text:
# Inverse fourier transform

• May 15th 2010, 02:25 PM
rebghb
Inverse fourier transform
Hello everyone!

I'm trying to IFT this fraction after I couldn't find it in a Fourier couples table.

$F(j\omega)=\frac{250\pi \delta(\omega)}{30+125\omega}$.
When I integrated the latter, I got: $\frac{1}{2\pi}\int_{0-}^{0+} \frac{250\pi \delta(\omega)}{30+125\omega}e^{-j\omega t}d\omega$ $\ =\frac{1}{2\pi}\int_{0-}^{0+} \frac{250\pi \delta(\omega)}{30}d\omega$ $=\frac{25}{6} u(t)$??

Is there ever a $u(t)$ in IFT ??
Thanks for any help!
• May 15th 2010, 11:55 PM
CaptainBlack
Quote:

Originally Posted by rebghb
Hello everyone!

I'm trying to IFT this fraction after I couldn't find it in a Fourier couples table.

$F(j\omega)=\frac{250\pi \delta(\omega)}{30+125\omega}$.
When I integrated the latter, I got: $\frac{1}{2\pi}\int_{0-}^{0+} \frac{250\pi \delta(\omega)}{30+125\omega}e^{-j\omega t}d\omega$ $\ =\frac{1}{2\pi}\int_{0-}^{0+} \frac{250\pi \delta(\omega)}{30}d\omega$ $=\frac{25}{6} u(t)$??

Is there ever a $u(t)$ in IFT ??
Thanks for any help!

For any function continuous in a neighbourhood of $0$ we have:

$\int_A f(\omega) \delta(\omega)\;d\omega=f(0)$

For any interval $A$ containing $0$ as an interior point.

CB
• May 16th 2010, 12:19 AM
rebghb
Isn't it that $\int_A f(\omega)\delta(\omega) d\omega$ $=\int_A f(0) \delta(\omega) d\omega = f(0)u(t)$
At least that's what I know from circuit analysis...


text:
# How do you graph y=x^2+4x+6?

Aug 8, 2015

Graph $y = {x}^{2} + 4 x + 6$

#### Explanation:

To graph a parabola, fins a few critical points, such as: vertex, axis of symmetry, y intercept and x-intercepts.
x-coordinate of vertex: $x = - \frac{4}{2} = - 2$
(y-coordinate of vertex: y = f(-2) = 4 - 8 + 6 = 2
Axis of symmetry: x = -2
y-intercept --> Make x = 0 --> y = 6
x-intercepts --> Solve y = 0.
D = 16 - 24 = - 8 < 0.
There are no x-intercepts. The parabola is completely above the x-axis.
Since a> 0, the parabola opens upward, there is a min at vertex.
graph{x^2 + 4x + 6 [-10, 10, -5, 5]}

format: plain


text:
# pymatgen.core.bonds module¶

class CovalentBond(site1, site2)[source]

Bases: object

Defines a covalent bond between two sites.

Initializes a covalent bond between two sites.

Parameters: site1 (Site) – First site. site2 (Site) – Second site.
static is_bonded(site1, site2, tol=0.2, bond_order=None)[source]

Test if two sites are bonded, up to a certain limit.

Parameters: site1 (Site) – First site site2 (Site) – Second site tol (float) – Relative tolerance to test. Basically, the code checks if the distance between the sites is less than (1 + tol) * typical bond distances. Defaults to 0.2, i.e., 20% longer. bond_order – Bond order to test. If None, the code simply checks against all possible bond data. Defaults to None. Boolean indicating whether two sites are bonded.
length

Length of the bond.

get_bond_length(sp1, sp2, bond_order=1)[source]

Get the bond length between two species.

Parameters: sp1 (Specie) – First specie. sp2 (Specie) – Second specie. bond_order – For species with different possible bond orders, this allows one to obtain the bond length for a particular bond order. For example, to get the C=C bond length instead of the C-C bond length, this should be set to 2. Defaults to 1. Bond length in Angstrom. If no data is available, the sum of the atomic radii is used.


text:
# Integration subscript question

1. Apr 27, 2013

### Intricacy

Simple Integration question

Hey: I am doing a practice exam question and it gives me two families of integrals, the first one being:

Cn= int(pi,0) xn cos(x) dx int n>= 0
It has asked for me to do integrate this as C0 I have assumed this means n = 0.
Do I change the n to a zero before or after integrating?

I can't find anywhere in my learning centre that tells me this. No urgency, this is just extra study for the exam in a few weeks. I may have additional questions that follow on this, but the integration itself seems straight forward, so do most of the future questions. Thanks in advance.

Last edited: Apr 27, 2013
2. Apr 27, 2013

### tiny-tim

Welcome to PF!

Hey Intricacy! Welcome to PF!
You mean Cn is defined as ∫0π xn cos(x) dx ?

Then C0 = ∫0π x0 cos(x) dx

3. Apr 27, 2013

### HomogenousCow

What does x underscript mean?

