text:
Circuit Theory/2Initially Excited

full circuit with charging section, discharging section and two switches
charging circuit (sw1 closed, sw2 open)
discharging circuit (sw1 open, sw closed)

Charging

The source voltage could be DC. In this case the initial capacitor voltage is going to be Vs and the derivative of the voltage is going to be zero.

The source voltage could be AC in which case the exact time or more precise phase angle of the switch over is going to determine Vs and the derivative of the voltage values.

Discharging

The initial inductor current is going to be zero and derivative is going to be zero.

After a long time nothing will be going on. The steady state particular solution is going to be zero.

The current is going to be the same throughout. Goal should be to find an expression for the current. However, any voltage could also be found and then the current. Doesn't matter.

In summary, there will be no initial current in the discharging circuit, there will be a voltage across the capacitor. The inductor is going to act like an open at t=0+ so this entire voltage will appear across the inductor and instantly (between t=0- and t=0+) cause a change in the derivative of the current through the inductor.

The equation for the current for first order circuits would be:

${\displaystyle V_{c}(t)=Ae^{-{\frac {t}{\tau }}}}$

However, this is a 2nd order circuit (with both capacitors and inductors), thus:

${\displaystyle V_{c}(t)=Ae^{-st}}$

The complex frequency s has replace tau. The time constant concept disappears. An intuitive interpretation of s exists, but is more complicated. Remember:

${\displaystyle s=\sigma +j\omega }$

The differential equation

Go back to the complex frequency domain to find the time constant. Start with the loop equation:

${\displaystyle V_{c}+V_{r}+V_{L}=0}$

Substitute terminal equation for Vr and VL but not for Vc because want to avoid an integral:

${\displaystyle V_{c}+i(t)R+L{di(t) \over dt}=0}$

From the terminal relation of a capacitor, know that:

${\displaystyle i(t)=C{dV_{c} \over dt}}$

So substituting in, have a differential equation for Vc:

${\displaystyle V_{c}+RC{dV_{c} \over dt}+LC{d^{2}V_{c} \over dt^{2}}=0}$

Can see the second order now.

Finding the time constant or zeros

A zero is like a time constant, but there are more than one zero's possible. They might each influence the circuit differently.

Need to find Vc. The particular solution (steady state) is going to be zero. (Where the name zero comes from).

The homogeneous solution is going to be of this form:

${\displaystyle V_{c}(t)=Ae^{st}}$

Plugging into the differential equation:

${\displaystyle Ae^{st}+RCAse^{st}+LCAs^{2}e^{st}=0}$

Dividing out Aest have:

${\displaystyle 1+RCs+LCs^{2}=0}$
${\displaystyle s^{2}+{\frac {Rs}{L}}+{\frac {1}{LC}}=0}$

The zero's of this equation are:

${\displaystyle s_{1,2}=-{\frac {R}{2L}}\pm {\sqrt {({\frac {R}{2L}})^{2}-{\frac {1}{LC}}}}=-(\sigma \pm j\omega )}$

From your experience with quadratics you know that 1,2 come in three types of pairs:

Pair Type Revised equation for Vc interpretation
real, not equal ${\displaystyle A_{1}e^{s_{1}t}+A_{2}e^{s_{2}t}}$ overdamped
equal ${\displaystyle A_{1}e^{st}+A_{2}te^{st}}$ critically damped
complex ${\displaystyle e^{-\sigma t}(A_{1}\cos \omega t+A_{2}\sin \omega t}$) underdamped

The revised equations (guesses) come from math proofs, and some courses derive these from Laplace transformations. Just remember this table.

There are two storage elements with two initial conditions each, so there could be four equations and four constants. A1 and A2 are two of them.

for undamped (blue), under-damped (green), critically damped (red), and over-damped (cyan) cases

To understand damping, think of a car driving over a curb. Undamped means the car bounces up and down for ever. Under-damped is what happens when the shocks start leaking (car bounces up and down for a while ... like riding in a boat). Critically damped is why people hate sports cars (hurts their bottom). An over damped response in a car gives feedback to a person driving a car. Circuits like that above have been used to model shock obsorbers. These circuits are called analog computers.

Almost Resonance (Underdamped)

Impact of R and L on resonance

The most interesting of the above three is Underdamped. Something like resonance occurs.

Satellites expand and contract depending upon which side is facing the sun or the 2.7 kelvin of space. Repeated expansion and contraction creates an oscillation that needs to be damped out by a motor that flings a weight around that dampens the vibration. Otherwise everything that can wiggle will wiggle more and more until parts begin to break off. (A car on earth is no different).

Most control circuit input is an unbalanced situation. The output models the desired behavior. The correcting (feedback control) circuits (not talked being studied here) kick in to make sure the actual behavior matches the desired behavior. Most often the objective is to design a critically damped circuit, rather than merely predict what is going to happen.

Resonance (or close to it) occurs when s1 and s2 are purely imaginary.

${\displaystyle s_{1,2}=-{\frac {R}{2L}}\pm {\sqrt {({\frac {R}{2L}})^{2}-{\frac {1}{LC}}}}=-(\sigma \pm j\omega )}$

So:

${\displaystyle \sigma ={\frac {R}{2L}}}$

What is interesting is the radical term. It determines the nature of the 2nd order response.

Pure resonance occurs when the real part of the quadratic solution is zero which would only happen when R << L.

Crystals are modeled with a capacitor in parallel with a series RLC network

The contents of the radical don't have to be zero, only the real part has to be zero in order to be overdamped:

${\displaystyle j\omega ={\sqrt {({\frac {R}{2L}})^{2}-{\frac {1}{LC}}}}}$
${\displaystyle j\omega =j{\sqrt {{\frac {1}{LC}}-({\frac {R}{2L}})^{2}}}}$

The j's cancel, and we have already assumed R << L, so:

${\displaystyle \omega ={\sqrt {{\frac {1}{LC}}-({\frac {R}{2L}})^{2}}}}$

Which called the damped, ringing, or undamped frequency.

When R << L, then this turns into the resonate frequency:

${\displaystyle \omega ={\sqrt {\frac {1}{LC}}}}$

Mechanical engineers try to predict ω for a given structure and materials. It could be a building or motor mount. The math proves that ω exists. Finding the exact ω usually requires testing. Instead of capacitors and inductors, mechanical engineers model everything with springs and dampers. Finding ω can be hard. This why there is an entire industry of shake tables.

The spring and damper values in a structure determine how narrow or muted the resonance is. When designing a building, mechanical engineers want the envelope to be as wide and flat (muted) as possible. When designing a notch filter the electrical engineer's design goal is the opposite: a tall and narrow envelope. Electrical engineers call this quality.

Most oscillators today are built from crystals.


text:
Question

# Prove that $2\cos \dfrac{\pi }{{13}}\cos \dfrac{{9\pi }}{{13}} + \cos \dfrac{{3\pi }}{{13}} + \cos \dfrac{{5\pi }}{{13}} = 0$

Verified
129.6k+ views
Hint: Here we use the property of trigonometry $2\cos A\cos B = \cos (A + B) + \cos (A - B)$ to solve first part of the equation and then we group together two pairs of values in cos on which we apply the formula $\cos A + \cos B = 2\cos (\dfrac{{A + B}}{2})\cos (\dfrac{{A - B}}{2})$.

