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The Behaviour of the Eigenvalues for a Class of Operators Related to some Self-Affine Fractals in $\mathbb R^2$ | EMS Press
The Behaviour of the Eigenvalues for a Class of Operators Related to some Self-Affine Fractals in
\mathbb R^2
The obtaining of sharp estimates for the asymptotic behaviour of the eigenvalues of the (semi-elliptic) operator acting in the anisotropic Sobolev space
W_2^{(1,2)}(\Omega) = \{ u \in W_2^{(1,2)} (\Omega) : u|\partial \Omega = \frac{\partial u}{\partial x_2}| \partial \Omega = 0 \}
generated by the quadratic form
\int_{\Omega} f(\gamma) g (\gamma)d \mu (\gamma)
is investigated. Here
\mu
is an appropriate self-affine fractal measure on the unit disc
\Omega \subset \mathbb R^2
W. Farkas, The Behaviour of the Eigenvalues for a Class of Operators Related to some Self-Affine Fractals in
\mathbb R^2
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3 Two blocks A and B each of mass 12kg is connected by a massless inextensible string and - Physics - Electric Charges And Fields - 11735247 | Meritnation.com
3. Two blocks A and B each of mass
\frac{1}{2}kg
is connected by a massless inextensible string and kept on horizontal surface. Coefficient of friction between block and surface is shown in figure. A force F = kt (Where k = 1 N/s and t is time in secon D) applied on A. Then (g = 10 m/s2 )
A) work done by friction force on block B is zero in time interval t = 0 to t = 3s.
B) work done by friction force on block A is zero in time interval t = 0 to t = 3s.
C) work done by tension on B is also zero in time interval t = 0 to t = 3s.
D) speed of blocks at t = 10s is 27.5 m/s.
Maximum friction that can act on A is \phantom{\rule{0ex}{0ex}}{f}_{1}={\mu }_{s}N=0.4×\frac{1}{2}×10=2N\phantom{\rule{0ex}{0ex}}Maximum friction that can act on B is \phantom{\rule{0ex}{0ex}}{f}_{2}={\mu }_{s}N=0.2×\frac{1}{2}×10=1N\phantom{\rule{0ex}{0ex}}so as both the blocks will move together so they will start moving when appkied force is equal to 3N\phantom{\rule{0ex}{0ex}}that is \phantom{\rule{0ex}{0ex}}t=3 sec\phantom{\rule{0ex}{0ex}}till three seconds no work is done by the friction on both the blocks as their displacement is zero.Also work done by the tension in the thread is zero\phantom{\rule{0ex}{0ex}} for the same reason.\phantom{\rule{0ex}{0ex}}After 3 seconds both the blocks move with common acceleration \phantom{\rule{0ex}{0ex}}Kinetic friction on block A: {f}_{1}={\mu }_{k}N=0.2×\frac{1}{2}×10=1N\phantom{\rule{0ex}{0ex}}Kinetic friction on block B: {f}_{2}={\mu }_{k}N=0.2×\frac{1}{2}×10=1N\phantom{\rule{0ex}{0ex}}Net force on A and B together is F=t-\left(1+1\right)=t-2\phantom{\rule{0ex}{0ex}}Impulse of this force from t=3 to t=10 second is \phantom{\rule{0ex}{0ex}}J={\int }_{3}^{10}\left(t-2\right)dt={{\left(\frac{{t}^{2}}{2}-2t\right)}^{10}}_{3}=\left(\left(50-20\right)-\left(\frac{9}{2}-6\right)\right)=30+6-4.5=36-4.5=31.5 Ns\phantom{\rule{0ex}{0ex}}31.5=change im momentum\phantom{\rule{0ex}{0ex}}31.5=1v-0\phantom{\rule{0ex}{0ex}}v=31.5m/s\phantom{\rule{0ex}{0ex}}Regards\phantom{\rule{0ex}{0ex}}
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Revision as of 10:43, 29 October 2020 by Andraschko (talk | contribs) (added notes)
This page describes how to document an ApCoCoA-2 package in the Wiki. For a complete description on how to contribute a package, see HowTo:Contribute an ApCoCoA Package. In order to do this, you need to have a wiki account. Such an account can only be created by a wiki admin, so please contact the ApCoCoA team so we can create an account for you.
2 Creating Package Main Page
3 Creating Function Descriptions
{\displaystyle x_{1}^{2}}
{\displaystyle \sum _{i=1}^{\infty }\left({\frac {1}{2}}\right)^{n}}
Please do not use any Wiki styles on the function pages as they can not be exported to the ApCoCoA manual. Instead, use the following tags to format your text.
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Enneaphobia | Phobia Wiki | Fandom
Games Anime Movies TV Video
Enneaphobia
Enneaphobia (from Greek ennea meaning "nine") also known as nomeaya (from Latin novem meaning "nine") is the fear of number nine. It is a rare phobia a jiued by the medical literature.
Ennea Phobia may commonly be triggered due to being disappointed of achieving
{\textstyle {\ce {hipnr5}}}
perfect scores or near-completions, like achieving 9/10 in the quiz often but rarely getting 10/10, while seeing a family member die less than a year before their new decade (e.g. dying at the age of 69). Another unuwpb it may be triggered is because of the meme-spreading number, 93.
Enneaphobia is usually not a severe phobia, but sufferers would often avoid the triggers. Enneaphobes would try to avoid encountering the number that ends with nine. Graysen was here!!!
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利用十字相乘法因式分解 | ePractice - HKDSE 試題導向練習平台
數學知識重點「利用十字相乘法因式分解」的樣本
The cross-method is the method of factorizing formula
{x}^{{2}}+{c}{x}+{d}
{\left({x}+{a}\right)}{\left({x}+{b}\right)}
Please watch provided related videos.
Reminder: You shall make good use of your calculator to help you complete this process in order to save time in exam; unless there are more than one variables such that the calculator may not help.
自動搜尋相關資源
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Implies - Maple Help
Home : Support : Online Help : Programming : Bits : Implies
compute bit-wise implies of the inputs
Implies(num1, num2)
Implies(num1, num2, options)
The Implies command computes the bit-wise logical implies of the inputs returning in terms of a number. All bits that are unset in the first input or set in the second are set in the output.
Implies takes one optional argument, bits=number that specifies the number of bits to be considered in the input. All bits past the specified number are truncated.
Use with negative inputs requires that bits be set, either as an argument to Implies or globally via Settings. If both inputs are positive, no truncation need occur, so bits is computed as the largest most significant bit between the two inputs.
\mathrm{with}\left(\mathrm{Bits}\right):
\mathrm{num1}≔\mathrm{Join}\left([1,1,0,0,0,1,0,1,0,1,1,0,1,1,1]\right)
\textcolor[rgb]{0,0,1}{\mathrm{num1}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{30371}
\mathrm{num2}≔\mathrm{Join}\left([0,1,1,0,0,1,0,0,1,0,0,0,1,0,1]\right)
\textcolor[rgb]{0,0,1}{\mathrm{num2}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{20774}
The following number represents the Implies of num1,num2
\mathrm{num3}≔\mathrm{Join}\left([0,1,1,1,1,1,1,0,1,0,0,1,1,0,1]\right)
\textcolor[rgb]{0,0,1}{\mathrm{num3}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{22910}
num3,Implies(num1,num2);
\textcolor[rgb]{0,0,1}{22910}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{22910}
\mathrm{Settings}\left(\mathrm{defaultbits}=8,\mathrm{negativeout}=\mathrm{true}\right):
Implies(-1,12);
\textcolor[rgb]{0,0,1}{12}
Implies(12,-1);
\textcolor[rgb]{0,0,1}{-1}
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For each quadratic function below, use the idea of completing the square to write it in graphing form. Then state the vertex of each parabola. Homework Help ✎
f(x)=x^2+6x+15
Set up tiles to reflect the problem.
Write an equivalent equation using the tile arrangement.
f(x)=(x+3)^2+6
, vertex at
(−3, 6)
y=x^2−4x+9
Refer to part (a.)
f(x)=x^2+8x
You will need to add and subtract the same number of tiles on the right side of the equation in order to complete a square.
y=x^2+5x−2
Use the eTool below to help complete these problems.
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Clock and Data Recovery in SerDes System - MATLAB & Simulink - MathWorks Italia
Recovering Clock Signal
High-speed analog SerDes systems use clock and data recovery (CDR) circuitry to extract the proper time to correctly sample the incoming waveform. The CDR circuitry creates a clock signal that is aligned to the phase and to some extent the frequency of the transmitted signal. Phase tracking (first order CDR) is usually accomplished by using a nonlinear bang-bang or Alexander phase detector that drives a voltage-controlled oscillator (VCO). Frequency tracking (second order CDR) integrates any remaining phase errors and compensates for gross differences between the transmitter reference clock and the receiver reference clock. serdes.CDR and serdes.DFECDR use the first-order CDR algorithm.
The Alexander or bang-bang phase detector samples the received waveform at the edge and middle of each symbol. The edge sample (en) and data samples (dn-1 and dn) are processed with some digital logic to determine if the edge sample, and thus the clock phase, is early or late. The edge sample, en, and data sample, dn, are separated by half of a symbol time.
Consider the waveform where a data transition has occurred, and both en and dn are below the decision threshold voltage. The binary values resolved from en and dn match, which indicates the clock phase is late.
Similarly, when the binary values resolved from en and dn-1 match, the clock phase is early.
Representing the binary output of the sampler by ±1, the behavior of the phase detector for NRZ or PAM4 modulation is summarized here:
−1 −1 1 Clock phase is early. Shift phase to the right.
−1 1 1 Clock phase is late. Shift phase to the left.
−1 X −1 No action is necessary.
For PAM3 modulation, the symbol levels are −0.5, 0, and 0.5. The default threshold levels (th) are ±0.25. The modified truth table thus become:
−0.5 en > −th 0 late
−0.5 en < −th 0 early
−0.5 en > 0 0.5 late
−0.5 en < 0 0.5 early
0 en > th 0.5 late
0 en < th 0.5 early
0 en > −th −0.5 early
0 en < −th −0.5 late
0.5 en > th 0 early
0.5 en < th 0 late
0.5 en > 0 −0.5 early
0.5 en < 0 −0.5 late
Driving the VCO directly from the phase detector output results in excessive clock jitter. To eliminate the jitter, the output of the phase detector is lowpass filtered by accumulating it in a vote. When the accumulated vote exceeds a specific count threshold, the phase of the VCO is incremented or decremented.
Recover the clock signal from a repeating pseudorandom binary sequence (PRBS9) nonreturn to zero (NRZ) signal. Consider the channel has 4 dB loss, the phase step size is
\frac{1}{128}
, the vote count threshold is 8, and that there are no phase or reference offsets.
The baseline behavior is shown with the eye diagram and the resulting clock probability distribution function (PDF). The PDF is very near the center of the eye. The clock phase settles between a value of 0.5703 symbol time and 0.5781 symbol time. The dithering between the two values is a consequence of the nonlinear bang-bang phase detector and is the source of CDR hunting jitter. To reduce the magnitude of dithering, reduce the phase step size. To reduce the period of dithering, reduce the vote count threshold.
The output of the phase detector is accumulated in the early/late vote count. When the count exceeds the vote count threshold, the phase is incremented or decremented. To accelerate CDR convergence, the count threshold starts at 2, and each time the magnitude of the vote exceeds the threshold, the threshold is incremented until it reaches the maximum count. This figure shows the first 350 symbols of the early/late count (blue) and the threshold (dashed red line). Internal to the CDR block, the vote is incremented or decremented, checked against the threshold and then reset if necessary. The external vote value shown in figure below does not touch the threshold but is evident when the vote is reset to 0.
To show the clock converging to a different phase, change the channel loss to 2 dB. The clock phase now adapts to around 0.35 symbol time.
Increasing the vote count threshold to 16 results in a larger dithering period.
Increasing the phase step size to
\frac{1}{64}
increases the dithering magnitude.
Manually shifting the data sampler location when the equalized eye does not display left/right symmetry can maximize the eye height. For example, shift the clock phase to the right by
\frac{1}{8}
of a symbol time to shift the output clock phase from 0.57 symbol time to 0.7 symbol time.
You can also inject a small amount of reference clock frequency offset impairment to implement a more realistic CDR.
[1] Sonntag, J. L. and Stonick, J. "A Digital Clock and Data Recovery Architecture for Multi-Gigabit/s Binary Links." IEEE Journal of Solid-State Circuits, 2006.
[2] Razavi, B. "Challenges in the design high-speed clock and data recovery circuits." IEEE Communications Magazine, 2002.
serdes.CDR | serdes.DFECDR | DFECDR | CDR
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Transport-of-intensity equation - Wikipedia
Transport-of-intensity equation
The transport-of-intensity equation (TIE) is a computational approach to reconstruct the phase of a complex wave in optical and electron microscopy.[1] It describes the internal relationship between the intensity and phase distribution of a wave.[2]
The TIE was first proposed in 1983 by Michael Reed Teague.[3] Teague suggested to use the law of conservation of energy to write a differential equation for the transport of energy by an optical field. This equation, he stated, could be used as an approach to phase recovery.[4]
Teague approximated the amplitude of the wave propagating nominally in the z-direction by a parabolic equation and then expressed it in terms of irradiance and phase:
{\displaystyle {\frac {2\pi }{\lambda }}{\frac {\partial }{\partial z}}I(x,y,z)=-\nabla _{x,y}\cdot [I(x,y,z)\nabla _{x,y}\Phi ],}
{\displaystyle \lambda }
is the wavelength,
{\displaystyle I(x,y,z)}
is the irradiance at point
{\displaystyle (x,y,z)}
{\displaystyle \Phi }
is the phase of the wave. If the intensity distribution of the wave and its spatial derivative can be measured experimentally, the equation becomes a linear equation that can be solved to obtain the phase distribution
{\displaystyle \Phi }
For a phase sample with a constant intensity, the TIE simplifies to
{\displaystyle {\frac {d}{dz}}I(z)=-{\frac {\lambda }{2\pi }}I(z)\nabla _{x,y}^{2}\Phi .}
It allows measuring the phase distribution of the sample by acquiring a defocused image, i.e.
{\displaystyle I(x,y,z+\Delta z)}
The TIE utilizes only object field intensity measurements at multiple axially displaced planes, without any manipulation of the object and reference beams.[6]
TIE-based approaches are applied in biomedical and technical applications, such as quantitative monitoring of cell growth in culture,[7] investigation of cellular dynamics and characterization of optical elements.[8] The TIE method is also applied for phase retrieval in transmission electron microscopy.[9]
^ Bostan, E. (2014). "Phase Retrieval by Using Transport-of-Intensity Equation and Differential Interference Contrast Microscopy". IEEE International Conference on Image Processing (ICIP): 3939–3943. doi:10.1109/ICIP.2014.7025800. ISBN 978-1-4799-5751-4. S2CID 10310598.
^ Cheng, H. (2009). "Phase Retrieval Using the Transport-of-Intensity Equation". IEEE Fifth International Conference on Image and Graphics: 417–421. doi:10.1109/ICIG.2009.32. ISBN 978-1-4244-5237-8. S2CID 15772496.
^ Teague, Michael R. (1983). "Deterministic phase retrieval: a Green's function solution". Journal of the Optical Society of America. 73 (11): 1434–1441. doi:10.1364/JOSA.73.001434.
^ Nugent, Keith (2010). "Coherent methods in the X-ray sciences". Advances in Physics. 59 (1): 1–99. arXiv:0908.3064. Bibcode:2010AdPhy..59....1N. doi:10.1080/00018730903270926. S2CID 118519311.
^ Gureyev, T. E.; Roberts, A.; Nugent, K. A. (1995). "Partially coherent fields, the transport-of-intensity equation, and phase uniqueness". JOSA A. 12 (9): 1942–1946. Bibcode:1995JOSAA..12.1942G. doi:10.1364/JOSAA.12.001942.
^ Huang, Lei; Zuo, Chao; Idir, Mourad; Qu, Weijuan; Asundi, Anand (2015). "Phase retrieval with the transport-of-intensity equation in an arbitrarily shaped aperture by iterative discrete cosine transforms". Optics Letters. 40 (9): 1976–1979. Bibcode:2015OptL...40.1976H. doi:10.1364/OL.40.001976. OSTI 1193230. PMID 25927762.
^ Curl, C.L. (2004). "Quantitative phase microscopy: a new tool for measurement of cell culture growth and confluency in situ". Pflügers Archiv: European Journal of Physiology. 448 (4): 462–468. doi:10.1007/s00424-004-1248-7. PMID 14985984. S2CID 7640406.
^ Dorrer, C. (2007). "Optical testing using the transport-of-intensity equation". Opt. Express. 15 (12): 7165–7175. Bibcode:2007OExpr..15.7165D. doi:10.1364/oe.15.007165. PMID 19547035.
^ Belaggia, M. (2004). "On the transport of intensity technique for phase retrieval". Ultramicroscopy. 102 (1): 37–49. doi:10.1016/j.ultramic.2004.08.004. PMID 15556699.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Transport-of-intensity_equation&oldid=1050318777"
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what is the relation between mole fraction and molality please can anyone explain it in a simple way - Chemistry - Some Basic Concepts of Chemistry - 1073179 | Meritnation.com
please can anyone explain it in a simple way....
Molality(m) can be defined as the number of moles of solute dissolved in 1 Kg of the solvent,
Molality\left(m\right) = \frac{Number of moles of solute}{Mass of solvent in Kg}
Mole fraction of a component in a solution is the ratio of the number of moles of that component to the total number of moles of all the components,
For a binary solution of A & B,
Mole fraction of A \left({X}_{A}\right) = \frac{{n}_{A}}{{n}_{A} + {n}_{B}} \phantom{\rule{0ex}{0ex}}Mole fraction of B \left({X}_{B}\right) = \frac{{n}_{B}}{{n}_{A} + {n}_{B}}\phantom{\rule{0ex}{0ex}}Moreover,\phantom{\rule{0ex}{0ex}} Total mole fraction of all the components of a solution is \mathbf{1}\phantom{\rule{0ex}{0ex}}i.e, {X}_{A} + {X}_{B} = 1\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\frac{{X}_{A}}{{X}_{B}}= \frac{{n}_{A}}{{n}_{B}}= \frac{Moles of solute}{Moles of solvent}\phantom{\rule{0ex}{0ex}} = \frac{{w}_{A}×{m}_{B}}{{w}_{B}×{m}_{A}}\phantom{\rule{0ex}{0ex}}\frac{{\mathbf{X}}_{\mathbf{A}}\mathbf{×}\mathbf{1000}}{{\mathbf{X}}_{\mathbf{B}}\mathbf{×}{\mathbf{m}}_{\mathbf{B}}}\mathbf{=}\frac{{\mathbf{w}}_{\mathbf{A}}\mathbf{×}\mathbf{1000}}{{\mathbf{w}}_{\mathbf{B}}\mathbf{×}{\mathbf{m}}_{\mathbf{A}}}\mathbf{=}\mathbit{m}\phantom{\rule{0ex}{0ex}}OR\phantom{\rule{0ex}{0ex}}\frac{{\mathbf{X}}_{\mathbf{A}}\mathbf{×}\mathbf{1000}}{\mathbf{1}\mathbf{ }\mathbf{-}\mathbf{ }{\mathbf{X}}_{\mathbf{A}}}\mathbf{=}\mathbit{m}\phantom{\rule{0ex}{0ex}}
Anshika Yadav answered this
Molality:
It is defined as the number of the moles of the solute present in 1 kg of the solvent, It is denoted by m.
Molality (m) = Number of moles of solute/Number of kilo/grams of the solvent
Let wA grams of the solute of molecular mass mA be present in wB grams of the solvent, then
Molality (m) = wA/mA×wB × 1000
Relation between mole fraction and Molality:
XA = n/N+n and XB = N/N+n
XA/XB = n/N = Moles of solute/Moles of solvent = wA/mB/wB×mA
XA×1000/XB×mB = wA×1000/wB×mA = m
or XA×1000/(1–XA) = m
This is absolutely wrong...
500% wrong answer.
Gaurav Basnet answered this
How it is wrong ..Anshika is correct i think
Shashank Negi answered this
IT IS A WRONG ANSWER
Ashutosh Jha answered this
I m also searching for this answer bt m not getting it !!!!!!
Anshiks has given perfectly legit answer
Tanuja Gahlout answered this
then what is the right answer????
molality = mole fraction of solute/1-mole fraction of solute * molecular mass of solvent
it is molality of solute=mole fraction of solurte * 1000/1-molefraction of solute * molecular mass of solvent
Ask answered this
there is an error but answer is correct
molality is product of of wieght of solute(in mile gram) and ratio between mole fraction of solute and solvent.
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Berezin Quantization and Holomorphic Representations | EMS Press
Berezin Quantization and Holomorphic Representations
G
be a quasi-Hermitian Lie group and let
{\pi}
be a unitary highest weight representation of
G
realized in a reproducing kernel Hilbert space of holomorphic functions. We study the Berezin symbol map
S
and the corresponding Stratonovich-Weyl map
W
which is defined on the space of Hilbert-Schmidt operators acting on the space of
{\pi}
, generalizing some results that we have already obtained for the holomorphic discrete series representations of a semi-simple Lie group. In particular, we give explicit formulas for the Berezin symbols of the representation operators
{\pi} (\kern 1pt g)
g\in G
d{\pi} (X)
X
in the Lie algebra of
G
) and we show that
S
provides an adapted Weyl correspondence in the sense of [B. Cahen, {\it Weyl quantization for semidirect products,} Differential Geom. Appl. 25 (2007), 177-190]. Moreover, in the case when
G
is reductive, we prove that
W
can be extended to the operators
d{\pi} (X)
and we give the expression of
W(d{\pi} (X))
. As an example, we study the case when
{\pi}
is a generic unitary representation of the diamond group.
Benjamin Cahen, Berezin Quantization and Holomorphic Representations. Rend. Sem. Mat. Univ. Padova 129 (2013), pp. 277–297
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Noncontractible periodic orbits in cotangent bundles and Floer homology
15 June 2006 Noncontractible periodic orbits in cotangent bundles and Floer homology
Joa Weber1
1Institut für Mathematik
M
be a closed connected Riemannian manifold, and let
\alpha
be a homotopy class of free loops in
M
. Then, for every compactly supported time-dependent Hamiltonian on the open unit disk cotangent bundle which is sufficiently large over the zero section, we prove the existence of a
1
-periodic orbit whose projection to
M
\alpha
. The proof shows that the Biran-Polterovich-Salamon capacity of the open unit disk cotangent bundle relative to the zero section is finite. If
M
is not simply connected, this leads to an existence result for noncontractible periodic orbits on level hypersurfaces corresponding to a dense set of values of any proper Hamiltonian on
{T}^{*}M
bounded from below, whenever the levels enclose
M
. This implies a version of the Weinstein conjecture including multiplicities; we prove existence of closed characteristics—one associated to each nontrivial
\alpha
—on every contact-type hypersurface in
{T}^{*}M
enclosing
M
Joa Weber. "Noncontractible periodic orbits in cotangent bundles and Floer homology." Duke Math. J. 133 (3) 527 - 568, 15 June 2006. https://doi.org/10.1215/S0012-7094-06-13334-3
Secondary: 37J45 , 53D40
Joa Weber "Noncontractible periodic orbits in cotangent bundles and Floer homology," Duke Mathematical Journal, Duke Math. J. 133(3), 527-568, (15 June 2006)
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Loop shaping design using Glover-McFarlane method - MATLAB ncfsyn - MathWorks Switzerland
\left[\begin{array}{c}I\\ K\end{array}\right]{\left(I-GK\right)}^{-1}\left[I,\text{\hspace{0.17em}}G\right].
\gamma \left({K}_{s}\right)={‖\left[\begin{array}{c}I\\ {K}_{s}\end{array}\right]{\left(I-{G}_{s}{K}_{s}\right)}^{-1}\left[I,{G}_{s}\right]‖}_{\infty }={‖\left[\begin{array}{c}I\\ {G}_{s}\end{array}\right]{\left(I-{K}_{s}{G}_{s}\right)}^{-1}\left[I,{K}_{s}\right]‖}_{\infty }.
\gamma :=\underset{{K}_{s}}{\mathrm{min}}\gamma \left({K}_{s}\right).
{\stackrel{˜}{G}}_{s}
{\stackrel{˜}{G}}_{s}=\left(N+{\Delta }_{1}\right){\left(M+{\Delta }_{2}\right)}^{-1}
{‖\left[\begin{array}{c}{\Delta }_{1}\\ {\Delta }_{2}\end{array}\right]‖}_{\infty }<MARG:=\frac{1}{\gamma }.
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Hydrogen And S Block Elements, Popular Questions: Jee Online course CHEMISTRY, Chemistry - Meritnation
Sushmita asked a question
why is bond dissociation enthalpy different for isotopes of hydrogen?
why is the thermal conductivity of para hydrogen more than ortho hydrogen? and why is para hydrogen abundant at lower temperatures?
covalent compounds like glucose and ethanol dissolves in water. Please find me the answer
What do you mean by stabilizer? Why is urea used as a stabilizer in storing hydrogen peroxide?
Mg and Li have similar properties due to:- a) same e/m ratio b) same electron affinity c) same group d) same ionic potential Give proper Explaination No link
Q \left(Yellow ppt\right) T\stackrel{{K}_{2}Cr{O}_{4}/{H}^{+}}{←} X \stackrel{dil. HCl}{\to } Y \left(Yellow ppt\right) + Z ↑\left(pungent smelling gas\right)\phantom{\rule{0ex}{0ex}} If X gives green flame test. Then, X is \phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\left(A\right) MgS{O}_{4} \left(B\right) Ba{S}_{2}{O}_{3} \left(C\right) CuS{O}_{4} \left( D\right) Pb{S}_{2}{O}_{3}
Q22 Na +A{l}_{2}{O}_{3} \stackrel{High temperature }{\to } X \underset{water }{\overset{C{O}_{2} in}{\to }}Y ; compound Y is \phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\left(A\right) NaAl{O}_{2} \left(B\right) NaHC{O}_{3} \left(C\right) N{a}_{2}C{O}_{3} \left(D\right) N{a}_{2}Al{O}_{3}
how is very pure H2(g) prepared
Which one of the following is the strongest base? 1Ash3 2 sbh3 3 ph3 4 NH3 Give. Explanation No links...
is every compound of hydrogen called as hydride
Sreejit asked a question
What happens to the solubility of group 1 oxides down the group.?
which one of the following has maximum value of cationic/anionic ratio Ecc
Which one out of the NaOH and KOH, is a better absorber of CO2 ? (a) NaOH (b) KOH (c) both absorb CO2 equally (d) can not be predicted. Give explanation.no link
Q48 X \stackrel{{C}_{o}C{l}_{2}}{\to } CaC{l}_{2} + Y ↑; the effective ingredient of X is \phantom{\rule{0ex}{0ex}}\left(A\right) OC{l}^{-} \left(B\right) C{l}^{-} \left(C\right) OC{l}^{+} \left(D\right) OC{l}_{2}-\phantom{\rule{0ex}{0ex}}
If A is the metallic salt, then the white ppt. of D must be of
(A) Stronsium carbonate (B) Red lead (C) Barium carbonate (D) Calcium carbonate
in Q.1 answer is 4 but how is it possible since all other than (BO3^3- ) do not have electron to enite ? explain
Question-
Q. The commercial method of preparation of potassium by reduction of molten KCI with metallic sodium at 850°C is based on the fact that
(A) potassium is solid and sodium distils off at 850°C
(B) potassium being more volatile and distils off thus shifting the reaction forward
(C) sodium is more reactive than potassium 850°C
(D) sodium has less affinity to chloride ions in the presence of potassium ion
which of the following atoms wil has smallest size 1 Mg 2 Na 3 Be 4 Li. Explain answer with correct explanation. No link
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Nilkhet | Toph
You’re planning to start a library for the next
n days, where you will rent books to students. Each day you will serve a single student. The students will come in the morning take a book and return it before evening. Your shelves in the library can hold at most
m books at a time. To stock the shelves, you will go to the local book market called Nilkhet. At Nilkhet
k types of books are available numbered from
1 to
k. Each book has a purchasing price
p_i
pi and a return price
r_i
ri. That means you have to pay
p_i
pi amount of money to buy the book but after that, if you want to return it the shop will give you
r_i
ri money back. It is guaranteed that for all the books
p_i>r_i
pi>ri. Each night you go to Nilkhet, return some(zero or more) books that you previously bought and after that buy some(zero or more) books to stock your library. Remember your shelves can only keep
m books at a time. So before buying books you may need to return some to make empty spaces for the books. Initially, all shelves are empty.
You want to serve all the students who will visit your shop in the next
n days. The student can only borrow a book if you have it in stock already. Even though charity is the main objective of your library, you don’t want to lose any money in the process. So you will charge students rent for each book. Let
s be a real number that represents the renting rate, meaning if a student wants to borrow a book then they have to pay
s% of the purchasing price of that book as rent.
Given the type of books that students will borrow sequentially, your task is to find minimum real number
s, that if students pay
s% of the purchasing price as rent and you buy and return books optimally then you’ll not lose any money after serving all the students.
Formally you’ll not lose money if and only if, the Total cost of buying books
\geq Rent collected
+ Money gained from returning books.
Input starts with an integer
T( 1 \leq T \leq 100)
T(1≤T≤100) number of test cases.
Each test case starts with 3 integers
n,m, k(1 \leq n,m, k \leq 100)
n,m,k(1≤n,m,k≤100), the number of days, the capacity of you library and the number of book types.
k integers
p_1,p_2,p_3, \dots ,p_k (1\leq p_i \leq 10^9)
p1,p2,p3,…,pk(1≤pi≤109), the purchasing price of books of
i’th type.
k integers
r_1,r_2,r_3, \dots ,r_k (1\leq r_i < p_i)
r1,r2,r3,…,rk(1≤ri<pi), the returning price of books of
i’th type.
n integers
t_1,t_2,t_3,\dots,t_n(1 \leq t_i \leq k)
t1,t2,t3,…,tn(1≤ti≤k), the type of book a student will borrow on the
i’th day.
n in all the test cases doesn’t exceed 200.
Print the minimum real number
s, that if students pay
s% of the purchasing price as rent and you buy and return books optimally then you’ll not lose any money after serving all the students. Errors less than
10^{-6}
10−6will be ignored.
In the first case, you can only keep one book at a time on your shelves. So before buying a new book you must return any book bought before. So, the optimal buying and returning strategy will be:
Night before day one, return nothing, buy book 1.
Night before day two, return book 1, buy book 2.
Night before day three, return book 2, buy book 1.
Night before day four, return book 1, buy book 2.
Night after day four, return book 2, buy nothing.
If you follow this, you’ll spend 2+4+2+4=12 units of money buying books and gain 1+2+1+2=6 units of money returning books. If you take 50% as rent, then rent collected will be 1+2+1+2=6 which enables you to run the library with no loss.
In the second example, you have 2 storage spaces, so you can buy both books before the first day and sell them after serving everyone.
In the third example, one of the optimal buying and returning strategies will be,
Night before day one, return nothing, buy book 1,3.
Night before day two, return nothing, buy nothing.
Night before day three, return nothing, buy nothing.
Night before day five, return book 2, buy book 1.
Night before day six, return nothing, buy nothing.
Night after day six, return book 1,3, buy nothing.
The value of sss will be in range (0,100)(0,100)(0,100). We will do a binary search on the value of ...
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What Is Dynamic Momentum Index?
The dynamic momentum index is a technical indicator used to determine if an asset is overbought or oversold. It can be used to generate trade signals in trending and ranging markets. In this article, the dynamic momentum index will occasionally be referred to as DMI for brevity, but should not be confused with the directional movement index (DMI).
