned above, fault proofs rely on a specific adversarial model around the existence of honest protocol
participants, while validity proofs rely on cryptographic assumptions.
The bisection protocol in the case of fault proofs, however, requires that each party involved in the protocol remains online throughout the course of the protocol. Furthermore, each party should
have sufficient time to exchange messages of which some have to be on-chain. This introduces a long period, often 7 days, to ensure that messages do not get censored by validators operating the base
chain. This long challenge period becomes particularly painful if users have to move any asset from the rollup to the base chain. Note that the base chain needs to wait for the challenge period to
elapse to be able to accept any incoming message from the rollup to the base chain.
It’s clear that both models currently have their own pros and cons. But, what if we could get the best of both worlds?
Bringing ZK to Optimistic Rollups via On-Demand ZK Proofs
The idea is rather intuitive: Can we upgrade fault proofs in optimistic rollups with cryptographic ZK proofs, whereby proofs are generated only when there is a challenge? Unlike a full ZK rollup,
where the operator needs to generate a ZK proof for every single state transition, the on-demand model will only require a ZK proof when there is a challenge.
This design of a rollup is still optimistic as a state produced by the rollup is considered valid unless someone challenges it. If no one challenges the state, the rollup operator does not need to
produce any cryptographic proof. However, in case, anyone does create a challenge, the operator will have to produce a proof of correctness for all transactions in the challenged block.
This ZK fraud proof architecture can be generalised and implemented across a wide variety of optimistic rollup SDKs.
Implementation by AltLayer using RISC Zero
The AltLayer and the RISC Zero team have been collaborating to implement this new ZK optimistic rollup model. The implementation comes in two variants:
Variant 1 (ZK fraud proof for a single disputed instruction): In this variant, the bisection protocol is kept mostly intact. The challenger engages with the rollup operator in a bisection protocol.
At the end of the protocol, the operator and the challenger identify a single disputed VM instruction. Now, instead of executing this single instruction on-chain, the rollup operator generates a ZK
proof of correct execution of this single instruction. The ZK proof is then verified on-chain.
Variant 2 (A full ZK validity proof in lieu of the bisection protocol): In this version, the bisection protocol is replaced entirely by a proof of correct execution or a proof of validity.
Leveraging RISC Zero’s zkVM
Both variants of this ZK optimistic rollup model are built on top of the RISC Zero zkVM which enables ZK proofs across a wide variety of general purpose languages such as Rust, WASM, and RISCV.
Additionally, proof generation is highly parallelizable thanks to continuations and Bonsai which allows the ZK system to achieve industry leading performance and complexity. Combined, this serves as
a platform which is highly upgradable and extensible, ready for the next generation of Optimistic rollups.
Benefits and Limitations
Variant 1 can be seen as the first step towards introducing cryptographic proofs in an optimistic rollup. Even though the performance benefits of this variant are somewhat limited as the operators
still have to engage in an off-chain bisection protocol, there are several advantages of this variant to optimistic rollup builders who wish to adopt validity proofs:
1. It requires minimal changes to the existing rollup design, is highly extensible and upgradable, and is backward compatible with existing fault proof methods.
1. Allows parties to perform bisection over any underlying instruction set as long as the execution proof of the single contested instruction can be verified on the base chain. For instance, one
could do bisection over WASM even if the base chain does not have a direct ability to run WASM instruction. As a result, the rollup does not need to emulate a corresponding interpreter in a
contract on the base chain.
Variant 2 however does lead to security benefits as there is no need to perform the complex bisection protocol where both parties need to stay online. This variant can further reduce a long 7 day
withdrawal period to hours. It also acts as a stepping stone enabling Optimistic rollups to transition into full ZK rollups removing the need to detect frauds and eventually reducing the withdrawal
period to mere minutes.
It’s important to note that there are a few caveats with both the variants. As with any optimistic rollup, these modified rollups still require the existence of at least one honest participant and
that this participant needs to have a long-enough time window to detect the fault and report it on the base chain. To this end, Altlayer and most optimistic rollups today have different types of
validators that are tasked to watch the network and report any fault. In the future, these ZK systems will enable Optimistic rollups to transition to full ZK rollups completely removing the fraud
dispute period.
Implementation Setup
Altlayer has implemented both these variants using an optimistic rollup implementation in Rust with fault proof over WASM. The system runs Sputnik VM (a Rust implementation of EVM) as the execution
engine. The rollup client compiles Sputnik VM (in Rust) to WASM instructions and then runs bisection protocol over WASM. The ZK proof is then done over WASM via RISC Zero’s Bonsai proving service
that in turn uses RISC Zero zkVM for proving. zkVM comes with recursive proofs, a general-purpose circuit (with a bespoke circuit compiler) and state continuations.
Concluding Remarks
This work introduces the idea of bringing ZK proofs to optimistic rollups by replacing fault proofs with on-demand ZK Fraud Proofs proofs. Altlayer has a working implementation that implements the
two variants described in the article and utilises a rollup instrumented by the AltLayer framework and the Bonsai proving service. The observed performance shows the viability of the solution.
Altlayer’s future work is to integrate it into all major rollup SDKs such as OP Stack and Arbitrum Orbit and make it available to all instantiations of optimistic rollups as an option. | {"url":"https://risczero.com/blog/altlayer-zkfraudproofs","timestamp":"2024-11-02T09:33:55Z","content_type":"text/html","content_length":"120517","record_id":"<urn:uuid:505940a7-95ff-4637-bbd0-ad80ed97a56e>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00029.warc.gz"} |
16 Bit Full Adder Vhdl Code For Serial Adder
full adder using half adder vhdl code, full adder using half adder structural vhdl code, vhdl code for half adder and full adder pdf, 4 bit adder using full adder vhdl code, half adder and full adder
vhdl code, 8 bit adder using full adder vhdl code, vhdl code for bcd adder using full adder, full adder vhdl code using half adders, half adder full adder vhdl code
carry-out signal from the full adder, and a finite state machine (FSM). The shift ... Figure 3. VHDL code for the top-level entity of the serial adder (Part a). 4.. 8. Figure 8. NORMAL.vhd Serial
Calculation Implementation Block Diagram . ... accommodate 32-bit integers and each multiplier must handle up to 16-bit integers. ... Like the RC adder, it takes two 32-bit binary numbers, A and B,
and produces their 32- bit sum, S ... For this architecture, VHDL code using entirely concurrent.. EXAMPLE 3.7 4-TO-2 BINARY ENCODER: VHDL MODELING USING ... EXAMPLE 7.2 VHDL TEST BENCH FOR A 4-BIT
RIPPLE CARRY ADDER USING ... WINDOW CONTROLLER IN VHDL: EXPLICIT STATE CODES . ... EXAMPLE 9.8 SERIAL BIT SEQUENCE DETECTOR IN VHDL: SIMULATION ... architecture, 16, 17.. Serial AdderEdit. library
IEEE; use IEEE.STD_LOGIC_1164.ALL; entity SA_VHDL is Port ( I : in std_logic_vector(15 downto 0); O : out std_logic_vector(7 downto.... Normally an N-bit adder circuit is implemented using N parallel
full adder circuits, simply connected next to each other. The advantage of this is.... 1991 - verilog code for 16 bit carry select adder. Abstract: ... Abstract: 4 bit parallel adder serial
correlator vhdl code for parallel to serial shift register vhdl code for.... I would start with a register (n bit) a full adder and than a flip flop as basic ... 1 downto 0)); The output must be
std_logic, because it is a serial output ... sum --full adder logic z. Designing a 16-bit carry-skip adder. Modeling and simulation of combinational logic using VHDL. ... A one-bit full adder is a
combinational circuit that forms the arithmetic sum of ... serial, with the carry output from each full adder connected to the carry input of ... Bits a0 and b0 are the LSB bits of the numbers to be
added.. A full adder circuit is central to most digital circuits that perform addition or subtraction. ... X and Y are the two bits to be added, Cin and Cout the carry-in and carry-out bits, and ...
The serial adder can also be used in the subtraction mode, as shown in Figure 12.13. ... Box 1.2 shows the VHDL code that describes a full-adder.. The latency of a 4-bit ripple carry adder can be ...
Signal Propagation in Pipelined serial Blocks ... the implemented VHDL codes for all the 64-bit adders.. Additional Key Words and Phrases: FPGA, serial arithmetic, SerDes, sliding window ...
multiply-accumulate (versus 1.2 and 2.8 for 16b and 32b ... two counter-addressed RAM structures, a 1-bit multiplier, and a full adder to compute ... closely follow Xilinx-recommended coding
practices to ensure high-quality bit-parallel.. -- This is a behavioral model of a multiplier for unsigned binary numbers. It multiplies a. -- 4-bit multiplicand by a 4-bit multiplier to give an
8-bit product. -- The.... A serial adder consists of three n-bit shift registers, a full- adder and a D flip-flop. Two parallel-in-serial-out (PISO) shift registers holds the numbers (A and B) to....
1. VHDL. Structural Modeling. EL 310. Erkay Sava. Sabanc University ... Full Adder. We assume that we have the description of half adders and or gate. Actually, we are not ... Bit Serial Adder
library IEEE; ... 16. Brent-Kung Formulation. An associative operation: (x, y)(w,z) = (x+yw, yz) ... add two numbers. Thus P. 1.. appropriate times. Block Diagram of a 4-bit Serial Adder with
Accumulator ... VHDL CODE for the 16 bit serial adder ... VHDL code for 4 X 4 Binary Multiplier.. 3.8 8-bit binary adder using 4-bit carry-look ahead adders ..................... 45. 3.9 Designing a
6-bit ... 8.8 Serial transmitter and receiver (USART) . ... The objective is to develop VHDL code and the final circuit for the 10-line to 4- line priority ... 16) Perform a gate-level simulation to
measure the worst-case propagation delay and.... A serial adder consists of a 1-bit full-adder and several shift registers. ... to Digital Systems: Modeling, Synthesis, and Simulation Using VHDL
[Book] ... Two right-shift registers are used to hold the numbers (A and B) to be added, while one.... Cout: std_logic. ); END;. The output of the testbench will be observe by the digital waveform of
the simulator. Page 16.... To add the contents of two register serially, bit by bit.. SREG2 are used to hold the four bit numbers to be added. ... Serial Adder. Sum. CarryOut ... should include a
full adder and a flip-flop to store carry, to be used at.
galactic monster quest hacked
Skanda Purana In Telugu Pdf Free 31
Dhoom 2 Movie Download 300 Mb Hindi Moviesk
Masterwriter 2.0 With Crack Torrent
Peemak Tagalog Full Movie Tagalog Version Cinema One Global
Aikman Series C.pdfl
poshida raaz book in urdu download
Baankey Ki Crazy Baraat Movie Download In Kickass Torrent
Underworld Awakening 2012 Tamil Dubbed Movie Download
world atlas of coffee epub | {"url":"https://chrysvehano.mystrikingly.com/blog/16-bit-full-adder-vhdl-code-for-serial-adder","timestamp":"2024-11-03T15:50:12Z","content_type":"text/html","content_length":"98539","record_id":"<urn:uuid:026f0a58-2f12-4007-9140-a4cd51a475e3>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00102.warc.gz"} |
How to use %in% and %notin% operators in R (with examples)
• %in% is a built-in infix operator, which is similar to value matching function match. %in% is an infix version of match. User defined infix operators can be created by creating a function and
naming it in between two % (e.g. %function_name%).
• %in% returns logical vector (TRUE or FALSE but never NA) if there is a match or not for its left operand. Output logical vector has the same length as left operand.
• If there are two vector x and y then the syntax of %in%: x %in% y
• %in% works only with vectors
• %notin% is not a built-in operator and can be created by negating the %in% operator (see below)
• Help syntax for %in% operator: ?"%in%"
Learn about $ operator for data extraction from Data Frame and list in R
Here are few examples of how to use %in% to manipulate vectors and Data Frames in R,
%in% to check the value in a vector
%in% is helpful to check any value in a vector. If there is a match to the value, it returns TRUE, otherwise FALSE
x <- c(1,5,10,20,20,24,45)
# check any number in x vector
20 %in% x
[1] TRUE
%in% to compare two vectors
%in% operator is useful to compare two vectors and identify common values in between the two vectors.
x <- c(1,5,45)
y <- c(5,45)
x %in% y
[1] FALSE TRUE TRUE
# find common values
x[x %in% y]
[1] 5 45
Check two sequences of numbers and identify common numbers
x <- 1:5
y <- 3:7
x %in% y
[1] FALSE FALSE TRUE TRUE TRUE
# find common letters
x[x %in% y]
[1] 3 4 5
Similarly, you can check two vectors containing letters and identify common letters
x <- LETTERS[1:5]
y <- LETTERS[4:7]
x %in% y
[1] FALSE FALSE FALSE TRUE TRUE
# find common numbers
x[x %in% y]
[1] "D" "E"
If you have a big vector (say vector with 1000 values), you can use any, all, or which functions with %in% operator
x <- 1:1000
y <- 900:2000
# check if there is any common values between a and b vectors
any(x %in% y)
[1] TRUE
# check if there are all values common between a and b vectors
all(x %in% y)
[1] FALSE
# get indexes of common values
a <- 1:10
b <- 6:200
which(a %in% b)
[1] 6 7 8 9 10
%in% to check the value in a Data Frames
%in% can be used for checking any value present in columns of Data Frames
Create a Data Frame,
df <- data.frame(col1 = c("A", "B", "C"),
col2 = c(1, 2, 3),
col3 = c(0.1, 0.2, 0.3))
# output
col1 col2 col3
1 A 1 0.1
2 B 2 0.2
3 C 3 0.3
Check if any value is present in Data Frame columns,
# check if 'B' is present in col1
'B' %in% df$col1
[1] TRUE
# to check if any value in col1 is B
df$col1 %in% 'B'
[1] FALSE TRUE FALSE
If you want to compare the vector with Data Frame columns, the %in% operator comes in handy. See below example,
# check if values of vector are present in a Data Frame columns
lapply(df, `%in%`, c(1, 4, 0.1))
# output
[1] FALSE FALSE FALSE
[1] TRUE FALSE FALSE
[1] TRUE FALSE FALSE
%in% to update the values in a Data Frame
%in% operator is useful when you want to scan Data Frame and update (replace) the existing values in Data Frame where there is a match with a given vector or values.
# search values in vector in Data Frame and replace with 0 where there is match
df[sapply(df, `%in%`, c(1, 4, 0.1))] <- 0
# output
col1 col2 col3
1 A 0 0.0
2 B 2 0.2
3 C 3 0.3
%in% and %notin% to filter (subset) Data Frames based on multiple values (with dplyr)
In filtering the Data Frame based on single or multiple values, the %in% operator is useful. You can use the %in% operator to filter the values in columns in Data Frame,
Filter (subset) Data Frame where multiple values from vector match to the values in col1,
df %>% filter(col1 %in% c('A', 'B')) # same as df[df$col1 %in% c('A', 'B'),]
# output
col1 col2 col3
1 A 1 0.1
2 B 2 0.2
Filter (subset) Data Frame where multiple values does not match to the values in col1 using %notin%,
For this example, you need to first create %notin% operator (see below at the end of this article)
df %>% filter(col1 %notin% 'C')
# output
col1 col2 col3
1 A 1 0.1
2 B 2 0.2
%in% and %notin% to remove column from Data Frames
%in% and %notin% operators can also be used for removing single or multiple columns from Data Frames
You need to first create the %notin% operator (see below at the end of this article)
# remove col2
df[ , !(names(df) %in% "col2")]
# output
col1 col3
1 A 0.1
2 B 0.2
3 C 0.3
# %notin% operator to remove columns.
# You need to first create %notin% operator (see below at the end of this article)
df[ , (names(df) %notin% "col2")]
# output
col1 col3
1 A 0.1
2 B 0.2
3 C 0.3
# to remove multiple column use column names vector such as c("col2", "col3")
%in% to select columns from Data Frames
%in% can be used to select single or multiple columns from Data Frames
# select single column
df[ ,(names(df) %in% "col3"), drop=FALSE]
# output
1 0.1
2 0.2
3 0.3
# select multiple columns
df[ ,(names(df) %in% c("col1", "col2"))]
# output
col1 col2
1 A 1
2 B 2
3 C 3
%in% can also be used for selecting specific columns where the column names match on a condition,
Select the columns from dataframe where column names matches row names of other dataframe
# create dataframe with rownames
df_row <- data.frame(M1 = c("X", "Y" ),
M2 = c(11, 22), row.names = c("col1", "col3"))
# get columns of dataframe df where its column names matches
# to rownames of dataframe df_row
df[, names(df) %in% rownames(df_row)]
# output
col1 col3
1 A 0.1
2 B 0.2
3 C 0.3
%in% to compare two Data Frames
%in% can be used to compare two Data Frames and subset Data Frames based on the column value match. This is more similar like left join query i.e. select all records from one Data Frame where column
values match to another Data Frame
Create another Data Frame,
df2 <- data.frame(col1 = c("A", "B", "D", "E"),
col4 = c(100, 200, 300, 400),
col5 = c("a", "b", "c", "d"))
# output
col1 col4 col5
1 A 100 a
2 B 200 b
3 D 300 c
4 E 400 d
Now compare df and df2 to get all records from df2 where col1 values match to col1 values in df (similar to left join of tables),
subset(df2, df2$col1 %in% df$col1)
# output
col1 col4 col5
1 A 100 a
2 B 200 b
Comparison of %in% and == operators
• == operator compares the value between two vectors element-wise (the first value of one vector compared with the first value of another vector), whereas %in% compares the value between two
vectors one by all (the first value of the first vector compared with all values of the second vector)
• With == operator, the length of the left and right operands must be the same. It is not necessary to have the same length for left and right operands for %in% operator.
x <- c(1, 2, 3)
y <- c(3, 2, 1)
x %in% y
[1] TRUE TRUE TRUE
x == y
[1] FALSE TRUE FALSE
# compare and get indexes of two vectors
a <- c(1, 2, 9, 2)
b <- c(1, 2, 3, 4, 5)
# == operator only found first two indexes
which(a == b)
[1] 1 2
Warning message:
In a == b : longer object length is not a multiple of shorter object length
# %in% operator found all matched value indexes
which(a %in% b)
[1] 1 2 4
Create %notin% operator (opposite to %in%)
%notin% operator is not built-in and can be created by applying Negate function to %in%.