First we solve $2\cos \dfrac{\pi }{{13}}\cos \dfrac{{9\pi }}{{13}}$by using the identity $2\cos A\cos B = \cos (A + B) + \cos (A - B)$
Where $A = \dfrac{\pi }{{13}},B = \dfrac{{9\pi }}{{13}}$
So, by substituting the values we can write.
$\Rightarrow 2\cos \dfrac{\pi }{{13}}\cos \dfrac{{9\pi }}{{13}} = \cos (\dfrac{\pi }{{13}} + \dfrac{{9\pi }}{{13}}) + \cos (\dfrac{\pi }{{13}} - \dfrac{{9\pi }}{{13}})$
Take LCM of the angles within the bracket.
$\Rightarrow 2\cos \dfrac{\pi }{{13}}\cos \dfrac{{9\pi }}{{13}} = \cos (\dfrac{{\pi + 9\pi }}{{13}}) + \cos (\dfrac{{\pi - 9\pi }}{{13}}) \\ \Rightarrow 2\cos \dfrac{\pi }{{13}}\cos \dfrac{{9\pi }}{{13}} = \cos (\dfrac{{10\pi }}{{13}}) + \cos (\dfrac{{ - 8\pi }}{{13}}) \\$
Since we know cosine is an even function, which means that $\cos ( - x) = \cos (x)$
Here value of $x = - \dfrac{{8\pi }}{{13}}$
So, substitute the value of $\cos ( - \dfrac{{8\pi }}{{13}}) = \cos (\dfrac{{8\pi }}{{13}})$in the equation.
$\Rightarrow 2\cos \dfrac{\pi }{{13}}\cos \dfrac{{9\pi }}{{13}} = \cos (\dfrac{{10\pi }}{{13}}) + \cos (\dfrac{{8\pi }}{{13}})$ … (i)
Now we substitute the value from equation (i) on the left hand side of the equation.
$\Rightarrow 2\cos \dfrac{\pi }{{13}}\cos \dfrac{{9\pi }}{{13}} + \cos \dfrac{{3\pi }}{{13}} + \cos \dfrac{{5\pi }}{{13}} = \cos \dfrac{{10\pi }}{{13}} + \cos \dfrac{{8\pi }}{{13}} + \cos \dfrac{{3\pi }}{{13}} + \cos \dfrac{{5\pi }}{{13}}$ … (ii)
Now we use the formula $\cos A + \cos B = 2\cos (\dfrac{{A + B}}{2})\cos (\dfrac{{A - B}}{2})$in two separate pairs.
We make pairs such that when we add the numerator it gets cancelled by the denominator and gives us an angle whose cosine is known to us.
We make a pair of $\left( {\cos \dfrac{{10\pi }}{{13}} + \cos \dfrac{{3\pi }}{{13}}} \right)$and other pair of $\left( {\cos \dfrac{{8\pi }}{{13}} + \cos \dfrac{{5\pi }}{{13}}} \right)$
First we solve $\left( {\cos \dfrac{{10\pi }}{{13}} + \cos \dfrac{{3\pi }}{{13}}} \right)$, where $A = \dfrac{{10\pi }}{{13}},B = \dfrac{{3\pi }}{{13}}$
$\Rightarrow \left( {\cos \dfrac{{10\pi }}{{13}} + \cos \dfrac{{3\pi }}{{13}}} \right) = 2\cos \left( {\dfrac{{\dfrac{{10\pi }}{{13}} + \dfrac{{3\pi }}{{13}}}}{2}} \right)\cos \left( {\dfrac{{\dfrac{{10\pi }}{{13}} - \dfrac{{3\pi }}{{13}}}}{2}} \right)$
Taking LCM in the numerator of the angles inside the bracket.
$\Rightarrow \left( {\cos \dfrac{{10\pi }}{{13}} + \cos \dfrac{{3\pi }}{{13}}} \right) = 2\cos \left( {\dfrac{{\dfrac{{10\pi + 3\pi }}{{13}}}}{2}} \right)\cos \left( {\dfrac{{\dfrac{{10\pi - 3\pi }}{{13}}}}{2}} \right) \\ \Rightarrow \left( {\cos \dfrac{{10\pi }}{{13}} + \cos \dfrac{{3\pi }}{{13}}} \right) = 2\cos \left( {\dfrac{{\dfrac{{13\pi }}{{13}}}}{2}} \right)\cos \left( {\dfrac{{\dfrac{{7\pi }}{{13}}}}{2}} \right) \\$
Cancelling the same term from numerator and denominator in the angle.
$\Rightarrow \left( {\cos \dfrac{{10\pi }}{{13}} + \cos \dfrac{{3\pi }}{{13}}} \right) = 2\cos \left( {\dfrac{\pi }{2}} \right)\cos \left( {\dfrac{{7\pi }}{{26}}} \right)$
Since we know $\cos \dfrac{\pi }{2} = 0$
$\Rightarrow \left( {\cos \dfrac{{10\pi }}{{13}} + \cos \dfrac{{3\pi }}{{13}}} \right) = 2 \times 0 \times \cos \left( {\dfrac{{7\pi }}{{26}}} \right) = 0$ … (iii)
Now we solve$\left( {\cos \dfrac{{8\pi }}{{13}} + \cos \dfrac{{5\pi }}{{13}}} \right)$, where $A = \dfrac{{8\pi }}{{13}},B = \dfrac{{5\pi }}{{13}}$
$\Rightarrow \left( {\cos \dfrac{{8\pi }}{{13}} + \cos \dfrac{{5\pi }}{{13}}} \right) = 2\cos \left( {\dfrac{{\dfrac{{8\pi }}{{13}} + \dfrac{{5\pi }}{{13}}}}{2}} \right)\cos \left( {\dfrac{{\dfrac{{8\pi }}{{13}} - \dfrac{{5\pi }}{{13}}}}{2}} \right)$
Taking LCM in the numerator of the angles inside the bracket.
$\Rightarrow \left( {\cos \dfrac{{8\pi }}{{13}} + \cos \dfrac{{5\pi }}{{13}}} \right) = 2\cos \left( {\dfrac{{\dfrac{{8\pi + 5\pi }}{{13}}}}{2}} \right)\cos \left( {\dfrac{{\dfrac{{8\pi - 5\pi }}{{13}}}}{2}} \right)$
$\Rightarrow \left( {\cos \dfrac{{8\pi }}{{13}} + \cos \dfrac{{5\pi }}{{13}}} \right) = 2\cos \left( {\dfrac{{\dfrac{{13\pi }}{{13}}}}{2}} \right)\cos \left( {\dfrac{{\dfrac{{3\pi }}{{13}}}}{2}} \right)$
Cancelling the same term from numerator and denominator in the angle.
$\Rightarrow \left( {\cos \dfrac{{8\pi }}{{13}} + \cos \dfrac{{5\pi }}{{13}}} \right) = 2\cos \left( {\dfrac{\pi }{2}} \right)\cos \left( {\dfrac{{3\pi }}{{26}}} \right)$
Since we know $\cos \dfrac{\pi }{2} = 0$
$\Rightarrow \left( {\cos \dfrac{{8\pi }}{{13}} + \cos \dfrac{{5\pi }}{{13}}} \right) = 2 \times 0 \times \cos \left( {\dfrac{{3\pi }}{{26}}} \right) = 0$ … (iv)
Now we substitute the values from equation (iii) and (iv) in equation (ii)
$\Rightarrow \cos \dfrac{{10\pi }}{{13}} + \cos \dfrac{{8\pi }}{{13}} + \cos \dfrac{{3\pi }}{{13}} + \cos \dfrac{{5\pi }}{{13}} = 0 + 0 = 0$
which is equal to the right hand side of the equation.
Hence Proved.

Note: Students many times make mistake of writing the value of $\cos ( - \dfrac{{8\pi }}{{13}}) = - \cos (\dfrac{{8\pi }}{{13}})$ because they think that negative sign comes out of the angle which is wrong, we always classify if the function is an odd or even function and then find if sign vanishes or comes out. In this case cosine is an even function, so negative sign vanishes.

format: plain


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# Force SSL with Name-based Virtual Host (possibly with .htaccess)

I have several subdomains served by Apache using Name-based Virtual Hosts. The way it's set up is that I have one directory holding all the site's pages, but it's accessible using several subdomains (a web application with a different theme for each customer). For each subdomain I have a symlink to the main dir, which matches the name-based virtual host mapping. So I just use $_SERVER['HTTP_HOST'] in my php scripts to determine which theme to serve, but the code base is the same. The problem is that the sites need to be SSL and I have trouble figuring out how to force it that way. I have seen this example for .htaccess: RewriteEngine On RewriteCond %{SERVER_PORT} 80 RewriteRule ^(.*)$ https://www.example.com/$1 [R,L]  But the trouble is that it requires you to hard code the domain www.example.com. How can I make it go to SSL version of whatever domain is actually requested, instead of hard coding it? I have seen {HTTP_HOST} variable getting used in RewriteCond portion of .htaccess, but it does not appear to work when I attempt to use it in RewriteRule. Tried the following and it gave me a 500 error: RewriteEngine On RewriteCond %{SERVER_PORT} 80 RewriteRule ^(.*)$ https://{HTTP_HOST}/$1 [R,L]  ## 2 Answers The ability to use those same conditional variables in the substitution string makes this a simple fix. Change RewriteEngine On RewriteCond %{SERVER_PORT} 80 RewriteRule ^(.*)$ https://www.example.com/$1 [R,L]  to RewriteEngine On RewriteCond %{SERVER_PORT} 80 RewriteRule ^(.*)$ https://%{HTTP_HOST}/\$1 [R,L]

Apache SSL dont work with name virtual host.. see:

http://wiki.apache.org/httpd/NameBasedSSLVHostsWithSNI

• This is a poor answer, your link shows exactly how Apache DOES work with named virtual hosts using SNI. – gparent May 9 '12 at 15:57


text: Consider the rotation $$T$$ of space with $$T(0,0,0)=(0,0,0)$$, $$T(6,6,3)=(4,-1,8)$$ and $$T(-6,3,6)=(4,8,-1)$$. If $$T(3,9,12)=(a,b,c)$$, find $$a+b+c$$.


text:
Multiple spatial scales in engineering and atmospheric low Mach number flows
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 39 (2005) no. 3, pp. 537-559.

The first part of this paper reviews the single time scale/multiple length scale low Mach number asymptotic analysis by Klein (1995, 2004). This theory explicitly reveals the interaction of small scale, quasi-incompressible variable density flows with long wave linear acoustic modes through baroclinic vorticity generation and asymptotic accumulation of large scale energy fluxes. The theory is motivated by examples from thermoacoustics and combustion. In an almost obvious way specializations of this theory to a single spatial scale reproduce automatically the zero Mach number variable density flow equations for the small scales, and the linear acoustic equations with spatially varying speed of sound for the large scales. Following the same line of thought we show how a large number of well-known simplified equations of theoretical meteorology can be derived in a unified fashion directly from the three-dimensional compressible flow equations through systematic (low Mach number) asymptotics. Atmospheric flows are, however, characterized by several singular perturbation parameters that appear in addition to the Mach number, and that are defined independently of any particular length or time scale associated with some specific flow phenomenon. These are the ratio of the centripetal acceleration due to the earth’s rotation vs. the acceleration of gravity, and the ratio of the sound speed vs. the rotational velocity of points on the equator. To systematically incorporate these parameters in an asymptotic approach, we couple them with the square root of the Mach number in a particular distinguished so that we are left with a single small asymptotic expansion parameter, $\epsilon$. Of course, more familiar parameters, such as the Rossby and Froude numbers may then be expressed in terms of $\epsilon$ as well. Next we consider a very general asymptotic ansatz involving multiple horizontal and vertical as well as multiple time scales. Various restrictions of the general ansatz to only one horizontal, one vertical, and one time scale lead directly to the family of simplified model equations mentioned above. Of course, the main purpose of the general multiple scales ansatz is to provide the means to derive true multiscale models which describe interactions between the various phenomena described by the members of the simplified model family. In this context we will summarize a recent systematic development of multiscale models for the tropics (with Majda).

DOI : https://doi.org/10.1051/m2an:2005022
Classification : 34E13,  76B99,  76Q05,  86A10
Mots clés : low Mach number flows, multiple scales asymptotics, atmospheric flows
@article{M2AN_2005__39_3_537_0,
author = {Klein, Rupert},
title = {Multiple spatial scales in engineering and atmospheric low Mach number flows},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
pages = {537--559},
publisher = {EDP-Sciences},
volume = {39},
number = {3},
year = {2005},
doi = {10.1051/m2an:2005022},
zbl = {1130.76326},
mrnumber = {2157149},
language = {en},
url = {www.numdam.org/item/M2AN_2005__39_3_537_0/}
}
Klein, Rupert. Multiple spatial scales in engineering and atmospheric low Mach number flows. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 39 (2005) no. 3, pp. 537-559. doi : 10.1051/m2an:2005022. http://www.numdam.org/item/M2AN_2005__39_3_537_0/

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[2] G.K. Batchelor, The conditions for dynamical similarity of motions of a frictionless perfect gas atmosphere. Quart. J. Roy. Meteorol. Soc. 79 (1953) 224-235.

[3] J.R. Biello and A.J. Majda, A new multiscale model for the madden julian oscillation. J. Atmosphere Sci. 62 (2005) in press.

[4] N. Botta, R. Klein and A. Almgren, Asymptotic analysis of a dry atmosphere. ENUMATH, Jyväskylä, Finland (1999). | Zbl 1057.86507

[5] C. Bretherton and A. Sobel. The gill model and the weak temperature gradient (wtg) approximation. J. Atmosphere Sci. 60 (2003) 451-460.

[6] J.G. Charney, A note on large-scale motions in the tropics. J. Atmosphere Sci. 20 (1963) 607-609.

[7] S.B. Dorofeev, V.P. Sidorov, A.E. Dvoinishnikov and W. Breitung, Deflagration to detonation transition in large confined volume of lean hydrogen-air mixtures. Combustion & Flame 104 (1996) 95-110.

[8] D.R. Durran, Improving the anelastic approximation. J. Atmosphere Sci. 46 (1989) 1453-1461.

[9] A.E. Gill, Some simple solutions for heat-induced tropical circulation. Quart. J. Roy. Meteorol. Soc. 87 (1980) 447-462.

[10] I.M. Held and B.J. Hoskins, Large-scale eddies and the general circulation of the troposphere. Adv. in Geophysics 28 (1985) 3-31.

[11] J.C.R. Hunt, K.J. Richards and P.W.M. Brighton, Stably stratified shear flow over low hills. Quart. J. Roy. Meteorol. Soc. 114 (1988) 859-886.

