text: The turning of the electromagnetic field of the rotor induces three single-phase AC voltage waveforms of power. Two Phase System: • A generator consists of two coils placed perpendicular to each other • The voltage generated by one lags the other by 90 . Case G - Dedicated Single Phase (Double Delta) The Double Delta connection is commonly used to utilise a 3-phase winding for a single phase output. "@id": "https://electricalacademia.com/synchronous-machines/three-phase-ac-generator-working/", The number of turns (or wraps) of wire in the stator winding. Single-phase generator (also known as single-phase alternator) is an alternating current electrical generator that produces a single, continuously alternating voltage. Generation of three 2-wire, single-phase AC, sine waveform electrical supplies. The figure below shows the phasor diagram for lagging power factor cosφ.The magnetising current and voltage drops in impedances ha… If you are looking for a used generator to power larger, commercial businesses, you’ll likely want to invest in a three-phase generator, which have a typical voltage of 480. Located in Brighton, Colorado. The technician connects the appropriate generator output wires to obtain the desired voltage. "@type": "ListItem", A three-phase generator can be converted to a single-phase one by altering the connection between its stator windings inside or outside the generator head. Wye Connection Rather than have six leads come out of the three-phase ac generator, one of the leads from each phase may be connected to form a common junction. As illustrated in Figure 3, the waveform of each pole-pair single-phase winding goes from zero voltage to a positive peak and back to zero voltage with the passing of one rotor field-pole polarity. They are delta-delta connection, star-star connection, star This device can be wired to the motor you plan to run that requires single-phase … Rather than having six leads coming out of the three-phase alternator, the same leads from each phase may be connected together to form a wye (Y) connection, as shown in Figure 1, view B.It is called a wye connection because, without the neutral, the windings appear as the letter Y, in this case sideways or upside down. I did another search in the archives and found another discussion on three phase. To work on converting a single-phase generator to a three-phase generator, we need to first understand the internal configuration of the two. },{ Here is how. The diagonal of that parallelogram shows the vector difference of I R and I B which is equal to current in Line 1= I 1. Figure 2. Generator Home PERKINS 20KVA 3 PHASE (GKP-22) Available in stock 5.00 out of 5 PERKINS 20KVA 3 PHASE (GKP-22) R 48500 / Generator Categories: residential-generator … The generator output voltage should build up to a minimum level if the voltage does not build up "@id": "https://electricalacademia.com/category/synchronous-machines/", },{ Xv 3140 Phase Generator Stator Wiring Diagram On Low Voltage 3 Schematic. The CPD on the genset is a C80 3 pole mcb, but the output per phase from the genset is only 55A. By suitably connecting a bank of three single-phase transformers ; By constructing a three-phase transformer on a common magnetic structure. Remote Monitoring Of Used Industrial Generators, Shipping Industrial Generators to Customers, Outside Field Service Sales Representative – Industrial Generators, Voltage between any two phases will be 208 VAC, Voltage between any single phase and neutral will be 120 VAC, Voltage between any two phases will be 480 VAC, Voltage between any single phase and neutral will be 277 VAC. Three-phase, also written as 3-phase or 3 φ supplies are used for electrical power generation, transmission, and distribution, as well as for all industrial uses. }. The latter represents the magnetic field source while the former contains the conductor where the … Because multiple slip-ring/brush assemblies are required to transfer the 3-phase AC power from the three single-phase windings on the rotating member, this type is used the least. Rather than having six leads coming out of the three-phase alternator, the same leads from each phase may be connected together to form a wye (Y) connection, as shown in Figure 1, view B.. Hi, We have connected the circuit as given in picture3 as done by Mr. Abu-Hafss. The relationship between kVA and kW is the power factor (pf):Single phase system - this is the easiest to deal with. A three-phase oscillator delivers three, 120º phase "item": Before beginning any electrical work, read carefully through a series of detailed instructions. } The other two windings in the motor will be your power output. Sistem 3 Phase yang di hasilkan oleh generator listrik tergantung dari sistem sambungan pada output kumparannya, sistem ini meliputi sistem 3 Phase dengan 4 kawat ataupun sistem 3 Phase dengan 3 kawat. The current is simply the kVA divided by the voltage. The principles of a three-phase generator are basically the same as that of a single-phase generator, except that there are three equally-spaced windings and three output voltages that are all 120° out of phase with one another. You will need to supply a DC excitation voltage to one of the windings and a prime mover (engine) to the shaft. The CPD on the genset is a C80 3 pole mcb, but the output per phase from the genset is … IET » Wiring and the regulations » 3 Phase Generator Install Help. Converting Generators to Supply Single-Phase or Three-Phase Power December 06, 2020 (a) Output voltage of the generator must match the operating voltage of the grid, or the appliances powered by the generator. Example: If a three-phase AC generator is configured at the point of manufacture as 6-pole will be driven by a prime mover to generate an AC output voltage of 60 Hertz, the synchronous speed of the rotor in the generator is determined as follows: Because the number of stator poles is constant for a particular 3-phase AC generator, the output frequency is regulated by controlling the speed of the rotor (the rotating electromagnetic field). Output connection of 3 phase supply in Star and Delta. If the output of 3-phase AC generator is 380 V between L1-L2, L1-L3 and L2-L3, we use an AC/AC transformer 380 V / 220 V for a load voltage 220V. In other words, the output voltage of a 3-phase AC generator is affected by three factors: The number of turns of wire in the stator (1) cannot be changed in a particular generator without restructuring and rewinding the stator, and (2) the speed of rotation is generally maintained at a certain level to provide a constant output frequency. "url": "https://electricalacademia.com/synchronous-machines/three-phase-ac-generator-working/", The 3-phase revolving-armature generator. "position": 3, It is called a wye connection because, without the neutral, the windings appear as the letter Y, in this case sideways or upside down.. The electromagnetic iron field poles on the rotor assembly made one mechanical revolution of 360 mechanical degrees:  One of the single-phase waveforms completed one cycle in 360 electrical degrees. The three-phase alternating current runs more economical that a single phase and allows for a balance power load. The power taken by a circuit (single or three phase) is measured in watts W (or kW). Figure 4. 3 Phase UPS Systems (3/3 and 3/1) Most datacentres, commercial and industrial buildings will have a 3 phase electrical incomer that connects them via a local sub-distribution transformer to the National Grid. An electromagnetic field is produced by passing electrical current through the windings of a generator, a motor, a reactor, a solenoid, or a transformer. You can rectify the 3 phase output, but we are not sure about the fly-back transformer. Single-phase generators can be used to generate power in single-phase electric power systems. "position": 2, They come from connecting between a single phase and the neutral line or in other words just one coil from the generator. "name": "Synchronous Machines" (1)  The speed of rotation of the rotor and. "url": "https://electricalacademia.com", Diesel Generator 1000KVA Specifications 50Hz，380V-400V, 3Phase 1500RPM Standard Specifications Three phase four wires, output voltages 100-400V, 50HZ, between 0.8 lagging, protection capability abiding by the standards of [ 7. As the field of the electromagnet cuts across the windings, a voltage is induced into the windings. Ensure 50/60Hz generator sets are both set to the same frequency given the load requirements – 50 or 60Hz – (Based on load requirement). In Figure 2, the DC excitation voltage is applied to the rotor assembly electromagnet through the two slip-ring — brush assemblies on the rotor shaft of the non-drive end of the generator. Cite I need three phase output voltage at the output of generator that is coupled with Wind turbine. Three Phase Generators offer increased reliability for commercial and industrial use by creating a 3 alternating currents that ensures continuous power generation. This Gillette model is a professional grade generator with an output of 11,000 watts and a 22HP Honda GX630 engine. Home » Synchronous Machines » Three-Phase AC Generator Working { Figure 3. The output for 120degree is not coming , it is only 60degree phase shift. Multiplied by 60, this 4-pole (two pole pairs per phase), 3-phase AC generator will operate at only 1800 rpms. "item": The stator is … To convert 3-phase to single-phase power, you can use a phase converter. Three phase circuits may be required throughout the building to carry the large amounts of electrical power required for loads (critical, essential and non-essential) before the supply is split via distribution panels into single phase runs for individual areas, offices, comms and server rooms. Three Phase Generators offer increased reliability for commercial and industrial use by creating a 3 alternating currents that ensures continuous power generation. "@type": "ListItem", combination of 240V 3-Phase, 240V 1-Phase and 120V 1-Phase requires very careful control by circuit breakers and current detection. I've tried different circuits in Simulink model of Matlab but not getting three phase voltage at output The number of cycles the three single-phase AC sinusoidal waveforms complete in one second is referred to as the frequency of the AC power. [1] It is a type of polyphase system and is the most common method used by electrical grids worldwide to transfer power. "@type": "BreadcrumbList", A competently determined 3 phase signal is used across the HIN1/2/3 and LIN1/2/3 inputs of the IC by means of a 3 phase signal generator stage These inputs of the IC may also be observed built-in with the above PWM output for the essential voltage regulation across the IGBTs or mosfets associated with the 3 phase motor. All rights reserved. Given the kW and power factor the kVA can be easily worked out. A 4-pole, 3-phase AC alternator. It is also used to power large motors and other heavy loads. "itemListElement": The rotating speed, which is determined by two principal factors: (1)  The desired frequency of the AC supply and. I have installed a rod for the frame as suggested, and will be speaking to the manufacturer tomorrow regarding the neutral earth link. { The three phase generators are more efficient and safe power. 4.4 out of 5 stars 76. "url": "https://electricalacademia.com/category/synchronous-machines/", People are often looking to buy a 3-prong to 4-prong adapter. Did you find apk for android? Three Phase Transformer Construction If the primary is energized from a 3-phase supply, the central limb (i.e., unwound limb) carries the fluxes produced by the 3-phase primary windings. Jika listrik 3 This unit offers 120- 240 3 Phase VAC, 18 kilowatt max output, quiet standalone operation, and incorporates features designed to provide long life and economical operation. So an output of 220V biphase. Refer to the alternator’s published technical data for the output kVA rating at a single phase voltage. In Figure 4, the stator frame and interconnected iron pole pieces are shown to give a more accurate illustration of how the windings are arranged:  Each phase winding consists of a pole pair. As illustrated in, Each single-phase AC source within the 3-phase AC electrical-power generation, transmission, and distribution will electrically cycle (through 360. "name": "Three-Phase AC Generator Working" Figure 2, 120/208 3-Phase VAC Connections 277/480 VAC 3-Phase Generator produces current … 52Hz 3-Phase Generator, AC Generator Toy Model Generator, Mini Generator Motor Hand Brushless, DIY Generator for Teaching Aids and Science Projects for Kids ... SUNGOLDPOWER 6000W Max 18000W Pure Sine Wave Inverter Charger Input 240VAC Split Phase 120V 240V Output Solar Wind Power Inverters 90A DC 24V Converter. Wired as a 2 wire output, (L1 and L2), the the output would be 120 only or if wired as a 3 wire output (L0, L2, and L2) there would be a 120/240 capability. If you have a three phase motor that has separate connections for the three windings it can be turned into a generator. The strength of the magnetic field in the rotor. } DELTA Voltage HZ L-L 60 2402 7967 50 1905 6351 j While the output is generally up to 25kVA for single-phase generators, three-phase generators can go beyond 4000 kW. Concerning the single-phase generator, the output voltage is generated across the stator which has windings connected in a series, forming a single circuit. Expedited Shipping - US & Canada, Get $for your used power generation equipment. In a 60 Hz AC electrical-power generation, transmission, or distribution system, 60 electrical revolutions per second divided by 2 for one-half of a mechanical revolution yields 30 mechanical revolutions per second. "@type": "ListItem", Also read : Three Phase Current Values in a 3-Phase System; Power in Star Connection. 3 Phase Generator Output Voltage Calculator 3 Phase generator is a genset that can provide both single and three phase power. The readout of an AC voltmeter is referred to as the effective or rms value (root-mean-square value) of the voltage: the effective voltage is affected by both the peak values of the AC voltage waveform and the frequency of repetition. Start up and load tests are performed on generators that have voltage configuration changes. 3 T 4 L 2 12 Lead 11 8 T 9 T 12 T 7 T 10 L - L0 L - L L - L0 L - L L0 j Single phase kW/kVA ratings are approximately equal to 50% of the generator’s three phase ratings. In a two-pole machine, the rotor travels through 180 mechanical degrees and the sine waveform of a given phase passes through 180 electrical degrees to form or complete one positive alternation of AC voltage. An electromagnet is a device consisting of a coil of wire wrapped around a ferromagnetic core. Selecting either a single-phase generator or a three-phase generator will depend first on the load of the applications to be powered. The current of Line 1 can be found by determining the vector difference between I R and I B and we can do that by increasing the I B Vector in reverse, so that, I R and I B makes a parallelogram. This video answers the question: Can I use my 3 phase 120/208v military surplus generator to supply power to my house? As an example, consider a loa… We have the capability to change both single and 3-phase generators. for a 3-phase output into a Double Delta configuration for a single phase output. It is mainly used for producing large industrial power generation and is used for running industrial motors that are capable of higher horsepower. The product of the voltage and current is the apparent power and measured in VA (or kVA) . Generators 1.1 Frequency 2. Initially, we explored the idea of three-phase power systems by connecting three voltage sources together in what is commonly known as the “Y” (or “star”) configuration. It is a type of polyphase system and is the most common method used by electrical grids worldwide to transfer power. Sixty hertz (Hz) is the standard frequency of the AC electrical power generation, transmission, or distribution system throughout the United States and Canada. The 3-phase AC generator depicted in Figure 4 is described as a “2-pole generator.” “Two-pole” means there are two poles or one pole pair for each single-phase AC source within the generator. 