4. Apr 27, 2013

### Staff: Mentor

Writing an index as a subscript is a way to indicate multiple variables using only a single letter. For example, x0, x1, and x2 represent three different values.

Subscripts are often used to represent sequences of numbers or the coefficients of polynomials of arbitrary degree, as in p(x) = c0 + c1x + c2x2 + ... + cnxn.
Here the coefficients of the terms are the numbers {c0, c1, c2, ... , cn}.

$$C_n = \int_0^{\pi} x_n~cos(x)~dx \text{?}$$

Or do you mean
$$C_n = \int_0^{\pi} x^n~cos(x)~dx \text{?}$$

5. Apr 28, 2013

### Intricacy

Oh, I am so sorry for that. Yes I meant x^n, so sorry.

Thank you for that as well TinyTim. So I can turn the x^0 straight to 1 prior to integration?
Cn=∫π0 x^n cos(x) dx? ----> C0=∫π0 cos(x)

6. Apr 28, 2013

### tiny-tim

Hi Intricacy!
Yes … that's the definition of Co.

7. Apr 28, 2013

### Intricacy

Thank you Tiny-Tim. Sorry for the obviously stupid question :P.


text:
The characteristic or fictitious diameter $D_x$ is the location at which the effect of the loss factor commences. Inside the circle of diameter $D_x$, the temperature changes according to the peak value of the losses. Outside this circle, it changes with the average losses. The characteristic diameter is a function of the diffusivity of the medium and the length of the loss cycle. In the majority of cases, the soil diffusivity $\delta_{soil}$ will not be known. In these cases, a value of 0.5 of 0.5*10-6 m²/s can be used. This value is based on a soil thermal resistivity of 1.0 K.m/W and a moisture content of about 7% of dry weight.

The calculation of the characteristic (or fictitious) diameter for sinusoidal load is based on the IEEE paper 'Ampacity calculation for deeply installed cables' by E. Dorison et al, dated 2010. Three different methods can be choses:

1. Neher McGrath: Neher evaluated constants epirically to best fit the temperature rises calculated over a range of cable sizes. Using measured data, Neher obtained the value 61200 when $\tau$ is expressed in hours and $\delta_{soil}$ is expressed in m²/s. Based on the evaluations by Neher, the characteristic diameter becomes 212 mm.
2. Dorison: The calculation uses the modified Bessel function K of order 0: k0() and the modified Bessel function of the first kind of order 1: k1(). The IEC standard 60853 uses an approximation with an exponential integral valid for an infinite thin wire whereas the formulas presented in the paper the cable diameter influences the diameter of the area affected by load variations. This is particularily important for cables in tunnels, where the tunnel diameter replaces the diameter of the cable.
3. Heinhold: Alternative expressions for $D_x$ are given in the book 'Kabel und Leitungen für Starkstrom' by L. Heinhold, 5th edition 1999. Heinhold defined three equations to calculate the characteristic diameter $D_x$ depending on the type of load curve (sinusoidal, rectilinear, and other load curve)
4. Symbol
$D_{\mathrm{x}}$
Unit
mm
Formulae
 $1020 \sqrt{\delta _{\mathrm{soil}} \tau}$ Method by Neher McGrath $\frac{\sqrt{6} k_{\mathrm{H}}}{720 \rho_{\mathrm{4}}^{0.4} \sqrt{\frac{1}{\tau}}}$ Method by Heinhold $D_{\mathrm{e}} e^{\frac{2000}{D_{\mathrm{e}} q_{\mathrm{x}}} \operatorname{abs}{\left (\frac{\operatorname{k_{0}}{\left (\frac{D_{\mathrm{e}} q_{\mathrm{x}}}{2000} \right )}}{\operatorname{k_{1}}{\left (\frac{D_{\mathrm{e}} q_{\mathrm{x}}}{2000} \right )}} \right )}}$ Method by Dorison
Related
$D_{\mathrm{e}}$
$\delta _{\mathrm{soil}}$
$\rho_{\mathrm{4}}$
$\tau$
$T_{\mathrm{4d}}$


text:
# Math Help - how do you solve this?

1. ## how do you solve this?

3/8 multiplied by x minus 1=4

2. Is $\displaystyle\frac{3}{8}x-1=4$ or $\displaystyle\frac{3}{8}(x-1)=4$ ?

3. its the first one you wrote

4. Multiply by 8:
$3x-8=32$
Can you continue?

5. yeah thanks!


