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# determine the sensitivity of lvdt

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indication of movement from the neutral position but will not indicate the direction of that The linear variable differential transformer ... A synchronous detector can determine a signed output voltage that relates to the displacement. A5 Mutual Induction Q6 Write any two advantages of LVDT. An output of 6.3 mV appears across the terminals of the LVDT when the core is displaced through a distance of 2.2 cm. Generate a plot of the output voltage vs. input displacement of the core. Diaphragm thickness and sensitivity are designed to suit the pressure range. It is very accurate inductive transducer as compared to other inductive transducers. Sensitivity: The LVDT sensitivity is virtually independent of changes in atmospheric temperature during high-frequency excitation. Sensitivity: Sensitivity usually ranges from 0.6 to 30 mV per 25 µm (0.001 in) under normal excitation of 3 to 6 V. Generally, the higher the frequency the higher the sensitivity. A small scale Linear Variable Differential Transducer (LVDT) was locally constructed and used to detect the levels of water, petroleum and gasoline. Generate a plot of the output voltage vs. input displacement of the core. Solution for 5. An LVDT, the cylindrical soft iron core is used to calculate the linear displacement, whereas cam-shaped rotary ferromagnetic core in RVDT for angular displacement measurements. LVDT (Linear Variable Differential Transformer) The linear variable differential transducer (LVDT) is a type of electrical transformer used for measuring linear displacement The transformer has three solenoid coils placed end-to-end around a tube. Provide details and share your research! Voltages induced in the coils. Ask Question Asked 10 months ago. A3 Soft iron Q4 LVDT is active transducer or passive? The output signals can be processed to recover … 1. The voltmeter scale has 164 divisions and the scale can be read to 1/4th of a division. OBSERVATION AND CALCULATION No. The coils on either side The LVDT models closely the ideal Zeroth-order displacement sensor structure at low frequency, where the output is a direct and linear function of the input. Now customize the name of a clipboard to store your clips. Units are in output units, often these are Volts. The LVDT can be used as an absolute position sensor. LVDTs are driven by an AC excitation source, typically a sine wave in the 1 to 10 kHz range, and generate an AC output signal that is proportional to the mechanical position of the core. LVDTs find uses in modern machine-tool, robotics, avionics, and computerized manufacturing. displacement transformer. Ask Question Asked 10 months ago. The result is a sensitivity coefficient of 0.1 Volts per Ampere. A counterpart to this device that is used for measuring rotary displacement is called a rotary The centre coil is the primary, and the two outer coils are the secondary. It is very accurate inductive transducer as compared to other inductive transducers. The Sensitivity Of LVDT Is 40 V Per Mm. Definition of LVDT. This wiring arrangement is known as ... where D is displacement of the iron core with respect to the transformer, and M is the sensitivity of the transformer (slope of the displacement-voltage curve). Determine the sensitivity of the LVDT. Define what the null position of an LVDT is. You can change your ad preferences anytime. many of the transducers If you continue browsing the site, you agree to the use of cookies on this website. The Voltmeter Scale Has 164 Divisions And The Scale Can Be Read To 1/4th Of A Division. Question: The Output Of A LVDT Is Connected To 22 V Voltmeter Through An Amplifier Of Amplification Factor 665. 2. A4 Passive Q5 what is the working principle of LVDT? The sensitivity of the LVDT due to linear displacement and induce Electromotive Force (EMF) in each case was investigated. Ratiometric Wiring LVDT: Another commonly used LVDT wiring is known as ratiometric wiring, as shown schematically below. Sensitivity The transducer must be sensitive enough to allow sufficient output ... LVDT (Linear Variable Differential Transformer) The linear variable differential transducer (LVDT) is a type of electrical transformer used for measuring linear displacement The transformer has three solenoid coils placed end-to-end around a tube. the coils. The LVDT linear transducer is mounted at right angles to the diaphragm with its core extension rod attached to the center of the disc. Procedure for finding the characteristics of LVDT:-Connect the terminals marked “Primary” on the DPM to the terminals marked “Primary” on LVDT transformer. FIGURE 1: THE PRIMARY WINDING IS ILLUSTRATED IN THE CENTER OF THE LVDT. Determine and graphically show the linear range of one side of the LVDT's null position. It determines the levels of water in a domestic's services tanks and the levels of fuel in our mobile and stationary Vehicles. Captive Armatures: These mechanisms are better for long working ranges. transformer} We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. and then may reduce for further movement from this maximum setting. A Linear Variable Differential Transformer (LVDT) is a sensor that uses a differential transformer with a sliding magnetic core to sense displacement. See our Privacy Policy and User Agreement for details. linear variable displacement transducer, or simply differential Sensitivity: The LVDT sensitivity is virtually independent of changes in atmospheric temperature during high-frequency excitation. The Linear Variable Differential Transformer (LVDT) is a displacement measuring instrument and is not a strain-based sensor. We observed that that changes in differential voltage from one specific reading to another is almost constant. Also it determines a combined set of information about the phase of the output associated with the magnitude of the output signal in a sinusoidal wave function. This feature is particularly useful in materials testing, vibration displacement measurements, and high resolution dimensional gaging systems. of rotation In left Output voltage No. The output of the LVDT is an alternating signal which is rectified and filtered to give fDC output (signal conditioner output). The sensitivity of the LVDT due to linear displacement and induce Electromotive Force (EMF) in each case was investigated. How to determine LVDT sensitivity? But … By using Equation 1, where V2-V1 is 10Vdc and X2-X1 is 1 inch, the sensitivity for the instrument is found to be ±10Vdc/inch. These sensitivities are usually given in terms of mVin −1 V −1 of excitation (or mVmm −1 V −1 of excitation) at a specified frequency and amplifier impedance. 2. This is report of the experiment that we did in our engineering college in the subject of the instrumentation. Active 10 months ago. The voltmeter scale has 156 divisions and the scale can be read to 1/5th of a division. The transformer contains one primary coil associated with two secondary coils. LINEAR VARIABLE DIFFERENTIAL TRANSFORMER LVDTs provide reliable position measurement for applications in subsea, power generation, industrial automation, aerospace, test and measurement, and more. Sensitivity: The ratio of the change in LVDT output to a change in the value of the measure and displacement. According to Howard (2003), LVDTs have certain characteristics features and benefits, most of The scale can be read to 1/5 of a division. Power Supply Most LVDT's are wired as shown in the schematic above. determine the linearity characteristics of the LVDT/RVDT over its specified operating range, and the transformation ratio of the device at maximum coupling. 2. LVDT-diagram(circuit-diagram) The generic LVDT symbol is shown in Fig.1. Why use an LVDT? The units of an LVDT measurement are typically in … Hysteresis: … Also a DC LVDT is more likely to be +/- 10V output for +/- full scale movement. The milli-voltmeter scale has 100 divisions. An LVDT, the cylindrical soft iron core is used to calculate the linear displacement, whereas the cam-shaped rotary ferromagnetic core in RVDT for angular displacement measurements. The voltmeter scale has 100 divisions and… In normal use, there is no mechanical contact between the LVDT's core and coil assembly, so there is no rubbing, dragging, or other source of friction. The highest value of the engine out (EOUT) happens at the highest core displacement from the middle position. The LVDT is designed with long slender coils to make the output voltage essentially linear over displacement up to several inches (several hundred millimetres) long. mech.3020 experiment lvdt accelerometer measurements introduction transducer is device which converts one form of energy to another. A1 Linear Variable Differential Transformer. Bias Intercept of the best-fit straight line (where it intersects with the O-axis). Two tenths of a division can be read with certainity. An Output Of 7.8 MV Appears Across The Terminals Of The LVDT When The Core Is Displaced Through A Distance Of 2.2 Cm. TWO … The LVDT … The term LVDT stands for the Linear Variable Differential Transformer.It is the most widely used inductive transducer that converts the linear motion into the electrical signal.. Then, divide the difference in volts by the difference in amperes. See our User Agreement and Privacy Policy. Define what the null position of an LVDT is. Determine the sensitivity required using the LVDT specification, the maximum expected displacement, and the 9600 excitation voltage of 1.1Vrms. Question: The Output Of A LVDT Is Connected To 21 V Voltmeter Through An Amplifier Of Amplification Factor 580. Sophisticated Phase-Sensitive LVDT Signal Conditioning Circuit. Another commonly used LVDT wiring is known as ratiometric wiring, as shown schematically below. Linear Variable Differential Transformer(LVDT) has the advantage of having high sensitivity, high range, ruggedness, low hysteresis, and low power consumption. The Bellows Is Deflected By 0.125 Mm By A Pressure Of 0.8 X 106 N Per M². In Figure 2 the bias is 1 The voltmeter scale has 174 divisions and the… An LVDT, the cylindrical soft iron core is used to calculate the linear displacement, whereas cam-shaped rotary ferromagnetic core in RVDT for angular displacement measurements. The center coil is the primary and is supplied from an AC supply. DYNA-1750 Note that the phase will are secondary coils and are labeled A & B in Fig 6.1.Coils A & B have equal number of turns An LVDT, the cylindrical soft iron core is used to calculate the linear displacement, whereas the cam-shaped rotary ferromagnetic core in RVDT for angular displacement measurements. How to Test an LVDT or RVDT The primary is excited with a sinewave - voltage that the frequency of which, depending on the design, may be from 200Hz to over 10kHz. The time response is dependent on the equipment to which the core is connected. Determine the sensitivity of the LVDT. 5mm. Semantic Scholar is a free, AI-powered research tool for scientific literature, based at the Allen Institute for AI. The value of EOUT at maximum core displacement from null depends upon the amplitude of the primary excitation voltage and the sensitivity factor of the particular LVDT, but is typically several volts RMS. An output of 6 mV appears across the terminals of the LVDT when the core is displaced through a distance of 1.4 cm. Using your recorded data, calculate the difference of the two voltage measurements and the two current set points. The result showed that while water was used as reference fluid, the transducer displayed (2.63V) output EMF due to (11.86 Weber) flux linkage while a ferrite core was displaced for about (12.86cm) at sensitivity of (2.99V/cm). Used in conjunction with a phase detector, an output can be obtained that is The output of a LVDT is connected to 5V voltmeter through an amplifier of amplification factor 250. Calculate the resolution of the instrument in mm. Sensitivity is usually specified in terms of milliVolt output per thousandths of an inch core displacement per Volt of excitation (mV/mil/Volt). an abrupt reversal when passing through the neutral position. LD400:Miniature DC Output Displacement Transducers with Acetal Bearings LVDT Sensors - determine whether you need to measure a relative current: C-in, AC-out, DC-in, DC-out; or measuring resonant frequencies of coils as a function of coil position, frequency based devices. The performance of sensory units drives the accuracy of the overall system. An amplitude only measurement of the output voltage, such as that provided by a meter, gives an 5. 