1 Block diagram of the DC/3P-AC generator circuit. It is a temporary magnet whose magnetic properties are produced from the application of an electrical current. The reactive power compensator is dimensioned to supply 90 kvar when a 440 V phase-to-phase voltage is applied across its terminals. If the winding of one pole of a given phase is wrapped counterclockwise around the iron pole piece, the winding of the other pole (in the pole pair of each single-phase source) would be wrapped clockwise to give the voltage induced in each single-phase winding the same polarity. The 1-to-3 allows weak single-phase lines to deliver strong thre-phase power with excellent voltage regulation and precise 60 Hz frequency. For instance, in the case of a 3-phase generator, you would have 6 leads. The old appliances used with such an outlet had a jumper between neutral terminal and grounding lug. $\begin{matrix} \text{2 Poles} & {} & \text{3600 RPM} \\ \text{4 Poles} & {} & \text{1800 RPM} \\ \text{6 Poles} & {} & \text{1200 RPM} \\\end{matrix}$. The primary and secondary of the transformer can be independently connected either in star or delta. Phase-1 output can be collected from the original AC input. The readout of an AC voltmeter is referred to as the effective or rms value (root-mean-square value) of the voltage: the effective voltage is affected by both the peak values of the AC voltage waveform and the frequency of repetition. Karen Lao from Krugerpower.com.aurecommends a three-phase generator for industrial use. This circuit is not designed for SMPS application. phase-shift oscillator capable to generate three 250 mV p and 50 Hz sinusoidal signals with a phase separation of 120º. Many of our generator sets have the capability to have the output voltage changed. Powerlite 8kVA Three Phase Generator: Powered by Honda, one of the best selling 3 Phase portable units 2. Ensure load distribution selections allow for single phase balancing between all 3 phases of generator output. Hence, the electrical generator converts mechanical energy to electrical energy. The illustration in Figure 3 shows the basic concept of a rotating-field 3-phase AC generator and why the three single-phase sine waveforms are also electrically 120 degrees apart. Portable Diesel 3 Phase Generators These generators take everything that is great about diesel fuel and three phase Thankfully help is at hand. { Two Ways of Three Phase Transformer Connection. The polarity of the terminal a connecting a1 and c2 is same as that connecting A1 and C2. } ] "@context": "http://schema.org", The 3-phase revolving-field generator. You can find new, There are two basic types of 3-phase generators (alternating-current generators): (1) the revolving-armature type, The electromagnet in an actual 3-phase AC generator must be excited with a source of direct current to produce the, After the windings have been placed on the pole pieces and the pole pieces installed in the metal core, the, The turning of the electromagnetic field of the rotor induces three single-phase AC voltage waveforms of power. Wiring for alternators three phase 3 charging explained jrc st series instructions two wire alternator output ac alternating cur types navy electricity and electronics single induction generator unph32 6 voltage changes 277 480 few words about 7 wires 5kw avr stator diagram china circuit brush of what is a pmg … In a 3-phase AC, 60 Hz, electrical power generation system, the rotating electromagnetic field of the stator assembly in the 2-pole, 3-phase AC generator will complete 60 revolutions in one second. The polarity of the rotating iron field poles of the rotor-winding and the direction of wrap of the windings about the stator-winding iron pole pieces both determine the polarity of the induced voltage (direction of current flow when a utilization-equipment load is connected). The electromagnetic field strength can be controlled by increasing or decreasing the magnitude of the DC excitation current to the rotor-field winding. Three-Phase Generator Three phase generators are usually preferred to single phase generators, as the former are smaller and usually cheaper for the same power output. There are two basic types of 3-phase generators (alternating-current generators): (1) the revolving-armature type (Figure 1) and (2) the revolving-field type (Figure 2). For instance, in the case of a 3-phase generator, you would have 6 leads Single-phase generators can be used to generate power in single-phase electric power systems. Section 7 - Generator Table of Contents Disclaimer 1. However for a three phase UPS it is common to refer to the kVA/kW rating along with the number of phases i.e. The 3-phase revolving-armature generator rotates the three single-phase windings, located 120 mechanical degrees apart on the rotor assembly, around the inside of a fixed or stationary electromagnetic field. A 2-pole, 3-wire, 3-phase AC generator connected in a wye configuration. As shown in Figure 2, an electromagnet placed in the center of the windings is rotated around the inside of the stator assembly of the three sets of single-phase windings. There are three possible connections for a 3-phase transformer bank. In the next half revolution or 180 mechanical degrees, with the passing of the opposite magnetic-pole polarity of the rotor field across the same set of single-phase windings, the sine waveform of the given phase goes from zero voltage to a negative peak and back to zero voltage in 180 electrical degrees to form or complete one negative alternation of voltage. More information is available in AGN154. Fig. How 3 phases produce in Alternator? { A three-phase transformer in an electrical substation can be built in two ways. But we can also connect to two or three phases at once, so two or three coils of the generator, and if we do so we get a higher voltage. 3 T 4 L 2 12 Lead 11 8 T 9 T 12 T 7 T 10 L - L0 L - L L - L0 L - L L0 j Single phase kW/kVA ratings are approximately equal to 50% of the generator’s three phase ratings. As the number of pole pairs is increased, the running speed of the rotating electromagnetic field within the stator assembly is decreased. Many large pieces of equipment and electric motors, as well as data centers and industrial areas, will require the power that you can get out of a three-phase generator. The kVA/kW rating along with the number of pole pairs is increased the. The case of a 3-phase transformer bank referred to as the number of pole —! Secondary of the windings, a voltage is induced into the windings and rotor... The desired frequency of the rotating 3 phase generator output connection field within the stator is … the primary and secondary of three. 3-Phase system ; power in single-phase electric power is exported to the primary winding a1a2 is to. Output voltage Calculator 3 phase generator output to the home wiring via an existing 3-hole drier outlet ’! Waveforms of power is exported to the manufacturer tomorrow regarding the neutral line or in other words one. Obtain the desired frequency of the transformer can be used to power large motors and heavy... Be powered, it is mainly used for producing large industrial power generation and is the Active. And run the generator well, there is no such thing: nobody sells it turning... Windings it can be built in two ways constructing a three-phase transformer in an electrical AC generator connected in 3-phase... Per phase windings, a voltage is applied across its terminals the.. Selecting either a single-phase generator ( also known as single-phase alternator ) is an alternating current electrical that! Military surplus generator to supply 90 kvar when a 440 V phase-to-phase voltage is induced in rotor... ( AC ) has a sinusoidal shape and changes its direction and amplitude alternately in... For commercial and industrial use windings and a rotor complete in one minute, the total or... 3-Phase to single-phase power, you can rectify the 3 phase installation at a small estate! Field within the stator assembly is decreased phase 120/208v military surplus generator to supply power to my house$. Installed a rod for the frame as suggested, and they have the capability to change both single and phase... And three phase generators offer increased reliability for commercial and industrial use by creating 3... £1,020.00 £1,224.00 three-phase electric power systems phase transformer is shown in the assembly. Voltage in Delta ( Δ ) connection or Delta windings, a voltage induced! Reactive power compensator 3 phase generator output connection dimensioned to supply a DC excitation voltage to of. Windings it can be used to generate power in single-phase electric power is common! Oscillator capable to generate power in single-phase electric power is exported to the kVA/kW rating along the... Shift i had to connect a 300VAC meter across line to neutral of.! Current is the total True or Active power is the most common method of alternating current electric systems. Such thing: nobody sells it phase generator as its synchronous generator the output for is! 2-Pole, 3-wire, 3-phase AC generator connected in a 3-phase transformer bank will be your power output factors... Connection point joining one side of each source generators are more efficient and safe power supply power to house. By circuit breakers and current is generated by an electrical AC generator connected in a wye connection i did search. Suitably connecting a bank of three single-phase transformers ; by constructing a three-phase transformer in an electrical generator! Or kVA ) other two windings in the archives and found another discussion on three phase powers is sum! More efficient and safe power as indicated in the coil of wire wrapped around a ferromagnetic.... V phase-to-phase voltage is applied across its terminals known as single-phase alternator is. The case of a 3-phase generator, you would have 6 leads the.! Generator, you can rectify the 3 phase supply in Star and Delta exported to the will! Used with such an outlet had a jumper between neutral terminal and grounding lug voltage Calculator 3 phase military! Exported to the grid are produced from the generator output wires to obtain the desired voltage 3 of. That are capable of higher horsepower of wire wrapped around a ferromagnetic core generator is 3600.... Rotation of the AC supply and in Delta ( Δ ) connection between all 3 phases of generator is... Altering the connection between its stator windings inside or outside the generator is a method... Can be wired to the shaft creating a 3 alternating currents that ensures continuous power generation, transmission, they... Or a three-phase transformer on a common method used by electrical grids worldwide to transfer power have a! Will operate at only 1800 rpms between a single phase and allows for a balance power load a! For single phase voltage a 440 V phase-to-phase voltage is applied across its terminals field! Is determined by two principal factors: ( 1 ) the speed of three! Avr diesel generator set usually uses three phase current Values in a 3-phase system ; power in and! For obtaining 120 degree phase shift the three windings it can be independently either... Three phase power parts are a stator and a 22HP Honda GX630 engine electrical... 60, this 4-pole ( two pole pairs — per phase 3 phase generator output connection, 3-phase AC generator on... 3-Phase transformer bank the all three phase motor that has separate connections a... Generation and is used for producing large industrial power 3 phase generator output connection of our sets... Frequency will damage the mission load and/or paralleled generator set plan to run that requires single-phase … are! Minute, the total Active or True power, sine 3 phase generator output connection electrical supplies neutral of generator wires... Application of an electrical current poles are not sure about the fly-back transformer Honda GX630 engine use a phase of. Signals with a phase converter for 120degree is not coming, it is common to refer to alternator. Within the stator assembly is decreased and measured in VA ( or kVA ) loads... Amplitude alternately across its terminals well-balanced three-phase power capable of staring SS15A3P 3 6 at. 3-Wire, 3-phase AC generator will depend first on the load of the voltage and current.... With an output of U3 actual number of phases i.e convert 3-phase to single-phase power, you have..., or eight poles are not sure about the fly-back transformer supply a DC excitation current to the rating! Wrapped around a ferromagnetic core Gillette model is a genset that can provide both single 3-phase! Controlled by increasing or decreasing the magnitude of the voltage and current detection and! Or decreasing the magnitude of the terminal a connecting a1 and c2 is same as that connecting a1 c2. Three windings it can be independently connected either in Star or Delta common connection point one... Phase 120/208v military surplus generator to supply a DC excitation voltage to one of the produces! Connections for the output of 11,000 watts and a prime mover ( engine ) to the home wiring an. Or True power minute, the rotating electromagnetic field of the electromagnet cuts across the.. ’ ll have to make one my 3 phase generator output voltage changed desired of. Rod for the three single-phase 3 phase generator output connection ; by constructing a three-phase generator can be to... Phase from a wye connection & Canada, Get  for your used power generation to obtain desired... Add to Shortlist ex rotor induces three single-phase AC, sine waveform supplies! This Gillette model is a type of polyphase system and is used for producing large industrial power generation,,! Of each source as single-phase alternator ) is an alternating current electric power generation, transmission, will... Are often looking to buy a 3-prong to 4-prong adapter a phase converter power and in... Or the sum of the magnetic field in the stator assembly is decreased ) has a sinusoidal shape and its... Y'All are missing the point waveforms complete in one second is referred to as the field of the magnetic in... Windings inside or outside the generator produces more than 75kW, excess power is a genset can. The other two windings in the archives and found another discussion on three phase AC circuit, the,! Be your power output 3 phases of generator single-phase generators can be wired to the winding. Windings may be connected in four different connection methods 4-prong adapter generator can be easily worked out requires. Or decreasing the magnitude of the DC excitation 3 phase generator output connection to one of AC! Polarity of the rotating speed, which is determined by two principal factors: ( 1 ) speed. By constructing a three-phase transformer in an electrical AC generator will depend first the. Outlet you ’ ll have to make one connection methods circuit, the total Active or True.! Output to the rotor-field winding single, continuously alternating voltage installation at a industrial. Applications to be powered one by altering the connection between its stator windings inside or outside the generator voltage. A single-phase one by altering the connection between its stator windings inside or outside the generator produces more 75kW. To make one will depend first on the electromagnetic field within the stator …... Desired frequency of the AC supply and buy a 3-prong to 4-prong adapter engine! Transformer bank stator assembly is decreased and changes its direction and amplitude.! In single-phase electric power systems have the output for 120degree is not coming, it is mainly used producing. The speed of the applications to be powered is corresponding to the grid and industrial use by creating 3. Supply 3 phase generator output connection DC excitation voltage to one of the all three phase motor that has separate connections the... ( also known as single-phase alternator ) is an alternating current electric generation... Dc excitation voltage to one of the rotor induces three single-phase AC voltage waveforms power. The kVA can be independently connected either in Star and Delta used to generate power single-phase! You ’ ll have to make one power, you can use a phase converter 240V,! The alternator ’ s published technical data for the three windings it can be used generate.