[12] R. Klein, Semi-implicit extension of a godunov-type scheme based on low mach number asymptotics I: One-dimensional flow. J. Comput. Phys. 121 (1995) 213-237. | Zbl 0842.76053

[13] R. Klein, Asymptotic analyses for atmospheric flows and the construction of asymptotically adaptive numerical methods. ZAMM 80 (2000) 765-777. | Zbl 1050.76056

[14] R. Klein, An applied mathematical view of theoretical meteorology, in Applied Mathematics Entering the 21st Century; Invited talks from the ICIAM 2003 Congress, SIAM Proceedings in Applied Mathematics 116 (2004). | MR 2296270 | Zbl pre05263216

[15] R. Klein, Multiple scales asymptotics for atmospheric flows, in Proceedings of the 4th European Conference on Mathematics, Stockholm, Sweden (2004). | Zbl 1137.86308

[16] R. Klein and S. Vater, Mathematical modelling in climate research. Technical report, Freie Universität, Berlin, Germany (2005).

[17] H. Lamb, Hydrodynamics. Dover Publishers, New York (1981).

[18] F. Lipps and R. Hemler, A scale analysis of deep moist convection and some related numerical calculations. J. Atmosphere Sci. 39 (1982) 2192-2210.

[19] F. Lipps and R. Hemler, Another look at the scale analysis of deep moist convection. J. Atmosphere Sci. 42 (1985) 1960-1964.

[20] R. Madden and P. Julian, Description of global cell circulation cells in the tropics with a 40-50 day period. J. Atmosphere Sci. 29 (1972) 1109-1123.

[21] A.J. Majda and J.A. Biello, A multiscale model for tropical intraseasonal oscillations. Proc. Natl. Acad. Sci. USA 101 (2004) 4736-4741. | Zbl 1063.86004

[22] A. Majda and R. Klein, Systematic multi-scale models for the tropics. J. Atmosphere Sci. 60 (2003) 393-408.

[23] A. Majda and J. Sethian, The derivation and numerical solution of the equations for zero Mach number combustion. Combust. Sci. Tech. 42 (1985) 185-205.

[24] T. Matsuno, Quasi-geostrophic motions in the equatorial area. J. Meteorol. Soc. Jap. 44 (1966) 25-43.

[25] T.M.J. Newley, H.J. Pearson and J.C.R. Hunt, Stably stratified rotating flow through a group of obstacles. Geophys. Astrophys. Fluid Dynam. 58 (1991) 147-171.

[26] Y. Ogura and N.A. Phillips, Scale analysis of deep moist convection and some related numerical calculations. J. Atmosphere Sci. 19 (1962) 173-179.

[27] J. Oliger and A. Sundstroem, Theoretical and practical aspects of some initial boundary value problems in fluid dynamics. NASA STI/Recon Technical Report N 77 25465 (November 1976). | Zbl 0397.35067

[28] J. Pedlosky, Ed., Geophysical Fluid Dynamics. Springer, Berlin, Heidelberg, New York, 2nd edition (1987). | Zbl 0429.76001

[29] V. Petoukhov, A. Ganopolski, V. Brovkin, M. Claussen, A. Eliseev, C. Kubatzki and S. Rahmstorf, Climber-2: A climate system model of intermediate complexity. Part I: Model description and performance for the present climate. Climate Dynamics 16 (2000) 1-17.

[30] A.S. Worlikar and O.M. Knio, Numerical study of oscillatory flow and heat transfer in a loaded thermoacoustic stack. Numer. Heat Transfer 35 (1999) 49-65.

[31] A.S. Worlikar, O.M. Knio and R. Klein, Numerical simulation of a thermoacoustic refrigerator. II: stratified flow around the stack. J. Comput. Fluids 144 (1998) 299-324. | Zbl 0920.76061

[32] Th. Schneider, N. Botta, R. Klein and K.J. Geratz, Extension of finite volume compressible flow solvers to multi-dimensional, variable density zero Mach number flow. J. Comput. Phys. 155 (1999) 248-286. | Zbl 0968.76054

[33] Th. Schneider, R. Klein, E. Besnoin and O. Knio, Computational analysis of a thermoacoustic refrigerator, in Proceedings of the joint EAA/ASA meeting (March 1999).

[34] S. Schochet, The mathematical theory of low mach number flows. ESAIM: M2AN 39 (2005) 441-458. | Numdam | Zbl 1094.35094

[35] V. Smiljanovski, V. Moser and R. Klein, A capturing-tracking hybrid scheme for deflagration discontinuities. J. Combustion Theory Modeling 1 (1997) 183-215. | Zbl 0956.76070

[36] A.A. Sobel, J. Nilsson and L. Polvani, The weak temperature gradient approximation and balanced tropical moisture waves. J. Atmosphere Sci. 58 (2001) 3650-3665.

[37] P.J. Webster, Response of the tropical atmosphere to local steady forcing. Monthly Weather Review 100 (1972) 518-541.

[38] A.A. White, A view of the equations of meteorological dynamics and various approximations, in Large Scale Atmosphere-Ocean Dynamics I: Analytical Methods and Numerical Models. J. Norbury and I. Roulstone, Eds., Cambridge University Press (2002). | MR 1947568 | Zbl 1036.86001

[39] G.B. Whitham, Linear and Non Linear Waves. John Wiley (1974). | Zbl 0373.76001

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text:
<lift:loc locid="stock.discuss"></lift:loc>
Curve Flows and Solitonic Hierarchies Generated by (Semi) Riemannian Metrics
2006.08.09
http://arxiv.org/abs/math-ph/0608024
We investigate bi-Hamiltonian structures and related mKdV hierarchy of solitonic equations generated by (semi) Riemannian metrics and curve flow of non-stretching curves. The corresponding nonholonomic tangent space geometry is defined by canonically induced nonlinear connections, Sasaki type metrics and linear connections. One yields couples of generalized sine-Gordon equations when the corresponding geometric curve flows result in hierarchies on the tangent bundle described in explicit form by nonholonomic wave map equations and mKdV analogs of the Schrodinger map equation.
Discussions
• Pls. be polite and constructive.
• You can input La|TeX for math formulas. E.g. $$x = {-b \pm \sqrt{b^2-4ac} \over 2a}$$
• Any attachment files should still be uploaded to arXiv.org


text:
# OEBuildSidechains¶

Attention

This is a preliminary API until 2020.Apr and may be improved based on user feedback. It is currently available in C++ and Python.

These functions build out partial sidechains using a rotamer library. They return false if all of the rotamers of a given residue clashes. The rotamer chosen is the one with the best interaction energy with the surrounding environment without taking solvent effects into account. If multiple sidechains are built that interact with one another, the energy chosen is the best collective interaction energy.

Note

Explicit hydrogens are not added to the structure during the build processes.

bool OEBuildSidechains(OEChem::OEMolBase &mol,
const unsigned rotamerLibrary=OEBio::OERotamerLibrary::Default,
const float rotamerCoverage=100.0,
const bool deleteClashingSolvent=true)

This function takes a molecule and build any partial sidechain of standard protein residues determined using the OEGetPartialResidues function. A rotamer library from the OERotamerLibrary namespace can be selected, otherwise the default library from that namespace is chosen. The number of rotamers used can be reduced by picking a rotamer coverage below the default 100%. The percentage is calculated cumulatively, so the most probably rotamer is tested first, up to the rotamer that takes it above the percentage limit. The function by default deletes water molecules that clash with the built out sidechains, but this behavior can be turned off. The function will return false if a sidechain can not be built due to clashes.

bool OEBuildSidechains(OEChem::OEMolBase &mol,
const std::vector<OEChem::OEResidue> residues,
const unsigned rotamerLibrary=OEBio::OERotamerLibrary::Default,
const float rotamerCoverage=100.0,
const bool deleteClashingSolvent=true)

This function works identically to the above function, but only builds the residues identified in the vector residues.


text:
# American Institute of Mathematical Sciences

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October  2010, 6(4): 911-927. doi: 10.3934/jimo.2010.6.911

## Analysis of renewal input bulk arrival queue with single working vacation and partial batch rejection

 1 School of Computer Application, KIIT University, Bhubaneswar - 751 024, India 2 Department of Applied Mathematics, Andhra University, Visakhapatnam - 530 003, India

Received  December 2009 Revised  July 2010 Published  September 2010

This paper analyzes a finite buffer bulk arrival queueing system with a single working vacation and partial batch rejection in which the inter-arrival and service times are, respectively, arbitrarily and exponentially distributed. Using the supplementary variable and the imbedded Markov chain techniques, we obtain the system length distributions at pre-arrival and arbitrary epochs. We also present Laplace-Stiltjes transform of the actual waiting time distribution in the system. Finally, several performance measures and a variety of numerical results in the form of tables and graphs are discussed.
Citation: Veena Goswami, Pikkala Vijaya Laxmi. Analysis of renewal input bulk arrival queue with single working vacation and partial batch rejection. Journal of Industrial & Management Optimization, 2010, 6 (4) : 911-927. doi: 10.3934/jimo.2010.6.911
##### References:
 [1] Y. Baba, Analysis of $GI$/$M$/$1$ queue with multiple working vacations,, Oper. Res. Lett., 33 (2005), 201.  doi: 10.1016/j.orl.2004.05.006.  Google Scholar [2] A. D. Banik, U. C. Gupta and S. S. Pathak, On the $GI$/$M$/$1$/$N$ queue with multiple working vacations - Anaytic analysis and computation,, Appl. Math. Model., 31 (2007), 1701.  doi: 10.1016/j.apm.2006.05.010.  Google Scholar [3] P. J. Burke, Delays in single-server queues with batch input,, Oper. Res., 23 (1975), 830.  doi: 10.1287/opre.23.4.830.  Google Scholar [4] K. C. Chae, D. E. Lim and W. S. Yang, The $GI$/$M$/$1$ queue and the $GI$/$Geo$/$1$ queue both with single working vacation,, Performance Evaluaton, 68 (2009), 356.  doi: 10.1016/j.peva.2009.01.005.  Google Scholar [5] K. C. Chae, S. M. Lee and H. W. Lee, On stochastic decomposition in the $GI$/$M$/$1$ queue with single exponential vacation,, Oper. Res. Lett., 34 (2006), 706.  doi: 10.1016/j.orl.2005.11.006.  Google Scholar [6] B. T. Doshi, Queueing systems with vacations - A survey,, Queueing Syst., 1 (1986), 29.  doi: 10.1007/BF01149327.  Google Scholar [7] B. T. Doshi, Single server queues with vacations,, Stochastic Analysis of Computer and Communication Systems, (1990), 217.   Google Scholar [8] F. Karaesmen and S. M. Gupta, The finite capacity $GI$/$M$/$1$ with server vacations,, Journal of the Operational Research Society, 47 (1996), 817.   Google Scholar [9] G. Latouche and V. Ramaswami, "Introduction to Matrix Analytic Methods in Stochastic Modelling,", SIAM $&$ ASA, (1999).   Google Scholar [10] J. H. Li, N. S. Tian and Z. Y. Ma, Performance analysis of $GI$/$M$/$1$ queue with working vacations and vacation interruption,, Appl. Math. Model., 32 (2008), 2715.  doi: 10.1016/j.apm.2007.09.017.  Google Scholar [11] W. Liu, X. Xu and N. Tian, Some results on the M/M/1 queue with working vacations,, Oper. Res. Lett., 35 (2007), 595.  doi: 10.1016/j.orl.2006.12.007.  Google Scholar [12] K. Sikdar, U. C. Gupta and R. K. Sharma, The analysis of a finite-buffer general input queue with batch arrival and exponential multiple vacations,, Int. J. Oper. Res., 3 (2008), 219.  doi: 10.1504/IJOR.2008.016162.  Google Scholar [13] L. D. Servi and S. G. Finn, $M$/$M$/$1$ queue with working vacations ($M$/$M$/$1$/$WV$),, Performance Evaluaton, 50 (2002), 41.  doi: 10.1016/S0166-5316(02)00057-3.  Google Scholar [14] H. Takagi, "Queueing Analysis - A Foundation of Performance Evaluation : Volume 2, Finite Systems,", North Holland, (1993).   Google Scholar [15] N. Tian and Z. G. Zhang, "Vacation Queueing Models: Theory and Applications,", Springer-Verlag, (2006).   Google Scholar [16] P. Vijaya Laxmi and U. C. Gupta, A unified approach to analyze the $GI^X$/$M$/$1$/$N$ and $GI$/$E_k$/$1$/$N$ queues,, Proceedings of the International Conference on Stochastic Processes and Their Applications (eds. A. Vijayakumar and M. Sreenivasan), (1998), 206.   Google Scholar [17] D. Wu and H. Takagi, $M$/$G$/$1$ queue with multiple working vacations,, Performance Evaluaton, 63 (2006), 654.  doi: 10.1016/j.peva.2005.05.005.  Google Scholar [18] M. M. Yu, Y. H. Tang and Y. H. Fu, Steady state analysis and computation of the $GI^{[x]}$/$M^b$/$1$/$L$ queue with multiple working vacations and partial batch rejection,, Computers & Industrial Engineering, 56 (2009), 1243.  doi: 10.1016/j.cie.2008.07.013.  Google Scholar