3. The deflection per unit force is 0.324mm/N. 4. CONCLUSION: i. This parameter is computed from the In-Phase ratio of the output as read on the Model 2250 display. of rotation in right Output voltage 1 0.012 1 0.010 2 0.019 2 0.017 3 0.026 3 0.025 4 0.030 4 0.032 5 0.0321 5 0.036 6 0.0321 6 0.039 7 0.0321 7 0.039 From Graph, Residual Voltage=3mV Then, Sensitiviy(S)= 0.030-019 4.2 =5.5mV/rotation RESULT AND CONCLUSION Hence the sensitivity of LVDT was found to be … However I would not expect a DC LVDT to have variable excitation, because it is internal, so why quote it like this. Linear transducers are available for operating temperatures up to 600°C. Connecting Leads TITLE: - TO FIND THE SENSITIVITY OF THE LVDT The output across secondary of this transformer is the differential thus it is called so. High Precision Linear Variable Differential Transformer. of three coils mounted on a common former and having a magnetic core that is movable within The voltage is read on a 9 V voltmeter having 100 divisions. Sensitivity Gradient of the straight line. 3. Actuating mechanisms used primarily for automatic control systems or as mechanical motion sensors in measurement technologies. The sensitivity range of LVDT is high, ranging 2.4mV per the voltage and per degree change in moments and 2 to 3 mV per voltage per degree rotation in RVDT. Multimeter Dependent on both magnitude and direction of movement from neutral Position. Definition of LVDT. The Voltmeter Scale Has 140 Divisions And The Scale Can Be Read To 1/4th Of A Division. The LVDT models closely the ideal Zeroth-order displacement sensor structure at low frequency, where the output is a direct and linear function of the input. Sensitivity is the output for a given movement, and that seems to mean for each volt of excitation the output is 75mV per mm. Pick a sensitivity setting that is as close as possible to the expected full scale input without going under it. It consists Several terms used to describe LVDTs can be confusing, such as nominal linear range, resolution, full-scale output, linearity range and repeatability. The sensitivity of the LVDT due to linear displacement and induce Electromotive Force (EMF) in each case…, Study of the Effect of Excitation Frequency Variation on, Artificial Neural Networks based Approach for Predicting LVDT Output Characteristics, Design of LVDT Based Digital Weighing System, Selection of Soft Magnetic Core Materials Used on an LVDT Prototype, American Sensor and Technology, LVDT Technology Elevates Differential Pressure Transducer, 2003, (Online) at www.LVDTtechnology.com, Fluid level gauge, University of London, 2010, (Online) at www.hydac.com 18/4/2013, 1:21pm, Micro sensors, LVDT Basis, Innovation in a Sensing Position, USA, 2003, (Online) at www.macrosensors.com, Principles of electrical engineering : electric circuits, electronics, instrumentation, energy conversion, control systems, computers, View 2 excerpts, references methods and background, By clicking accept or continuing to use the site, you agree to the terms outlined in our. If the LVDT has a linear input range of +/- 1 inch with a corresponding output range of +/- 10Vdc, what nominal sensitivity should this instrument have? An LVDT with a sensitivity of 0.4 V/mm is used. It is an amplitude function of the main side excitation voltage as well as the sensitivity factor of the specific type of LVDT. Some features of the site may not work correctly. The sensitivity range of LVDT is high, ranging 2.4mV per the voltage and per degree change in moments and 2 to 3 mV per voltage per degree rotation in RVDT. Untangling Some Confusing LVDT Terms. The sensitivity range of LVDT is high, ranging 2.4mV per the voltage and per degree change in moments and 2 to 3 mV per voltage per degree rotation in RVDT. It is a common type of electromechanical transducer that can convert the rectilinear motion of an object to which it is coupled mechanically into a corresponding electrical signal. Conclude on the accuracy of the experiment and ways of improving it. The construction and circuit arrangement of an LVDT are as shown in Fig 6.1. Q2 Uses of LVDT A2 Measurement of displacement, thickness measurement, level indicators Q3 Core of LVDT is made up of which material? Solution for The output of a LVDT is connected to 10 V voltmeter through an amplifier of amplification factor 515. The coil assembly includes three wire-wound coils on the hollow shape. THOERY:- If you continue browsing the site, you agree to the use of cookies on this website. APIdays Paris 2019 - Innovation @ scale, APIs as Digital Factories' New Machi... No public clipboards found for this slide. The voltmeter scale has 100 divisions and… Typical excitation voltages range … For example, in the example shown in Figure 2, the sensitivity is 10 Volts/mm. In general, it is quite considerable at RMS. Here, the core is protected by the thing whose location is being calculated, while the coil assembly is increased to a stationary structure. Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. 4. Determine and graphically show the linear range of one side of the LVDT's null position. and are connected in series opposing so that the output voltage is the difference between the Most thermal-sensitivity shift errors result from the significant thermal coefficient of resistance (TCR) of the copper transformer windings. Assuming this function is is n= f (A, lmbda), Then the amplitude sensitivity S at a given lambda= (dn/n) / (dA/A), one can calculate S at different lambdas and plot S versus lambda. Clipping is a handy way to collect important slides you want to go back to later. An LVDT (linear variable differential transformer= is an electromechanical sensor used to convert mechanical motion or vibrations, specifically rectilinear motion, into a variable electrical current, voltage or electric signals, and the reverse. Sensitivity of LVDT $\text{Sensitivity} =\frac{\text { Output voltage }}{\text { displacement }}$ Instructions to use calculator. Learn how any measuring instrument in physics has a certain sensitivity, range and linearity depending on its design. This voltage increases with the movement from the neutral position to a maximum value One of the most important features of an LVDT is its friction-free operation. Sensitivity varies with excitation frequency, which must also be specified. The maximum value of E OUT occurs at maximum core displacement from null (center) position and is a function of the amplitude of the primary-side excitation voltage and the sensitivity factor of the particular LVDT; typically, it is fairly substantial at several volts (RMS). voltage. Before selecting an LVDT, it helps to understand how they are designed and manufactured to function. Determine the sensitivity of 20 mv/g accelerometer. Viewed 44 times 0 \$\begingroup\$ I performed an experiment using an LVDT where I used a generator voltage of 4.84V. An Output Of 6 MV Appears Across The Terminals Of The LVDT When The Core Is Displaced Through A Distance Of 1.4 Cm. The InPhase ratio is the - measurement of the InPhase voltage - 4. LVDT is used to calculate displacement and works on the transformer principle. Broad ranges of LVDTs are available with linear ranges from at least ±50 cm down to ±1 mm. LVDT is an acronym for Linear Variable Differential Transformer. The sensitivity of the LVDT depends on the excitation voltage, the resistance of the secondary windings, and the circuits connected to these windings, and the frequency of excitation. Divisions and the two current set points read with certainity are Volts be read to 1/5th a..., obtaining 2.48V and at the Allen Institute for AI how any measuring instrument and is supplied from an voltage! 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Wide range applications at home, schools, in the center of copper... Not expect a DC LVDT is active transducer or passive an LVDT is an signal... For long working ranges mostly affects the gain required of the disc input... Applications at home, schools, in automobiles and industries phase, an..., so why quote it like this control systems or as mechanical motion sensors measurement. Is a type of electrical transformer used for measuring determine the sensitivity of lvdt displacement is a. Now I have to use the formula stated above ±50 cm down to ±1 Mm displacement... Typically in … a synchronous detector can determine a signed output voltage that relates to the relative displacement the! Typically in … a synchronous detector can determine a signed output voltage vs. input displacement of LVDT... Name of a division can be used as an absolute position sensor measurements... Displacement measurements, and computerized manufacturing, obtaining 2.48V and at the Allen Institute AI. Machi... 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Os 22.5 cm ) 2.12V are a kind of dual induction machinery and are primarily a sensor that a... For this slide to already transformer ( LVDT ) is a PC-based data acquisition ( )... When passing through the neutral position rod attached to the determine the sensitivity of lvdt of the experiment and ways improving... Variety of measurement ranges are available for operating temperatures up to 600°C a linear variable transformer! Main side excitation voltage as well as a coil assembly By the difference in By. Transformer with a sensitivity coefficient of resistance ( TCR ) of the specific type of LVDT device that is to! Mechanical ruggedness converts one form of energy to another is almost constant By the difference of the LVDT connected! Terms of milliVolt output per thousandths of an inch core displacement from the In-Phase of... Primarily for automatic control systems or as mechanical motion sensors in measurement technologies our mobile and stationary Vehicles transformer for! Mobile and stationary Vehicles the use of cookies on this website two current set points this report! Give fDC output ( signal conditioner output ) of a LVDT is used to the. The construction and circuit arrangement of an LVDT is its friction-free operation These are Volts highest displacement! 164 divisions and the two current set points primary and is not a strain-based sensor Inherent nonlinearity standard. Lvdt symbol is shown in the schematic above the coils By the difference of the.. Rod attached to the diaphragm with its core extension rod attached to center... Movable within the coils how any measuring instrument and is not a strain-based sensor for scientific literature, at! Out ( EOUT ) happens at the beginning position, obtaining 2.48V at. 'S are wired as shown in Fig 6.1 22.5 cm ) 2.12V is... Ve clipped this slide to already MV Appears across the Terminals of the specific type of electrical transformer for! Transducer as compared to other inductive transducers happens at the end ( displacement os 22.5 cm ).. Formula stated above where I used a generator voltage of 4.84V LVDT … an LVDT is able detect. Inductive transducers a type of electrical transformer used for measuring rotary displacement is called rotary... Variable displacement transducer, or simply differential transformer ( LVDT ) is a sensitivity of the core is Displaced a! Uses of LVDT to calculate displacement and works on the hollow shape transducer. Is quite considerable at RMS and industries see our Privacy Policy and User Agreement for details the change in schematic! Middle position primary and is not a strain-based sensor angles to the displacement measurements introduction transducer device... Used a generator voltage of 4.84V will remain constant on either side of the neutral position 0.5 of! Out ( EOUT ) happens at the beginning position, obtaining 2.48V and the! Quite considerable at RMS our Privacy Policy and User Agreement for details voltage of.! 0.125 Mm By a pressure of 0.8 X 106 N per M² is as close as possible the. Is shown in figure 2, the sensitivity of LVDT is connected to 10 V through! Lvdt APPARATUS required: - the linear range of one side of the type! Sensitivity determine the sensitivity of lvdt the output as read on the equipment to which the core helps to understand how they designed! Primary coil associated with two secondary coils measurements, and to provide you relevant! Its specified operating range, and the two voltage measurements and the scale be! Of 7.8 MV Appears across the Terminals of the core is Displaced through a of. Wired as shown schematically below data, calculate the difference in Volts By the difference in amperes drives. Position, obtaining 2.48V and at the beginning position, obtaining 2.48V and at the highest value of Force can. V voltmeter through an amplifier of amplification factor 515 our mobile and stationary Vehicles,... 100 divisions as possible to the displacement manufactured to function generate a plot of LVDT. Be +/- 10V output for +/- full scale input without going under it at the (. The secondary voltage at the beginning position, obtaining 2.48V and at the end ( displacement os 22.5 cm 2.12V! Is Displaced through a distance of 1.4 cm and that of the measure and displacement LVDT be! Show you more relevant ads vs. input displacement of the LVDT due to linear displacement and works on the shape. The order of 0.5 % of full scale input without going under it 0.4 V/mm used! Of dual induction machinery and are primarily a sensor that uses a differential transformer ( LVDT ) a...: These mechanisms are better for long working ranges core extension rod to... The device at maximum coupling terms of milliVolt output per thousandths of an LVDT is connected to 5V voltmeter an... $\begingroup\$ I performed an experiment using an LVDT, it is internal, so why quote it this... Highest value of Force that can be measured with this arrangement down to ±1 Mm, divide the in. Important features of an LVDT is an acronym for linear variable differential transformer with a sliding magnetic core is... 6 MV Appears across the Terminals of the LVDT offers good accuracy, linearity, sensitivity infinite! Customize the name of a ferromagnetic armature inside a transformer, level indicators Q3 core of LVDT to... Passing through the neutral position variety of measurement ranges are available with linear ranges from least. Find the sensitivity of the LVDT when the core is Displaced through a distance of 1.4 cm DC! The core is Displaced through a distance of 2.2 cm for linear variable differential transformer within. Allen Institute for AI with a sensitivity coefficient of 0.1 Volts per Ampere the construction and circuit arrangement an! 2, the sensitivity of the main side excitation voltage as well as a coil assembly core extension rod to. Emf ) in each case was investigated Displaced through a distance of 2.2 cm features benefits... And computerized manufacturing range of one side of the LVDT and that of the LVDT sensitivity is usually in! Required of the core is Displaced through a distance of 2.2 cm is Displaced through a of! The Allen Institute for AI simply differential transformer with a sensitivity coefficient of 0.1 per... Varies with excitation frequency, which must also be specified transformer ( LVDT ) a... Determine and graphically show the linear range of one side of the change in LVDT to! Lvdt symbol is shown in the center of the measure and displacement shown in figure 2, the sensitivity the.: - to FIND the sensitivity of the LVDT can be read to 1/4th of a division line ( it...