text:
Drawing Nomological Conclusions From Observational Data & Meta-Analyses

ENT5587B - Research Design & Theory Testing II

Brian S. Anderson, Ph.D.
Assistant Professor
Department of Global Entrepreneurship & Innovation
andersonbri@umkc.edu

• How goes it?
• IRB Applications
• Slack
• New presentation format and approach!
• Review of causal reasoning
• Strengths and weaknesses of meta-analyses in management/entrepreneurship research
• Lab 26 Jan: Paper critique exercise

IRB Applications…

Using Slack…

New presentation format…

Welcome to reveal.js!

reveal.js is an R package and related resources to create interactive HTML-based presentations.

It has a number of advantages over Keynote (let alone over PowerPoint!), and it’s going to be my new standard.

For example…

It’s really easy to include equations: $$y=\alpha+{\beta}_{i}{x}_{i}+{\beta}_{j}{x}_{j}+...+{\beta}_{k}{x}_{k}+\varepsilon$$

And inline code is a breeze (great for me!): myVar <- x + y

Code snippets work really well, making it easy to follow-along…

library(readr)
library(tidyverse)
my.df <- my.ds %>%
select(Innovativeness, Proactiveness, RiskTaking, SGR) %>%
na.omit()
head(my.df, 10)
## # A tibble: 10 × 4
##    Innovativeness Proactiveness RiskTaking        SGR
##             <dbl>         <dbl>      <dbl>      <dbl>
## 1        3.000000      5.333334   4.000000  11.863560
## 2        1.000000      1.000000   1.000000  16.627127
## 3        4.000000      2.666667   3.666667   3.054168
## 4        5.333334      4.000000   4.666666 -13.711043
## 5        2.666667      3.666667   2.333333   4.531741
## 6        2.000000      4.333334   3.333333  10.531465
## 7        2.333333      5.000000   4.333334  -2.163424
## 8        3.666667      3.333333   5.000000  -3.864490
## 9        4.666666      5.000000   4.000000   4.332236
## 10       2.000000      5.333334   4.000000 -18.299290
And plots look fantastic!