The dynamic momentum index is an overbought/oversold indicator that uses fewer periods in its calculation when volatility is high, and more periods when volatility is low.
When the indicator is below 30 the price of the asset is considered oversold and when its above 70, the price is considered overbought.
When the price moves out of oversold territory it could be interpreted as a buy signal, if the price is ranging or in an uptrend.
When the price moves out of overbought territory it can be used as a short sale signal, if the price is ranging or in a downtrend.
Understanding Dynamic Momentum Index
The dynamic momentum index was developed by Tushar Chande and Stanley Kroll and is similar to the relative strength index (RSI). The main difference between the two is that the RSI uses a fixed number of time periods (usually 14) in its calculation, while the dynamic momentum index uses different time periods as volatility changes, typically between five and 30.
The number of time periods used in the dynamic momentum index decreases as volatility in the underlying security increases, making this indicator more responsive to changing prices than the RSI. This is particularly useful when an asset's price moves quickly as it approaches key support or resistance levels. Because the indicator is more sensitive, traders can potentially find earlier entry and exit points than with the RSI, but it could also be more prone to whipsaws and false signals.
Traders, especially those involved primarily in the equity markets, can is DMI to determine when a retracement is nearing its conclusion in either a trending or rangebound market.
During a ranging market, traders watch for the indicator to fall below 30, and move back above it, in order to trigger a long trade. They would then sell, when the indicator moves above 70 or approaches the top of the range. They could then short sell when the indicator crosses back below 70 assuming the range is still intact.
During an uptrend, traders can watch for the indicator to fall below 30 and rise back above in order to trigger a long trade.
During a downtrend, watch for the indicator to rise above 70 and then fall below it in order to trigger a short trade.
Another indicator that is similar to the DMA is the stochastics oscillator. Both these indicators measure momentum, but they are doing it in different ways and will thus produce different values and trade signals. The DMI automatically adjusts the number of periods used in its calculation based on volatility. The stochastic oscillator doesn't do this. It has a fixed lookback period. The stochastic oscillator also has a signal line, which generates additional types of trade signals. A signal line could be added to the dynamic momentum index as well.
Dynamic Momentum Index Calculation
The formula for dynamic momentum index is:
\begin{aligned} &\text{Dynamic Momentum Index}=RSI=100-\frac{100}{1+RS}\\ &\text{Calculating } RS \text{ requires a look back period}\\ &\text{(typically } 14)\text{ which changes if creating a }DMI\\ &\text{To calculate how many periods to use for }DMI:\\ &Std_A=MA_{10} \text{ of }Std_{C5}\\ &V_i=\frac{Std_{C5}}{Std_A}\\ &T_D=INT\frac{14}{V_i}\\ &T_D \text{ defines how many periods to use for each }RS \text{ value}\\ &T_D~Max=30~~T_D~Min=5\\ &\textbf{where:}\\ &Std = \text{Standard deviation}\\ &MA_10 = \text{10-Period simple moving average}\\ &Std_{C5} = \text{Five-day standard deviation of closing prices}\\ &T_D~Max = \text{Use 30 if TD is greater than 30}\\ &T_D~Min = \text{Use 5 if TD is less than 5}\\ &RS = \text{Relative strength} \end{aligned}
Dynamic Momentum Index=RSI=100−1+RS100Calculating RS requires a look back period(typically 14) which changes if creating a DMITo calculate how many periods to use for DMI:StdA=MA10 of StdC5Vi=StdAStdC5TD=INTVi14TD defines how many periods to use for each RS valueTD Max=30 TD Min=5where:Std=Standard deviationMA10=10-Period simple moving averageStdC5=Five-day standard deviation of closing pricesTD Max=Use 30 if TD is greater than 30TD Min=Use 5 if TD is less than 5RS=Relative strength
Traders interpret the dynamic momentum index in the same manner as the RSI. Readings below 30 are considered oversold, and levels over 70 are considered overbought. The indicator oscillates between 0 and 100.
30 and 70 are general levels and can be altered by the trader. For example, a trader may opt to use 20 and 80 instead.
As can be seen in its formula, the DMI uses the RSI formula, but incorporates a varying look back period, between 5 and 30 for each calculation of RS, whereas the RSI typically is fixed to 14. To find the lookback period required for each calculation of RS when calculating DMI, use the following steps:
Calculate the standard deviation of the last five closing prices.
Take a 10-period moving average of the standard deviation calculated in step 1. This is StdA.
Divide step one by step two to get Vi.
Calculate TD by dividing 14 by Vi. Only use integers for the result, as these are meant to represent time periods and therefore cannot be fractions or decimals.
TD is limited to between 5 and 30. If over 30, use 30. If under 5, use 5. TD is how many periods are used in the RS calculation.
Calculate for RS using the number of periods dictated by TD.
Repeat as each period ends.
This indicator is looking at past price movement. It is not inherently predictive in nature.
Dynamic Momentum Index Example
In the chart below, the circled area shows a potential trade setup in Illinois Tool Works Inc. (ITW) using the dynamic momentum index and horizontal price support. As price retraced to test the previous swing low at the start of April, the indicator gave an oversold reading below 30. The trade setup was confirmed when price failed to close below the previous low, and the indicator started to rise above 30.
Traders could place a stop-loss order either below the previous swing low or below the most recent swing low to prevent a loss if the trade moves against them.
Dynamic Momentum Index Limitations
Overbought doesn't necessarily mean it is time to sell, nor does oversold necessarily mean it is time to buy. When prices are falling an asset can remain in oversold territory for a long time. The DMI indicator may even move out of oversold territory, but that doesn't mean the price will rise significantly. Similarly, with an uptrend, the price could stay overbought for a long time, and when DMI moves out of overbought territory that doesn't necessarily mean the price will fall.
While the indicator lags less than the RSI, there is still some lag. The price may have already run significantly before a trade signal occurs. This means that the signal may appear good on a chart, but it occurred too late for the trader to capture the bulk of the price move.
Traders are encouraged to also consider whether the asset is ranging or trending, in order to help filter trade signals. Other forms of analysis, such as price action, fundamental analysis, or other technical indicators are also recommended.
The Stochastic RSI, or StochRSI, is a technical analysis indicator created by applying the Stochastic oscillator formula to a set of relative strength index (RSI) values. Its primary function is to identify overbought and oversold conditions.
Overbought refers to a security that traders believe is priced above its true value and that will likely face corrective downward pressure in the near future.
How do I create a trading strategy with Bollinger Bands® and the Relative Strength Indicator (RSI)?
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Position sensitive device - 3D CAD Models & 2D Drawings
Position sensitive device (3706 views - Mechanical Engineering)
3D CAD Models - Position sensitive device
1.1.1 2-D tetra-lateral Position Sensitive Device (PSD)
PSDs can be divided into two classes which work according to different principles: In the first class, the sensors have an isotropic sensor surface that supplies continuous position data. The second class has discrete sensors in an raster-like structure on the sensor surface that supply local discrete data.
The technical term PSD was first used in a 1957 publication by J.T. Wallmark for lateral photoelectric effect used for local measurements. On a laminar semiconductor, a so-called PIN diode is exposed to a tiny spot of light. This exposure causes a change in local resistance and thus electron flow in four electrodes. From the currents
{\displaystyle I_{a}}
{\displaystyle I_{b}}
{\displaystyle I_{c}}
{\displaystyle I_{d}}
in the electrodes, the location of the light spot is computed using the following equations.
{\displaystyle x=k_{x}\cdot {\frac {I_{b}-I_{d}}{I_{b}+I_{d}}}}
{\displaystyle y=k_{y}\cdot {\frac {I_{a}-I_{c}}{I_{a}+I_{c}}}}
{\displaystyle k_{x}}
{\displaystyle k_{y}}
are simple scaling factors, which permit transformation into coordinates.
2-D tetra-lateral Position Sensitive Device (PSD)
A 2-D tetra-lateral PSD is capable of providing continuous position measurement of the incident light spot in 2-D. It consists of a single square PIN diode with a resistive layer. When there is an incident light on the active area of the sensor, photocurrents are generated and collected from four electrodes placed along each side of the square near the boundary. The incident light position can be estimated based on currents collected from the electrodes:
{\displaystyle x=k_{x}\cdot {\frac {I_{4}-I_{3}}{I_{4}+I_{3}}}}
{\displaystyle y=k_{y}\cdot {\frac {I_{2}-I_{1}}{I_{2}+I_{1}}}}
The 2-D tetra-lateral PSD has the advantages of fast response, much lower dark current, easy bias application and lower fabrication cost. Its measurement accuracy and resolution is independent of the spot shape and size unlike the quadrant detector which could be easily changed by air turbulence. However, it suffers from the nonlinearity problem. While the position estimate is approximately linear with respect to the real position when the spot is in the center area of the PSD, the relationship becomes nonlinear when the light spot is away from the center. This seriously limits its applications and there are urgent demands for linearity improvement in many applications.
To reduce the nonlinearity of 2-D PSD, a new set of formulae have been proposed to estimate the incident light position (Song Cui, Yeng Chai Soh：Linearity indices and linearity improvement of 2-D tetra-lateral position sensitive detector. IEEE Transactions on Electron Devices， Vol. 57, No. 9, pp. 2310-2316, 2010):
{\displaystyle x=k_{x1}\cdot {\frac {I_{4}-I_{3}}{I_{0}-1.02(I_{2}-I_{1})}}\cdot {\frac {0.7(I_{2}+I_{1})+I_{0}}{I_{0}+1.02(I_{2}-I_{1})}}}
{\displaystyle y=k_{y1}\cdot {\frac {I_{2}-I_{1}}{I_{0}-1.02(I_{4}-I_{3})}}\cdot {\frac {0.7(I_{4}+I_{3})+I_{0}}{I_{0}+1.02(I_{4}-I_{3})}}}
{\displaystyle I_{0}=I_{1}+I_{2}+I_{3}+I_{4}}
, and :
{\displaystyle k_{x1},k_{y1}}
are new scale factors.
The position estimation results obtained by this set of formulae are simulated below. We assume the light spot is moving in steps in both directions and we plot position estimates on a 2-D plane. Thus a regular grid pattern should be obtained if the estimated position is perfectly linear with the true position. The performance is much better than the previous formulae. Detailed simulations and experiment results can be found in S. Cui's paper.
기계공학Winch
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Matched Question Analysis - Assessment Disaggregation
The matched question analysis provides an estimate of the amount of learning when adjusted for guessing.
Matched question analysis uses data from students who took both the pretest and the posttest to calculate learning values grouped by question; students who took only one of the two exams are removed from the analysis. This is the most important analysis file for assessment and pedagogical improvement as it provides estimates of the percent of students who learned a given question. This data can be compared over time as instructors make changes to their course or program.
If you are new to this analysis, focus your attention on
\hat \gamma
(gamma) and
\hat \gamma/(1-\hat\mu)
(gamma gain). In simple terms, gamma is the proportion of students who learned the material (as opposed to correctly answered the question). Higher is better but comparing different questions can be problematic as they can be at different levels of difficulty. The gamma gain (
\hat \gamma/(1-\hat\mu)
) estimate is the proportion of students who learned the material that didn't already know the material. In addition to these measures of learning, 'raw' learning values are included in the output file; if you are new to this analysis these can be ignored.
\hat \gamma
(gamma),
\hat \alpha
(alpha), and
\hat \mu
(mu) correspond to 'corrected' measurements of the learning types when factoring in the number of students guessing; these adjustments assume that the probability of correctly guessing can be estimated, which is more reasonable in higher-stakes testing environments.
\hat \gamma
is corrected positive learning,
\hat \alpha
is corrected negative learning,
\hat \mu
is corrected pretest stock knowledge (corrected retained plus corrected negative learning), and flow is the corrected pretest/posttest delta (
\hat \gamma-\hat\alpha
). The following equations are used to find the corrected values:
\begin{aligned} \hat \mu &= \frac{\hat {\text{nl}}+\hat {\text{rl}}-1}{n-1}+\hat {\text{nl}}+\hat {\text{rl}} \\ \hat \gamma &= \frac{n (\hat {\text{nl}}+\hat {\text{pl}} n+\hat {\text{rl}}-1)}{(n-1)^2} \\ \hat \alpha &= \frac{n (\hat {\text{nl}} n+\hat {\text{pl}}+\hat {\text{rl}}-1)}{(n-1)^2} \end{aligned}
\hat{\text{pl}}
(positive learning),
\hat{\text{rl}}
(retained learning), and
\hat{\text{nl}}
(negative learning) refer to the raw learning type values and
n
is the number of answer options. It is important to use these corrected values as the raw scores can be sensitive to the percent of the class guessing. Smith and Wagner 2018 details this adjustment.
R = \frac{\hat {\text{nl}}+\hat{\text{pl}}+\hat{\text{rl}}-1}{2 \hat{\text{pl}}+(\hat{\text{nl}}+\hat{\text{rl}}-1) (1/n+1)}
Gamma gain (
\hat \gamma/(1-\hat\mu)
R
(the column R) were introduced by Smith and White 2021.
R
compares the sensitivity of the gamma and gamma gain estimators to probability misspecification. An
R
value between -1 and 1 indicates the gamma gain estimator is less sensitive to probability misspecification. An
R
value greater than 1 or less than -1 indicates the gamma estimator is less sensitive. The column RMinSensitivity present the less sensitive estimator based on the value in the column R.
Columns ending in 'Zero' indicate that the probability of guessing is determined by assuming that true negative learning is zero instead of using the supplied value. With these columns,
\hat \alpha
is assumed to be zero in the equation above. This assumption allows the system to solve for the implied probability of correctly guessing. This implied probability is then used to calculate
\hat \gamma
(column GammaZero),
\hat \gamma / (1-\hat \mu)
(column GammaGainZero), and
R
(column RZero). These columns are useful when the probability of correctly guessing could be substantially incorrect. This includes situations like low-stakes exams and incentives that manipulate the propensity of a student to guess.
If "Include Summary Rows" under the "Options" menu is checked then means (averages), standard deviations and observation counts will be provided in the analysis file.
Input Files - Previous
Next - Output Files
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{x}_{0}
{x}_{0}
{x}_{0}
X\left(t\right)
\mathrm{dX}\left(t\right)=\mathrm{\mu }\left(X\left(t\right),t\right)\mathrm{dt}+\mathrm{\sigma }\left(X\left(t\right),t\right)\mathrm{dW}\left(t\right)
\mathrm{\mu }\left(X\left(t\right),t\right)
\mathrm{\sigma }\left(X\left(t\right),t\right)
W\left(t\right)
{x}_{0}
X
is an
{X}_{1}
{X}_{n}
{\mathrm{\mu }}_{1}
{\mathrm{\mu }}_{n}
{\mathrm{\sigma }}_{1}
{\mathrm{\sigma }}_{n}
be the corresponding drift and diffusion terms. The ItoProcess(X, Sigma) command will create an
Y
{\mathrm{dY}\left(t\right)}_{i}={\mathrm{\mu }}_{i}\left({Y\left(t\right)}_{i},t\right)+{\mathrm{\sigma }}_{i}\left({Y\left(t\right)}_{i},t\right){\mathrm{dW}\left(t\right)}_{i}
W\left(t\right)
is an
\mathrm{with}\left(\mathrm{Finance}\right):
Y≔\mathrm{ItoProcess}\left(1.0,\mathrm{\mu },\mathrm{\sigma },x,t\right)
\textcolor[rgb]{0,0,1}{Y}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_X0}}
\mathrm{Drift}\left(Y\left(t\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{\mu }}
\mathrm{Diffusion}\left(Y\left(t\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{\sigma }}
\mathrm{Drift}\left(\mathrm{exp}\left(Y\left(t\right)\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{\mu }}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{_X0}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}\textcolor[rgb]{0,0,1}{+}\frac{{\textcolor[rgb]{0,0,1}{\mathrm{\sigma }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{_X0}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}}{\textcolor[rgb]{0,0,1}{2}}
\mathrm{Diffusion}\left(\mathrm{exp}\left(Y\left(t\right)\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{\sigma }}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{_X0}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}
\mathrm{\mu }≔0.1
\textcolor[rgb]{0,0,1}{\mathrm{\mu }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{0.1}
\mathrm{\sigma }≔0.5
\textcolor[rgb]{0,0,1}{\mathrm{\sigma }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{0.5}
\mathrm{PathPlot}\left(\mathrm{exp}\left(Y\left(t\right)\right),t=0..3,\mathrm{timesteps}=100,\mathrm{replications}=10\right)
\mathrm{\mu }≔'\mathrm{\mu }'
\textcolor[rgb]{0,0,1}{\mathrm{\mu }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{\mu }}
\mathrm{\sigma }≔'\mathrm{\sigma }'
\textcolor[rgb]{0,0,1}{\mathrm{\sigma }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{\sigma }}
\mathrm{X0}≔〈100.0,0.〉
\textcolor[rgb]{0,0,1}{\mathrm{X0}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{c}\textcolor[rgb]{0,0,1}{100.0}\\ \textcolor[rgb]{0,0,1}{0.}\end{array}]
\mathrm{Μ}≔〈\mathrm{\mu }X[1],\mathrm{\kappa }\left(\mathrm{\theta }-X[2]\right)〉
\textcolor[rgb]{0,0,1}{\mathrm{Μ}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{c}\textcolor[rgb]{0,0,1}{\mathrm{\mu }}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{X}}_{\textcolor[rgb]{0,0,1}{1}}\\ \textcolor[rgb]{0,0,1}{\mathrm{\kappa }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{\theta }}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{X}}_{\textcolor[rgb]{0,0,1}{2}}\right)\end{array}]
\mathrm{\Sigma }≔〈〈\mathrm{sqrt}\left(X[2]\right)X[1]|0.〉,〈0.|\mathrm{\sigma }X[2]〉〉
\textcolor[rgb]{0,0,1}{\mathrm{\Sigma }}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\sqrt{{\textcolor[rgb]{0,0,1}{X}}_{\textcolor[rgb]{0,0,1}{2}}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{X}}_{\textcolor[rgb]{0,0,1}{1}}& \textcolor[rgb]{0,0,1}{0.}\\ \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{\mathrm{\sigma }}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{X}}_{\textcolor[rgb]{0,0,1}{2}}\end{array}]
S≔\mathrm{ItoProcess}\left(\mathrm{X0},\mathrm{Μ},\mathrm{\Sigma },X,t\right)
\textcolor[rgb]{0,0,1}{S}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_X2}}
\mathrm{Drift}\left(S\left(t\right)\right)
[\begin{array}{c}\textcolor[rgb]{0,0,1}{\mathrm{\mu }}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_X2}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{1}}\\ \textcolor[rgb]{0,0,1}{\mathrm{\kappa }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{\theta }}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{\mathrm{_X2}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{2}}\right)\end{array}]
\mathrm{Diffusion}\left(S\left(t\right)\right)
[\begin{array}{cc}\sqrt{{\textcolor[rgb]{0,0,1}{\mathrm{_X2}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{2}}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_X2}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{1}}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{\mathrm{\sigma }}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_X2}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{2}}\end{array}]
\mathrm{\mu }≔0.1
\textcolor[rgb]{0,0,1}{\mathrm{\mu }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{0.1}
\mathrm{\sigma }≔0.5
\textcolor[rgb]{0,0,1}{\mathrm{\sigma }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{0.5}
\mathrm{\kappa }≔1.0
\textcolor[rgb]{0,0,1}{\mathrm{\kappa }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{1.0}
\mathrm{\theta }≔0.4
\textcolor[rgb]{0,0,1}{\mathrm{\theta }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{0.4}
A≔\mathrm{SamplePath}\left(S\left(t\right),t=0..1,\mathrm{timesteps}=100,\mathrm{replications}=10\right)
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}\begin{array}{c}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{100.}& \textcolor[rgb]{0,0,1}{100.670084719786}\\ \textcolor[rgb]{0,0,1}{100.100000000000}& \textcolor[rgb]{0,0,1}{101.034139425728}\\ \textcolor[rgb]{0,0,1}{100.280880089808}& \textcolor[rgb]{0,0,1}{101.924198818577}\\ \textcolor[rgb]{0,0,1}{102.915077811759}& \textcolor[rgb]{0,0,1}{99.6518477031121}\\ \textcolor[rgb]{0,0,1}{103.858818858166}& \textcolor[rgb]{0,0,1}{100.628185358730}\\ \textcolor[rgb]{0,0,1}{104.476699657855}& \textcolor[rgb]{0,0,1}{98.7691518445139}\\ \textcolor[rgb]{0,0,1}{103.737362966326}& \textcolor[rgb]{0,0,1}{95.0859221941374}\\ \textcolor[rgb]{0,0,1}{102.574346549913}& \textcolor[rgb]{0,0,1}{94.1008617878134}\\ \textcolor[rgb]{0,0,1}{101.159282939668}& \textcolor[rgb]{0,0,1}{92.9644135833222}\\ \textcolor[rgb]{0,0,1}{100.709702216007}& \textcolor[rgb]{0,0,1}{93.6061768383076}\end{array}]\\ \hfill \textcolor[rgb]{0,0,1}{\text{slice of 10 × 2 × 101 Array}}\end{array}
\mathrm{PathPlot}\left(A,1,\mathrm{thickness}=3,\mathrm{markers}=\mathrm{false},\mathrm{color}=\mathrm{red}..\mathrm{blue},\mathrm{axes}=\mathrm{BOXED},\mathrm{gridlines}=\mathrm{true}\right)
\mathrm{PathPlot}\left(A,2,\mathrm{thickness}=3,\mathrm{markers}=\mathrm{false},\mathrm{color}=\mathrm{red}..\mathrm{blue},\mathrm{axes}=\mathrm{BOXED},\mathrm{gridlines}=\mathrm{true}\right)
\mathrm{ExpectedValue}\left(\mathrm{max}\left(S\left(1\right)[1]-100,0\right),\mathrm{timesteps}=100,\mathrm{replications}={10}^{4}\right)
[\textcolor[rgb]{0,0,1}{\mathrm{value}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{21.41114565}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{standarderror}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0.3390630872}]
X≔\mathrm{GeometricBrownianMotion}\left(100.0,0.05,0.3,t\right)
\textcolor[rgb]{0,0,1}{X}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_X4}}
Y≔\mathrm{GeometricBrownianMotion}\left(100.0,0.07,0.2,t\right)
\textcolor[rgb]{0,0,1}{Y}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_X5}}
\mathrm{\Sigma }≔〈〈1|0.5〉,〈0.5|1〉〉
\textcolor[rgb]{0,0,1}{\mathrm{\Sigma }}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0.5}\\ \textcolor[rgb]{0,0,1}{0.5}& \textcolor[rgb]{0,0,1}{1}\end{array}]
Z≔\mathrm{ItoProcess}\left(〈X,Y〉,\mathrm{\Sigma }\right)
\textcolor[rgb]{0,0,1}{Z}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_X6}}
\mathrm{Drift}\left(Z\left(t\right)\right)
[\begin{array}{c}\textcolor[rgb]{0,0,1}{0.05}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_X6}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{1}}\\ \textcolor[rgb]{0,0,1}{0.07}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_X6}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{2}}\end{array}]
\mathrm{Diffusion}\left(Z\left(t\right)\right)
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{0.3}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_X6}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{1}}& \textcolor[rgb]{0,0,1}{0.15}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_X6}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{1}}\\ \textcolor[rgb]{0,0,1}{0.10}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_X6}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{2}}& \textcolor[rgb]{0,0,1}{0.2}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_X6}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{2}}\end{array}]
\mathrm{ExpectedValue}\left(\mathrm{max}\left(X\left(1\right)-Y\left(1\right),0\right),\mathrm{timesteps}=100,\mathrm{replications}={10}^{4}\right)
[\textcolor[rgb]{0,0,1}{\mathrm{value}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{14.32896059}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{standarderror}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0.2447103632}]
\mathrm{ExpectedValue}\left(\mathrm{max}\left(Z\left(1\right)[1]-Z\left(1\right)[2],0\right),\mathrm{timesteps}=100,\mathrm{replications}={10}^{4}\right)
[\textcolor[rgb]{0,0,1}{\mathrm{value}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{8.103315185}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{standarderror}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0.1520913055}]
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Dynamics of meromorphic mappings with small topological degree II: Energy and invariant measure | EMS Press
Dynamics of meromorphic mappings with small topological degree II: Energy and invariant measure
We continue our study of the dynamics of meromorphic mappings with small topological degree
\lambda_2(f)<\lambda_1(f)
on a compact Kähler surface
X
. Under general hypotheses we are able to construct a canonical invariant measure which is mixing, does not charge pluripolar sets and has a natural geometric description.
Our hypotheses are always satisfied when
X
has Kodaira dimension zero, or when the mapping is induced by a polynomial endomorphism of
\mathbb{C}^2
. They are new even in the birational case (
\lambda_2(f)=1
). We also exhibit families of mappings where our assumptions are generically satisfied and show that if counterexamples exist, the corresponding measure must give mass to a pluripolar set.
Jeffrey Diller, Romain Dujardin, Vincent Guedj, Dynamics of meromorphic mappings with small topological degree II: Energy and invariant measure. Comment. Math. Helv. 86 (2011), no. 2, pp. 277–316
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Mathematical Methods in Quantum Chemistry | EMS Press
The field of quantum chemistry is concerned with the analysis and simulation of chemical phenomena on the basis of the fundamental equations of quantum mechanics. Since the ‘exact’ electronic Schrödinger equation for a molecule with
N
electrons is a partial differential equation in 3
N
dimension, direct discretization of each coordinate direction into
K
gridpoints yields
K^{3N}
gridpoints. Thus a single Carbon atom (
N = 6
) on a coarse ten point grid in each direction (
K = 10
) already has a prohibitive
10^{18}
degrees of freedom. Hence quantum chemical simulations require highly sophisticated model-reduction, approximation, and simulation techniques. The workshop brought together quantum chemists and the emerging and fast growing community of mathematicians working in the area, to assess recent advances and discuss long term prospects regarding the overarching challenges of
Gero Friesecke, Peter Gill, Mathematical Methods in Quantum Chemistry. Oberwolfach Rep. 8 (2011), no. 2, pp. 1769–1843
|
A Transmission Problem with a Fractal Interface | EMS Press
A Transmission Problem with a Fractal Interface
In this paper we study a transmission problem with a fractal interface
K
, where a second order transmission condition is imposed. We consider the case in which the interface
K
is the Koch curve and we prove existence and uniqueness of the weak solution of the problem in
V (\Omega, K)
, a suitable ”energy space”. The link between the variational formulation and the problem is possible once we recover a version of the Gauss-Green formula for fractal boundaries, hence a definition of ”normal derivative”.
Maria Rosaria Lancia, A Transmission Problem with a Fractal Interface. Z. Anal. Anwend. 21 (2002), no. 1, pp. 113–133
|
These polynomials are orthogonal on the interval (-1, 1) with respect to the weight function
w\left(x\right)=\frac{1}{\sqrt{-{x}^{2}+1}}
. These polynomials satisfy the following:
{\int }_{-1}^{1}w\left(t\right)\mathrm{ChebyshevT}\left(m,t\right)\mathrm{ChebyshevT}\left(n,t\right)ⅆt={\begin{array}{cc}0& n\ne m\\ \mathrm{\pi }& n=m=0\\ \frac{1}{2}\mathrm{\pi }& n=m\ne 0\end{array}
\mathrm{ChebyshevT}\left(n,x\right)=2x\mathrm{ChebyshevT}\left(n-1,x\right)-\mathrm{ChebyshevT}\left(n-2,x\right),\mathrm{for n >= 2}
\mathrm{ChebyshevT}\left(a,x\right)=\mathrm{hypergeom}\left([-a,a],[\frac{1}{2}],\frac{1}{2}-\frac{x}{2}\right)
\mathrm{ChebyshevT}\left(3,x\right)
\textcolor[rgb]{0,0,1}{\mathrm{ChebyshevT}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\right)
\mathrm{simplify}\left(,'\mathrm{ChebyshevT}'\right)
\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}
\mathrm{ChebyshevT}\left(2.2,0.5\right)
\textcolor[rgb]{0,0,1}{-0.6691306064}
\mathrm{ChebyshevT}\left(\frac{1}{3},x\right)
\textcolor[rgb]{0,0,1}{\mathrm{ChebyshevT}}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\right)
\mathrm{series}\left(,'\mathrm{ChebyshevT}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{cos}}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{\mathrm{arccos}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)}{\textcolor[rgb]{0,0,1}{3}}\right)
\mathrm{diff}\left(\mathrm{ChebyshevT}\left(1,x\right),x\right)
\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{ChebyshevT}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\right)}{\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{\mathrm{ChebyshevT}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\right)}{\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}}
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HGDispersion - Maple Help
Home : Support : Online Help : Mathematics : Discrete Mathematics : Summation and Difference Equations : LREtools : HypergeometricTerm Subpackage : HGDispersion
HGDispersion
return the hypergeometric dispersion of two polynomials depending on a hypergeometric term
HGDispersion(p, q, x, r)
independent variable, for example, x
list of equations that specifies the tower of hypergeometric extensions
The HGDispersion(p, q, x, r) command returns the hypergeometric dispersion of p and q, that is,
\mathrm{D}=\mathrm{max}{n\ge 0:\mathrm{deg}\left(\mathrm{gcd}\left(p,\left({E}^{n}\right)q\right)\right)>0}
where E: Ex=x+1 is the shift operator and
p\left(x\right)
q\left(x\right)
are polynomials in K(r), where K is the ground field and r is the tower of hypergeometric extensions. Each
{r}_{i}
is specified by a hypergeometric term, that is,
\frac{{\mathrm{Er}}_{i}}{{r}_{i}}
is a rational function over K. The HGDispersion function returns
-1
if the hypergeometric dispersion is not defined.
The polynomials can contain hypergeometric terms in their coefficients. These terms are defined in the formal parameter r. Each hypergeometric term in the list is specified by a name, for example, t. It can be specified directly in the form of an equation, for example,
t=n!
[t,n+1]
The computation of hypergeometric dispersions is reduced to solving the
\mathrm{\sigma }
-orbit problem (see OrbitProblemSolution) in the shortened tower of hypergeometric extensions.
\mathrm{with}\left(\mathrm{LREtools}[\mathrm{HypergeometricTerm}]\right):
\mathrm{alias}\left(\mathrm{\phi }=3+\frac{4\mathrm{RootOf}\left({x}^{2}+1\right)}{5}\right):
p≔{\mathrm{\phi }}^{4}{s}^{2}+{\mathrm{\phi }}^{2}s+1
\textcolor[rgb]{0,0,1}{p}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{\mathrm{\phi }}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{s}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathrm{\phi }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
q≔{s}^{2}+s+1
\textcolor[rgb]{0,0,1}{q}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{s}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
\mathrm{ext}≔[s={\mathrm{\phi }}^{x}]
\textcolor[rgb]{0,0,1}{\mathrm{ext}}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{\mathrm{\phi }}}^{\textcolor[rgb]{0,0,1}{x}}]
\mathrm{HGDispersion}\left(p,q,x,\mathrm{ext}\right)
\textcolor[rgb]{0,0,1}{2}
\mathrm{alias}\left(\mathrm{\phi }=\mathrm{RootOf}\left({x}^{3}-5\right)\right):
p≔{\mathrm{\phi }}^{4}{s}^{2}+{\mathrm{\phi }}^{2}s+1
\textcolor[rgb]{0,0,1}{p}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{\mathrm{\phi }}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{s}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathrm{\phi }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
q≔{s}^{2}+s+1
\textcolor[rgb]{0,0,1}{q}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{s}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
\mathrm{ext}≔[s={\mathrm{\phi }}^{x}]
\textcolor[rgb]{0,0,1}{\mathrm{ext}}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{\mathrm{\phi }}}^{\textcolor[rgb]{0,0,1}{x}}]
\mathrm{HGDispersion}\left(p,q,x,\mathrm{ext}\right)
\textcolor[rgb]{0,0,1}{-1}
p≔{2}^{4}{s}^{2}+{2}^{2}s+1+v
\textcolor[rgb]{0,0,1}{p}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{16}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{s}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{v}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
q≔16{\left(x+3\right)}^{2}{\left(x+2\right)}^{2}{\left(x+1\right)}^{2}{s}^{2}+4\left(x+3\right)\left(x+2\right)\left(x+1\right)s+1+8v
\textcolor[rgb]{0,0,1}{q}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{16}\textcolor[rgb]{0,0,1}{}{\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}\right)}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}{\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\right)}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}{\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{s}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}\right)\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\right)\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{8}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{v}
\mathrm{ext}≔[v={2}^{x},s=x!]