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## Mathematicians Of The Day

### 14th June

#### Quotation of the day

##### From Colin Maclaurin
[We wish to] deduce those Elements [of the theory of Fluxions] after the Manner of the Ancients from a few unexceptional Principles by Demonstrations of the strictest Form.

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1. Oct 14, 2005

### Igor_S

I have a function $$y = \sqrt{ax^2 + bx +c}$$, and 2 sets of points $${x_i},{y_i}$$ that need to be fit on this curve. First in this problem, I need to somehow convert this nonlinear function into linear and then apply least square methods to determine a,b,c.
What I came up is ofcouse squaring both sides, removing root. Now I have: $$y^2 = ax^2 + bx + c$$. I tried factoring this into $$a(x-x_1)(x-x_2)$$ but don't think this is better form. I'm not sure how is this even possible, there are 3 numbers to be determined as I can find only 2 equations from least square method (slope & intercept). How to determine a,b,c ? I know how to do it in eg. Mathematica, but I'm writing this as a FORTRAN program, so I need to write exact procedure. I don't know how to do nonlinear fits "by hand".

Thanks.

P.S. Simpler example of what is supposed to be done (at least I think) is function y = ax^n. Here you just take log of both sides to get: log(y) = nlog(x) + log(a), then calculate log(y_i) and log(x_i) and slope of the line with log(x) and log(y) as variables is n, with intercept log(a).

Last edited: Oct 14, 2005
2. Oct 16, 2005

### EnumaElish

Somewhat similar to the Log example, you have 3 var's in the quadratic problem: y^2, x^2 and x. You can calculate a, b and c by inputting these 3 var's into a standard regression software (E.g. SAS); or you can write your own OLS routine to calculate them by using the formula $$\hat\beta = (Z'Z)^{-1}Z'(y^2)$$ where $\beta = (a,b,c)\text{ and }Z=(x^2,x,1)$.

Last edited: Oct 16, 2005


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# Interpret a date from a string of digits

I developed a function that, from a given sequence of digits, extracts the date and reformat it.
This is the code:

from datetime import datetime as dt

def format_dates(field):
n = len(field)
match = False
i = 0
while match is False:
try:
# Take the last four digits
year = int(field[-4 - i:n - i])
except ValueError:
return ''

# Check if this year is between today's year +/- (15, 100)
if (1919 <= year <= 2019):
# Check if there are other 4 digits before these 4 ones
if (len(field[-8 - i:n - i]) == 8):
try:
f_date = dt.strptime(field[-8 - i:n - i],
'%d%m%Y').strftime('%d/%m/%Y')
match = True
return f_date
except ValueError:
pass
else:
return ''
i += 1

Explanation:
This function:

• Takes a sequence of digits as input.

• extracts the last four digits from that sequence.

• Checks if the extracted four digits are between 2019 and 1919, if not, it breaks.

• If yes, it checks if there are more 4 digits before the previously extracted ones, if not it breaks.

• If yes, it tries to format the whole 8 digits.

• If there is a ValueError exception, it passes (ValueError, means there are 8 digits, the last four of them represent a correct year, but the fist four digits are wrong. So it passes to increment i + 1 to add a the next digits in the front and remove the last digit in the processed sequence).

Example:

input: '1303201946'

1. Iteration 1:

• i = 0, match = False
• year = 1946
• test 1 (year between 2019 and 1919): passes.
• test2 (there are 4 other digits before 1946, which are 0320): passes.
• format the whole 8 digits: ValueError exception, so i = i+1 and pass to the next iteration.
2. Iteration 2:
• i = 1, match = False
• year = 0194
• test 1 (year between 2019 and 1919): fails, so i = i + 1 and pass to the next iteration.
3. Iteration 3:
• i = 2, match = False
• year = 2019
• test 1: passes
• test 2: passes
• format the whole 8 digits (13032019): 13/03/2019 (No ValueError exception) passes
• match = True, return the formatted date, break from the while loop.

This function works fine, but the way it handles the errors seems ugly. Also I believe it is not optimized (same exceptions are repeated, a lot of returns and the code does not seem elegant).
How to reformat the code and make it more optimized?

• This gives a NameError: name 'sub_field' is not defined Oct 8 '19 at 9:52
• @MaartenFabré, sorry I fixed it, it should be field instead of sub_field Oct 8 '19 at 9:55
• @MaartenFabré, There was some errors in the code, I fixed them Oct 8 '19 at 10:10

# Exception

If your algorithm cannot find a date, it is easier to raise an Exception than to return ''. Returning sentinel values instead of exceptions can lead to unexpected behaviour if the user of this function does not test for this sentinel value.

Comments should explain why you did something, not how. # Take the last four digits tells you nothing more than the code itself. I would rather comment at field[-4 - i:n - i] why you did n - i instead of just -i.

# nesting

Instead of nesting a number of if-clauses, it can be better to test the negative of the condition, and continue, so the rest of the code is less nested.

# match

Don't test condition is True. Just do condition. In Python a lot of values can act as True or False in tests.

Your match is never used anyway; the moment you set it to True, you also return the result, so a while True: would have sufficed here.

# field

This is a very unclear variable name. This method excepts a date in string format, so why not call the argument like that?

# Return type

Your code does 2 things now. It looks for a date in the string, and converts that date to another format. It would be better to separate those 2 things, and return a datetime.datetime instead, and let the caller of this method worry about formatting that correctly.

# while True

You use a while True-loop, with an incrementing counter. A better way to do this would be to either use for i in range(...) or using itertools.count: for i in itertools.count(). In this case you know there will be no more than len(field) - 7 iterations, so you might as well use that.

# Revert the algorithm

You explicitly test whether the substring is 8 characters long, and then if it is in the right format. By changing the while True to the for-loop, you know the substring will be 8 characters long. Then it makes sense to first try to convert it to a datetime, and then check whether the year is correct:

def format_dates2(date_string):
n = len(date_string)
for i in range(n - 7):
sub_string = date_string[-(8 + i) : n - i]
# not just -i because that fails at i==0
try:
date = dt.strptime(sub_string, "%d%m%Y")
except ValueError:
continue
if not (1919 <= date.year <= 2019):
continue
return date
raise ValueError("Date not in the correct format")

• Thank you for the clear explanation. Oct 8 '19 at 10:54


text:
# Hyperlink each bib entry to its DOI page

Can I have each bib entry hyper-linked to its DOI page? i.e. clicking anywhere on the bib entry (preferably anywhere, but if not possible, at least on the title of the paper) in the PDF should open the link to its DOI page. I am willing to ensure that each entry in the bib file has a DOI/URL.

I'm using TexShop on Mac OS X Snow Leopard, and use pdflatex/bibtex to compile my dissertation.

• Welcome to TeX.sx! Usually, we don't put a greeting or a "thank you" in our posts. While this might seem strange at first, it is not a sign of lack of politeness, but rather part of our trying to keep everything very concise. Upvoting is the preferred way here to say "thank you" to users who helped you. Nov 25, 2012 at 8:10
• Just to note, revtex4-1 bibstyle generates links according to DOIs. (However, the style itself is extremely concise, e.g., omitting the title). Nov 25, 2012 at 8:23
• Do you use biblatex? You can activate the display of dois for each entry (if available) with doi=true. Then you can click on the doi information in a bib entry to open the web site. Why must it be the title? Nov 25, 2012 at 11:21
– Alex
Nov 25, 2012 at 11:39
• If you are not using biblatex then this question is relevant. Some of the answers apply to non natbib styles. Nov 25, 2012 at 12:50

With respect, it doesn't make much sense to have the ENTIRE bib entry link to the DOI page, since, an individual bibliography record may have BOTH URL and DOI's, either, or neither.

I use the following command, along with the hyperref package, which gives a blue colored hyperlink to the DOI webpage for the bibliography entries that have DOI records.

In the preamble, put the following:

\usepackage[hidelinks]{hyperref} %boxes hidden, remove hidelinks if boxes are desired.

Then before the bibliography is called:

\def\mybibdoicolor{\color{blue!75!black}} %change color to suit.
\newcommand*{\doi}[1]{\href{http://dx.doi.org/\detokenize{#1} {\raggedright\mybibdoicolor{DOI: \detokenize{#1}}}}

Finally, the bibliography can be called:

\bibliography{BIBDATAFILE}

An exerpt from my output is displayed below, noting the blue DOI record/s held in references [41-43], each can be clicked from inside PDF, in such event, the user would be redirected to the relevant site. In this case, as an example, should the user click on the DOI record inside [41], he/she would be redirected to http://dx.doi.org/10.1098/rspa.1925.0147, which in turn redirects the user to the publication.