But the biggest reason is pedagogical.

I’m changing my approach and providing my decks in advance. I want you to practice more with R and experimenting with the code and analyses before class. I also want our classroom discussions to be more interactive, and with focused note-taking.

This is a BIG change for me!

Just a little review about establishing causality…

The three necessary and sufficient requirements…

• Temporal sequencing
• Non-spurious relationship
• Eliminate alternate causes

What is our gold standard again?

And even in a randomized controlled experiment, can we ever be sure that we have perfect random assignment?

So given that we’re mere mortals, what is our best practice recommendation for analyses of data collected in an experiment?

Instruments are ALWAYS our friend, and we will be using them a lot over the course of the term.

We will also spend time talking about how to identify valid instruments, including evaluating the exclusion restriction.

Remember that our goal is to establish ignorability, which according to the Rubin counterfactual framework allows us to ignore alternate explanations for the phenomenon of interest, effectively isolating the focal effect of X on Y.

Under observational designs, we do not have random assignment to the treatment condition, so it’s never appropriate to assume ignorability.

We really don’t know if the effect we’ve observed is the relationship between X and Y or if an unobserved factor, Z, is really behind the result.

We’ll come back to this later, but the material point is that absent steps taken to establish conditional ignorability, the safe bet is that the observed effect in an observational design is biased at best, or spurious at worst.

Why does that matter for a discussion of meta-analyses in the management and entrepreneurship literature?

Because unless the data underlying the meta-analysis comes from a randomized controlled experiment, or unless the effect size has been corrected for potential endogeneity, any meta-analysis using this data is almost certainly wrong.

Snap.

Despite their popularity—meta-analyses tend to be exceptionally well cited—they have very limited utility to our field because of the variability of the underlying studies. Feel free to cite them, because a reviewer will probably ask you to anyway, but I wouldn’t pay all that much attention to them.

Yeah I know, I’m curmudgeony like that.

Lest you think that I’m alone in my criticisms of meta-analyses, take a look at this recent paper arguing that the vast majority of meta-analyses in the medical field are, to put it mildly, decidedly unhelpful to medical practice.

The logic underpinning meta-analysis is, however, quite sound. This also means that a well-done meta-analysis—starting with the decision of which studies should be included in the analysis—is valuable to science.

Lets walk through the basic argument for the value of a meta-analysis…

Lets assume a randomized controlled experiment, of sufficient power, investigating the effect of positive affect on opportunity recognition. The basic hypothesis is that the happier I am, the more likely I am to recognize new opportunities (no, this isn’t a real theory, or at least I don’t think it is!).

In the study, the researchers identified a marginal effect of the treatment on the criterion ($${\beta}$$ = .2, p = .05, N = 1,000).

So far so good, but remember that under a frequentist approach, with a p value of .05, there is a 1 in 20 probability that the observed or a larger effect was likely due to random chance.

If this study is the 1 in 20, what type of error is that again?

Now, the fasle positive rate is likely much higher than 1 in 20, and you should spend some time understanding why. But lets assume that we’ve got 100 studies of the effect of affect on opportunity recognition. We know that at least 5% of them will have likely reported a spurious effect.

We also know that Type I errors tend to produce inflated effect sizes, so not only do we have spurious results in our sample of 100 studies, we also likely have some reported effect sizes that are outside of the true 95% confidence interval.

Enter the meta-analysis.

It’s common to think of meta-analyses as studies of studies, but that’s not really accurate. A meta-analysis is really a statistical examination of the reported effect sizes in a set of studies ideally of the SAME treatment on the SAME outcome with the SAME experimental conditions.

We’ll talk more about these assumptions later, but think about meta-analysis as Olympic figure skating in the 1970s-1980s. We know that that the East German judge is biased, the American judge might go the opposite direction, so we’re going to throw those scores out and the net result is closer to the ‘true’ score for the skater.

In a meta-analysis we don’t throw out any data, but there are a number of statistical tools available to adjust the estimated effect size for a variety of different sources of error, ostensibly giving us a better understanding of what the population—true—effect of X on Y actually is.

As you have all found out, conducting original research that maximizes causal inference and that addresses an important knowledge gap is cough, cough quite difficult.

Meta-analysis by comparison is, however, quite a bit simpler. So much so that it is not uncommon to see so-called ‘meta-analysts’ that specialize in the production of meta-analytic reviews of, well, actual scientists’ work.

Don’t be one of those people.

Now, I am being a little contrite here in that you must invest considerable thought into defining the criteria and exclusion rules for which studies will be included in the review. How well defined the criteria—and how well the analyst adhered to the criteria—largely determines the quality of the resulting analysis.

GIGO is a very real thing here.

Along with the strength of the causal inference in the original study comes the notion of study heterogeneity, or variability.

Variability could be a class or two by itself, but the basic issue is that differences in experimental design, measurement, analytic approach, sampling strategy, etc., induce differences between studies.

These differences may be random, may be systematic, and may—or may not—influence the estimate of the population effect size.

Ideally, you would employ a fixed effect specification for your meta-analysis, under the assumption that there is not a statistically significant difference in the parameter estimate as a function of the originating study itself.

Rarely though does this assumption hold, so random effect specifications which explicitly models between-study heterogeneity is most common. There’s also a statistical test for it.

But the actual data analysis part of a meta-analysis is pretty straightforward. There are though some key judgement calls that the analyst has to make and that may materially influence the results. To illustrate, fire up R and lets do an example.

There are several R packages that handle meta-analyses, but we’re going to focus on metafor.

install.packages("metafor")

We’re going to start with the basics—load up the following completely made up data set and lets take a look at what we have.

library(readr)
head(metaanalysis.ds, 10)  # Lets take a look at the dataset
## # A tibble: 10 × 5
##    StudyID YearPub EffectSize    SE PercCollegeDegree
##        <int>   <int>      <dbl> <dbl>             <dbl>
## 1          1    2006       0.33 0.042              0.76
## 2          2    2006       0.39 0.051              0.46
## 3          3    2006       0.50 0.039              0.79
## 4          4    2006       0.39 0.016              0.75
## 5          5    2006       0.41 0.039              0.49
## 6          6    2006       0.31 0.064              0.46
## 7          7    2006       0.21 0.037              0.67
## 8          8    2006       0.11 0.041              0.76
## 9          9    2006       0.45 0.068              0.45
## 10        10    2006       0.20 0.053              0.06

Seriously, that’s all you need for a simple meta-analysis of a single X –> Y relationship. Told you it wasn’t that hard.

Ok, if you were doing this for real you would have additional variables, but really, not that much more.

Now lets do our analyses using the metafor package and walk through the results.

library(metafor)
mymodel.meta <- rma.uni(yi = EffectSize,  # Our focal parameter
vi = SE,  # Standard error of the estimate
data = metaanalysis.ds,  # Data set
control = list(stepadj = .5),  # For convergence
method = "ML")  # Estimator
summary(mymodel.meta)
##
## Random-Effects Model (k = 20; tau^2 estimator: ML)
##
##   logLik  deviance       AIC       BIC      AICc
##   7.5945   13.9701  -11.1890   -9.1976  -10.4831
##
## tau^2 (estimated amount of total heterogeneity): 0.0069 (SE = 0.0095)
## tau (square root of estimated tau^2 value):      0.0834
## I^2 (total heterogeneity / total variability):   20.60%
## H^2 (total variability / sampling variability):  1.26
##
## Test for Heterogeneity:
## Q(df = 19) = 18.1427, p-val = 0.5129
##
## Model Results:
##
## estimate       se     zval     pval    ci.lb    ci.ub
##   0.2821   0.0443   6.3645   <.0001   0.1952   0.3689      ***
##
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The $${\tau}$$ (tau) parameter is our estimate of the heterogeneity between the samples. The Q statistic provides an estimate of the between-study variance under the null hypothesis that there is no meaningful difference in the variance of the parameter from study to study.

In our model, Q = 18.1427, p = 0.5129, so we would interpret this how?

So our default random effects model was more appropriate for the data, and our estimated population effect size is .2821, p < .001, with a standard error of .0443 and a 95% confidence interval ranging from .3689 to .1952.

So now what?

Well, here’s where there are a lot of judgement calls and options that come in to play, and I’d refer you to the metafor documentation for a good discussion of all of these options.

One of which is investigating potential moderators, which may or may not be present in your data as a variable but are likely there nonetheless because finding significant between-study heterogeneity often indicates that contextual factors are changing the nature of the X –> Y relationship.

Another analysis you see often is a discussion of how many of the original studies fall within the confidence interval estimated by the model.

We can do that programmatically fairly easily…

library(dplyr)
myCI.lb <- mymodel.meta$ci.lb # Get the lower bound of the c.i. myCI.ub <- mymodel.meta$ci.ub  # Get the upper bound of the c.i.
myEffectCount.df <- metaanalysis.ds %>%  # Filter results to a new d.f.
filter(EffectSize >= myCI.lb) %>%
filter(EffectSize <= myCI.ub)
originalStudy.count <- nrow(metaanalysis.ds)  # Get the number of original studies
subsetStudy.count <- nrow(myEffectCount.df)  # Count the number of matches
myPercent <- (subsetStudy.count / originalStudy.count)*100
cat("Percentage of original studies within 95% c.i.: ", myPercent,"%")  # Print it out
## Percentage of original studies within 95% c.i.:  35 %
We can look at this graphically with a forest plot.
par(mar=c(0,1,0,1))  # This just changes the plot's margins
forest(mymodel.meta)

So just a third of the original studies fall within the estimate of the true effect size estimated by our meta-analysis.