%notin% is opposite to %in% operator
You can also use %notin% as by putting ! in front of the %in% expression (!%in%)
`%notin%` <- Negate(`%in%`)
Check the value in a vector using %notin%
x <- c(1,5,10,20,20,24,45)
# check any number in x vector
50 %notin% x # same as !(50 %in% x)
[1] TRUE
Update values of Data Frame to NA where values do not match,
# create data frame
df <- data.frame(col1 = c("A", "B", "C"),
col2 = c(1, 2, 3),
col3 = c(0.1, 0.2, 0.3))
# update values of Data Frame to NA where values does not match to c(2, 3)
df[sapply(df, `%notin%`, c(2, 3))] <- NA # df[sapply(!(df, `%in%`, c(2, 3)))] <- NA
# output
col1 col2 col3
1 <NA> NA NA
2 <NA> 2 NA
3 <NA> 3 NA
Enhance your skills with courses R
• https://stackoverflow.com/questions/42637099/difference-between-the-and-in-operators-in-r/42637186
If you have any questions, comments or recommendations, please email me at reneshbe@gmail.com
This work is licensed under a Creative Commons Attribution 4.0 International License
Some of the links on this page may be affiliate links, which means we may get an affiliate commission on a valid purchase. The retailer will pay the commission at no additional cost to you. | {"url":"https://www.reneshbedre.com/blog/in-operator-r.html","timestamp":"2024-11-07T06:25:45Z","content_type":"text/html","content_length":"115696","record_id":"<urn:uuid:b164b253-ad91-4094-b8c5-121d3cb1ece2>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00267.warc.gz"} |
Cite as
Gabriel Bathie, Panagiotis Charalampopoulos, and Tatiana Starikovskaya. Longest Common Extensions with Wildcards: Trade-Off and Applications. In 32nd Annual European Symposium on Algorithms (ESA
2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 19:1-19:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
Copy BibTex To Clipboard
author = {Bathie, Gabriel and Charalampopoulos, Panagiotis and Starikovskaya, Tatiana},
title = {{Longest Common Extensions with Wildcards: Trade-Off and Applications}},
booktitle = {32nd Annual European Symposium on Algorithms (ESA 2024)},
pages = {19:1--19:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-338-6},
ISSN = {1868-8969},
year = {2024},
volume = {308},
editor = {Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.19},
URN = {urn:nbn:de:0030-drops-210904},
doi = {10.4230/LIPIcs.ESA.2024.19},
annote = {Keywords: Longest common prefix, longest common extension, wildcards, Boolean matrix multiplication, approximate pattern matching, periodicity arrays} | {"url":"https://drops.dagstuhl.de/search/documents?author=Starikovskaya,%20Tatiana","timestamp":"2024-11-14T15:36:54Z","content_type":"text/html","content_length":"197815","record_id":"<urn:uuid:e7f084a2-4e02-404a-a2d5-eb92c878266e>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00053.warc.gz"} |
Truncated SVD - (Inverse Problems) - Vocab, Definition, Explanations | Fiveable
Truncated SVD
from class:
Inverse Problems
Truncated Singular Value Decomposition (SVD) is a dimensionality reduction technique that approximates a matrix by using only the largest singular values and their corresponding singular vectors.
This method is particularly useful in filtering noise from data and improving computational efficiency in inverse problems, allowing for better handling of ill-posed situations and enhancing the
stability of numerical algorithms.
congrats on reading the definition of Truncated SVD. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Truncated SVD retains only a specified number of the largest singular values, which can significantly reduce the complexity of computations.
2. This technique is often employed in image compression, natural language processing, and recommendation systems to filter out noise and enhance relevant features.
3. In inverse problems, truncated SVD aids in obtaining more stable solutions by mitigating the effects of noise and ensuring better convergence properties.
4. Numerical stability is improved with truncated SVD, making it a preferred choice for handling large datasets with inherent uncertainties.
5. The choice of how many singular values to retain can be critical; retaining too few may lead to loss of important information, while retaining too many can lead to overfitting.
Review Questions
• How does Truncated SVD improve computational efficiency when solving inverse problems?
□ Truncated SVD enhances computational efficiency in inverse problems by reducing the dimensionality of the data. By focusing only on the largest singular values and their corresponding
vectors, it simplifies matrix operations while preserving essential features. This reduction helps to combat noise and improves convergence rates of numerical algorithms, allowing for more
robust solutions in situations where traditional methods may struggle.
• Discuss the implications of using Truncated SVD for filtering noise in data analysis. What are some potential challenges?
□ Using Truncated SVD for filtering noise can significantly improve data analysis by highlighting relevant patterns and diminishing irrelevant fluctuations. However, one potential challenge is
determining the optimal number of singular values to retain; too few may discard valuable information while too many can reintroduce noise. Additionally, reliance on this method may overlook
important aspects of the original data structure, leading to misinterpretations if not carefully applied.
• Evaluate how Truncated SVD relates to regularization techniques in managing ill-posed problems. What makes it a viable option?
□ Truncated SVD serves as an effective regularization technique in managing ill-posed problems by limiting the influence of smaller singular values that are often associated with noise. By
selectively retaining significant components, it stabilizes solutions and prevents overfitting. This makes it a viable option as it balances accuracy and robustness, ensuring that the
reconstructed data or solutions remain interpretable while effectively addressing uncertainties inherent in ill-posed situations.
"Truncated SVD" also found in:
ยฉ 2024 Fiveable Inc. All rights reserved.
APยฎ and SATยฎ are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website. | {"url":"https://library.fiveable.me/key-terms/inverse-problems/truncated-svd","timestamp":"2024-11-07T18:52:12Z","content_type":"text/html","content_length":"156989","record_id":"<urn:uuid:f4359ba8-72b0-4ad2-8dc9-7c1b7371669f>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00886.warc.gz"} |
Mastering Float Ranges in Python: Numpy Generators and More - Adventures in Machine Learning
Generating a Range of Floating-Point Numbers in Python
Are you struggling with generating a range of floating-point numbers in Python? Don’t worry, you’re not alone. There are multiple ways to generate a range of floating-point numbers in Python, and in
this article, we’ll explore two popular methods that can make your life a lot easier. These methods are:
1. Using NumPy’s arange() and linspace() functions
2. Using a Python generator to produce a range of float numbers
Let’s dive into the details of each approach.
Using NumPy’s arange() and linspace() functions
NumPy is a popular library in the Python ecosystem that provides support for multi-dimensional arrays, matrices, and mathematical functions. It also offers two functions, arange() and linspace(), for
generating a range of floating-point numbers.
arange() Function
The arange() function is quite similar to Python’s built-in range() function, but it takes floating-point arguments and returns a NumPy array. Here’s the basic syntax of the arange() function:
np.arange(start, stop, step, dtype=None)
The start parameter specifies the starting point of the range (inclusive), the stop parameter specifies the ending point of the range (exclusive), the step parameter specifies the increment between
values, and the dtype parameter specifies the data type of the output array.
For instance, assume that you want to generate a range of floating-point numbers from 1.5 to 6.5 in steps of 0.5. Here’s how to do it using the arange() function:
import numpy as np
nums = np.arange(1.5, 6.5, 0.5)
[1.5 2. 2.5 3. 3.5 4. 4.5 5. 5.5 6. ]
You can see that the arange() function returns an array containing a sequence of floating-point numbers with the specified increment. Note that the value of the stop parameter is not included in the
Therefore, if you want to include the ending point, you can set it as follows:
nums = np.arange(1.5, 7, 0.5)
In this case, the output will be:
[1.5 2. 2.5 3. 3.5 4. 4.5 5. 5.5 6. 6.5]
linspace() Function
The linspace() function is similar to the arange() function but specifies the number of samples instead of specifying the step size. Here’s the basic syntax of the linspace() function:
np.linspace(start, stop, num=50, endpoint=True, retstep=False, dtype=None)
The start parameter specifies the starting point of the range (inclusive), the stop parameter specifies the ending point of the range (inclusive), the num parameter specifies the number of samples to
generate, the endpoint parameter specifies whether to include the ending point, the retstep parameter specifies whether to return the step size, and the dtype parameter specifies the data type of the
output array.
For example, if you want to generate ten floating-point numbers between 1.0 and 5.0, you can use the linspace() function as follows:
nums = np.linspace(1.0, 5.0, 10)
[1. 1.44444444 1.88888889 2.33333333 2.77777778 3.22222222
3.66666667 4.11111111 4.55555556 5. ]
As you can see, the linspace() function returns an array with ten evenly spaced floating-point numbers between 1.0 and 5.0.
Using a Python generator to produce a range of float numbers
Generators are a powerful feature in Python that allows you to create iterable objects on-the-fly. A generator function is a special type of function that returns an iterator object.
The yield statement in a generator function is used to produce a series of values, one at a time. To generate a range of floating-point numbers using a Python generator, you can follow these simple
Step 1: Define a generator function that takes the start, stop, and step parameters.
def float_range(start, stop, step):
while start < stop:
yield start
start += step
Step 2: Call the generator function with the desired parameters.
nums = list(float_range(1.5, 6.5, 0.5))
[1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5, 6.0]
In this example, we define a generator function float_range(), which takes the start, stop, and step parameters and produces a sequence of floating-point numbers. The while loop continues until the
next value would be greater than or equal to the stop value.
The yield keyword is used to return each value in turn across successive calls to the generator function. Finally, we pass the parameters into the generator function, convert the generator object
into a list of floating-point numbers, and print the output.
Generating a range of floating-point numbers in Python is a common requirement in many data analysis and scientific computing tasks. In this article, we explored two popular methods to accomplish
this task: using NumPy’s arange() and linspace() functions and using a Python generator to produce a range of float numbers.
Both methods have their advantages and disadvantages, and choosing the appropriate method depends on the specific use case. However, by understanding these techniques, you can efficiently work with
ranges of floating-point numbers in Python and overcome the most common hurdles along the way.
Reverse float range using NumPy’s arange()
Creating a range of floating-point numbers is a common task in several scientific and data analysis fields. Python’s NumPy library provides various functions to generate sequences of numbers,
including the arange() function, which can be used to create an array of evenly spaced floating-point numbers.
However, in some cases, you might require a descending order range of floating-point numbers instead of ascending. To generate a descending float sequence, NumPy provides an easy solution which is
using the reversed() function.
The reversed() function can be used to reverse a range of floating-point numbers generated by the arange() function. Here’s an example showing how to use the reversed() function to display a float
range sequence in descending order:
import numpy as np
nums = np.arange(1.5, 6.5, 0.5)
reverse_nums = list(reversed(nums))
[6.0, 5.5, 5.0, 4.5, 4.0, 3.5, 3.0, 2.5, 2.0, 1.5]
In the example above, we first generate the sequence using the arange() function. We then pass the sequence into the reversed() function to display a descending sequence of floating-point numbers.
As you can see, we get a sequence starting from the maximum value down to the minimum. A key advantage of the reversed() function is that it returns the values in the same type as the input.
This means that if the input sequence was float, the returned sequence will also be float with the floating-point precision retained.
Range for negative float numbers using NumPy’s arange()
In many cases, you might require to generate a range of negative floating-point numbers in Python. NumPy’s arange() function can also handle negative float numbers efficiently.
Let’s see how we can use the np.arange() function to generate a range of negative float numbers:
import numpy as np
nums = np.arange(-2.0, 0.0, 0.2)
[-2.0000000e+00 -1.8000000e+00 -1.6000000e+00 -1.4000000e+00
-1.2000000e+00 -1.0000000e+00 -8.0000000e-01 -6.0000000e-01
-4.0000000e-01 -2.0000000e-01 -8.8817842e-16]
In the example above, we use negative float numbers in the start and stop parameters of the arange() function to specify the range of the sequence. We use an increment of 0.2 to determine the spacing
between numbers in the range.
As a result, the arange() function generates a sequence of floating-point numbers in the range [-2.0, 0.0) with an increment of 0.2. The output displays the array of negative floats ranging from -2.0
to 0.0.
It is worth noting that when the generated sequence contains a value equivalent to zero or close to zero, the precision of the floating-point number can be affected by the machine’s floating-point
representation. In the example above, we can see number ” -8.8817842e-16″ which has a very small value close to 0.
In this article, we have discussed two additional methods of generating range of float values using Python’s NumPy library. First, we looked at how to reverse a sequence of floating-point numbers
generated by the arange() function using the reversed() function.
Reversing the sequence is sometimes required for creating an array of numbers in descending order, which can be useful in certain applications. Additionally, we explored how to generate a range of
negative float numbers using the arange() function by passing negative values as the start and stop parameters.
By having a deeper understanding of these techniques, you can work more efficiently with range of floating-point numbers in Python, making scientific computation and data analysis tasks more
Range of floats using NumPy’s linspace()
NumPy’s linspace() function is useful for creating a linear sequence of floating-point numbers. It is somewhat different from the arange() function as it takes the number of samples to generate and
provides the endpoints.
This means we can specify exactly the number of values we want without worrying about the endpoint or spacing between values. The basic syntax of np.linspace() function is:
np.linspace(start, stop, num=50, endpoint=True, retstep=False, dtype=None)
The start parameter specifies the starting point of the range (inclusive), the stop parameter specifies the ending point of the range (inclusive), the num parameter specifies the number of samples to
generate, the endpoint parameter specifies whether or not to include the endpoint value, the retstep parameter specifies whether or not to return the step size between values, and the dtype parameter
specifies the data type of the output array.
Here is an example of how to use the np.linspace() function:
import numpy as np
x = np.linspace(0, 1, num=5)
In the example above, we generate a sequence of floating-point numbers from 0 to 1 with five evenly spaced values. By default, the endpoint parameter is set to True, which means the stop value is
included in the output.
Setting endpoint parameter to False excludes the stop value from the output:
x = np.linspace(0, 1, num=5, endpoint=False)
As you can see, when the endpoint parameter is set to False, the last value (in this case, 1) is excluded from the output.
Range of floats using generator and yield
Creating custom functions in Python can help write more efficient code and minimize errors. In certain situations, we might need to create a range of float values using a specific algorithm.
Python provides generators, which are an efficient way to create sequences of values on-the-fly. Here’s an example of how to create a custom frange() function using Python generators and the yield
def frange(start, stop, step):
i = 0
while start < stop:
yield start
i += 1
start = round(start + step, i)
In the frange() custom function, we pass the start and stop parameters to specify the range of float values we want to generate, and also provide the size of the increment with the step parameter.
The while loop continues until the start value is less than the stop value. The yield keyword returns the current start value and adds the step value until the stop value is reached.
To test our frange() function, we can use it to generate a range of floating-point values:
for f in frange(0.1, 0.6, 0.1):
In the example above, we generate a sequence of floating-point numbers ranging from 0.1 to 0.6 with an increment of 0.1. The values are then printed using a for loop that iterates through the
generated sequence. Another benefit of using a custom function is that it enables you to modify the algorithm’s internals for any specific use case.
Therefore, you can tweak the implementation to achieve the desired outcome.
In conclusion, generating a range of floating-point numbers is a routine task when working with data analysis and scientific computing fields. We’ve covered two additional methods to create float
ranges using NumPy’s linspace() function and Python generators and yield.
We’ve emphasized the differences between the two methods and demonstrated how to use them in different situations. The np.linspace() function provides simplicity and flexibility when generating a
sequence of precisely spaced values.
On the other hand, creating a custom frange() function using Python generators and yield offers more control and customization possibilities for specific use cases. Whether you need more control over
the sequence or require a ready-to-use function with a simple syntax, Python’s inbuilt and custom functions can serve the purpose.
Range of floats using list comprehension
Python’s list comprehensions are a simple and practical way to create lists based on existing sequences. It provides a concise way to create lists by using a single line of code.
A list comprehension creates a new list by iterating through each item of a sequence and applying a set of conditions or transformations. List comprehension can also be used to create a range of
floating-point numbers, similar to other techniques we’ve discussed in this article.
Here’s an example of how to create a range of float values using list comprehension:
start = 1.5
stop = 6.5
step = 0.5
float_sequence = [i for i in (start, stop, step)]
In the example above, we define the start, stop, and step values to generate a range of float numbers. We then pass these values as arguments to the list comprehension syntax to create a new list
called ‘float_sequence‘.
The output displays the float sequence. The list comprehension approach provides a simplified way to generate a range of float values, although it does not provide the sequence values themselves.
Therefore, you may need to use additional code to produce the required sequence.
Range of floats using itertools
Python’s itertools library provides a set of tools for efficient, iterative data processing. One of those tools is the ‘count()‘ function.
The count() function generates an infinite sequence of numbers with a constant increment. Combining this with the ‘islice()‘ function allows us to generate a range of floating-point numbers with
specific conditions such as starting point, stopping point, and the increment.
Here’s an example of how to generate a float range using itertools‘ count() and islice() functions:
import itertools
start = 1.5
stop = 6.5
step = 0.5
float_sequence = list(itertools.islice(itertools.count(start, step), int((stop-start)//step)))
[1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5, 6.0]
In the example above, we define the start, stop, and step values to generate a range of float numbers. We then use the count() function | {"url":"https://www.adventuresinmachinelearning.com/mastering-float-ranges-in-python-numpy-generators-and-more/","timestamp":"2024-11-11T20:19:17Z","content_type":"text/html","content_length":"104707","record_id":"<urn:uuid:bb2c8940-1369-45cf-a5d9-e43c829171e3>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00805.warc.gz"} |
Master of Science (Applied Mathematics) | Institute of Mathematics
Master of Science (Applied Mathematics)
The Master of Science in Applied Mathematics Program is designed for students seeking training in applied mathematics to pursue a career in the academe or in research and development in industry. Its
curriculum is designed in such a way that graduates of this program are equipped with enhanced mathematical, analytical and critical thinking skills to solve complex problems in life and physical
sciences, mathematical finance, etc. The program also provides the students with a solid mathematical foundation for doctoral studies in mathematics or related fields.
The program has four tracks:
• Mathematical Finance (MF)
• Mathematics in Life and Physical Sciences (MLPS)
• Numerical Analysis of Differential Equations (NADE)
• Optimization and Approximation (OA)
The MSAM program aims to provide and equip the students:
1. the expertise in formulating and analyzing mathematical models. In particular, the application of mathematical methods to problems in life and physical science, finance, and other related
2. sufficient knowledge in the underlying mathematical theories in order to create new mathematical tools and models.
3. training in the use of computational algorithms in solving mathematical problems arising from applications.
4. the production of relevant and substantial research works in various fields of applied mathematics and their applications that would possibly lead to published works in peer-reviewed scientific
• Admission into the Program
Applications for admission to the program are processed by the College of Science (more information here). Students can apply for admission during the 1st semester or 2nd semester.
Aside from the general requirements for admission set forth by the College of Science, applicants of the MSAM program must have a bachelor’s degree from a recognized institution of higher
learning, and completion of Advanced Calculus and Linear Algebra courses, among others.
For more inquiries, please send an email to ddapr@math.upd.edu.ph.
• Program Curriculum
A student may opt to take the thesis option or the non-thesis option. The maximum residence of any student under the MSAM program is five (5) years.
For the thesis option, students are required to take 24 units of formal graduate courses, 1-unit graduate seminar, and 6 units of thesis. The thesis which will have to be defended before an
Examination Committee. Submission of bound copies of the thesis will also be required.
For the non-thesis option, students are required to take 33 units of formal graduate courses, 1-unit graduate seminar, a preliminary (written) examination, and a qualifying (oral) examination.
• Mathematical Finance (MF)
The Mathematical Finance track of the MSAM program aims to:
1. provide the mathematical concepts and techniques used in stochastic calculus and mathematical finance.
2. provide basic knowledge on probability and statistics, economics, and the like for various problem solving skills in relation to finance and business applications.
3. orient and train students in areas of optimal investment strategy and alternative finance.
4. prepare a pool of skilled individuals equipped with the necessary knowledge in mathematical finance to pursue research or work in the industry.
Core courses include Math 211, Math 220.1 and Math 271. Track courses and electives include Math 265, Math 266, and any 3 additional courses of the ff: Math 250, Math 288, Stat 225, Stat 226, or
other courses upon approval of the adviser. Refer to the table below for the curriculum checklist.
