Section Definition and 680 Threads

I am reading Paolo Aluffi's book, Algebra: Chapter 0. I am, at present, focused on Chapter 1, Section 5: Universal Properties. I subsection 5.3 on Quotients, Aluffi writes the following: I am uncertain of the nature of the category that Aluffi is constructing. in particular, do the functions...

In terms of unit material spent my impression is that an I beam uses it most efficiently at least for non-torsional loads? i.e. putting most material far from the axis into the flanges giving higher moments. If so, what are situations when an angle or channel cross sections are preferred in...

I've got a 300 liter water heater that has an electrical element (14 kW). The issue is that the element is positioned in the top third of the heater. I fear the water will become segregated so that only water in the top third will be heated, and the lower section will remain cold. The target...

So I'm working On hydrostatic pressure. The problem #8 in stewart Is a trapezoid with the base above water. 4 \------------------------/ 1(above watee) \ / 1(in watee) \ / --------------- 2 So if the...

As we know RCS is measure of how detectable an object is with a radar. But I don't understand one thing. If we take a sphere as an example, we can see that RCS is the largest in forward direction (forward scattering). If the sphere is to big (radius>>lambda), still this forward scattering is...

I am reading Dummit and Foote, CH 10 Section 10.5, Exact Sequences - Projective, Injective and Flat Modules. As they introduce split sequences, D&F write the following: I am concerned at the following statement: "In this case the module $$ B $$ contains a sub-module $$C'$$ isomorphic to $$C$$...

I have this problem in my review packet and it is making me crazy because it wasn't covered in lecture and the book seems to just skip over it. The final is tomorrow and I don't really have a clue about phase relationship between current and voltage in a simple ac circuit and filtering...

What are three different ways that i can produce a hole with a square cross section in a plate with a thickness of at least 1"?

Hi, I'm doing a project to see the feasibility of transmuting tungsten into platinum on a small scale (maximum output of 1 kg per month). However, some of the cross section data that I need seems to be hard to find. Now because I am new to this, I don't know if its because no body has...

Hi all, I think that the off shell decay dosn't decrease the decay cross section, I mean if a particle X decays into a jet pair has mass for example Mjj ~ 130 GeV, being X has mass M ~ 130 GeV so decay in resonance or on shell into jj or being MX even much greater ~ 2000 GeV , this...

Hi all I was trying to understand the Born-Bethe approximation related to cross sections for atomic and molecular collisions. All the stuffs that i got are explaining in complicated way which am not able to follow. Can anyone explain in simple terms what the theory explains? It will be of...

Homework Statement Find the flux of F=<y,-x,z> through the piece of ρ=2 that lies above z=1 and is oriented up. Homework Equations The Attempt at a Solution S = < x, y, \sqrt{4-x^{2}-y^{2}} > Take Find Sx and Sy, cross them and end up with: dS = <...

Homework Statement Using the Eikonal approximation (1) Determine the expression for the total scattering cross section of a particle in a potential V(r) (2) Using this result, compute the total scattered cross section for the following potential. V(r)= \begin{cases} V_0, \text{for } r < a \\...

What does "cross section area" mean when dealing with stress/strain? Homework Statement For clarification, here is an example problem: A circular steel wire 2 m long must stretch no more than 0.25 cm when a tensile force of 400 N is applied to each end of the wire. What minimum diameter is...

Hi, I am self-teaching Quantum Elctrodynamics, and have come across something which I do not understand. I would appreciate feedback from anyone on this specific issue from Atchison & Hey, "Guage Theories in Particle Physics" pg 238-239: In calculating the u-channel electron-muon...

I am reading Dummit and Foote Section 10.5 Exact Sequences - Projective, Injective and Flat Modules. I need some help in understanding D&F's proof of Proposition 27, Section 10.5, page 386 (see attachment). Proposition 27 reads as follows: (see attachment)...

Homework Statement For a laser, assume single mode operation, only lifetime broadening due to A21 and A1. Write out the rate equations for the atomic densitions N1 and N2 and the flux \Phi. Then there's a diagram where there's 2 stats. Pump rate is P, stimulated emission between 1 and 2 is...

Homework Statement I've been reading through Quantum Optics by Scully and Zubairy and have been stuck in section 6.2 getting from the definition of the atomic inversion given by equation (6.2.20, pg 199) W(t)=\sum_{n}\left[|c_{a,n}(t)|^2-|c_{b,n}(t)|^2\right] to the atomic inversion for the...