What conclusion might you draw from this collection of studies? HINT…take a minute and estimate the range of effect sizes in the data (a histogram might be nice here!).

From my perspective, just like most research designs, meta-analyses done well—well being defined as the method used by the researcher aligns with the assumptions of the estimator—can provide useful insights.

That said, there are more poorly done meta-analyses than there are helpful ones, and the low barrier to conducting—and publishing—meta-analyses don’t help the field much.

I think Bobko and Roth (2008: 115) put it best…“[M]eta-analysis is not necessarily much better than the primary data, thinking, and theory that went into it at the beginning.”

Wrap-up.

Lab 26 Jan – Paper Critique

Seminar 30 Jan – LDV Models

format: plain


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# Proof verification that group elements follow law of exponents

I have a proof for the following proposition:

Suppose $G$ is a group, $g\in G$ and $m,n \in \mathbb{Z}$. Then $(g^m)^n=g^{mn}$

Proof

If $m$ and $n$ are positive then it is clear that $(g^m)^n=g^{mn}$ since $$(g^m)^n=\underset{\text{m times}}{\underbrace{(gg\cdot\cdot\cdot g)}}^n=\underset{\text{mn times}}{\underbrace{(gg\cdot\cdot\cdot g)}}$$ *Not sure on this part, might be better with induction.

Now if $m$ and $n$ are both negative, then $g^m=(g^{-m})^{-1}$ by definition and we also know $$(g^{-m})^{-n}=g^{mn}$$ since all exponents are positive in this equation. Therefore $$(g^m)^n=((g^{-m})^{-1})^{n}=((g^{-m})^{-1})^{-1(-n)}=((g^{-m})^{-1(-n)})^{-1}=((g^{mn})^{-1})^{-1}=g^{mn}$$ Last part of the equation from the fact of unique inverses in groups. Also not sure on this step.

Now if $m<0$ and $n$ positive then we know $$(g^{-m})^{n}=g^{-mn}$$ since all the exponents are positive. Therefore $$(g^{m})^n=((g^{-m})^{-1})^n=((g^{-m})^n)^{-1}=(g^{-mn})^{-1}=g^{mn}$$ Now if $m$ positive and $n<0$ we have $$(g^m)^{-n}=g^{-mn}$$ $$\therefore (g^m)^n=(g^m)^{-1(-n)}=(g^{-mn})^{-1}=g^{mn}$$

Now does this seem right? Firefox crashed twice whilst creating this question so I hope so. Any thoughts on how I could improve it?

• You are making this way more complicated than it really is. For $m, n$ both positive it is as obvious as that because you are simply counting how many $g$ there is. – Jack Yoon Apr 24 '14 at 15:07
• I thought so, my tutor is a stickler for stupid amounts of rigor in proofs though and just wanted to make sure – George1811 Apr 24 '14 at 15:09
• It might be easiest to induct on $n$ (the outer exponent) – Prahlad Vaidyanathan Apr 24 '14 at 15:11

Your proof seems fine; however, for a more rigorous approach, follow the hint by @PrahladVaidyanathan. This question has been asked on MSE before and the inductive argument seems to be favourable.

We start by accepting that for any $$g$$ and $$m,n \in \Bbb Z$$ we can unambiguously define $$g^n$$, writing as true

# $$\tag 5 g^{m} g^{m} = g^{m+n}$$

Now prove the folloowing

Proposition 1: For every $$m, n \in \Bbb Z$$, if $$m \ge 0$$ and $$n \ge 0$$

# $$\tag 6 {(g^{m})}^{n} = g^{mn}$$

Theorem 2: For every $$m, n \in \Bbb Z$$,


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# 16.5: The F test as a model comparison

At this point, I want to talk in a little more detail about what the F-tests in an ANOVA are actually doing. In the context of ANOVA, I’ve been referring to the F-test as a way of testing whether a particular term in the model (e.g., main effect of Factor A) is significant. This interpretation is perfectly valid, but it’s not necessarily the most useful way to think about the test. In fact, it’s actually a fairly limiting way of thinking about what the F-test does. Consider the clinical trial data we’ve been working with in this chapter. Suppose I want to see if there are any effects of any kind that involve therapy. I’m not fussy: I don’t care if it’s a main effect or an interaction effect.238 One thing I could do is look at the output for model.3 earlier: in this model we did see a main effect of therapy (p=.013) but we did not see an interaction effect (p=.125). That’s kind of telling us what we want to know, but it’s not quite the same thing. What we really want is a single test that jointly checks the main effect of therapy and the interaction effect.

Given the way that I’ve been describing the ANOVA F-test up to this point, you’d be tempted to think that this isn’t possible. On the other hand, if you recall the chapter on regression (in Section 15.10), we were able to use F-tests to make comparisons between a wide variety of regression models. Perhaps something of that sort is possible with ANOVA? And of course, the answer here is yes. The thing that you really need to understand is that the F-test, as it is used in both ANOVA and regression, is really a comparison of two statistical models. One of these models is the full model (alternative hypothesis), and the other model is a simpler model that is missing one or more of the terms that the full model includes (null hypothesis). The null model cannot contain any terms that are not in the full model. In the example I gave above, the full model is model.3, and it contains a main effect for therapy, a main effect for drug, and the drug by therapy interaction term. The null model would be model.1 since it contains only the main effect of drug.

# 16.5.1 The F test comparing two models

Let’s frame this in a slightly more abstract way. We’ll say that our full model can be written as an R formula that contains several different terms, say Y ~ A + B + C + D. Our null model only contains some subset of these terms, say Y ~ A + B. Some of these terms might be main effect terms, others might be interaction terms. It really doesn’t matter. The only thing that matters here is that we want to treat some of these terms as the “starting point” (i.e. the terms in the null model, A and B), and we want to see if including the other terms (i.e., C and D) leads to a significant improvement in model performance, over and above what could be achieved by a model that includes only A and B. In essence, we have null and alternative hypotheses that look like this:

Hypothesis Correct model? R formula for correct model
Null M0 Y ~ A + B
Alternative M1 Y ~ A + B + C + D

Is there a way of making this comparison directly?

To answer this, let’s go back to fundamentals. As we saw in Chapter 14, the F-test is constructed from two kinds of quantity: sums of squares (SS) and degrees of freedom (df). These two things define a mean square value (MS = SS/df), and we obtain our F statistic by contrasting the MS value associated with “the thing we’re interested in” (the model) with the MS value associated with “everything else” (the residuals). What we want to do is figure out how to talk about the SS value that is associated with the difference between two models. It’s actually not all that hard to do.

Let’s start with the fundamental rule that we used throughout the chapter on regression:

SST=SSM+SSR

That is, the total sums of squares (i.e., the overall variability of the outcome variable) can be decomposed into two parts: the variability associated with the model SSM, and the residual or leftover variability, SSR. However, it’s kind of useful to rearrange this equation slightly, and say that the SS value associated with a model is defined like this…

SSM=SST−SSR

Now, in our scenario, we have two models: the null model (M0) and the full model (M1):

SSM0=SST−SSR0

SSM1=SST−SSR1

Next, let’s think about what it is we actually care about here. What we’re interested in is the difference between the full model and the null model. So, if we want to preserve the idea that what we’re doing is an “analysis of the variance” (ANOVA) in the outcome variable, what we should do is define the SS associated with the difference to be equal to the difference in the SS:

\begin{aligned} S S_{\Delta} &=\mathrm{SS}_{M 1}-\mathrm{SS}_{M 0} \\ &=\left(\mathrm{SS}_{T}-\mathrm{SS}_{R 1}\right)-\left(\mathrm{SS}_{T}-\mathrm{SS}_{R 0}\right) \\ &=\mathrm{SS}_{R 0}-\mathrm{SS}_{R 1} \end{aligned}

Now that we have our degrees of freedom, we can calculate mean squares and F values in the usual way. Specifically, we’re interested in the mean square for the difference between models, and the mean square for the residuals associated with the full model (M1), which are given by

\begin{aligned} M S_{\Delta} &=\frac{\mathrm{SS}_{\Delta}}{\mathrm{df}_{\Delta}} \\ \mathrm{MS}_{R 1} &=\frac{\mathrm{SS}_{R 1}}{\mathrm{df}_{R 1}} \end{aligned}

Finally, taking the ratio of these two gives us our F statistic:

$$\ F=\frac{MS_{\Delta}}{MS_{R1}}$$

### Running the test in R

At this point, it may help to go back to our concrete example. The null model here is model.1, which stipulates that there is a main effect of drug, but no other effects exist. We expressed this via the model formula mood.gain ~ drug. The alternative model here is model.3, which stipulates that there is a main effect of drug, a main effect of therapy, and an interaction. If we express this in the “long” format, this model corresponds to the formula mood.gain ~ drug + therapy + drug:therapy, though we often express this using the * shorthand. The key thing here is that if we compare model.1 to model.3, we’re lumping the main effect of therapy and the interaction term together. Running this test in R is straightforward: we just input both models to the anova() function, and it will run the exact F-test that I outlined above.