text:
# Inferring BRAM with unused addresses efficiently

What is a correct way to infer a RAM with some unused higher addresses (using block RAMs)?
Using the code below (default values for generics, Xilinx synth and map) I get a RAM sized the same as if depth was set to 2**ADDRWIDTH:

entity foo is
generic (
DATAWIDTH : positive := 8;
);
port (
clk_a  : in std_logic;
we_a   : in std_logic;
di_a   : in std_logic_vector(DATAWIDTH-1 downto 0);
do_a   : out std_logic_vector(DATAWIDTH-1 downto 0)
);
end foo;

architecture bar of foo is
type myram_type is array (DATADEPTH-1 downto 0) of std_logic_vector(DATAWIDTH-1 downto 0); --! type for ram content
shared variable myram : myram_type; --! ram
begin
process (clk_a)
begin
if rising_edge(clk_a) then
if we_a = '1' then
end if;
end if;
end process;
end bar;

For example, I want a RAM with DATAWIDTH = 8 and DATADEPTH = 5000, so the address has to be ADDRWIDTH = 13 because ADDRWIDTH = 12would only allow to address 4096 RAM locations. Lets assume one block RAM ressource on my FPGA can hold 8192 bits. If I handcoded this I required 5000*8/8192 rounded upwards = 5 block RAM ressources. However, with the code above, synthesis and map of Xilinx results in 8 block RAM ressources being used, because thats what can be addressed by 13 bit wide addresses...
Nontheless, this is not really efficient use of ressources since 3 of the 8 block RAMs will never be used.
I tried to check if the address at the input is larger than DATADEPTH and then assign don't cares for the data, but that results in the whole ram being implemented as distributed RAM / LUTRAM.
Am I missing something important or do I have to use one big ugly generate for this?

Actually, using 8 BRAMs in an 8K×1 configuration, rather than 5 BRAMs in a 1K×8 configuration, is more efficient in several important ways.

With the 8 BRAMs, you can simply connect all of the address and control lines to all of the BRAMs, and one bit from the data input and data output buses to each of the BRAMs. No other logic is required at all.

On the other hand, with the 5-BRAM configuration, you'll need extra logic to decode the upper 3 address bits to enable one BRAM at a time, and you'll also need a 5:1 multiplexer on the data output bus to select the data from correct BRAM when reading. This uses extra resources within the FPGA, and it also adversely affects the timing, reducing the maximum clock frequency you can use.

If you really need to use the BRAM capacity as efficiently as possible, and you don't care about the timing and resource issues, then you'll have to explicitly code your memory as a module that uses five 1K×8 memories internally.

• A better 5-BRAM configuration in this case would be four 4Kx2 and one 1Kx8, with a 2:1 output mux and a single address bit decoder for writing. Oct 24, 2013 at 13:31

Inferring hard modules (like block rams) from code is a rather fragile proposition, so synthesizers usually provide coding guidelines, of which I presume you used Xilinx's "RAMs Hardware Description Language (HDL) Coding Guidelines".

When reading those guidelines and realizing that certain coding styles that would seem perfectly valid (like using signals instead of shared variables for a dual port ram) won't actually work, you get a feeling of how limited inference techniques are.

Therefore I think either there is such a limitation with Xilinx, or the tool is choosing not to minimize BRAM usage in favor of other optimizations (as mentioned in Dave Tweed's answer).

So it seems you are unfortunately limited to:

• Explicitly cascading the required BRAMs.
• Cascading multiple memories coded in the simplistic 'guideline' way.
• Using the CORE generator.
• Letting the extra BRAMs go to waste and move on - maybe when the conditions are right the tool will do what you expected and use 5 BRAMs.

If you need to use 13 address bits for your RAM then those 13 physical address signals for a BRAM must be allocated to your RAM. The address lines can't be easily shared with some other module, so you effectively get all of the physical RAM that is accessed by those 13 address lines. If you need 5000 words then you will get 8192 words of BRAM. As you noted, if you synthesize the RAM from LUTs you can make a RAM with only 5000 words but you lose the efficiency of the BRAM.


text:
489 views
If a language(L1) is Recursive enumerable  (RE)  and  L2 is Recursive   (REC)  , then  what  will be L1 - L2  ? Can we  directly use set difference property   or do the so called  intersection method  i.e  L1 - L2 = L1 ∩ L2 .
| 489 views
0

Always remember that a language is a set of strings, so yes you can apply set difference property. See Arjun Sir's answer: https://gateoverflow.in/2190/gate2010-17?show=2381#a2381

+1 vote

L1 = RE
L2 = REC
L1-L2 = L1 ∩ L2c = RE ∩ REC = RE ∩ RE = RE

As, REC is closed under complement and RE is closed under intersection.

by Boss (44.1k points)
0
L1 = RE  , L2 = REC , all REC Recursive are Recursive enumerable  so L2 = RE

now as per closure property of languages  in case of SET DIFFERENCE , RE ARE NOT CLOSED UNDER SET DIFFERENCE   .so  RE - RE   NOT EQUAL  TO Recursive enumerable . SO how you get RE - RE = RE
0
0

Why you are using L1 - L2 = L1 ∩ L2 .  , why not directly apply SET DIFFERENCE PROPERTY , THAT IS IN CASE OF Recursive enumerable  RE- RE =  NOT EQUAL TO RE.