I am reading Dummit and Foote Section 10.4: Tensor Products of Modules. I would appreciate some help in understanding Example (8) on page 366 concerning viewing the quotient ring $$R/I $$ as an $$ (R/I, R) $$-bimodule. Example (8) D&F page 370 reads as follows: (see attachment)...

I am reading Dummit and Foote Section 10.4: Tensor Products of Modules. I am currently studying Example 2, page 363 (see attachment) and I am trying to closely relate the example to Theorem 8 and the D&F text on extension of the scalars preceding Theorem 8 on pages 359-362) In Example 2 (see...

I am reading Dummit and Foote, Section 10.4: Tensor Products of Modules. I am studying Corollary 9 and attempting to fully understand the Corollary and it proof. (For details see the attachement page 362 in which Theorem 8 is stated and proved. This is followed by the statement and proof of...

I am reading Dummit and Foote, Section 10 on tensor products of modules. I am at present trying to understand the use of Theorem 6 (D&F, page 354 - see attachment) in Theorem 8 (D&F page 362, see attachment). The proof of Theorem 8 in D&F Chapter 10 (see attachment) reads as follows...

In Dummit and Foote, Section 10.4: Tensor Products of Modules, on pages 359 - 364 (see attachment) the authors deal with a process of 'extension of scalars' of a module, whereby we construct a left $$S$$-module $$ S \oplus_R N $$ from an $$R$$-module $$N$$. In this construction the unital ring...

Dear Forum ： I'm looking for proton-proton inelastic cross section in MeV. There are lots of results in GeV, TeV. But I can not find the results in MeV in ENDF nor TENDL. https://www-nds.iaea.org/exfor/endf.htm ftp://ftp.nrg.eu/pub/www/talys/tendl2011/proton_html/proton.html Please advise...

Homework Statement See the problem attached in this post. Homework Equations See the problem attached in this post. The Attempt at a Solution I set my limits of integration with respect to z axis and got an upper limit of 2 since that's the vertex point/height of the pyramid...

I am reading Dummit and Foote, Example 2 of Section 15.4, page 708. Rewriting the assertions of the example as exercise style questions, EXAMPLE 2, reads as follows: ------------------------------------------------------------------------------------- Let R be any commutative ring with 1 and...

Hi, I've looked under the mathematics textbook section on this site and I can see algebra and geometry but not a single trigonometry book... Is trigonometry normally covered in algebra books something? My trigonometry is really quite terrible and I know the absolute basics so I'm looking for...

Dear Forum: I would like to know where could I find charged particle interaction cross section for oxygen-18? I know I can get some of it from ENDF. https://www-nds.iaea.org/exfor/endf.htm Where can I get reference,like the threshold energy for 18O(p,n)18F is 2.5MeV and its cross...

Homework Statement I want to know the section modulus for a beam which has a UDL of 13.5 kn/m across 10 metres. Its simply supported at both ends. max allowable stress = 255 MPa Homework Equations Z= M / max allowable stress [cm^3] The Attempt at a Solution max bending...

Homework Statement Homework Equations The Attempt at a Solution Hello, Comparing these two problems, 9.6 says to calculate the elastic curve for portion BC. However, in the solution they include the contribution from P which is not contained in portion BC in the representative cut to calculate...

Homework Statement I have a 13 foot long beam supported by a pin at x = 0 feet and a roller at x = 9 feet. There is a triangular distributed load of 50 lb/ft from 0 ft to 9 ft. (Increasing as it approaches 9 ft) At the end of the beam there is a moment of 200 lb-ft counter-clockwise.Homework...

Hi all, Could anyone help for calculating the hadronc production cross section for example for tree level : p p > t t~ process, I try to calculate, but the first problem I meet is a negative value of the matrix element amplitude (and so cross section ) and a negative ## \hat{t} ## Mandelstam...

I'm currently doing some research at the moment for my professor, and he gave me a list of things to look at. Before he had me calculate the "real center of mass cross section." Now, starting at point 0 for publication, he's having me go through and do some other things. In his list of things to...

Dear Forum : I'm reading a cross section data of 20MeV proton + 16O reaction from ICRU 63. ( as attachment and link http://ppt.cc/qf-9 ) The total cross setion of (p,n) reaction is 4.372mb However, the cross section of emitting neutron of energy between 0 to 1.5MeV is 0.91 mb / MeV. The...