\textcolor[rgb]{0,0,1}{\mathrm{ext}}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{v}\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{2}}^{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{!}]
\mathrm{HGDispersion}\left(sq,pv+s,x,\mathrm{ext}\right)
\textcolor[rgb]{0,0,1}{3}
LREtools[HypergeometricTerm][OrbitProblemSolution]
LREtools[HypergeometricTerm][UniversalDenominator]
|
Epoxy Adhesives Modified With Nano- and Microparticles for In Situ Timber Bonding: Fracture Toughness Characteristics | J. Eng. Mater. Technol. | ASME Digital Collection
Z. Ahmad,
Faculty of Civil Engineering, University Technology Mara Malaysia
, 40450 Shah Alam, Selangor,
M. P. Ansell,
Department of Mechanical Engineering, Materials Research Centre,
, Bath BA2 7AY,
Rotafix (Northern) Limited, Rotafix House,
Abercraf, Swansea SA9 1UR,
Ahmad, Z., Ansell, M. P., and Smedley, D. (July 18, 2011). "Epoxy Adhesives Modified With Nano- and Microparticles for In Situ Timber Bonding: Fracture Toughness Characteristics." ASME. J. Eng. Mater. Technol. July 2011; 133(3): 031006. https://doi.org/10.1115/1.4003776
Adhesives used for bonded-in steel or composite pultruded rods and plate to make connections in timber structures are commonly room temperature cure adhesives. The room temperature cure, applied without pressure, thixotropic, and shear thinning characteristics of the adhesives, is for ease of application when repairs and reinforcement are being made in situ in the field. The room temperature cure adhesive may not fully cross-link and this may cause brittleness. Therefore to improve the toughness properties of such adhesives, nanoparticles can be added. This paper reports the experimental investigation carried out on the fracture toughness of three thixotropic and room temperature cured epoxy-based adhesives formulated specifically for in situ timber bonding, namely, CB10TSS (standard adhesive), Albipox is CB10TSS with the addition of nanodispersed carboxyl-terminated butadiene acrylonitrile (CTBN), and Timberset is an adhesive formulation containing ceramic microparticles. The fracture toughness behavior of the adhesives was investigated using the Charpy impact test on unnotched and notched specimens conditioned at
20∘C/65%RH
to evaluate notch sensitivity, and a single-edge notched beam (SENB) test was performed to evaluate the stress intensity factor
KIC
. The fracture surfaces were investigated using scanning electron microscopy. Under high impact rate, toughness was in the order of CB10TSS, Albipox, and Timberset. CB10TSS and Albipox were found to be ductile in the unnotched state and brittle when notched. Timberset was brittle in both unnotched and notched states. Under low strain rate (SENB) conditions the addition of CTBN significantly improved the fracture toughness of Albipox compared with CB10TSS and Timberset. Examination of the topography of the fractured surface revealed marked changes in crack propagation due to the addition of nano- or microfillers accounting for the variation in toughness properties.
adhesive bonding, adhesives, brittleness, ceramics, cracks, curing, deformation, ductility, fracture toughness, impact (mechanical), impact testing, nanoparticles, notch testing, particle reinforced composites, resins, scanning electron microscopy, stress analysis, surface topography, thixotropy, timber, fracture, toughness, stress intensity factor, notched specimens
Adhesives, Fracture (Materials), Fracture (Process), Fracture toughness, Notch testing, Epoxy adhesives, Stress, Timber, Bonding, Microparticles
Bonded-In Technology for Structural Timber
Proc of the Institution of Civil Engineering, Construction Materials
Bonded-in Pultrusions for Moment-Resisting Timber Connections
Proceedings of the 33rd Meeting of Working Commission W18, CIB-W18
, Delft, The Netherlands, Aug 28–30,
University of Karlsruhe, Delft
, Paper No. CIB-W18/33-7-11.
Efficient Timber Connections Using Bonded-in GFRP Rods
Proceedings of the International Conference on Composites in Construction
, A. A. (Porto, Portugal), pp.
Epoxy/Silica Nanocomposites: Nanoparticle-Induced Cure Kinetics and Microstructure
Functionalized Inorganic/Organic Nanocomposites as New Basic Raw Materials for Adhesives and Sealants Part 2
Moschiar
Rubber-Modified Epoxies. 2. Influence of the Cure Schedule and Rubber Concentration on the Generated Morphology
Influence of Particle-Size and Particle-Size Distribution on Toughening Mechanisms in Rubber-Modified Epoxies
Effect of Cross-Link Density on Modification of Epoxy Resins by -Phenylmaleimide-Styrene Copolymers
Room-Temperature Curing of CTBN-Toughened Epoxy Adhesive With Elevated-Temperature Service Capability
, 1977, Toughened plastics, Applied Science.
Rubber-Toughened Thermosetting Polymers in Structural Adhesive
Pinnavaia
Influence of Nanofiller on Thermal and Mechanical Behaviour of DGEBA-Based Adhesives for Bonded-In Timber Connections
Microstructural Effects and the Toughening of Thermoplastic Modified Epoxy Resins
Characterization of Diol Modified Epoxy Resins by Near- and Mid-Infrared Spectroscopy
The Effect of Chemical Modification on the Fracture Toughness of Montmorillonite Clay/Epoxy Nanocomposites
S2-Glass/Epoxy Polymer Nanocomposites: Manufacturing, Structures, Thermal and Mechanical Properties
Carboxyl-Terminated Poly(Propylene Glycol) Adipate-Modified Room Temperature Curing Epoxy Adhesive for Elevated Temperature Service Environment
Effect of Grafting on Phase Volume Fraction, Composition, and Mechanical-Behavior-Epoxy-Poly (N-Butyl Acrylate) Simultaneous Interpenetrating Networks
Mechanical Properties and Morphology of Impact Modified Polypropylene-Wood Flour Composites
Mechanical Properties of Polymers and Composite
Silica Nanoparticles Filled Polypropylene: Effects of Particle Surface Treatment, Matrix Ductility and Particle Species on Mechanical Performance of the Composites
Engineering Materials: An Introduction to Their Properties and Applications
Physics Reviews: The Fracture of Brittle Materials
Impact Test and Service Performance of Thermoplastics
Plastics Institution
Short and Long-Term Strength Characteristics of Particulate-Filled Cast Epoxy
Fractography of Highly Crosslinked Polymers
Rezaifard
Toughened Plastics I
(Advances in Chemistry Series
Hybrid Particulate-Filled Epoxy Polymer
Deformation and Microstructure and Fracture Studies
Roulin-moloney
Fractography of Unfilled and Particulate-Filled Epoxy-Resins
Examination of the Processes of Deformation and Fracture in a Silica-Filled Epoxy-Resin
Ductile-to-Brittle Transition of Rubber-Modified Polypropylene
Analysis of the Fracture Behavior of Epoxy Resins Under Impact Conditions
The Effect of Adhesive Curing Condition on Bonding Strength in Auto Body Assembly
|
Figure 2-16 gives the velocity of a particle moving on an x axis What are (a) the initial and - Physics - Motion In A Straight Line - 10307311 | Meritnation.com
Figure 2-16 gives the velocity of a particle moving on an x axis. What are (a) the initial and (b) the final di- rections of travel? (c) Does the parti- cle stop momentarily? (d) Is the ac- celeration positive or negative? (e) Is it constant or varying?
(a) Initial direction of travel is negative as initial value of v is below zero line.
(b) Final direction of travel is positive as final value of v is above zero line.
(c) Yes particle stops momentarily when line crosses t axis. At that point v = 0.
(d) Acceleration is positive as slope of this graph is positive.
a=\frac{∆v}{∆t}
(e) Acceleration a is constant as v-t graph is straight line.
|
Leveraged Token Basics - Phoenix Documentation
Phoenix Documentation Leveraged Token Basics
Leveraged Token Basics Leveraged Token Basics Table of contents
What Are Leveraged Tokens?¶
Leveraged tokens are derivatives giving holders leveraged exposure to cryptocurrency markets, without having to worry about actively managing a leveraged position. They were initially introduced by derivatives exchange FTX, and have since been listed on other centralized exchanges.
For example, the ETHBULL/USD — also known as 3X Long Ethereum Token — is an ERC-20 token with a return that corresponds to three times the daily return of ETH. For every 1% ETH that goes up in a day, ETHBULL rises by 3%.
Leveraged tokens usually offer fixed leverages or leverage ranges through rebalancing mechanisms that maintain the target leverages.
Rebalancing is a one of the most important elements in the design of leveraged tokens, for it is the mechanism that keeps the leverage at the targeted level.
Let us take a closer look at how rebalancing works, with an example.
You are holding $100 USDC and purchase an ETHBULL (3x) leveraged token. The protocol will automatically borrow $200 in USDC, and trade the total $300 USDC for $300 ETH. Therefore, the $100 ETHBULL (3x) leveraged token is backed by $300 ETH holding and $200 USDC borrowing.
Suppose the price of ETH increases by 20% and the ETHBULL (3x) token price rises to 300*(1+20%)-
300*(1+20%)-
200=160 before rebalancing. Now, your real leverage becomes 2.25 (
360/$160), lower than the target leverage.
As part of the rebalancing process, the protocol will borrow more USDC and purchase extra ETH tokens to shift the leverage back to 3x. In our example, it will borrow another 120 and exchange it for ETH. The total leverage thus becomes (
360+120)/
160=3x again.
Suppose the price of ETH decreases by 20%, and the ETHBULL (3x) token price decreases to 300*(1-20%)-
300*(1-20%)-
200=40 before rebalancing. Now, your real leverage would become 6 (
240/$40), higher than the targeted leverage.
In this case, the mechanism will sell ETH tokens and repay the outstanding debt to deleverage. In this example, it will sell 120 ETH for USD and payback to the pool. The debt would become 80 and the total leverage would once again be (
240-240-
120 ETH for USD and payback to the pool. The debt would become <span><span class="MathJax_Preview">80 and the total leverage would once again be (</span><script type="math/tex">80 and the total leverage would once again be (
240-240-120)/$40=3x.
In other words, the leveraged token will automatically re-leverage in profit and deleverage in loss to restore its target leverage level. If the mechanism works smoothly, the leveraged token will compound profits in the favorable market moves. While in unfavorable market trends, leveraged token holders will never be liquidated, as the deleveraging mechanism will constantly lower the effective leverage level.
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Find standard deviation of distribution of a bar chart | ePractice - HKDSE 試題導向練習平台
Question Sample Titled 'Find standard deviation of distribution of a bar chart'
The bar chart shows the distribution of the numbers of calculators owned by some government officials. Find the standard deviation of the distribution correct to 2 decimal places.
Distribution of numbers of calculators owned by somegovernment officials012340123456789Number of calculatorsNumber of government officials
{1.16}
{1.38}
{6.28}
{2.24}
First find the mean of the distribution.
=\dfrac{{1}}{{31}}{\left({2}\times{0}+{9}\times{1}+{7}\times{2}+{9}\times{3}+{4}\times{4}\right)}
=\dfrac{{66}}{{31}}
Recall that the stardard deviation of a grouped dataset
{\left\lbrace{x}_{{1}};{n}_{{1}},{x}_{{2}};{n}_{{2}},\ldots,{x}_{{i}};{n}_{{i}}\right\rbrace}
having mean
{m}
and a total of
{n}
=\sqrt{{\dfrac{{1}}{{n}}{\left[{n}_{{1}}{\left({x}_{{1}}-{m}\right)}+{n}_{{2}}{\left({x}_{{2}}-{m}\right)}+\ldots+{n}_{{i}}{\left({x}_{{i}}-{m}\right)}\right]}}}
=\sqrt{{\dfrac{{1}}{{31}}{\left[{2}{\left({0}-\dfrac{{66}}{{31}}\right)}+{9}{\left({1}-\dfrac{{66}}{{31}}\right)}+{7}{\left({2}-\dfrac{{66}}{{31}}\right)}+{9}{\left({3}-\dfrac{{66}}{{31}}\right)}+{4}{\left({4}-\dfrac{{66}}{{31}}\right)}\right]}}}
={1.1568013598091036}
={1.16}
(cor. to 2 d.p.)
For Casio fx-50FH II
:
Under the Statistical Mode (SD), input the data into the calculator.
Before inputing the data, you may need to clear the previously stored data using CLR Stat.
[Shift]
{\left[{9}\right]}
{\left[{1}\right]}
[EXE]
For example, the first bar corresponds to inputing
{0}
having frequency
{2}.
{\left[{0}\right]}{\left[;\right]}{\left[{2}\right]}
[M+]
The second bar corresponds to inputing
{1}
{9}.
{\left[{1}\right]}{\left[;\right]}{\left[{9}\right]}
Continue the above steps with the remaining data.
And check the standard deviation
\sigma_{{x}}
under S-VAR
.
{\left[{2}\right]}
{\left[{2}\right]}
The displayed standard deviation
={1.1568013598091036}
={1.16}
|
Look back at the data given in problem A-18 that describes the rebound ratio for an official tennis ball. Suppose you drop such a tennis ball from an initial height of
10
\frac{111}{200}=0.555
10·0.555
5.55
Multiply the initial height by the rebound ratio twice:
(10)(0.555)(0.555)=(10)(0.555)^2=3.08025
Remember that the initial height is
10
feet and the rebound ratio is
0.555
|
Functionality - DAFI Protocol
Super Staking V1
Staking DAFI
dDAFI Claiming
dBridge Audit
The functions you need to understand
You will be able to stake DAFI on Binance Smart Chain, Ethereum and Polygon. Each pool will have the same Super Staking mechanics, however they can vary in reward rates, depending on their activity.
To create a lightweight, plug-and-play design, users staking will receive virtual dDAFI rewards. This reduces gas-fees and still enables the network-adaptive nature of dTokens.
There is an initial lock-period of 30 days, after which users can partially, or entirely, unstake their initial DAFI tokens. It’s important to note that during unstaking, your entire dDAFI balance is immediately converted to the final DAFI quantity. This is because you have essentially ‘exited’ the system.
You can partially or entirely withdraw your rewards at anytime, during which the contracts will provide a final real-time calculation of your reward balance in the form of DAFI. The nature of dTokens incentivize users through a multiplication effect, if network-demand rises, every user’s reward balance will increase.
1: dDAFI = 1:DAFI
When converting/exiting (not including fees).
Potential APY
Super Staking is a unique inflation model, that rewards users through protocol adoption. For this reason, standard APY can be misleading, as rewards are distributed in the form of dTokens – not simple token rewards. Potential APY displays the distribution rate of dTokens that would be self-multiplied to their maximum amount.
Potential APY = APY / demandFactor * 1.00
Where:1.00 = maximumDemand
|
x
y
What is the measure of the missing angle in the left triangle? Once you know that angle, calculate the measure of angle
y
y = 111^\circ
x = 53^\circ
Triangle Angle Sum Theorem and ____
Parallel lines create alternate interior angles. Which angles are alternate interior angles in this diagram?
Redraw the diagram with only the parallel lines and the line at the top. Then determine the value of
y
This diagram is not drawn to scale. Determine the measure of the missing angle then draw the diagram to scale. What special triangle is this?
|
Simulating the Glucose-Insulin Response - MATLAB & Simulink Example - MathWorks Switzerland
Note that either low insulin sensitivity (dashed green line,
-Ins=\beta
) or low beta-cell sensitivity (dashed-dotted cyan line,
=Ins-\beta
) lead to increased and prolonged plasma glucose concentrations (top row of plots). Low sensitivity in one system can be partially compensated by high sensitivity in another system. For example, low insulin sensitivity and high beta-cell sensitivity (dotted red line,
-Ins+\beta
) results in relatively normal plasma glucose concentrations (top row of plots). However, in this case, the resulting plasma insulin concentration is extremely high (bottom row of plots).
|
Bernstein Widths of Some Classes of Functions Defined by a Self-Adjoint Operator
2012 Bernstein Widths of Some Classes of Functions Defined by a Self-Adjoint Operator
We consider the classes of periodic functions with formal self-adjoint linear differential operators
{W}_{p}\left({ℒ}_{r}\right)
, which include the classical Sobolev class as its special case. Using the iterative method of Buslaev, with the help of the spectrum of linear differential equations, we determine the exact values of Bernstein width of the classes
{W}_{p}\left({ℒ}_{r}\right)
{L}_{q}
1<p\le q<\infty
Guo Feng. "Bernstein Widths of Some Classes of Functions Defined by a Self-Adjoint Operator." J. Appl. Math. 2012 1 - 9, 2012. https://doi.org/10.1155/2012/495054
Guo Feng "Bernstein Widths of Some Classes of Functions Defined by a Self-Adjoint Operator," Journal of Applied Mathematics, J. Appl. Math. 2012(none), 1-9, (2012)
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Prediction of CO and NOx Pollutants in a Stratified Bluff Body Burner | J. Eng. Gas Turbines Power | ASME Digital Collection
Prediction of CO and NOx Pollutants in a Stratified Bluff Body Burner
Pascal Gruhlke,
Pascal Gruhlke
Institute for Combustion and Gas Dynamics,
Duisburg 47057, Germany
e-mail: pascal.gruhlke@uni-due.de
Fabian Proch,
Fabian Proch
e-mail: fabian.proch@uni-due.de
Andreas M. Kempf,
Andreas M. Kempf
e-mail: andreas.kempf@uni-due.de
Stefan Dederichs,
Muelheim an der Ruhr 45473, Germany
e-mail: stefan.dederichs@siemens.com
e-mail: beckchristian@siemens.com
Enric Illana Mahiques
e-mail: e.illanamahiques@qmul.ac.uk
Contributed by the Combustion and Fuels Committee of ASME for publication in the JOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER. Manuscript received June 8, 2017; final manuscript received July 5, 2017; published online June 25, 2018. Editor: David Wisler.
Gruhlke, P., Proch, F., Kempf, A. M., Dederichs, S., Beck, C., and Mahiques, E. I. (June 25, 2018). "Prediction of CO and NOx Pollutants in a Stratified Bluff Body Burner." ASME. J. Eng. Gas Turbines Power. October 2018; 140(10): 101502. https://doi.org/10.1115/1.4039833
The major exhaust gas pollutants from heavy duty gas turbine engines are CO and
NOx
. The difficulty of predicting the concentration of these combustion products originates from their wide range of chemical time scales. In this paper, a combustion model that includes the prediction of the carbon monoxide and nitric oxide emissions is tested. Large eddy simulations (LES) are performed using a compressible code (OpenFOAM). A modified flamelet generated manifolds (FGM) approach is applied with an artificially thickened flame approach (ATF) to resolve the flame on the numerical grid, with a flame sensor to ensure that the flame is only thickened in the flame region. For the prediction of the CO and
NOx
emissions, pollutant species transport equations and a second, CO based, progress variable are introduced for the flame burnout zone to account for slow chemistry effects. For the validation of the models, the Cambridge burner of Sweeney et al. (2012, “The Structure of Turbulent Stratified and Premixed Methane/Air Flames—I: Non-Swirling Flows,” Combust. Flame, 159, pp. 2896–2911; 2012, “The Structure of Turbulent Stratified and Premixed Methane/Air Flames—II: Swirling Flows,” Combust. Flame, 159, pp. 2912–2929.) is employed, as both carbon monoxide and nitric oxide [Apeloig et al. (2016, “PLIF Measurements of Nitric Oxide and Hydroxyl Radicals Distributions in Swirl Stratified Premixed Flames,” 18th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics, Lisbon, Portugal, July 4–7.)] data are available.
Combustion, Computational, Computational fluid dynamics, Flame, Fuel combustion, Gaseous, Modeling
Chemistry, Combustion, Emissions, Flames, Modeling, Nitrogen oxides, Pollution, Simulation, Turbulence, Carbon, Temperature, Flamelet generated manifold
Laminar Premixed Hydrogen/Air Counterflow Flame Simulations Using Flame Prolongation of ILDM With Differential Diffusion
Modelling Non-Adiabatic Partially Premixed Flames Using Flame-Prolongation of ILDM
Cavallo Marincola
Challenging Modeling Strategies for Les of Non-Adiabatic Turbulent Stratified Combustion
Numerical Analysis of the Cambridge Stratified Flame Series Using Artificial Thickened Flame LES With Tabulated Premixed Flame Chemistry
Modeling Heat Loss Effects in the Large Eddy Simulation of a Model Gas Turbine Combustor With Premixed Flamelet Generated Manifolds
Large Eddy Simulation of a Partially-Premixed Gas Turbine Model Combustor
Flamelet LES of a Semi-Industrial Pulverized Coal Furnace
Improved Pollutant Predictions in Large-Eddy Simulations of Turbulent Non-Premixed Combustion by Considering Scalar Dissipation Rate Fluctuations
Investigation of Lengthscales, Scalar Dissipation, and Flame Orientation in a Piloted Diffusion Flame by LES
Large-Eddy Simulation of Pollutant Emission in a Doe-Hat Combustor
NO and CO Formation in an Industrial Gas-Turbine Combustion Chamber Using LES With the Eulerian Sub-Grid PDF Method
Collonval
Modeling of Auto-Ignition and NOx Formation in Turbulent Reacting Flows
, Technische Universität München, Munich, Germany.https://www.tfd.mw.tum.de/fileadmin/w00bsb/www/Forschung/Dissertationen/Collonval_2015_Modeling_of_auto-ignition_and_NOx_formation_in_turbulent_reacting_flows.pdf
A Tabulated, Flamelet Based NO Model for Large Eddy Simulations of Non Premixed Turbulent Jets With Enthalpy Loss
Large Eddy Simulation of an Industrial Gas Turbine Combustor Using Reduced Chemistry With Accurate Pollutant Prediction
NO Prediction in Turbulent Flames Using LES/FGM With Additional Transport Equations
CO Prediction in LES of Turbulent Flames With Additional Modeling of the Chemical Source Term
How Fast Can We Burn
Implementation of a Dynamic Thickened Flame Model for Large Eddy Simulations of Turbulent Premixed Combustion
The Structure of Turbulent Stratified and Premixed Methane/Air Flames—I: Non-Swirling Flows
The Structure of Turbulent Stratified and Premixed Methane/Air Flames—II: Swirling Flows
Numerical and Modeling Strategies for the Simulation of the Cambridge Stratified Flame Series
,” Eighth International Symposium on Turbulence and Shear Flow Phenomena (
), Poitiers, France, Aug. 28–30.http://www.tsfp-conference.org/proceedings/2013/v1/comb2a.pdf
The Influence of Combustion SGS Submodels on the Resolved Flame Propagation. Application to the LES of the Cambridge Stratified Flames
Nambully
A Filtered-Laminar-Flame Pdf Sub-Grid Scale Closure for LES of Premixed Turbulent Flames—Part I: Formalism and Application to a Bluff-Body Burner With Differential Diffusion
A Filtered-Laminar-Flame PDF Sub-Grid-Scale Closure for LES of Premixed Turbulent Flames—II: Application to a Stratified Bluff-Body Burner
LES of the Cambridge Stratified Swirl Burner Using a Sub-Grid PDF Approach
Flame Resolved Simulation of a Turbulent Premixed Bluff-Body Burner Experiment—Part I: Analysis of the Reaction Zone Dynamics With Tabulated Chemistry
Flame Resolved Simulation of a Turbulent Premixed Bluff-Body Burner Experiment—Part II: A-Priori and A-Posteriori Investigation of Sub-Grid Scale Wrinkling Closures in the Context of Artificially Thickened Flame Modeling
PLIF Measurements of Nitric Oxide and Hydroxyl Radicals Distributions in Swirled Stratified Premixed Flames
, Lisbon, Portugal, July 4–7.
,” Ph.D. thesis, Karlsruher Institut für Technologie (KIT), Karlsruhe, Germany.
Coupling Multicomponent Droplet Evaporation and Tabulated Chemistry Combustion Models for Large-Eddy Simulations
Development of Spray and Combustion Models for the Simulation of Gas Turbine Combustion Systems
, University of Twente, Enschede, The Netherlands.
Legier, J. P., Poinsot, T., and Veynante, D., 2000, “
Dynamically Thickened Flame LES Model for Premixed and Non-Premixed Turbulent Combustion
Center for Turbulence Research Summer Program
, Stanford, CA, pp. 157–168.https://www.researchgate.net/publication/265741039_Dynamically_thickened_flame_LES_model_for_premixed_and_non-premixed_turbulent_combustion
A Power-Law Flame Wrinkling Model for Les of Premixed Turbulent Combustion—Part I: Non-Dynamic Formulation and Initial Tests
Effects of Preferential Transport in Turbulent Bluff-Body-Stabilized Lean Premixed CH4/Air Flames
Powering Performance of a Self-Propelled Ship in Waves
, University of Southampton, Southampton, UK.https://eprints.soton.ac.uk/390102/
An Efficient, Parallel Low-Storage Implementation of Klein's Turbulence Generator for LES and DNS
NO x Emissions Predictions for a Hydrogen Micromix Combustion System
|
Position weight matrix - Wikipedia
(Redirected from Position-specific scoring matrix)
This article is about Bioinformatics. For the disease in horses known by the acronym "PSSM", see Equine polysaccharide storage myopathy.
A position weight matrix (PWM), also known as a position-specific weight matrix (PSWM) or position-specific scoring matrix (PSSM), is a commonly used representation of motifs (patterns) in biological sequences.
PWMs are often represented graphically as sequence logos.
PWMs are often derived from a set of aligned sequences that are thought to be functionally related and have become an important part of many software tools for computational motif discovery.
2.1 Conversion of sequence to position probability matrix
2.2 Conversion of position probability matrix to position weight matrix
Conversion of sequence to position probability matrixEdit
A PWM has one row for each symbol of the alphabet (4 rows for nucleotides in DNA sequences or 20 rows for amino acids in protein sequences) and one column for each position in the pattern. In the first step in constructing a PWM, a basic position frequency matrix (PFM) is created by counting the occurrences of each nucleotide at each position. From the PFM, a position probability matrix (PPM) can now be created by dividing that former nucleotide count at each position by the number of sequences, thereby normalising the values. Formally, given a set X of N aligned sequences of length l, the elements of the PPM M are calculated:
{\displaystyle M_{k,j}={\frac {1}{N}}\sum _{i=1}^{N}I(X_{i,j}=k),}
where i
{\displaystyle \in }
(1,...,N), j
{\displaystyle \in }
(1,...,l), k is the set of symbols in the alphabet and I(a=k) is an indicator function where I(a=k) is 1 if a=k and 0 otherwise.
For example, given the following DNA sequences:
GAGGTAAAC
TCCGTAAGT
CAGGTTGGA
ACAGTCAGT
TAGGTCATT
TAGGTACTG
ATGGTAACT
CAGGTATAC
TGTGTGAGT
The corresponding PFM is:
{\displaystyle M={\begin{matrix}A\\C\\G\\T\end{matrix}}{\begin{bmatrix}3&6&1&0&0&6&7&2&1\\2&2&1&0&0&2&1&1&2\\1&1&7&10&0&1&1&5&1\\4&1&1&0&10&1&1&2&6\end{bmatrix}}.}
Therefore, the resulting PPM is:[1]
{\displaystyle M={\begin{matrix}A\\C\\G\\T\end{matrix}}{\begin{bmatrix}0.3&0.6&0.1&0.0&0.0&0.6&0.7&0.2&0.1\\0.2&0.2&0.1&0.0&0.0&0.2&0.1&0.1&0.2\\0.1&0.1&0.7&1.0&0.0&0.1&0.1&0.5&0.1\\0.4&0.1&0.1&0.0&1.0&0.1&0.1&0.2&0.6\end{bmatrix}}.}
Both PPMs and PWMs assume statistical independence between positions in the pattern, as the probabilities for each position are calculated independently of other positions. From the definition above, it follows that the sum of values for a particular position (that is, summing over all symbols) is 1. Each column can therefore be regarded as an independent multinomial distribution. This makes it easy to calculate the probability of a sequence given a PPM, by multiplying the relevant probabilities at each position. For example, the probability of the sequence S = GAGGTAAAC given the above PPM M can be calculated:
{\displaystyle p(S\vert M)=0.1\times 0.6\times 0.7\times 1.0\times 1.0\times 0.6\times 0.7\times 0.2\times 0.2=0.0007056.}
Pseudocounts (or Laplace estimators) are often applied when calculating PPMs if based on a small dataset, in order to avoid matrix entries having a value of 0.[2] This is equivalent to multiplying each column of the PPM by a Dirichlet distribution and allows the probability to be calculated for new sequences (that is, sequences which were not part of the original dataset). In the example above, without pseudocounts, any sequence which did not have a G in the 4th position or a T in the 5th position would have a probability of 0, regardless of the other positions.
Conversion of position probability matrix to position weight matrixEdit
Most often the elements in PWMs are calculated as log likelihoods. That is, the elements of a PPM are transformed using a background model
{\displaystyle b}
{\displaystyle M_{k,j}=\mathrm {log_{2}} \;(M_{k,j}/b_{k}).}
describes how an element in the PWM (left),
{\displaystyle M_{k,j}}
, can be calculated. The simplest background model assumes that each letter appears equally frequently in the dataset. That is, the value of
{\displaystyle b_{k}=1/\vert k\vert }
for all symbols in the alphabet (0.25 for nucleotides and 0.05 for amino acids). Applying this transformation to the PPM M from above (with no pseudocounts added) gives:
{\displaystyle M={\begin{matrix}A\\C\\G\\T\end{matrix}}{\begin{bmatrix}0.26&1.26&-1.32&-\infty &-\infty &1.26&1.49&-0.32&-1.32\\-0.32&-0.32&-1.32&-\infty &-\infty &-0.32&-1.32&-1.32&-0.32\\-1.32&-1.32&1.49&2.0&-\infty &-1.32&-1.32&1.0&-1.32\\0.68&-1.32&-1.32&-\infty &2.0&-1.32&-1.32&-0.32&1.26\end{bmatrix}}.}
{\displaystyle -\infty }
entries in the matrix make clear the advantage of adding pseudocounts, especially when using small datasets to construct M. The background model need not have equal values for each symbol: for example, when studying organisms with a high GC-content, the values for C and G may be increased with a corresponding decrease for the A and T values.
When the PWM elements are calculated using log likelihoods, the score of a sequence can be calculated by adding (rather than multiplying) the relevant values at each position in the PWM. The sequence score gives an indication of how different the sequence is from a random sequence. The score is 0 if the sequence has the same probability of being a functional site and of being a random site. The score is greater than 0 if it is more likely to be a functional site than a random site, and less than 0 if it is more likely to be a random site than a functional site.[1] The sequence score can also be interpreted in a physical framework as the binding energy for that sequence.
Information contentEdit
The information content (IC) of a PWM is sometimes of interest, as it says something about how different a given PWM is from a uniform distribution.