Closure Properties of Languages

 Property Regular CFL DCFL CSL Recursive RE Union Yes Yes No Yes Yes Yes Intersection Yes No No Yes Yes Yes Set Difference Yes No No Yes Yes No Complementation Yes No Yes Yes Yes No Intersection with a regular lang. Yes Yes Yes Yes Yes Yes Concatenation Yes Yes No Yes Yes Yes Kleen Closure Yes Yes No Yes Yes Yes Kleen Plus Yes Yes No Yes Yes Yes Reversal Yes Yes Yes Yes Yes Yes Homomorphism Yes Yes No No No Yes ϵ-free Homomorphism Yes Yes No Yes Yes Yes Inverse Homomorphism Yes Yes Yes Yes Yes Yes Substitution Yes Yes No No No Yes ϵ-free Substitution Yes Yes No Yes Yes Yes

0

Is A-B = A∩Bc correct?

0
It means that  RE - RE  is not equal to RE is FALSE . IT SHOULD BE , RE - RE = RE ,

So Recursive enumerable  (RE)  ARE CLOSED UNDER SET DIFFERENCE  , Simply say it is correct or not
0
This conclusion is wrong!! RE-RE=RE is not true because RE is not closed under Complement.

in the original question we are subtracting REC and REC is closed under Complement.
0

Closure Properties of Languages

 Property Regular CFL DCFL CSL Recursive RE Union Yes Yes No Yes Yes Yes Intersection Yes No No Yes Yes Yes Set Difference Yes No No Yes Yes No Complementation Yes No Yes Yes Yes No Intersection with a regular lang. Yes Yes Yes Yes Yes Yes Concatenation Yes Yes No Yes Yes Yes Kleen Closure Yes Yes No Yes Yes Yes Kleen Plus Yes Yes No Yes Yes Yes Reversal Yes Yes Yes Yes Yes Yes Homomorphism Yes Yes No No No Yes ϵ-free Homomorphism Yes Yes No Yes Yes Yes Inverse Homomorphism Yes Yes Yes Yes Yes Yes Substitution Yes Yes No No No Yes ϵ-free Substitution Yes Yes No Yes Yes Yes
0
A-B = A∩Bc is correct and RE is correct answer
0

Ok , So we cannot take  SET DIFFERENCE   DIRECTLY ,  ALWAYS TO USE  L1- L2 = L1 ∩ L2c.

+1
@Sandeep no need to memorize, try to understand the difference why is REC closed under set difference but not RE. REC is closed under set difference because it is closed under INTERSECTION and COMPLEMENT and RE is not closed under set difference because RE is not closed under COMPLEMENT.
by Active (2.6k points)