show all references

##### References:
 [1] Y. Baba, Analysis of $GI$/$M$/$1$ queue with multiple working vacations,, Oper. Res. Lett., 33 (2005), 201.  doi: 10.1016/j.orl.2004.05.006.  Google Scholar [2] A. D. Banik, U. C. Gupta and S. S. Pathak, On the $GI$/$M$/$1$/$N$ queue with multiple working vacations - Anaytic analysis and computation,, Appl. Math. Model., 31 (2007), 1701.  doi: 10.1016/j.apm.2006.05.010.  Google Scholar [3] P. J. Burke, Delays in single-server queues with batch input,, Oper. Res., 23 (1975), 830.  doi: 10.1287/opre.23.4.830.  Google Scholar [4] K. C. Chae, D. E. Lim and W. S. Yang, The $GI$/$M$/$1$ queue and the $GI$/$Geo$/$1$ queue both with single working vacation,, Performance Evaluaton, 68 (2009), 356.  doi: 10.1016/j.peva.2009.01.005.  Google Scholar [5] K. C. Chae, S. M. Lee and H. W. Lee, On stochastic decomposition in the $GI$/$M$/$1$ queue with single exponential vacation,, Oper. Res. Lett., 34 (2006), 706.  doi: 10.1016/j.orl.2005.11.006.  Google Scholar [6] B. T. Doshi, Queueing systems with vacations - A survey,, Queueing Syst., 1 (1986), 29.  doi: 10.1007/BF01149327.  Google Scholar [7] B. T. Doshi, Single server queues with vacations,, Stochastic Analysis of Computer and Communication Systems, (1990), 217.   Google Scholar [8] F. Karaesmen and S. M. Gupta, The finite capacity $GI$/$M$/$1$ with server vacations,, Journal of the Operational Research Society, 47 (1996), 817.   Google Scholar [9] G. Latouche and V. Ramaswami, "Introduction to Matrix Analytic Methods in Stochastic Modelling,", SIAM $&$ ASA, (1999).   Google Scholar [10] J. H. Li, N. S. Tian and Z. Y. Ma, Performance analysis of $GI$/$M$/$1$ queue with working vacations and vacation interruption,, Appl. Math. Model., 32 (2008), 2715.  doi: 10.1016/j.apm.2007.09.017.  Google Scholar [11] W. Liu, X. Xu and N. Tian, Some results on the M/M/1 queue with working vacations,, Oper. Res. Lett., 35 (2007), 595.  doi: 10.1016/j.orl.2006.12.007.  Google Scholar [12] K. Sikdar, U. C. Gupta and R. K. Sharma, The analysis of a finite-buffer general input queue with batch arrival and exponential multiple vacations,, Int. J. Oper. Res., 3 (2008), 219.  doi: 10.1504/IJOR.2008.016162.  Google Scholar [13] L. D. Servi and S. G. Finn, $M$/$M$/$1$ queue with working vacations ($M$/$M$/$1$/$WV$),, Performance Evaluaton, 50 (2002), 41.  doi: 10.1016/S0166-5316(02)00057-3.  Google Scholar [14] H. Takagi, "Queueing Analysis - A Foundation of Performance Evaluation : Volume 2, Finite Systems,", North Holland, (1993).   Google Scholar [15] N. Tian and Z. G. Zhang, "Vacation Queueing Models: Theory and Applications,", Springer-Verlag, (2006).   Google Scholar [16] P. Vijaya Laxmi and U. C. Gupta, A unified approach to analyze the $GI^X$/$M$/$1$/$N$ and $GI$/$E_k$/$1$/$N$ queues,, Proceedings of the International Conference on Stochastic Processes and Their Applications (eds. A. Vijayakumar and M. Sreenivasan), (1998), 206.   Google Scholar [17] D. Wu and H. Takagi, $M$/$G$/$1$ queue with multiple working vacations,, Performance Evaluaton, 63 (2006), 654.  doi: 10.1016/j.peva.2005.05.005.  Google Scholar [18] M. M. Yu, Y. H. Tang and Y. H. Fu, Steady state analysis and computation of the $GI^{[x]}$/$M^b$/$1$/$L$ queue with multiple working vacations and partial batch rejection,, Computers & Industrial Engineering, 56 (2009), 1243.  doi: 10.1016/j.cie.2008.07.013.  Google Scholar
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2019 Impact Factor: 1.366


text:
## Continuous Uniform Distribution Calculator

Use Continuous Uniform Distribution calculator to find the probability density and cumulative probabilities for continuous Uniform distribution with parameter $a$ and $b$.

The continuous uniform distribution is the simplest probability distribution where all the values belonging to its support have the same probability density. It is also known as rectangular distribution.

Uniform Distribution Calculator
Minimum Value $a$:
Maximum Value $b$
Value of x
Results
Probability density : f(x)
Probability X less than x: P(X < x)
Probability X greater than x: P(X > x)

## Definition of Uniform Distribution

A continuous random variable $X$ is said to have uniform distribution with parameter $\alpha$ and $\beta$ if its p.d.f. is given by

$$f(x; \alpha,\beta) = \dfrac{1}{\beta-\alpha}; \alpha< x< \beta$$

## Distribution function of Uniform Distribution

Distribution function of continuous uniform distribution is

$$F(x) = \dfrac{x-\alpha}{\beta-\alpha}; \alpha< x< \beta$$

## Mean of Uniform Distribution

The mean of uniform distribution is $E(X) = \dfrac{\alpha+\beta}{2}$.

## Variance of Uniform Distribution

The variance of uniform distribution is $V(X) = \dfrac{(\beta - \alpha)^2}{2}$.

## Continous Uniform Distribution Example

The waiting time at a bus stop is uniformly distributed between 1 and 10 minute.

a. What is the probability density function?

b. What is the probability that the rider waits 8 minutes or less?

c. What is the expected waiting time?

d. What is standard deviation of waiting time?

### Solution

Let $X$ denote the waiting time at a bust stop. The waiting time at a bus stop is uniformly distributed between 1 and 10 minute. That is $X\sim U(1,10)$.

a. The probability density function of $X$ is

\begin{aligned} f(x) & = \frac{1}{10-1},\; 1\leq x \leq 10\\ & = \frac{1}{9},\; 1\leq x \leq 10. \end{aligned}

The probability density function : 0.1111

b. The probability that the rider waits 8 minutes or less is

\begin{aligned} P(X\leq 8) & = \int_1^8 f(x) \; dx\\ & = \frac{1}{9}\int_1^8 \; dx\\ & = \frac{1}{9} \big[x \big]_1^8\\ &= \frac{1}{9}\big[ 8-1\big]\\ &= \frac{7}{9}\\ &= 0.7778. \end{aligned}

lower cumulative distribution : 0.7778

c. The expected wait time is $E(X) =\dfrac{\alpha+\beta}{2} =\dfrac{1+10}{2} = 5.5$

d. The variance of waiting time is $V(X) =\dfrac{(\beta-\alpha)^2}{10} =\dfrac{(10-1)^2}{10} = 8.1$.

Hope you like Continous Uniform Distribution Calculator. Click on Examples button to understand several examples solved using uniform distribution.


text:
# Slow compilation without sudo [closed]

With any *.tex file, even with almost empty template, `tex`, `latex`, `pdflatex` commands takes ages to compile it. `strace` shows excessive amount of `stat` calls to all the files in my home directory, which is the cause of slow behaviour. If I run a compilation with `sudo` command all fine and quick as it supposed to be. How can I fix this?

`strace` output consists of many many lines like this:

``````stat("/home/...", {st_mode=S_IFDIR|0755, st_size=4096, ...}) = 0
open("/home/...", O_RDONLY|O_NONBLOCK|O_DIRECTORY|O_CLOEXEC) = -1 ENOENT (No such file or directory)
getdents(8, /* 7 entries */, 32768)     = 208
close(8)
``````

EDIT: I'm using RHEL 6

`env | grep TEX`

`kpsewhich -a texmf.cnf`

/usr/share/texmf/web2c/texmf.cnf

`kpsewhich -var-value=TEXINPUTS`

• Can you add the output of the shell commands `env | grep TEX` and `kpsewhich -a texmf.cnf` (as normal user)? – egreg Jun 8 '15 at 11:58


text:
# Correct term in english to call the symbols: “=” / “≥” / “≤”

I'm not speak English very well, and I have a question about how I can call the symbols = / ≥ / ≤.

To describe them with a word in Portuguese, we use sinal (signal).

Which is the correct term in English? I want a single word to describe all three symbols, not three separate words.