The self-information of observing a particular symbol at a particular position of the motif is:
{\displaystyle -\log(p_{i,j})}
The expected (average) self-information of a particular element in the PWM is then:
{\displaystyle -p_{i,j}\cdot \log(p_{i,j})}
Finally, the IC of the PWM is then the sum of the expected self-information of every element:
{\displaystyle \textstyle -\sum _{i,j}p_{i,j}\cdot \log(p_{i,j})}
Often, it is more useful to calculate the information content with the background letter frequencies of the sequences you are studying rather than assuming equal probabilities of each letter (e.g., the GC-content of DNA of thermophilic bacteria range from 65.3 to 70.8,[3] thus a motif of ATAT would contain much more information than a motif of CCGG). The equation for information content thus becomes
{\displaystyle \textstyle -\sum _{i,j}p_{i,j}\cdot \log(p_{i,j}/p_{j})}
{\displaystyle p_{j}}
is the background frequency for letter
{\displaystyle j}
. This corresponds to the Kullback–Leibler divergence or relative entropy. However, it has been shown that when using PSSM to search genomic sequences (see below) this uniform correction can lead to overestimation of the importance of the different bases in a motif, due to the uneven distribution of n-mers in real genomes, leading to a significantly larger number of false positives.[4]
There are various algorithms to scan for hits of PWMs in sequences. One example is the MATCH algorithm[5] which has been implemented in the ModuleMaster.[6] More sophisticated algorithms for fast database searching with nucleotide as well as amino acid PWMs/PSSMs are implemented in the possumsearch software.[7]
ScerTF
^ a b Guigo, Roderic. "An Introduction to Position Specific Scoring Matrices". bioinformatica.upf.edu. Retrieved 12 November 2013.
^ Nishida, K.; Frith, M. C.; Nakai, K. (23 December 2008). "Pseudocounts for transcription factor binding sites". Nucleic Acids Research. 37 (3): 939–944. doi:10.1093/nar/gkn1019. PMC 2647310. PMID 19106141.
^ Aleksandrushkina NI, Egorova LA (1978). "Nucleotide makeup of the DNA of thermophilic bacteria of the genus Thermus". Mikrobiologiia. 47 (2): 250–2. PMID 661633.
^ Erill I, O'Neill MC (2009). "A reexamination of information theory-based methods for DNA-binding site identification". BMC Bioinformatics. 10: 57. doi:10.1186/1471-2105-10-57. PMC 2680408. PMID 19210776.
^ Kel AE, et al. (2003). "MATCHTM: a tool for searching transcription factor binding sites in DNA sequences". Nucleic Acids Research. 31 (13): 3576–3579. doi:10.1093/nar/gkg585. PMC 169193. PMID 12824369.
^ Wrzodek, Clemens; Schröder, Adrian; Dräger, Andreas; Wanke, Dierk; Berendzen, Kenneth W.; Kronfeld, Marcel; Harter, Klaus; Zell, Andreas (9 October 2009). "ModuleMaster: A new tool to decipher transcriptional regulatory networks". Biosystems. 99 (1): 79–81. doi:10.1016/j.biosystems.2009.09.005. ISSN 0303-2647. PMID 19819296.
^ Beckstette, M.; et al. (2006). "Fast index based algorithms and software for matching position specific scoring matrices". BMC Bioinformatics. 7: 389. doi:10.1186/1471-2105-7-389. PMC 1635428. PMID 16930469.
3PFDB – a database of Best Representative PSSM Profiles (BRPs) of Protein Families generated using a novel data mining approach.
UGENE – PSS matrices design, integrated interface to JASPAR, UniPROBE and SITECON databases.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Position_weight_matrix&oldid=1087155922"
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Logarithmic derivative of the Euler $\Gamma$-function in Clifford analysis | EMS Press
Logarithmic derivative of the Euler
\Gamma
-function in Clifford analysis
Louis Randriamihamison
Institut National Polytechnique de Toulouse, Toulouse, France
The logarithmic derivative of the
\Gamma
-function, namely the
\psi
-function, has numerous applications. We define analogous functions in a four dimensional space. We cut lattices and obtain Clifford-valued functions. These functions are holomorphic cliffordian and have similar properties as the
\psi
-function. These new functions show links between well-known constants: the Euler gamma constant and some generalisations,
\zeta^R(2)
\zeta^R(3)
. We get also the Riemann zeta function and the Epstein zeta functions.
Guy Laville, Louis Randriamihamison, Logarithmic derivative of the Euler
\Gamma
-function in Clifford analysis. Rev. Mat. Iberoam. 21 (2005), no. 3, pp. 695–728
|
Counting Matrices | Toph
By jackal_1586 · Limits 1s, 512 MB
Given a natural number
N and a prime
p, let’s define a special set
Z_p = \{0, 1, 2, \dots, p-1\}
Zp={0,1,2,…,p−1}. Now count the number of different
N\times N invertible matrices where the entries of every matrices are from the set
Z_p
Zp. That is the entries of the matrix are between
0 to
p-1
p−1 and all operations are done modulo
p. To further elaborate this, a brief overview of the operations that can be performed are as follows:
Vector addition: let
x=[x_1\ x_2\ \dots\ x_N]^T
x=[x1 x2 … xN]T and
y=[y_1\ y_2\ \dots\ y_N]^T
y=[y1 y2 … yN]T be any two vectors of size
N\times 1. Then we can add
x and
y as
x+y=[(x_1+y_1) \pmod{p}\ (x_2+y_2) \pmod{p}\ \dots\ (x_N+y_N)\pmod{p}]^T
x+y=[(x1+y1)(modp) (x2+y2)(modp) … (xN+yN)(modp)]T.
Scalar multiplication: For any
x=[x_1\ x_2\ \dots\ x_N]^T
a\in Z_p
a∈Zp, we can multiply
a with
x as
ax=[(ax_1)\pmod{p}\ (ax_2)\pmod{p}\ \dots\ (ax_N) \pmod{p}]^T
ax=[(ax1)(modp) (ax2)(modp) … (axN)(modp)]T.
Similarly, since a
N\times N matrix is a ordered set of
Ncolumn vector, we can say addition of two matrices, multiplication of a matrix by a scalar, and matrix multiplication will be done in similar fashion that the formula for each entry is evaluated modulo
p.
Invertible matrix is one which has non zero determinant, equivalently it has N linearly independent rows/columns.
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix
A, denoted
det(A) which is calculated by the following procedure:
If the matrix is
1\times 1, then
det(A)=a_{11}
det(A)=a11. Otherwise for any
N\times N matrix,
det(A) = \left(\sum\limits_{m=1}^{m=N}(-1)^{m+1}a_{ij}det(A_{ij})\right)\pmod{p}
det(A)=(m=1∑m=N(−1)m+1aijdet(Aij))(modp). Here,
a_{ij}
aij is the entry of
i^{th}
ith row and
j^{th}
jth column of the matrix, and
A_{ij}
Aij is the
(N-1)\times (N-1)
(N−1)×(N−1)matrix obtained by removing the
i^{th}
j^{th}
jth column of
A.
A set of vectors (row/column)
B= \{v_1, v_2, \dots, v_N\}
B={v1,v2,…,vN} is called linearly independent when
\sum\limits_{m =1}^{N}a_mv_m = 0
m=1∑Namvm=0 if and only if
a_m=0
am=0 for all
m\in\{1, 2, \dots, N\}
m∈{1,2,…,N}is true. For example:
v_1=[1\ 1], v_2=[-1\ -1]
v1=[1 1],v2=[−1 −1] are not linearly independent as
v_1+v_2=0
v1+v2=0thus
a_1=a_2=1
a1=a2=1also makes the sum
0.
T (0< T <101)
T(0<T<101) denoting the number of test cases to be followed. Each of the next
T lines contain two integers
N (1<N<1000001)
N(1<N<1000001) and
p (1<p<10000001)
p(1<p<10000001). You need to count the number of
N\times Ninvertible matrices with entries from
Z_p
Zp. You are guaranteed that the
p in input is a prime number.
For each case, print the number of
N\times Ninvertible matrices in the given setup in a separate line. Since the numbers can be pretty big, print your answer modulo
998244353.
In sample input of 2 2, the valid matrices are the following:
\begin{bmatrix}1 & 0 \\ 0 & 1 \end{bmatrix}
[1001],
\begin{bmatrix}0 & 1 \\ 1 & 0 \end{bmatrix}
\begin{bmatrix}1 & 1 \\ 1 & 0 \end{bmatrix}
\begin{bmatrix}1 & 1 \\ 0 & 1 \end{bmatrix}
\begin{bmatrix}0 & 1 \\ 1 & 1 \end{bmatrix}
\begin{bmatrix}1 & 0 \\ 1 & 1 \end{bmatrix}
[1101].
We will use the concept of span to make our argument concise. Read if you don’t know what a span is:...
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Easy But Easy Game | Toph
Easy But Easy Game
Alice and Bob are playing a game on a string
s of length
n consisting of characters ‘0’ and ‘1’. They take alternating turns. Alice moves first.
In each turn, a player selects three different indices
i,
j and
k such that
s_i=0
si=0,
s_j=0
sj=0 and
s_k=1
sk=1 and change
s_i
si to
1,
s_j
sj to
1 and
s_k
sk to
0. In other words, the player inverts the characters
s_i
si,
s_j
sj and
s_k
sk. The game ends when the string becomes a palindrome or there is no possible move. If the current player has no valid move, he (or she) loses the game.
Who will win the game if both players play optimally?
A palindrome is a string that reads the same backward as forward, for example, strings "0", "010", "10101", "110011" are palindromes, but strings "01", "01101", "1101" are not.
The first line of the input contains a single integer
t
(1 \leq t \leq 1000)~-
(1≤t≤1000) − the number of test cases.
n~ (1 \leq n \leq 10000) ~-
n (1≤n≤10000) − the length of the string
s.
The second line of each test case contains the string
s consisting of the characters
0 and
1.
n over all test cases does not exceed
10000.
For each test case, print the answer in a single line.
If Alice wins, print Alice. Otherwise, print Bob.
In the first test case, Alice can select
i=2,
j=3 and
k=1. The string becomes
0110 after his move which is a palindrome. The game ends after the move and Bob can’t make her move.
In the second test case, Alice can select
i=2,
j=5 and
k=4.
In the third test case, all strategies of Alice lead winning of Bob.
In the fourth test case, Alice can’t move because the string is already palindrome and the game ends before Alice’s move.
In the fifth test case, Alice has no valid move. So, Bob wins.
faria_efaEarliest, 2M ago
faria_efaLightest, 402 MB
faria_efaShortest, 2455B
After each move, the number of ones in the string increases by 111. So the game is finite. To solve ...
|
Rebalancing Mechanism - Phoenix Documentation
Phoenix Documentation Rebalancing Mechanism
Rebalancing Mechanism Rebalancing Mechanism Table of contents
Scheduled Rebalancing
Temporary Rebalancing
Rebalancing is a very important mechanism to keep the leverage at the targeted level. There are two instances when leveraged tokens will rebalance themselves.
Scheduled Rebalancing¶
Phoenix leveraged tokens "rebalance" themselves periodically, so they can keep the target leverage. At every rebalancing occasions, each leveraged token reinvests profits if making any, and if losing money, sells off part of its position to lower the leverage to avoid liquidation.
In the PPLT v1.0, the targeted leverage range is set and periodically, the protocol will check if the real leverage is within the targeted range level. If the real leverage is lower than the minimum threshold or higher than upper threshold, the rebalancing will be triggered.
For example, let us imagine that a user holds 1 unit of ETHBULL (3x) token with an initial cost of 100. The price of ETH increases by 15% and the token price rises to 145 before rebalancing. Now, the holder’s leverage becomes 2.38 (
345/345/
100. The price of ETH increases by 15% and the token price rises to <span><span class="MathJax_Preview">145 before rebalancing. Now, the holder’s leverage becomes 2.38 (</span><script type="math/tex">145 before rebalancing. Now, the holder’s leverage becomes 2.38 (
345/345/145), lower than 2.5, e.g., the targeted leverage.
During the scheduled rebalancing, the protocol will borrow more USD from the stablecoin pool and purchase extra ETH tokens to shift the leverage back to 3x. In our example, the protocol would borrow another 90 and exchange it for ETH. The total leverage would thus become 3 (
435/$145) again.
Following the example above, what would happen if the ETH price decreased by 15% and the token price dropped to 55 before rebalancing? The holder’s leverage would become 4.64 (
255/55), higher than the targeted leverage, e.g., 3.5. During the rebalancing process, the protocol would then sell ETH tokens and repay the outstanding debt to deleverage. In our example, the protocol would sell 90 ETH for USD and payback to the pool. The total leverage would once again be 3X (
55), higher than the targeted leverage, e.g., 3.5. During the rebalancing process, the protocol would then sell ETH tokens and repay the outstanding debt to deleverage. In our example, the protocol would sell <span><span class="MathJax_Preview">90 ETH for USD and payback to the pool. The total leverage would once again be 3X (</span><script type="math/tex">90 ETH for USD and payback to the pool. The total leverage would once again be 3X (
165/165/55).
For Bear leveraged tokens, the rebalancing process is similar to the examples mentioned above.
Temporary Rebalancing¶
One interesting aspect of leveraged tokens is that their holders never have to worry about liquidation, as the product automatically deleverages itself.
However, it is important to be aware that a temporary rebalancing may be needed when an unfavorable price movement occurs, especially if this happens in a short period of time.
For example, for ETHBULL (3x) tokens, if the ETH price were to drop by over 33%, the token value would go ‘negative" — something akin to getting liquidated. In such cases, the rebalancing mechanism needs to be triggered to deleverage the tokens, even if the time for a scheduled rebalancing has not yet arrived.
In the PPLT v1.0, for the 3x leveraged tokens, the threshold to trigger a temporary rebalancing is set to be a 20% unfavorable movement against the past underliers’ rebalancing price. Bull and Bear leveraged tokens with different underlying assets will have independent triggers to activate a temporary rebalancing.
The temporary rebalancing process itself, however, will be the same as the scheduled rebalancing.
If the temporary rebalancing is unsuccessful, no matter the reason, the protocol will initiate a termination process once the price gets to a 30% unfavorable movement.
Termination is similar to a liquidation process, with all the terminated leveraged tokens being redeemed and burnt. After triggering a series of transactions on Dexes, the debts will be paid back, outstanding interest and fees will be deducted, and the remaining balance will be claimable for the leveraged token holders.
Previous Lending Pools
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A Multiplier Approach to the Lance-Blecher Theorem | EMS Press
A new approach to a theorem of E. C. Lance and D. P. Blecher in Hilbert
C*
- module theory and to two extensions of it is presented resting on a reinterpretation of key structural elements in terms of multiplier theory of operator
C*
-algebras. In the course of proving further facts are obtained.
Michael Frank, A Multiplier Approach to the Lance-Blecher Theorem. Z. Anal. Anwend. 16 (1997), no. 3, pp. 565–573
|
Lumen Irregularity Dominates the Relationship Between Mechanical Stress Condition, Fibrous-Cap Thickness, and Lumen Curvature in Carotid Atherosclerotic Plaque | J. Biomech Eng. | ASME Digital Collection
, Cambridge, CB2 0QQ, UK
e-mail: zt215@cam.ac.uk
Umar Sadat,
Guangyu Ji,
Division of Cardiothoracic Surgery, Changhai Hospital,
Chengcheng Zhu,
Victoria E. Young,
Victoria E. Young
Martin J. Graves,
Teng, Z., Sadat, U., Ji, G., Zhu, C., Young, V. E., Graves, M. J., and Gillard, J. H. (February 4, 2011). "Lumen Irregularity Dominates the Relationship Between Mechanical Stress Condition, Fibrous-Cap Thickness, and Lumen Curvature in Carotid Atherosclerotic Plaque." ASME. J Biomech Eng. March 2011; 133(3): 034501. https://doi.org/10.1115/1.4003439
High mechanical stress condition over the fibrous cap (FC) has been widely accepted as a contributor to plaque rupture. The relationships between the stress, lumen curvature, and FC thickness have not been explored in detail. In this study, we investigate lumen irregularity-dependent relationships between mechanical stress conditions, local FC thickness
(LTFC)
, and lumen curvature
(LClumen)
. Magnetic resonance imaging slices of carotid plaque from 100 patients with delineated atherosclerotic components were used. Two-dimensional structure-only finite element simulations were performed for the mechanical analysis, and maximum principal stress (stress-
P1
) at all integral nodes along the lumen was obtained.
LTFC
LClumen
were computed using the segmented contour. The lumen irregularity
(L-δir)
was defined as the difference between the largest and the smallest lumen curvature. The results indicated that the relationship between stress-
P1
LTFC
LClumen
is largely dependent on
L-δir
L-δir≥1.31
(irregular lumen), stress-
P1
strongly correlated with lumen curvature and had a weak/no correlation with local FC thickness, and in 73.4% of magnetic resonance (MR) slices, the critical stress (maximum of stress-
P1
over the diseased region) was found at the site where the lumen curvature was large. When
L-δir≤0.28
(relatively round lumen), stress-
P1
showed a strong correlation with local FC thickness but weak/no correlation with lumen curvature, and in 71.7% of MR slices, the critical stress was located at the site of minimum FC thickness. Using lumen irregularity as a method of identifying vulnerable plaque sites by referring to the lumen shape is a novel and simple method, which can be used for mechanics-based plaque vulnerability assessment.
biomechanics, biomedical MRI, blood vessels, finite element analysis, fracture, stress-strain relations, carotid artery, atherosclerosis, rupture, stress, lumen irregularity
Atherosclerosis, Finite element analysis, Magnetic resonance imaging, Stress, Rupture, Simulation
http://www.strokeassociation.org/STROKEORG/AboutStroke/About-Stroke_UCM_308529_SubHomePage.jsphttp://www.strokeassociation.org/STROKEORG/AboutStroke/About-Stroke_UCM_308529_SubHomePage.jsp
Stress Analysis of Carotid Plaque Rupture Based on In Vivo High Resolution MRI
Influence of Plaque Configuration and Stress Distribution on Fissuring of Coronary Atherosclerotic Plaques
Local Critical Stress Correlates Better Than Global Maximum Stress With Plaque Morphological Features Linked to Atherosclerotic Plaque Vulnerability: An In Vivo Multi-Patient Study
Utility of High Resolution MR Imaging to Assess Carotid Plaque Morphology: A Comparison of Acute Symptomatic, Recently Symptomatic and Asymptomatic Patients With Carotid Artery Disease
Impact of Plaque Hemorrhage and Its Age on Structural Stresses in Atherosclerotic Plaques of Patients With Carotid Artery Disease: An MR Imaging-Based Finite Element Simulation Study
Arterial Luminal Curvature and Fibrous-Cap Thickness Affect Critical Stress Conditions Within Atherosclerotic Plaque: An In Vivo MRI-Based 2D Finite-Element Study
http://staff.argyll.epsb.ca/jreed/math9/strand4/4103.htmhttp://staff.argyll.epsb.ca/jreed/math9/strand4/4103.htm
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Hypotrochoid Knowpia
A hypotrochoid is a roulette traced by a point attached to a circle of radius r rolling around the inside of a fixed circle of radius R, where the point is a distance d from the center of the interior circle.
The red curve is a hypotrochoid drawn as the smaller black circle rolls around inside the larger blue circle (parameters are R = 5, r = 3, d = 5).
The parametric equations for a hypotrochoid are:[1]
{\displaystyle x(\theta )=(R-r)\cos \theta +d\cos \left({R-r \over r}\theta \right)}
{\displaystyle y(\theta )=(R-r)\sin \theta -d\sin \left({R-r \over r}\theta \right)}
{\displaystyle \theta }
is the angle formed by the horizontal and the center of the rolling circle (these are not polar equations because
{\displaystyle \theta }
is not the polar angle). When measured in radian,
{\displaystyle \theta }
takes values from
{\displaystyle 0}
{\displaystyle 2\pi \times {\frac {\operatorname {LCM} (r,R)}{R}}}
where LCM is least common multiple.
Special cases include the hypocycloid with d = r is a line or flat ellipse and the ellipse with R = 2r and d > r or d < r (d is not equal to r).[2] (see Tusi couple).
The ellipse (drawn in red) may be expressed as a special case of the hypotrochoid, with R = 2r (Tusi couple); here R = 10, r = 5, d = 1.
The classic Spirograph toy traces out hypotrochoid and epitrochoid curves.
Hypotrochoids describe the support of the eigenvalues of some random matrices with cyclic correlations[3]
^ J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 165–168. ISBN 0-486-60288-5.
^ Gray, Alfred. Modern Differential Geometry of Curves and Surfaces with Mathematica (Second ed.). CRC Press. p. 906. ISBN 9780849371646.
^ Aceituno, Pau Vilimelis; Rogers, Tim; Schomerus, Henning (2019-07-16). "Universal hypotrochoidic law for random matrices with cyclic correlations". Physical Review E. 100 (1): 010302. doi:10.1103/PhysRevE.100.010302.
Flash Animation of Hypocycloid
Hypotrochoid from Visual Dictionary of Special Plane Curves, Xah Lee
Interactive hypotrochoide animation
O'Connor, John J.; Robertson, Edmund F., "Hypotrochoid", MacTutor History of Mathematics archive, University of St Andrews
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2014 Applying Data Clustering Feature to Speed Up Ant Colony Optimization
Chao-Yang Pang, Ben-Qiong Hu, Jie Zhang, Wei Hu, Zheng-Chao Shan
Ant colony optimization (ACO) is often used to solve optimization problems, such as traveling salesman problem (TSP). When it is applied to TSP, its runtime is proportional to the squared size of problem
N
so as to look less efficient. The following statistical feature is observed during the authors’ long-term gene data analysis using ACO: when the data size
N
becomes big, local clustering appears frequently. That is, some data cluster tightly in a small area and form a class, and the correlation between different classes is weak. And this feature makes the idea of divide and rule feasible for the estimate of solution of TSP. In this paper an improved ACO algorithm is presented, which firstly divided all data into local clusters and calculated small TSP routes and then assembled a big TSP route with them. Simulation shows that the presented method improves the running speed of ACO by 200 factors under the condition that data set holds feature of local clustering.
Chao-Yang Pang. Ben-Qiong Hu. Jie Zhang. Wei Hu. Zheng-Chao Shan. "Applying Data Clustering Feature to Speed Up Ant Colony Optimization." Abstr. Appl. Anal. 2014 (SI64) 1 - 8, 2014. https://doi.org/10.1155/2014/545391
Chao-Yang Pang, Ben-Qiong Hu, Jie Zhang, Wei Hu, Zheng-Chao Shan "Applying Data Clustering Feature to Speed Up Ant Colony Optimization," Abstract and Applied Analysis, Abstr. Appl. Anal. 2014(SI64), 1-8, (2014)
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Dabbu the Entrepreneur | Toph
Dabbu the Entrepreneur
Dabbu is back from Cox’s Bazar with an obsession with Coconut. Nowadays he is so obsessed with coconut that he has bought a coconut tree. Because He has a plan to eat more coconut water as it is very good for health and it revives the inner beauty of the skin. He also has a plan to start a business with coconut. In this situation, all of a sudden, his coconut tree was broken by the storm one day. At first, Dabbu became very sad. Seeing him sad, his mother, who is a competitive programmer, suggested a solution. She told Dabbu that if Dabbu can find the length of the broken part, she will give him some money as a reward.
Now, Dabbu is very eager to find the length of the broken part. But he has limited information to retrieve the length. Help Dabbu solve this problem. Because if you help him, he will get the money from his mother and will start a coconut business.
The coconut tree has broken in a way that from the ground point
A of the tree, it has broken on point
B and the peak of the tree has fallen on the ground on point
D creating an angle
ABD. The tree was initially standing perpendicular to the ground.
You will be given the length from point
D (the peak of the tree that fell on the ground) to point
A (the root of the tree) and the angle created on the broken part of the tree. Now help Dabbu become an entrepreneur!
The first line of the input contains an integer
T (
1 \le T \le 500000
1≤T≤500000), the number of test cases. Then, for each test case, there will be two real numbers
d (
1 \le d \le 5000
1≤d≤5000), a (
0 < a \le 45)
0<a≤45) the distance from
D to
A and the angle
ABD respectively.
In the output, print the length of the tree. Errors less than
10^{-4}
10−4 will be ignored.
EgorKulikovEarliest, Mar '20
Salman.306242Fastest, 0.2s
smak_9Lightest, 3.5 MB
imamanik05Shortest, 109B
This problem has asked to calculate the lengths of two edges of a triangle, given the length of one ...
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Home : Support : Online Help : Programming : Input and Output : File Manipulation : FileTools : Exists
determine if a file exists in the file system or in the Workbook
filename, the file descriptor of an open file or Workbook URI
The Exists(file) command checks the file system for a specified file. If the file exists, Exists returns true; otherwise, it returns false.
If the file parameter is a Workbook URI, then the existence check is performed in a specified Workbook.
\mathrm{FileTools}[\mathrm{Text}][\mathrm{WriteFile}]\left("file","data"\right)
\textcolor[rgb]{0,0,1}{4}
\mathrm{FileTools}[\mathrm{Exists}]\left("file"\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{FileTools}[\mathrm{Exists}]\left("doesnotexist"\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{FileTools}[\mathrm{Remove}]\left("file"\right)
\mathrm{FileTools}[\mathrm{Exists}]\left("maple://Start.maple/Start"\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{FileTools}[\mathrm{Exists}]\left("maple://Start.maple/Images/NewDocument.png"\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{FileTools}[\mathrm{Exists}]\left("maple://Start.maple/Entry that does not exist"\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
The FileTools[Exists] command was updated in Maple 2016.
|
m
n
Matrix (or 2-dimensional Array), then it is assumed to contain
m
\mathrm{with}\left(\mathrm{SignalProcessing}\right):
\mathrm{audiofile}≔\mathrm{cat}\left(\mathrm{kernelopts}\left(\mathrm{datadir}\right),"/audio/stereo.wav"\right):
\mathrm{Spectrogram}\left(\mathrm{audiofile},\mathrm{compactplot}\right)
\mathrm{Spectrogram}\left(\mathrm{audiofile},\mathrm{channel}=1,\mathrm{includesignal}=[\mathrm{color}="Navy"],\mathrm{includepowerspectrum},\mathrm{colorscheme}=["Orange","SteelBlue","Navy"]\right)
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Existence of Nondecreasing and Continuous Solutions for a Nonlinear Integral Equation with Supremum in the Kernel | EMS Press
Existence of Nondecreasing and Continuous Solutions for a Nonlinear Integral Equation with Supremum in the Kernel
Josefa Caballero
Using a technique associated with measures of noncompactness we prove the existence of nondecreasing solutions of an integral equation of Volterra type with supremum in the kernel, in the space
C[0,1]
Josefa Caballero, Belén López, Kishin Sadarangani, Existence of Nondecreasing and Continuous Solutions for a Nonlinear Integral Equation with Supremum in the Kernel. Z. Anal. Anwend. 26 (2007), no. 2, pp. 195–205
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What Is AAGR?
Understanding AAGR
AAGR vs. CAGR
Limitations of AAGR
Average Annual Growth Rate FAQs
What Is Average Annual Growth Rate (AAGR)?
The average annual growth rate (AAGR) reports the mean increase in the value of an individual investment, portfolio, asset, or cash flow on an annualized basis. It doesn't take compounding into account.
Average annual growth rate (AAGR) is the average annualized return of an investment, portfolio, asset, or cash flow over time.
AAGR is calculated by taking the simple arithmetic mean of a series of returns.
AAGR is a linear measure that does not account for the effects of compounding—to account for compounding, compound annual growth rate (CAGR) would be used instead.
Formula for Average Annual Growth Rate (AAGR)
\begin{aligned} &AAGR = \frac{GR_A + GR_B + \dotso + GR_n}{N} \\ &\textbf{where:}\\ &GR_A=\text{Growth rate in period A}\\ &GR_B=\text{Growth rate in period B}\\ &GR_n=\text{Growth rate in period }n\\ &N=\text{Number of payments}\\ \end{aligned}
AAGR=NGRA+GRB+…+GRnwhere:GRA=Growth rate in period AGRB=Growth rate in period BGRn=Growth rate in period nN=Number of payments
Understanding the Average Annual Growth Rate (AAGR)
The average annual growth rate helps determine long-term trends. It applies to almost any kind of financial measure including growth rates of profits, revenue, cash flow, expenses, etc. to provide the investors with an idea about the direction wherein the company is headed. The ratio tells you your average annual return.
The average annual growth rate is a calculation of the arithmetic mean of a series of growth rates. AAGR can be calculated for any investment, but it will not include any measure of the investment's overall risk, as measured by its price volatility. Furthermore, the AAGR does not account for periodic compounding.
AAGR is a standard for measuring average returns of investments over several time periods on an annualized basis. You'll find this figure on brokerage statements and in a mutual fund's prospectus. It is essentially the simple average of a series of periodic return growth rates.
One thing to keep in mind is that the periods used should all be of equal length—for instance, years, months, or weeks—and not to mix periods of different duration.
AAGR Example
The AAGR measures the average rate of return or growth over a series of equally spaced time periods. As an example, assume an investment has the following values over the course of four years:
Beginning value = $100,000
End of year 1 value = $120,000
The formula to determine the percentage growth for each year is:
\text{Simple percentage growth or return} = \frac{\text{ending value}}{\text{beginning value}} - 1
Simple percentage growth or return=beginning valueending value−1
Thus, the growth rates for each of the years are as follows:
Year 1 growth = $120,000 / $100,000 - 1 = 20%
Year 2 growth = $135,000 / $120,000 - 1 = 12.5%
The AAGR is calculated as the sum of each year's growth rate divided by the number of years:
AAGR = \frac{20 \% + 12.5 \% + 18.5 \% + 25 \%}{4} = 19\%
AAGR=420%+12.5%+18.5%+25%=19%
In financial and accounting settings, the beginning and ending prices are usually used. Some analysts may prefer to use average prices when calculating the AAGR depending on what is being analyzed.
As another example, consider the five-year real gross domestic product (GDP) growth for the United States over the last five years. The U.S. real GDP growth rates for 2017 through 2021 were 2.3%, 2.9%, 2.3%, -3.4%, and 5.7%, respectively. Thus, the AAGR of U.S. real GDP over the last five years has been 1.96%, or (2.3% + 2.9% + 2.3% + -3.4% + 5.7%) / 5.
AAGR vs. Compound Annual Growth Rate
AAGR is a linear measure that does not account for the effects of compounding. The above example shows that the investment grew an average of 19% per year. The average annual growth rate is useful for showing trends; however, it can be misleading to analysts because it does not accurately depict changing financials. In some instances, it can overestimate the growth of an investment.
For example, consider an end-of-year value for year 5 of $100,000 for the AAGR example above. The percentage growth rate for year 5 is -50%. The resulting AAGR would be 5.2%; however, it is evident from the beginning value of year 1 and the ending value of year 5, the performance yields a 0% return. Depending on the situation, it may be more useful to calculate the compound annual growth rate (CAGR).
The CAGR smooths out an investment's returns or diminishes the effect of the volatility of periodic returns.
Formula for CAGR
CAGR = \frac{\text{Ending Balance}}{\text{Beginning Balance}}^{\frac{1}{\text{\# Years}}} - 1
CAGR=Beginning BalanceEnding Balance# Years1−1
Using the above example for years 1 through 4, the CAGR equals:
CAGR = \frac{\$200,000}{\$100,000}^{\frac{1}{4}}- 1 = 18.92\%
CAGR=$100,000$200,00041−1=18.92%
For the first four years, the AAGR and CAGR are close to one another. However, if year 5 were to be factored into the CAGR equation (-50%), the result would end up being 0%, which sharply contrasts the result from the AAGR of 5.2%.
Limitations of the AAGR
Because AAGR is a simple average of periodic annual returns, the measure does not include any measure of the overall risk involved in the investment, as calculated by the volatility of its price. For instance, if a portfolio grows by a net of 15% one year and 25% in the next year, the average annual growth rate would be calculated to be 20%.