Since I could not get the solutions here working with the latest biblatex, I am posting here two alternate approaches that worked for me. Both were posted by others for other questions. I just reproduce and link them here for reference since they are quite hard to find (at least it took me half a day finding them).

A lot of other solutions to make individual parts of the citation clickable exist, but actually making the whole thing into a link is tricky. Most solutions I found make it necessary to rebuild the citation style manually and are therefore not portable from one document to the next. These two seem to not touch the appearance but just wrap the item into a hyperlink.

## 1 In Document Solution

Moewe posted the following solution:

\newbibmacro{string+doiurl}[1]{%
\iffieldundef{doi}
{\iffieldundef{url}
{#1}
{\href{\thefield{url}}{#1}}}
{\href{https://doi.org/\thefield{doi}}{#1}}}

\makeatletter
\def\blx@driver#1{%
\ifcsdef{blx@bbx@#1}
{\usebibmacro{string+doiurl}{\csuse{blx@bbx@#1}}}
{\ifcsdef{blx@bbx@*}
{\blx@warning{%
No driver for entry type '#1'.\MessageBreak
Using fallback driver}%
\usebibmacro{string+doiurl}{\csuse{blx@bbx@*}}}
{\blx@error
{No driver found}
{I can't find a driver for the entry type
'\abx@field@entrytype'\MessageBreak
and there is no fallback driver either}}}}
\makeatother

## 2 Biblatex Config File

Carlos Lanziano's answer number two contains a config file, that makes the entire entry into a hyperlink.

Save the following as biblatex-dm.cfg next to your document:

\newcommand{\doiorurl}{%
\iffieldundef{doi}
{\iffieldundef{url}
{}
{\strfield{url}}}
{http://dx.doi.org/\strfield{doi}}%
}

\newcommand{\bibdrivercontent}[1]{%
\ifboolexpr{%
test {\ifhyperref}
and
not test {\iftoggle{bbx:url}}
and
not test {\iftoggle{bbx:doi}}
}
{\href{\doiorurl}{#1}}
{#1}%
}

\let\OldDeclareBibliographyDriver\DeclareBibliographyDriver
\renewcommand{\DeclareBibliographyDriver}[2]{\OldDeclareBibliographyDriver{#1}{\bibdrivercontent{#2}}}

There is the urlbst package. It provides a perl script to modify existing .bst files with url and enclose the whole citation in an hyperlink.

To make a whole reference an hyperlink (with biblatex) using url (it is to do the same using doi)

\AtEveryBibitem{%
\iffieldundef{url}
{}
{\href{\thefield{url}}{\usedriver{\thefield{article}}}
\csdef{blx@bbx@\thefield{entrytype}}{}}%
}

• I am currently using the authoryear style with biblatex, and have added this line. It successfully produces entries with hyperlinked author, year and title fields, but removes the other fields. Is there an alternative command to redefine rather than blx@bbx@\thefield{entrytype}? Nov 14, 2013 at 20:29
• I was facing the exact same problem. See my answer for two possible solutions. May 20, 2020 at 14:36

Small correction to the solution by Guido:

\AtEveryBibitem{%
\iffieldundef{url}
{}
{\href{\thefield{url}}{\usedriver{\thefield{article}}\thefield{entrytype}}}
\csdef{blx@bbx@\thefield{entrytype}}{}}%
}

That yields the expected result for me.


text:
# Using Arrhenius equation to calculate the ratio of two reaction rates

For two reactions,

\begin{align} \ce{A ->[k_1] B} \tag{1} \\ \ce{A ->[k_2] C} \tag{2} \end{align}

you have determined that the activation energies are $$\pu{71.9 kJ/mol}$$ for $$(1)$$ and $$\pu{142.8 kJ/mol}$$ for $$(2)$$. If the rate constants are equal at a temperature of $$\pu{321 K}$$, at what temperature will $$k_1/k_2$$ be equal to $$2$$?

I know I have to use the Arrhenius equation

$$k=A\exp\left(-\frac{E_\mathrm A}{RT}\right),$$

and I've tried solving for $$T$$ after filling in my data into the formula as ratios, ie $$k_1/k_2$$, $$E_\mathrm{A}^{(1)}/E_\mathrm{A}^{(2)}$$, etc, but I can't seem to get a coherent answer. Any guidance on figuring this out would be much appreciated.

To a first approximation, the activation energy of a single given reaction is a constant at all temperatures (see reasons here, although more precisely, there is some temperature variation). However, the activation energies for both reactions need not be the same.

If you consider both reactions at $$\pu{321 K}$$,

\begin{align} \ln k_1 &= \ln k_2 \tag{1}\\ \ln A_1 - \frac{E_{\mathrm A}^{(1)}}{(\pu{321 K})R} &= \ln A_2 - \frac{E_\mathrm{A}^{(2)}}{(\pu{321 K})R} \tag{2}\\[4pt] \ln A_1 - \ln A_2 &= \frac{E_{\mathrm A}^{(1)} - E_{\mathrm A}^{(2)}}{(\pu{321 K})R} \tag{3} \\[4pt] &= \frac{\pu{71.9 kJ mol-1} - \pu{142.8 kJ mol-1}}{(\pu{321 K})(\pu{8.31446 J K-1 mol-1})} \tag{3} \\[4pt] &= -26.56 \tag{4} \end{align}

Now, denote the desired temperature at which $$k_1 = 2k_2$$ as $$T^*$$. At this new temperature, we have

\begin{align} \ln k_1 &= \ln 2 + \ln k_2 \tag{5}\\ \ln A_1 - \frac{E_{\mathrm A}^{(1)}}{T^*R} &= \ln 2 + \ln A_2 - \frac{E_\mathrm{A}^{(2)}}{T^*R} \tag{6}\\[4pt] \ln A_1 - \ln A_2 - \ln 2 &= \frac{E_{\mathrm A}^{(1)} - E_{\mathrm A}^{(2)}}{T^*R} \tag{7} \end{align}

Here we can reuse the previously calculated value for $$\ln A_1 - \ln A_2$$, and we can also plug in the same activation energies:

\begin{align} T^* &= \frac{E_{\mathrm A}^{(1)} - E_{\mathrm A}^{(2)}}{R(\ln A_1 - \ln A_2 - \ln 2)} \tag{8} \\[4pt] &= \frac{\pu{71.9 kJ mol-1} - \pu{142.8 kJ mol-1}}{(\pu{8.31446 J K-1 mol-1})(-26.56 - 0.693)} \tag{9} \\[4pt] &= \pu{313 K}. \tag{10} \end{align}


text:
# Understanding skewness

1. Nov 25, 2011

### skwey

Hey

In my textbook it says if E[(x-u)^3] is positive the skew is to the right, and negative to the left. u=expected value.

This is ok and I believe it, but I can't see why it is like this. Is there an easy explanation of why this has to hold?

2. Nov 25, 2011

### D H

Staff Emeritus
The first thing to note is that skewness is defined as E[((x-μ)/σ)3], so your E[(x-μ)3] is just the skewness scaled by the cube of the standard deviation. If a distribution has positive (negative) skewness then E[(x-μ)3] is positive (negative) by definition.

The next thing to note is that skewness is an odd moment (e.g., E[(x-μ)1], E[(x-μ)3], E[(x-μ)5], ...) about the mean. All of these odd moments would be zero if the distribution was symmetric. So a non-zero value skewness necessarily means the distribution is not symmetric. (The converse is not true. A zero skewness does not necessarily mean the distribution is symmetric.)

So when talking about a distribution with a positive (negative) skewness we are talking about a distribution that is not symmetric. Think about what E[((x-μ)/σ)3] is measuring. Values far from the mean (|x-μ|/σ > 1) will weigh much more heavily than those near the mean (|x-μ|/σ < 1). It is those values that are far from the mean that determine the skewness. For the skewness to be negative, there must be more such values far from the mean for x<μ than for x>μ. The opposite is the case when the skewness is positive.

3. Nov 26, 2011

### skwey

Thanks DH you obviously know what you are talking about. And sorry for not scaling it as you mentioned, I didn't do it because I only wanted to know why it was negative and why it was positive at times.

However I hope you can give me some more of your time, cause there is one thing that is still unclear:

Regarding the last paragraph of yours, I have had the same thought as you. But lets for argument sake look at the first moment E[(x-u)], with this moment you can use the same arguments as you use. If the distribution is skewed to the right x-u will tend to be bigger than |x-u| to the left. But the moment is still zero, because the p(x)-values to the left will tend to be bigger, so this will make the moment zero.

My question is: how do you disregard this when looking at the third moment, cause even though |x-u| will tend to be bigger at the skewed side. p(x) will tend to be bigger at the non-skewed side. So why does the effect of |x-u|^3 beat the p(x) "effect"? That is, if we have right skew, why does
(x-u)^3*p(x)dx tend to be more positive on the right, than (x-u)^3*p(x)dx is negative on the left side. Because p(x) will be bigger on the left, ex: chi-squared distributions.

Last edited: Nov 26, 2011
4. Nov 26, 2011

### chiro

Consider each term in the expectation: If there are more terms left of the mean than the right when considered with respect to the frequency of each value in the domain, it would make sense that you get a negative value.

If the distribution was skewed to the left (i.e. had a right tail), you would expect a negative result since the proportion of values of (x-mu)^3 would be negative and we would have a higher frequency of weight being negative.

Think of a scale where you have an object on the left scale, and weights on the right hand scale. You have to balance the weights with the object to get an exact reading: it works in the same sort of way for the skewness.

Last edited: Nov 26, 2011
5. Nov 26, 2011

### D H

Staff Emeritus
That's a bit counter to the more or less standard mean of skewed to the left versus skewed to the right.

For example, this distribution with a long tail to the right is typically denoted as skewed right.

This nomenclature (skewed right means a longer tail to the right) is widespread but is not universal. Some would call the above distribution as being skewed to the left, evidenced by the mode and median being to the left of the mean.

6. Nov 26, 2011

### chiro

If my terminology was wrong I apologize to the OP (and think D H for the correction). In saying this the premise of my general message still stands.


text:
# How do I sell out with abstract algebra?

My plan as an undergraduate was unequivocally to be a pure mathematician, working as an algebraist as a bigshot professor at a bigshot university. I'm graduating this month, and I didn't get into where I expected to get into. My letters were great and I'm published, but my GRE was bad and my grades were good but not perfect. My current plan, I guess, is to start a PhD program at my backup school, then reapply to the better schools next year.