text:
# zbMATH — the first resource for mathematics

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Fischer’s diagonal condition for lattice-valued convergence spaces. Zbl 1168.54005
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The spectral theory of commutative $$C^*$$-algebras: The constructive Gelfand-Mazur theorem. Zbl 0977.18004
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Combined sinh-cosh-Gordon equation: symmetry reductions, exact solutions and conservation laws. Zbl 1397.35152
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The disjunctive domination number of a graph. Zbl 1404.05150
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Core-EP pre-order of Hilbert space operators. Zbl 1442.15005
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On atoms of the lattice of supernilpotent radicals. Zbl 0618.16004
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An oscillation theorem for a two-parameter system of differential equations. Zbl 0425.34035
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Strong zero-dimensionality of biframes and bispaces. Zbl 0722.54019
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Algebraic entropy of shift endomorphisms on abelian groups. Zbl 1194.37015
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Weaker forms of the Menger property. Zbl 1274.54074
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Function spaces and $$d$$-separability. Zbl 1091.54008
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Localification of variable-basis topological systems. Zbl 1274.06043
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On cotype and a Grothendieck-type theorem for absolutely summing multilinear operators. Zbl 1274.46092
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Some properties and expansions associated with the $$q$$-digamma function. Zbl 1274.41061
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1994
MV-algebra valued filter theory. Zbl 0865.54004
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Fibonacci numbers which are concatenations of two repdigits. Zbl 07340199
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2021
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2021
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2021
Spectra of composition operators on Fock-type spaces. Zbl 07340164
Mengestie, Tesfa; Seyoum, Werkaferahu
2021
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2021
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Wang, Weidong; Zhao, Xia
2021
Remarks on normal distributive nearlattices. Zbl 07357117
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Certain new factor theorems for infinite series and trigonometric Fourier series. Zbl 1451.40005
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2020
A note on the Appell matrix functions. Zbl 1435.15006
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On the signed Roman $$k$$-domination in graphs. Zbl 1452.05115
2020
Iterative algorithm for a family of monotone inclusion problems in $$\mathrm{CAT}(0)$$ spaces. Zbl 07311174
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2020
The inner site-perimeter of compositions. Zbl 1432.05008
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2020
Kantorovich-Bernstein $$\alpha$$-fractal function in $$\mathcal{L}^P$$ spaces. Zbl 1436.28006
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2020
Explicit expressions for the related numbers of higher order Appell polynomials. Zbl 07265481
Hu, Su; Komatsu, Takao
2020
Variations of classical selection principles: an overview. Zbl 1450.54010
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2020
The Fischer-Marsden conjecture on almost Kenmotsu manifolds. Zbl 1433.53077
De, U. C.; Mandal, Krishanu
2020
Nonlinear diameter preserving maps on function spaces. Zbl 1435.47043
Font, Juan J.; Hosseini, Maliheh
2020
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Dwivedi, Ravi; Sahai, Vivek
2020
Blow-up of smooth solutions of the time-fractional Burgers equation. Zbl 1439.35231
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2020
Some identities involving Appell polynomials. Zbl 1443.05019
Mihoubi, Miloud; Taharbouchet, Said
2020
Shifted Cauchy numbers. Zbl 1444.11041
Komatsu, Takao; Pita-Ruiz, Claudio
2020
Large characteristic subgroups and abstract group classes. Zbl 1457.20024
de Giovanni, Francesco; Trombetti, Marco
2020
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2020
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2020
Some applications of exponentially separable spaces. Zbl 07307653
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2020
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2020
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Ma, Haifeng
2020
New spaces of matrices with operator entries. Zbl 07311593
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Associated weights for spaces of $$p$$-integrable entire functions. Zbl 07311597
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2020
Extension and integral representation of the finite Hilbert transform in rearrangement invariant spaces. Zbl 1467.44002
Curbera, Guillermo P.; Okada, Susumu; Ricker, Werner J.
2020
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Braha, Naim L.
2019
Joint discrete universality for periodic zeta-functions. Zbl 1441.11235
Laurinčikas, Antanas
2019
A new factor theorem for absolute matrix summability. Zbl 1432.40007
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2019
Connectedness for lattice-valued subsets in lattice-valued convergence spaces. Zbl 07048630
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Colouring games based on autotopisms of Latin hyper-rectangles. Zbl 1420.05114
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Limit-point criteria for $$q$$-Sturm-Liouville equations. Zbl 1427.39003
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2019
Stochastic dynamics of an HIV/AIDS epidemic model with treatment. Zbl 1417.92194
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Multiplicative $$\ast$$-Lie triple higher derivations of standard operator algebras. Zbl 07111860
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2019
Global Italian domination in graphs. Zbl 1425.05116
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2019
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Sturm Liouville equations in the frame of fractional operators with exponential kernels and their discrete versions. Zbl 1429.34007
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 294 Quaestiones Mathematicae 205 Topology and its Applications 165 Fuzzy Sets and Systems 154 Journal of Mathematical Analysis and Applications 82 Communications in Algebra 77 Journal of Pure and Applied Algebra 74 Applied Categorical Structures 63 Discrete Mathematics 54 Discrete Applied Mathematics 48 Journal of Computational and Applied Mathematics 45 Positivity 36 Bulletin of the Australian Mathematical Society 36 Rocky Mountain Journal of Mathematics 34 Acta Mathematica Hungarica 32 Proceedings of the American Mathematical Society 32 Mediterranean Journal of Mathematics 30 Applied Mathematics and Computation 29 Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 28 Cahiers de Topologie et Géométrie Différentielle Catégoriques 28 Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM 27 Linear and Multilinear Algebra 27 Journal of Algebra and its Applications 26 Algebra Universalis 26 Monatshefte für Mathematik 25 Abstract and Applied Analysis 24 Results in Mathematics 24 Linear Algebra and its Applications 23 Bulletin of the Malaysian Mathematical Sciences Society. Second Series 22 Archiv der Mathematik 21 Journal of Functional Analysis 21 Indagationes Mathematicae. New Series 18 Journal of Mathematical Physics 17 Czechoslovak Mathematical Journal 17 Journal of Algebra 17 Transactions of the American Mathematical Society 17 Communications in Nonlinear Science and Numerical Simulation 16 Journal of Approximation Theory 16 Mathematische Nachrichten 15 Computers & Mathematics with Applications 15 Soft Computing 14 International Journal of Mathematics and Mathematical Sciences 14 Journal of Combinatorial Optimization 14 Journal of Mathematics 13 Periodica Mathematica Hungarica 13 Glasgow Mathematical Journal 13 Rendiconti del Circolo Matemàtico di Palermo. Serie II 13 The Ramanujan Journal 12 Applicable Analysis 12 Indian Journal of Pure & Applied Mathematics 12 Journal of Number Theory 12 Proceedings of the Edinburgh Mathematical Society. Series II 12 Order 12 Acta Mathematica Sinica. English Series 12 Central European Journal of Mathematics 11 Mathematical Methods in the Applied Sciences 11 Journal of Differential Equations 11 Theoretical Computer Science 11 Bulletin of the Iranian Mathematical Society 11 Graphs and Combinatorics 11 Journal of Mathematical Sciences (New York) 11 Advances in Difference Equations 11 Applied General Topology 10 Integral Equations and Operator Theory 10 Proceedings of the Indian Academy of Sciences. Mathematical Sciences 10 Mathematical Problems in Engineering 10 Iranian Journal of Fuzzy Systems 10 Asian-European Journal of Mathematics 10 Journal of Function Spaces 10 Advances in Operator Theory 9 Advances in Mathematics 9 Annali di Matematica Pura ed Applicata. Serie Quarta 9 Information Sciences 9 Semigroup Forum 9 Acta Applicandae Mathematicae 9 Journal of the Australian Mathematical Society 9 Banach Journal of Mathematical Analysis 9 Afrika Matematika 9 Open Mathematics 8 Mathematical Proceedings of the Cambridge Philosophical Society 8 Mathematica Slovaca 8 Numerical Functional Analysis and Optimization 8 International Journal of Computer Mathematics 8 Computational and Applied Mathematics 8 Turkish Journal of Mathematics 8 Journal of Applied Mathematics 8 Boundary Value Problems 7 International Journal of Theoretical Physics 7 Demonstratio Mathematica 7 Mathematische Zeitschrift 7 Annals of Pure and Applied Logic 7 Discussiones Mathematicae. Graph Theory 7 Journal of Inequalities and Applications 7 Journal of Nonlinear Mathematical Physics 7 Tbilisi Mathematical Journal 6 Mathematical Notes 6 Collectanea Mathematica 6 Journal of Graph Theory 6 Ricerche di Matematica 6 Advances in Applied Mathematics 6 Applied Mathematics Letters ...and 394 more Journals
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text:
# explicit UES for $D$-regular graphs over $N$ vertices through the line graph