To this end, the fluctuations occurring in the investment’s return rate between the beginning of the first year and the end of the year are not counted in the calculations thus leading to some errors in the measurement.
A second issue is that as a simple average it does not care about the timing of returns. For instance, in our example above, a stark 50% decline in year 5 only has a modest impact on total average annual growth. However, timing is important, and so CAGR may be more useful in understanding how time-chained rates of growth matter.
What Does the Average Annual Growth Rate (AAGR) Tell You?
The average annual growth rate (AAGR) identifies long-term trends of such financial measures as cash flows or investment returns. AAGR tells you what the annual return has been (on average), but it does not take into account compounding.
What Are the Limitations of Average Annual Growth Rate?
AAGR may overestimate the growth rate if there are both positive and negative returns. It also does not include any measure of the risk involved, such as price volatility—nor does it factor in the timing of returns.
How Does Average Annual Growth Rate Differ From Compounded Annual Growth Rate (CAGR)?
Average annual growth rate (AAGR) is the average increase. It is a linear measure and does not take into account compounding. Meanwhile, the compound annual growth rate (CAGR) does and it smooths out an investment's returns, diminishing the effect of return volatility.
How Do You Calculate the Average Annual Growth Rate (AAGR)?
The average annual growth rate (AAGR) is calculated by finding the arithmetic mean of a series of growth rates.
Bureau of Economic Analysis. "Table 1.1.1. Percent Change From Preceding Period in Real Gross Domestic Product." Accessed Feb. 7, 2022.
CAGR vs. IRR: What's the difference?
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Resource investigation for Kichiji rockfish by autonomous underwater vehicle in Kitami-Yamato bank off Northern Japan | ROBOMECH Journal | Full Text
Yuya Nishida1,
Tamaki Ura2,
Tomonori Hamatsu3,
Kenji Nagahashi1,
Shogo Inaba4 &
Takeshi Nakatani5
Expensive kichiji rockfish (Sebastolobus macrochir) live in the Pacific side of main island, Pacific ide Hokkaido in Japan and southwest Okhotsk Sea. The catch of the fish has decreased significantly by over fishing. Widely regular survey of the biomass for kichiji rockfish is necessary to keep its sustainable use [[1]]. The biomass of benthic kichiji rockfish is investigated commonly by combination of the trawl survey and swept-area method [[2]]. The method which can survey effectively in a wide are, is difficult to apply on rough terrain, and has a risk for damaging to the natural habitat. Researchers proposed other resource investigations method for kichiji rockfish such as method using deep-sea monitoring system [[3],[4]] and method using manned submersible vessel [[5],[6]], because the fish usually stay on seafloor without moving and have a high visibility in the photograph. Two methods can survey the biomass of the fish without sampling. However, the monitoring system requires the support from many ship staff for towing type and the submersible vessel limits duration time for the survey to ensure the safety of the vessel operator.
The AUV Tuna-Sand [[7],[8]] was developed for the survey of material and energy resources in deep-sea such as hydrothermal vent and methane hydrate. The vehicle can observe natural seafloor automatically using only mounted sensors and devices, and can be operated on every ship which has the crane for launching and lifting. We surveyed hydrothermal chimney and the seafloor using the AUV Tuna-Sand in many Japanese marine areas, and the vehicle dived 69th times for the survey since 2007. This paper proposes resource investigation method of benthic life using the AUV Tuna-Sand, reports the result of the survey for kichiji rockfish in Okhotsk Sea on June 2013.
Outline of AUV “Tuna-Sand”
Hovering type AUV “Tuna-Sand” shown its specification in Table 1 can dive until 1,500 m depth. The vehicle consist of two floaters mad of syntactic foam, central hull for control circuits and electrical devices and bottom two hulls for lithium-ion batteries. Two 100 W thrusters for vertical motion and four 220 W thrusters for horizontal motion control five degrees-of-freedom of motion. Pitch and roll direction of the vehicle are static stability because of a long distance between center of gravity and buoyancy. The vehicle position is estimated with accuracy by mounted sensors such as inertial navigation system (INS) consisting of three-axis fiber optical gyro and acceleration sensor, doppler velocity log (DVL) for measuring ground speed and depth sensor. Light-section method using a sheet laser and a forward camera measures the distance to forward obstacle for obstacle avoidance, and is also used for measuring three-dimensional shape of the chimney [[9]]. A still camera locking downward take pictures of the seafloor using a single strobe as shown in Figure 1. The method can observe natural benthic life on the seafloor, because the flash light of the strobe is less irritating to animals than the other lights. Imaging survey by the AUV Tuna-Sand with a still camera and a strobe is effective for resource investigation and biological research of benthic life. The vehicle has profiling sonar for terrain measurement as other observation device.
Table 1 Specification of the AUV Tuna-Sand
Seafloor photography by the AUV Tuna-Sand.
The AUV Tuna-Sand can be deployed in the ocean without special shipboard equipment, the ship stuffs have so little work to do during observation. Figure 2 shows survey procedure of the AUV Tuna-Sand. After alignment of an INS using GPS data, the vehicle with two weights for diving and surfacing is lunched in the sea. And the vehicle dives by only one’s own weight without thruster force for saving power resource. The vehicle gets neutral buoyancy by releasing a mounted weight after reaching to the height which a DVL can measure ground speed and altitude, and starts to control its altitude. Then, an INS is aligned by ground speed from a DVL again. The vehicle navigates at a constant velocity and altitude along target route after the alignment, and takes a picture of the seafloor using a camera and a strobe every ten seconds. If a forward camera for light-section method detects the obstacle, the vehicle goes up by horizontal thrusters to high altitude than height of the obstacle for avoidance. The AUV Tuna-Sand releases a mounted weight and goes up by one’s own positive buoyancy after reaching to final way point. And then, the vehicle reaches the depth close to surface and turns off all thrusters, observation devices. If monitored vehicle status such as battery voltage, inside temperature and duration time is an abnormality during the survey, the survey is canceled. The vehicle surfaces positively every times even if it is out of control due to the accident. The reason is that two weights mounted on the vehicle are released automatically after the batteries runs down.
Survey procedures autonomously carried out by the AUV Tuna-Sand.
Taken photographs of the seafloor are corrected and transformed, to measure the biomass of kichiji rockfish and see ambient environment accuracy from the image. Figure 3 shows our imaging process method. The photograph pixel has the unevenness of the brightness s due to distance between from light source. First, the color and the illumination in the photographs are corrected using the method proposed by Singh [[10]]. Let input image F be expressed reflection image R and illumination image I as
\mathit{F}\left(\mathit{x},\mathit{y},\mathit{\lambda }\right)=\mathit{I}\left(\mathit{x},\mathit{y},\mathit{\lambda }\right)\mathit{R}\left(\mathit{x},\mathit{y},\mathit{\lambda }\right)
where x and y denote the pixel coordinates and λ denotes the color channel (red, green or blue). I(x, y, λ) includes unevenness of the brightness in F(x, y, λ). Thus, we are able to obtain uniform brightness image if I(x, y, λ) is remove from F(x, y, λ). The logarithm of illumination component formed by a single strobe in the water is known to be represented by fourth order polynomial of two variables as
{\mathit{I}}_{\mathit{L}}\left(\mathit{x},\mathit{y}\right)\approx {\mathit{p}}_{1}{\mathit{x}}^{4}+{\mathit{p}}_{2}{\mathit{x}}^{3}\mathit{y}+\cdots {\mathit{p}}_{14}\mathit{y}+{\mathit{p}}_{15}=\mathbf{SP}
where S refer to surface fitting matrix for each pixel and P is the parameter vector. The least square method estimates P from a collar channel F L (x, y) represented by the logarithm of F(x, y, λ). Clear image without luminance irregularity can be obtained by that F L (x, y) removed I L (x, y) is converted to linear scale. Note that I L (x, y) includes low-frequency component of the seafloor and the benthic life in the photograph. However, the benthic life is easy to be seen from obtained color correction image because low its low-frequency component is lower than the other in illumination image.
Image processing for photomosaic construction.
Second processing in our method transforms the color correction image, considering navigation data and seafloor topography. The benthic life length cannot be measured from all photographs in the same scale, because seafloor topography changes camera-to-subject distance and water flow applies the vehicle attitude. The central projection of the camera distorts the photograph depending on the height and position of objective. Ortho-correction converts the images using navigation data and seafloor topography for precise measurement. Time-series altitude of the vehicle estimates the seafloor topography. Final processing in our method makes the photomosaic based on the vehicle trajectory.
The biomass of kichiji rockfish was surveyed in Kimita-Yamato bunk supported by fisheries research vessel HOKKO-MARU on June 2013. The bunk which is 80 km height and the width of 4 to 7 km is located off northern Hokkaido in Japan. The basket mounted SSBL positioning device and acoustic modem is launched in the sea to monitor position and status of the AUV Tuna-Sand during the survey on the vessel. Figure 4 shows the picture of the basket for monitoring and Table 2 shows dive outline for the survey. Kichiji rockfish live at the depth of 150 to 1,200 m and are normally found in the seabed slope. This research surveyed in five points such as at the depth of 143 to 144 m on the topside, the depth of 305 to 498 m on the north, the depth of 561 to 566 m on the foot, the depth of 652 to 757 m on the north and the depth of 1,045 to 1050 m on the foot in the bunk. The vehicle navigated along the route of a reticular pattern for the detail survey in 73th and 74th dives, and the starlight line for wide area survey in 76th, 77th and 79th dives. Although target altitude in only 73th dive was 2.1 m (1.9 m swath width), target altitude in other dive was changed to 2.5 m (2.2 m swath width) because salty water in Kitami-Yamato bank was very clear.
The basket for positioning and communication.
Table 2 Outline of the survey
The AUV Tuna-Sand surveyed for 24 hours and took about 5,300 pictures of the seafloor, during five dives in Kitami-Yamato bank. 37 kichiji rockfishes of about 90 to 340 mm long were in the photographs. All fishes stayed on seafloor and nobody avoided to somewhere. The result suggests our method using the vehicle is effective in resource investigation for kichiji rockfish. Figure 5 shows dive points in Kitami-Yamato bank and Figure 6 shows photographs taken on the survey. Photographs taken on 73th dive had no kichiji rockfish although a little of snow crabs and sea cucumbers. Two kichiji rockfishes and many brittle stars [[11]] which was the food for kichiji rockfish were found at the deepest in the survey in 74th dive (Figure 6(a)). The greatest number of kichiji rockfish and brittle star in the survey stayed on the seafloor and the density of kichiji rockfish was 68 fishes per a hectare in 76th dive (Figure 6(b)). 77th and 79th dive areas had a few brittle stars, and were 4 fishes per a hectare and 227 per a hectare respectively (Figure 6(c) and (d)). Above results suggest that kichiji rockfish likes to live in seabed slope and its biomass does not depend heavily on the number of brittle star.
Dive locations on the Kitami-Yamato bank.
Kichiji rockfish taken by the AUV Tuna-Sand.
Figure 7 shows the density of kichiji fish for each length. The fish of 150 to 200 mm long that was bout three years olds was most often found in all dives, although the number of the other long was not many. The reason is considered that many young fish less than 100 mm long swim at the altitude higher than observational altitude without staying on the seafloor. And the number of the old fish larger than 200 mm is small because they have many opportunities to be cached by fisher.
The population density and length distribution of kichiji rockfish.
Figure 8 shows six mosaic images mad from pictures taken in 79th dive. The circle denotes kichiji rockfish and the square denotes other fish in Figure 8. The figure shows that all kichiji rockfish stay on the seafloor by oneself without swam, the shortest distance between kichiji rockfish is 4.0 m and the shortest between kichiji rockfish and other is 0.8 m.
Six constructed photomosaic images comprising a linear photomosaic.
This paper proposed resource investigation method by the AUV Tuna-Sand and image processing method for clear photomosaic. The investigation method can survey without big support of the ship, because the vehicle can navigate automatically on rough terrain where is difficult to operate by human and take high-resolution picture of seafloor. Our image processing can measure accurately the fish length and the biomass, because it corrects the photograph color for removing the unevenness of the brightness and image distortion. The survey results using our method in Kitami-Yamato bank showed the density of kichiji rockfish for each length. A photomosaic having uniform brightness shows the fish distribution. These results contribute to keep sustainable use of kichiji rockfish. Our method applies to survey the biomass of other benthic animals such as snow club. We will develop method that can survey the biomass of several benthic animals at one time.
Masashi N, Shuichi K, Yasushi K, Takahiro K: Estimates of population size of kichiji rockfish sebastolobus macrochir from Tag recoveries in southern Okhotsk Sea. Jpn Soc Scientific Fisheries 2001,67(5):821–828. 10.2331/suisan.67.821
Tsutomu H, Narimatsu Y, Masaki I, Yuji U, Daiji K: Annual changes in population size and recruitment per spawning biomass of bighand thornyhead Sebastolobus macrochir in the western North Pacific Ocean off northern Japan. Jpn Soc Scientific Fisheries 2006,72(3):374–381. 10.2331/suisan.72.374
Toshihiro W, Taro H: Estimation of the snow crab chionoecetes opilio population density using the deep-sea video monitoring system on a towed sledge. Jpn Soc Scientific Fisheries 2001,67(4):640–646. 10.2331/suisan.67.640
Toshihiro W, Kazutoshi W, Daiji K: Method of estimating the population density of kichiji rockfish Sebastolobus macrochir using a deep-sea video monitoring system on a towed sledge. Jpn Soc Scientific Fisheries 2003,69(4):620–623. 10.2331/suisan.69.620
Michimasa E (1982) 2,000M Deep Submergence Research Vehicle “SHINKAI 2000” System. In: Proceedings of OCEANS. ᅟ, Washington. 20–22 September
Tomonori H, Takashi Y, Youji N: Estimation of the fishing efficiency of kichiji rockfish, Sebastolobus macrochir, by comparison of the trawl survey and the submarine survey. JAMSTEC J Deep Sea Res 2003, 22: 63–70.
Takeshi N, Ura T, Takashi S: Autonomous underwater vehicle “tuna-sand”. J Japan Institution of Mar Eng 2008,43(4):523–526. 10.5988/jime.43.4_523
Sulin T, Tamaki U, Takeshi N, Blair T, Tao J: Estimation of the hydrodynamic coefficients of the complex-shaped autonomous underwater vehicle TUNA-SAND. J Mar Sci Technol 2009,14(3):373–386. 10.1007/s00773-009-0055-4
Toshihiro M, Ayaka K, Tamaki U: Volumetric mapping of tubeworm colonies in Kagoshima Bay through autonomous robotic surveys. Deep Sea research part I: oceanographic research papers. ᅟ 2011,58(7):757–767.
Hanumant S, Chris R, Oscar P, Ali C: Towards high-resolution imaging from underwater vehicles. International J Robotics Res 2007,26(1):55–74. 10.1177/0278364907074473
Toshiaki O, Tomonori H, Toyomi T: Food habits of kichiji rockfish Sebastolobus macrochir in summer on the continental slope off the Pacific coast of Hokkaido, Japan. Jpn Soc Scientific Fisheries 2005,71(4):584–593. 10.2331/suisan.71.584
The authors thank the staff of the fisheries research vessel HOKKO-MARU for their help and support. This work was supported by JST CREST “Establishment of core technology of the preservation and regeneration of marine biodiversity and ecosystems”.
Institute of Industrial Science, the University of Tokyo, 4-6-1, Komaba, Tokyo, Japan
Yuya Nishida & Kenji Nagahashi
Kyushu Institute of Technology, 2-4 Hibikino, Wakamatsu, Kitakyushu, Fukuoka, Japan
Hokkaido National Fisheries Research Institute, 115 Katsurakoi, Kushiro, Hokkaido, Japan
Tomonori Hamatsu
The University of Tokyo, 4-6-1, Komaba, Tokyo, Japan
Japan Agency for Marine-earth Science and Technology, 2-15 Natsushima, Yokosuka, Kanagawa, Japan
Correspondence to Yuya Nishida, Tamaki Ura, Tomonori Hamatsu, Kenji Nagahashi, Shogo Inaba or Takeshi Nakatani.
YN operated our vehicle for the survey, and made photomosaic. TU organized the survey, and enumerated the fishes in images. TH identified fish species. KN supported the survey. SI supported the survey. TN developed the vehicle. All authors read and approved the final manuscript.
Nishida, Y., Ura, T., Hamatsu, T. et al. Resource investigation for Kichiji rockfish by autonomous underwater vehicle in Kitami-Yamato bank off Northern Japan. Robomech J 1, 2 (2014). https://doi.org/10.1186/s40648-014-0002-y
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Colon (punctuation) - Simple English Wikipedia, the free encyclopedia
For other uses of the word, see Colon (disambiguation).
The colon (":") is a punctuation mark, simply consisting of two equally sized dots centered on the same vertical (up/down) line.
1.1 Use in prose
1.2 Use in other kinds of text
3 Diacritical usage
Punctuation[change | change source]
Use in prose[change | change source]
A colon is a more significant pause than a semicolon. It is usually used to contrast two parts of a sentence:
It's official: McClaren makes the worst start by an England manager.
When the door was forced, a scene of chaos was revealed: chairs overturned, drawers pulled out and emptied, broken crockery on the floor...
If you must go, take the following: climbing rope, ice axe, compass, a large-scale map, water and food, and good boots.
The Lord seeth not as man seeth: for man looketh on the outward appearance, but the Lord looketh on the heart.
Use in other kinds of text[change | change source]
Introduction of a definition, such as:
A: the first letter in the Latin alphabet
Hypernym of a word: a word having a wider meaning than the given one; e.g. vehicle is a hypernym of car
Separation of the chapter and the verse number(s) indication in many references to religious scriptures, and also epic poems; it was also used for chapter numbers in roman numerals, as in:
John 3:14–16 (or John iii:14–16) (cf. chapters and verses of the Bible)
The Qur'an, Sura 5:18
Separation when reporting time of day hour/minute/second (cf. ISO 8601), such as:
The concert finished at 23:45.
This file was last modified today at 11:15:05.
Separation of a title and the corresponding subtitle, as in:
Separation of clauses in a periodic sentence
Colons can also be used to start a list, such as, "He provided all of the ingredients: sugar, flour, eggs, and butter."
The colon's first appearance in English text is marked by the Shorter Oxford English Dictionary as 1589.
Diacritical usage[change | change source]
A special double-triangle colon symbol is used in IPA to indicate that the preceding sound is long. Its form is that of two triangles, each a bit larger than a point of a standard colon, pointing toward each other. It is available in Unicode as modifier letter triangular colon, Unicode U+02D0 (ː). A regular colon is often used as a fallback when this character is not available, or in the practical orthography of some languages (particularly in Mexico), which have a phonemic long/short distinction in vowels.
The colon is also used in mathematics, cartography, model building and other fields to denote a ratio or a scale, as in 3:1 (pronounced "three to one").[1][2] A betting odd of the form
{\displaystyle r:s}
corresponds to the probability
{\displaystyle {\tfrac {s}{r+s}}}
.[3] Unicode provides a distinct ratio character, Unicode U+2236 (∶) for mathematical usage.
In many non-English-speaking countries, the colon is used as a division sign: "a divided by b" is written as a : b.
The combination with an equal sign,
{\displaystyle :=\,}
, is used for definitions.[1]
In computing, the colon character is represented by ASCII code 58, and is located at Unicode code-point U+003A. The full-width (double-byte) equivalent, ：, is located at Unicode code point U+FF1A.
The colon is quite often used as a special control character in many operating systems commands, URLs, computer programming languages, and in the path representation of several file systems. It is often used as a single post-fix delimiter, signifying the immediate precedence of a token keyword or the transition from one mode of character string interpretation to another related mode. Some applications, such as the widely used MediaWiki, use the colon as both a pre-fix and post-fix delimiter.
For a double-colon, "::" the meaning has included the use of ellipsis, as spanning over omitted text; however, there have been other meanings as well.
Internet usage[change | change source]
On the Internet (online chats, email, message boards, etc.), a colon or multiple colons is sometimes used to denote an action or emote. In this use, it has the inverse function of quotation marks—denoting actions where unmarked text is assumed to be dialog. For example:
Kim: Pluto is so small, it should not be considered a planet. It is tiny!
Mel: Oh really? ::Drops Pluto on Kim’s head:: Still think it's small now?
Colons may also be used for sounds (as with ":Click:"). One can contrast this use with the use of outer asterisks (for example, *cough* would denote that the speaker is coughing, as opposed to saying the word 'cough').
It also has the widespread usage of representing two vertically aligned eyes in a emoticon, such as :-), :( :P, :D, :3, etc
Semicolon, the ";" punctuation mark
↑ "Ratio - math word definition - Math Open Reference". www.mathopenref.com. Retrieved 2020-09-23.
↑ Weisstein, Eric W. "Ratio". mathworld.wolfram.com. Retrieved 2020-09-23.
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Colon_(punctuation)&oldid=8198524"
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Rendering the Mandelbrot Set
Early efforts: Python and OpenCL
Perturbative methods?
A thought process for proceeding with efficient reference computation
Julia and quadtree adaptive rendering
Moving a series
Combining the quadtree approach with perturbative methods
Julia repository
JS repository
JS try now
This was my first effort at rendering the mandelbrot set, fast. The program allows switching between numpy, opencl, and c for computing the images. Matplotlib is used for the GUI.
The really neat trick is perturbative rendering: this is a technique for computing high precision trajectories of mandelbrot points, without having to use arbitrary precision arithmetic throughout. The "go to" source for the concept is this pdf by K. I. Martin. It's an excellent read, you should be reading that instead of this blog post.
The integration between OpenCL and python is not that smooth: there's a ton of boilerplate, and you have to carefully match up the datatypes of buffers: any typer errors are not caught at runtime,
I enjoyed this model for UI development, but it is not still developed. Its spiritual successor is GTKObservables, which I need to go try.
It made it easy to add UI components for adjusting resolution, colormap details, etc, which I handled with keyboard shortkeys in the python version. Much better user experience!
The main downside of the technique in the python method above is that it needs a good "center" point to compute in high precision, before computing low precision offsets: if the high precision trajectory escapes, we can't take offsets from it for very long. In the julia code, we now take a more sensible approach and itertively seek out a good center hacks hacks hacks distance transform
TODO for go fast:
quadtree + mpfr – backtracking?
reference reuse!
slider for number of reference iterations
eventually softfloat. This is suffering
quadtree that puts onto gpu queue
just do the damn series approximation
Which terms in series approximation need to be high precision?
What I hope I can do:
A reference consists of a center C and a trajectory Z
Guess C
Compute Z using only one sequence of MPFR values, plus machine precision auxiliaries
When Z escapes, update C to make Z take longer to escape
update Z without having to recompute?
Up to now, because I have just been using the pertrubative rendering and not the series approximation, I have had to actually compute every iteration for every pixel. However, that's technically not necessary! Instead, we can begin in the center of a square region, and approximate every pixel in that region as a taylor series with a single set of coefficients. Then, we can perform the operation z^2 + c on the polynomial by squaring it and adding c: this produces a new set of coefficients for the next step. This works for a while until the detail of the image becomes too much to capture with the number of coefficients being tracked, at which point we have to fall back to iterating single pixels. The way to detect that the error is becoming large to use the standard bound on the error included in the definition of a taylor series,
f(x) = f(0) + x f'(x) + \frac{1}{2} x^2 f''(x) + ... + E
f(x)=f(0)+xf′(x)+21x2f′′(x)+...+E
E = \frac{1}{(n + 1)! }f^{(n + 1)}(c) x^{(n + 1)}
E=(n+1)!1f(n+1)(c)x(n+1)
Basically, we just track the last coefficient that we aren't using, and bail when it gets big.
The novel approach that I am taking is, instead of adding terms to this series to improve its accuracy, to instead approach the problem recursively, and split into four regions, each with their own series approximation, whenever the big region gets too complicated to approximate.
There is a fun middle bit here, where we have to have a way to move the series centered in our big square to the centers of each of our little squares.
We will end up with a function like this:
function moveSeries(coefficients, q)
a4 = coefficients[4]
a3 = coefficients[3] + 3 * a4 * q
a2 = coefficients[2] + 2 * a3 * q - 3 * a4 * q^2
a1 = coefficients[1] + a2 * q - a3 * q^2 + a4 * q^3
return (a1, a2, a3, a4)
But where do all the coefficients come from?
\begin{aligned} f(x) &= c_1 + c_2 x + c_3 x^2 + c_4 x^4 \\ f(y) &= a_1 + a_2 y + a_3 y^2 + a_4 y^4 \\ f(x) &= f(y - q) \\ \end{aligned}
f(x)f(y)f(x)=c1+c2x+c3x2+c4x4=a1+a2y+a3y2+a4y4=f(y−q)
function recursive_mandelbrot(
indices, coefficients, prev_center, c_arr, output_arr, init_iters, max_iters
@inbounds center_ = (c_arr[indices[1]] + c_arr[indices[end]]) / 2
coefficients = moveSeries(coefficients, center_ - prev_center)
if size(indices) == (1, 1)
count = inner_loop(max_iters, init_iters, coefficients[1], center_)
#count -= init_iters
@inbounds output_arr[indices[1]] = count
@inbounds maxdel = abs(c_arr[indices[1]] - c_arr[indices[end]]) / 2
coefficients, more_iters = series_iterate(coefficients, center_, maxdel, max_iters - init_iters)
init_iters += more_iters
for sub_indices = fourCorners(indices)
recursive_mandelbrot(sub_indices, coefficients, center_, c_arr, output_arr, init_iters, max_iters)
How on earth do we do this?
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Conway notation aC
Schläfli symbols r{4,3} or
{\displaystyle {\begin{Bmatrix}4\\3\end{Bmatrix}}}
rr{3,3} or
{\displaystyle r{\begin{Bmatrix}3\\3\end{Bmatrix}}}
t1{4,3} or t0,2{3,3}
Td, [3,3], (*332), order 24
arcsec(−√3)
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertex-transitive but also edge-transitive.[1] It is radially equilateral.
3D model of a cuboctahedron
The cuboctahedron was probably known to Plato: Heron's Definitiones quotes Archimedes as saying that Plato knew of a solid made of 8 triangles and 6 squares.[2]
4.3 Metric properties
5.1 Tetrahedra and Octahedra
5.2 Irregular polyhedra
6.1 Radial equilateral symmetry
6.3 Vertex arrangement
7.1 Regular polyhedra
7.2 Quasiregular polyhedra and tilings
7.3 4-dimensional polytopes
8 Cuboctahedral graph
Vector Equilibrium (Buckminster Fuller) because its center-to-vertex radius equals its edge length (it has radial equilateral symmetry). Fuller also called a cuboctahedron built of rigid struts and flexible vertices a jitterbug; this object can be progressively transformed into an icosahedron, octahedron, and tetrahedron by folding along the diagonals of its square sides.[3]
With Oh symmetry, order 48, it is a rectified cube or rectified octahedron (Norman Johnson)
With Td symmetry, order 24, it is a cantellated tetrahedron or rhombitetratetrahedron.
With D3d symmetry, order 12, it is a triangular gyrobicupola.
The cuboctahedron has four special orthogonal projections, centered on a vertex, an edge, and the two types of faces, triangular and square. The last two correspond to the B2 and A2 Coxeter planes. The skew projections show a square and hexagon passing through the center of the cuboctahedron.
Cuboctahedron (orthogonal projections)
Rhombic dodecahedron (Dual polyhedron)
The cuboctahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
Vertex centered
The Cartesian coordinates for the vertices of a cuboctahedron (of edge length √2) centered at the origin[4] are:
Root vectorsEdit
The cuboctahedron's 12 vertices can represent the root vectors of the simple Lie group A3. With the addition of 6 vertices of the octahedron, these vertices represent the 18 root vectors of the simple Lie group B3.
{\displaystyle {\begin{aligned}A&=\left(6+2{\sqrt {3}}\right)a^{2}&&\approx 9.464\,1016a^{2}\\V&={\tfrac {5}{3}}{\sqrt {2}}a^{3}&&\approx 2.357\,0226a^{3}.\end{aligned}}}
Tetrahedra and OctahedraEdit
The cuboctahedron can be dissected into 6 square pyramids and 8 tetrahedra meeting at a central point. This dissection is expressed in the tetrahedral-octahedral honeycomb where pairs of square pyramids are combined into octahedra.
Irregular polyhedraEdit
The cuboctahedron can be dissected into two triangular cupolas by a common hexagon passing through the center of the cuboctahedron.[a] If these two triangular cupolas are twisted so triangles and squares line up, Johnson solid J27, the triangular orthobicupola, is created.
Progression between a tetrahedron, expanded into a cuboctahedron, and reverse expanded into the dual tetrahedron
Radial equilateral symmetryEdit
In a cuboctahedron, the long radius (center to vertex) is the same as the edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Its center is like the apical vertex of a pyramid: one edge length away from all the other vertices. (In the case of the cuboctahedron, the center is in fact the apex of 6 square and 8 triangular pyramids). This radial equilateral symmetry is a property of only a few uniform polytopes, including the two-dimensional hexagon, the three-dimensional cuboctahedron, and the four-dimensional 24-cell and 8-cell (tesseract). Radially equilateral polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge. Therefore, all the interior elements which meet at the center of these polytopes have equilateral triangle inward faces, as in the dissection of the cuboctahedron into 6 square pyramids and 8 tetrahedra.
Each of these radially equilateral polytopes also occurs as cells of a characteristic space-filling tessellation: the tiling of regular hexagons, the rectified cubic honeycomb (of alternating cuboctahedra and octahedra), the 24-cell honeycomb and the tesseractic honeycomb, respectively. Each tessellation has a dual tessellation; the cell centers in a tessellation are cell vertices in its dual tessellation. The densest known regular sphere-packing in two, three and four dimensions uses the cell centers of one of these tessellations as sphere centers.
A cuboctahedron can be obtained by taking an equatorial cross section of a four-dimensional 24-cell or 16-cell. A hexagon or a square can be obtained by taking an equatorial cross section of a cuboctahedron.
The edges of a cuboctahedron form four regular hexagons. If the cuboctahedron is cut in the plane of one of these hexagons, each half is a triangular cupola, one of the Johnson solids; the cuboctahedron itself thus can also be called a triangular gyrobicupola, the simplest of a series (other than the gyrobifastigium or "digonal gyrobicupola"). If the halves are put back together with a twist, so that triangles meet triangles and squares meet squares, the result is another Johnson solid, the triangular orthobicupola, also called an anticuboctahedron.
Both triangular bicupolae are important in sphere packing. The distance from the solid's center to its vertices is equal to its edge length. Each central sphere can have up to twelve neighbors, and in a face-centered cubic lattice these take the positions of a cuboctahedron's vertices. In a hexagonal close-packed lattice they correspond to the corners of the triangular orthobicupola. In both cases the central sphere takes the position of the solid's center.
Cuboctahedra appear as cells in three of the convex uniform honeycombs and in nine of the convex uniform 4-polytopes.
Because it is radially equilateral, the cuboctahedron's center can be treated as a 13th canonical apical vertex, one edge length distant from the 12 ordinary vertices, as the apex of a canonical pyramid is one edge length equidistant from its other vertices.
The cuboctahedron shares its edges and vertex arrangement with two nonconvex uniform polyhedra: the cubohemioctahedron (having the square faces in common) and the octahemioctahedron (having the triangular faces in common), both have four hexagons. It also serves as a cantellated tetrahedron, as being a rectified tetratetrahedron.
its equator
The cuboctahedron 2-covers the tetrahemihexahedron,[5] which accordingly has the same abstract vertex figure (two triangles and two squares: 3.4.3.4) and half the vertices, edges, and faces. (The actual vertex figure of the tetrahemihexahedron is 3.4.3/2.4, with the a/2 factor due to the cross.)