EDIT

I want knows the better term to use in my function name getConstraintSign.

export const getConstraintSign = function(constraint) {
return constraint.greater ? '≥' : constraint.less ? '≤' : '=';
}

• I think the word you're looking for is "sign," though "symbol" is perfectly good. – saulspatz Mar 21 '18 at 18:04
• I think "sign" will do. But to be sure, show us a complete sentence where you want to use it. – Ethan Bolker Mar 21 '18 at 18:05
• "sign" is overloaded, though, so be aware of the context. It also refers to the unary plus or minus sign indicating whether a number is positive or negative. – Bungo Mar 21 '18 at 18:09
• In this context, I would call it a relational operator, but if you don't like getRelationalOperator I would suggest getOperator. – saulspatz Mar 21 '18 at 18:35
• Comparisons or relations. – copper.hat Mar 21 '18 at 18:43

In the program context you provide,

getConstraintSign

is fine.

getConstraintSymbol

would work too, and is a little more precise, since it's really a glyph (symbol) you're returning.

• thanks for the answer, but the @saulspatz suggests use getConstraintOperator, my question continues the sames, how is the best term? I believe isn't a better term and all this terms are fine. Am I correct? – FabianoLothor Mar 21 '18 at 18:43
• There is no "best term". Any one of the three is clear enough so that someone reading your code will know just what the function does. I would use "symbol" but wouldn't object (wouldn't even comment) on the other choices in a code review. – Ethan Bolker Mar 21 '18 at 18:47


text:
# Cabibbo angle enhanced and doubly suppressed D ---> K pi weak decays and PCAC

Authors
Publication Date
Jan 01, 1995
Identifiers
DOI: 10.1142/S0217732395002866
OAI: oai:inspirehep.net:407782
Source
INSPIRE-HEP
The observed branching ratio of D°→K+π− to K−π+ is obtained using current algebra-PCAC consistency. The latter approach also sets the scale for the three Cabibbo angle-enhanced $D \to \bar K{\rm{\pi }}$ decays in a manner compatible with data.


text:
Publication

Monte Carlo calculation and verification of the geometrical factors for the NPDGamma experiment

Authors

Blyth, D.; Fry, J.; Fomin, N.; Alarcon, R.; Alonzi, L.; Askanazi, E.; Baeßler, S.; Balascuta, S.; Barron-Palos, L.; Barzilov, A.

Abstract

The NPDGamma experiment measures the parity-violating asymmetry in gamma-ray emission in the capture of polarized neutrons on liquid parahydrogen. The sensitivity to the asymmetry for each detector in the array is used as a parameter in the extraction of the physics asymmetry from the measured data. The detector array is approximately cylindrically symmetric around the target and a step-wise sinusoidal function has been used for the sensitivity in the previous iteration of the NPDGamma experiment, but deviations from cylindrical symmetry necessitate the use of a Monte Carlo model to determine corrections to the geometrical factors. For the calculations, source code modifications to MCNPX were done in order to calculate the sensitivity of each cesium iodide detector to the physics asymmetry. We describe the MCNPX model and results from calculations and how the results are validated through measurement of the parity violating asymmetry of gamma-rays from neutron capture on chlorine.

HEP

2018

Article


text:
Waiting for answer This question has not been answered yet. You can hire a professional tutor to get the answer.

QUESTION

# How do you calculate molar volume of nitrogen gas?

Molar volume is the volume of the gas under the conditions T=273K, P=1.0 atm (101.3 kPa). These conditions are known as STP (standard temperature and pressure)

You can use the equation: PV = nRT. If you isolate the variable "V" (by dividing both sides by "P"), you will be left with:

V= nRT/P -- where R is the universal gas constant (8.31 J/K/mol).

Using the , you would find the volume of 1 mole of a gas at Standard Temperature and Pressure (STP).

STP = 1 atm of pressure and 273 K for temperature

P = 1 atm V = ??? n = 1 mole R = 0.0821 atm L/mol K K = 273 K

P V = n R T  solves to V = (n R T )/ P

V = (1 mol (0.0821 (atm L))/(mol K) 273 K)/(1 atm)

V = 22.41 L of gas @ STP This is also known as Avogadro's number for @ STP

I hope this was helpful. SMARTERTEACHER


text:
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• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-10-26
Yuliang Li, Jianguo Wang, Benjamin Pullman, Nuno Bandeira, Yannis Papakonstantinou

Given a database of vectors, a cosine threshold query returns all vectors in the database having cosine similarity to a query vector above a given threshold θ. These queries arise naturally in many applications, such as document retrieval, image search, and mass spectrometry. The paper considers the efficient evaluation of such queries, as well as of the closely related top-k cosine similarity queries

更新日期：2020-10-30
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-10-24
Dekel Tsur

In the Cluster Vertex Deletion problem the input is a graph G and an integer k. The goal is to decide whether there is a set of vertices S of size at most k such that the deletion of the vertices of S from G results in a graph in which every connected component is a clique. We give an algorithm for Cluster Vertex Deletion whose running time is O∗(1.811k).

更新日期：2020-10-30
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-10-23
Costas Busch, Maurice Herlihy, Miroslav Popovic, Gokarna Sharma

We investigate scheduling algorithms for distributed transactional memory systems where transactions residing at nodes of a communication graph operate on shared, mobile objects. A transaction requests the objects it needs, executes once those objects have been assembled, and then possibly forwards those objects to other waiting transactions. Minimizing execution time in this model is known to be NP-hard

更新日期：2020-10-30
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-10-21
Bireswar Das, Shivdutt Sharma

The isomorphism problem for groups, when the groups are given by their Cayley tables is a well-studied problem. This problem has been studied for various restricted classes of groups. Kavitha gave a linear time isomorphism algorithm for abelian groups (JCSS 2007). Although there are isomorphism algorithms for certain nonabelian group classes represented by their Cayley tables, the complexities of those

更新日期：2020-10-30
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-10-09
Ludmila Glinskih, Dmitry Itsykson

We show that any nondeterministic read-once branching program that decides a satisfiable Tseitin formula based on an n × n grid graph has size at least 2Ω(n). Then using the Excluded Grid Theorem by Robertson and Seymour we show that for an arbitrary graph G(V, E) any nondeterministic read-once branching program that computes a satisfiable Tseitin formula based on G has size at least $$2^{\Omega (\text 更新日期：2020-10-11 • Theory Comput. Syst. (IF 0.497) Pub Date : 2020-10-04 Floris Geerts There is a long tradition in understanding graphs by investigating their adjacency matrices by means of linear algebra. Similarly, logic-based graph query languages are commonly used to explore graph properties. In this paper, we bridge these two approaches by regarding linear algebra as a graph query language. More specifically, we consider MATLANG, a matrix query language recently introduced, in 更新日期：2020-10-04 • Theory Comput. Syst. (IF 0.497) Pub Date : 2020-09-15 Antoine Mottet, Karin Quaas We investigate the complexity of the containment problem “Does \(L(\mathcal {A})\subseteq L({\mathscr{B}})$$ hold?” for register automata and timed automata, where $${\mathscr{B}}$$ is assumed to be unambiguous and $$\mathcal {A}$$ is arbitrary. We prove that the problem is decidable in the case of register automata over $$(\mathbb N,=)$$, in the case of register automata over $$(\mathbb Q,<)$$ when

更新日期：2020-09-15
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-09-12
Eric Allender, Rahul Ilango, Neekon Vafa

The Minimum Circuit Size Problem (MCSP) has been the focus of intense study recently; MCSP is hard for SZK under rather powerful reductions (Allender and Das Inf. Comput. 256, 2–8, 2017), and is provably not hard under “local” reductions computable in TIME(n0.49) (Murray and Williams Theory Comput. 13(1), 1–22, 2017). The question of whether MCSP is NP-hard (or indeed, hard even for small subclasses

更新日期：2020-09-13
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-09-10
Stefan Mengel, Sebastian Skritek

We study the complexity of evaluating well-designed pattern trees, a query language extending conjunctive queries with the possibility to define parts of the query to be optional. This possibility of optional parts is important for obtaining meaningful results over incomplete data sources as it is common in semantic web settings. Recently, a structural characterization of the classes of well-designed

更新日期：2020-09-10
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-08-22
Z. Donovan, K. Subramani, V. Mkrtchyan

As the modern integrated circuit continues to grow in complexity, the design of very large-scale integrated (VLSI) circuits involves massive teams employing state-of-the-art computer-aided design (CAD) tools. An old yet significant CAD problem for VLSI circuits is physical design automation. In physical design automation, we need to compute the best physical layout of millions to billions of circuit

更新日期：2020-08-22
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-08-17
Beate Bollig, Martin Farenholtz

Structured d-DNNFs and SDDs are restricted negation normal form circuits used in knowledge compilation as target languages into which propositional theories are compiled. Structuredness is imposed by so-called vtrees. By definition SDDs are restricted structured d-DNNFs. Beame and Liew (2015) as well as Bova and Szeider (2017) mentioned the question whether structured d-DNNFs are really more general

更新日期：2020-08-18
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-08-17
Marco Calautti, Andreas Pieris

The chase procedure is a fundamental algorithmic tool in database theory with a variety of applications. A key problem concerning the chase procedure is all-instances termination: for a given set of tuple-generating dependencies (TGDs), is it the case that the chase terminates for every input database? In view of the fact that this problem is undecidable, it is natural to ask whether known well-behaved

更新日期：2020-08-18
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-08-14
Moses Ganardi, Danny Hucke, Markus Lohrey

In the sliding window streaming model the goal is to compute an output value that only depends on the last n symbols from the data stream. Thereby, only space sublinear in the window size n should be used. Quite often randomization is used in order to achieve this goal. In the literature, one finds two different correctness criteria for randomized sliding window algorithms: (i) one can require that

更新日期：2020-08-14
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-08-14
Olivier Carton, Joseph Vandehey

We give two different proofs of the fact that non-oblivious selection via regular group sets preserves normality. Non-oblivious here means that whether or not a symbol is selected can depend on the symbol itself. One proof relies on the incompressibility of normal sequences, the other on the use of augmented dynamical systems.