I am reading Dummit and Foote Section 3.1: Quotient Groups and Homomorphisms. Exercise 17 in Section 3.1 (page 87) reads as follows: ------------------------------------------------------------------------------------------------------------- Let G be the dihedral group od order 16. G = <...

I am studying Dummit and Foote Section 15.2. I am trying to understand the proof of Proposition 19 Part (5) on page 682 (see attachment) Proposition 19 Part (5) reads as follows...

I am studying Dummit and Foote Section 15.2. I am trying to understand the proof of Proposition 19 Part (5) on page 682 (see attachment) Proposition 19 Part (5) reads as follows...

Dummit and Foote Section 15.1, Exercise 24 reads as follows: --------------------------------------------------------------------------------------------------------- Let V = \mathcal{Z} (xy - z) \subseteq \mathbb{A}^3 . Prove that V is isomorphic to \mathbb{A}^2 and provide an explicit...

Dummit and Foote Section 15.1, Exercise 24 reads as follows: --------------------------------------------------------------------------------------------------------- Let V = \mathcal{Z} (xy - z) \subseteq \mathbb{A}^3 . Prove that $$ V $$ is isomorphic to \mathbb{A}^2 and provide an...

Dummit and Foote (D&F), Ch15, Section 15.1, Exercise 15 reads as follows: ---------------------------------------------------------------------------------------------------- If k = \mathbb{F}_2 and V = \{ (0,0), (1,1) \} \subset \mathbb{A}^2 , show that \mathcal{I} (V) is the...

Dummit and Foote (D&F), Ch15, Section 15.1, Exercise 15 reads as follows: ---------------------------------------------------------------------------------------------------- If k = \mathbb{F}_2 and V = \{ (0,0), (1,1) \} \subset \mathbb{A}^2 , show that \mathcal{I} (V) is the...

I am reading Dummit and Foote Ch 15, Commutative Rings and Algebraic Geometry. In Section 15.1 Noetherian Rings and Affine Algebraic Sets, Example 2 on page 660 reads as follows: (see attachment) ----------------------------------------------------------------------------------------------...

I am reading Dummit and Foote Ch 15, Commutative Rings and Algebraic Geometry. In Section 15.1 Noetherian Rings and Affine Algebraic Sets, Example 3 on page 660 reads as follows: (see attachment) ----------------------------------------------------------------------------------------------...

Homework Statement A particularly beautiful note reaching your ear from a rare Stradivarius violin has a wavelength of 39.1 cm. The room is slightly warm, so the speed of sound is 344 m/s. If the string's linear density is 0.560g/m and the tension is 160N , how long is the vibrating section...

I am reading Nicholson: Introduction to Abstract Algebra, Section 6.3 Splitting Fields. Example 1 reads as follows: (see attachment) -------------------------------------------------------------------------------------------------- Example 1. Find an extension E \supseteq \mathbb{Z}_2 in...

I am reading Nicholson: Introduction to Abstract Algebra, Section 6.2 - Algebraic Extensions. Example 15 on page 282 (see attachment) reads as follows: --------------------------------------------------------------------------------------------------------------------------------- Example 15...

xample 13 from Nicholson: Introduction to Abstract Algebra, Section 6.2, page 282 reads as follows: (see attachment) ------------------------------------------------------------------------------------------------------------------ Example 13: If u = \sqrt[3]{2} show that \mathbb{Q}(u) =...

I am reading Nicholson: Introduction to Abstract Algebra, Section 6.2 Algebraic Extensions. Example 14 on page 282 (see attachment) reads as follows: ------------------------------------------------------------------------------------------- Example 14. Let E \supseteq F be fields and let...

Homework Statement (a) Design a π section symmetrical attenuator to provide a voltage attenuation of 15 dB and a characteristic impedance of 600 Ω. (b) Construct and test the π section attenuator. Measure and record the input & output voltages of the attenuator and determine the...

I am reading Nicholson: Introduction to Abstract Algebra, Section 6.2 Algebraic Extensions. Example 13 on page 282 (see attachment) reads as follows: "If u = \sqrt[3]{2} show that \mathbb{Q}(u) = \mathbb{Q}(u^2) " In the third line of the explanation - see page 282 of attachment - we...