First of all observe that if we have $G$, a $D$-regular graph over $N$ vertices that is equipped with a consistent labeling $\ell$ then we can induce a consistent labeling for $L(G)$ the line graph of $G$.

This is done in the following way, every edge $e$ in $G$ has a label $\ell(e)$.

Every edge $e'$ in $L(G)$ connects two adjacent edges $e_1,e_2$ in $G$, thus the new labeling for $L(G)$ can be $\ell'(e')=\ell(e_1)-\ell(e_2)\ (\text{mod}\ D)$.

It is not hard to verify that if $\ell$ is consistent then $\ell'$ is also consistent.

Now, how is the above fact that consistent labeling for $G$ induces a consistent labeling for $L(G)$ implies explicit $\text{UES}$.

More concretely, is it possible (using the above fact directly) to give a log-space algorithm that on input $1^{N+D}$ outputs a $\text{UES}$ for $D$-regular graphs over $N$ vertices.

Note: I am familiar with transitions from $\text{UTS}$ to $\text{UES}$, but I am more concerned about the connections with the line graph properties.


text:
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

## Sorting floating numbers

INTRODUCTION

During a CS251: Data Structure & Algorithms lecture about various sorting algorithms, Professor Aliaga mentioned a sorting algorithm of linear running time - radix sort. While showing us how radix sort can be used to sort binary numbers, he added that radix sort is not a general-purpose sorting algorithm, and our lower-bound for comparison sorts are still set at O(nlogn). But I wonder - all digital objects are represented as 0/1 in memory. Can I use radix sort to sort objects like floating point number whose binary structure is well defined?

PRELIMINARIES

It's well known that the comparison sorting algorithms have a theoretical lower bound of nlogn. Therefore, we can have something radix sort, which fall outside the category of of comparison sort, that attains status of linear time sorting algorithm. To illustrate the maneuvers of radix sort, I first introduce bucket sort with an illustrative example:

Suppose you have a collection of marbles. You want to sort the marbles out categorically, say its color. You can easily sort out the marbles by going through each marble and placing them inside a "bucket" labeled blue, red, orange, yellow, etc, and Viola! your marbles are sorted.

I concede that the above example is incredibly simple, but it is only so because radix sort itself stems from a simple method. Now consider an example of radix sort:

Say you would like to sort 4 numbers, each represented by 4 binary digits: 0101 0111 1011 1000 We prepare two buckets, labeled "1" and "0" and do the following:

1. Look at the 1st digit (count from the least significant) and throw the number into corresponding bucket

"0" = (1000) "1" = (0101, 0111, 1011) Note that the order in which the numbers are stored is of importance (I chose (-) instead of {-} or [-] to emphasize this)

2. Repeat 1 for kth digit from k = 1:4, maintaining its relative order. Elements in bucket "0" is given the lower ordering than the items in bucket "1".

k = 2: "0" = (1000, 0101) "1" = (0111, 1011)

k = 3: "0" = (1000, 1011) "1" = (0101, 0111)

k = 4: "0" = (0101, 0111) "1" = (1000, 1011)

3. Collect the elements, starting from bucket "0" and maintaining its relative order. Viola! The numbers are now sorted.

Since it takes n*constant steps to categorize and throw each item into a bucket for each bit, the total time it takes to sort n number is given by n*constant *k where k = number of bits. Since k is usually 32 or 64 bits and therefore a constant, radix sort can sort n integers in O(n).