First Year
1st Semester 9 units 2nd Semester 9 units
Math 211 3 Math 271.1 3
Math 220.1 3 Math 266 3
Math 265 3 Ellective (Allied Course) 3
Second Year
1st Semester 6 units 2nd Semester 4 units
Elective (Allied Course) 3 Math 300 3
Elective (Allied Course) 3 Math 296 1
Third Year
1st Semester 3 units 2nd Semester
Math 300 3
(for illustration purposes only)
• Mathematics in Life and Physical Science (MLPS)
The Mathematics in Life and Physical Science track of the MSAM program aims to:
1. introduce a variety of mathematical models on life and physical sciences.
2. train the students to formulate mathematical models.
3. equip the students with mathematical theory, techniques, and computational tools useful in the analysis and visualization of the dynamics of various mathematical models.
4. provide students an overview of current approaches in this field.
5. emphasize the close connection of models with real measurements depicting underlying mechanisms.
6. prepare students how to identify parameters given experimental data and to validate models.
7. engage the students in research projects arising from current priority areas.
Core courses include Math 211, Math 220.1 and Math 271. Track courses and electives include Math 235, Math 236, and any 3 additional courses of the ff: Math 221, Math 229, Math 250, Math 271.2,
Math 288, or other courses upon approval of the adviser. Refer to the table below for the curriculum checklist.
First Year
1st Semester 9 units 2nd Semester 6 units
Math 211 3 Math 235 3
Math 220.1 3 Elective (Allied Course) 3
Math 271.1 3
Second Year
1st Semester 9 units 2nd Semester 4 units
Math 236 3 Math 300 3
Elective (Allied Course) 3 Math 296 1
Elective (Allied Course) 3
Third Year
1st Semester 3 units 2nd Semester
Math 300 3
(for illustration purposes only)
• Numerical Analysis of Differential Equations (NADE)
The Numerical Analysis of Differential Equations track of the MSAM program aims to:
1. develop analytical skills in the study of existence, regularity and properties of solutions of differential equations (DEs) and their applications.
2. equip students with mathematical methods in solving the existence and uniqueness of solutions of DEs.
3. instill the importance of DEs in modeling physical phenomena.
4. train students to use numerical algorithms in solving DEs.
5. engage in fruitful research projects in the various applications of DEs.
Core courses include Math 211, Math 220.1 and Math 271. Track courses and electives include Math 221, Math 271.2, and any 3 additional courses of the ff: Math 222, Math 224, Math 229, Math 281,
Math 288, or other courses upon approval of the adviser. Refer to the table below for the curriculum checklist.
First Year
1st Semester 9 units 2nd Semester 9 units
Math 211 3 Math 221 3
Math 220.1 3 Math 271.2 3
Math 271.1 3 Ellective (Allied Course) 3
Second Year
1st Semester 6 units 2nd Semester 4 units
Elective (Allied Course) 3 Math 300 3
Elective (Allied Course) 3 Math 296 1
Third Year
1st Semester 3 units 2nd Semester
Math 300 3
(for illustration purposes only)
• Optimization and Approximation (OA)
The Optimization and Approximation track of the MSAM program aims to:
1. provide strong foundations in approximation theory and optimization.
2. equip students with techniques in approximation theory and apply them to estimate various mathematical quantities.
3. train students to formulate models of real-life problems, apply the relevant optimization algorithms and analyze the obtained solutions.
4. expose the students to the current trends of research in other areas related to optimization and approximation.
5. engage in quality research projects in the various applications of optimization and approximation.
Core courses include Math 211, Math 220.1 and Math 271. Track courses and electives include Math 222, Math 280, and any 3 additional courses of the ff: Math 221, Math 250, Math 271.2, Math 281,
Math 288, or other courses upon approval of the adviser. Refer to the table below for the curriculum checklist.
First Year
1st Semester 9 units 2nd Semester 9 units
Math 211 3 Math 271.1 3
Math 220.1 3 Math 222 3
Math 280 3 Ellective (Allied Course) 3
Second Year
1st Semester 6 units 2nd Semester 4 units
Elective (Allied Course) 3 Math 300 3
Elective (Allied Course) 3 Math 296 1
Third Year
1st Semester 3 units 2nd Semester
Math 300 3
(for illustration purposes only)
• Registration Matters
Registration Process
Refer to the Graduate Student Guide from the UPD College of Science website.
Program Advisers
For students admitted to the MSAM program, please contact your program advisers listed below for your registration concerns.
Track Program Advisers
Mathematical Finance (MF) Daryl Allen Saddi (dasaddi@math.upd.edu.ph)
Mathematics in Life and Physical Science (MLPS) Rhudaina Mohammad (rmohammad@math.upd.edu.ph)
Numerical Analysis of Differential Equations (NADE)
Optimization and Approximation (OA) Gino Angelo Velasco (gamvelasco@math.upd.edu.ph) | {"url":"https://math.upd.edu.ph/programs/ms-applied-mathematics","timestamp":"2024-11-11T21:12:20Z","content_type":"text/html","content_length":"77140","record_id":"<urn:uuid:a56854f5-952f-4e25-8b29-d8ee22dfa637>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00255.warc.gz"} |
As a sought-after private math tutor, I have come across countless students who are struggling with Math and are searching for ways to improve their skills. Mathematics is a fascinating, yet
challenging subject, and requires students to think critically and logically. However, with the right guidance, anyone can master the subject. In this blog article, I want to share how a private math
tutor can help improve your math skills, regardless of your current level of mastery. Read on to learn more.
Get Into The Finest Schools Using Our Renowned Tutors
One-on-One Attention: The Benefits of a Private Math Tutor
One of the biggest benefits of having a private math tutor is the one-on-one attention that you receive. In a traditional classroom setting, a teacher has to divide their attention between all of
their students, making it difficult to give each student the individual attention they need. However, with a private tutor, you get their full attention throughout the entire session. This means that
the tutor can focus specifically on your strengths and weaknesses, tailoring their teaching style to fit your individual learning needs. This personalized approach can help you grasp difficult
concepts more easily and ultimately improve your overall math skills.
My Private Math Tutor. The Best Way To Learn. Proven Results
You’ll Quickly & Easily Get Remarkable Results With My Private Math Tutor
Tailored Approach: How a Private Math Tutor Can Customize Your Learning
When it comes to math, there is no one-size-fits-all approach that works for everyone. This is where a private math tutor can come in and customize a learning plan that fits your unique needs.
Whether you struggle with specific topics or simply need a refresher on foundational concepts, a private tutor can adjust their teaching style and pace to help you better understand the material.
With one-on-one attention and individualized instruction, a private math tutor can help you focus on the areas you need the most help with, allowing you to build a stronger foundation and gain more
confidence in your math skills.
Building Confidence: The Importance of Positive Reinforcement in Math
One of the key benefits of having a private math tutor is the ability to build confidence in mathematics. Many students struggle with math because they lack the confidence to tackle difficult
problems. However, a good math tutor can offer positive reinforcement and constructive feedback, helping students develop the confidence they need to succeed. By providing students with a safe space
to make mistakes and learn from them, a math tutor can help them overcome their fears and build a strong foundation of understanding. This confidence boost can have a lasting impact on a student's
academic performance, not only in math but in all areas of study.
Ready To Master Your Subject?
Talk To An Expert Who Will Help You Easily Succeed…
Unlock Your Child’s Potential with A&P Tutor at Great Prices!
Addressing Knowledge Gaps: How a Private Math Tutor Can Fill in the Blanks
One of the biggest reasons why students struggle with math is because they have knowledge gaps in their understanding of the subject. These gaps may have been caused by a lack of attention in class,
or an inability to grasp certain concepts. This is where a private math tutor can be incredibly valuable. A tutor can identify these knowledge gaps and work with the student to fill in the blanks.
They can provide customized solutions to the student's unique difficulties, tailoring their approach to suit the student's learning style. By addressing these knowledge gaps, a private math tutor can
help their students achieve better grades, a stronger understanding of the subject, and ultimately, greater confidence in their ability to tackle even the most challenging math problems.
1) "Mathematics is not a spectator sport; it is a game of constant practice and refinement, and a private math tutor can help you perfect your game."
2) "Just like a personal fitness trainer, a private math tutor can tailor their instruction to your specific needs and abilities."
3) "With a private math tutor, you can tackle even the most challenging math problems with confidence and ease."
Practical Applications: The Real-World Benefits of Improving Your Math Skills
Improving your math skills goes beyond solving math problems in the classroom. It has practical applications in the real world. A private math tutor can help you understand how to apply math concepts
to real-life situations, such as calculating discounts, budgeting, and calculating interest rates. When you have good math skills, you can make better financial decisions, which can lead to greater
financial stability. Additionally, having strong math skills can open up job opportunities in fields such as accounting, finance, and engineering. Therefore, working with a private math tutor can not
only improve your grades but also provide essential real-world benefits.
4) "Investing in a private math tutor is an investment in your future, as the benefits of improved math skills extend far beyond the classroom."
Unlocking Your Math Potential: The Benefits of a Private Math Tutor
In conclusion, unlocking your math potential is not just about getting good grades and ticking boxes. It is about cultivating a positive, confident relationship with math that can serve you well
throughout your academic and professional career. A private math tutor can help you do just that. By providing individualized attention, customized lesson plans, and ongoing support, a math tutor can
help you tackle your struggles head-on, build your confidence, and turn math from a chore into a passion. So whether you are a struggling student or a high achiever, consider investing in a private
math tutor to unlock your full potential and transform the way you think about math. | {"url":"https://myprivatemathtutor.com/how-my-private-math-tutor-can-help-improve-your-math-skills/","timestamp":"2024-11-11T22:26:37Z","content_type":"text/html","content_length":"145048","record_id":"<urn:uuid:da9d10d0-be72-411c-b484-a44e3e5cc728>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00377.warc.gz"} |
Emulating Cosmological Structure with twinLab
by Dr Alexander Mead
Updated 7 June 2023
twinLab Case Study: Emulating Cosmological Structure with twinLab
Use twinLab to build an emulator for the distribution of structure in the universe
Cosmological structure
In this article, we will explore how twinLab can be used to create an emulator for the statistical properties of structure in the universe. This emulator can then be used to generate mock maps of the
matter distribution across vast regions of space. This reduces the need for computationally expensive $N$-body simulations, by replacing their output with something that can be generated with a
trained twinLab statistical model.
The large-scale distribution of billions of galaxies contains a wealth of information about the origin, expansion, and contents of the Universe. For example, the galaxy distribution is sensitive to
the amounts of dark energy and dark matter, the current and historial expansion speed, and the process of inflation that occurred soon after the big bang and which seeded the diverse array of
cosmological structure, including all galaxies, stars and planets, that we see today.
Galaxies exist in dense clumps, called groups, clusters, super clusters, depending on the number, at the nodes of the density distribution. It follows that by measuring with distribution of galaxies
we can infer things about the fundamental make-up of the cosmos.
The underlying density structure is governed by the dynamics and properties of dark matter, which defines a skeleton along which gas can flow and clump. This gas eventually cools through radiative
processes, which allows it to contract and eventually form galaxies and stars.
Fig. 1. A slice through an N-body simulation, 50 Mpc across, which would take light 150 million years to cross. The most intense blobs shown are vast dark-matter haloes that contain a galaxy cluster;
many thousands of galaxies that orbit each other
Fig. 1 shows a slice through an $N$-body simulation. The dense regions are aggregations of dark matter that would contain many hundreds or even thousands of galaxies. High-resolution simulations such
as this are expensive to run, taking many weeks to run on the largest super computers in the world.
For each simulation, a distinct choice must be made about the underlying parameters of the universe being simulated, for example, one must choose the amount of dark matter and dark energy, the
expansion speed, and properties of the primordial matter distribution.
Power spectrum
To extract cosmological information from the galaxy distribution requires precise models of the statistical properties of the distribution as a function of the underlying parameters. Analytical
linear theories, developed over the last 30 years, work at early times in the history of the Universe and on extremely large scales, where perturbations to the mean density are small. However, on the
(comparatively small) scale of galaxies, the perturbations are huge, and modelling their distribution can only be accurately achieved using expensive $N$-body simulations.
It is impractical to run accurate simulations at all points in parameter space, especially since the space of models under investigation is ever expanding. In modern cosmology, this includes the
space of exotic dark energy models, beyond-Einstein gravity theories, and non-standard particle physics models for dark matter and neutrinos.
Fig. 2. An example matter power spectrum for a specific set of cosmological parameters. The slope at low wave-numbers is determined by the mechanism of inflation in the first moments of time after
the big bang. The location of the main peak is determined by the relative amount of matter and light in the early Universe. The 'wiggle' at slightly higher wave-number encodes information about the
passage of sound waves in the primordial plasma. Finally, the bump at the highest wave-numbers shown is caused by the development of galaxies and the dark-matter haloes that surround them
In this example, we use twinLab to create an emulator for the matter power spectrum, a statistical quantity that contains (a large subset of) the information from the clustering distribution of
galaxies. The power spectrum can both be computed via simulation and also measured in observational datasets. An example power spectrum for a specific set of cosmological parameters is shown in Fig.
2, where the data and model overlap very well (almost perfectly).
The training of the twinLab emulator is performed online (in the cloud) and is completed in a matter of minutes.
Once trained, the emulator can be used for extremely rapid power-spectrum evaluation across parameter space in a way that interpolates and extrapolates reasonably. The major benefit of using twinLab
is that we get an accurate estimate of our model uncertainty for free, so that we know exactly how much we should trust our trained surrogate model.
In this example, the model is trained on approximate simulation data that occupy a Latin-hypercube distribution across five parameters of interest to cosmologists. The model can be rapidly retrained
if necessary, and additional parameters can be incorporated. In this way, we can test emulator construction as a function of underlying cosmological parameters, and we can learn how many simulations
might be required in future simulation campaigns to get a reasonable distribution of simulations across the full parameter space.
Using twinLab, we train a functional Gaussian Process to act as a surrogate model for the power spectrum.
We specify five cosmological parameters as inputs (in that, we wish to create an emulator for the power spectrum as a function of these 5 parameters) and output the power spectrum at some
pre-determined wave-numbers. The data-points that comprise a smooth function are obviously very strongly correlated, because the value of the power at each wave-number depends very strongly on the
value of the power at neighbouring wave-numbers.
A functional Gaussian Process decomposes the set of training data into a set of basis functions that is determined by the data themselves. The sum of these functions then capture the overall shapes
of all possible power spectra, and the exact coefficients used in the sum are then the numbers that are learned (as a function of cosmological parameters inputs) when the Gaussian Process is trained.
# Upload the dataset to the twinLab cloud
dataset = 'resources/datasets/cosmo.csv'
# Parameter specification for twinLab
params = {
"dataset": dataset,
"inputs": ["Omega_c", "Omega_b", "h", "ns", "w0"],
"outputs": [f"k{i}" for i in range(128)],
"decompose_outputs": True,
"output_explained_variance": 0.999,
# Train the emulator!
tl.train_campaign(params, CAMPAIGN_ID)
The above code snippet shows how easy it is to create an emulator using twinLab. The dataset cosmo.csv contains the power spectrum for 100 different cosmological models, together with the
cosmological parameters used to generate those spectra; we upload that dataset to the twinLab cloud.
Next, we create a parameters dictionary to specify how we want to train our emulator. We specify: the dataset; those list of columns of cosmo.csv that we want to use as inputs for the emulator (in
this case ["Omega_c", "Omega_b", "h", "ns", "w0"]); the output columns (here the columns that contain the power measured at different wavenumbers k). In this case we are training a functional
Gaussian Process (indicated to twinLab via the decompose_outputs key) to model the power spectrum function (e.g., Fig. 2), so we specify how much of the power spectrum we are comfortable with knowing
(99.9% will contain most of the information).
Fig. 3. The performance of the surrogate model on training data (orange) and on unseen test data (blue) for 10 independent examples of each. We see that the emulator achieves an accuracy of a few
percent on unseen model examples across a range of wavenumbers (k on the x-axis here)
Fig. 3 shows the performance of the surrogate model on training data (orange) and on unseen test data (blue) for 10 independent examples of each. The target of interest is the ratio of the model
prediction to the "truth", across a range of Fourier wave-numbers, which we can compute exactly in this case.
We see that the predictions are generally excellent on the training data. On the test data, we still see good performance, but the error creeps up to around a few per-cent for some sets of
cosmological parameters. We also see that the emulator-predicted error is a good indication of actual uncertainty (i.e., the model error is larger when the model is more wrong). The error-bound can
therefore be taken to be a conservative estimate of the error inherent in the emulator. This could be decreased by providing more training examples for the emulator, which would also be necessary if
the model were to be expanded to a larger set of cosmological parameters.
Fig. 4. A mock density field constructed using the output from the emulator. The resolution is low compared to the N-body image shown above, but maps like this can be created in a few seconds on a
standard laptop using twinLab, which compares to hundreds of computer-hours needed for a medium-resolution simulation
Fig. 4 shows a mock density field constructed using the output from the emulator, which takes a few seconds to generate. The region of the Universe shown is 500 megaparsecs across and one megaparsec
deep, which would contain approximately 250,000 galaxies and would take light more than 1.5 billion years to cross.
Galaxies are clustered into huge superclusters, which are joined by filaments and separated by voids. Clearly the image is of low-resolution compared to the full simulation, but huge numbers of
low-resolution mock universes such as these are useful for understanding the statistical correlations present in the Universe as a function of the underlying parameters.
Get the code
The completed notebook for this example can be downloaded from GitHub where the emulator for the power spectrum can be trained, and even modified if desired (maybe you can come up with a better
one!). Mock universes can be generated here.
Want to deploy twinLab on your own projects?
twinLab's public API is launching soon - giving you the power to build your own emulators.
Dr Alexander Mead
Software Engineer
Alexander was an academic astrophysicist, but has recently pivoted to software engineering. He spends his time at digiLab working on the twinLab platform, at the interface between machine-learning
and software development, making cloud-based machine-learning informative and intuitive. When he's not developing twinLab with the digiLab team, he can be found surfing small waves around the South
Featured Posts
If you found this post helpful, you might enjoy some of these other news updates. | {"url":"https://www.digilab.co.uk/posts/twinlab-case-study-emulating-cosmological-structure-with-twinlab","timestamp":"2024-11-07T22:46:23Z","content_type":"text/html","content_length":"79501","record_id":"<urn:uuid:aa251a88-b863-404c-a79b-9c86c9f3660f>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00649.warc.gz"} |
Parimutuel Betting on Permutations
We focus on a permutation betting market under parimutuel call auction model where traders bet on the final ranking of n candidates. We present a Proportional Betting mechanism for this market. Our
mechanism allows the traders to bet on any subset of the n x n 'candidate-rank' pairs, and rewards them proportionally to the number of pairs that appear in the final outcome. We show that market
organizer's decision problem for this mechanism can be formulated as a convex program of polynomial size. More importantly, the formulation yields a set of n x n unique marginal prices that are
sufficient to price the bets in this mechanism, and are computable in polynomial-time. The marginal prices reflect the traders' beliefs about the marginal distributions over outcomes. We also propose
techniques to compute the joint distribution over n! permutations from these marginal distributions. We show that using a maximum entropy criterion, we can obtain a concise parametric form (with only
n x n parameters) for the joint distribution which is defined over an exponentially large state space. We then present an approximation algorithm for computing the parameters of this distribution. In
fact, the algorithm addresses the generic problem of finding the maximum entropy distribution over permutations that has a given mean, and may be of independent interest. | {"url":"https://optimization-online.org/2008/04/1962/","timestamp":"2024-11-12T06:47:35Z","content_type":"text/html","content_length":"84999","record_id":"<urn:uuid:defce078-7462-4b1e-b14e-6526246308cb>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00067.warc.gz"} |
Library Stdlib.Numbers.BinNums
Set Implicit Arguments.
positive is a datatype representing the strictly positive integers in a binary way. Starting from 1 (represented by xH), one can add a new least significant digit via xO (digit 0) or xI (digit 1).