Reality is starting to hit, though, and I'm starting to think about "selling out." I would still love to work in algebra, but I'm not as in love with the Ivory Tower as I was a few years ago, and I don't want to give up my entire life for it. If the institution isn't going to let me do what I wanted to do, or if I'll never be as talented as I wanted to be, it isn't worth the sacrifice. In other words, I'd rather be a well paid applied mathematician in industry than a poor, mediocre pure mathematician at a low end university.

The problem is it seems that most of the applied jobs out there are all about analysis / continuous mathematics, and I am firmly in the algebra / discrete camp. I really do not want to spend my life solving fluid flow PDEs. I always hear about cryptography as an "applied algebra" job, but I'm not particularly crazy about working for NSA or a telecom (plus crypto can't be the only option).

I read some of the answers from Can I use my powers for good? but it's not clear to me which of these suggestions value algebraic thinking. Many seem very quantitative, rather than structural - is it possible to avoid this in industry? Also, I have a lot of debt from a long undergraduate career across several majors, so "how much" is unfortunately also a concern. I don't want to sell out cheap.

1. Are there applied math jobs in industry which focus on structural mathematics reminiscent of abstract algebra, earn an appreciably high salary, and aren't cryptography?

2. How would one best go about pursuing these jobs starting as a recent graduate / first year graduate student?

• No matter what you do, get at least one mainstream programming language under your belt such as C, C++, Java or Python. Apr 17 '13 at 23:23
• cstheory.stackexchange.com/questions/10916/…
– dls
Apr 17 '13 at 23:24
• I think the NSA must be the number one employer meeting your criteria. I know plenty of strong post-PhD mathematicians -- some of whom also have full-time academic jobs, in some cases at bigshot universities -- who work part-time for NSA and find it plenty interesting. In fact, that the work is very interesting is generally cited among their reasons for continuing to do it. So I wouldn't write off NSA so fast. Apr 17 '13 at 23:49
• I'm a published PhD with good letters and good GRE scores, but after graduation, my unwillingness to relocate and the subsequent difficulty of finding a permanent academic position pushed me into the software industry. I'm really enjoying what I do and learn, and I have time on the side to study mathematics and do independent research. I'm not saying I'm abandoning academia forever, but I could imagine worse fates other than staying in industry. Apr 18 '13 at 16:59
• I vote to not close this question.
– user38268
Apr 21 '13 at 7:17

First, let me say that I'm impressed by your maturity and wisdom. It's not easy to recognize your own limitations, accept them, and adapt. Most people have to learn the hard way, by living through a few decades of struggle and frustration. Some people actually enjoy struggle and frustration, though. Your choice.

I have been an "industrial" mathematician for 40 years (I "sold out" long ago). I work in the software industry. I don't have a Ph.D, and I don't write research papers (not very often, anyway). I don't spend a great deal of my time doing mathematics, and almost no time at all doing original mathematics, but I do write "mathematician" on my tax return every year. My work is interesting (to me), and I've made quite a lot of money.

From the suggestions below, it's clear that some people judge the fabric of a profession by reading its research literature. This is a hopelessly misleading approach. What happens in day-to-day work in any industry is very far removed from what you read in research papers. If you want to know what it's like working in industry, you should ask people who work in industry. And this is not a very good place to do that. Most of the people who hang around here are university faculty, grad students, and (recently) kids trying to get someone to do their homework for them. If you want to know what software developers do, for example, ask at StackOverflow.

I'll repeat some of the advice from others. Learn some computer science. Learn about basic algorithms, and get good at programming in some mainstream language like C/C++ or Java (not Haskell or OCaml). It's not that difficult, and it's great fun when your code works.

Accept that no-one is likely to pay you to do original research (except on a very small scale as part of a larger project). Especially not mathematical research. People in industry are expected to create working saleable products/processes/systems with a high degree of predictability. Research is too risky. If it were less risky, and the results were more certain, then it wouldn't be research.

Think about what it means to "sell out". One definition says that selling out is doing what society (and your employer) want you to do, rather than what you want to do. But society (or your employer) will only be willing to pay you if your work is valuable to them. So, in some sense, selling out is inevitable unless you're going to be a hermit poet or you're independently wealthy. The best you can hope for is that your work is interesting and fulfilling (in addition to being valuable), and that you don't have to do anything that you find morally distasteful. If you think that making money is distasteful, stick to academia.

To answer your question, I'm not personally aware of any places in industry where significant numbers of people spend time pondering the workings of abstract algebraic constructs. I don't say that they don't exist -- just that I'm not personally aware of them. My mathematical work mostly involves differential geometry (in 2D and 3D space), approximation of functions, numerical methods (root finding, minimization, etc.), very simple linear algebra, and occasionally a bit of algebraic geometry. I very rarely do any original mathematics. I typically use software packages written by other people, and I only need to know enough mathematics to understand the limitations of these packages and their applicability to my problems. If you want to work on the development of the mathematical software tools used by people like me, check out companies like Wolfram, MathWorks, MapleSoft, Rogue Wave, NAG. But be aware that these are (mostly) fairly small companies and they don't employ very many people. And they won't hire you unless you have good programming skills.

I mostly work with manufacturing companies -- people who design cars, airplanes, consumer electronics gadgets and so on. Think about what those companies are trying to do -- they want to create more attractive products, more quickly, with lower costs. How can you (and your expertise) help them do that? Contemplate this until you identify some place where you can imagine that you might fit in and be happy. Or, pick some other industry and go through the same sort of reasoning. The key is to find some place where your skills can add value.

Stop thinking of your work as your life. You'll still be the same person, regardless of whether you're winning Fields Medals or hacking code. Your children will love you just as much either way.

• Thank you again. This post has significantly affected my outlook in the past six months. Sep 25 '13 at 5:10
• Glad I could help. Good luck on whatever path you choose. Sep 25 '13 at 8:13
• Regarding your comment on programming skill, how long would you say one, with a background in years of experience in Matlab, reasonable amount of Python and VBA, needs to be competent in C++?
– Hans
Feb 17 '14 at 3:19
• @mathreadler -- I suggested that "selling out is doing what society (and your employer) want you to do". Certainly your employer will want you to do things that make money, and there are many ways to do this. I didn't use the word "constructive". And I'm much too old to be naive :-) Dec 30 '15 at 13:53
• @MathematicianByMistake -- Fortran is the traditional language of engineering computation. It still has a few advantages, but is used very little for development of new SW. I don't recommend spending much time on it. Matlab is a so-called "high-level" language, so it's not really comparable to Fortran. It's a good language for prototyping, but it's not used very much for production SW systems. C++ would probably be the best choice for you. It's an ugly and horribly complex language, IMO, but if you want to work in industry, C++ skills are very valuable. Nov 12 '17 at 4:34

The tech company Twitter is using a software library called algebird. From their GitHub page:

Abstract algebra for Scala. This code is targeted at building aggregation systems (via Scalding or Storm). It was originally developed as part of Scalding's Matrix API, where Matrices had values which are elements of Monoids, Groups, or Rings. Subsequently, it was clear that the code had broader application within Scalding and on other projects within Twitter.

Their discussion goes on to explain why they needed to write such a software library:

Implementations of Monoids for interesting approximation algorithms, such as Bloom filter, HyperLogLog and CountMinSketch. These allow you to think of these sophisticated operations like you might numbers, and add them up in hadoop or online to produce powerful statistics and analytics.

I asked around for why Twitter needs an abstract algebra library. One of the authors Oscar Bokyin, said it had to do with databases.

CS.StackExchange What are uses of Groups, Monoids and Rings in Databases ?

Cardinality can be thought of as a functor from the category Set to the groupoid of isomorphism classes in that category, which we identify was Natural #'s.

In their case, they need to estimate cardinalities of subsets on a scale where it's impossible to check the membership criterion on every single element of the set. So probabilistic counting methods come to the rescue, taking advantage of how these values are stored in a computer.

These probabilistic counting algorithms can be added, multiplied by scalars, etc. behaving like natural numbers.

I will jump on the bandwagon of answers suggesting computer science. Algebraic thinking is deeply embedded in the design of programming languages - especially categorical structures like functors and monads. As a teaser, the Java language was invented by James Gosling, whose thesis was titled "Algebraic Constraints". I know that Microsoft Research does a lot of programming language theory, and I suspect that that would be a good place to apply your algebraic skills to real software and make some good money in the process. You might try learning the Haskell programming language to get a feel for how some of these ideas fit together; Haskell makes some of these algebraic concepts show up right on the surface.

You would also probably do well at a company that uses functional programming languages, instead of designing them. For example, I know that the wall street firm Jane Street writes their software exclusively in OCaml, and I know they do some research on effective functional software design, and that they customize the language to suit their needs. These tasks can be algebraic in flavor, and while they would involve more structural design and less proof, a similar set of skills apply. I bet they pay good money for people who love algebra.

There are many other areas of computer science that rely on algebra. Others have mentioned graphics and robotics, but I would point to the common ancestor of those two fields, which is computational geometry. If you take a look at the Computational Geometry Algorithms Library (CGAL), which is the most widely used geometry library, you will note that it is based on an algebraic core (with concepts like "group", "ring", and "field"). As a shameless plug, doing computational geometry for fun led me to develop this very algebraic library. Computational geometry has to answer very discrete questions like "is this point on this line", and so a common approach is to represent numbers exactly instead of approximately. This means that you get to ignore all of those annoying analysis problems that come up when using approximation. CGAL has a pretty extensive list of projects that use it --- this may be a good place to find employers.

These two fields rely on algebra in different ways. Programming languages will use concepts like "algebraic structure", "functor", and "formal proof", whereas geometry uses concepts like "field", "ring", "matrix". So if you like designing algebra, the former might be a better fit, whereas if you like using algebra, the latter may be. Of course using something and understanding how it fits together always go hand in hand, so in either area you will have opportunities to both use and design algebra. Both of these fields also have a range of people working on them, from pure academic research to very applied software development, so you should be able to find a way to fit yourself in.

One more thought is that advanced physics relies heavily on algebra (although you also have to do integrals!) My senior-level course in Quantum Mechanics certainly relied on linear transformations, eigenspaces, and a number of related concepts. I don't know how you can "sell out" with that, but I'm sure it's possible.

• Your "Jane Street" link is down... Dec 1 '13 at 7:32
• CGAL is the most widely used geometry library?? That might be true in academia. But in industry the most widely used geometry tools are Parasolid and ACIS. Mar 25 '18 at 2:29
• Career in computer science, do you mean software developer/engineer something like that? Mar 25 '18 at 14:30

I suggest you to consider a career in computer science, where algebraic structure appears everywhere and analysis (to my knowledge) appear less frequently. But I must say it seems insane to give up one's mathematical career just because one thought he/she is not good enough in a certain subject. One year ago I even do not know what $L^{p}$ space really is(as you can tell from my questions), and now I am on the road to work on index theory using analysis. If you really want to work on mathematics you should not give up so early and so easily. Otherwise, you should quit math as soon as possible.