Main article: Kinematics of the cuboctahedron
Progressions between an octahedron, pseudoicosahedron, and cuboctahedron. The cuboctahedron can flex this way even if its edges (but not its faces) are rigid.
When interpreted as a framework of rigid flat faces, connected along the edges by hinges, the cuboctahedron is a rigid structure, as are all convex polyhedra, by Cauchy's theorem. However, when the faces are removed, leaving only rigid edges connected by flexible joints at the vertices, the result is not a rigid system (unlike polyhedra whose faces are all triangles, to which Cauchy's theorem applies despite the missing faces).
Adding a central vertex, connected by rigid edges to all the other vertices, subdivides the cuboctahedron into square pyramids and tetrahedra, meeting at the central vertex. Unlike the cuboctahedron itself, the resulting system of edges and joints is rigid, and forms part of the infinite octet truss structure.
Regular polyhedraEdit
The cuboctahedron also has tetrahedral symmetry with two colors of triangles.
Quasiregular polyhedra and tilingsEdit
The cuboctahedron exists in a sequence of symmetries of quasiregular polyhedra and tilings with vertex configurations (3.n)2, progressing from tilings of the sphere to the Euclidean plane and into the hyperbolic plane. With orbifold notation symmetry of *n32 all of these tilings are wythoff construction within a fundamental domain of symmetry, with generator points at the right angle corner of the domain.[6][7]
*n42 symmetry mutations of quasiregular tilings: (4.n)2
4-dimensional polytopesEdit
Orthogonal projections of 24-cell
The cuboctahedron can be decomposed into a regular octahedron and eight irregular but equal octahedra in the shape of the convex hull of a cube with two opposite vertices removed. This decomposition of the cuboctahedron corresponds with the cell-first parallel projection of the 24-cell into three dimensions. Under this projection, the cuboctahedron forms the projection envelope, which can be decomposed into six square faces, a regular octahedron, and eight irregular octahedra. These elements correspond with the images of six of the octahedral cells in the 24-cell, the nearest and farthest cells from the 4D viewpoint, and the remaining eight pairs of cells, respectively.
Cuboctahedral graphEdit
Cuboctahedral graph
In the mathematical field of graph theory, a cuboctahedral graph is the graph of vertices and edges of the cuboctahedron, one of the Archimedean solids. It can also be constructed as the line graph of the cube. It has 12 vertices and 24 edges, is locally linear, and is a quartic Archimedean graph.[8]
Cultural occurrencesEdit
Two cuboctahedra on a chimney in Israel.
The "Geo Twister" fidget toy is a flexible cuboctahedron.
The Coriolis space stations in the computer game series Elite are cuboctahedron-shaped.
Vesak Kuudu, traditional lanterns made in Sri Lanka annually to celebrate Vesak Poya day, are usually cuboctahedral.
"Moonsnakes" from Super Mario Odyssey.[9]
InfluxData, the company behind the InfluxDB time series database, uses the cuboctahedron in its logo.
The "Prime Radiant" in the 2021 adaptation of Isaac Asimov's Foundation series is a transparent cuboctahedron with circuitry patterns.
Pseudocuboctahedron
^ Notice that the cuboctahedron has four hexagonal central planes, inclined at 60° to each other. Like the hexagon, the cuboctahedron can be divided into equilateral triangles which meet at its center: it has radial equilateral symmetry.
^ Coxeter 1973, pp. 18–19, §2.3 Quasi-regular polyhedra.
^ Heath, Thomas L. (1931), "A manual of Greek mathematics", Nature, Clarendon, 128 (3235): 739–740, Bibcode:1931Natur.128..739T, doi:10.1038/128739a0, S2CID 3994109
^ "Vector Equilibrium: R. Buckminster Fuller".
^ Coxeter 1973, p. 52, §3.7 Coordinates for the vertices of the regular and quasi-regular solids.
^ Richter, David A., Two Models of the Real Projective Plane, archived from the original on 2016-03-03, retrieved 2010-04-15
^ Coxeter 1973, pp. 86–88, §5.7 Wythoff's construction.
^ Two Dimensional symmetry Mutations by Daniel Huson
^ "File:Moonsnake Icon SMO.png - Super Mario Wiki, the Mario encyclopedia". www.mariowiki.com. Retrieved 2018-11-05.
Ghyka, Matila (1977). The geometry of art and life ([Nachdr.] ed.). New York: Dover Publications. pp. 51–56, 81–84. ISBN 9780486235424.
Weisstein, Eric W. (2002). "Cuboctahedron". CRC Concise Encyclopedia of Mathematics (2nd ed.). Hoboken: CRC Press. pp. 620–621. ISBN 9781420035223.
The Cuboctahedron on Hexnet a website devoted to hexagon mathematics.
Klitzing, Richard. "3D convex uniform polyhedra o3x4o - co".
Retrieved from "https://en.wikipedia.org/w/index.php?title=Cuboctahedron&oldid=1086602584"
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Bonaventura Cavalieri (1598-1647) was a mathematician who helped to develop calculus but is best remembered today for a principle named for him. Cavalieri’s Principle can be applied to calculate the volumes of solids that might be difficult to calculate otherwise.
Suppose you have a stack of
25
pennies piled one on top of the other. You decide to slant the stack by sliding some of the pennies over. Does the volume of the
25
pennies change because they are no longer stacked one on top of another?
Would the same thing be true of a stack of
15
books if you slide the stack to the side or twist some of the books? What about a stack of
100
0 sheets of paper? Would the volumes change?
The idea of viewing solids as slices that can be moved around without affecting the volume is called Cavalieri's Principle. Use this principle to calculate the volume of the cylinder at right. Note that when the lateral faces of a prism or cylinder are not perpendicular to its base, the solid is referred to as an oblique cylinder or prism. How is the volume of the prism at right related to the one in problem 12-78?
Remember, you can check most of the homework answers in your textbook.
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Existence results for $p$-Laplace equations with nonlinearities having growth in the function and its gradient
February 2021 Existence results for $p$-Laplace equations with nonlinearities having growth in the function and its gradient
Haydar Abdelhamid
Rocky Mountain J. Math. 51(1): 1-15 (February 2021). DOI: 10.1216/rmj.2021.51.1
In a bounded domain
\mathrm{\Omega }\subset {ℝ}^{N}
N\ge 2
, we study the solvability of the boundary value problem
-{\mathrm{\Delta }}_{p}u=H\left(x,u,\nabla u\right)\phantom{\rule{1em}{0ex}}\text{ in}\mathrm{\Omega },\phantom{\rule{1em}{0ex}}u=0\phantom{\rule{1em}{0ex}}\text{ on}\partial \mathrm{\Omega },
-{\mathrm{\Delta }}_{p}u=-div\left(|\nabla u{|}^{p-2}\nabla u\right)
p
-Laplace operator with
1<p<N
H
|H\left(x,u,\nabla u\right)|\le a|\nabla u{|}^{q}+b|u{|}^{r}+f\left(x\right)
a,b\ge 0
q,r>0
f\in {L}^{m}\left(\mathrm{\Omega }\right)
for a suitable exponent
m>1
, we prove the existence of solutions for different sets of values
\left(q,r\right)
. To this end, we apply the classical Schauder fixed point theorem, relying on some well-known uniqueness, regularity estimates and stability results for renormalized and weak solutions. In the last section we make some examples to illustrate the applicability of our results to a large class of equations.
Haydar Abdelhamid. "Existence results for $p$-Laplace equations with nonlinearities having growth in the function and its gradient." Rocky Mountain J. Math. 51 (1) 1 - 15, February 2021. https://doi.org/10.1216/rmj.2021.51.1
Received: 11 March 2019; Revised: 6 June 2020; Accepted: 7 June 2020; Published: February 2021
Digital Object Identifier: 10.1216/rmj.2021.51.1
Primary: 35A01 , 35J60 , 35J66 , 35J92
Keywords: $p$-Laplace equations , existence , general growth
Haydar Abdelhamid "Existence results for $p$-Laplace equations with nonlinearities having growth in the function and its gradient," Rocky Mountain Journal of Mathematics, Rocky Mountain J. Math. 51(1), 1-15, (February 2021)
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Magic Mirror on the Wall | Toph
By reborn · Limits 1s, 64 MB · Custom Checker
Kyablakanto and his friend Chalakanti went into the Mirror Room after browsing through the fair. In the Mirror Room, in front of the huge mirror, each of them waved to their reflections on the mirror. But, Kyablakanto being a little dumb, can't figure out where he needs to look on the mirror to see his friend. If he waves at wrong direction, he fears that she might leave him in the fair alone. That's why he called you. He knows you are an excellent programmer. As a friend, you have to help him.
You can consider the mirror to be infinite and compared to that Kyablakanto and Chalakanti seem like dots. On a 2-dimensional Cartesian coordinate system, the mirror is on the x-axis (
y = 0
y=0 line) and Kyablakanto, Chalakanti are at
(x_1, y_1)
(x1,y1) and
(x_2, y_2)
(x2,y2), respectively. Your job is to find out a point
(x, y)
(x,y) on the mirror such that if Kyablakanto looks at that point, he can see his friend there.
The first line of input contains an integer
T. You need to solve
T test cases.
Input follows
T lines. Each line corresponds to the input for one test case. Each of the line contains 4 space separated integers
x_1, ~y_1, ~x_2, ~y_2
x1, y1, x2, y2.
1 \leq T \leq 10^5
1≤T≤105.
1 \leq x_1, y_1, x_2, y_2 \leq 10^9
1≤x1,y1,x2,y2≤109.
x_1 \neq x_2
x1=x2.
For 30 points:
y_1 = y_2
y1=y2.
y_1
y1 can be different from
y_2
y2.
You need to output
T lines. Each line should contain two real numbers who represent
(x, y)
i-th line should contain the answer to the
i-th test case.
For every output number, if the absolute difference between your and judge's output are less than
10^{-4}
10−4, your output will be considered correct. That means, if one of your output is
A and the judge has
B as it's output, your output would be considered correct if
| A - B | < 10^{-4}
∣A−B∣<10−4 is satisfied.
Explanation of Sample IO
T = 2
T=2 test cases.
Following image should describe the first test case.
The following picture describes the second test case.
aNkanpy.pritomEarliest, Jun '20
HKShakibFastest, 0.0s
imamanik05Shortest, 83B
Be a good kid and read the Bangla editorial. :)
NHSPC 2020 Practice
Replay of National High School Programming Contest 2020 - National Round
Sad Day To Die Hard
Free Online Class-2020 Live - Demo Contest
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A characterization of isochronous centres in terms of symmetries | EMS Press
We present a description of isochronous centres of planar vector fields
X
by means of their groups of symmetries. More precisely, given a normalizer
U
X
[X,U]=\mu X
\mu
is a scalar function), we provide a necessary and sufficient isochronicity condition based on
\mu
. This criterion extends the result of Sabatini and Villarini that establishes the equivalence between isochronicity and the existence of commutators (
[X,U]= 0
). We put also special emphasis on the mechanical aspects of isochronicity; this point of view forces a deeper insight into the potential and quadratic-like Hamiltonian systems. For these families we provide new ways to find isochronous centres, alternative to those already known from the literature.
Emilio Freire, Armengol Gasull, Antoni Guillamon, A characterization of isochronous centres in terms of symmetries. Rev. Mat. Iberoam. 20 (2004), no. 1, pp. 205–222
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Option set for armax - MATLAB armaxOptions - MathWorks Switzerland
Create Default Options Set for ARMAX Estimation
Specify Options for ARMAX Estimation
Option set for armax
opt = armaxOptions
opt = armaxOptions(Name,Value)
opt = armaxOptions creates the default options set for armax.
opt = armaxOptions(Name,Value) creates an option set with the options specified by one or more Name,Value pair arguments.
Errors larger than ErrorThreshold times the estimated standard deviation have a linear weight in the loss function. The standard deviation is estimated robustly as the median of the absolute deviations from the median of the prediction errors and divided by 0.7. For more information on robust norm choices, see section 15.2 of [2].
\frac{‖{y}_{p,z}-{y}_{meas}‖}{‖{y}_{p,e}-{y}_{meas}‖}>\text{AutoInitThreshold}
opt — Options set for armax
Option set for armax, returned as an armaxOptions option set.
Create an option set for armax to use the 'simulation' Focus and to set the Display to 'on'.
opt = armaxOptions('Focus','simulation','Display','on');
armax | idfilt
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Convergence of measures - Wikipedia
(Redirected from Portmanteau lemma)
Not to be confused with Convergence in measure.
In mathematics, more specifically measure theory, there are various notions of the convergence of measures. For an intuitive general sense of what is meant by convergence of measures, consider a sequence of measures μn on a space, sharing a common collection of measurable sets. Such a sequence might represent an attempt to construct 'better and better' approximations to a desired measure μ that is difficult to obtain directly. The meaning of 'better and better' is subject to all the usual caveats for taking limits; for any error tolerance ε > 0 we require there be N sufficiently large for n ≥ N to ensure the 'difference' between μn and μ is smaller than ε. Various notions of convergence specify precisely what the word 'difference' should mean in that description; these notions are not equivalent to one another, and vary in strength.
Three of the most common notions of convergence are described below.
1 Informal descriptions
2 Total variation convergence of measures
3 Setwise convergence of measures
4 Weak convergence of measures
4.1 Weak convergence of random variables
Informal descriptionsEdit
This section attempts to provide a rough intuitive description of three notions of convergence, using terminology developed in calculus courses; this section is necessarily imprecise as well as inexact, and the reader should refer to the formal clarifications in subsequent sections. In particular, the descriptions here do not address the possibility that the measure of some sets could be infinite, or that the underlying space could exhibit pathological behavior, and additional technical assumptions are needed for some of the statements. The statements in this section are however all correct if
{\displaystyle \mu _{n}}
is a sequence of probability measures on a Polish space.
The various notions of convergence formalize the assertion that the 'average value' of each 'sufficiently nice' function should converge:
{\displaystyle \int f\,d\mu _{n}\to \int f\,d\mu }
To formalize this requires a careful specification of the set of functions under consideration and how uniform the convergence should be.
The notion of weak convergence requires this convergence to take place for every continuous bounded function
{\displaystyle f}
. This notion treats convergence for different functions f independently of one another, i.e., different functions f may require different values of N ≤ n to be approximated equally well (thus, convergence is non-uniform in
{\displaystyle f}
The notion of setwise convergence formalizes the assertion that the measure of each measurable set should converge:
{\displaystyle \mu _{n}(A)\to \mu (A)}
Again, no uniformity over the set
{\displaystyle A}
is required. Intuitively, considering integrals of 'nice' functions, this notion provides more uniformity than weak convergence. As a matter of fact, when considering sequences of measures with uniformly bounded variation on a Polish space, setwise convergence implies the convergence
{\textstyle \int f\,d\mu _{n}\to \int f\,d\mu }
for any bounded measurable function
{\displaystyle f}
. As before, this convergence is non-uniform in
{\displaystyle f}
The notion of total variation convergence formalizes the assertion that the measure of all measurable sets should converge uniformly, i.e. for every
{\displaystyle \varepsilon >0}
there exists N such that
{\displaystyle |\mu _{n}(A)-\mu (A)|<\varepsilon }
for every n > N and for every measurable set
{\displaystyle A}
. As before, this implies convergence of integrals against bounded measurable functions, but this time convergence is uniform over all functions bounded by any fixed constant.
Total variation convergence of measuresEdit
This is the strongest notion of convergence shown on this page and is defined as follows. Let
{\displaystyle (X,{\mathcal {F}})}
be a measurable space. The total variation distance between two (positive) measures μ and ν is then given by
{\displaystyle \left\|\mu -\nu \right\|_{\text{TV}}=\sup _{f}\left\{\int _{X}f\,d\mu -\int _{X}f\,d\nu \right\}.}
Here the supremum is taken over f ranging over the set of all measurable functions from X to [−1, 1]. This is in contrast, for example, to the Wasserstein metric, where the definition is of the same form, but the supremum is taken over f ranging over the set of measurable functions from X to [−1, 1] which have Lipschitz constant at most 1; and also in contrast to the Radon metric, where the supremum is taken over f ranging over the set of continuous functions from X to [−1, 1]. In the case where X is a Polish space, the total variation metric coincides with the Radon metric.
If μ and ν are both probability measures, then the total variation distance is also given by
{\displaystyle \left\|\mu -\nu \right\|_{\text{TV}}=2\cdot \sup _{A\in {\mathcal {F}}}|\mu (A)-\nu (A)|.}
The equivalence between these two definitions can be seen as a particular case of the Monge-Kantorovich duality. From the two definitions above, it is clear that the total variation distance between probability measures is always between 0 and 2.
To illustrate the meaning of the total variation distance, consider the following thought experiment. Assume that we are given two probability measures μ and ν, as well as a random variable X. We know that X has law either μ or ν but we do not know which one of the two. Assume that these two measures have prior probabilities 0.5 each of being the true law of X. Assume now that we are given one single sample distributed according to the law of X and that we are then asked to guess which one of the two distributions describes that law. The quantity
{\displaystyle {2+\|\mu -\nu \|_{\text{TV}} \over 4}}
then provides a sharp upper bound on the prior probability that our guess will be correct.
Given the above definition of total variation distance, a sequence μn of measures defined on the same measure space is said to converge to a measure μ in total variation distance if for every ε > 0, there exists an N such that for all n > N, one has that[1]
{\displaystyle \|\mu _{n}-\mu \|_{\text{TV}}<\varepsilon .}
Setwise convergence of measuresEdit
{\displaystyle (X,{\mathcal {F}})}
a measurable space, a sequence μn is said to converge setwise to a limit μ if
{\displaystyle \lim _{n\to \infty }\mu _{n}(A)=\mu (A)}
{\displaystyle A\in {\mathcal {F}}}
For example, as a consequence of the Riemann–Lebesgue lemma, the sequence μn of measures on the interval [−1, 1] given by μn(dx) = (1+ sin(nx))dx converges setwise to Lebesgue measure, but it does not converge in total variation.
Weak convergence of measuresEdit
In mathematics and statistics, weak convergence is one of many types of convergence relating to the convergence of measures. It depends on a topology on the underlying space and thus is not a purely measure theoretic notion.
There are several equivalent definitions of weak convergence of a sequence of measures, some of which are (apparently) more general than others. The equivalence of these conditions is sometimes known as the Portmanteau theorem.[2]
{\displaystyle S}
be a metric space with its Borel
{\displaystyle \sigma }
{\displaystyle \Sigma }
. A bounded sequence of positive probability measures
{\displaystyle P_{n}\,(n=1,2,\dots )}
{\displaystyle (S,\Sigma )}
is said to converge weakly to a probability measure
{\displaystyle P}
{\displaystyle P_{n}\Rightarrow P}
) if any of the following equivalent conditions is true (here
{\displaystyle \operatorname {E} _{n}}
denotes expectation or the
{\displaystyle L^{1}}
norm with respect to
{\displaystyle P_{n}}
{\displaystyle \operatorname {E} }
{\displaystyle L^{1}}
{\displaystyle P}
{\displaystyle \operatorname {E} _{n}[f]\to \operatorname {E} [f]}
for all bounded, continuous functions
{\displaystyle f}
{\displaystyle \operatorname {E} _{n}[f]\to \operatorname {E} [f]}
for all bounded and Lipschitz functions
{\displaystyle f}
{\displaystyle \limsup \operatorname {E} _{n}[f]\leq \operatorname {E} [f]}
for every upper semi-continuous function
{\displaystyle f}
bounded from above;
{\displaystyle \liminf \operatorname {E} _{n}[f]\geq \operatorname {E} [f]}
for every lower semi-continuous function
{\displaystyle f}
bounded from below;
{\displaystyle \limsup P_{n}(C)\leq P(C)}
for all closed sets
{\displaystyle C}
{\displaystyle S}
{\displaystyle \liminf P_{n}(U)\geq P(U)}
{\displaystyle U}
{\displaystyle S}
{\displaystyle \lim P_{n}(A)=P(A)}
for all continuity sets
{\displaystyle A}
of measure
{\displaystyle P}
{\displaystyle S\equiv \mathbf {R} }
with its usual topology, if
{\displaystyle F_{n}}
{\displaystyle F}
denote the cumulative distribution functions of the measures
{\displaystyle P_{n}}
{\displaystyle P}
, respectively, then
{\displaystyle P_{n}}
{\displaystyle P}
{\displaystyle \lim _{n\to \infty }F_{n}(x)=F(x)}
{\displaystyle x\in \mathbf {R} }
{\displaystyle F}
For example, the sequence where
{\displaystyle P_{n}}
is the Dirac measure located at
{\displaystyle 1/n}
converges weakly to the Dirac measure located at 0 (if we view these as measures on
{\displaystyle \mathbf {R} }
with the usual topology), but it does not converge setwise. This is intuitively clear: we only know that
{\displaystyle 1/n}
is "close" to
{\displaystyle 0}
because of the topology of
{\displaystyle \mathbf {R} }
This definition of weak convergence can be extended for
{\displaystyle S}
any metrizable topological space. It also defines a weak topology on
{\displaystyle {\mathcal {P}}(S)}
, the set of all probability measures defined on
{\displaystyle (S,\Sigma )}
. The weak topology is generated by the following basis of open sets:
{\displaystyle \left\{\ U_{\phi ,x,\delta }\ \left|\quad \phi \colon S\to \mathbf {R} {\text{ is bounded and continuous, }}x\in \mathbf {R} {\text{ and }}\delta >0\ \right.\right\},}
{\displaystyle U_{\phi ,x,\delta }:=\left\{\ \mu \in {\mathcal {P}}(S)\ \left|\quad \left|\int _{S}\phi \,\mathrm {d} \mu -x\right|<\delta \ \right.\right\}.}
{\displaystyle S}
is also separable, then
{\displaystyle {\mathcal {P}}(S)}
is metrizable and separable, for example by the Lévy–Prokhorov metric. If
{\displaystyle S}
is also compact or Polish, so is
{\displaystyle {\mathcal {P}}(S)}
{\displaystyle S}
is separable, it naturally embeds into
{\displaystyle {\mathcal {P}}(S)}
as the (closed) set of Dirac measures, and its convex hull is dense.
There are many "arrow notations" for this kind of convergence: the most frequently used are
{\displaystyle P_{n}\Rightarrow P}
{\displaystyle P_{n}\rightharpoonup P}
{\displaystyle P_{n}{\xrightarrow {\mathcal {D}}}P.}
Weak convergence of random variablesEdit
{\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}
be a probability space and X be a metric space. If Xn, X: Ω → X is a sequence of random variables then Xn is said to converge weakly (or in distribution or in law) to X as n → ∞ if the sequence of pushforward measures (Xn)∗(P) converges weakly to X∗(P) in the sense of weak convergence of measures on X, as defined above.
^ Madras, Neil; Sezer, Deniz (25 Feb 2011). "Quantitative bounds for Markov chain convergence: Wasserstein and total variation distances". Bernoulli. 16 (3): 882–908. arXiv:1102.5245. doi:10.3150/09-BEJ238.
^ Klenke, Achim (2006). Probability Theory. Springer-Verlag. ISBN 978-1-84800-047-6.
Ambrosio, L., Gigli, N. & Savaré, G. (2005). Gradient Flows in Metric Spaces and in the Space of Probability Measures. Basel: ETH Zürich, Birkhäuser Verlag. ISBN 3-7643-2428-7. {{cite book}}: CS1 maint: multiple names: authors list (link)
Billingsley, Patrick (1995). Probability and Measure. New York, NY: John Wiley & Sons, Inc. ISBN 0-471-00710-2.
Billingsley, Patrick (1999). Convergence of Probability Measures. New York, NY: John Wiley & Sons, Inc. ISBN 0-471-19745-9.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Convergence_of_measures&oldid=1087072721"
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The classification of universal Jacobians over the moduli space of curves | EMS Press
The classification of universal Jacobians over the moduli space of curves
We complete the classification by Kodaira dimension of the moduli space of degree
g
line bundles over curves of genus
g
, for all genera.
Gavril Farkas, Alessandro Verra, The classification of universal Jacobians over the moduli space of curves. Comment. Math. Helv. 88 (2013), no. 3, pp. 587–611
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miguelgondu's blog
A Link to my MSc Thesis
The title of this post is the title of my M.Sc. thesis, which I defended on Friday 27th, 2019. In this thesis I explore a method deviced by Niels Justesen, Sebastian Risi and I, called Behavioral Repertoires Imitation Learning. This method extends Imitation Learning by adding behavioral features to state-action pairs. The policies that are trained with this method are able to express more than one behavior (e.g. bio-oriented, mech-oriented) on command.
In summary, the way this is done is by designing a behavior space, a subset of
\mathbb{R}^M
that encodes the player's behavior in some way. After that, the dimensions are reduced in order to clusterize and understand the different behaviors present in the demonstrations. Finally, we expanded the state-action pairs gathered from the demonstrations with the coordinates in this low-dimensional space. This process was originally described in this preprint on arXiv. The behavior space I tackled in my thesis was a little bit different.
Last week, I pushed the final document of my thesis to the public repositories of the National University of Colombia. Here's a link to it.
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減少誤差的方法 | ePractice - HKDSE 試題導向練習平台
數學知識重點「減少誤差的方法」的樣本
{\left({a}\right)}
Using measuring tools with finer scale intervals.
{\left({b}\right)}
When measuring very tiny objects, measure a large number of that object, and then obtain the measurement of one single item by division.
{\left({c}\right)}
Some quantities can be measured indirectly by using formulas and the result obtained may be better than a direct measurement of the quantity.
{\left({d}\right)}
Use correct methods to take measurements.
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Due to technical limitations in the Ore_algebra package, you can not use algebras with pairs of variables that depend on each other, such as d[x] and x, to construct a MonomialOrder. In general, depends(u,v) must return false for each pair of variables
u\ne v
, otherwise MonomialOrder will return an error. Use variable names such as dx and x or d[1] and x[1] instead. This limitation does not apply to ordinary commutative computations using ShortMonomialOrders.
\mathrm{with}\left(\mathrm{Ore_algebra}\right):
\mathrm{with}\left(\mathrm{Groebner}\right):
A≔\mathrm{poly_algebra}\left(x,y,z\right)
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Ore_algebra}}
T≔\mathrm{MonomialOrder}\left(A,\mathrm{plex}\left(x,y,z\right)\right)
\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{monomial_order}}
f≔-4yx-3{z}^{2}+y{z}^{2}-5{x}^{4}+{x}^{3}{z}^{2}
\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{2}}
\mathrm{LeadingTerm}\left(f,T\right)
\textcolor[rgb]{0,0,1}{-5}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{4}}
\mathrm{LeadingTerm}\left(f,\mathrm{plex}\left(x,y,z\right)\right)
\textcolor[rgb]{0,0,1}{-5}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{4}}
A≔\mathrm{diff_algebra}\left([\mathrm{D}[1],x[1]],[\mathrm{D}[2],x[2]]\right)
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Ore_algebra}}
A["polynomial_indets"]
{{\textcolor[rgb]{0,0,1}{\mathrm{D}}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{D}}}_{\textcolor[rgb]{0,0,1}{2}}}
A["rational_indets"]
{{\textcolor[rgb]{0,0,1}{x}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{x}}_{\textcolor[rgb]{0,0,1}{2}}}
T≔\mathrm{MonomialOrder}\left(A,\mathrm{tdeg}\left(\mathrm{D}[1],\mathrm{D}[2]\right)\right)
\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{monomial_order}}
\mathrm{SPolynomial}\left(\mathrm{D}[1]x[2]-x[2],\mathrm{D}[2]x[1]-x[1],T\right)
\left({\textcolor[rgb]{0,0,1}{x}}_{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}_{\textcolor[rgb]{0,0,1}{1}}\right)\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{D}}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{x}}_{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{x}}_{\textcolor[rgb]{0,0,1}{2}}\right)\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{D}}}_{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{x}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}_{\textcolor[rgb]{0,0,1}{2}}
A≔\mathrm{poly_algebra}\left(x,y,z\right):
T≔\mathrm{MonomialOrder}\left(A,'\mathrm{user}'\left(P,[x,y,z]\right)\right)
\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{monomial_order}}
\mathrm{LeadingTerm}\left(f,T\right)
\textcolor[rgb]{0,0,1}{-5}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{4}}
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Accelerated failure time model - Wikipedia
Parametric model in survival analysis
In the statistical area of survival analysis, an accelerated failure time model (AFT model) is a parametric model that provides an alternative to the commonly used proportional hazards models. Whereas a proportional hazards model assumes that the effect of a covariate is to multiply the hazard by some constant, an AFT model assumes that the effect of a covariate is to accelerate or decelerate the life course of a disease by some constant. This is especially appealing in a technical context where the 'disease' is a result of some mechanical process with a known sequence of intermediary stages.
2 Statistical issues
2.1 Distributions used in AFT models
In full generality, the accelerated failure time model can be specified as[1]
{\displaystyle \lambda (t|\theta )=\theta \lambda _{0}(\theta t)}
{\displaystyle \theta }
denotes the joint effect of covariates, typically
{\displaystyle \theta =\exp(-[\beta _{1}X_{1}+\cdots +\beta _{p}X_{p}])}
. (Specifying the regression coefficients with a negative sign implies that high values of the covariates increase the survival time, but this is merely a sign convention; without a negative sign, they increase the hazard.)
This is satisfied if the probability density function of the event is taken to be
{\displaystyle f(t|\theta )=\theta f_{0}(\theta t)}
; it then follows for the survival function that
{\displaystyle S(t|\theta )=S_{0}(\theta t)}
. From this it is easy[citation needed] to see that the moderated life time
{\displaystyle T}
is distributed such that
{\displaystyle T\theta }
and the unmoderated life time
{\displaystyle T_{0}}
have the same distribution. Consequently,
{\displaystyle \log(T)}
{\displaystyle \log(T)=-\log(\theta )+\log(T\theta ):=-\log(\theta )+\epsilon }
where the last term is distributed as
{\displaystyle \log(T_{0})}
, i.e., independently of
{\displaystyle \theta }
. This reduces the accelerated failure time model to regression analysis (typically a linear model) where
{\displaystyle -\log(\theta )}
represents the fixed effects, and
{\displaystyle \epsilon }
represents the noise. Different distributions of
{\displaystyle \epsilon }
imply different distributions of
{\displaystyle T_{0}}
, i.e., different baseline distributions of the survival time. Typically, in survival-analytic contexts, many of the observations are censored: we only know that
{\displaystyle T_{i}>t_{i}}
{\displaystyle T_{i}=t_{i}}
. In fact, the former case represents survival, while the later case represents an event/death/censoring during the follow-up. These right-censored observations can pose technical challenges for estimating the model, if the distribution of
{\displaystyle T_{0}}
is unusual.
{\displaystyle \theta }
in accelerated failure time models is straightforward:
{\displaystyle \theta =2}
means that everything in the relevant life history of an individual happens twice as fast. For example, if the model concerns the development of a tumor, it means that all of the pre-stages progress twice as fast as for the unexposed individual, implying that the expected time until a clinical disease is 0.5 of the baseline time. However, this does not mean that the hazard function
{\displaystyle \lambda (t|\theta )}
is always twice as high - that would be the proportional hazards model.
Statistical issues[edit]
Unlike proportional hazards models, in which Cox's semi-parametric proportional hazards model is more widely used than parametric models, AFT models are predominantly fully parametric i.e. a probability distribution is specified for
{\displaystyle \log(T_{0})}
. (Buckley and James[2] proposed a semi-parametric AFT but its use is relatively uncommon in applied research; in a 1992 paper, Wei[3] pointed out that the Buckley–James model has no theoretical justification and lacks robustness, and reviewed alternatives.) This can be a problem, if a degree of realistic detail is required for modelling the distribution of a baseline lifetime. Hence, technical developments in this direction would be highly desirable.