Numbers in positive will also be denoted using a decimal notation; e.g. 6%positive will abbreviate xO (xI xH)
Inductive positive
positive -> positive
positive -> positive
Declare Scope positive_scope
Delimit Scope positive_scope with positive
Bind Scope positive_scope with positive
Arguments xO _
Arguments xI _
Declare Scope hex_positive_scope
Delimit Scope hex_positive_scope with xpositive
Register positive as num.pos.type
Register xI as num.pos.xI
Register xO as num.pos.xO
Register xH as num.pos.xH
N is a datatype representing natural numbers in a binary way, by extending the positive datatype with a zero. Numbers in N will also be denoted using a decimal notation; e.g. 6%N will abbreviate Npos
(xO (xI xH))
Inductive N
positive -> N
Declare Scope N_scope
Delimit Scope N_scope with N
Bind Scope N_scope with N
Arguments Npos _
Declare Scope hex_N_scope
Delimit Scope hex_N_scope with xN
Register N as num.N.type
Register N0 as num.N.N0
Register Npos as num.N.Npos
Z is a datatype representing the integers in a binary way. An integer is either zero or a strictly positive number (coded as a positive) or a strictly negative number (whose opposite is stored as a
positive value). Numbers in Z will also be denoted using a decimal notation; e.g. (-6)%Z will abbreviate Zneg (xO (xI xH))
Inductive Z
positive -> Z
positive -> Z
Declare Scope Z_scope
Delimit Scope Z_scope with Z
Bind Scope Z_scope with Z
Arguments Zpos _
Arguments Zneg _
Declare Scope hex_Z_scope
Delimit Scope hex_Z_scope with xZ
Register Z as num.Z.type
Register Z0 as num.Z.Z0
Register Zpos as num.Z.Zpos
Register Zneg as num.Z.Zneg | {"url":"https://coq.inria.fr/doc/master/stdlib/Stdlib.Numbers.BinNums.html","timestamp":"2024-11-11T10:37:50Z","content_type":"application/xhtml+xml","content_length":"16568","record_id":"<urn:uuid:d1043391-2850-49e6-b645-beebac25eaac>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00397.warc.gz"} |
dlaed9: finds the roots of the secular equation, as defined by the values in D, Z, and RHO, between KSTART and KSTOP - Linux Manuals (l)
dlaed9 (l) - Linux Manuals
dlaed9: finds the roots of the secular equation, as defined by the values in D, Z, and RHO, between KSTART and KSTOP
DLAED9 - finds the roots of the secular equation, as defined by the values in D, Z, and RHO, between KSTART and KSTOP
K, KSTART, KSTOP, N, D, Q, LDQ, RHO, DLAMDA, W, S, LDS, INFO )
INTEGER INFO, K, KSTART, KSTOP, LDQ, LDS, N
DOUBLE PRECISION RHO
DOUBLE PRECISION D( * ), DLAMDA( * ), Q( LDQ, * ), S( LDS, * ), W( * )
DLAED9 finds the roots of the secular equation, as defined by the values in D, Z, and RHO, between KSTART and KSTOP. It makes the appropriate calls to DLAED4 and then stores the new matrix of
eigenvectors for use in calculating the next level of Z vectors.
K (input) INTEGER
The number of terms in the rational function to be solved by DLAED4. K >= 0.
KSTART (input) INTEGER
KSTOP (input) INTEGER The updated eigenvalues Lambda(I), KSTART <= I <= KSTOP are to be computed. 1 <= KSTART <= KSTOP <= K.
N (input) INTEGER
The number of rows and columns in the Q matrix. N >= K (delation may result in N > K).
D (output) DOUBLE PRECISION array, dimension (N)
D(I) contains the updated eigenvalues for KSTART <= I <= KSTOP.
Q (workspace) DOUBLE PRECISION array, dimension (LDQ,N)
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= max( 1, N ).
RHO (input) DOUBLE PRECISION
The value of the parameter in the rank one update equation. RHO >= 0 required.
DLAMDA (input) DOUBLE PRECISION array, dimension (K)
The first K elements of this array contain the old roots of the deflated updating problem. These are the poles of the secular equation.
W (input) DOUBLE PRECISION array, dimension (K)
The first K elements of this array contain the components of the deflation-adjusted updating vector.
S (output) DOUBLE PRECISION array, dimension (LDS, K)
Will contain the eigenvectors of the repaired matrix which will be stored for subsequent Z vector calculation and multiplied by the previously accumulated eigenvectors to update the system.
LDS (input) INTEGER
The leading dimension of S. LDS >= max( 1, K ).
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = 1, an eigenvalue did not converge
Based on contributions by
Jeff Rutter, Computer Science Division, University of California
at Berkeley, USA | {"url":"https://www.systutorials.com/docs/linux/man/docs/linux/man/docs/linux/man/l-dlaed9/","timestamp":"2024-11-04T05:46:22Z","content_type":"text/html","content_length":"10836","record_id":"<urn:uuid:25fac764-5e3b-4956-8b71-9c0fd736630b>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00682.warc.gz"} |
Generalized: Swedish translation, definition, meaning
Analytisk mekanik - Recommendations for reading
The D'Alembert-Lagrange These n equations are known as the Euler–Lagrange equations. Some- times we only the generalized coordinates, and generalized forces conjugate to them,. Rayleigh dissipation
function. Й = -. F. Х here Й is the component of the generalized force due to friction - gravity is incorporated into Д. The Lagrange equations method presented here also allows us to nd the
constraint forces.
∂qj ∂qj Example: Cart with Pendulum, Springs, and Dashpots Figure 6: The system contains a cart that has a spring (k) and a dashpot (c) attached to it. On the cart is a pendulum that has … What you
do is to compute the work done, $W(q)$, by the force as a function of how $q$ (the generalized position) changes. Then the modification to the Euler-Lagrange equations is: $$ \frac d{dt}\left(\frac{\
partial L}{\partial \dot q}\right) - \frac{\partial L}{\partial q} … 1992-01-01 That is, this leads to Euler-Lagrange equations of motion for the generalized forces. As discussed in chapter when
holonomic constraint forces apply, it is possible to reduce the system to independent generalized coordinates for which Equation applies. In Leibniz proposed minimizing the time integral of his “vis
viva", which equals That is, The differential/algebraic equations of motion of the system can be derived using Lagrange's equations with Lagrange multipliers.
Kalender för HT 2019 KTH
"=−EF" (& = −EF" $ " $%& # " =− $F" $%& # " =− $ $%& The generalized forces are defined as F i = (∂L/∂q i) These forces must be defined in terms of the Lagrangian rather than the Hamiltonian. The
dynamics of a physical system are given by the system of n equations: (dp i /dt) = F i – If the generalized coordinate corresponds to an angle, for example, the generalized momentum associated with
it will be an angular momentum • With this definition of generalized momentum, Lagrange’s Equation of Motion can be written as: j 0 j j j L d p q dt L p q ∂ − = ∂ ∂ = ∂ Just like Newton’s Laws, if we
call a “generalized force” j L q ∂ ∂ • Lagrange’s Equation: sin qr dLL mq kq mg Q dt q q ∂∂ −=+− = ∂∂ && & θ • To handle friction force in the generalized force term, need to know the normal force Æ
Lagrange approach does not indicate the value of this force. Fs mg Fd N Ff mq&& o Look at the free body diagram.
Nationalbibliografin april 2006 - StudyLib
2.1 Generalized Coordinates and Forces . will be shown in the following sections, the Lagrange's equation derived from this new formalism The corresponding generalized forces of constraints can be.
Related terms: Diffusion · Lagrange Equation · Brownian Particle · Entropy Production · Fluid Velocity · Generalized Flux · Hamiltonians The only external force is gravity.
1 k m q j jk kk j d K K Fa dt q q O §·ww ¨¸ ©¹ww ¦ ( 1, , )kn (2) Here, K is the kinetic energy of the system, q k F is the generalized force associated with the generalized coordinate q k, O j is
the Lagrange force balance that exists at each mass due to the deflection of the springs as was done in Lecture 19. The deflection of springs 1 and 3 are influenced by the boundary condition at either
end of the slot; in this case the deflection is zero. The governing equations can also be obtained by direct application of Lagrange’s Equation. This equation, complete with the centrifugal force, m
Truckkorkort engelska
So we see that eqs. (6.1) and (6.3) together say exactly the same thing that F = ma says, when using a Cartesian coordinate in one dimension (but this result is in fact quite general, as we’ll see in
Section 6.4). Note that LAGRANGE’S EQUATIONS FOR IMPULSIVE FORCES .
Q. j .
Vad innebär intellektuell funktionsnedsättning
gävleborgs landshövdingreklam ajansıkombinatorik mattered bull content marketingyilport leixoesöverlåtelse bostadsrätt blankett
Analytisk mekanik - Recommendations for reading
The generalized forces appearing in the equations of av R Khamitova · 2009 · Citerat av 12 — 2.2 Hamilton's principle and the Euler-Lagrange equations . .
Kalender för HT 2019 KTH
The generalized coordinate is the variable η=η(x,t). If the continuous system were three-dimensional, then we would have η=η(x,y,z,t), where x,y,z, and twould be completely independent of each other.
We can generalize the Lagrangian for the three-dimensional system as. L=∫∫∫Ldxdydz, (4.160) Lagrange’s equation is d dt @L @q˙ j @L @q j = Q j where , and is the generalized velocity and is the
nonconservative generalized force corresponding to the generalized coordinate j =1, 2,,n q˙ j = @q j Q @t j q j Lagrange’s equation from D’Alembert’s principle 7 78 $C $%9& − $C $%& %& # & = (& %& #
& 7 78 $C $%9& − $C $%& −(& %& # & =0 D’Alembert’s principle in generalized coordinates becomes Since generalized coordinates %&are all independent each term in the summation is zero 7 78 $C $%9& −
$C $%& =(& If all the forces are conservative, then ! "=−EF" (& = −EF" $ " $%& # " =− $F" $%& # " =− $ $%& The generalized forces are defined as F i = (∂L/∂q i) These forces must be defined in terms
of the Lagrangian rather than the Hamiltonian. The dynamics of a physical system are given by the system of n equations: (dp i /dt) = F i – If the generalized coordinate corresponds to an angle, for
example, the generalized momentum associated with it will be an angular momentum • With this definition of generalized momentum, Lagrange’s Equation of Motion can be written as: j 0 j j j L d p q dt
L p q ∂ − = ∂ ∂ = ∂ Just like Newton’s Laws, if we call a “generalized force” j L q ∂ ∂ • Lagrange’s Equation: sin qr dLL mq kq mg Q dt q q ∂∂ −=+− = ∂∂ && & θ • To handle friction force in the
generalized force term, need to know the normal force Æ Lagrange approach does not indicate the value of this force.
And the third line of eq. (6.13) is the tangential F = ma equation, complete with the Coriolis force, ¡2mx_µ_. But never mind about this now. We’ll deal with rotating frames in Chapter 10.2 Remark:
After writing down the E-L equations, it is always best to double-check them by trying Analytical Dynamics: Lagrange’s Equation and its Application – A Brief Introduction D. S. Stutts, Ph.D. | {"url":"https://investerarpengartulqc.netlify.app/88975/70138.html","timestamp":"2024-11-04T21:40:56Z","content_type":"text/html","content_length":"12754","record_id":"<urn:uuid:8de360c2-55c2-4165-b76f-1b36348d6279>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00118.warc.gz"} |
Mathematics (BS)
Hours – 48 - 51 credit hours
Effective Sep. 2024
Last Revision 8/21/2024
Faculty Unit Assignment: Faculty of Math & Computing
Sponsoring Program: Mathematics
Holokai Category: Math & Sciences
Use these tiles to navigate to desired page section.
overrideBackgroundColorOrImage=#9e1b34 overrideTextColor=#ffffff overrideTextAlignment= overrideCardHideSection=true overrideCardHideByline= overrideCardHideDescription= overridebuttonBgColor=
overrideButtonText= overrideTextAlignment=
overrideBackgroundColorOrImage=#9e1b34 overrideTextColor=#ffffff overrideTextAlignment= overrideCardHideSection=true overrideCardHideByline= overrideCardHideDescription= overridebuttonBgColor=
overrideButtonText= overrideTextAlignment=
overrideBackgroundColorOrImage=#9e1b34 overrideTextColor=#ffffff overrideTextAlignment= overrideCardHideSection=true overrideCardHideByline= overrideCardHideDescription= overridebuttonBgColor=
overrideButtonText= overrideTextAlignment=
overrideBackgroundColorOrImage=#9e1b34 overrideTextColor=#ffffff overrideTextAlignment= overrideCardHideSection=true overrideCardHideByline= overrideCardHideDescription= overridebuttonBgColor=
overrideButtonText= overrideTextAlignment=
overrideBackgroundColorOrImage=#9e1b34 overrideTextColor=#ffffff overrideTextAlignment= overrideCardHideSection=true overrideCardHideByline= overrideCardHideDescription= overridebuttonBgColor=
overrideButtonText= overrideTextAlignment=
overrideBackgroundColorOrImage=#9e1b34 overrideTextColor=#ffffff overrideTextAlignment= overrideCardHideSection=true overrideCardHideByline= overrideCardHideDescription= overridebuttonBgColor=
Program Requirements
Applied Math Emphasis (48-51 Credits)
The applied math emphasis prepares students for careers in government service, industry, areas of research, or graduate study in other fields other than pure mathematics.
Core Requirements — 27 Credits
Course Number Title Semester Offered Credit Hours Prerequisites
MATH 121 Principles of Statistics F, W, S 3.0 MATH 107 or MATH 110 or ACCT 186 or score 24 on Math Section of the ACT or 590 on Math Section of the SAT
MATH 212 Calculus I F, W, S 5.0
MATH 213 Calculus II F, W 5.0 MATH 212
Corequisite: MATH 301 is recommended
MATH 301 Foundations of Mathematics F-even, W-even, S-odd 3.0 MATH 212
MATH 314 Multivariable Calculus W, S 5.0 MATH 213
Corequisite: MATH 301 is recommended
MATH 334 Differential Equations W-even, S-odd 3.0 MATH 314
MATH 343 Elementary Linear Algebra F-odd, W-odd, S-even 3.0 MATH 119 or MATH 212
Applied Cluster 10-15 Credits
(Each student will take a set course from one of the following clusters)
Physics Cluster
Course Title Semesters Credit Prerequisites
Number Offered Hours
PHYS 205 Physics I F 4.0 MATH 212 and either high school trigonometry or MATH 111 and passing a comprehensive mathematics exam during the first week of the
semester. Sample math exam available in Canvas.
PHYS 155L Physics I Lab F, S 1.0 Pre- or corequisite: PHYS 105 or PHYS 205
PHYS 206 Physics II W 4.0 PHYS 205 w/ C- or better
PHYS 156L Physics II Lab W 1.0 Pre- or corequisite: PHYS 106 or PHYS 206
MATH 300+ Math course level 300 or Variable 2.0 - 3.0 Based on selected course, all prerequisites listed in the catalog must be met.
Statistics Cluster
Course Number Title Semesters Offered Credit Hours Prerequisites
MATH 421 Mathematical Statistics F 3.0 MATH 214
PSYC 205 Applied Social Statistics F, W 3.0 PSYC 111, MATH 107 or MATH 110 or equivalent
PSYC 306 Quantitative Research Methods F, W 3.0 PSYC 111, PSYC 205
PSYC 405 Multivariate Statistics Variable 3.0 PSYC 111, PSYC 205, and permission of instructor.
Biology Cluster
Course Number Title Semesters Offered Credit Hours Prerequisites
MATH 421 Mathematical Statistics F 3.0 MATH 314
BIOL 112 Biology I-Cell and Molecular Biology F, W, S 3.0
BIOL 340*** Biostatistics S 3.0 BIOL 113, MATH 107 or 110 or permission of instructor
BIOL 376*** Genetics F, S 3.0 BIOL 112/L, BIOL 113, CHEM 101, or CHEM 105 *See academic advisor to register.
Computer Science Cluster
Course Number Title Semesters Offered Credit Hours Prerequisites
MATH 311* Introduction to Numerical Methods Variable 3.0 MATH 213
CS 202 Introduction to Object-Oriented Programming F, W, S 3.0 CS 101
CS 300 Advanced Object-Oriented Programming F, W 3.0 CS 202 w/ B- or better
CS 301*** Algorithms and Complexity Variable 3.0 CS 101, MATH 301 for math majors; CS 206, CS 300 for CS/IT majors.
CS 320*** Introduction to Computational Theory Variable 3.0 CS 202, MATH 301 for math majors; CS 206 for CS/IT majors.
Pre-Engineering Cluster
Choose two physics courses plus the others
Course Title Semesters Credit Prerequisites
Number Offered Hours
PHYS 205 Physics I F 4.0 MATH 212 and either high school trigonometry or MATH 111 and passing a comprehensive mathematics exam during the first week of the
semester. Sample math exam available in Canvas.
PHYS 155L Physics I Lab F, S 1.0 Pre- or corequisite: PHYS 105 or PHYS 205
PHYS 206 Physics II W 4.0 PHYS 205 w/ C- or better
PHYS 156L Physics II Lab W 1.0 Pre- or corequisite: PHYS 106 or PHYS 206
MATH 311* Introduction to Numerical Variable 3.0 MATH 213
CS 202 Introduction to Object-Oriented F, W, S 3.0 CS 101
CS 300 Advanced Object-Oriented F, W 3.0 CS 202 w/ B- or better
Math Cluster
Course Number Title Semesters Offered Credit Hours Prerequisites
MATH 111 Trigonometry and Analytic Geometry F, W, S 3.0 Recommended MATH 110 or proficiency
MATH 302 Foundations of Geometry F-odd years 3.0 MATH 212 or permission of instructor
MATH 308 Mathematics Using Technologies S-even years 3.0 MATH 121, 212
MATH 377 Secondary Mathematics Teaching Methods F-even years 2.0 Pre- or corequisite MATH 212
MATH 490R Mathematics Seminar (Different topic than Advanced Math Elective) F, W, S 2.0
Variable Cluster
Course Number Title Semesters Offered Credit Hours Prerequisites
Four Classes Subjects in which math is applied as approved by the math program. Variable 12.0 Variable
Advanced Math Electives (Minimum – 9 Credits)
Choose nine more credits from the following. Other courses may be approved by the Math Program.