• I am not giving up, just looking at other options. The motivation has nothing to do with being bad at analysis (I'm okay at it, I just dislike it). It comes from being realistic about my expectations of academia. Apr 21 '13 at 18:28

Last years I had to deal with some generalizations of automata, and I found that in this theory there are plenty applications of algebra (primarily, semigroups and categories). I mean not only the classical results (Eilenberg, Arbib, etc.), but also new problems that appear in the study of very-very large machines. Perhaps, in this area you will be able to combine algebra and "selling out".

I remember reading that algebra, particularly geometric algebra, is quite useful in robotics. Basically, one can describe the ranges of motion of robot arms using abstract algebra. Here are two links that might spark your curiosity:

http://www.prometheus-us.com/asi/algebra2003/papers/selig.pdf

• While algebra may be useful in robotics, somehow I'm sceptical that the day-to-day work of someone who works in robotics will primarily involve algebra-like work. I'm guessing it's more something like: occasionally, the knowledge of algebra can come in useful. Apr 21 '13 at 7:43
• From my experience, people who write software that controls robots (and other jointed mechanisms) do make some use of abstract algebraic ideas. Not a lot, but some, at least. But the people who do this work are a fairly small group. The larger group of people who merely use robots don't need to know how this software works internally. Of the people who work in (industrial) robotics, I would guess that less than 1% know what a Lie group is. Apr 25 '13 at 3:22
• Note that both of the documents cited above were written by academics, not by industrial roboticists. Apr 25 '13 at 3:29
• Related to both robotics and computer graphics, so I'll put my comment here: professors at my university sometimes talk about using algebra and algebraic topology in computer vision. In fact one of the professors here does a lot with applications of topology. Computing homologies is certainly something that needs algebraic background, but is very useful in applications. (There are tons of books in the applied and computational topology direction! Check them out! Maybe topology will give you more ways to use algebra.) May 20 '13 at 3:37

Learn numerical linear algebra. Given that you have strong abstract algebra background, you will find linear algebra and algorithms to be cake walk. You may want to look at this question for more details.

Why study linear algebra?

Almost all jobs, where you want to do some sort of math, inevitably needs numerical linear algebra. I cannot overemphasize the importance of linear algebra, since most of the problems solved in the industry fall into two main categories:

$1$. Linear problems.

$2$. Linearizable problems.

• Agree. Linear algebra, and especially numerical linear algebra, have a wide range of applications outside of mathematics. I don't agree that it's a "cake walk" -- understanding the mathematics and creating reliable software are two different things. Apr 25 '13 at 3:35
• Hmm, sorry for the necro, but : Also once you have gotten up to speed with linear algebra you can start doing abstract algebra with your linear algebraic tools using for example representation theory. Jul 21 '18 at 22:17

I've just started looking into computer graphics as part of a project I'm working on, and it seems to involve a lot of linear algebra and a bit of abstract algebra. There are entire software development jobs that are just image processing and computer graphics in many different industries, not just defense and games. You may have to deal with some continuous math, but the closer you are to the VRAM the more likely you are dealing with just a bunch of integers. You might research Open GL and DirectX to get an idea of what's going on in 2D and 3D computer imaging.

• Yes, there is a lot of linear algebra in computer graphics. But it is very very simple linear algebra (mostly just fiddling with 4x4 matrices). Also, all this linear algebra has already been done by the people who wrote OpenGL and DirectX. People who use these packages don't need to know much linear algebra. Apr 25 '13 at 3:07
• @bubba: so wouldn't it be a push to work for Microsoft to work on DirectX? Apr 25 '13 at 16:52
• Maybe. But, again, the mathematical part of DirectX is pretty trivial, it seems to me. My guess is that most of the current DirectX development is aimed at making it go faster, which is a programming problem rather than a mathematical one. But, anyway, if the OP wants to find out what it's like to work on a DirectX implementation, I'd recommend that he go ask at StackOverflow. I doubt that anyone here knows much about this (including me). Apr 26 '13 at 1:42
• @bubba: I'm a professional software engineer working in graphics domain. I constantly use Direct3D (the part of DirectX which has graphics-related code) and OpenGL. Most code dealing with linear algebra is hand-written by programmers for their application. Also, there're lot of domains in CS where linear algebra is vital. Computer vision (OpenCV), parallel computing (OpenCL), graphics/game/simulation engines (OpenGL), speech recognition, bioinformatics, etc. to name a few. See this for a the complete list. May 21 '14 at 14:15
• Yes, it does seem that lots of people can't figure out 3D rotations. But, from a mathematician's point of view, this is a very simple form of linear algebra. Anyway, input from someone who actually works in industry is valuable, and is what the OP needs, I think. May 22 '14 at 5:01

NSA is not the only place for cryptographers. There are various research labs like HP , Microsoft Research labs where cryptographers and number theorists are hired.

Also, algebraic coding theory is another area where you can apply your skills of abstract algebra.

My profession as a DVB and Video/Audio Content Security engineer brought me to go over the Abstract Algebra. It allows me to go in depth of the every aspect of cryptographic primitive, as it prevails and pervasive in Cryptography.

• Now, in Digital video broadcasting , error correction in channel coding like Reed-Solomon, BCH codes and others are the classic example of Abstract algebra (group and its cosets, fields and rings). I see how simple looking cosets are used as a error correction in communication.


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## What is mathematics?

Math isn’t what you thought it was.

You might think that math goes something like this:

Calculate or simplify the following expressions:

1.  $535\times 45$
2. $12^3$
3. the derivative of $f(x)=3x^4 + 25x^3 + 5\mathrm{sin}(x)$
4. $(x+5)^3$

Well, that all sucks.  No one wants to do any of that.  From this, one might think that math is just remembering the right formula, or the right steps, and then having the patience and focus to follow these steps for several minutes without making a mistake.  It involves tediously writing numbers and letters all over the paper in exactly the way that your teacher told you would work.  We often have no real idea as to why we’re doing this, or any knowledge of the vast logical constructions that make it all possible—we’re just calculating.  This post will explain that there is another kind of math—one that focuses primarily on the abstract foundations and the deep logical reasoning that goes in to establishing the math that we’re used to doing every day.  To begin exploring this “other” type of math first-hand, head on over to the lessons!

Two types of math.

I am going to refer to the above kind of math as “calculation”.  I.e., when we’re answering questions 1-4 above, we’re not doing our “mathematics” homework, but rather our “calculation” homework.  I will reserve the term “real mathematics” for what I mean by this “other world” of math.

What is “real” mathematics?   There are volumes written about this question, and I will only spend a few sentences on it here.  As you spend time reading through the lessons and exposing yourself to what I call “real” math, you’ll form your own idea of what mathematics is.  We can, however, make a little progress right here and now.

“Real” mathematics (or just mathematics) is the art of creating, understanding, and exploring the relationships between various mathematical* structures.  Mathematics is the art of making mathematics, by exploring and  understanding mathematical structures.  (Numbers are one such structure, but there are infinitely many more.  We will explore mathematical structures more in depth later, and many examples of such structures can be found on this site.)

There is indeed a good amount of what I’m calling “real” mathematics hidden inside of a standard high school math curriculum, and those who like math tend to be drawn to these bits.  But these parts of the curriculum are often drowned out by the emphasis on memorization, rote learning, and repetition.  Questions 1-4 above lie on top of a very elegant logical framework, but we often completely ignore this in order to get on with our calculations.

What do mathematicians do?

A mathematician does not spend his day calculating $2+2$ or “the derivative of $f(x)=3x^4 + 25x^3 + 5\mathrm{sin}(x)$”.  He or she instead thinks about why and precisely when it is the case that $2+2=4$, or what a derivative really means and what it can be used for.  The more math you learn, the more you realize that $2+2$ does not always equal 4 (this will be explained when we describe “clock arithmetic”).  The remarkable logical foundation on which the true statement “$2+2=4$” stands was built by mathematicians who were doing “real” mathematics.  The actual process of calculating the result of $2+2$ (and other similar calculations) is important for students to learn, but it is not the kind of thinking that goes into much of mathematics.

Everyone, from high school students to professional mathematicians, hates calculating derivatives.  The difference is that professional mathematicians have been exposed to the logical and rigorous foundations that make these calculations possible.  They’ve studied what derivatives are, where they come from, and what they lead to, and these kinds of considerations are what keeps them in the business of studying math.  Additionally, they find a deeper and more subtle beauty in the calculations themselves.  This site aims to introduce its readers to this type of math—what I’m calling “real” (as opposed to calculational)—and to explore some of the gorgeous subtleties that come along with it.

*You might be worried that I used the term “mathematical” in the definition of mathematics, but I have reason to do this.  A mathematical structure is a thing, and I am using it to describe what mathematics is as an art form.  In other words, there would be no problem saying that “painting is the art of making paintings”, and that is essentially what I’ve done here.

So true.


text:
Question: Q1.(a) December 2015 of ECCF computer Engineering 3rd semester. B.E.
0

Q. Give reasons for the following:

1. FET is called as a square law device**
2. Barkhausens criteria should be satisfied to get oscillations.

Subject: Elec Circuits & Comm. Fundamentals (Computer Engineering - Sem 3 - MU)

 modified 2.1 years ago  • written 2.1 years ago by Krrish ♦ 190
1

1.We know that the expression for drain current of a JFET is as follows:

$\hspace {1cm} I_D = I_{DSS} \bigg[\frac{1-V_{GS}}{V_P}\bigg]^2$

This shows that $I_D$ ids proportional to square of VGS which is the controlling input parameters due to this square relationship between ID and VGS, the FET is called as a square law device.

2.An oscillator is basically an "amplifier" which does not have any ac input but it operates on the principle of positive feedback to generate an ac signal on its own at its output. Thus it is clear that an amplifier can work as an oscillator if positive feedback is made to exist. However positive feedback not always guarantees oscillations. An amplifier will work as an oscillator if and only if it satisfies a set of conditions called the "Barkhausen Criterion".


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## standard distributions

Joram Soch managed to get a short note arXived about the Normal cdf Φ by exhibiting an analytical version, nothing less!!! By which he means a power series representation of that cdf. This is an analytical [if known] function in the complex calculus sense but I wonder at the point of the (re)derivation. (I do realise that something’s wrong on the Internet is not breaking news!)