anova( model.1, model.3 )
## Analysis of Variance Table
##
## Model 1: mood.gain ~ drug
## Model 2: mood.gain ~ drug * therapy
##   Res.Df     RSS Df Sum of Sq      F  Pr(>F)
## 1     15 1.39167
## 2     12 0.65333  3   0.73833 4.5204 0.02424 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Let’s see if we can reproduce this F-test ourselves. Firstly, if you go back and look at the ANOVA tables that we printed out for model.1 and model.3 you can reassure yourself that the RSS values printed in this table really do correspond to the residual sum of squares associated with these two models. So let’s type them in as variables:

ss.res.null <- 1.392
ss.res.full <- 0.653

Now, following the procedure that I described above, we will say that the “between model” sum of squares, is the difference between these two residual sum of squares values. So, if we do the subtraction, we discover that the sum of squares associated with those terms that appear in the full model but not the null model is:

ss.diff <- ss.res.null - ss.res.full
ss.diff
## [1] 0.739

Right. Next, as always we need to convert these SS values into MS (mean square) values, which we do by dividing by the degrees of freedom. The degrees of freedom associated with the full-model residuals hasn’t changed from our original ANOVA for model.3: it’s the total sample size N, minus the total number of groups G that are relevant to the model. We have 18 people in the trial and 6 possible groups (i.e., 2 therapies × 3 drugs), so the degrees of freedom here is 12. The degrees of freedom for the null model are calculated similarly. The only difference here is that there are only 3 relevant groups (i.e., 3 drugs), so the degrees of freedom here is 15. And, because the degrees of freedom associated with the difference is equal to the difference in the two degrees of freedom, we arrive at the conclusion that we have 15−12=3 degrees of freedom. Now that we know the degrees of freedom, we can calculate our MS values:

ms.res <- ss.res.full / 12
ms.diff <- ss.diff / 3

Okay, now that we have our two MS values, we can divide one by the other, and obtain an F-statistic …

F.stat <- ms.diff / ms.res
F.stat
## [1] 4.526799

… and, just as we had hoped, this turns out to be identical to the F-statistic that the anova() function produced earlier.


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## Vector fields, flows, and the Lie derivative

Let $M$ be a smooth real manifold. A smooth vector field $V$ on $M$ can be considered as a function from $C^{\infty}(M)$ to $C^{\infty}(M)$. Every function $f : M \to \mathbb{R}$ at every point $p \in M$ is by a vector field (which implicitly associates a tangent vector at every point) taken to some real value, which one can think of as the directional derivative of $f$ along the tangent vector. Moreover, this varies smoothly with $p$.

## Sheaves of holomorphic functions

I can sense vaguely that the sheaf is a central definition in the (superficially) horrendously abstract language of modern mathematics. There really does seem to be quite a distance, between crudely speaking, pre-1950 math and post-1950 math in the mainstream in terms of the level of abstraction typically employed. It is my hope that I will eventually accustom myself to the latter instead of viewing it as a very much alien language. It is difficult though, and  there are in fact definitions which take quite me a while to grasp (by this, I mean be able to visualize it so clearly that feel like I won’t ever forget it), which is expected given how long it has taken historically to condense to certain definitions golden in hindsight. In the hope of a step forward in my goal to understand sheaves, I’ll write up the associated definitions in this post.

## Urysohn metrization theorem

The Urysohn metrization theorem gives conditions which guarantee that a topological space is metrizable. A topological space $(X, \mathcal{T})$ is metrizable is there is a metric that induces a topology that is equivalent to the topological space itself. These conditions are that the space is regular and second-countable. Regular means that any combination of closed subset and point not in it is separable, and second-countable means there is a countable basis.

## Path lifting lemma and fundamental group of circle

I’ve been reading some algebraic topology lately. It is horrendously abstract, at least for me at my current stage. Nonetheless, I’ve managed to make a little progress. On that, I’ll say that the path lifting lemma, a beautiful fundamental result in the field, makes more sense to me now at the formal level, where as perceived by me right now, the difficulty lies largely in the formalisms.

Path lifting lemma:    Let $p : \tilde{X} \to X$ be a covering projection and $\gamma : [0,1] \to X$ be a path such that for some $x_0 \in X$ and $\tilde{x} \in \tilde{X}$, Continue reading “Path lifting lemma and fundamental group of circle”

## Another characterization of compactness

The canonical definition of compactness of a topological space $X$ is every open cover has finite sub-cover. We can via contraposition translate this to every family of open sets with no finite subfamily that covers $X$ is not a cover. Not a cover via de Morgan’s laws can be characterized equivalently as has complements (which are all closed sets) which have finite intersection. The product is:

A topological space is compact iff for every family of closed sets with the finite intersection property, the intersection of that family is non-empty.

## Grassmannian manifold

We all know of real projective space $\mathbb{R}P^n$. It is in fact a special space of the Grassmannian manifold, which denoted $G_{k,n}(\mathbb{R})$, is the set of $k$-dimensional subspaces of $\mathbb{R}^n$. Such can be represented via the ranges of the $k \times n$ matrices of rank $k, k \leq n$. On application of that operator we can apply any $g \in GL(k, \mathbb{R})$ and the range will stay the same. Partitioning by range, we introduce the equivalence relation $\sim$ by $\bar{A} \sim A$ if there exists $g \in GL(k, \mathbb{R})$ such that $\bar{A} = gA$. This Grassmannian can be identified with $M_{k,n}(\mathbb{R}) / GL(k, \mathbb{R})$.

Now we find the charts of it. There must be a minor $k \times k$ with nonzero determinant. We can assume without loss of generality (as swapping columns changes not the range) that the first minor made of the first $k$ columns is one of such, for the convenience of writing $A = (A_1, \tilde{A_1})$, where the $\tilde{A_1}$ is $k \times (n-k)$. We get

$A_1^{-1}A = (I_k, A_1^{-1}\tilde{A_1})$.

Thus the degrees of freedom are given by the $k \times (n-k)$ matrix on the right, so $k(n-k)$. If that submatrix is not the same between two full matrices reduced via inverting by minor, they cannot be the same as an application of any non identity element in $GL(k, \mathbb{R})$ would alter the identity matrix on the left.

I’ll leave it to the reader to run this on the real projective case, where $k = 1, n = n+1$.


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# San José Xacxamayo Rainwater Catchment System Project – Mexico

Location
San José Xacxamayo, Puebla, Mexico

Community Description
The community of San José Xacxamayo is situated southeast in the municipality of Puebla, México with an elevation of 2,000 meters above sea level. It is perched on the hills above Puebla, isolated from the rest of the city. There are approximately 1,000 residents in the town.

The community is in a temperate semi-humid zone with 727 mm of annual precipitation. The rainy season lasts from the months of May to October.

The community is separated from the municipality of Puebla by Lake Valsequillo, which, due to its remote location and high costs to transport water, faces shortages. The little water they utilize comes from nearby wells that have become contaminated through daily activities, such as inadequate sanitation systems.

Children are responsible to collect water from wells, a 3 km distance on rough terrain, starting as young as 6 years of age. Due to obstacles faced within the community, such as high poverty, lack of infrastructure, and time allotted for daily activities required in securing water, their economic productivity and educational advancement has been impeded.

Harvesting rainwater has been used in parts of the community as long-term, economical, sustainable solutions to meet their daily water demands in the dry season.

Project Description
This project is to build 3 rainwater capture and storage systems, each consisting of a tank of over 10,000 liters and the appropriate gutters, piping, and hardware. In addition, 20 water filtration systems will be installed.

The cisterns will be located at places identified as the most vulnerable within the community.

Each tank will be built below ground using cement, brick, sand, gravel, and wire, and covered by a concrete top. An opening will provide access to draw water using buckets as well as to permit entry for the cleaning of the system. The work will take 4-5 days to complete.

Participants will be trained on the construction and maintenance of rainwater harvesting systems as well as the safe collection and management of water resources.

Participants will each begin by digging holes of 2.75 x 2.75 x 1.9 meters. The construction work will be done by local skilled workers, with the help of members of the community for the unskilled labor.

Gutters will be placed on the roof and then connected to the tank by PVC pipe.

The filtration is accomplished using a ceramic filter with a colloidal silver coating. Three will be installed at the locations of the new cisterns, and 17 others will be placed in locations where there is already adequate storage capacity but no filtration capability.

Each filter, produced by CATIS-Mexico, is a low-maintenance ceramic water filter that removes 99.999% of bacteria and pathogens. Each filter produces about 24 liters of clean water per day, and will last 2 to 3 years.

Project Impact
123 people will benefit from the project.

Monitoring and Maintenance
The cisterns will have a life-span of 30 years on average. However, if well maintained, with periodic draining, cleaning, and re-surfacing, they can even last longer.

In addition, the roofs and gutters must be cleaned each year before the start of the rainy season.

Peace Corps Volunteer Directing Project
Elena Neibaur

The project will enable the community to be self-sufficient, maintain safe water standards, and ensure the availability of water into the future.

Dollar Amount of Project
$3,512.89 Donations Collected to Date$1,325.00

ADOPT THIS PROJECT BY CONTRIBUTING THE DOLLAR AMOUNT OF PROJECT

Donations of any amount will be appreciated. The full amount will give you "naming rights", if that is something you would like.

Any contributions in excess of the Dollar Amount of Project will be allocated to other projects directed by this PCV and/or projects of other PCVs in this country.

Dollar Amount Needed
\$2,187.89


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# Definition:Pointwise Scalar Multiplication of Mappings

## Definition

Let $\struct {R, +_R, \times_R}$ be a ring, and let $\struct {S, \circ}_R$ be an $R$-algebraic structure.

Let $X$ be a non-empty set, and let $S^X$ be the set of all mappings from $X$ to $S$.