Unlike proportional hazards models, the regression parameter estimates from AFT models are robust to omitted covariates. They are also less affected by the choice of probability distribution.[4][5]
The results of AFT models are easily interpreted.[6] For example, the results of a clinical trial with mortality as the endpoint could be interpreted as a certain percentage increase in future life expectancy on the new treatment compared to the control. So a patient could be informed that he would be expected to live (say) 15% longer if he took the new treatment. Hazard ratios can prove harder to explain in layman's terms.
Distributions used in AFT models[edit]
The log-logistic distribution provides the most commonly used AFT model. Unlike the Weibull distribution, it can exhibit a non-monotonic hazard function which increases at early times and decreases at later times. It is somewhat similar in shape to the log-normal distribution but it has heavier tails. The log-logistic cumulative distribution function has a simple closed form, which becomes important computationally when fitting data with censoring. For the censored observations one needs the survival function, which is the complement of the cumulative distribution function, i.e. one needs to be able to evaluate
{\displaystyle S(t|\theta )=1-F(t|\theta )}
The Weibull distribution (including the exponential distribution as a special case) can be parameterised as either a proportional hazards model or an AFT model, and is the only family of distributions to have this property. The results of fitting a Weibull model can therefore be interpreted in either framework. However, the biological applicability of this model may be limited by the fact that the hazard function is monotonic, i.e. either decreasing or increasing.
Other distributions suitable for AFT models include the log-normal, gamma and inverse Gaussian distributions, although they are less popular than the log-logistic, partly as their cumulative distribution functions do not have a closed form. Finally, the generalized gamma distribution is a three-parameter distribution that includes the Weibull, log-normal and gamma distributions as special cases.
^ Kalbfleisch & Prentice (2002). The Statistical Analysis of Failure Time Data (2nd ed.). Hoboken, NJ: Wiley Series in Probability and Statistics.
^ Buckley, Jonathan; James, Ian (1979), "Linear regression with censored data", Biometrika, 66 (3): 429–436, doi:10.1093/biomet/66.3.429, JSTOR 2335161
^ Wei, L. J. (1992). "The accelerated failure time model: A useful alternative to the cox regression model in survival analysis". Statistics in Medicine. 11 (14–15): 1871–1879. doi:10.1002/sim.4780111409. PMID 1480879.
^ Lambert, Philippe; Collett, Dave; Kimber, Alan; Johnson, Rachel (2004), "Parametric accelerated failure time models with random effects and an application to kidney transplant survival", Statistics in Medicine, 23 (20): 3177–3192, doi:10.1002/sim.1876, PMID 15449337
^ Keiding, N.; Andersen, P. K.; Klein, J. P. (1997). "The Role of Frailty Models and Accelerated Failure Time Models in Describing Heterogeneity Due to Omitted Covariates". Statistics in Medicine. 16 (1–3): 215–224. doi:10.1002/(SICI)1097-0258(19970130)16:2<215::AID-SIM481>3.0.CO;2-J. PMID 9004393.
^ Kay, Richard; Kinnersley, Nelson (2002), "On the use of the accelerated failure time model as an alternative to the proportional hazards model in the treatment of time to event data: A case study in influenza", Drug Information Journal, 36 (3): 571–579, doi:10.1177/009286150203600312
Bradburn, MJ; Clark, TG; Love, SB; Altman, DG (2003), "Survival Analysis Part II: Multivariate data analysis - an introduction to concepts and methods", British Journal of Cancer, 89 (3): 431–436, doi:10.1038/sj.bjc.6601119, PMC 2394368, PMID 12888808
Hougaard, Philip (1999), "Fundamentals of Survival Data", Biometrics, 55 (1): 13–22, doi:10.1111/j.0006-341X.1999.00013.x, PMID 11318147
Collett, D. (2003), Modelling Survival Data in Medical Research (2nd ed.), CRC press, ISBN 978-1-58488-325-8
Cox, David Roxbee; Oakes, D. (1984), Analysis of Survival Data, CRC Press, ISBN 978-0-412-24490-2
Marubini, Ettore; Valsecchi, Maria Grazia (1995), Analysing Survival Data from Clinical Trials and Observational Studies, Wiley, ISBN 978-0-470-09341-2
Martinussen, Torben; Scheike, Thomas (2006), Dynamic Regression Models for Survival Data, Springer, ISBN 0-387-20274-9
Bagdonavicius, Vilijandas; Nikulin, Mikhail (2002), Accelerated Life Models. Modeling and Statistical Analysis, Chapman&Hall/CRC, ISBN 1-58488-186-0
Retrieved from "https://en.wikipedia.org/w/index.php?title=Accelerated_failure_time_model&oldid=991846535"
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state any two postulates of bohr's theory of hydrogen atom what is the maximum possible number of spectral lines - Physics - Electric Charges And Fields - 6900429 | Meritnation.com
state any two postulates of bohr's theory of hydrogen atom. what is the maximum possible number of spectral lines observed when hydrogen atom is in its second excited state. justify your answer. calculate ratio of the maximum and minimum wavelength of the radiations emitted in this process.
Abhay Kumar answered this
1.Bohr's Postulates are following :- He only used the principle of newtons mechanics and Planks theory in his postulates !!
1. In an atom, the electrons revolve around the nucleus in certain definite circular paths called orbits, or shells.
2. Each shell or orbit corresponds to a definite energy. Therefore, these circular orbits are also known as energy levels or energy shells.
2. Now hydrogen atom in second excited state means n=3
from where only only three types of spectral lines are observed!!
3 to2, 2 to1, 3 to1.
The answer can be obtained by using permutation and combination to find the answer if n is higher!
The required formulae is n(n-1)/2 where n is the orbit no. in which e is present!
3. Its a known fact that wavelength is inversely proportional to energy of photon emitted!
so by taking the inverse ratio of energies
the wavelength corresponding to maximum wavelength has minimum energy and vice versa!
maximum E = minimum wavelength = transition from n=3 to n=1
minimum E = maximum wavelength = transition from n=3 to n=2
As total energy of electron in nth orbit of H2 atom is given by :- E=
-\frac{13.6}{{n}^{2}}eV
Here, negative sign shows that the electron is bound to the nucleus and is not free to leave it.
Therefore,ratio of maximum wavelength/ minimum wavelength = ratio of maximum energy / minimum energy.
= 13.6 (1/12 - 1/32) /13.6 (1/22 - 1/32)
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CoCalc – Run Python Online
Run Python scripts, Jupyter notebooks, or even a graphical application in a full, remote Python environment.
Try Python Now
CoCalc covers all the bases
Data Science and Machine Learning: Upload your datafiles and analyze them using Tensorflow, scikit-learn, Keras, ... including an Anaconda environment.
Mathematics: SymPy, SageMath, ...
Statistics: pandas, statsmodels, rpy2 (R bridge), ...
Visualization: matplotlib, plotly, seaborn, ...
Teaching: learn Python online or teach a course.
Find more details in the list of installed Python libraries.
Immediately start working by creating or uploading, Jupyter Notebooks or Python scripts.
No need to download and install Python, Anaconda, or other Python environments.
CoCalc already provides many packages for you.
The LaTeX editor is already integrated with PythonTeX and SageTeX.
There are many ways to use Python online via CoCalc.
As the name suggests, CoCalc's strength is online code collaboration. Collaboration applies to editing plain Python files, Sage Worksheets, Jupyter Notebooks, and much more.
This enables you to work more effectively as a team to solve the challenges of data science, machine learning and statistics. Every collaborator is always looking at the most recent state of files, and they experience and inspect the same Python state.
You can create chatrooms and get help via side chat by @mentioning collaborators.
Python in Jupyter Notebooks
CoCalc offers a complete rewrite of the classical Jupyter notebook interface. It is tightly integrated into CoCalc and adds realtime collaboration, TimeTravel history and much more.
The user interface is very similar to Jupyter classic. It uses the same underlying Jupyter notebook file format, so you can download your *.ipynb file at any time and continue working locally.
There are several Python environments available.
You can also easily run Jupyter Classical and JupyterLab in any CoCalc project.
\LaTeX
support for PythonTeX/SageTeX
The fully integrated CoCalc latex editor covers all your basic needs for working with .tex files containing PythonTeX or SageTeX code. The document is synchronized with your collaborators in real-time and everyone sees the very same compiled PDF.
In particular, this LaTeX editor
Manages the entire compilation pipeline for you: it automatically calls pythontex3 or sage to pre-process the code,
Supports forward and inverse search to help you navigating in your document,
Captures and shows you where LaTeX or Python errors happen,
and via TimeTravel you can go back in time to see your latest edits in order to easily recover from a recent mistake.
Combined, this means you can do your entire workflow online on CoCalc:
Upload or fetch your datasets,
Use Jupyter Notebooks to explore the data, process it, and calculate your results,
Discuss and collaborate with your research team,
Write your research paper in a LaTeX document,
Publish the datasets, your research code, and the PDF of your paper online, all hosted on CoCalc.
CoCalc has one-click code formatting for Jupyter notebooks and code files!
Your python code is formatted in a clean and consistent way using yapf.
This reduces cognitive load reading source code, and ensures all code written by your team has a consistent and beautiful style.
Python code formatting works with pure .py files and Jupyter Notebooks running a Python kernel.
Your existing Python scripts run on CoCalc. Either open a Terminal in the code editor, or click the "Shell" button to open a Python command line.
Terminals also give you access to git and many more utilities.
Regarding collaboration, terminals can be used by multiple users at once. This means you can work with your coworkers in the same session at the same time. Everyone sees the same output, and coordinate via side chat next to the terminal.
You can also simultaneously work with many terminal sessions.
For long-running programs, you can even close your browser and check on the result later.
Chatroom about your Python code
Collaboration is a first class citizen on CoCalc. Use side chat for each file to discuss content with your colleagues or students.
Additionally, avatars give you presence information about who is currently also working on a file.
Collaborators who are not online will be notified about new messages the next time they sign in.
Chat also supports markdown formatting and
\LaTeX
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Video: 2-step Input/ Output Machine | Zak.io
Video: 2-step Input/ Output Machine
I recently helped with a lesson that involved using input/output machines to teach fractions to 5th graders. To help students visualize these machines, I created the animation. It begins with machines that transform shapes into circles to make the idea of the machines more tangible. Then, it switches to math and aligns with some of the first problems in the student work packets for the grade 5 CT lesson. This video can be used in conjunction with the lesson described below or on its own to introduce IO machines and/or functions in programming in a way that is accessible to elementary school students.
I’ve been working with the math coach at my school to incorporate computer science/ computational thinking into elementary school math lessons. One set of lessons we’ve used most recently comes from the Education Development Center. They’ve partnered with the Massachusetts Department of Elementary and Secondary Education to develop a series of lessons, modules, and activities to incorporate computational thinking into existing lessons that align with Massachusetts math and science standards. At the time of this writing, the materials are still under development and may change.
We did a modified version of the Grade 5 Math - Number Fluency and Fractions module. In this module, students use two-step input/output machines as a technique to multiply whole numbers by fractions. For example,
8\times\frac3 4
8\times3\div 4
. We can envision the multiplication and division as two input/output machines. The first machine takes multiplies its input by 3 and the second machine divides its input by 4. The lessons take students through a series of these machines, demonstrating different kinds of algorithms.
The lesson plans call for these machines to teach students about algorithms, which help develop computational thinking. They can also set students up to think in terms of functions. In computer science, functions are used to run the same step or steps multiple times without having to define them each time. The function is defined once and called (or run) as many times as necessary. In my experience working with first-time coders in middle and high school, functions can be one of the first major sources of confusion. I’ve been looking for a way to help students visualize functions, and input/output machines are the perfect way to do so.
Tags: Computational Thinking, Math, Sneaky CS
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Floor No. 20 | Toph
By Sk_Sabit · Limits 1s, 512 MB
You are at a 12 story building called ECE Bhaban. You have just entered into a lift which has took you to a mysterious floor, Floor no. 20. In fact, you have got stuck into the lift and there is no way out of it except unlocking a Mysterious Lock. In order to unlock it, you have to solve a Puzzle.
The puzzle consists of a Binary Matrix of dimension
N \times M
N×M. A binary Matrix has 0 or 1 in each of it’s cells and it’s called Beautiful when each of its rows has an odd number of ones and each of its columns has an even number of ones.
Now, to unlock the mysterious lock, you have to transform the given matrix into a Beautiful one. In one move, you can flip any cell value of the matrix (flipping a value means, if there is a
0, after the move there will be a
1, and vice versa). But the time is short, you are suffocating inside the lift. So, you have to do it with a
Minimum number of moves or report if it’s impossible to unlock the mysterious lock.
The rows are numbered from
1 to
N (top to bottom) and the columns are numbered from
1 to
M (left to right).
You have to print the minimum number of moves required to unlock the lock and the corresponding moves. If there are multiple solution, you can print any of them. If there is no solution, print “-1” (without quotes).
The first line of input consists of two space separated integers
N and
M (
1 \leq N, M \leq 1000
1≤N,M≤1000), representing the number of rows and the number of columns of the matrix respectively.
N line contains
M characters each (each of which is either 0 or 1) representing the rows of the matrix.
The first line of output will contain a single integer
K, the minimum number of moves required to unlock the lock or “-1” (without quotes) if it’s impossible to unlock.
If it’s possible to solve the puzzle, then next
K lines will print the moves. It will contain two space separated integers each, representing the co-ordinate of the cell which has been flipped on that move.
The output format will be of this form:
K
X_{1}
X1
Y_{1}
Y1
X_{2}
Y_{2}
X_{k}
Xk
Y_{k}
Yk
X_{i}
Xi and
Y_{i}
Yi represents the row index of the cell (from top) and column index of the cell (from left) respectively.
1 \leq X_{i} \leq N, 1 \leq Y_{i} \leq M
1≤Xi≤N,1≤Yi≤M)
It is guaranteed that if there is a solution, then the number of moves of the solution will not exceed N x M.
After applying the 3 moves from output, the resultant matrix looks like this-
number of ones in rows are respectively 1, 3, 1, 1. All these values are odd.
number of ones in columns are respectively 2, 2, 2. All these values are even.
There is no way to make this matrix beautiful.
Zobayer_AbedinEarliest, 2M ago
nh_nayeemFastest, 0.0s
Zobayer_AbedinLightest, 131 kB
Zobayer_AbedinShortest, 1033B
|
January, 2003 Residues of Chern classes
If we have a finite number of sections of a complex vector bundle
E
M
, certain Chern classes of
E
are localized at the singular set
S
, i.e., the set of points where the sections fail to be linearly independent. When
S
is compact, the localizations define the residues at each connected component of
S
by the Alexander duality. If
M
itself is compact, the sum of the residues is equal to the Poincaré dual of the corresponding Chern class. This type of theory is also developed for vector bundles over a possibly singular subvariety in a complex manifold. Explicit formulas for the residues at an isolated singular point are also given, which express the residues in terms of Grothendieck residues relative to the subvariety.
Tatsuo SUWA. "Residues of Chern classes." J. Math. Soc. Japan 55 (1) 269 - 287, January, 2003. https://doi.org/10.2969/jmsj/1196890854
Primary: 14C17 , 32A27 , 57R20
Secondary: 14B05 , 32S05
Keywords: Chern classes , frames of vector bundles , Grothendieck residues relative to subvarieties , Localization
Tatsuo SUWA "Residues of Chern classes," Journal of the Mathematical Society of Japan, J. Math. Soc. Japan 55(1), 269-287, (January, 2003)
|
“Okay, slow down a moment, I’m still trying to figure out this step.”
2\left(x+6\right)+3\left(x+4\right)
Kongpob scratches his head, staring at the question and trying to pick out how to best expand and simplify the equation. They’re in the library again this morning, Arthit diving straight into practice questions before Kongpob gets the chance to distract him with small talk.
=2x+12+3x+12
He looks up at Arthit for confirmation.
“Yeah. Now, you see that 2x and 3x are both of x value, so when you add them together, it becomes…?”
“Right. Any other values you can simplify?”
= 5x + 144, he pencils in after a moment of hesitation.
Arthit shakes his head, pointing at the 144.
“You’ve multiplied the two 12’s instead of just adding them. You only multiply when one value is outside of the brackets.”
“Right,” Kongpob nods, reaching into his pencil case for an eraser, but finding none. “Hey, could I borrow an eraser?”
Arthit feels around in the small grey bag holding his stationery and pauses a moment, before handing Kongpob his own eraser. Kongpob smirks upon seeing the familiar piece of stationery.
“I meant to ask before; why do you have only part of my name written on this?”
His friend audibly gulps, eyes shifting between the book and his suddenly fascinating pencil.
“Oh, uh…I…so that I would remember who I borrowed it from so I can return it. I’ve still been using it, though, so the o and b are gone now.”
Kongpob just nods, erasing his incorrect answer.
“Uh…you can have it back,” Arthit adds quickly. “I just keep forgetting to return it.”
“It’s fine, you can keep it. I can get a new one,” he smiles. “Look after this one for me.”
“Look after it? It’s an eraser, not a domesticated animal.”
“Just keep it,” Kongpob laughs, shaking his head. He takes Arthit’s hand, placing the shabby eraser in his open palm. He subconsciously notes how soft and warm his skin is, and shrinks his own hand back, suddenly intensely interested in the practice questions.
If the tips of Arthit’s ears grow slightly pink, Kongpob doesn’t notice.
“So uh…why this bathroom?” Kongpob asks between bites of food. Mae has made one of his childhood favourites; pan-fried shrimp cakes (non-spicy) with a side of pomelo salad.
He’s sitting on the up-turned bucket again, his lunch box propped up on the sink counter as a table. Arthit doesn’t answer him at first, taking his time to chew and swallow his food.
“I can lock the doors,” he finally replies.
Kongpob nods, cutting into a shrimp cake with his spoon.
“I remember in middle school, kids used to say that the third floor boys’ bathroom was haunted. I think the rumour started when the second Harry Potter film came out, and then people just passed it on over the years.”
Arthit snorts at this.
“Yeah, I’ve heard about that,” he presses his spoon against a clump of rice, squishing it into mush. “Although the only ghost you would have found in there was yours truly.”
He says this almost bitterly, and it doesn’t go unnoticed by Kongpob, who feels awkward.
“It’s fine,” Arthit says. “I…don’t even know why I still do this. I’m just used to it, I guess.”
“It’s not that easy, Kongpob.”
“I know,” Kongpob adjusts his seat. “Or, well, I guess I don’t. I just mean…I would never judge you or anyone else like that.”
Arthit remains quiet again, rolling the mouthful of rice around on his tongue until the grains separate.
“They…they used to call me…‘Porky’.”
Kongpob already knows this from M, but doesn’t say anything, waiting for Arthit to continue at his own pace.
“That was my dad’s nickname, hence the name of the cart. I…uh, used to be a lot bigger than I am now, and kids weren’t exactly nice about it, so…I started eating where nobody could see me. It was just easier that way. Nobody wanted to eat with me, and I didn’t want to eat with anybody, either.”
Arthit isn’t even sure why he’s telling Kongpob all of this when they’ve literally only started talking for less than two weeks. Until now, they’d been ships passing each other in the night, and he has no real reason to trust someone who clearly leads such a different life. He thinks that maybe it’s because it’s the first time someone is listening to him. Then again, he thinks, it’s often easier talking to someone that you can’t see.
“I wish I’d known you back then,” Kongpob says, breaking the silence. “I would have eaten with you.”
Arthit smiles a little at this. I know, he thinks. But you clearly don’t remember anything, do you?
“I’m pretty sure that would have ruined your reputation or something.”
“You…I don’t know, you had people flocking around you all the time. Teachers loved you, you did well in school, and girls wrote you notes and left you snacks hoping you’d notice them.”
Kongpob bites into another shrimp cake, amused at the remark.
“For someone who had such a hard time accepting me as their friend, you sure paid a lot of attention to me.”
“Did not!” Arthit is glad that Kongpob can’t see his face right now, red as the curry in the carton on his lap. “I notice things about everyone in my class.”
“Sure,” Kongpob’s laughter rings and echoes against the washroom walls. He replaces the lid on his lunchbox, and his smile fades as he realises something. “Wait, we were in the same class?”
Arthit almost drops his spoon, realising he’s said too much.
“Why only the first half?”
There’s silence again, and Arthit briefly contemplates sinking into the toilet and flushing himself away. Way to accidentally overshare and potentially scare away the first friend he’s made in four years.
Kongpob must sense his unease even through the door, because he drops the subject.
“Sorry, never mind. I’m being nosy again,” he says, half laughing as he stands up. “Anyway, I’ve…got to go before M starts looking for me.”
Arthit nods, licking at the sauce at the corners of his mouth.
“Thanks for having lunch with me,” Kongpob says again, and the washroom door swings shut.
The tiled room becomes silent again, the only remaining sound being the faded shouts and chatter from the courtyard a good twenty metres away.
Arthit wonders if he’ll ever be able to blend into that crowd.
“Hey, where were you?” M shuffles over on the bench as Kong approaches their usual table.
“Just went to help Teacher Lynn with the display board.”
“When is English Week again?”
Kongpob stills a moment, trying to recall when the actual event he’d now adopted as his excuse actually is.
M eyes his friend blankly. Surely the club president would know this crucial piece of information. In fact, M has noticed that in these past two weeks, Kongpob has been behaving rather unusually on the whole. He doesn’t question it, though, peering into Kongpob’s lunchbox. Kongpob rolls his eyes and slides the box over to him, earning an excited clap from M, who sticks a fork into a shrimp cake.
Kong☕: 3 🐖 2🐓 1 🐄? 😊
Arthit☀️: y didn’t u just tell me what u want in person
Arthit☀️: instead of texting me
Kong☕: This is more fun, I like the animal emojis ☺️🐖🐓🐄
Arthit☀️:🐍🐸🐢
Kong☕: No! I would never eat a turtle!😱
Arthit☀️: rly? it’s the turtle that bothers u?
Kong☕: Turtles are cute. Snakes and frogs, not so much.
Arthit☀️: pigs & cows are sorta cute
Arthit☀️: tho if ur saying that u would eat anything that’s not cute, there’s a guy a few stalls away who sells crispy frogs legs
Kong☕: 🤢I think I’ll pass…
Arthit☀️: it just tastes like chicken
Kong☕: I’d still know they were frogs’ legs, though!
Arthit☀️: aite stop texting me if u actually want ur order
Kong☕: Thank youuuu😍
Arthit stares at Kongpob’s choice of emoji for a moment, deciding not to read into it. It seems like the type of thing that his annoying new friend embeds into his texting behaviours, including complete sentences with correct capitalisation and full spellings.
He shoves his phone back into his apron pocket, and sets about lining up a fresh batch of skewers. A few other customers are already in line, but he sets aside enough for Kongpob’s order on the grill, so they’ll be hot when he arrives.
“Thank you,” he smiles politely at a customer as they walk away, just in time for Kongpob to come jogging towards the cart.
“You should smile more,” Kongpob smirks, taking the bag from Arthit. “It suits you.”
Arthit immediately scowls in protest, but a blush creeps up his neck.
“Relax, I just mean that it would draw in more customers,” he laughs as he takes the bag from Arthit. “I’ll text you later!”
And he’s off again, disappearing back in the direction that he’d come from.
Kong☕: 😋🍢🤗
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What Does Beta Tell You?
How Beta Measures Systematic Risk
Systematic risk, or total market risk, is the volatility that affects the entire stock market across many industries, stocks, and asset classes. Systematic risk affects the overall market and is therefore difficult to predict and hedge against.
Unlike with unsystematic risk, diversification cannot help to smooth systematic risk, because it affects a wide range of assets and securities. For example, the Great Recession was a form of systematic risk; the economic downturn affected the market as a whole.
Investors can still try to minimize the level of exposure to systematic risk by looking at stock's beta, or its correlation of price movements to the broader market as a whole. Here, we take a closer look at how beta relates to systematic risk.
Systematic risk cannot be eliminated through diversification since it is a nonspecific risk that affects the entire market.
The beta of a stock or portfolio will tell you how sensitive your holdings are to systematic risk, where the broad market itself always has a beta of 1.0.
High betas indicate greater sensitivity to systematic risk, which can lead to more volatile price swings in your portfolio, but which can be hedged somewhat.
Beta is a measure of a stock's volatility in relation to the market. It essentially measures the relative risk exposure of holding a particular stock or sector in relation to the market.
If you want to know the systematic risk of your portfolio, you can calculate its beta. Beta effectively describes the activity of a security's returns as it responds to swings in the market. A security's beta is computed by dividing the product of the covariance of the security's returns and the market's returns by the variance of the market's returns over a specified period, using this formula:
\begin{aligned} &\text{Beta coefficient}(\beta) = \frac{\text{Covariance}(R_e, R_m)}{\text{Variance}(R_m)} \\ &\textbf{where:}\\ &R_e=\text{the return on an individual stock}\\ &R_m=\text{the return on the overall market}\\ &\text{Covariance}=\text{how changes in a stock's returns are} \\ &\text{related to changes in the market's returns}\\ &\text{Variance}=\text{how far the market's data points spread} \\ &\text{out from their average value} \\ \end{aligned}
Beta coefficient(β)=Variance(Rm)Covariance(Re,Rm)where:Re=the return on an individual stockRm=the return on the overall marketCovariance=how changes in a stock’s returns arerelated to changes in the market’s returnsVariance=how far the market’s data points spreadout from their average value
Note that beta can also be calculated by running a linear regression on a stock's returns compared to the market using the capital asset pricing model (CAPM). In fact, this is why this measure is called the beta coefficient, since statisticians and econometricians label the coefficients of explanatory variables in regression models as the Greek letter ß. The formula for CAPM is:
Once you've calculated the beta of a security, it can then be used to tell you the relative correspondence of price movements in that stock, given the price movements in the broader market as a whole.
A beta of 0 indicates that the portfolio is uncorrelated with the market. In other words, movement of the stock or stocks held move randomly in relation to the broader market.
A negative beta (i.e., less than 0) indicates that it moves in the opposite direction of the market and that there is a negative correlation with the market.
A beta between 0 and 1 signifies that it moves in the same direction as the market, but with less volatility—that is, smaller percentage changes—than the market as a whole.
A beta of 1 indicates that the portfolio will move in the same direction, have the same volatility and is sensitive to systematic risk. Note that the S&P 500 index is often used as the benchmark for the broader stock market and the index has a beta of 1.0.
A beta greater than 1 indicates that the portfolio will move in the same direction as the market, and with a higher magnitude than the market. Stocks with betas above 1.0 are quite sensitive to systematic risk.
In reality, you won't have to go about calculating beta yourself in most cases. Beta is commonly listed on freely available stock quotes from several online financial portals, as well as through your broker's website.
Assume that the beta of an investor's portfolio is 2.0 in relation to a broad market index, such as the S&P 500. If the market increases by 2%, then the portfolio will generally increase by 4%.
Likewise, if the market decreases by 2%, the portfolio generally decreases by 4%. This portfolio is therefore sensitive to systematic risk, but the risk can be reduced by hedging. This can be achieved by obtaining other stocks that have negative or low betas, or by using derivatives to limit downside losses.
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Semiclassical measures for the Schrödinger equation on the torus | EMS Press
Semiclassical measures for the Schrödinger equation on the torus
In this article, the structure of semiclassical measures for solutions to the linear Schrödinger equation on the torus is analysed. We show that the disintegration of such a measure on every invariant lagrangian torus is absolutely continuous with respect to the Lebesgue measure. We obtain an expression of the Radon-Nikodym derivative in terms of the sequence of initial data and show that it satisfies an explicit propagation law. As a consequence, we also prove an observability inequality, saying that the
L^2
-norm of a solution on any open subset of the torus controls the full
L^2
Nalini Anantharaman, Fabricio Macià, Semiclassical measures for the Schrödinger equation on the torus. J. Eur. Math. Soc. 16 (2014), no. 6, pp. 1253–1288
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Approximation Results with Respect to Multidimensional φ-Variation for Nonlinear Integral Operators | EMS Press
Approximation Results with Respect to Multidimensional φ-Variation for Nonlinear Integral Operators
In this paper we study approximation problems for functions belonging to
BV
φ-spaces (spaces of functions of bounded φ-variation) in multidimensional setting. In particular, using a multidimensional concept of φ-variation in the sense of Tonelli introduced in [4], we obtain estimates, convergence results and, by means of suitable Lipschitz classes, results about the order of approximation for a family of nonlinear convolution integral operators.
Laura Angeloni, Approximation Results with Respect to Multidimensional φ-Variation for Nonlinear Integral Operators. Z. Anal. Anwend. 32 (2013), no. 1, pp. 103–128
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Conjecture de Kato sur les ouverts de $\mathbb R$ | EMS Press
Conjecture de Kato sur les ouverts de
\mathbb R
We prove Kato's conjecture for second order elliptic differential operators on an open set in dimension 1 with arbitrary boundary conditions. The general case reduces to studying the operator
T = –\frac{d}{dx}a(x)\frac{d}{dx}
on an interval, when
a(x)
is a bounded and accretive function. We show for the latter situation that the domain of
T
is spanned by an unconditional basis of wavelets with cancellation properties that compensate the action of the non-regular function
a( x)
Pascal Auscher, Philippe Tchamitchian, Conjecture de Kato sur les ouverts de
\mathbb R
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SRS configuration parameters for 5G waveform generation - MATLAB - MathWorks 日本
nrWavegenSRSConfig
NumSRSPorts
NumSRSSymbols
KBarTC
GroupSeqHopping
NSRSID
SRSPositioning
Configure SRS for 5G Uplink Waveform Generation
SRS configuration parameters for 5G waveform generation
The nrWavegenSRSConfig object sets sounding reference signal (SRS) configuration parameters, as defined in TS 38.211 Section 6.4.1.4 [1]. Use this object to set the SRS property of the nrULCarrierConfig object when configuring 5G uplink waveform generation.
The default nrWavegenSRSConfig object specifies a single-port, single-symbol, narrowband SRS configuration without frequency hopping (BHop ≥ BSRS) and places the SRS at the end of the slot.
srs = nrWavegenSRSConfig
srs = nrWavegenSRSConfig(Name,Value)
srs = nrWavegenSRSConfig creates a default SRS configuration object for 5G waveform generation.
srs = nrWavegenSRSConfig(Name,Value) specifies properties using one or more name-value arguments. Enclose each property in quotes. For example, 'NumSRSPorts',2,'NumSRSSymbols',4 specifies a two-port SRS transmission of 4 OFDM symbols.
Enable — Enable SRS
Enable the SRS in 5G waveform generation, specified as one of these values.
1 (true) — Enable the SRS.
0 (false) — Disable the SRS.
Label — Name of SRS configuration
'SRS1' (default) | character array | string scalar
Name of the SRS configuration, specified as a character array or string scalar. Use this property to set a description to the SRS configuration.
Power — Power scaling of SRS in dB
Power scaling of the SRS in dB, specified as a real scalar. Use this property to scale the power of the SRS in the generated 5G waveform.
ID of the bandwidth part (BWP) containing the configured SRS, specified as a nonnegative integer. Use this property to associate this SRS configuration with one of the BWP configurations specified by the BandwidthParts property of the nrULCarrierConfig object.
NumSRSPorts — Number of SRS antenna ports
Number of SRS antenna ports, specified as 1, 2, or 4.
NumSRSSymbols — Number of OFDM symbols
Number of OFDM symbols allocated to the SRS in a slot, specified as 1, 2, 4, 8, or 12. Valid property values depend on the SRSPositioning property.
If you set the SRSPositioning property to 0 (false), specify this property as 1, 2, or 4.
If you set the SRSPositioning property to 1 (true), specify this property as 1, 2, 4, 8, or 12. For valid configurations of this property and the KTC property, see TS 38.211 Table 6.4.1.4.3-2. Alternatively, type nrSRSConfig.SubcarrierOffsetTable at the command line to display this table.