更新日期：2020-08-14
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-08-14

We study the problem of exploration by a mobile entity (agent) of a class of dynamic networks, namely constantly connected dynamic graphs. This problem has already been studied in the case where the agent knows the dynamics of the graph and the underlying graph is a ring of n vertices (Ilcinkas and Wade 2018). In this paper, we consider the same problem and we suppose that the underlying graph is a

更新日期：2020-08-14
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-08-13
Sergey Dudakov, Boris Karlov

This paper is dedicated to studying decidability properties of theories of regular languages with classical operations: union, concatenation, and the Kleene star. The theory with union only is a theory of some Boolean algebra, so it is decidable. We prove that the theory of regular languages with the Kleene star only is decidable. If we use union and concatenation simultaneously, then the theory becomes

更新日期：2020-08-14
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-08-11
Yeganeh Bahoo, Prosenjit Bose, Stephane Durocher, Thomas C. Shermer

Two points p and q in a simple polygon P are k-visible when the line segment pq crosses the boundary of P at most k times. Given a query point q, a positive integer k, and a polygon P, we design an algorithm that computes the region of P that is k-visible from q in O(nk) time, where n denotes the number of vertices of P. This region is called the k-visibility region of q. This is the first algorithm

更新日期：2020-08-11
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-07-27
Atsuki Nagao, Kazuhisa Seto, Junichi Teruyama

The satisfiability of a given branching program is to determine whether there exists a consistent path from the root to 1-sink. In a syntactic read-k-times branching program, each variable appears at most k times in any path from the root to a sink. In a preliminary version of this paper, we provide a satisfiability algorithm for syntactic read-k-times branching programs with n variables and m edges

更新日期：2020-07-27
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-07-20
Faith Ellen, Rati Gelashvili, Philipp Woelfel, Leqi Zhu

Many shared memory algorithms have to deal with the problem of determining whether the value of a shared object has changed in between two successive accesses of that object by a process when the responses from both are the same. Motivated by this problem, we define the signal detection problem, which can be studied on a purely combinatorial level. Consider a system with n + 1 processes consisting

更新日期：2020-07-20
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-06-30
Florian Stober, Armin Weiß

MergeInsertion, also known as the Ford-Johnson algorithm, is a sorting algorithm which, up to today, for many input sizes achieves the best known upper bound on the number of comparisons. Indeed, it gets extremely close to the information-theoretic lower bound. While the worst-case behavior is well understood, only little is known about the average case. This work takes a closer look at the average

更新日期：2020-06-30
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-06-30
Paraschos Koutris, Jef Wijsen

We study the complexity of consistent query answering on databases that may violate primary key constraints. A repair of such a database is any consistent database that can be obtained by deleting a minimal set of tuples. For every Boolean query q, CERTAINTY(q) is the problem that takes a database as input and asks whether q evaluates to true on every repair. In Koutris and Wijsen (ACM Trans. Database

更新日期：2020-06-30
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-06-27
Erik D. Demaine, John Iacono, Grigorios Koumoutsos, Stefan Langerman

We revisitself-adjustingexternal memory tree data structures, which combine the optimal (and practical) worst-case I/O performances of B-trees, while adapting to the online distribution of queries. Our approach is analogous to undergoing efforts in the BST model, where Tango Trees (Demaine et al., SIAM J. Comput. 37(1), 240–251, 2007) were shown to be $$O(\log \log N)$$-competitive with the runtime

更新日期：2020-06-27
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-06-25
Shmuel T. Klein, Shoham Saadia, Dana Shapira

Huffman coding is known to be optimal, yet its dynamic version may yield smaller compressed files. The best known bound is that the number of bits used by dynamic Huffman coding in order to encode a message of n characters is at most larger by n bits than the size of the file required by static Huffman coding. In particular, dynamic Huffman coding can also generate a larger encoded file than the static

更新日期：2020-06-25
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-06-22
Jin-Yi Cai, Pinyan Lu, Mingji Xia

Holant problems are a general framework to study counting problems. Both counting constraint satisfaction problems (#CSP) and graph homomorphisms are special cases. We prove a complexity dichotomy theorem for $$\text {Holant}^{*}(\mathcal {F})$$, where $${\mathcal {F}}$$ is a set of constraint functions on Boolean variables and taking complex values. The constraint functions need not be symmetric functions

更新日期：2020-06-23
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-06-09
Evangelos Markakis, Apostolos Ntokos, Orestis Telelis

We study the computation of revenue-maximizing envy-free outcomes, in a monopoly market with budgeted buyers. Departing from previous works, we focus on buyers with combinatorial valuation functions, that are defined over subsets of distinct goods. Our results concern several classes of valuation functions. For single-minded budgeted buyers, we show a best possible polynomial-time approximation of

更新日期：2020-06-09
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-06-06
Michael A. Henning, Arti Pandey, Vikash Tripathi

For a graph G = (V, E) with no isolated vertices, a set $$D\subseteq V$$ is called a semipaired dominating set of G if (i)D is a dominating set of G, and (ii)D can be partitioned into two element subsets such that the vertices in each two element set are at distance at most two. The minimum cardinality of a semipaired dominating set of G is called the semipaired domination number of G, and is denoted

更新日期：2020-06-06
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-05-23
Matthias Bentert, Roman Haag, Christian Hofer, Tomohiro Koana, André Nichterlein

We investigate parameterized algorithms for the NP-hard problem Min-Power Asymmetric Connectivity (MinPAC) that has applications in wireless sensor networks. Given a directed arc-weighted graph, MinPAC asks for a strongly connected spanning subgraph minimizing the summed vertex costs. Here, the cost of each vertex is the weight of its heaviest outgoing arc in the chosen subgraph. We present linear-time

更新日期：2020-05-23
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-05-15
David Barina

This article presents a new multiplication algorithm based on the Collatz function. Assuming the validity of the Collatz conjecture, the time complexity of multiplying two n-digit numbers is O(kn), where the k is the number of odd steps in the Collatz trajectory of the first multiplicand. Most likely, the algorithm is only of theoretical interest.

更新日期：2020-05-15
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-05-09
Kiichi Watanabe, Yuto Nakashima, Shunsuke Inenaga, Hideo Bannai, Masayuki Takeda

For a string S, a palindromic substring S[i..j] is said to be a shortest unique palindromic substring (SUPS) for an interval [s,t] in S, if S[i..j] occurs exactly once in S, the interval [i,j] contains [s,t], and every palindromic substring containing [s,t] which is shorter than S[i..j] occurs at least twice in S. In this paper, we study the problem of answering SUPS queries on run-length encoded strings

更新日期：2020-05-09
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-04-28
Suthee Ruangwises, Toshiya Itoh

Given a set A of n people, with each person having a preference list that ranks a subset of A as his/her acceptable partners in order of preference, we consider the Roommates Problem (rp) and the Marriage Problem (mp) of matching people with their partners. In rp there is no further restriction, while in mp only people of opposite genders can be acceptable partners. For a pair of matchings X and Y

更新日期：2020-04-28
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-04-25
Shaohua Li, Marcin Pilipczuk

We study the Independent Feedback Vertex Set problem — a variant of the classic Feedback Vertex Set problem where, given a graph G and an integer k, the problem is to decide whether there exists a vertex set $$S\subseteq V(G)$$ such that G ∖ S is a forest and S is an independent set of size at most k. We present an $$\mathcal {O}^{\ast }((1+\varphi ^{2})^{k})$$-time FPT algorithm for this problem,

更新日期：2020-04-25
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-04-23
Dmitry Gavinsky; Pavel Pudlák

The notion of semi-random sources, also known as Santha-Vazirani (SV) sources, stands for a sequence of n bits, where the dependence of the i’th bit on the previousi − 1 bits is limited for every i ∈ [n]. If the dependence of the i’th bit on the remainingn − 1 bits is limited, then this is a strongSV-source. Even the strong SV -sources are known not to admit (universal) deterministic extractors, but

更新日期：2020-04-23
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-04-21
Elena A. Petrova, Arseny M. Shur

We study cube-free words over arbitrary non-unary finite alphabets and prove the following structural property: for every pair (u, v) of d-ary cube-free words, if u can be infinitely extended to the right and v can be infinitely extended to the left respecting the cube-freeness property, then there exists a “transition” word w over the same alphabet such that uwv is cube free. The crucial case is the

更新日期：2020-04-21
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-04-08
Jacob Carson; Douglas Cenzer; Jeffrey B. Remmel

We study automatic equivalence and nested equivalence structures. The goal is to compare and contrast these automatic structures with computable equivalence and nested equivalence structures. Equivalence structures $$\mathcal {A}$$ may be characterized by their characters $$\chi ({\mathcal {A}})$$ which encodes the number of equivalence classes of any given size. The characters of computably categorical

更新日期：2020-04-08
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-04-08
Juho Lauri, Christodoulos Mitillos

We strengthen a result by Laskar and Lyle (Discrete Appl. Math. 157, 330–338 2009) by proving that it is NP-complete to decide whether a bipartite planar graph can be partitioned into three independent dominating sets. In contrast, we show that this is always possible for every maximal outerplanar graph with at least three vertices. Moreover, we extend their previous result by proving that deciding

更新日期：2020-04-08
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-04-07
Federico Corò; Gianlorenzo D’Angelo; Vahan Mkrtchyan

In the Maximum Connectivity Improvement (MCI) problem, we are given a directed graph G = (V,E) and an integer B and we are asked to find B new edges to be added to G in order to maximize the number of connected pairs of vertices in the resulting graph. The MCI problem has been studied from the approximation point of view. In this paper, we approach it from the parameterized complexity perspective in

更新日期：2020-04-07
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-04-04
Shaull Almagor, Joël Ouaknine, James Worrell

Orbit Problems are a class of fundamental reachability questions that arise in the analysis of discrete-time linear dynamical systems such as automata, Markov chains, recurrence sequences, and linear while loops. Instances of the problem comprise a dimension $$d\in \mathbb {N}$$, a square matrix $$A\in \mathbb {Q}^{d\times d}$$, and a query regarding the behaviour of some sets under repeated applications

更新日期：2020-04-04
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-02-28
Pallavi Jain; Lawqueen Kanesh; Pranabendu Misra

Let π be a family of graphs. In the classical π-Vertex Deletion problem, given a graph G and a positive integer k, the objective is to check whether there exists a subset S of at most k vertices such that G − S is in π. In this paper, we introduce the conflict free version of this classical problem, namely Conflict Free π-Vertex Deletion (CF-π-VD), and study this problem from the viewpoint of classical

更新日期：2020-02-28
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-02-28
Leah Epstein, Thomas Erlebach

The original version of the article was inadvertently published with misinterpretations on author’s proof corrections on mathematical expressions.