Floating Point Numbers

Integers can be represented as binary pretty readily (think two's complement), but how can we represent decimal numbers? Fixing the decimal point greatly limits the range and the precision of decimal point numbers we can represent. Hence the formulation of floating point numbers. Because floating point number is incredibly delicate and dangerous subject to touch upon, I leave the explanation to the experts- there are many articles written about floating point numbers [1][2], and I conclude my discussion with the following diagram from Professor Comer's textbook, Essentials of Computer Architecture[3]. The diagram outlines how floating point numbers are stored in memory.

PROJECT OUTLINE

Since I know how floating point numbers are represented in memory, can I tweak general implementation of radix sort to sort floating point numbers?

1. Implement radix sort for integers (CHECK)

2. Investigate how floating point is represented in memory and device a method to sort floating point numbers

3. Implement modified radix sort for floating point numbers

4. Naive analysis of running time?

RESULTS

I elected to build a queue with linked list to implement bucket. My first attempt at radix sort takes in an array of short ints. Download the source code here (upload coming this weekend). Compile using the command:

Input file should be a list of short ints, each number separated by a space. Simply run:

The program will print out the sorted list.

Task 2: Manipulating floating-point number in memory

Since bit-wise operators that proved to be so useful in manipulating integer number is no longer in my toolkit (compiler treats bit-wise operator on floating point numbers as explicit compile-time error). Since I do not know much about compilers (yet), I needed to work with the C language to access the binary representation of floating point numbers in memory.

I defined a new data type called bitFloat as follows:

typdef union {

float num;
struct {
unsigned char lsb: 4;
unsigned char msb: 4;
} bit[4];

} bitFloat;

"bitFloat" stores "float num", and its binary representation is stored in memory in the following way:

And the following code

int main(void) {
bitFloat bF1, bF2;
bF1.num = 1;
bF2.num = -1;
printf("sign bit of %2.0f:\t%d\n", bF1.num, bF1.bit[3].msb >> 3);
printf("sign bit of %2.0f:\t%d\n", bF2.num, bF2.bit[3].msb >> 3);
}

produces

 >> sign bit of  1: 0
>> sign bit of -1: 1

</br>

which is exactly what I sought after.

My strategy reduces to three steps:

1. Do radix sort according to the binary value of each number's mantissa

2. Do radix sort according to the binary value of each number's exponent

3. Separate negative and positive numbers, and print accordingly

To briefly explain what I mean by "print accordingly" in step 3, consider that 1000 > 100 while -1000 < -100. Then, if I define a function to print the positive numbers in a particular order, I need call the function in the reverse fashion for the negative numbers.

This project is taking too long! Will return later

REFERENCE

[1] Smith, Steven W. The Scientist and Engineer's Guide to Digital Signal Processing. <http://www.dspguide.com/ch4/3.htm>

[2] Schmalz, M.S. Organization of Computer Systems. http://www.cise.ufl.edu/~mssz/CompOrg/CDA-arith.html

[3] Comer, Douglas. Essentials of Computer Architecture.

Last Edit: 2 February 2013

## Alumni Liaison

Correspondence Chess Grandmaster and Purdue Alumni

Prof. Dan Fleetwood


text:
zbMATH — the first resource for mathematics

Schurity of S-rings over a cyclic group and generalized wreath product of permutation groups. (English. Russian original) Zbl 1278.20002
St. Petersbg. Math. J. 24, No. 3, 431-460 (2013); translation from Algebra Anal. 24, No. 3, 84-127 (2012).
Summary: With the help of the generalized wreath product of permutation groups introduced in the paper, the automorphism group of an S-ring over a finite cyclic group $$G$$ is studied. Criteria for the generalized wreath product of two such S-rings to be Schurian or non-Schurian are proved. As a byproduct, it is shown that the group $$G$$ is a Schur one (i.e., any S-ring over it is Schurian) whenever the total number $$\Omega(n)$$ of prime factors of the integer $$n=|G|$$ does not exceed 3. Moreover, the structure of a non-Schurian S-ring over $$G$$ is described in the case where $$\Omega(n)=4$$. In particular, the last result implies that if $$n=p^3q$$, where $$p$$ and $$q$$ are primes, then $$G$$ is a Schur group.