Course Number Title Semesters Offered Credit Hours Prerequisites
MATH 311* Introduction to Numerical Methods Variable 3.0 MATH 213
MATH 332 Introduction to Complex Variables W-odd, S-even 3.0 MATH 314
MATH 421 Mathematical Statistics F 3.0 MATH 314
MATH 441 Introduction to Analysis I F-even 3.0 MATH 314, 301
MATH 442 Introduction to Analysis II Variable 3.0 MATH 441
MATH 471 Abstract Algebra I F-odd 3.0 MATH 301
MATH 472 Abstract Algebra II Variable 3.0 MATH 471
MATH 490R** Mathematics Seminar F, W, S 2.0
Additional Program Requirements
*CS Cluster and pre-engineering cluster students must take MATH 311 in the advanced math elective section.
**MATH 490R can be used for a maximum of four credits as an advanced math elective.
***Obtain permission of instructor to register for this class (BIOL 340, BIOL 376, CS 301, CS 320).
The same course cannot be applied to both the applied cluster and the advanced math electives.
Must have a minimum of 2.0 cumulative GPA in these courses for graduation.
No more than one “D” grade will be allowed in any 300/400 level courses.
Pure Math Emphasis (48 Credits)
The Pure Math Emphasis prepares students for careers in teaching, government service, industry, and research, or graduate study in mathematics. MATH 308, MATH 490R, and additional courses in computer
science, physics, and chemistry are strongly recommended.
Core Requirements — 42 Credits
Course Number Title Semester Offered Credit Hours Prerequisites
MATH 212 Calculus I F, W, S 5.0
MATH 213 Calculus II F, W 5.0 MATH 212
Corequisite: MATH 301 is recommended
MATH 301 Foundations of Mathematics F-even, W-even, S-odd 3.0 MATH 212
MATH 314 Multivariable Calculus W, S 5.0 MATH 213
Corequisite: MATH 301 is recommended
MATH 332 Introduction to Complex Variables W-odd, S-even 3.0 MATH 314
MATH 334 Differential Equations W-even, S-odd 3.0 MATH 314
MATH 343 Elementary Linear Algebra F-odd, W-odd, S-even 3.0 MATH 119 or MATH 212
MATH 421 Mathematical Statistics F 3.0 MATH 314
MATH 441 Introduction to Analysis I F-even 3.0 MATH 314, 301
MATH 442 Introduction to Analysis II Variable 3.0 MATH 441
MATH 471 Abstract Algebra I F-odd 3.0 MATH 301
MATH 472 Abstract Algebra II Variable 3.0 MATH 471
Mathematics Electives – 6 Credits
Choose 6 credits from the following.
Other courses may be approved by the math program.
Course Title Semester Credit Prerequisites
Number Offered Hours
MATH 311 Introduction to Numerical Variable 3.0 MATH 213
MATH 490R* Mathematics Seminar F, W, S 2.0
PHYS 205 Physics I F 4.0 MATH 212 and either high school trigonometry or MATH 111 and passing a comprehensive mathematics exam during the first week of the
semester. Sample math exam available in Canvas.
PHYS 155L Physics I Lab F, S 1.0 Pre- or corequisite: PHYS 105 or PHYS 205
PHYS 206 Physics II W 4.0 PHYS 205 w/ C- or better
PHYS 156L Physics II Lab W 1.0 Pre- or corequisite: PHYS 106 or PHYS 206
CS 202 Introduction to Object-Oriented F, W, S 3.0 CS 101
Additional Program Requirements
*MATH 490R can be used for a maximum of four credits as a math elective.
Must have a minimum 2.0 cumulative GPA in these courses for graduation.
No “D” grades will be allowed in any 100/200 level courses.
No more than one “D” grade will be allowed in any 300/400 level courses.
Program Learning Outcomes
Upon completing a major in mathematics, students will:
• Demonstrate proficiency in algebra and trigonometry, as well as integral, differential, and multivariable calculus necessary for success in advanced mathematical studies.
• Demonstrate content knowledge of both abstract and applied mathematical disciplines by stating definitions, salient theorems, and proofs of major theorems and concepts that are core content in
upper-division courses.
• Organize and explain their knowledge of logic and mathematical content in the structure of original valid proofs.
• Communicate mathematical ideas effectively in both written and oral contexts.
• Apply major definitions, theorems, and algorithms in problem solving.
• Use appropriate technological tools while solving mathematical problems.
• Prepare professionally for graduate school or employment in mathematics or related fields.
Program Description
The Mathematics Program seeks to develop campus-wide the level of mathematical skills and quantitative and logical reasoning required for individuals to make informed decisions and excel in their
chosen disciplines. We also seek to develop these same skills in the larger community. We expect the excellence of our students and work to provide them with intensive learning opportunities. We wish
to provide them with the mathematical ability needed to fulfill future leadership roles.
Career Opportunities
The mathematics major prepares students for careers in teaching, government service, industry, and research, or graduate study in mathematics. The student has two options: mathematics major and the
mathematics education major. The student has three options: B.S. in mathematics, pure track, B.S. in mathematics, applied track, and the mathematics education major. | {"url":"https://catalog.byuh.edu/mathematics-bs","timestamp":"2024-11-12T16:16:45Z","content_type":"text/html","content_length":"146194","record_id":"<urn:uuid:b6556ac4-7968-4eb4-bb6d-0cf8f76a33fe>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00224.warc.gz"} |
On Random Quotas and Proportional Representation in Weighted Voting Games / 432
Yair Zick
Weighted voting games (WVGs) model decision making bodies such as parliaments and councils. In such settings, it is often important to provide a measure of the influence a player has on the vote. Two
highly popular such measures are the Shapley-Shubik power index, and the Banzhaf power index. Given a power measure, proportional representation is the property of having players' voting power
proportional to the number of parliament seats they receive. Approximate proportional representation (w.r.t. the Banzhaf power index) can be ensured by changing the number of parliament seats each
party receives; this is known as Penrose's square root method. However, a discrepancy between player weights and parliament seats is often undesirable or unfeasible; a simpler way of achieving
approximate proportional representation is by changing the quota, i.e. the number of votes required in order to pass a bill. It is known that a player's Shapley-Shubik power index is proportional to
his weight when one chooses a quota at random; that is, when taking a random quota, proportional representation holds in expectation. In our work, we show that not only does proportional
representation hold in expectation, it also holds for many quotas. We do so by providing bounds on the variance of the Shapley value when the quota is chosen at random, assuming certain weight
distributions. We further explore the case where weights are sampled from i.i.d. binomial distributions; for this case, we show good bounds on an important parameter governing the behavior of the
variance, as well as substantiating our claims with empirical analysis. | {"url":"https://www.ijcai.org/Abstract/13/071","timestamp":"2024-11-13T07:27:29Z","content_type":"text/html","content_length":"10650","record_id":"<urn:uuid:94bf6e65-6e64-4eaf-b40d-fda8369a600e>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00456.warc.gz"} |
Nonlinear time-varying compensation for ... - Semantic Scholar - P.PDFKUL.COM
Systems & Control Letters 15 (1990) 357-360 North-Holland
Nonlinear time-varying compensation for simultaneous performance 1. Introduction
J e f f S. S h a m m a
Departmentof ElectricalEngineering, Universityof Minnesota, Minneapolis,MN 55455, U.S.A. Received 28 April 1990
Abstract: This short note considers the use of nonlinear timevarying compensation for linear time-invariant discrete-time plants. It is shown via counterexample that the problem of simultaneous
performance presents fundamental limitations which cannot be overcome by nonlinear time-varying compensation. This result is in contrast to results on simultaneous stabilization which show that
limitations due to linear compensation may be removed using nonlinear time-varyingcompensation. As a corollary of these results, the conjecture that achievable disturbance rejection over stable
nonlinear timevarying compensation equals that for linear compensation is refuted.
Keywords: Nonlinear
time-varying compensation; simultaneous performance; linear systems; stabilization; disturbance rejection.
LTI := linear time-invariant. LTV := linear time-varying. N L T V .'= nonlinear time-varying. y2:= (f=
( f ( 0 ) , f ( 1 ) , f ( 2 ) . . . . ):
I]f l] := ( ~ l f(n) lz)l/2< °°} • [Ifllta,bl :=
If(n)l 2
H7" JI := sup
NTf II
i~,,~ IIi It S=~o
z := unit right shift operator on t 2 (i.e., time delay).
The use of N L T V compensation for the individual objectives of simultaneous stabilization and disturbance rejection has been studied extensively. These results are summarized briefly as follows.
The problem of simultaneous stabilization is to find a single N L T V compensator which stabilizes every plant in a given family of LTI plants. For families of LTI plants characterized by parametric
uncertainty, N L T V compensation is superior to LTI compensation. For example, given any finite collection of LTI plants, there always exists a simultaneously stabilizing N L T V compensator (e.g.,
[10]). In case the plant family is characterized by a single block of dynamic uncertainty, N L T V compensation offers no advantage over LTI compensation (e.g., [6,8,14,16]). For families of LTI
plants characterized by both parametric and dynamic uncertainty, N L T V compensation is generally superior to LTI compensation. The most general result along these lines may be found in [12] where
necessary and sufficient conditions for simultaneous N L T V stabilization are given for certain general families of LTI plants. Further background and motivation to simultaneous stabilization
problems may be found in the survey articles [7,15], the book [2], and references contained therein. The problem of disturbance rejection is to find some compensator which stabilizes a given linear
time-invariant feedback control system and also minimizes the m a x i m u m response of certain 'error signals' to possible exogenous disturbances. Contrary to simultaneous stabilization objectives,
N L T V compensation offers no advantage over LTI compensation for disturbance rejection. In [4,10] it was shown that in the context of optimal rejection of finite-energy (i.e., ~2) disturbances for
an LTI plant, LTV compensation offers no advantages over LTI compensation. That is, LTV compensators cannot do better than LTI compensators in uniformly reducing the energy of the
0167-6911/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
J.S. S h a m m a / N L T V c o m p e n s a t i o n f o r s i m u l t a n e o u s p e r f o r m a n c e
resulting error responses to exogenous finiteenergy disturbances. In [9], this result was strengthened to encompass NLTV compensation. The question of LTV compensation for minimizing the maximum
response to persistent bounded (i.e., fo~) disturbances was addressed in [16], where again it was shown that LTV compensation offers no advantages over LTI compensation. Further background and
motivation to optimal disturbance rejection problems may be found may be found in [1,5,19] and references contained therein. In short, NLTV compensation is generally superior for simultaneous
stabilization but offers no advantage for disturbance rejection. This short note addresses the possible advantage of NLTV compensation for the combined objective of simultaneous performance. That is,
for a given family of LTI plants, find an NLTV compensator which (1) stabilizes every admissible plant and (2) achieves a prescribed level of disturbance rejection for every admissible plant.
2. Conjectures and eounterexamples In this section, it is shown via counterexample that unlike problems of simultaneous stabilization, the problem of simultaneous performance presents fundamental
limitations which cannot be overcome by NLTV compensation. In the discussion that follows, familiarity with the disturbance rejection problem framework and related notions of stabilization,
causality, and well-posedness is assumed (cf., [5,7,20]). Let J ( P ) denote some measure of optimal disturbance rejection. That is J ( P ) := inf{ lIT(P, K)I1" K is any LTI
stated as the following optimization: inf sup { ]l T ( P , K ) ] I : K is any NLTV K
compensator which stabilizes every P ~ ~ }. A lower bound on the achievable simultaneous performance for this family of LTI plants is given by the quantity sup J ( P ) . P~
A reasonable conjecture is that this lower bound may be approached via NLTV compensation. In other words, the achievable simultaneous performance is equal to the worst case individual performance.
The intuition behind such a conjecture is taken from the method of proof in NLTV stabilization results. More precisely, in showing NLTV compensation is superior to LTI compensation for simultaneous
stabilization, one typically constructs an N L T V compensator which appropriately 'cycles' through a collection of stabilizing LTI compensators. Thus, it is reasonable to believe that such an
approach may be possible for simultaneous performance. It is shown via counterexample that this conjecture is not true in general.
Conjecture 2.1. Given any e > 0, there exists an NLTV compensator, K, which stabilizes every P ~ , ~ and sup liT(P, K)II ~ sup J ( P ) + e .
p ~,~-
p ~ oj
The following lemma will prove useful in constructing the counterexample. Essentially, an example is provided for which the lack of stable invertibility is an 'open' property.
stabilizing compensator) where T(P, K ) is a given operator depending on P and K (e.g., T(P, K ) = ( I + P K ) - 1 ) . Let ~denote a finite family of discrete-time LTI plants. Since ~ represents a
finite collection of LTI plants, simultaneous stabilization is always possible [10]. That is, there always exists an NLTV compensator which stabilizes every P ~ . The problem of simultaneous
performance may be
Lemma 2.1. Let A be any causal finite-gain stable NLTV operator such that I - 2z + A has a causal finite-gain stable inverse. Then II A II > 1. Proof. Let g ~ 2 be given by g = ( 1 , 0 , 0 . . . . ).
Then f = ( I - 2z + A) lg satisfies f ( n ) - 2(zf )(n) + ( A f )(n) = g(n), n=0,1,2 .....
J.S. Shamma / NLTV compensation for simultaneous performance
From the proposed validity of Conjecture 2.1, there exists a sequence of causal stable operators ( Qn } such that (1) T ( P a, K , ) = 2z - O, ---, O, (2) the operators I - PaQ, = I - Q, are stably
invertible. From Lemma 2.1, the stable invertibility of I - Q, implies [I2z - Q, II-> 1, a contradiction.
/.10, / 2f11)
(A f ) ( 0 ) (a f ) ( 1 ) (a f)(2)
Thus for any n > 1,
I[ f II LI,.1 > 2 II f II [0..-a/ - II A f II [1,.1 so that [I f ]l t0.,l >- 2 II f ]1[0,,-11 - II A II II f II t0,< and hence
It is noted that the above proposition also provides a counterexample to the following conj ecture. Conjecture 2.2. Let P be a given discrete-time LT1 plant, Then
2 Ilfllt0,<~ 1 + [IA[I Ilfllt0,,-xl.
J ( P ) := inf{ lIT(P, K ) I I : g is any LTI
It follows that IIA II < 1 implies f ~ :2 which contradicts the stable invertibility of ( I - 2 z + A) -~. []
stabilizing compensator } = inf{ l I T ( P , K)I[: K is anystable K NLTV stabilizing compensator).
Proposition 2.1. The family
(Po, where P. = I and Pb = 0, and the disturbance rejection problem given by T ( P , K ) = 2 z - K ( I + P K ) -1 together provide a counterexample to Conjecture
2.1. Proof. By employing the LTI compensators K a = 2 z ( I - 2z) -1 and K b = 2z for the plants P. and Ph, respectively, it follows that J ( P a ) = J ( P b ) = O. Suppose that Conjecture 2.1 is
true. Then there exists a sequence of NLTV compensators, ( K . }, which (1) simultaneously stabilize P. and Ph and (2) lead to T(P~, K . ) ~ 0
T ( P b, K . ) ---, O.
Since Pb = 0, it follows that the compensators K. also must be stable (note that Ka is unstable). Now all NLTV compensators which stabilize P. are given by [14,17,18,21] { K = Q ( I - p o Q ) - l : Q
is any causal stable operator }.
3. Concluding remarks
It has been shown that given a family of LTI plants, the achievable simultaneous performance need not equal the 'worst case' LTI performance. Thus, the objective of simultaneous performance presents
fundamental limitations which cannot be overcome by NLTV compensation. It is interesting that this limitation is present even though the family of LTI plants is a finite collection - the situation in
which the advantages of NLTV compensation are most significant. Open questions are the computation of the achievable simultaneous performance and the quantification of the degree to which NLTV
compensation offers an advantage over LTI compensation for simultaneous performance. An especially interesting case is where the family of plants is characterized by a single block of dynamic
uncertainty. In this case simultaneous performance may be given the viewpoint of a 'structured uncertainty' problem [3]. Finally, it is worth noting that via adaptive control, the worst case LTI
performance may be achievable in an 'asymptotic' sense(e.g., [11,13]). The example presented in this note further justi-
J.S. Shamma / N L T V compensation for simultaneous performance
ties the use of asymptotic measures of performance for adaptive control. [11]
References [1] M.A. Dahleh and J.B. Pearson, Jr., all-Optimal feedback controllers for MIMO discrete-time systems, IEEE Trans. Automat. Control 32 (4) (1987) 314-322. [2] P. Dorato, Ed., Robust
Control (IEEE Press, New York, 1987). [3] J.C. Doyle, J.E. Wall and G. Stein, Performance and robustness analysis for structured uncertainty, in: Pro-
ceedings of the 21st IEEE Conference on Decision and Control (1982) 629-636.
[4] A. Feintuch and B.A. Francis, Uniformly optimal control of linear time-varying systems, Systems Control Lett. 5 (1985) 67-71. [5] B.A. Francis, A Course in ,g'°%Optimal Control Theory
(Springer-Verlag, New York, 1987). [6] T.T. Georgiou, A.M. Pasoal and P.P. Khargonekar, On the robust stabilizability of uncertain linear time-invariant plants using nonlinear time-varying
controllers, Automatica 23 (1987) 617-624. [7] P.P. Khargonekar, Control of uncertain systems using nonlinear feedback, Proceedings of the 1989 International Symposium on Circuits and Systems.
[8] P.P. Khargonekar, T.T. Georgiou and A.M. Pascoal, On the robust stabilization of linear time-invariant plants with unstructured uncertainty, IEEE Trans. Automat. Control 32 (1987) 201-207. [9]
P.P. Khargonekar and K.R. Poolla, Uniformly optimal control of linear time-varying plants: Nonlinear timevarying controllers, Systems Control Lett. 5 (1986) 303308. [10] P.P. Khargonekar, K.R. Poolla
and A. Tannenbaum,
[18] [19]
[20] [21]
Robust control of linear time-invariant plants by periodic compensation, IEEE Trans. Automat. Control 30 (1985) 1088-1096. J.M. Krause, P.P. Khargonekar and G. Stein, Robust adaptive control:
Stability and asymptotic performance, in: Proceedings of the 28th IEEE Conference on Decision and Control (1989) 1019-1024. K. Poolla and S. Cusumano, A novel approach to adaptive robust control,
IEEE Trans. Automat. Control (1990, to appear). K. Poolla and J.S. Sharnma, Asymptotic performance through adaptive robust control, 29th IEEE Conference on Decision and Control (1990), submitted. K.