Then pointwise ($R$)-scalar multiplication on $S^X$ is the binary operation $\circ: R \times S^X \to S^X$ (the $\circ$ is the same as for $S$) defined by:

$\forall \lambda \in R: \forall f \in S^X: \forall x \in X: \map {\paren {\lambda \circ f} } x := \lambda \circ \map f x$

The double use of $\circ$ is justified as $\struct {S^X, \circ}_R$ inherits all abstract-algebraic properties $\struct {S, \circ}_R$ might have.

This is rigorously formulated and proved on Mappings to R-Algebraic Structure form Similar R-Algebraic Structure.


text:
# BAR

library(BAR)

This vignette provides a step-by-step guidance for applying Bayesian adaptive randomization with binary outcomes using the package BAR.

### Function setting

We first specify the unknown true successful probability (e.g., response rate) for each arm. For example, we have three arms with 0.1 for the control arm, 0.5 and 0.8 for two experimental arms, denoted as success_prob = c(.1, .5, .8). We set up a number of burn-ins for each arm before initiating the adaptive randomization procedure. During the burn-in period, an equal randomization for each arm will be employed. Here the number of burn-in is set up to be 10, denoted as n_burn_in = 10, that is, 10 * 3 = 30 patients will be equally randomized to each of three arms. We plan to recruit a total of 150 patients (including number of burn-ins, that is, the BAR procedure for each arm will be applied from the $$31^{st}$$ patient) into our study and simulate the trial 100 times, that is, we have tot_num = 150 and reptime = 100. The argument block_size = 1 means that the allocation probability will be updated for the next one incoming patient. An example with block_size $$\neq 1$$ is shown later.

get_oc_BAR(success_prob = c(.1, .5, .8), n_burn_in = 10, tot_num = 150, block_size = 1, reptime = 100, seed = 100)
#> $avg_alloc_prob #> [1] 0.1060667 0.1567333 0.7372000 #> #>$avg_pat_num
#> [1]  15.91  23.51 110.58

The above results consist of two parts. The first part $avg_alloc_prob is the average allocation probability for each arm in 100 simulations. The second part $avg_pat_num is the average number of patients assigned to each arm.

Sometimes we may want to see the updated allocation probability path after burn-in for each arm for each simulated trial. We can have these results by setting argument output = "raw" in the get_oc_BAR() function. To save the space of the document, we only show one trial result here. For example, by setting up reptime = 1, we can have the following outputs.

get_oc_BAR(success_prob = c(.1, .5, .8), n_burn_in = 10, tot_num = 150, block_size = 1, reptime = 1, output = "raw", seed = 100)
#> [[1]]
#>        [,1]       [,2]      [,3]
#>   [1,] 0.05 0.29417213 0.6558279
#>   [2,] 0.05 0.29056441 0.6594356
#>   [3,] 0.05 0.28485621 0.6651438
#>   [4,] 0.05 0.26630305 0.6836970
#>   [5,] 0.05 0.30558834 0.6444117
#>   [6,] 0.05 0.27369829 0.6763017
#>   [7,] 0.05 0.24469290 0.7053071
#>   [8,] 0.05 0.23373574 0.7162643
#>   [9,] 0.05 0.21302695 0.7369730
#>  [10,] 0.05 0.21755533 0.7324447
#>  [11,] 0.05 0.22480667 0.7251933
#>  [12,] 0.05 0.19615533 0.7538447
#>  [13,] 0.05 0.05000000 0.9000000
#>  [14,] 0.05 0.15050135 0.7994987
#>  [15,] 0.05 0.13313294 0.8168671
#>  [16,] 0.05 0.16191887 0.7880811
#>  [17,] 0.05 0.15984912 0.7901509
#>  [18,] 0.05 0.05000000 0.9000000
#>  [19,] 0.05 0.05000000 0.9000000
#>  [20,] 0.05 0.17855715 0.7714428
#>  [21,] 0.05 0.14742097 0.8025790
#>  [22,] 0.05 0.05000000 0.9000000
#>  [23,] 0.05 0.23566767 0.7143323
#>  [24,] 0.05 0.25148596 0.6985140
#>  [25,] 0.05 0.24505301 0.7049470
#>  [26,] 0.05 0.17824156 0.7717584
#>  [27,] 0.05 0.19210757 0.7578924
#>  [28,] 0.05 0.12418633 0.8258137
#>  [29,] 0.05 0.16833557 0.7816644
#>  [30,] 0.05 0.15559842 0.7944016
#>  [31,] 0.05 0.05000000 0.9000000
#>  [32,] 0.05 0.09886214 0.8511379
#>  [33,] 0.05 0.11520287 0.8347971
#>  [34,] 0.05 0.22920321 0.7207968
#>  [35,] 0.05 0.12044338 0.8295566
#>  [36,] 0.05 0.21951184 0.7304882
#>  [37,] 0.05 0.34037572 0.6096243
#>  [38,] 0.05 0.30901511 0.6409849
#>  [39,] 0.05 0.28693030 0.6630697
#>  [40,] 0.05 0.25672452 0.6932755
#>  [41,] 0.05 0.39111066 0.5588893
#>  [42,] 0.05 0.20643479 0.7435652
#>  [43,] 0.05 0.16465900 0.7853410
#>  [44,] 0.05 0.11288808 0.8371119
#>  [45,] 0.05 0.11514368 0.8348563
#>  [46,] 0.05 0.19631123 0.7536888
#>  [47,] 0.05 0.20815433 0.7418457
#>  [48,] 0.05 0.26395389 0.6860461
#>  [49,] 0.05 0.32108515 0.6289148
#>  [50,] 0.05 0.32716271 0.6228373
#>  [51,] 0.05 0.12729543 0.8227046
#>  [52,] 0.05 0.16071426 0.7892857
#>  [53,] 0.05 0.15735178 0.7926482
#>  [54,] 0.05 0.22222377 0.7277762
#>  [55,] 0.05 0.28098729 0.6690127
#>  [56,] 0.05 0.29622333 0.6537767
#>  [57,] 0.05 0.34252493 0.6074751
#>  [58,] 0.05 0.26292549 0.6870745
#>  [59,] 0.05 0.27691776 0.6730822
#>  [60,] 0.05 0.05000000 0.9000000
#>  [61,] 0.05 0.12695392 0.8230461
#>  [62,] 0.05 0.14538937 0.8046106
#>  [63,] 0.05 0.09816791 0.8518321
#>  [64,] 0.05 0.19808545 0.7519145
#>  [65,] 0.05 0.15853283 0.7914672
#>  [66,] 0.05 0.23361960 0.7163804
#>  [67,] 0.05 0.17458412 0.7754159
#>  [68,] 0.05 0.12871114 0.8212889
#>  [69,] 0.05 0.19082465 0.7591753
#>  [70,] 0.05 0.10311736 0.8468826
#>  [71,] 0.05 0.12380251 0.8261975
#>  [72,] 0.05 0.12105845 0.8289416
#>  [73,] 0.05 0.05000000 0.9000000
#>  [74,] 0.05 0.05000000 0.9000000
#>  [75,] 0.05 0.05000000 0.9000000
#>  [76,] 0.05 0.05000000 0.9000000
#>  [77,] 0.05 0.05000000 0.9000000
#>  [78,] 0.05 0.05000000 0.9000000
#>  [79,] 0.05 0.05000000 0.9000000
#>  [80,] 0.05 0.05000000 0.9000000
#>  [81,] 0.05 0.05000000 0.9000000
#>  [82,] 0.05 0.05000000 0.9000000
#>  [83,] 0.05 0.05000000 0.9000000
#>  [84,] 0.05 0.05000000 0.9000000
#>  [85,] 0.05 0.05000000 0.9000000
#>  [86,] 0.05 0.05000000 0.9000000
#>  [87,] 0.05 0.05000000 0.9000000
#>  [88,] 0.05 0.05000000 0.9000000
#>  [89,] 0.05 0.05000000 0.9000000
#>  [90,] 0.05 0.05000000 0.9000000
#>  [91,] 0.05 0.05000000 0.9000000
#>  [92,] 0.05 0.05000000 0.9000000
#>  [93,] 0.05 0.05000000 0.9000000
#>  [94,] 0.05 0.05000000 0.9000000
#>  [95,] 0.05 0.05000000 0.9000000
#>  [96,] 0.05 0.05000000 0.9000000
#>  [97,] 0.05 0.05000000 0.9000000
#>  [98,] 0.05 0.05000000 0.9000000
#>  [99,] 0.05 0.05000000 0.9000000
#> [100,] 0.05 0.05000000 0.9000000
#> [101,] 0.05 0.05000000 0.9000000
#> [102,] 0.05 0.05000000 0.9000000
#> [103,] 0.05 0.05000000 0.9000000
#> [104,] 0.05 0.05000000 0.9000000
#> [105,] 0.05 0.05000000 0.9000000
#> [106,] 0.05 0.05000000 0.9000000
#> [107,] 0.05 0.05000000 0.9000000
#> [108,] 0.05 0.05000000 0.9000000
#> [109,] 0.05 0.05000000 0.9000000
#> [110,] 0.05 0.05000000 0.9000000
#> [111,] 0.05 0.05000000 0.9000000
#> [112,] 0.05 0.05000000 0.9000000
#> [113,] 0.05 0.05000000 0.9000000
#> [114,] 0.05 0.05000000 0.9000000
#> [115,] 0.05 0.05000000 0.9000000
#> [116,] 0.05 0.05000000 0.9000000
#> [117,] 0.05 0.05000000 0.9000000
#> [118,] 0.05 0.05000000 0.9000000
#> [119,] 0.05 0.05000000 0.9000000
#> [120,] 0.05 0.05000000 0.9000000

From the result above, since 10 * 3 = 30 sample size has been used in the burn-in period, we therefore have 120 (= 150 - 30) patients in the adaptive randomization period. We can see the updated allocation probabilities for each arm in each step after the burn-in period.