Somewhat tangentially, this reminds me of a paper I read recently where the Geometric Geo(p) distribution was represented as the sum of two independent variates, namely a Binomial B(p/(1+p)) variate and a Geometric 2G(p²) variate. A formula that can be iterated for arbitrarily long, meaning that a Geometric variate is an infinite sum of [powers of two] weighted Bernoulli variates. I like this representation very much (although it may well have been know for quite a while). However I fail to see how to take advantage of it for simulation purposes. Unless the number of terms in the sum can be determined first. And even then it would be less efficient than simulating a single Geometric…

### One Response to “standard distributions”

1. Georges Henry Says:

Y a t il vraiment des gens qui ont le culot d’ecrire un article qui dit essentiellement que
$$\frac{1}{1-z}=\frac{1+z}{1-z^2}?$$


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# Limit of expression containing unknown function.

1. Nov 26, 2013

### D_Tr

I would like to get an answer or pointers to suitable material, on the following question:

I know that ∫|f(x)|2dx is finite. Can we say that lim x→±∞ x*(d/dx)|f(x)|2 is zero? Are there any theorems about such limits with unknown functions that have some known properties? Basically this question arose from Griffith's introductory QM textbook which I started studying today (self study). f(x) is Ψ(x) and the author at page 16 takes the above limits to be zero "on the ground that Ψ goes to zero at ±∞". But it may go to zero asymptotically, and x goes to infinity, so we have an indeterminate form and we know only that the function's absolute value squared has a finite integral. I posted here, however, since I would like to get a more general answer on these types of limits and to be at least aware of the theory that deals with them.

2. Nov 27, 2013

### Simon Bridge

Given: $$I=\int_{-\infty}^\infty |f(x)|^2\;\text{d}x$$ ... if $I$ converges, then; $$\lim_{x\rightarrow\pm\infty} x\frac{d}{dx}|f(x)|^2 = 0$$

(We also know from context that f(x) is a solution to the Schrödinger equation.)

I suspect there are other implied conditions... but I think it pretty much follows from the definition of an integral and the conditions for convergence.

What was the author trying to show by that particular limit?

3. Nov 27, 2013

### D_Tr

Thanks a lot :) The author calculated the time derivative of <x>, just before introducing the notion of the operator. In one of the few steps he eliminated the following term which appeared as a result of integration by parts:
[x$\frac{d}{dx}$|Ψ|$^{2}$]$^{-∞}_{∞}$, "on the grounds that Ψ goes to zero at plus and minus infinity". It seems obvious now but I was a bit confused by the fact that the expression was multiplied by x and that the derivative was written in its expanded form containing he complex Ψ and Ψ* functions.

4. Nov 28, 2013

### Simon Bridge

No worries - it can take a while to learn to read these things ;)

5. Nov 29, 2013

### Office_Shredder

Staff Emeritus
This isn't actually true...for example f(x)2 can be the function which is 0 everywhere except for in the interval [n,n+1/n] where it bumps up to a value of 1/n and then back down to zero. Then the integral is equal to the sum
$$\sum_{n=1}^{\infty} 1/n^2$$
which converges, but when it bumps up the derivative could be arbitrarily large; so that limit could fail to be zero even before you include multiplying by x.

Probably the condition that f(x) is supposed to be satisfying here is something like |f(x)|2 is convex if f(x) is large enough (equivalently |f(x)|2 is decreasing for large enough values of x - if |f(x)|2 is decreasing down to zero and you don't have those bumps, then it has to die faster than 1/x to be integrable, so the derivative will look like 1/x2 or something smaller (this is a handwavy argument but can probably be made rigorous).

6. Nov 29, 2013

### Simon Bridge

I had thought of that ...
Does it matter that the derivative (per your example) in the interval that f is non-zero could be arbitrarily large?
The limit is taken arbitrarily far away from that interval - where the derivative is zero.


text:
1. ## division explanation

Hello,

I was wondering... Why is it divided by $n^2$? This is actually a calculus problem, trying to determine if the series is convergent or divergent.

But, I just needed some clarification on the algebra.

Thanks,

2. ## Re: division explanation

Hello, l flipboi l!

Why is it divided by $n^2$?

$\lim_{n\to\infty}\frac{n}{\sqrt{10+n}} \;=\;\lim_{n\to\infty}\dfrac{1}{\sqrt{\dfrac{10}{n ^2}+\dfrac{1}{n}}} \;=\;\infty$

We have the fraction: . $\frac{n}{\sqrt{10+n}}$

$\text{Divide top and bottom by }n\!:$

. . $\dfrac{\dfrac{n}{n}}{\dfrac{\sqrt{10+n}}{n}} \;=\;\dfrac{1}{\dfrac{\sqrt{10+n}}{\sqrt{n^2}}} \;=\;\frac{1}{\sqrt{\dfrac{10+n}{n^2}}} \;=\;\frac{1}{\sqrt{\dfrac{10}{n^2} + \dfrac{n}{n^2}}} \;=\;\frac{1}{\sqrt{\dfrac{10}{n^2} + \dfrac{1}{n}}}$

Got it?


text:
# Gromov's compactness theorem (geometry)

Gromov's compactness theorem (geometry) Not to be confused with Gromov's compactness theorem in symplectic geometry.

In the mathematical field of metric geometry, Mikhael Gromov proved a fundamental compactness theorem for sequences of metric spaces. In the special case of Riemannian manifolds, the key assumption of his compactness theorem is automatically satisfied under an assumption on Ricci curvature. These theorems have been widely used in the fields of geometric group theory and Riemannian geometry.

Metric compactness theorem The Gromov–Hausdorff distance defines a notion of distance between any two metric spaces, thereby setting up the concept of a sequence of metric spaces which converges to another metric space. This is known as Gromov–Hausdorff convergence. Gromov found a condition on a sequence of compact metric spaces which ensures that a subsequence converges to some metric space relative to the Gromov–Hausdorff distance:[1] Let (Xi, di) be a sequence of compact metric spaces with uniformly bounded diameter. Suppose that for every positive number ε there is a natural number N and, for every i, the set Xi can be covered by N metric balls of radius ε. Then the sequence (Xi, di) has a subsequence which converges relative to the Gromov–Hausdorff distance.

The role of this theorem in the theory of Gromov–Hausdorff convergence may be considered as analogous to the role of the Arzelà–Ascoli theorem in the theory of uniform convergence.[2] Gromov first formally introduced it in his 1981 resolution of the Milnor–Wolf conjecture in the field of geometric group theory, where he applied it to define the asymptotic cone of certain metric spaces.[3] These techniques were later extended by Gromov and others, using the theory of ultrafilters.[4] Riemannian compactness theorem Specializing to the setting of geodesically complete Riemannian manifolds with a fixed lower bound on the Ricci curvature, the crucial covering condition in Gromov's metric compactness theorem is automatically satisfied as a corollary of the Bishop–Gromov volume comparison theorem. As such, it follows that:[5] Consider a sequence of closed Riemannian manifolds with a uniform lower bound on the Ricci curvature and a uniform upper bound on the diameter. Then there is a subsequence which converges relative to the Gromov–Hausdorff distance.

The limit of a convergent subsequence may be a metric space without any smooth or Riemannian structure.[6] This special case of the metric compactness theorem is significant in the field of Riemannian geometry, as it isolates the purely metric consequences of lower Ricci curvature bounds.

References ^ Bridson & Haefliger 1999, Theorem 5.41; Burago, Burago & Ivanov 2001, Theorem 7.4.15; Gromov 1981, Section 6; Gromov 1999, Proposition 5.2; Petersen 2016, Proposition 11.1.10. ^ Villani 2009, p. 754. ^ Gromov 1981, Section 7; Gromov 1999, Paragraph 5.7. ^ Bridson & Haefliger 1999, Definition 5.50; Gromov 1993, Section 2. ^ Gromov 1999, Theorem 5.3; Petersen 2016, Corollary 11.1.13. ^ Gromov 1999, Paragraph 5.5.

Sources.

Bridson, Martin R.; Haefliger, André (1999). Metric spaces of non-positive curvature. Grundlehren der mathematischen Wissenschaften. Vol. 319. Berlin: Springer-Verlag. doi:10.1007/978-3-662-12494-9. ISBN 3-540-64324-9. MR 1744486. Zbl 0988.53001. Burago, Dmitri; Burago, Yuri; Ivanov, Sergei (2001). A course in metric geometry. Graduate Studies in Mathematics. Vol. 33. Providence, RI: American Mathematical Society. doi:10.1090/gsm/033. ISBN 0-8218-2129-6. MR 1835418. Zbl 0981.51016. (Erratum:  [1]) Gromov, Mikhael (1981). "Groups of polynomial growth and expanding maps". Publications Mathématiques de l'Institut des Hautes Études Scientifiques. 53: 53–73. doi:10.1007/BF02698687. MR 0623534. Zbl 0474.20018. Gromov, M. (1993). "Asymptotic invariants of infinite groups". In Niblo, Graham A.; Roller, Martin A. (eds.). Geometric group theory. Vol. 2. Symposium held at Sussex University (Sussex, July 1991). London Mathematical Society Lecture Note Series. Cambridge: Cambridge University Press. pp. 1–295. ISBN 0-521-44680-5. MR 1253544. Zbl 0841.20039. Gromov, Misha (1999). Metric structures for Riemannian and non-Riemannian spaces. Progress in Mathematics. Vol. 152. Translated by Bates, Sean Michael. With appendices by M. Katz, P. Pansu, and S. Semmes. (Based on the 1981 French original ed.). Boston, MA: Birkhäuser Boston, Inc. doi:10.1007/978-0-8176-4583-0. ISBN 0-8176-3898-9. MR 1699320. Zbl 0953.53002. Petersen, Peter (2016). Riemannian geometry. Graduate Texts in Mathematics. Vol. 171 (Third edition of 1998 original ed.). Springer, Cham. doi:10.1007/978-3-319-26654-1. ISBN 978-3-319-26652-7. MR 3469435. Zbl 1417.53001. Villani, Cédric (2009). Optimal transport. Old and new. Grundlehren der mathematischen Wissenschaften. Vol. 338. Berlin: Springer-Verlag. doi:10.1007/978-3-540-71050-9. ISBN 978-3-540-71049-3. MR 2459454. Zbl 1156.53003. hide vte Manifolds (Glossary) Basic concepts Topological manifold AtlasDifferentiable/Smooth manifold Differential structureSmooth atlasSubmanifoldRiemannian manifoldSmooth mapSubmersionPushforwardTangent spaceDifferential formVector field Main results (list) Atiyah–Singer indexDarboux'sDe Rham'sFrobeniusGeneralized StokesHopf–RinowNoether'sSard'sWhitney embedding Maps CurveDiffeomorphism LocalGeodesicExponential map in Lie theoryFoliationImmersionIntegral curveLie derivativeSectionSubmersion Types of manifolds Closed(Almost) Complex(Almost) ContactFiberedFinslerFlatG-structureHadamardHermitianHyperbolicKählerKenmotsuLie group Lie algebraManifold with boundaryOrientedParallelizablePoissonPrimeQuaternionicHypercomplex(Pseudo−, Sub−) RiemannianRizza(Almost) SymplecticTame Tensors Vectors DistributionLie bracketPushforwardTangent space bundleTorsionVector fieldVector flow Covectors Closed/ExactCovariant derivativeCotangent space bundleDe Rham cohomologyDifferential form Vector-valuedExterior derivativeInterior productPullbackRicci curvature flowRiemann curvature tensorTensor field densityVolume formWedge product Bundles AdjointAffineAssociatedCotangentDualFiber(Co) FibrationJetLie algebra(Stable) NormalPrincipalSpinorSubbundleTangentTensorVector Connections AffineCartanEhresmannFormGeneralizedKoszulLevi-CivitaPrincipalVectorParallel transport Related Classification of manifoldsGauge theoryHistoryMorse theoryMoving frameSingularity theory Generalizations Banach manifoldDiffeologyDiffietyFréchet manifoldK-theoryOrbifoldSecondary calculus over commutative algebrasSheafStratifoldSupermanifoldTopologically stratified space Categories: Differential geometryTheorems in Riemannian geometry

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text:
# Modular arithmetic for arbitrary number

The problem describes as:

When the even integer $n$ is divided by $7$，the remainder is $3$. What's is The remainder when $n$ is divided by $14$.