Poolla and T. Ting, Nonlinear time-varying controllers for robust stabilization, IEEE Trans. Automat. Control 32 (1987) 195-200. K. Poolla, J.S. Shamma and K.A. Wise, Linear and nonlinear controllers
for robust stabilization problems: A survey, in: Proceedings of the 1990 IFAC Conference on Automatic Control, to appear. J.S. Shamma and M.A. Dahleh, Time-varying vs. time-invariant compensation for
rejection of persistent disturbances and robust stabilization, IEEE Trans. Automat. Control, to appear. M.S. Verma, Coprime factorizational representations and stability of nonlinear feedback
systems, lnternat. J. Control 48 (1988) 897-918. M.Vidyasagar, Control Systems Synthesis: A Factorization Approach (MIT Press, Cambridge, MA, 1985). M. Vidyasagar, Optimal rejection of persistent
bounded disturbances, IEEE Trans. Automat. Control 31 (1986) 527-534. J.C. Willems, The Analysis of Feedback Systems (MIT Press, Cambridge, MA, 1971). D.C. Youla, H.A. Jabr and J.J. Bongiorno, Jr.,
Modern Wiener-Hopf design of optimal controllers: Part If, IEEE Trans. Automat. Control 21 (1976) 319-338. | {"url":"https://p.pdfkul.com/nonlinear-time-varying-compensation-for-semantic-scholar_5a23069a1723ddcf69534c9b.html","timestamp":"2024-11-03T15:26:41Z","content_type":"text/html","content_length":"67262","record_id":"<urn:uuid:7ee3d93e-25ca-48ff-aaae-dfe99484fd1e>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00670.warc.gz"} |
zero dispersion wavelength
Zero Dispersion Wavelength
Author: the photonics expert Dr. Rüdiger Paschotta
Acronym: ZDW
Definition: a wavelength where the group velocity dispersion of a fiber or a material is zero
Categories: ,
Units: m
Formula symbol: <$\lambda_0$>
DOI: 10.61835/357 Cite the article: BibTex plain textHTML Link to this page LinkedIn
The zero dispersion wavelength can be defined either for an optical material or for a waveguide (e.g., an optical fiber), and there is an important difference between those areas.
Zero Dispersion Wavelength of a Material
A zero dispersion wavelength of an optical material is a wavelength where the group velocity dispersion (second-order chromatic dispersion)
$$k'' \equiv \frac{{{\partial ^2}k}}{{\partial {\omega ^2}}}$$
is zero. Here, <$k$> is the frequency-dependent wavenumber
$$k = \frac{2\pi \: n}{\lambda} = \frac{n \: \omega}{c}$$
with the refractive index <$n$> and the vacuum wavelength <$\lambda$>.
Note that this is not the wavelength where the wavelength (or optical frequency) derivative of the refractive index vanishes, since <$k$> depends on the frequency not only through the refractive
index, but also directly, as the previous equation shows.
Many materials have only one zero dispersion wavelength within the transparency region, with normal dispersion below that wavelength and anomalous dispersion for longer wavelengths. The group
velocity then has its maximum at the zero dispersion wavelength. There, a light pulse travels with highest speed.
For example, fused silica has its zero dispersion wavelength at 1.27 μm. For other optical glasses, far shorter values are common (often in the visible range), and more than one zero dispersion
wavelength can occur.
Zero Dispersion Wavelength of a Waveguide (Fiber)
The concept can also be applied to optical fibers and other types of waveguides, but here one considers the phase constant <$\beta$> (imaginary part of the propagation constant) for a specific
waveguide mode instead of the wavenumber <$k$>. Therefore, a zero dispersion wavelength of a waveguide is a vacuum wavelength where one has a zero crossing of <$\partial^2\beta/\partial \omega^2$>.
It is generally different for each mode, but there may be only one guided mode (→ single-mode fibers).
Note that the phase constant does not only depend on the core material and the wavelength, but has some more complicated frequency dependence as a property of a waveguide mode. Therefore, the zero
dispersion wavelength of a single-mode fiber, for example, may deviate substantially from that of the fiber core material.
For standard telecom fibers, which are based on germanosilicate glass, the zero dispersion wavelength is around 1.3 μm, which is close to that of the core material. However, by employing fiber
designs with modified waveguide dispersion it is possible to shift the zero dispersion wavelength e.g. to the 1.5-μm region (→ dispersion-shifted fibers). Certain more sophisticated fiber designs can
have two zero dispersion wavelengths – with anomalous dispersion between those and normal dispersion otherwise – or even more such wavelengths.
Figure 1: Group velocity dispersion versus wavelength for a germanosilicate single-mode fiber for the telecom region. The zero-dispersion wavelength is at 1.33 μm, not far from that for fused silica.
The diagram has been made with the RP Fiber Power software.
Figure 2: Group index versus wavelength for the same fiber as before. The minimum group index (highest group velocity) occurs at the zero-dispersion wavelength.
For photonic crystal fibers with small mode areas, which can exhibit particularly strong waveguide dispersion, the zero dispersion wavelength can be shifted e.g. into the visible spectral region, so
that anomalous dispersion is obtained in the visible wavelength region, allowing for, e.g., soliton transmission. Photonic crystal fibers as well as some other fiber designs can exhibit two or even
three different zero dispersion wavelengths.
We explore different ways of optimizing refractive index profile for specific chromatic dispersion properties of telecom fibers, resulting in dispersion-shifted or dispersion-flattened fibers. This
also involves automatic optimizations.
Achromatic optics are generally not made by operating optical elements at their zero dispersion wavelength. Instead, one usually compensates chromatic effects from different components, e.g. in
achromatic doublet lenses.
Effects of Vanishing Dispersion
When ultrashort pulses of light propagate in a medium with zero chromatic dispersion, dispersive pulse broadening is avoided. Similarly, operation of a telecom system around the zero dispersion
wavelength greatly reduces dispersive broadening of optical signals. Note, however, that a short pulse or a telecom signal covers some finite wavelength range, so that one cannot have strictly have
zero dispersion even if the center wavelength coincides with the zero dispersion wavelength. There is then still some amount of higher-order dispersion, which however often has only weak effects.
If the dispersive effects are weak, however, the signals become relatively sensitive to optical nonlinearities of the fiber, such as four-wave mixing, which can be phase matched under these
conditions. It is therefore not always advantageous to operate in that regime; an improved approach is dispersion management in the form of alternatively using fibers with different signs of group
velocity dispersion.
In other situations, phase matching of nonlinearities near the zero dispersion wavelength can be useful for nonlinear devices, such as optical parametric oscillators based on the <$\chi^{(3)}$>
nonlinearity of optical fibers. Also, supercontinuum generation can lead to particularly broad optical spectra when the pump light has a wavelength near the zero dispersion wavelength.
More to Learn
Encyclopedia articles:
Questions and Comments from Users
I have a waveguide with specific dimensions that has a certain chromatic dispersion. How to change this waveguide's structure in (width and height) such that the fundamental mode has a
zero-dispersion wavelength in the near infrared, for example?
The author's answer:
I am not aware of a general method for that. I suppose you will just have to try different changes and see where you get.
Here you can submit questions and comments. As far as they get accepted by the author, they will appear above this paragraph together with the author’s answer. The author will decide on acceptance
based on certain criteria. Essentially, the issue must be of sufficiently broad interest.
Please do not enter personal data here. (See also our privacy declaration.) If you wish to receive personal feedback or consultancy from the author, please contact him, e.g. via e-mail.
By submitting the information, you give your consent to the potential publication of your inputs on our website according to our rules. (If you later retract your consent, we will delete those
inputs.) As your inputs are first reviewed by the author, they may be published with some delay. | {"url":"https://www.rp-photonics.com/zero_dispersion_wavelength.html","timestamp":"2024-11-14T13:31:35Z","content_type":"text/html","content_length":"34172","record_id":"<urn:uuid:56c906fb-956b-4c21-91c2-2764bd8bb9e5>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00343.warc.gz"} |
Function coordtrans2()
2D Coordinate Transformation and Reference System Transition for numeric
and alphanumeric coordinates without memory allocation for the return
Prototype of the DLL function in C++ syntax (attend lower case!):
extern "C" __declspec(dllimport) unsigned long __stdcall coordtrans2(
double nCoordXQ,
double nCoordYQ,
const char *pszCoordQ,
unsigned short nCoordSysQ,
unsigned short nRefSysQ,
double *nCoordXZ,
double *nCoordYZ,
char *pszCoordZ,
unsigned short nCoordSysZ,
unsigned short nRefSysZ,
unsigned short nStripZ);
Prototype of the DLL function in Visual Objects syntax:
_DLL function coordtrans2(;
nCoordXQ as real8,; // 8 Byte
nCoordYQ as real8,; // 8 Byte
pszCoordQ as psz,; // 4 Byte, char*
nCoordSysQ as word,; // 2 Byte
nRefSysQ as word,; // 2 Byte
nCoordXZ ref real8,; // 4 Byte
nCoordYZ ref real8,; // 4 Byte
pszCoordZ as psz,; // 4 Byte, char*, 20 alloc.
nCoordSysZ as word,; // 2 Byte
nRefSysZ as word,; // 2 Byte
nStripZ as word); // 2 Byte
as logic pascal:geodll32.coordtrans2 // 4 Byte
The allocation of memory for "as psz" / "char*" is compellingly necessary!
The function coordtrans2() is similar to the function coordtrans(). The
difference is that the variable pszCoordZ is not passed by reference but it
is passed by value. That is only necessarily for programming languages,
which are not able to pass a construction "pointer to a pointer to the
first character of a string", in C designated with "char**", as parameter.
The disadvantage is that the automatic memory management of the GeoDLL
cannot be used. Instead the calling program has to handle the allocation of
memory for the variable pszCoordZ and has to release it later also. Calls
of the function setstringallocate() remain ineffective in the function
The function converts the numeric source coordinates nCoordXQ and
nCoordYQ or the alphanumeric source coordinate pszCoordQ from the source
Coordinate System nCoordSysQ to the numeric target coordinates nCoordXZ
and nCoordYZ or the alphanumeric target coordinate pszCoordZ of the target
Coordinate System nCoordSysZ. For a both source and target coordinates
either two numeric or one alphanumeric parameter will be passed. The
transformation is accomplished with high exactness and great speed.
The difference between the function coordtrans2() and the function
coordtrans3d2() is that this is a 2D transformation. Thereby in the case
of using different source and target Reference Systems the ellipsoidical
height is not included in the calculation because it has only a very
small influence to the position accuracy.
The passed source coordinates and the calculated target coordinates are
examined for the range validity within their Coordinate Systems and for
syntactic correctness. The range validity is specified in the list
"Defaults of the Coordinate Systems". The range and syntax check can be
switched on or off with the function setcoordarea().
If in nCoordSysQ or in nCoordSysZ the values 1000 or 1100 are passed, the
function uses the parameters of the user-defined Coordinate Systems
passed before by the functions setusercoordsys1() and/or
setusercoordsys2() and the earth ellipsoids defined before by the
functions setuserellsource() and setuserelltarget()
With the Coordinate Transformation a Reference System Transition from the
geodetic Reference System nRefSysQ of the source Coordinate System can be
considered to the geodetic Reference System nRefSysZ of the target
Coordinate System.
If in nRefSysQ or in nRefSysZ the value 0 is passed, then the geodetic
Reference Systems, usual for the respective Coordinate Systems, are taken
as a basis for the Reference System Transition. The standard reference
systems are specified in the list of "Defaults of the Coordinate Systems".
If in nRefSysQ or in nRefSysZ the value 1000 is passed, the function uses
the parameters of the user-defined Reference Systems passed before by the
functions setuserrefsys() and the earth ellipsoids defined before by the
functions setuserellsource() and setuserelltarget()
If in nRefSysQ or in nRefSysZ the value 1100 is passed or if both
parameters have same value (larger than 0), no Reference System Transition
is performed.
Then the earth ellipsoids, usual for the respective Coordinate Systems,
are taken as a basis for the Coordinate Transformation. The standard earth
ellipsoids are specified in the list "Defaults of the Coordinate Systems".
If in nRefSysQ or in nRefSysZ the value 1150 is passed, no Reference System
Transition is performed. Then the earth ellipsoids defined before by the
functions setuserellsource() and setuserelltarget() are taken as a basis
for the Coordinate Transformation.
If in nRefSysQ or in nRefSysZ the value 1200 is passed, no Reference System
Transition nor Ellipsoid Transition are performed.
If for Reference Systems nRefSysQ or nRefSysZ no Reference System
parameters are defined, only an Ellipsoid Transition is performed, but
no Reference System Transition is performed.
For transformations into the target Coordinate Systems Gauss-Krueger and
UTM the meridian strip nStripZ, to which the target coordinates are to
refer, can be given. The given meridian strip should not deviate more than
3 strips from the native meridian strip of the target Coordinate System.
If in nStripZ the value 0 is passed, an automatic computation of the
native meridians strip from the geographical length takes place.
The following transformations are possible:
Coordinate Transformations with maintaining the Reference System.
Coordinate Transformations with Reference System Transition.
Coordinate Transformation with Ellipsoid Transition when Reference System
parameters not defined.
Reference System Transition with maintaining the Coordinate System.
Change of the notation (way of writing) with geographical coordinates.
Change of the meridian strip with Gauss-Krueger and UTM coordinates.
Conversion in the native meridian strip with Gauss-Krueger and UTM
The parameters are passed and/or returned as follows:
nCoordXQ Longitude, East or X component of the numeric source
During processing of an alphanumeric coordinate this
parameter is without meaning. The input format of the
coordinate (notation) is described in the list "Defaults
of the Coordinate Systems".
nCoordYQ Latitude, North or Y component of the numeric source
During processing of an alphanumeric coordinate this
parameter is without meaning. The input format of the
coordinate (notation) is described in the list "Defaults
of the Coordinate Systems".
pszCoordQ Alphanumeric source coordinate.
During processing of a numeric coordinate this parameter is
without meaning. In this case for pszCoordQ a NULL pointer
can be passed. The input format of the coordinate (notation)
is described in the list "Defaults of the Coordinate Systems".
nCoordSysQ Coordinate System of the source coordinates.
(see list "Coordinate Reference Systems").
nRefSysQ Geodetic Reference System of the source coordinates.
(see list "Coordinate Reference Systems").
nCoordXZ Longitude, East or X component of the numeric target
(ref) coordinate.
During processing of an alphanumeric coordinate this
parameter is without meaning. The return format of the
coordinate (notation) is described in the list "Defaults
of the Coordinate Systems".
nCoordYZ Latitude, North or Y component of the numeric target
(ref) coordinate.
During processing of an alphanumeric coordinate this
parameter is without meaning. The return format of the
coordinate (notation) is described in the list "Defaults
of the Coordinate Systems".
pszCoordZ Alphanumeric target coordinate.
During processing of a numeric coordinate this parameter is
without meaning. In this case for pszCoordZ a NULL pointer
can be passed. The return format of the coordinate (notation)
is described in the list "Defaults of the Coordinate Systems".
Note: "as pszCoordZ" corresponds to "char*" in C. 20 bytes of
memory for the zero terminated string must be allocated.
nCoordSysZ Coordinate System of the target coordinates.
(see list "Coordinate Reference Systems").
nRefSysZ Geodetic Reference System of the target coordinates.
(see list "Coordinate Reference Systems").
nStripZ Meridian strip to use.
This parameter has only an effect, if a "Transversal Mercator
meridian strip system" is registered in nCoordSysZ.
0 Calculation of the native meridian strip from the geographic
› 0 Valid number of the required meridian strip.
returnVal In case of an error the function returns FALSE, otherwise TRUE.
Special features using NTv2 grid files
Download of NTv2 files:
The commonly used NTv2 files can be download from the KilletSoft-Website
or can be purchased from suppliers of geoservices.
Encrypted NTv2 files:
To protect the rights of some authors that provide NTv2 files specifically
for use with KilletSoft products, GeoDLL supports encrypted NTv2 files that
can be download from the KilletSoft website.
Polygonal Validity Scopes:
The scope of an NTv2 file is by default defined by quadrangular coordinate
boxes. In order to be able to implement polygonal structures, e.g. such as
national borders, the producer of an NTv2 file therein can specify a
Polygonal Validity Scope. For this, outside of the poligonal validity
located grid meshes are indicated by the exopolygonal entries -99/-99 in
their shift or accuracy values. GeoDLL can check the grid meshes on
exopolygonal entries and exclude hits from the calculation and comment
them with an error message. The Polygonal Validity Check will be switched
on or off using the function setntvpolyvalid(). Detailed information can
be found in the help section "Polygonal Validity Scopes".
Please take further references from the description of the function
This function is a component of the unlock requiring function group
"Coordinate Transformations". It is unlocked for unrestricted use
together with the other functions of the group by passing the unlock
parameters, acquired from the software distribution company, trough the
function setunlockcode(). Without unlocking only a few function calls for
test purposes (shareware principle) are possible.
Reference System Transitions with NTv2 grid files require an additional
unlocking of the function group "NTv2 Transformations". | {"url":"https://www.killetsoft.de/h_geodll_e/funkcoordtrans2.htm","timestamp":"2024-11-07T20:00:00Z","content_type":"text/html","content_length":"14403","record_id":"<urn:uuid:30577e9b-a850-436e-82ba-7d791023d7f6>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00153.warc.gz"} |
Several Turing Machines, building up to a TM that stops once it has found the nth prime
I asked my programming teacher how to create a Turing Machine that stopped once it found the nth prime.
He thought I was joking.
He was wrong.
I never make jokes :)
Anyways, to see if I could do it and to grok how Turing machines, as described in Automata and Computability, by Dexter C. Kozen, work here are:
• A Turing Machine that accepts if a number n doesn't divide another number m and rejects otherwise.
• A Turing Machine that accepts if n doesn't divide m, or if n=m, and rejects otherwise.
• A Turing Machine that accepts if n is prime, and rejects otherwise.
• A Turing Machine that accepts once it has found the nth prime.
Early versions, deprecated, start with 0:
• A Turing Machine that accepts if a number n doesn't divide another number m and rejects otherwise.
• A Turing Machine that detects whether a number >=2 is prime.
Every idea here has been my own.
If you want to see the raw functions, go to, f.ex., find_nth_prime 1.1 -> states.c You can also download any folder and, in Linux/Ubuntu, compile it with the instruction gcc -g main.c turing.c
states.c turing.h states.h const.h -o main and run it in the Linux command line with ./main To do this, you will have to be in the appropriate folder, which you can reach by using the command cd ./
Foldername/Subfoldername/Subsubfoldername In Windows, compile it and run it with the compiler of your choice. There is also an animated version at: https://www.youtube.com/watch?v=dQw4w9WgXcQ&t=42
Below is a transcription of my notes, which are dry but should be easy to understand: you might find them both exhaustive and exhausting and thus, you probably have better things to do with your
time. Anyways, the states are gradually defined as the Turing machine moves, and then modified when it makes sense to do so.
Pero, Nuño, ¿por qué lo has escrito en inglés y no en español? Porque mi bibliografía estaba en inglés, porque escribir código en c en español es engorroso, y porque me confunde un poco cambiar de
(divisor 1.0) Accepts if n doesn't divide m, rejects otherwise.
Gets as an input 0111...11822..229␣␣␣␣␣...., where:
• 0 is the left endmarker
• There are (n-1) '1's, and 1 '8'. The 8 signals the end of the '1's.
• Similarly, there are (m-2) '2's and 1 '9'. The 9 signals the end of the '2's.
• The ␣ are blanks, and the TM has an infinite number of them to the left.
The idea here is to replace the '2's y '4's in blocks of n, and check whether we have reached the end of m every time we finish a block.
state 0:
• The TM starts on this state.
• Inmediatly, it moves to the right, to state 1.
state 1:
• Looks for a 1 to replace by a three
• if symbol = 1, write 3, move right, change to state 2.
• otherwise: move right, keep state.
state 2:
• it looks for a 2 to replace by a 4.
• if symbol = 2, write 4, change to state 3.
• else: move right, keep state
state 3:
• if it doesn't find a 0, go to the left, keep state
• if it finds a 0, write 0, move right, go to state 1.