更新日期：2020-02-28
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-02-20
Barbara Geissmann

We study the problem of computing a longest increasing subsequence in a sequence S of n distinct elements in the presence of persistent comparison errors. In this model, Braverman and Mossel (Noisy sorting without resampling, SODA 2008, pages 268–276, 2008) every comparison between two elements can return the wrong result with some fixed (small) probability p, and comparisons cannot be repeated. Computing

更新日期：2020-02-20
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-02-03
Jarosław Byrka; Mateusz Lewandowski; Joachim Spoerhase

We study the problem of finding a minimum weight connected subgraph spanning at least k vertices on planar, node-weighted graphs. We give a (4 + ε)-approximation algorithm for this problem. We achieve this by utilizing the recent Lagrangian-multiplier preserving (LMP) primal-dual 3-approximation for the node-weighted prize-collecting Steiner tree problem by Byrka et al. (SWAT’16) and adopting an approach

更新日期：2020-02-03
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-01-25
Antoine Amarilli, Florent Capelli, Mikaël Monet, Pierre Senellart

The article title in the original publication contains an error. The correct title is presented in this Erratum. The online version of the original article can be found at: https://doi.org/10.1007/s00224-019-09930-2

更新日期：2020-01-25
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-01-08
Patrick Gardy; Patricia Bouyer; Nicolas Markey

Strategy Logic (SL) is a very expressive temporal logic for specifying and verifying properties of multi-agent systems: in SL, one can quantify over strategies, assign them to agents, and express LTL properties of the resulting plays. Such a powerful framework has two drawbacks: first, model checking SL has non-elementary complexity; second, the exact semantics of SL is rather intricate, and may not

更新日期：2020-01-08
• Theory Comput. Syst. (IF 0.497) Pub Date : 2020-01-06
Yossi Azar; Amit Jacob Fanani

We consider the online Minimum-Cost Perfect Matching with Delays (MPMD) problem introduced by Emek et al. (STOC 2016), in which a general metric space is given, and requests for points in space are submitted in different times in this space by an adversary. The goal is to match requests, while minimizing the sum of distances between matched pairs in addition to the time intervals passed from the moment

更新日期：2020-01-06
• Theory Comput. Syst. (IF 0.497) Pub Date : 2019-12-27
Ilkka Törmä

As a variant of the equal entropy cover problem, we ask whether all multidimensional sofic shifts with countably many configurations have SFT covers with countably many configurations. We answer this question in the negative by presenting explicit counterexamples. We formulate necessary conditions for a vertically periodic shift space to have a countable SFT cover, and prove that they are sufficient

更新日期：2019-12-27
• Theory Comput. Syst. (IF 0.497) Pub Date : 2019-12-20
Raffaello Carosi; Gianpiero Monaco

We investigate pure Nash equilibria in generalized graphk-coloring games where we are given an edge-weighted undirected graph together with a set of k colors. Nodes represent players and edges capture their mutual interests. The strategy set of each player consists of k colors. The utility of a player v in a given state or coloring is given by the sum of the weights of edges {v, u} incident to v such

更新日期：2019-12-20
• Theory Comput. Syst. (IF 0.497) Pub Date : 2019-12-20
Funda Ergün; Elena Grigorescu; Erfan Sadeqi Azer; Samson Zhou

We investigate the problem of detecting periodic trends within a string S of length n, arriving in the streaming model, containing at most k wildcard characters, where k = o(n). A wildcard character is a special character that can be assigned any other character. We say that S has wildcard-period p if there exists an assignment to each of the wildcard characters so that in the resulting stream the

更新日期：2019-12-20
• Theory Comput. Syst. (IF 0.497) Pub Date : 2019-12-04
T.-H. Hubert Chan; Fei Chen; Li Ning

Stable and balanced outcomes of network bargaining games have been investigated recently, but the existence of such outcomes requires that the linear program relaxation of a certain maximum matching problem have integral optimal solution. We propose an alternative model for network bargaining games in which each edge acts as a player, who proposes how to split the weight of the edge among the two incident

更新日期：2019-12-04
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- Current Issue - Ahead of Print Articles - All Issues - Search - Open Access - Information for Authors - Downloads - Guideline - Regulations ㆍPaper Submission ㆍPaper Reviewing ㆍPublication and Distribution - Code of Ethics - For Authors ㆍOnlilne Submission ㆍMy Manuscript - For Reviewers - For Editors
 Tensor products of log-hyponormal operators Bull. Korean Math. Soc. 2005 Vol. 42, No. 2, 269-277 Published online June 1, 2005 In Hyoun Kim Kyungpook National University Abstract : The tensor product $A\otimes B$ of Hilbert space operators $A$ and $B$ will be shown to be log-hyponormal if and only if $A$ and $B$ are log-hyponormal. Some general comments about generalized hyponormality are also made. Keywords : tensor product, log-hyponormal, $w$-hyponormal, $p$-quasi-hyponormal MSC numbers : Primary 47A80; Secondary 47B20 Downloads: Full-text PDF


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For $M\otimes_R N$, why is it so imperative that $M$ be a right $R$-module and $N$ a left $R$-module?

I was reading about the construction of a tensor product of any right $R$-module $M$ and left $R$-module $N$ over a ring $R$.

Why is it required that $M$ and $N$ be right and left modules, respectively? How does the construction not work otherwise?

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Well, if they were both left modules, say, you'd get stuff like $r(sx)\otimes y = sx \otimes ry = x \otimes s(ry) = x \otimes (sr)y = (sr)x \otimes y$, which is problematic if $R$ is not commutative. –  Omar Antolín-Camarena Mar 8 '12 at 4:19

Try tensoring a left $R$-module $M$ with a left $R$-module $N$. This tensor product should be comprised of "tensors" $m\otimes n$ (sorry for denoting them by the same notation as usual tensors; they don't really deserve this) satisfying $rm\otimes n = m\otimes rn$ for all $r\in R$.

Now, for any $r,s\in R$, we must have $rsm\otimes n=sm\otimes rn=m\otimes srn$, but at the same time $rsm\otimes n=m\otimes rsn$ (because we can move the $rs$ as a whole across the tensor sign). Thus, $m\otimes srn=m\otimes rsn$. In other words, $m\otimes\left(rs-sr\right)n=0$. Similarly, $\left(rs-sr\right)m\otimes n=0$.

But this means that our fake tensor product $M\otimes N$ does not depend on the left $R$-modules $M$ and $N$, but only on the left $R / \left(\mathrm{Ab} R\right)$-modules $M / \left(\left(\mathrm{Ab} R\right) M\right)$ and $N / \left(\left(\mathrm{Ab} R\right) N\right)$, where $\mathrm{Ab} R$ is the commutator ideal of $R$ (that is, the ideal generated by all differences of the form $rs-sr$ with $r,s\in R$). And it is actually exactly the tensor product of these two $R / \left(\mathrm{Ab} R\right)$-modules over the commutative ring $R / \left(\mathrm{Ab} R\right)$. This tensor product, of course, can be reinterpreted as a tensor product of a left module with a right module (since over a commutative ring, modules can be switched from left to right at will).

So the notion of the tensor product of a right module with a left module is more reasonable and completely encompasses the notion of a "tensor product" of two left modules.

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So unless $M$ is an $(R,R)$-bimodule, it only makes sense to have the relation $mr\otimes n=m\otimes rn$ one the construction is finished? –  Tents Mar 8 '12 at 4:32
I don't understand this question, sorry. –  darij grinberg Mar 8 '12 at 22:35

I don't understand the possible confusion. The tensor product of $M$ And $N$ is defined to be the object that takes $R$-biadditive maps $M\times N\to G$ to additive maps $M\otimes_R N\to G$. In other words, the entire function of a tensor product, its entire universal reason for being is to be a machine that turns $R$-biadditive maps into additive ones. With this said it's clear that the notion of $R$-biadditive maps are important, no? What are $R$-biadditive maps? They're maps $f:M\times N\to G$ (where $G$ is some abelian group) wiht the property that they are additive in each slot and are "balanced" in the sense that $f(mr,n)=f(m,rn)$ fora ll $r\in R$ and $(m,n)\in M\times N$. To even say this last sentence, to make sense of $f(mr,n)=f(m,rn)$ we need $M$ to be a right $R$-module and $N$ a left $R$-module.

Now, if $R$ happens to be commutative so that distinguishing between left and right modules is just a formality we can really take the tensor product of any two modules.

This all said, there is no problem in just taking to abelian groups $M$ and $N$ and forming the quotient $\mathbb{Z}[M\times N]/S$ where $S$ is the subgroup generated by making $\otimes:M\times N\to\mathbb{Z}[M\times N]$ bilinear. That said, you are secretly just computing the tensor product of two $\mathbb{Z}$-modules.

The moral of the story is that we need some kind of $R$ structure on $M$ and $N$ since tensor products were born out of a desire to transfer one $R$ property to another ($R$-bilinear to $R$-linear for commutative $R$).

EDIT: This is to address KCd's remarks. He is most likely right about me misinterpreting your question, for that I apologize.

So, why exactly can we not consider the tensor product of two left $R$-modules? Let's first see why the question that tensor products attempt to answer doesn't make sense if we consider two left $R$-modules, say. In this case it's clear that the analogue we are looking for is additive in each variable maps $f:M\times N\to R$ such that $f(rm,n)=f(m,rn)$. The issue with this that if $R$ is not commutative we start getting screwy things--things we probably didn't anticipate. For example,

$$f((rs)m,n)=f(r(sm),n)=f(sm,rn)=f(m,srn)$$

which if this is to be true for all $r,s$ strongly suggests that there is some kind of commutativity thing going on here. Of course, if our rings are commutative then everythings fine and we don't care if $M$ or $N$ are left or right $R$-modules, but if our ring isn't commutative then we seem like we might have issues. For example, a let's assume that our $M,N$ are actually algebras over $R$. A very natural map then is to consider the multiplication map $M\times M\to M$--our hope is that this would fit the kind of map we'd like to consider, but we see that if this map were "$R$-biadditive" in this new sense then we'd have $rs=(rs1)1=1(sr1)=sr$ for all $s,r$ so that $M$ would need to be commutative.

This all said, I guess you could theoretically make the same construction with both left $R$-modules, I guess the problem is what the construction would correspond to. It's easy to think of $R$-biadditive maps with a left and right $R$-modules (for example think about the map $\text{Mat}_{n\times m}(R)\times\text{Mat}_{m\times n}(R)\to \text{Mat}_n(R)$ given by matrix multiplication--this is is a $\text{Mat}_m(R)$-biadditive map) but I can't imagine (off the top of my head) where the kind of maps we'd be discussing when we have two left $R$-modules where $R$ is noncommuative (since any map would have to satisfy that weird "commutative condition")--and of course, if $R$ is commutative then thinking about them as left $R$ modules and doing it the way you suggest is the same as thinking about them as a left and right $R$-modules respectively.

Moral of the story: You could make such a construction, but why it would be meaningful is not at all clear.

About why anyone would care about biadditive maps I don't think I can beat the explanation given by Gowers here.

I hope that I was able to help a little bit.

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Concerning the first paragraph, it is not at all clear why a notion (like R-biadditive maps) is important just because it is related to a universal mapping problem. But the more significant misunderstanding in this answer is that you didn't highlight what goes wrong if $M$ and $N$ are, say, both left $R$-modules. My sense of the intent behind the question is why one should want to take $M$ and $N$ to be $R$-modules on different sides in general. –  KCd Mar 8 '12 at 4:21
@KCd Thank you for pointing that out. I'm sorry that I initially misinterpreted things. I'm not as confident about my answer now that I have understood the question, but hopefully I was able to add something. P.S. Thanks for the great notes. –  Alex Youcis Mar 8 '12 at 4:48

Now, although I will use language suitable for bimodules here, what I say is also applicable to:

• sets with a monoid or group action,
• bifunctors $\mathcal{A} \times \mathcal{B}^\op \to \textbf{Set}$, where $\mathcal{A}$ and $\mathcal{B}$ are small categories, and more generally,
• bifunctors of categories enriched over a suitable monoidal category.