MSC:
 20C05 Group rings of finite groups and their modules (group-theoretic aspects) 20E22 Extensions, wreath products, and other compositions of groups 16S34 Group rings 05E30 Association schemes, strongly regular graphs 20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
Full Text:
References:
 [1] S. A. Evdokimov and I. N. Ponomarenko, On a family of Schur rings over a finite cyclic group, Algebra i Analiz 13 (2001), no. 3, 139 – 154 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 13 (2002), no. 3, 441 – 451. [2] S. A. Evdokimov and I. N. Ponomarenko, Characterization of cyclotomic schemes and normal Schur rings over a cyclic group, Algebra i Analiz 14 (2002), no. 2, 11 – 55 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 14 (2003), no. 2, 189 – 221. · Zbl 1056.20001 [3] S. A. Evdokimov and I. N. Ponomarenko, Recognition and verification of an isomorphism of circulant graphs in polynomial time, Algebra i Analiz 15 (2003), no. 6, 1 – 34 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 15 (2004), no. 6, 813 – 835. [4] R. A. Bailey, Cheryl E. Praeger, C. A. Rowley, and T. P. Speed, Generalized wreath products of permutation groups, Proc. London Math. Soc. (3) 47 (1983), no. 1, 69 – 82. · Zbl 0496.20018 · doi:10.1112/plms/s3-47.1.69 · doi.org [5] Sergei Evdokimov and Ilia Ponomarenko, Permutation group approach to association schemes, European J. Combin. 30 (2009), no. 6, 1456 – 1476. · Zbl 1228.05311 · doi:10.1016/j.ejc.2008.11.005 · doi.org [6] -, Schur rings over a product of Galois rings (submitted to Beiträge zur Algebra und Geometrie). · Zbl 1291.05227 [7] V. V. Ishkhanov, B. B. Lur$$^{\prime}$$e, and D. K. Faddeev, The embedding problem in Galois theory, Translations of Mathematical Monographs, vol. 165, American Mathematical Society, Providence, RI, 1997. Translated from the 1990 Russian original by N. B. Lebedinskaya. · Zbl 0883.12002 [8] M. H. Klin and R. Pöschel, The König problem, the isomorphism problem for cyclic graphs and the method of Schur rings, Algebraic methods in graph theory, Vol. I, II (Szeged, 1978) Colloq. Math. Soc. János Bolyai, vol. 25, North-Holland, Amsterdam-New York, 1981, pp. 405 – 434. [9] Ka Hin Leung and Shing Hing Man, On Schur rings over cyclic groups. II, J. Algebra 183 (1996), no. 2, 273 – 285. · Zbl 0860.20005 · doi:10.1006/jabr.1996.0220 · doi.org [10] R. Pöschel, Untersuchungen von \?-Ringen, insbesondere im Gruppenring von \?-Gruppen, Math. Nachr. 60 (1974), 1 – 27 (German). · Zbl 0273.20005 · doi:10.1002/mana.19740600102 · doi.org [11] Helmut Wielandt, Finite permutation groups, Translated from the German by R. Bercov, Academic Press, New York-London, 1964. · Zbl 0138.02501 [12] -, Permutation groups through invariant relations and invariant functions, Lecture Notes Dept. Math. Ohio State Univ., Columbus, Ohio, 1969.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
## xartaan one year ago Given a matrix A such that $A = \left[\begin{matrix}5 & -1 & 3 \\ -4 & 3 & -6 \\ -3 & 1 & 2\end{matrix}\right]$ I am asked to compute:$A-5I_3$ Now, since I_3 is the identity matrix, it looks like this right?: $\left[\begin{matrix}5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5\end{matrix}\right]$

1. xartaan

$\left[\begin{matrix}0 & -1 & 3 \\ -4 & -2 & -6 \\ -3 & 1 & -3\end{matrix}\right]$ should be my solution right?

2. KingGeorge

Looks good to me.

3. xartaan

Great, thanks!

4. KingGeorge

You're welcome.


text:
# A researcher wishes to research the health outcomes for Blackadults that grew up with a grandparent in their household versusBlack

###### Question:

A researcher wishes to research the health outcomes for Black adults that grew up with a grandparent in their household versus Black adults that grew up without a grandparent in their household. She knows that these populations are under-represented. Which of the following sampling procedures should she conduct if she wishes to over-sample for her populations? a.proportionate stratified sample b.systematic random sample c.disproportionate stratified sample d.simple random sample

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##### Yuestion 3 (3 points)The capacitors shown have all the same plate separation and the same plate depths The plate heights are the only way they are different from each other Capacitors shown are of three values: 1.0,or 2.0, or 3.0 microfarads The effective capagitance of the combination between the terminals a and b is measured with a capacitance meter: The value the meter reads in microfarads is going to be:4/303/201/2
Yuestion 3 (3 points) The capacitors shown have all the same plate separation and the same plate depths The plate heights are the only way they are different from each other Capacitors shown are of three values: 1.0,or 2.0, or 3.0 microfarads The effective capagitance of the combination between the ...
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##### Prove that if $x$ and $y$ are in the same conjugacy class of a group, then $|C(x)|=|C(y)|$. (This exercise is referred to in the discussion on the probability that two elements from a group commute.)
Prove that if $x$ and $y$ are in the same conjugacy class of a group, then $|C(x)|=|C(y)|$. (This exercise is referred to in the discussion on the probability that two elements from a group commute.)...