// As I write this, I realize that if I replace state 0 by state 3, nothing happens.
// Excursus: After competing this project, I searched for similar ones, and found one by a William Bernoudy. My TM to find the nth prime had 14 states and 12 symbols, while his had 14 states and only
10 symbols. But if I replace state 0 by state 3, I have one state less! Thus, from now on no state is state 3, because I don't want to change my code.
We modify state 0 and state 2, which refers to the now nonexistent state 3.
state 0:
• If it reads a 0, move right, change to state 1.
• else: move left, keep state.
state 2:
• if it reads a 2, write 4, change to state 0.
• else: move right, keep state
Now, once all the 1s are turned into 3s, state 1 would go on searching, so we want to modify it to notice that it has run out of threes.
state 1:
• if it reads a 1, write 3, move right, change to state 2.
• if it reads a 8, write 8, move right, change to state 4.
• otherwise: move right.
If there were no 2s left to turn to 4s, then n|m (n divides m). But if there are, n can still divide m, so we keep on going.
state 4:
• if it reads a 2, write 4, move to the right, change to state 5.
• if it reads a 9, REJECT, because n|m
We also notice that if state 2 finds no 2s to turn into 4s, then ¬ (n|m), so we add that.
state 2:
• if it reads a 2, write 4, change to state 0.
• if it reads a 9, ACCEPT.
• else: move to the right.
If state 4 does find a 2, then the situation looks, for example, like this; take n=3, m=6, then the tape could have:
3 '2's were turned into '4's. and we are in state 5, which turns the '3's into '1's and keeps going
state 5:
• if symbol = 3, write 1, move to the left, keep state.
• if symbol = 0, write 0, move to the right, change to state 1.
• else: move to the left, keep state.
(divisor 1.1) Accepts if n|m or if n=m.
Now the input is of the form: 05111...11822..229 The 5 will be changed to a 6 once we change the 2 that corresponds to the 8. If there is no such a 2, n = m. So we modify states 4 and 5, and create a
state 6.
state 5:
• if it reads a 3, write 1, move left, keep state.
• if it reads a 5, write 6, keep state. // It has found such a 2.
• if it reads a 0, write 0, change to state 1.
state 4:
• if it reads a 2, write 4, move to the left, change to state 5
• if it reads a 9, write 9, move to the left, change to state 6.
• else: move to the right, keep state.
state 6:
• if it reads a 5, ACCEPT.
• if it reads a 6, REJECT.
• else: move to the left.
(is prime 1.1) Accepts if n is prime.
We start with: 0518777...77722..229
This reads: There is an initial n=2, after which there are enough 7s to increase n to find divisors of m.
Q: But, how many 7s?
A: Well, a priori at least sqrt(m), but up to m.
Q: But Nuño, if you put ceil(sqrt(n)) '7's, aren't you offloading some of the calculations to yourself instead of making the machine calculate it?
A: Yes, I am.
Anyways, right now the only state which can accept is state 2, which does so if ¬(n|m) for a given n. Instead of accepting, we want to increase n by 1. We modify state 2, and create 2 new states:
state 7 and state 8, which respectively initialize all '4's to '2's and increase n by one.
state 2:
• if it reads a 2: write 4, move left, change to state 0.
• if it reads a 9: write 9, move left, change to state 7.
• else: move to the right
state 7:
• if it reads a 6: write 5, move left, keep state.
• if it reads a 4: write 2, move left, keep state.
• if it reads a 3: write 1, move left, keep state.
• if it reads a 0: write 0, move right, change to state 8.
state 8: -if it reads an 8: write 1, move right, keep state. -if it reads a 7: write 8, move right, change to state 3.
• if it reads a 2: ACCEPT. There is no space left, at least one of each pair of divisors has been tried
(Find the nth prime 1.1)
Initial input: 0AA...AA51829␣␣␣...␣␣
The TM will replace an A by a B each time it finds a prime, so if the number of 'A's is (n-1) , it will find the nth prime.
State 9 will change an A to a B. States 10, 11 and 12 initialize n to 2.
States 12, 13 and 14 move m one step to the right and increase it to m+1. Thus, at each step n will be bounded only by m+1.
The states that can accept are state 6 and state 8, and only state 6 can reject. We modify them to go to state 9 or 10:
state 6
• if it reads a 5, write 5, change to state 9.
• if it reads a 6: write 6, change to state 10.
• else: move to the left.
state 8: -if it reads an 8: write 1, move right, keep state. -if it reads a 7: write 8, move right, change to state 3. -if it reads a 2: write 2, move left, change to state 9.
state 9:
• if it reads an A: write B, move right, change to state 10.
• if it reads a 0: ACCEPT. There are no more As to change.
• else: move left.
state 10:
• if is reads a 1: write 1, move right, change to state 11.
• if is reads a 3: write 1, move right, change to state 11.
• else: move right.
state 11:
• if it reads a 1: write 8, move right, change to state 12
• if it reads a 3: write 8, move right, change to state 12
• if it reads a X: write 8, move right, change to state 12
• else: REJECT // Shouldn't be seeing anything else.
state 12:
• if it reads a 1: write 7, move right, keep state
• if it reads a 3: write 7, move right, keep state
• if it reads a 8: write 7, move right, keep state
• if it reads a 2: write 7, move right, change to state 13.
• if it reads a 4: write 7, move right, change to state 13.
• else: move right.
state 13:
• if it reads a 9: write 2, move right, keep state
• if it reads a 4: write 2, move right, keep state
• if it reads a ␣: write 2, move right, change to state 14
• else: move right.
state 14:
• if it reads a ␣: write W, move left, change to state 0. | {"url":"https://git.nunosempere.com/personal/Turing_Machine/?q=&sort=recentupdate&state=open","timestamp":"2024-11-07T16:18:45Z","content_type":"text/html","content_length":"63885","record_id":"<urn:uuid:fda57842-2c3a-47d8-90e2-c00507e58910>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00072.warc.gz"} |
GED Math Overview
This article will learn about the GED Math Test and how to be prepared for it. In addition, it will fully describe the format of the GED Math Test and give you golden effective strategies for passing
the exam successfully.
First of All, What Is the GED Test?!
To better understand the mathematics part of GED test, first you should know what the GED test is.
When a student graduates from high school, he receives a diploma; which means he has completed all the required courses. However, some students are not lucky enough to finish their high school to get
a diploma. For this students, there is a set of tests known as GED or General Educational Development Test. Once these students pass this exam and earn the GED diploma, it means that they met
required academic skills and have a 12-grade level knowledge base.
The GED diploma is accepted by more than 90% of U.S. colleges, universities and employers. Passing the GED Tests can also provide you a better job opportunity or a pathway to higher education.
56% Off*
Includes GED Math Prep Books, Workbooks, and Practice Tests
GED Test Format
GED test comprises of four main areas of testing, on four different topics, including mathematical reasoning, reasoning through language arts, science, and social studies.
To earn the GED diploma, you must get a certain score in each part. In addition, the GED tests are administered both in-person and online.
The Mathematical Reasoning Test
Whether you are a genius in math or it feels like a foreign language to you, mathematics test is an essential part of the GED test and you have no option but to pass it!
You will have 115 minutes to complete the mathematical reasoning test and the minimum score to pass the exam is 145 points.
In the following, we will explain different parts of the test and introduce strategies for walking into the test well-prepared.
48% Off*
The Ultimate Step by Step Guide to Preparing for the GED Math Test
A. What to Study?!
It will be much easier to study for an exam, when you know what exactly the material is.
Breaking down the materials into understandable parts is a successful strategy for studying an exam; therefore, we divided the materials into four main categories and explained them separately.
Types of GED Math Test
GED test consists of four main types of math, including:
Basic Math, Geometry, Algebra, Graphs and Functions. It has 46 questions. These questions can be in multiple-choice, drag-and-drop, multiple-select, fill-in-the-blank, matching, and table entry. Each
types of GED math test is explained in more detail below.
1. Basic Math
This type of questions test your ability to perform subtraction, addition, multiplication, and division. It also examine your knowledge of different kinds of numbers such as decimals, whole
numbers, percentages and fractions.
2. Geometry
This is about understanding basic geometry concepts and using formulas relating to different shapes and objects. You may be asked to calculate the surface area of circles, triangles and squares.
It is necessary for you to be able to use a provided formula to calculate volume, radius, diameter, and so on.
3. Algebra
In this part, finding the value of a variable in an equation is important. Algebra part involves simplifying algebraic expressions, converting numbers to scientific notation, and finding the
absolute value of a rational number.
4. Graphs and Functions
Your ability to read and analyze information in graphs and charts is tested in this part. You should be able to put data into tables and organize them. In addition, you must be familiar with the
concepts of median, mode mean, range and so on.
B. How to Be Prepared For the GED Math Test?
Now that you get familiar with the format and the content of the GED math test, you can focus on studying each type.
The followings are useful ideas and tips for preparing for the GED Math test:
1. Practice Tests
Taking practice tests before you take the actual test is extremely helpful.
By practicing the practice tests, you get familiar with the types of questions, and the subject areas. Try to practice as much as you can before test day.
2. Reading Books
After practicing enough, and when you recognized your strengths and weaknesses, you can find books that address your weaknesses to increase your chances to pass the GED math test.
3. Participating in a GED test preparation class
Having a teacher to guide you through your preparation, solve and explain your math problems, is just one benefit of enrolling in a GED test preparation class. Today, there are many in-person and
online GED test preparation classes you can attend.
4. Planning
It is vital to schedule for your study. You should study regularly and Focus on the subjects you’re weakest in.
Remember that only through regular study and enough practice, you’ll be well prepared to pass the GED math exam.
56% Off*
Includes GED Math Prep Books, Workbooks, and Practice Tests
You May Also Like to Read
More Articles | {"url":"https://testinar.com/article/ged_math_overview","timestamp":"2024-11-03T18:37:58Z","content_type":"text/html","content_length":"63432","record_id":"<urn:uuid:e507a72a-b835-4f32-aef2-b06cfeae010f>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00658.warc.gz"} |
Vincent Bouchard | 2023 CMS Winter Mtg
top of page
Vincent Bouchard
University of Alberta
Vincent Bouchard is Professor in the Department of Mathematical and Statistical Sciences at University of Alberta. He obtained his D.Phil. in Mathematics from University of Oxford in 2005, as a
Rhodes scholar. He held postdoctoral fellowships in University of Pennsylvania, Mathematical Science Research Institute in Berkeley, Perimeter Institute for Theoretical Physics in Waterloo, and
Harvard University, before joining University of Alberta in 2009.
His research focuses on exploring new mathematical structures physically motivated by modern physics, which often give rise to unexpected connections between mathematical objects that appear a priori
unrelated. He is also passionate about teaching and creating an active learning environment in the classroom. Outside of math, Vincent likes to apply the perseverance and grit required of research in
mathematics to the pursuit of long-distance sports and mountain adventures.
Monday, December 4, 2023 | 11am - 12pm
Airy structures: a new connection between geometry, algebra and physics
Modern physics involves beautiful and intricate mathematics, and entirely new mathematical structures often emerge from physical theories. An example of this is the concept of Airy structures, which
was first introduced by Kontsevich and Soibelman in 2017 as an algebraic reformulation and extension of the Chekhov-Eynard-Orantin topological recursion. One can also think of Airy structures as a
wide generalization of Witten's conjecture; as such, it provides a fascinating new connection between enumerative geometry, algebra and integrable systems. In this talk I will introduce the concept
of Airy structures, mention some recent applications of the theory to enumerative geometry, vertex operator algebras and gauge theories, and discuss potential generalizations and open questions. My
hope with this talk is to convey why I believe that the formalism of Airy structures (and topological recursion) should be in the toolbox of all geometers, algebraists and mathematical physicists!
bottom of page | {"url":"https://cmssmc.wixsite.com/winter23/vincent-bouchard","timestamp":"2024-11-02T02:47:22Z","content_type":"text/html","content_length":"696663","record_id":"<urn:uuid:99fdf5e0-ef70-4a89-ad12-b441bc7b93c2>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00652.warc.gz"} |
Advanced Order Of Operations Worksheet Answer Key Worksheet Resume | Order of Operation Worksheets
Advanced Order Of Operations Worksheet Answer Key Worksheet Resume
Advanced Order Of Operations Worksheet Answer Key Worksheet Resume
Advanced Order Of Operations Worksheet Answer Key Worksheet Resume – You may have listened to of an Order Of Operations Worksheet, yet what specifically is it? In addition, worksheets are a terrific
way for trainees to exercise brand-new skills and also testimonial old ones.
What is the Order Of Operations Worksheet?
An order of operations worksheet is a type of math worksheet that needs trainees to perform mathematics operations. These worksheets are divided into 3 main sections: multiplication, addition, as
well as subtraction. They also consist of the examination of parentheses and exponents. Trainees that are still discovering how to do these tasks will certainly discover this type of worksheet
The main function of an order of operations worksheet is to assist pupils discover the appropriate means to solve math equations. If a trainee doesn’t yet recognize the concept of order of
operations, they can evaluate it by describing an explanation web page. In addition, an order of operations worksheet can be separated right into a number of groups, based upon its trouble.
One more crucial function of an order of operations worksheet is to educate students exactly how to perform PEMDAS operations. These worksheets begin with basic problems associated with the basic
regulations and also develop to extra intricate problems involving all of the guidelines. These worksheets are a wonderful way to present young students to the excitement of resolving algebraic
Why is Order of Operations Important?
One of the most essential points you can discover in math is the order of operations. The order of operations guarantees that the math problems you solve are consistent. This is necessary for
examinations as well as real-life computations. When resolving a math issue, the order should begin with backers or parentheses, complied with by multiplication, subtraction, as well as addition.
An order of operations worksheet is a fantastic way to show students the right means to address mathematics formulas. Before trainees start utilizing this worksheet, they might need to evaluate
concepts associated to the order of operations.
An order of operations worksheet can help students establish their abilities in addition and subtraction. Natural born player’s worksheets are an excellent method to aid pupils find out concerning
the order of operations.
Advanced Order Of Operations Worksheets
Printables Order Of Operations Worksheets 7th Grade Tempojs Thousands
Advanced Order Of Operations Worksheets
Advanced Order Of Operations Worksheets provide a fantastic source for young students. These worksheets can be easily personalized for details demands. They can be located in three degrees of
difficulty. The very first level is simple, requiring pupils to practice utilizing the DMAS technique on expressions containing four or more integers or 3 operators. The second level calls for
students to use the PEMDAS technique to simplify expressions using inner and external parentheses, braces, as well as curly braces.
The Advanced Order Of Operations Worksheets can be downloaded absolutely free as well as can be printed out. They can then be reviewed utilizing addition, division, multiplication, and also
subtraction. Trainees can likewise make use of these worksheets to evaluate order of operations as well as the use of backers.
Related For Advanced Order Of Operations Worksheets | {"url":"https://orderofoperationsworksheet.com/advanced-order-of-operations-worksheets/advanced-order-of-operations-worksheet-answer-key-worksheet-resume-5/","timestamp":"2024-11-11T13:49:50Z","content_type":"text/html","content_length":"28456","record_id":"<urn:uuid:cac91c74-4cef-4558-9ba8-d79e23415458>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00150.warc.gz"} |
JavaScript Puzzles
I found my new grind. Goodbye Hearthstone. Goodbye World of Warcraft.
Hello, CodeFights.
Arcade Mode was a lot of fun for me today. In The Labyrinth of Nested Loops, I found this little gem :
We define the weakness of number x as the number of positive integers smaller than x that have more divisors than x.
It follows that the weaker the number, the greater overall weakness it has. For the given integer n, you need to answer two questions:
□ what is the weakness of the weakest numbers in the range [1, n]?
□ how many numbers in the range [1, n] have this weakness?
Return the answer as an array of two elements, where the first element is the answer to the first question, and the second element is the answer to the second question.
What fun!
This feels like I could brute-force it with a nested loop is and suffer the $O(n^2)$ performance. If I manage to solve it using only sequential loops on the other hand, it would be $O(n)$ no matter
how many loops I used. I think I scored somewhere in between, maybe $O(n \log(n))$ depending on what you believe about the inner loops in the following sections.
The problem can be decomposed by first finding the number of divisors, then counting weakness counts (not a typo – counts of counts!), and finally finding the weakest among those.
Finding divisors can be done in many ways. I use variable-stride loops to increment every spot where a given number is a divisor.
// Find the number of divisors
divisors = new Array(n+1)
for (var i=1; i<=n; i++)
for (var j=i; j<=n; j+=i)
Next, I want a count of all numbers that have a given number of divisors. This segment of code will end with a variable called divisorsCounts, where divisorsCounts[3] = 4 if there are exactly 4
numbers in range that have 3 divisors.
// Count the counts
divisorsCounts = new Array(n) // probably overkill
// Compute the weakness
weakness = new Array(n+1)
greatestWeakness = 0
The array divisorsCounts does not need to be that large, but it’s only memory, right? Apparently something called the Divisor Function $\sigma_0(i)$ has an upper limit of $n/2$. I’m no number
theorist, but I buy it. A factor of two is not even worth typing Math.floor() for, so I’ll leave the code alone as it now looks clean.
for (var me=1; me<=n; me++) {
for (var j=divisorsCounts.length-1; j>divisors[me]; j--)
weakness[me] += divisorsCounts[j]
It is key here that I fill the divisorsCounts at the same time as I calculate weakness. At any given point in time I have only counted the divisors up to the number I am examining. For a given number
m, weakness is the number of divisor counts greater than divisors[m]found in numbers smaller than m.
I can optimize these loops by shrinking divisorsCount. JavaScript creates sparse arrays by default, and their length is determined by the highest-indexed element. The inner nested loop will be
// Count the counts
divisorsCounts = []
// Compute the weakness
weakness = new Array(n+1)
greatestWeakness = 0
for (var me=1; me<=n; me++) {
divisorsCounts[divisors[me]] =
(divisorsCounts[divisors[me]] | 0) + 1
for (var j=divisorsCounts.length-1; j>divisors[me]; j--)
weakness[me] = (divisorsCounts[j] | 0) + weakness[me]
The pipes handle undefined elements (note that x + undefined will never produce a number). With n=500, divisorsCount is only 25, so this may improve performance quite a bit.
Finally, I must track the greatestWeakness and how many times it is found.