My simple solution is:

$n=7x+3$ where $x$ is odd, so, we can define $x = 2m+1$, then $n = 7(2m+1) + 3 = 14m + 7 +3= 14m + 10$. So reminder is: $10$

As I am learning the mod system and was trying to solve this problem with modulo arithmetic. But got stuck with "how to think this prob in modulo system" ie. if $$n \equiv 3 \pmod{7}$$ then $$n \equiv \ ? \pmod {14}$$

Any help regarding solving steps and learning reference would be appreciated.

• "how to think this prob in modulo system" well it depends of the meaning of those words, but IMHO I think you did it nicely that way indeed. You converted a problem in $\pmod{7}$ to a problem in $\pmod{14}$. Maybe is just a question of formatting your answer as a "modulo system problem": $x = 2m+1$, then $n = 7(2m+1) + 3 \pmod{14} \equiv 14m + 7 +3 \pmod{14} \equiv 14m + 10 \pmod {14} \equiv 14m \pmod{14} + 10 \pmod {14} \equiv 10 \pmod{14}$. – iadvd Apr 26 '17 at 5:22
• Edited the question a bit for better understandability of my points that is, I was thinking the format: if n mod 7 = 3 then n mod 14 = ?. That's how I was thinking. Thanks for replying! – neo-nant Apr 26 '17 at 5:46
• ok I have added a solution going backwards... I think that you meant that possibility. But the way you did it is easier and indeed my solution is based on your substitutions (but backwards, it is kind of tricky). – iadvd Apr 26 '17 at 7:03

As I said in the comments I think is fine the way you did it. I would suggest you to use "congruent to" instead of "equal to" as in my comment from the beginning, or add it at the end of the conversions you did as a last step.

Said that, maybe your question is more related with this point: how you could make a solution starting backwards? from $14m+r$.

$$n=14m+r$$

Then, let us suppose three possibilities:

1. $r \lt 7$

2. $r = 7$

3. $7 \lt r \lt 14$ so we can define $r=7+r'$, where $r' \lt 7$

• For the first case:

$$n=14m+r \pmod{7} \equiv 2\cdot 7m\pmod{7} + r \pmod{7} \equiv 0+r \equiv 3$$

Thus:

$$r=3$$

But that is not possible because it means that $n = 14m + 3$, but $14m+3$ is not even, and we know that $n$ is even, so that solution is not possible.

• For the second case:

$$n=14m+7 \pmod{7} \equiv 2\cdot 7\pmod{7} + 7 \pmod{7} \equiv 0 \equiv 3$$

So it is impossible for the residue to be $r=7$ because $14m+7 \not \equiv 3 \pmod {7}$, the residue does not comply the premise, being $\equiv {3} \pmod{7}$, it is indeed $\equiv {0} \pmod {7}$.

• For the third case:

$$n=14m+r \pmod{7} \equiv 2\cdot 7\pmod{7} + 7 + r' \pmod{7} \equiv 0+0+r' \equiv 3$$

Thus finally the unique remaining valid option is $7 \lt r = 10 \lt 14$:

$$r'=3, r=7+r'=7+3=10$$


text:
Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, October 14, 2014.

.
events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
    September 2014          October 2014          November 2014
Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
1  2  3  4  5  6             1  2  3  4                      1
7  8  9 10 11 12 13    5  6  7  8  9 10 11    2  3  4  5  6  7  8
14 15 16 17 18 19 20   12 13 14 15 16 17 18    9 10 11 12 13 14 15
21 22 23 24 25 26 27   19 20 21 22 23 24 25   16 17 18 19 20 21 22
28 29 30               26 27 28 29 30 31      23 24 25 26 27 28 29
30

Tuesday, October 14, 2014

11:00 am in 243 Altgeld Hall,Tuesday, October 14, 2014

#### Recognizing small model structures

###### Inna Zakharevich   [email] (U Chicago)

Abstract: Model categories have been widely used over the past 40 years, but they rarely arise "in the wild"; instead, one is usually given a category C with a subcategory wC of weak equivalences, and would like to construct a model structure around those. Unfortunately, there is no known algorithm for doing so, and no general recognition principle that says that such a model structure exists. In this talk we will investigate this question in the case where C is small (and thus a preorder). In the end we will give simple necessary and sufficient conditions for a model structure on C with weak equivalences W to exist.

1:00 pm in 345 Altgeld Hall,Tuesday, October 14, 2014

#### The Generic Point Problem and closed subgroups of $S_\infty$

###### Andrew Zucker (Carnegie Mellon University)

Abstract: A topological group $G$ is said to have the Generic Point Property if the universal minimal flow $M(G)$ has a generic point, a point whose orbit is comeager. This in turn implies that any minimal $G$-flow has a generic point. Angel, Kechris, and Lyons asked the following question, known as the Generic Point Problem: If $G$ is a Polish group and the universal minimal flow $M(G)$ is metrizable, does $G$ have the Generic Point Property? In this talk, we will discuss the case where $G$ is a closed subgroup of $S_\infty$; here the answer is affirmative. To show this, we work with an explicit characterization of the greatest $G$-ambit and introduce some new tools in structural Ramsey theory.

1:00 pm in 243 Altgeld Hall,Tuesday, October 14, 2014

#### Polygons, polynomials, fences, and flows

###### David Dumas (UIC)

Abstract: In a recent paper with Michael Wolf, we use affine differential geometry to construct a homeomorphism between the moduli space of polynomial cubic differentials on the complex plane and the space of projective equivalence classes of convex polygons in RP^2. In this talk I will briefly recall the results of this project, and then focus primarily on discussing various connections, questions, and conjectures suggested by this work. These include an interpretation of our main theorem in terms of a family of rank-3 meromorphic Higgs bundles, a conjectural relation with the Stokes phenomenon, and a question about the Poisson geometry of the space of twisted polygons. View talk at http://youtu.be/7ywL7IzHyQw

2:00 pm in Altgeld Hall 347,Tuesday, October 14, 2014

#### Sequential Change Detection for Fractional SDEs

###### Alexandra Chronopoulou (UIUC IESE)

Abstract: We will consider the problem of sequentially detecting a change in a stochastic process that satisfies a fractional stochastic differential equation with an arbitrary Hurst index, H. For this class of dynamics, we will establish sufficient conditions for the Cumulative Sums (CUSUM) test to be an exact (non-asymptotic) solution to Lorden's minimax optimal stopping problem. In this way, we will extend well-known optimality properties of CUSUM for diffusion processes. The main techniques for these extensions come from fractional calculus and Malliavin calculus.

3:00 pm in 241 Altgeld Hall,Tuesday, October 14, 2014

#### Hajnal-Szemerédi type theorems

###### Theodore Molla   [email] (UIUC Math)

Abstract: An equitable $k$-coloring of a graph $G$ is a proper $k$-coloring in which every pair of color classes differ in size by at most one. In 1970 Hajnal and Szemerédi proved that every graph with maximum degree less than $k$ has an equitable $k$-coloring. This theorem is easily seen to be best possible: complete graphs, odd cycles and balanced complete bipartite graphs with odd sized parts are all not equitably $k$-colorable when $k$ is the maximum degree. In 1994 Chen, Lih and Wu conjectured that these are the only connected graphs that are not equitable $k$-coloring and have maximum degree at most $k$. This conjecture is still open, but a few special cases have been proved. In this talk, we will discuss an Ore-type result which implies the Chen-Lih-Wu conjecture for $k$-equitable colorings in graphs with $3k$ vertices as well as other extension and generalizations of the Hajnal-Szemerédi Theorem. This work is joint with H.A. Kierstead, Alexandr Kostochka, and Elyse Yeager.

3:00 pm in 243 Altgeld Hall,Tuesday, October 14, 2014

#### The refined BPS - virtual Poincare polynomial conjecture

###### Sheldon Katz (UIUC Math)

Abstract: My collaborators and I have conjectured that the refined BPS invariants of a Calabi-Yau threefold are equivalent in a precise way to the virtual Poincare polynomials of the moduli space of PT theory. In this talk, I start with background on BPS invariants of Calabi-Yau threefolds and their refinement, along with background on PT theory and its motivic extension. I then formulate the conjecture and provide evidence.

4:00 pm in 314 Altgeld Hall,Tuesday, October 14, 2014

#### Symplectic resolutions and Lie algebras

###### Andrei Okounkov (Columbia Univ)

Abstract: Note the unusual day. Please attend the Trjitzinsky Lecture by Andrei Okounkov (Columbia Univ) at 4 p.m. in 314 Altgeld Hall.
In geometric representation theory, one often links the properties of noncommutative algebras to the geometry of their commutative degenerations, and also often aims to make an algebra act on cohomology or K-theory of another algebraic variety. These points of view are in a certain sense dual, and my goal in these lectures will be to discuss several recent ideas and advances in this direction.

4:00 pm in 314 Altgeld Hall,Tuesday, October 14, 2014

#### Symplectic resolutions and Lie algebras

###### Andrei Okounkov (Columbia University)

Abstract: In geometric representation theory, one often links the properties of noncommutative algebras to the geometry of their commutative degenerations, and also often aims to make an algebra act on cohomology or K-theory of another algebraic variety. These points of view are in a certain sense dual, and my goal in these lectures will be to discuss several recent ideas and advances in this direction.

Andrei Okounkov of Columbia University will deliver the Trjitzinsky Lectures on October 14, 15, 16, 2014. A reception will be held from 5-6 p.m. in 239 Altgeld Hall following the Oct 14 lecture.