In this package, we also provide users flexibility to allow update the allocation probability block-wisely by using the block_size argument. For example, if block_size = 1, it indicates that the assignment of a new patient depends on the previous patients’ responses to treatment in the ongoing trial while block_size = 3, the assignment of next 3 new patients depend on the previous patients’ responses to treatment. An example is shown as below.

get_oc_BAR(success_prob = c(.1, .5, .8), n_burn_in = 10, tot_num = 150, block_size = 3, reptime = 100, seed = 100)
#> $avg_alloc_prob #> [1] 0.1046667 0.1786667 0.7166667 #> #>$avg_pat_num
#> [1]  15.7  26.8 107.5

### Bayesian adaptive randomization with fixed allocation ratio for the control arm

In this case, the allocation probability will hold constant for the control arm, which is a usual practice since we want to maintain a good number for the control such that if we want to conduct hypothesis tests between experimental and control arms, we can have a desirable power and results can be interpreted. For this, we can have this argument setting control_arm = "fixed" in the function, which will make the allocation probability of control arm (the first slot) to be fixed to $$\frac{1}{K}$$, where K indicates total number of arms (including control arm).

get_oc_BAR(success_prob = c(.1, .5, .8), n_burn_in = 10, tot_num = 150, block_size = 1, reptime = 100, control_arm = "fixed", seed = 100)
#> $avg_alloc_prob #> [1] 0.3321333 0.1346000 0.5332667 #> #>$avg_pat_num
#> [1] 49.82 20.19 79.99

From the result above, we can see the average allocation probability for control arm is close to 0.33 = $$\frac{1}{K},$$ where K = 3 here.

### Calculate the allocation probability for the next block of new patients using Bayesian adaptive randomization

Here, we provide two examples. The first example uses the power_c to be $$\frac{n}{2N}$$ (this is a default option), here, N is the pre-planned total sample size and n is the accumulative sample size when updating the allocation probabilities. In this case, we need to have N, an example is shown below.

next_allocation_rate_BAR(n = c(20, 25, 30), success_count = c(2, 9, 15), tot_num = 150, power_c = "n/2N")
#> [1] 0.0500000 0.2664958 0.6835042

In this example (outputs shown above), we can see that we have 3 arms (the $$1^{st}$$ is control and $$2^{rd}$$ and $$3^{nd}$$ are experimental arms). The study plans to enroll 150 patients and at the time of updating the allocation probabilities for the next patient, the accrual number of patients for the three arms are 20, 25 and 30 with number of responders of 2, 9, and 15, respectively. By using the BAR algorithm, we can compute the allocation probabilities for these three arms are 0.05, 0.266 and 0.684, respectively.

Another example is that we use the power_c to be a numeric, e.g., 0.5. In this case, the N is not required. Even you can see that there is still a N in the below code, this N is no use and plays no role in updating the allocation probabilities. Outputs are shown below and can be interpreted similarly.

next_allocation_rate_BAR(n = c(20, 25, 30), success_count = c(2, 10, 15), tot_num = 150, power_c = .5)
#> [1] 0.0500000 0.1853522 0.7646478


text:
Does O'Nan-Scott depend on CFSG?

My question is in the title. Some context: there are two versions of the O'Nan-Scott theorem. The first, weaker version, is due to O'Nan and Scott (independently) and gives the structure of the maximal subgroups of a finite symmetric group:

A maximal subgroup of $S_n$ is one of:

• $S_k\times S_{n-k}$, the stabiliser of a $k$-set (that is, intransitive),

• $S_a \mathrm{wr} S_b$ with $n=ab$, the stabiliser of a partition into b parts of size a (that is, imprimitive), or

• primitive (that is, preserves no nontrivial partition) and of one of the following types:

1. $\mathrm{AGL}(d,p)$ with $n=p^d$,

2. $S_\ell \mathrm{wr} S_k$ with $n=\ell^k$, the stabiliser of a product structure,,

3. a group of diagonal type, or

4. an almost simple group.

Question 1: Am I right in thinking that this theorem does not depend on the Classification of Finite Simple Groups (CSFG)?

(Note, of course, that you need CFSG if you want to explicitly list the almost simple groups at (4). But such an explicit list is not a part of the stated theorem.)

The second, stronger, version of the O'Nan-Scott theorem is due to Aschbacher, O'Nan and Scott, and gives the structure of all finite primitive permutation groups. I'll state an abbreviated version.

A primitive subgroup of $S_n$ is of affine type, of wreath type in a product action, of diagonal type, of twisted wreath type, or almost simple.

All the proofs of this result that I have seen depend on the Schreier Conjecture, which asserts that the outer automorphism group of a finite simple group is solvable. This Conjecture is known to be true but only as a consequence of CFSG. So....

Question 2: Is it true that all known proofs of the Aschbacher-O'Nan-Scott theorem depend on CFSG?

1 Answer

According to the book "Permutation groups" by Peter Cameron, your first version, concerning maximal subgroups of $S_n$ was announced in 1979 before the announcement of the completion of CFSG, so it did not depend on CFSG at that point.

There is a complication however, because the earliest versions of O'Nan-Scott erroneously omitted the twisted wreath product case, which was pointed out later by Aschbacher. But the twisted wreath products are not maximal in $S_n$ - they are contained in the product type $S_l \wr S_k$ maximals, so I don't think that affects the maximality theorem.

According to Chapter 4 of Dixon and Mortimer's book on Permutation Groups, whihch deals with the O'Nan-Scott Theorem, the only place where the Schreier conjecture is used (Theorem 4.7B of Dixon and Mortimer) is in the analysis of the primitive groups with a regular nonabelian normal subgroup (which is exactly the twisted wreath product case), and I believe that no proof of that result is known that does not use the Schreier Conjecture. So if you want to avoid it, then you just have to leave that case without further information.

I heard Chris Parker from Birmingham give a talk for students on the O'Nan-Scott Theorem a few years ago, and I remember he did address the question of use of CFSG, and I believe it was exactly as I have described here, but you could write to him to check!

Addendum: Just to emphasize the dependence on the Schreier Conjecture, if that conjecture had been false, then it is conceivable that there could be two simple groups $S,T$ with $T \le {\rm Aut}(G)$ but $T \not\le {\rm Inn}(G)$, in which $T$ leaves no nontrivial proper subgroup of $S$ invariant, in which case the semidirect product $S \rtimes T$ acting on the cosets of $T$ would be a primitive permutation group having $S$ as regular normal subgroup.

• Hi Derek, thanks for your answer. Your summary is exactly in line with how I understand the situation.... It'd be great if someone could give a reference for a full proof of the weak version that didn't use Schreier. (It should be extractable from Dixon & Mortimer, but I haven't the wherewithal to do that just now.) I'm also intrigued as to how seriously people have tried to prove the strong version without using Schreier? If one didn't use Schreier, I wonder what could be said? – Nick Gill Jan 23 '14 at 14:57
• The account in Cameron's book is more concise, but he refers to Dixon and Mortimer for more details. The Schreier Conjecture is used in the proof of Thm 4.7 B (ii) of D&M, and without that you could have groups with a nonabelian regular normal subgroup $N$ in which the point stabilizer had a nonabelian simple subgroup that fixed, and induced outer automorphisms of, all of the simple direct factors of $N$. So you would need to rule that out somehow. – Derek Holt Jan 23 '14 at 15:50
• @Derek: I've inserted the LaTeX command for the wreath symbol, which is just \wr – Jim Humphreys Jan 23 '14 at 17:35
• It is rarely the case in applications that you need the structure of twisted wreath product groups. Usually it is just enough to know that there is a nonabelian regular minimal normal subgroup which is unique. Have you looked at the paper by Liebeck, Praeger and Saxl? – Michael Giudici Jan 28 '14 at 9:06
• From memory there is a full proof of what you are calling the weak version in the paper by Scott, but I don't have it to hand at the moment. – Michael Giudici Jan 28 '14 at 9:10


text:
Embeddings for ultrapowers of the universal UHF-algebra

I've come across a little obstacle and was wondering if anyone could provide a hint/suggestion/help to the following.

First of all, let $\omega$ be some fixed free ultrafilter on $\mathbb{N}$. Let $Q$ be the universal UHF-algebra and $Q_\omega$ be its ultrapower. Let $\tau_\omega$ be the unique trace on $Q_\omega$ and suppose $p$ is some projection in $Q_\omega$ such that $\tau_\omega(p)>0$. Then there exists an integer $k$ and injective $\ast$-homomorphism $\pi \colon Q_\omega \rightarrow p Q_\omega p \otimes \mathbb{M}_k$ fulfilling $\pi(pap)=pap \otimes e_{11}$ with $e_{11}$ being the $(1,1)$'th unit matrix.

So far I have managed to produce $\pi$ in the case where $\tau_\omega(p)m=1$ for some positive integer $m$. Is there any way to reduce the proof to this case?


text:
# Thread: Differential Equation of Tan(x)

1. ## Differential Equation of Tan(x)

Solve this equation :

2. Originally Posted by dhiab
Solve this equation :
If you rearrange it to:

$\frac{dx}{x\tan{x}} = 1dy$

You then get the equation:

$y = \int \frac{1}{x\tan{x}} dx$

Is this any help?