Here it is in all of its beauty.
function weakNumbers(n) {
// Find the number of divisors
divisors = new Array(n+1)
for (var i=1; i<=n; i++)
for (var j=i; j<=n; j+=i)
// Count the counts
divisorsCounts = []
// Compute the weakness
weakness = new Array(n+1)
greatestWeakness = 0
greatestWeaknessCount = 0
for (var me=1; me<=n; me++) {
divisorsCounts[divisors[me]] =
(divisorsCounts[divisors[me]] | 0) + 1
for (var j=divisorsCounts.length-1; j>divisors[me]; j--)
weakness[me] = (divisorsCounts[j] | 0) + weakness[me]
if (greatestWeakness < weakness[me]) {
greatestWeakness = weakness[me]
greatestWeaknessCount = 1
} else if (greatestWeakness == weakness[me]) {
return [greatestWeakness, greatestWeaknessCount]
CodeFights is awesome! Come find me! Come fight me! | {"url":"http://mikejfromva.com/2016/10/06/javascript-puzzles/","timestamp":"2024-11-08T20:09:20Z","content_type":"text/html","content_length":"76140","record_id":"<urn:uuid:d4127e46-0a25-496d-9a25-4e927f5068ee>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00561.warc.gz"} |
dsbgv: computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite banded eigenproblem, of the form A*x=(lambda)*B*x - Linux Manuals (l)
dsbgv (l) - Linux Manuals
dsbgv: computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite banded eigenproblem, of the form A*x=(lambda)*B*x
DSBGV - computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite banded eigenproblem, of the form A*x=(lambda)*B*x
JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, Z, LDZ, WORK, INFO )
CHARACTER JOBZ, UPLO
INTEGER INFO, KA, KB, LDAB, LDBB, LDZ, N
DOUBLE PRECISION AB( LDAB, * ), BB( LDBB, * ), W( * ), WORK( * ), Z( LDZ, * )
DSBGV computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite banded eigenproblem, of the form A*x=(lambda)*B*x. Here A and B are assumed to be
symmetric and banded, and B is also positive definite.
JOBZ (input) CHARACTER*1
= aqNaq: Compute eigenvalues only;
= aqVaq: Compute eigenvalues and eigenvectors.
UPLO (input) CHARACTER*1
= aqUaq: Upper triangles of A and B are stored;
= aqLaq: Lower triangles of A and B are stored.
N (input) INTEGER
The order of the matrices A and B. N >= 0.
KA (input) INTEGER
The number of superdiagonals of the matrix A if UPLO = aqUaq, or the number of subdiagonals if UPLO = aqLaq. KA >= 0.
KB (input) INTEGER
The number of superdiagonals of the matrix B if UPLO = aqUaq, or the number of subdiagonals if UPLO = aqLaq. KB >= 0.
AB (input/output) DOUBLE PRECISION array, dimension (LDAB, N)
On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first ka+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if
UPLO = aqUaq, AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j; if UPLO = aqLaq, AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka). On exit, the contents of AB are destroyed.
LDAB (input) INTEGER
The leading dimension of the array AB. LDAB >= KA+1.
BB (input/output) DOUBLE PRECISION array, dimension (LDBB, N)
On entry, the upper or lower triangle of the symmetric band matrix B, stored in the first kb+1 rows of the array. The j-th column of B is stored in the j-th column of the array BB as follows: if
UPLO = aqUaq, BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j; if UPLO = aqLaq, BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb). On exit, the factor S from the split Cholesky factorization B = S**T*S,
as returned by DPBSTF.
LDBB (input) INTEGER
The leading dimension of the array BB. LDBB >= KB+1.
W (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.
Z (output) DOUBLE PRECISION array, dimension (LDZ, N)
If JOBZ = aqVaq, then if INFO = 0, Z contains the matrix Z of eigenvectors, with the i-th column of Z holding the eigenvector associated with W(i). The eigenvectors are normalized so that
Z**T*B*Z = I. If JOBZ = aqNaq, then Z is not referenced.
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >= 1, and if JOBZ = aqVaq, LDZ >= N.
WORK (workspace) DOUBLE PRECISION array, dimension (3*N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, and i is:
<= N: the algorithm failed to converge: i off-diagonal elements of an intermediate tridiagonal form did not converge to zero; > N: if INFO = N + i, for 1 <= i <= N, then DPBSTF
returned INFO = i: B is not positive definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed. | {"url":"https://www.systutorials.com/docs/linux/man/l-dsbgv/","timestamp":"2024-11-04T08:00:06Z","content_type":"text/html","content_length":"11428","record_id":"<urn:uuid:73cd7a43-0b72-4a77-b956-bb4757be7024>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00725.warc.gz"} |
Anomaly and Topology | KTH
Anomaly and Topology
On the axial anomaly, domain wall dynamics, and local topological markers in quantum matter
Time: Wed 2024-02-21 09.00
Location: FB53, Roslagstullsbacken 21, Stockholm
Language: English
Doctoral student: Julia D. Hannukainen , Kondenserade materiens teori
Opponent: Professor Teemu Ojanen,
Supervisor: Docent Jens H. Bardarson, Kondenserade materiens teori; Chargé de recherche Adolfo G. Grushin,
QC 2024-01-31
Chiral anomalies and topological phases of matter form the basis of the research presented in this dissertation. The chiral anomaly is considered both in the context of magnetic Weyl semimetals and
in the context of non-Hermitian Dirac actions. Topological phases of matter play a role in this work through the research on Weyl semimetals and in the formulation of local topological markers.
The simplest example of magnetic Weyl semimetals consist of two Weyl cones separated in momentum space by a magnetisation vector which acts as an axial gauge field. We describe the emergence of axial
electromagnetic fields by considering a magnetic field driven domain wall in this magnetisation. The parallel axial magnetic and axial electric fields give rise to the axial anomaly, and in turn to
the chiral magnetic effect; a nonequilibrium current located at the domain wall. The chiral magnetic effect is a source of electromagnetic radiation, and a measurement of this radiation would provide
evidence of the existence of the axial anomaly.
Electronic manipulation of domain walls is a central objective in spintronics. We describe how the axial anomaly, in terms of external electromagnetic fields, acts as a torque on the domain wall, and
allows for electric control of the equilibrium configuration of the domain wall. We show how the axial anomaly is used to flip the chirality of the domain wall by tuning the electric field. Measuring
the change in domain wall chirality constitutes a signal of the axial anomaly. We also describe how the Fermi arc boundary states of the Weyl semimetal at the domain wall result in an effective hard
axis anisotropy which allows for large domain wall velocities irrespective of the intrinsic anisotropy of the material.
Our interest in non-Hermitian chiral anomalies stems from the existence of topological phases of matter in non-Hermitian models. We evaluate the chiral anomaly for a non-Hermitian Dirac theory with
massless fermions with complex Fermi velocities coupled to non-Hermitian axial and vector gauge fields. The anomaly is compared with the corresponding anomaly of a Hermitianised and an
anti-Hermitianised action derived from the non-Hermitian action. We find that the non-Hermitian anomaly does not correspond to the combined anomalous terms derived from the Hermitianised and
anti-Hermitianised theory, as would be expected classically, resulting in new anomalous terms in the conservation laws for the chiral current.
Local topological markers are real space expressions of topological invariants evaluated by local expectation values and are important for characterising topology in noncrystalline structures. We
derive analytic expressions for local topological markers for strong topological phases of matter in odd dimensions, by generalising the formulation of the even dimensional local Chern marker. This
is not a straightforward task since the topological invariants in odd dimensions are basis dependent. Our solution is to express the invariants in terms of a family of parameter dependent projectors
interpolating between a trivial state and the topological state of interest. The odd dimensional invariant is therefore expressed as a Chern character integrated over the combined space of the odd
dimensional Brillouin zone and the one dimensional parameter space. As a result, we provide an easy-to-use chiral marker for symmetry classes with a chiral constraint, and a Chern-Simons marker for
symmetry classes with either time reversal symmetry (in three dimensions) or particle hole symmetry (in one dimension). These markers are readily extended to interacting systems by considering the
topological equivalence between a gapped one-particle density matrix of the interacting state and a projector corresponding to a free fermion state. | {"url":"https://www.physics.kth.se/2.51534/disputation/anomaly-and-topology-1.1313096?date=2024-02-21&orgdate=2024-02-21&length=1&orglength=1","timestamp":"2024-11-02T07:58:55Z","content_type":"text/html","content_length":"49529","record_id":"<urn:uuid:eca1a51f-cea6-4b91-9034-56126978ae76>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00497.warc.gz"} |
Formula to calculate gdp growth rate
But how is this actually done? Nominal GDP is a measure of the value of output produced in a country or
11 Jan 2008 Formula used by BEA to calculate the average annual growth. where. GDPt is the level of activity in the later period;. GDP0 is the level of activity 29 Jan 2016 Calculating gross
domestic product (GDP) data of any country is a complex Raising doubts over the new GDP growth rate methodology, RBI What is GDP growth rate and how to calculate it. Since the media often talk about
the growth rate of an economy, it is important to clarify and to correctly define The proxy variable for the GDP calculation is GNI in US dollars. When we show GDP aggregate growth rates over a
period (e.g., 1990-2004), they are derived The data for real GDP are measured in constant US dollars to facilitate the calculation of country growth rates and aggregation of the country data.
Rationale :. The proper formula is where GDP 1 is the GDP of the later period GDP 2 is the GDP of the earlier period p= Periodicity of the data. (1 for annual data, 4 for Calculating real GDP by
weighting final goods and services by their prices in a base year can lead to an overstatement of real GDP growth because the prices of
Nominal GDP. Nominal GDP is the total dollar value of all goods and services produced in an economy. There are only two goods, wine and cheese, in our assumed economy. The formula for nominal GDP is
as such: Where is the price of wine, is the quantity of wine, is the price of cheese and is the quantity of cheese.
9 Sep 2019 The NDA government launched the first set of data, giving out levels of GDP and growth rates from 2011-12. What are the main differences in the Less attention has been paid to how the
accelerated growth of real GDP will be method of calculating the growth rate of potential GDP over the next decade how well one can predict the real GDP growth rate around 1980. The predicted and
The relative growth rate is calculated with a changing baseline value. principles to how monthly growth relates to quarterly growth rates as well as Annual average growth rates are calculated mainly
by statistical agencies. real gross domestic product (GDP) and the consumer price index (CPI), annual growth real (or constant price) GDP estimates are crucial agencies to calculate and present
quarterly growth rates. This method calculates quarterly growth rates as formula : a = (1 + r)4 – 1 where a = annualised quarter-on-quarter growth rate.
How to Calculate the Growth Rate of Nominal GDP - Calculating Nominal GDP Understand the distinction between nominal and real GDP. Add together that period's consumer spending or consumption. Sum all
investments. Add together all government spending. Determine the net exports. Calculate the GDP
The Gross Domestic Product (GDP) for a country is a total market value of all domestically produced goods and services. The GDP growth rate indicates the current growth trend of the economy. When
calculating GDP growth rates, the U.S. Bureau of Economic Analysis uses real GDP, which equalizes the actual figures to filter out the effects of
How to Calculate Real GDP Growth Rates 1) Find the Real GDP for Two Consecutive Periods. 2) Calculate the Change in GDP. Once we know the real GDP values for two consecutive periods, 3) Divide the
Change in GDP by the Initial GDP. 4) Multiply the Result by 100 (Optional) Finally, to convert
9 Sep 2019 The NDA government launched the first set of data, giving out levels of GDP and growth rates from 2011-12. What are the main differences in the Less attention has been paid to how the
accelerated growth of real GDP will be method of calculating the growth rate of potential GDP over the next decade how well one can predict the real GDP growth rate around 1980. The predicted and
The relative growth rate is calculated with a changing baseline value. principles to how monthly growth relates to quarterly growth rates as well as Annual average growth rates are calculated mainly
by statistical agencies. real gross domestic product (GDP) and the consumer price index (CPI), annual growth
This guide explains how GDP is measured, as well as which things GDP doesn't or simply “growth” – is a key measure of the overall strength of the economy.
real (or constant price) GDP estimates are crucial agencies to calculate and present quarterly growth rates. This method calculates quarterly growth rates as formula : a = (1 + r)4 – 1 where a =
annualised quarter-on-quarter growth rate. This guide explains how GDP is measured, as well as which things GDP doesn't or simply “growth” – is a key measure of the overall strength of the economy.
No matter how we measure economic growth, it needs to be pursued in a smart way. But how is this actually done? Nominal GDP is a measure of the value of output produced in a country or 13 Jan 2016
To visualize those growth rates, and to do some crude analysis, we invariably plot real GDP per capita in logs. When I say log, I mean the natural Economic and Social Research Institute provides
Real GDP in local currency, at chain linked 2011 prices. Real GDP Growth prior to Q1 1995 is calculated from
Economic and Social Research Institute provides Real GDP in local currency, at chain linked 2011 prices. Real GDP Growth prior to Q1 1995 is calculated from View the annual rate of economic output,
or the inflation-adjusted value of all new Q4 2019: 19,219.767 | Billions of Chained 2012 Dollars | Quarterly | Updated: Jan 30, 2020 Or calculate the spread between 2 interest rates, a and b, by
using the formula a - b. Real GDP Growth Rate GDP: Does It Measure Up? | {"url":"https://bestexmosoyqp.netlify.app/mccullan72973tuc/formula-to-calculate-gdp-growth-rate-ge","timestamp":"2024-11-14T18:46:03Z","content_type":"text/html","content_length":"33415","record_id":"<urn:uuid:fd4be2a9-fc6f-4317-9ab2-dd9a7a08dac3>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00609.warc.gz"} |
Time delay in optimal control loops for wave equations
Leugering G, Gugat M (2015)
Publication Language: English
Publication Status: Published
Publication Type: Journal article, Original article
Publication year: 2015
Publisher: EDP Sciences
Article Number: 038
DOI: 10.1051/cocv/2015038
In optimal control loops delays can occur, for example through transmission via digital communication channels. Such delays influence the state that is generated by the implemented control. We study
the effect of a delay in the implementation of L-norm minimal Neumann boundary controls for the wave equation. The optimal controls are computed as solutions of problems of exact optimal control,
that is if they are implemented without delay, they steer the system to a position of rest in a given finite time T. We show that arbitrarily small delays δ > 0 can have a destabilizing effect in the
sense that we can find initial states such that if the optimal control u is implemented in the form yx(t, 1) = u(t - δ) for t > δ, the energy of the system state at the terminal time T is almost
twice as big as the initial energy. We also show that for more regular initial states, the effect of a delay in the implementation of the optimal control is bounded above in the sense that for
initial positions with derivatives of BV -regularity and initial velocities with BV -regularity, the terminal energy is bounded above by the delay δ multiplied with a factor that depends on the
BV-norm of the initial data. We show that for more general hyperbolic optimal exact control problems the situation is similar. For systems that have arbitrarily large eigenvalues, we can find
terminal times T and arbitrarily small time delays δ, such that at the time T + δ, in the optimal control loop with delay the norm of the state is twice as large as the corresponding norm for the
initial state. Moreover, if the initial state satisfies an additional regularity condition, there is an upper bound for the effect of time delay of the order of the delay with a constant that depends
on the initial state only.
Authors with CRIS profile
How to cite
Leugering, G., & Gugat, M. (2015). Time delay in optimal control loops for wave equations. Esaim-Control Optimisation and Calculus of Variations. https://doi.org/10.1051/cocv/2015038
Leugering, Günter, and Martin Gugat. "Time delay in optimal control loops for wave equations." Esaim-Control Optimisation and Calculus of Variations (2015).
BibTeX: Download | {"url":"https://cris.fau.de/publications/122296724/","timestamp":"2024-11-07T07:32:36Z","content_type":"text/html","content_length":"10923","record_id":"<urn:uuid:62361cde-79ed-4567-b297-c89e798d5731>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00083.warc.gz"} |
pKa to KA Calculator - Convert pKa to KA
A pKa to KA calculator is a useful tool in chemistry that helps convert between two important measures of acid strength: pKa and Ka.
pKa, or the negative logarithm of the acid dissociation constant, is a measure of how easily an acid gives up a proton. On the other hand, Ka, or the acid dissociation constant, directly represents
the extent to which an acid dissociates in an aqueous solution.
Convert pKa to KA Calculator
pKa Ka
4.76 1.74 × 10^(-5)
3.00 1.00 × 10^(-3)
7.50 3.16 × 10^(-8)
9.25 5.62 × 10^(-10)
2.15 7.08 × 10^(-3)
To calculate these values, we used the formula:
Ka = 10^(-pKa)
For example, for the first row: Ka = 10^(-4.76) = 1.74 × 10^(-5)
More Chemistry Calculators : – Moles to Atoms Calculator
pKa to KA Conversion Formula
The relationship between pKa and Ka is defined by a simple logarithmic equation:
pKa = -log₁₀(Ka)
Conversely, to calculate Ka from pKa, we use the inverse logarithmic function:
Ka = 10^(-pKa)
These formulas form the basis of any pKa to KA calculator, allowing for quick and accurate conversions between the two values.
How do you convert pKa to KA?
Converting pKa to Ka involves a few simple steps:
1. Start with the pKa value you want to convert.
2. Recognize the relationship: pKa = -log₁₀(Ka)
3. Rearrange the equation to solve for Ka: Ka = 10^(-pKa)
4. Use a calculator to compute 10 raised to the power of the negative pKa value.
For example, if you have a pKa of 4.5: Ka = 10^(-4.5) = 3.16 × 10^(-5)
This process can be easily automated using a KA to pKa calculator, which performs these calculations instantly.
What is the pKa of 4.76 to Ka?
Let’s apply the conversion process to a specific example:
Given pKa = 4.76, we need to find Ka.
Using the formula Ka = 10^(-pKa):
Ka = 10^(-4.76) Ka = 1.74 × 10^(-5)
Therefore, a pKa of 4.76 corresponds to a Ka of 1.74 × 10^(-5).
This value indicates that the acid is relatively weak, as it has a small Ka value (much less than 1), meaning it doesn’t dissociate extensively in solution.
How is pKa different to Ka?
pKa and Ka both describe acid strength, they differ in several key aspects:
1. Scale: Ka is a direct measure of the acid dissociation constant, while pKa is a logarithmic scale.
2. Range: Ka values can span many orders of magnitude (e.g., 10^(-15) to 10^(15)), while pKa values typically range from about -2 to 16 for common acids.
3. Interpretation: A higher Ka indicates a stronger acid, whereas a lower pKa indicates a stronger acid.
4. Ease of use: pKa values are often easier to work with due to their more manageable range.
5. Logarithmic nature: pKa represents the negative logarithm of Ka, similar to how pH represents the negative logarithm of hydrogen ion concentration.
Is Ka directly proportional to pKa?
No, Ka is not directly proportional to pKa. Instead, they have an inverse logarithmic relationship.
As pKa increases:
• Ka decreases exponentially
• The acid becomes weaker
As pKa decreases:
• Ka increases exponentially
• The acid becomes stronger
How to Calculate Ka and pKa from pH and Concentration
Let’s focus on weak acids for this explanation.
• pH = 4.5
• Initial concentration (C₀) = 0.1 M
• Calculate [H⁺]: [H⁺] = 10^(-4.5) = 3.16 × 10^(-5) M
• Calculate [A⁻]: [A⁻] = [H⁺] = 3.16 × 10^(-5) M
• Calculate [HA]: [HA] = 0.1 – (3.16 × 10^(-5)) ≈ 0.1 M
• Calculate Ka: Ka = (3.16 × 10^(-5))² / 0.1 = 1.00 × 10^(-8)
• Calculate pKa: pKa = -log₁₀(1.00 × 10^(-8)) = 8.00
Therefore, for this weak acid solution:
• Ka = 1.00 × 10^(-8)
• pKa = 8.00 | {"url":"https://ctrlcalculator.com/chemistry/pka-to-ka-calculator/","timestamp":"2024-11-04T21:28:51Z","content_type":"text/html","content_length":"103248","record_id":"<urn:uuid:f5ea3b03-3e60-431e-96b6-bae026c2bed1>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00459.warc.gz"} |