Firstly, let us examine the hom objects, since tensor products are defined in terms of them. If $A$ and $B$ are left $R$-modules, then they are also abelian groups, and there is a natural monomorphism $$\Hom_R(A, B) \hookrightarrow \Hom(A, B)$$ Indeed, observe that the $R$-actions of $B$ and $A$ (respectively) induce a $(R, R^\op)$-bimodule structure on $\Hom(A, B)$, as follows: if $f : A \to B$ is a homomorphism of abelian groups, we define $r \cdot f \cdot s : A \to B$ by $$(r \cdot f \cdot s)(m) = r f(s m)$$ It is easily checked that this makes $\Hom(A, B)$ into a $(R, R^\textrm{op})$-bimodule. But what is $\Hom_R(A, B)$? It is the subgroup of those homomorphisms $f : A \to B$ such that $$r \cdot f \cdot 1 = 1 \cdot f \cdot r$$ Thus, we get an equaliser diagram of the form $$\Hom_R(A, B) \longrightarrow R \otimes \Hom(A, B) \mathrel{\rlap{\lower{0.5ex}{\longrightarrow}}\raise{0.5ex}{\longrightarrow}} \Hom(A, B)$$ where the upper map is given by the left $R$-action of $A$: $$r \otimes f \mapsto 1 \cdot f \cdot r$$ and the lower map is given by the left $R$-action of $B$: $$r \otimes f \mapsto r \cdot f \cdot 1$$

More generally, we need to form a special kind of limit called an ‘end’: $$\Hom_R(A, B) = \int_R \Hom(A, B)$$ In the case of two functors $A, B : \mathcal{R} \to \textbf{Set}$, this amounts to the well-known formula $$\textrm{Nat}(A, B) \cong \int_{r : \mathcal{R}} \textrm{Hom}(A r, B r)$$

Now, what about tensor products? They are formed in an essentially dual fashion. We take it for granted that there is a tensor product of abelian groups satisfying $$\Hom(A \otimes B, C) \cong \Hom(A, \Hom(B, C))$$ and we want to understand how to get the corresponding constructions over $R$. Let $A, B , C$ be bimodules as in the first paragraph. First, consider the hom-object $\Hom(A, \Hom_{T^\op}(B, C))$. Because $\Hom$ is left exact in the second variable, we can extract the end to obtain $$\int_{T^\op} \Hom(A, \Hom(B, C)) \cong \Hom(A, \Hom_{T^\op}(B, C))$$ Ends commmute with ends, so we can replace the outer $\Hom$ with $\Hom_{(S, R^\op)}$ as follows: $$\int_S \int_{R^\op} \int_{T^\op} \Hom(A, \Hom(B, C)) \cong \Hom_{(S, R^\op)}(A, \Hom_{T^\op}(B, C))$$ We have assumed that there is a tensor-hom adjunction for abelian groups, so we can replace the integrand on the left hand side to obtain $$\int_S \int_{R^\op} \int_{T^\op} \Hom(A \otimes B, C) \cong \Hom_{(S, R^\op)}(A, \Hom_{T^\op}(B, C))$$ and we know that $$\Hom_{(S, T^\op)}(A \otimes B, C) \cong \int_S \int_{T^\op} \Hom(A \otimes B, C)$$ so we obtain $$\int_{R^\op} \Hom_{(S, T^\op)}(A \otimes B, C) \cong \Hom_{(S, R^\op)}(A, \Hom_{T^\op}(B, C))$$ and now, applying the Yoneda lemma, we may push the end into the first argument of the integrand to get the coend formula for the tensor product: $$A \otimes_R B = \int^R A \otimes B$$ Unpacking the definition of coend, this means there is a coequaliser diagram of the form $$A \otimes R \otimes B \mathrel{\rlap{\lower{0.5ex}{\longrightarrow}}\raise{0.5ex}{\longrightarrow}} A \otimes B \longrightarrow A \otimes_R B$$ where the upper arrow is the map given by the right $R$-action of $A$: $$a \otimes r \otimes b \mapsto a r \otimes b$$ and the lower arrow is the map given by the left $R$-action of $B$: $$a \otimes r \otimes b \mapsto a \otimes r b$$ Thus, we see that the requirement that $A$ be a right $R$-module and $B$ be a left $R$-module is forced on us by the fact that $\Hom(B, -)$ has a natural right $R$-module structure when $B$ is a left $R$-module!

Moreover, it is clear from this construction that $\Hom_R$ and $\otimes_R$ are just two dual ways of neutralising two opposite $R$-actions:

• $\Hom_R(B, C)$ extracts the universal $R$-equivariant subobject of $\Hom(B, C)$, while
• $A \otimes_R B$ extracts the universal $R$-equivariant quotient of $A \otimes B$.
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text: Let's say that both Ben's Shoes and Jason's Shoes are publicly traded companies. And all that means is that both of their shares are traded on exchanges. Maybe it's the NASDAQ, or the New York Stock Exchange, or some other exchange. And the going price on those exchanges-- the last closing price for Ben's stock was $21.50 per share, and the last closing price for Jason's stock is$12.00 per share. What I want to explore in this video, and probably the next few is, what is that saying for what the market thinks these businesses are worth? So in both of these situations, they have 10,000 shares. And remember, the shares are a split of the owner's equity. It's not a split of the assets. It's a split of just the equity part, right over here. So if shareholders are willing to pay 21.50 per share for Ben's stock, and there are 10,000 shares in Ben's company. So you take 21.50 times-- I'll do it this way-- times 10,000 shares gives us a market cap. So 21.50 times 10,000 gives us $215,000. And what this says is, look, if each of those 10,000 slices of the equity is worth 21.50, then the entire equity portion is going to be-- the market is valuing it at$215,000. And this calculation, this multiplication of the market price per share times the number of shares, this is called the market cap. Short for market capitalization. The market cap of the company. And all it is, is what is the market valuing the equity part of Ben's company worth? Let's do the same thing for Jason's company. You have $12.00 per share times 10,000 shares. That gives us$120,000 market cap. So the market is telling us that even though on the books, Ben's equity-- based on how he valued his assets and his liabilities-- is 135,000. The market is actually valuing this at 215,000. And in the next video, we'll think about what that means for how the market is actually valuing the business. In the case of Jason's business, instead of $35,000 of equity-- just straight up from what's on the books-- the market is valuing this piece right here at$120,000. So hopefully that gives you a little sense of one, what shares are a share of. They're a share of the owner's equity, not of the assets. And also gives you a good sense of what market cap is. It's the market's value of the owner's equity. And notice, in both cases-- and it's usually the case-- it's going to be a different number than the book value, the number that's actually on the books.


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# Geometry homework

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In triangle ABC the acute angles are b=60 and c=30. The centoride of abc is G. If ab=1 what is ag

Help asap.

Jan 14, 2019
edited by Mathguru  Jan 14, 2019

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pls be more descriptive when the say the centroid is...?

Jan 14, 2019
#2
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fixed it

Mathguru  Jan 14, 2019
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Since $$\triangle ABC$$ is a 30-60-90 triangle and $$\overline{AB}$$ is 1, $$\overline{AC}$$ is $$\sqrt{3}$$ and $$\overline{BC}$$ is 2. Let the intersection of the ray $$\overline{AG}$$ through $$\overline{BC}$$ be D. Becuase $$\overline{AD}$$ is a median, it divides $$\overline{BC}$$ into 2 equal parts. Therefore, $$\overline{BD}$$ is 1. Now, we can say that $$\triangle BAD$$ is equilateral because it is isoceles, and one of the angles is 60. As such, $$\overline{AD}$$ is 1.

The centroid divides medians into 2 parts ratio 1:2, so $$\overline{AG}$$ is $$\boxed{\frac{2}{3}}$$.

First post, How'd I do?

-24

Jan 15, 2019
#4
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Good job, TwentyFour.....Welcome aboard  !!!!

CPhill  Jan 15, 2019


text:
## Journal of the Mathematical Society of Japan

### On Alperin's weight conjecture for $p$-blocks of $p$-solvable groups

Masafumi MURAI

#### Abstract

For $p$-solvable groups, a strong form of Alperin's weight conjecture has been proved by T. Okuyama (unpublished). L. Barker has refined this theorem by taking Green correspondence into account. We prove here a relative version of Barker's theorem.

#### Article information

Source
J. Math. Soc. Japan, Volume 65, Number 4 (2013), 1037-1054.

Dates
First available in Project Euclid: 24 October 2013

https://projecteuclid.org/euclid.jmsj/1382620184

Digital Object Identifier
doi:10.2969/jmsj/06541037

Mathematical Reviews number (MathSciNet)
MR3127815

Zentralblatt MATH identifier
1290.20009

Subjects
Primary: 20C20: Modular representations and characters

#### Citation

MURAI, Masafumi. On Alperin's weight conjecture for $p$-blocks of $p$-solvable groups. J. Math. Soc. Japan 65 (2013), no. 4, 1037--1054. doi:10.2969/jmsj/06541037. https://projecteuclid.org/euclid.jmsj/1382620184

#### References

• J. L. Alperin and M. Broué, Local methods in block theory, Ann. of Math. (2), 110 (1979), 143–157.
• L. Barker, On $p$-soluble groups and the number of simple modules associated with a given Brauer pair, Quart. J. Math. Oxford Ser. (2), 48 (1997), 133–160.
• D. W. Burry, A strengthened theory of vertices and sources, J. Algebra, 59 (1979), 330–344.
• E. C. Dade, Endo-permutation modules over $p$-groups. I, Ann. of Math. (2), 107 (1978), 459–494.
• W. Feit, The Representation Theory of Finite Groups, North-Holland Math. Library, 25, North-Holland, Amsterdam, 1982.
• B. Huppert and N. Blackburn, Finite Groups. II, Grundlehren Math. Wiss., 242, Springer-Verlag, Berlin, 1982.
• B. Huppert and N. Blackburn, Finite Groups. III, Grundlehren Math. Wiss., 243, Springer-Verlag, Berlin, 1982.
• I. M. Isaacs, Character Theory of Finite Groups, Pure Appl. Math. (Amst.), 69, Academic Press, New York, 1976.
• I. M. Isaacs and G. Navarro, Weights and vertices for characters of $\pi$-separable groups, J. Algebra, 177 (1995), 339–366.
• A. Laradji, On normal subgroups and simple modules with a given vertex in a $p$-solvable group, J. Algebra, 308 (2007), 484–492.
• M. Murai, Normal subgroups and heights of characters, J. Math. Kyoto Univ., 36 (1996), 31–43.
• H. Nagao and Y. Tsushima, Representations of Finite Groups, Academic Press, New York, 1989.
• L. Puig, Local block theory in $p$-solvable groups, In: The Santa Cruz Conference on Finite Groups, Univ. California, Santa Cruz, Calif., 1979, (Eds. B. Cooperstein and G. Mason), Proc. Sympos. Pure Math., 37, Amer. Math. Soc., Providence, RI, 1980, pp.,385–388.
• J. Thévenaz, $G$-algebras and Modular Representation Theory, Oxford Math. Monogr., Clarendon Press, Oxford, 1995.
• A. Watanabe, Normal subgroups and multiplicities of indecomposable modules, Osaka J. Math., 33 (1996), 629–635.


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