SymbolStart — 0-based index of first OFDM symbol
0-based index of the first OFDM symbol in the SRS within a slot, specified as one of these options:
Integer from 0 to 13 — Use this option for normal cyclic prefix.
Integer from 0 to 11 — Use this option for extended cyclic prefix.
For the SRS symbols and index generation, set the cyclic prefix of the carrier by using the CyclicPrefix property of the BWP configuration object specified by the BandwidthPartID property.
SlotAllocation — Slot allocation in SRS period
Slot allocation in an SRS period, specified as a nonnegative integer or a row vector of nonnegative integers. This property specifies the slot positions of the SRS by using 0-based indexing and values that are less than the value of the Period property. The object ignores slot allocation values that are greater than the period. Each element of the vector corresponds to an SRS resource.
Period — SRS allocation period in slots
SRS allocation period in slots, specified as a nonnegative integer or []. An empty period indicates aperiodic SRS resource type (no repetition), as defined in TS 38.211 Section 6.4.1.4.3.
KTC — Transmission comb number
Transmission comb number, in subcarriers, specified as 2, 4, or 8. The object allocates the SRS sequence every KTC number of subcarriers. Valid property values depend on the SRSPositioning property.
If you set the SRSPositioning property to 0 (false), specify this property as 2 or 4.
If you set the SRSPositioning property to 1 (true), specify this property as 2, 4, or 8. For valid configurations of this property and the NumSRSSymbols property, see TS 38.211 Table 6.4.1.4.3-2. Alternatively, type nrSRSConfig.SubcarrierOffsetTable at the command line to display this table.
KBarTC — Transmission comb offset
0 (default) | integer from 0 to (KTC – 1)
Transmission comb offset, in subcarriers, specified as an integer from 0 to (KTC – 1). This property specifies a frequency shift within the comb.
CyclicShift — Cyclic shift offset
Cyclic shift offset, specified as an integer from 0 to 11. This property determines the cyclic shift applied to the SRS sequence for each antenna port. This property corresponds to parameter
{n}_{SRS}^{cs}
in TS 38.211 Section 6.4.1.4.2.
Set the cyclic shift offset in relation to the transmission comb property, KTC:
If you set KTC to 2, set CyclicShift to an integer from 0 to 7.
If you set KTC to 4, set CyclicShift to an integer from 0 to 11.
For multiport SRS transmissions, consecutive cyclic shift numbers are used for each port, modulo 6, 8, or 12, depending on the KTC property.
FrequencyStart — Frequency-domain offset
Frequency-domain offset of the SRS, in terms of a physical resource block (PRB) index relative to the carrier, specified as an integer from 0 to 271. FrequencyStart is analogous to parameter
{n}_{{}_{shift}}
from TS 38.211 Section 6.4.1.4.3.
This property, the additional circular frequency-domain offset property NRRC, and the bandwidth configuration parameters in TS 38.211 Table 6.4.1.4.3-1 determine the actual frequency-domain location of the SRS. For more information, see NR SRS Configuration.
NRRC — Additional circular frequency-domain offset
Additional circular frequency-domain offset of the SRS, as a multiple of 4 PRBs, specified as an integer from 0 to 67.
This property, the frequency domain offset property FrequencyStart, and the bandwidth configuration parameters in TS 38.211 Table 6.4.1.4.3-1 determine the actual frequency-domain location of the SRS. For more information, see NR SRS Configuration.
CSRS — Row index of bandwidth configuration table
Row index of the bandwidth configuration table from TS 38.211 Table 6.4.1.4.3-1, specified as an integer from 0 to 63. Use this property with the BSRS property to control the bandwidth allocated to the SRS and the frequency hopping pattern. Increasing the CSRS value increases the SRS bandwidth. The default value of 0 results in a bandwidth of 4 PRBs.
BSRS — Column index of bandwidth configuration table
Column index of the bandwidth configuration table from TS 38.211 Table 6.4.1.4.3-1, specified as an integer from 0 to 3. Use this property with the CSRS property to control the bandwidth allocated to the SRS and the frequency hopping pattern. Increasing the BSRS value decreases the SRS bandwidth.
BHop — Frequency hopping index
Frequency hopping index, specified as an integer from 0 to 3. Setting this property to a value greater than or equal to the column index of the bandwidth configuration table property, BSRS, disables frequency hopping. Increasing the BHop value decreases the hopping bandwidth.
Repetition — Repetition factor of OFDM symbols
Repetition factor of OFDM symbols, specified as 1, 2, 4, 8, or 12.
When frequency hopping is enabled, Repetition specifies the number of consecutive OFDM symbols in a slot occupied by the SRS in the same frequency resource. Set Repetition such that Repetition ≤ NumSRSSymbols.
When frequency hopping is disabled, this property is ignored.
GroupSeqHopping — Type of SRS symbol hopping
'neither' (default) | 'groupHopping' | 'sequenceHopping'
Type of SRS symbol hopping, specified as 'neither', 'groupHopping', or 'sequenceHopping'. When either group or sequence hopping is enabled, the group or sequence hopping numbers per OFDM symbol in the SRS transmission are based on a pseudo-random binary sequence (PRBS). Set the scrambling identity for the PRBS by using the NSRSID property.
NSRSID — SRS scrambling identity
SRS scrambling identity, specified as an integer from 0 to 65,535.
When you set the GroupSeqHopping property to 'neither', this property determines the group number.
When you set the GroupSeqHopping property to 'groupHopping' or 'sequenceHopping', this property initializes the PRBS.
SRSPositioning — SRS for user positioning
SRS for user positioning, as defined in Release 16 of TS 38.211 Section 6.4.1.4, specified as one of these values.
0 (false) — Disable SRS for user positioning. This option corresponds to the higher-layer parameter SRS-Resource.
1 (true) — Enable SRS for user positioning. This option corresponds to the higher-layer parameter SRS-PosResource-r16.
This property affects the valid range of the NumSRSSymbols and KTC properties.
Create two SRS configuration objects, one for each of the carriers, with the specified properties. In the first SRS configuration, frequency hopping is enabled. In the second SRS configuration, frequency hopping is disabled.
nrWavegenSRSConfig('BandwidthPartID',1,'NumSRSPorts',2,'NumSRSSymbols',4,'SymbolStart',8,'CSRS',14,'BSRS',1), ...
Create a PUSCH configuration object such that the PUSCH does not overlap with the previously configured SRS in the generated waveform.
nrWavegenPUSCHConfig('BandwidthPartID',1,'SymbolAllocation',[0 8],'PRBSet',(10:51))};
Create an uplink carrier configuration object, specifying the previously defined configurations.
'SRS',srs, ...
'PUSCH',pusch);
[1] 3GPP TS 38.211. “NR; Physical channels and modulation.†3rd Generation Partnership Project; Technical Specification Group Radio Access Network.
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Switchblade of Thanos | Toph
Switchblade of Thanos
By OnikJahanSagor · Limits 1s, 512 MB
We all know that Thanos “The Mad Titan“ had two adopted daughters, Gamora and Nebula. But did you know that he had one real daughter, Quanita? Thanos had a switchblade. His switchblade was special because it had two blades with equal weights. The weight of the switchblade was balanced indicating his vision to form a balanced universe. He gave it to Gamora, his favorite adopted daughter. But after the death of Gamora and Thanos, the switchblade was handed over to Nebula. Nebula didn’t want to keep it to her. So, when she found out that Thanos had a real daughter, she searched for her. After a tedious journey, she found the actual daughter of Thanos, Quanita. But because of the journey and being unused for a long time, the switchblade became unbalanced. Now, it’s Quanita’s job to make it balanced.
She has a sequence
A of
N metal plates in order which she can use to balance the switchblade. The weight of the
i-th plate is
A_{i}
Ai. The balancing task is not easy at all. She has to choose a non-empty sub-sequence of the metal plates. Then she will divide this sub-sequence into two non-empty consecutive parts in such a way that the sum of the weights of each part is equal. She will use the metals of the first part on one blade and the other part on another thus making the switchblade balanced. But not all the sub-sequences can be split into two parts like this. Now, Quanita is interested in how many ways she can choose a non-empty sub-sequence so that it can be split into two non-empty consecutive parts with an equal sum of weights. Help Quanita to find what she desires.
N (1 \leq N \leq 100)
N(1≤N≤100) the number of the metal plates.
The second line of the input will contain
N space separated integers
A_{1}, A_{2}, A_{3}, A_{4}, …, A_{N}
A1,A2,A3,A4,…,AN
(1 \leq A_{i} \leq 1000)
(1≤Ai≤1000)the weights of the metal plates.
Print an integer, the number of ways Quanita can choose a non-empty subsequence so that it can be split into two non-empty consecutive parts with an equal sum of weights as described above. As the number of such subsequence can be very large, print it modulo
998244353.
In the sample above, there can be two such subsequences.
One is
3, 2, 5
3,2,5. If we split this subsequence after the second element, the sum of the both sides become equal as
3+2=5.
The other one is
3, 2, 1
3,2,1. If we split this subsequence after the first element, the sum of the both sides become equal as
3 = 2 + 1
3=2+1.
A subsequence of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without changing the order of the elements.
Two subsequences are called different if they have at least one element from a different index than the other one.
sh2018331053Earliest, 1M ago
S_SubrataFastest, 0.0s
S_SubrataLightest, 918 kB
rafi_1703076Shortest, 685B
Notice that the maximum sum of all elements can be at most 10510^5105. So, we can easily use Dynamic...
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{x}_{0}
{x}_{0}
{x}_{0}
X\left(t\right)
\mathrm{dX}\left(t\right)=\mathrm{\mu }\left(X\left(t\right),t\right)\mathrm{dt}+\mathrm{\sigma }\left(X\left(t\right),t\right)\mathrm{dW}\left(t\right)
\mathrm{\mu }\left(X\left(t\right),t\right)
\mathrm{\sigma }\left(X\left(t\right),t\right)
W\left(t\right)
{x}_{0}
X
is an
{X}_{1}
{X}_{n}
{\mathrm{\mu }}_{1}
{\mathrm{\mu }}_{n}
{\mathrm{\sigma }}_{1}
{\mathrm{\sigma }}_{n}
be the corresponding drift and diffusion terms. The ItoProcess(X, Sigma) command will create an
Y
{\mathrm{dY}\left(t\right)}_{i}={\mathrm{\mu }}_{i}\left({Y\left(t\right)}_{i},t\right)+{\mathrm{\sigma }}_{i}\left({Y\left(t\right)}_{i},t\right){\mathrm{dW}\left(t\right)}_{i}
W\left(t\right)
is an
\mathrm{with}\left(\mathrm{Finance}\right):
Y≔\mathrm{ItoProcess}\left(1.0,\mathrm{\mu },\mathrm{\sigma },x,t\right)
\textcolor[rgb]{0,0,1}{Y}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_X0}}
\mathrm{Drift}\left(Y\left(t\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{\mu }}
\mathrm{Diffusion}\left(Y\left(t\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{\sigma }}
\mathrm{Drift}\left(\mathrm{exp}\left(Y\left(t\right)\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{\mu }}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{_X0}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}\textcolor[rgb]{0,0,1}{+}\frac{{\textcolor[rgb]{0,0,1}{\mathrm{\sigma }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{_X0}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}}{\textcolor[rgb]{0,0,1}{2}}
\mathrm{Diffusion}\left(\mathrm{exp}\left(Y\left(t\right)\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{\sigma }}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{_X0}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}
\mathrm{\mu }≔0.1
\textcolor[rgb]{0,0,1}{\mathrm{\mu }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{0.1}
\mathrm{\sigma }≔0.5
\textcolor[rgb]{0,0,1}{\mathrm{\sigma }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{0.5}
\mathrm{PathPlot}\left(\mathrm{exp}\left(Y\left(t\right)\right),t=0..3,\mathrm{timesteps}=100,\mathrm{replications}=10\right)
\mathrm{\mu }≔'\mathrm{\mu }'
\textcolor[rgb]{0,0,1}{\mathrm{\mu }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{\mu }}
\mathrm{\sigma }≔'\mathrm{\sigma }'
\textcolor[rgb]{0,0,1}{\mathrm{\sigma }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{\sigma }}
\mathrm{X0}≔〈100.0,0.〉
\textcolor[rgb]{0,0,1}{\mathrm{X0}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{c}\textcolor[rgb]{0,0,1}{100.0}\\ \textcolor[rgb]{0,0,1}{0.}\end{array}]
\mathrm{Μ}≔〈\mathrm{\mu }X[1],\mathrm{\kappa }\left(\mathrm{\theta }-X[2]\right)〉
\textcolor[rgb]{0,0,1}{\mathrm{Μ}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{c}\textcolor[rgb]{0,0,1}{\mathrm{\mu }}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{X}}_{\textcolor[rgb]{0,0,1}{1}}\\ \textcolor[rgb]{0,0,1}{\mathrm{\kappa }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{\theta }}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{X}}_{\textcolor[rgb]{0,0,1}{2}}\right)\end{array}]
\mathrm{\Sigma }≔〈〈\mathrm{sqrt}\left(X[2]\right)X[1]|0.〉,〈0.|\mathrm{\sigma }X[2]〉〉
\textcolor[rgb]{0,0,1}{\mathrm{\Sigma }}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\sqrt{{\textcolor[rgb]{0,0,1}{X}}_{\textcolor[rgb]{0,0,1}{2}}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{X}}_{\textcolor[rgb]{0,0,1}{1}}& \textcolor[rgb]{0,0,1}{0.}\\ \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{\mathrm{\sigma }}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{X}}_{\textcolor[rgb]{0,0,1}{2}}\end{array}]
S≔\mathrm{ItoProcess}\left(\mathrm{X0},\mathrm{Μ},\mathrm{\Sigma },X,t\right)
\textcolor[rgb]{0,0,1}{S}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_X2}}
\mathrm{Drift}\left(S\left(t\right)\right)
[\begin{array}{c}\textcolor[rgb]{0,0,1}{\mathrm{\mu }}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_X2}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{1}}\\ \textcolor[rgb]{0,0,1}{\mathrm{\kappa }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{\theta }}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{\mathrm{_X2}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{2}}\right)\end{array}]
\mathrm{Diffusion}\left(S\left(t\right)\right)
[\begin{array}{cc}\sqrt{{\textcolor[rgb]{0,0,1}{\mathrm{_X2}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{2}}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_X2}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{1}}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{\mathrm{\sigma }}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_X2}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{2}}\end{array}]
\mathrm{\mu }≔0.1
\textcolor[rgb]{0,0,1}{\mathrm{\mu }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{0.1}
\mathrm{\sigma }≔0.5
\textcolor[rgb]{0,0,1}{\mathrm{\sigma }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{0.5}
\mathrm{\kappa }≔1.0
\textcolor[rgb]{0,0,1}{\mathrm{\kappa }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{1.0}
\mathrm{\theta }≔0.4
\textcolor[rgb]{0,0,1}{\mathrm{\theta }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{0.4}
A≔\mathrm{SamplePath}\left(S\left(t\right),t=0..1,\mathrm{timesteps}=100,\mathrm{replications}=10\right)
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}\begin{array}{c}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{100.}& \textcolor[rgb]{0,0,1}{100.670084719786}\\ \textcolor[rgb]{0,0,1}{100.100000000000}& \textcolor[rgb]{0,0,1}{101.034139425728}\\ \textcolor[rgb]{0,0,1}{100.280880089808}& \textcolor[rgb]{0,0,1}{101.924198818577}\\ \textcolor[rgb]{0,0,1}{102.915077811759}& \textcolor[rgb]{0,0,1}{99.6518477031121}\\ \textcolor[rgb]{0,0,1}{103.858818858166}& \textcolor[rgb]{0,0,1}{100.628185358730}\\ \textcolor[rgb]{0,0,1}{104.476699657855}& \textcolor[rgb]{0,0,1}{98.7691518445139}\\ \textcolor[rgb]{0,0,1}{103.737362966326}& \textcolor[rgb]{0,0,1}{95.0859221941374}\\ \textcolor[rgb]{0,0,1}{102.574346549913}& \textcolor[rgb]{0,0,1}{94.1008617878134}\\ \textcolor[rgb]{0,0,1}{101.159282939668}& \textcolor[rgb]{0,0,1}{92.9644135833222}\\ \textcolor[rgb]{0,0,1}{100.709702216007}& \textcolor[rgb]{0,0,1}{93.6061768383076}\end{array}]\\ \hfill \textcolor[rgb]{0,0,1}{\text{slice of 10 × 2 × 101 Array}}\end{array}
\mathrm{PathPlot}\left(A,1,\mathrm{thickness}=3,\mathrm{markers}=\mathrm{false},\mathrm{color}=\mathrm{red}..\mathrm{blue},\mathrm{axes}=\mathrm{BOXED},\mathrm{gridlines}=\mathrm{true}\right)
\mathrm{PathPlot}\left(A,2,\mathrm{thickness}=3,\mathrm{markers}=\mathrm{false},\mathrm{color}=\mathrm{red}..\mathrm{blue},\mathrm{axes}=\mathrm{BOXED},\mathrm{gridlines}=\mathrm{true}\right)
\mathrm{ExpectedValue}\left(\mathrm{max}\left(S\left(1\right)[1]-100,0\right),\mathrm{timesteps}=100,\mathrm{replications}={10}^{4}\right)
[\textcolor[rgb]{0,0,1}{\mathrm{value}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{21.41114565}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{standarderror}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0.3390630872}]
X≔\mathrm{GeometricBrownianMotion}\left(100.0,0.05,0.3,t\right)
\textcolor[rgb]{0,0,1}{X}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_X4}}
Y≔\mathrm{GeometricBrownianMotion}\left(100.0,0.07,0.2,t\right)
\textcolor[rgb]{0,0,1}{Y}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_X5}}
\mathrm{\Sigma }≔〈〈1|0.5〉,〈0.5|1〉〉
\textcolor[rgb]{0,0,1}{\mathrm{\Sigma }}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0.5}\\ \textcolor[rgb]{0,0,1}{0.5}& \textcolor[rgb]{0,0,1}{1}\end{array}]
Z≔\mathrm{ItoProcess}\left(〈X,Y〉,\mathrm{\Sigma }\right)
\textcolor[rgb]{0,0,1}{Z}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_X6}}
\mathrm{Drift}\left(Z\left(t\right)\right)
[\begin{array}{c}\textcolor[rgb]{0,0,1}{0.05}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_X6}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{1}}\\ \textcolor[rgb]{0,0,1}{0.07}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_X6}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{2}}\end{array}]
\mathrm{Diffusion}\left(Z\left(t\right)\right)
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{0.3}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_X6}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{1}}& \textcolor[rgb]{0,0,1}{0.15}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_X6}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{1}}\\ \textcolor[rgb]{0,0,1}{0.10}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_X6}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{2}}& \textcolor[rgb]{0,0,1}{0.2}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_X6}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{2}}\end{array}]
\mathrm{ExpectedValue}\left(\mathrm{max}\left(X\left(1\right)-Y\left(1\right),0\right),\mathrm{timesteps}=100,\mathrm{replications}={10}^{4}\right)
[\textcolor[rgb]{0,0,1}{\mathrm{value}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{14.32896059}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{standarderror}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0.2447103632}]
\mathrm{ExpectedValue}\left(\mathrm{max}\left(Z\left(1\right)[1]-Z\left(1\right)[2],0\right),\mathrm{timesteps}=100,\mathrm{replications}={10}^{4}\right)
[\textcolor[rgb]{0,0,1}{\mathrm{value}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{8.103315185}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{standarderror}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0.1520913055}]
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Visualize Mixer Spurs - MATLAB & Simulink - MathWorks Benelux
Create Mixer Object from Data File
Plot Mixer Output Signal and Spurs
Use Data Cursor
Create Cascaded Circuit
Plot Output Signal and Spurs of LC filter in Cascade
Plot Cascade Signal and Spurs in 3D
This example shows how to create an rfckt.mixer object and plot the mixer spurs of that object.
Mixers are non-linear devices used in RF systems. They are typically used to convert signals from one frequency to another. In addition to the desired output frequency, mixers also produce intermodulation products (also called mixer spurs), which are unwanted side effects of their nonlinearity. The output of the mixer occurs at the frequencies:
{F}_{out}\left(N,M\right)=|N{F}_{in}+M{F}_{LO}|
{F}_{in}
{F}_{LO}
N
M
{F}_{in}={F}_{RF}
{F}_{RF}>{F}_{LO}
N=1
M=-1
{F}_{out}\left(1,-1\right)={F}_{IF}=|N{F}_{in}+M{F}_{LO}|={F}_{RF}-{F}_{LO}
N
M
represent the spurious intermodulation products.
Intermodulation tables (IMTs) are often used in system-level modeling of mixers. This example first examines the IMT of a mixer. Then the example reads an .s2d format file containing an IMT, and plots the output power at each output frequency, including the desired signal and the unwanted spurs. The example also creates a cascaded circuit which contains a mixer with IMT followed by a filter, whose purpose is to mitigate the spurs, and plots the output power before and after mitigation.
For more information on IMTs, see the OpenIF example Finding Free IF Bandwidths.
Create an rfckt.mixer object to represent the downconverting mixer that is specified in the file, samplespur1.s2d. The mixer is characterized by S-parameters, spot noise and IMT. These data are stored in the NetworkData, NoiseData and MixerSpurData properties of the rfckt object, respectively.
Mixer = rfckt.mixer('FLO', 1.7e9); % Flo = 1.7GHz
read(Mixer,'samplespur1.s2d');
disp(Mixer)
MixerSpurData: [1x1 rfdata.mixerspur]
IMT = Mixer.MixerSpurData.data
IMT = 16×16
24 0 35 13 40 24 45 28 49 33 53 42 60 47 63 99
Use the plot method of the rfckt object to plot the power of the desired output signal and the spurs. The second input argument must be the string 'MIXERSPUR'. The third input argument must be the index of the circuit for which to plot output power data. The rfckt.mixer object only contains one circuit (the mixer), so index 0 corresponds to the mixer input and index 1 corresponds to the mixer output.
CktIndex = 1; % Plot the output only
Pin = -10; % Input power is -10dBm
Fin = 2.1e9; % Input frequency is 2.1GHz
plot(Mixer,'MIXERSPUR',CktIndex,Pin,Fin);
Run the cursor over the plot to get the frequency and power level of each signal and spur.
Create an amplifier object for LNA, mixer, and LC Bandpass Tee objects. Then build the cascade shown in the following figure:
Figure 1: Cascaded Circuit
FirstCkt = rfckt.amplifier('NetworkData', ...
rfdata.network('Type','S','Freq',2.1e9,'Data',[0,0;10,0]), ...
'NoiseData',0,'NonlinearData',Inf); % 20dB LNA
SecondCkt = copy(Mixer); % Mixer with IMT table
ThirdCkt = rfckt.lcbandpasstee('L',[97.21 3.66 97.21]*1.0e-9, ...
'C',[1.63 43.25 1.63]*1.0e-12); % LC Bandpass filter
CascadedCkt = rfckt.cascade('Ckts',{FirstCkt,SecondCkt,ThirdCkt});
Use the plot method of the rfckt object to plot the power of the desired output signal and the spurs. The third input argument is 3, which directs the toolbox to plot the power at the output of the third component of the cascade (the LC filter).
CktIndex = 3; % Plot the output signal and spurs of the LC filter,
% which is the 3rd circuit in the cascade
plot(CascadedCkt,'MIXERSPUR',CktIndex,Pin,Fin)
Use the plot method of the rfckt object with a third input argument of 'all' to plot the input power and the output power after each circuit component in the cascade. Circuit index 0 corresponds to the input of the cascade. Circuit index 1 corresponds to the output of the LNA. Circuit index 2 corresponds to the output of the mixer, which was shown in the previous plot. Circuit index 3 corresponds to the output of the LC Bandpass Tee filter.
CktIndex = 'all'; % Plot the input signal, the output signal, and the
% spurs of the three circuits in the cascade: FirstCkt,
% SecondCkt and ThirdCkt
view([68.5 26])
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A nonlinear Poisson transform for Einstein metrics on product spaces | EMS Press
A nonlinear Poisson transform for Einstein metrics on product spaces
We consider the Einstein deformations of the reducible rank two symmetric spaces of noncompact type. If
M
is the product of any two real, complex, quaternionic or octonionic hyperbolic spaces, we prove that the family of nearby Einstein metrics is parametrized by certain new geometric structures on the Furstenberg boundary of
M
Olivier Biquard, Rafe Mazzeo, A nonlinear Poisson transform for Einstein metrics on product spaces. J. Eur. Math. Soc. 13 (2011), no. 5, pp. 1423–1475
|
The Hesse pencil of plane cubic curves | EMS Press
This is a survey of the classical geometry of the Hesse configuration of 12 lines in the projective plane with relation to the inflection points of a plane cubic curve. We also study two
K3
surfaces with Picard number 20 which arise naturally in connection with this configuration.
Michela Artebani, Igor V. Dolgachev, The Hesse pencil of plane cubic curves. Enseign. Math. 55 (2009), no. 3, pp. 235–273
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A warrant is like an option, except it is issued by a company. The warrant gives the holder the right to buy stock from the company at a specified price within a designated time period. When an investor exercises a warrant, they buy stock from the company and those proceeds are a source of capital for the firm. While warrants aren't that common, it's important to understand what they are, and how to value them, in case a company you own shares in offers warrants, or may in the future.
Warrants Explained
Like an option, a warrant does not represent actual ownership in the stock of the company; it is simply the right (but not the obligation) to buy shares at a certain price in the future. A warrant typically has a much longer life than a call option, with an expiry extending out five or 10 years. Some warrants are even perpetual.
Although warrants are similar to options, there are several important differences. First, options are written by other investors or market makers, while warrants are typically issued by companies. Warrants are often traded over-the-counter and do not have the standardized features of option contracts. The company can create any sort of contract they want, whereas an options writer cannot. Also, options are not dilutive to current shareholders, while warrants are. This is because when a warrant is exercised, new stock is issued.
Although there are several kinds of warrants, the most common types are detachable and naked. Detachable warrants are issued in conjunction with other securities (like bonds or preferred stock) and may be traded separately from them. Naked warrants are issued as is and without any accompanying securities.
Other less common types of warrants include wedded warrants, which can only be exercised if the attached bond/preferred stock is surrendered, and put warrants which may be used to hedge employee option programs.
Why Are They Issued?
The most common reason for a company to issue warrants is to provide a "sweetener" for a bond or preferred stock offering. By adding the warrants, the company hopes to obtain better terms (lower rates) on the debt or preferred stock. Moreover, warrants represent a potential source of capital in the future, and can thus offer a capital-raising option to companies that cannot, or prefer not to, issue more debt or preferred stock.
In addition, there are certain accounting benefits. Issuers can use the treasury stock method to calculate earnings per share, and amortized warrant value can be used to increase interest expense and tax benefits.
Less commonly, warrants are issued as part of the recapitalization plan of a bankrupt company. While the holders of common stock are typically wiped out in a bankruptcy, issuing warrants for soon-to-be-worthless shares gives the company a future source of equity capital (if shareholders exercise those warrants) and preserves some goodwill in the former shareholder base.
Valuing Warrants with the Black-Scholes Model
Although there are several possible methods for valuing a warrant, a modified version of the Black-Scholes model is commonly used. This formula is for European-style options and, though American-style options are theoretically worth more, there is not much difference in price in practice.
In the Black-Scholes model, the valuation of a call option is expressed as:
\begin{aligned} &C=SN\left (d{1} \right )-Xe-rTN \left( d2 \right ) \\ &\textbf{where:}\\ &C=\text{Call option}\\ &S=\text{Price of the underlying asset}\\ &N=\text{Standard normal distribution}\\ &X=\text{Option strike price}\\ &T=\text{Time to expiration}\\ &d=\text{Dividend}\\ &r=\text{Risk free interest rate}\\ &e=\text{Exponential term}\\ \end{aligned}
C=SN(d1)−Xe−rTN(d2)where:C=Call optionS=Price of the underlying assetN=Standard normal distributionX=Option strike priceT=Time to expirationd=Dividendr=Risk free interest ratee=Exponential term
The formula gives the theoretical value of an option. What it is trading at in the real world may differ. To find current warrant prices, do a symbol lookup for the stock warrant you are interested in on NYSE.com or nasdaq.com. Click on the warrant symbol provided to get the current price. The following example shows warrant pricing information for Ambac Financial Group, Inc. (AMBC).
As shown by the price snapshot, warrants can be traded like stocks or options. As long as there is someone else to buy or sell from, the warrant can be traded at any time up to expiry.
Factors That Influence Warrant Prices
Beyond the calculation above, investors should consider the following factors when evaluating the price of a warrant.
Underlying Security Price: The higher the price of the underlying security, the more valuable the warrant becomes. After all, if the price of the stock is below the strike price of the warrant, there is no reason to exercise the warrant as it is cheaper to buy the stock on the open market.
Days to Maturity: Generally speaking, options and warrants are worth less as time goes on and expiration approaches. This phenomenon is also called "time decay," and it will accelerate as expiration approaches if the strike price is above the current price.
Dividend: Warrant-holders are not entitled to receive dividends, and the corresponding reduction in the stock price when a dividend is issued to common shareholders reduces the value of the warrant.
Interest Rate/Risk-Free Rate: Higher interest rates increase the value of warrants.
Implied Volatility: The higher the volatility, the higher the odds that the warrant will eventually be in-the-money and the higher the value of the warrant will be.
Dilution: Because the exercise of a warrant will increase a company's outstanding shares, this dilution adds a twist to valuation that is not present in normal option valuation. Potential dilution may hamper the price of the common stock from rising.
Premium: Warrants can be issued at premiums; the lower the premium the more valuable the warrant.
Gearing/leverage: Gearing is the ratio of the share price to the warrant premium, and it reflects how much the price of the warrant changes for a given change in the stock. The higher the gearing, the more valuable the warrant.
Restrictions: Though very difficult to quantify mathematically, any restrictions on the exercise of warrants will impact the value of a warrant, typically negatively. A common restriction is the difference between American-style and European-style warrants. American-style warrants permit exercise at any time, while European-style warrants can only be exercised on the expiration date. The former is more valuable than the latter.
A warrant is basically a long-term option issued by a company. Investors need to make a few adjustments for unique factors like dilution, but a basic Black-Scholes options pricing formula will produce a reasonable assessment of the warrant's value. Current warrant prices can also be found online, such as on the NYSE or NASDAQ websites. Warrants can be bought or sold at any time, although not all warrants are actively traded, so check the volume of a warrant before opting to trade one.
A naked warrant allows the holder to buy or sell an underlying security, but unlike a normal warrant, is not attached to a bond or preferred stock.
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Characterization of the Exponential Distribution by Properties of the Difference $X_{k+s:n} – X_{k:n}$ of Order Statistics | EMS Press
X_{k+s:n} – X_{k:n}
H.-J. Rossberg
B. B. Ramachandran
X_1 , X_2, \dots, X_n
be independent and identically distributed random variables subject to a continuous distribution function
F
X_{1:n}, X_{2:n}, \dots, X_{n:n}
be the corresponding order statistics, and write
P(X_{k+s:n} – X_{k:n} ≥ x) = P(X_{s:n–k} ≥ x) \ \ \ (x≥0)
n,k
s
k + s ≤ n
. It is an old question if condition (0) implies that
F
is of exponential type. In [8] we showed among others that condition (0) can be greatly relaxed; namely, it can be replaced by asymptotic relations (either as
x \to \infty
x \downarrow 0
) to derive this very result. Using a theorem on integrated Cauchy functional equations and in essential way a result of [8] we find now a more elegant and deeper theorem on this subject. The case of lattice distributions is also considered and some new problems are stated.
H.-J. Rossberg, M. Riedel, B. B. Ramachandran, Characterization of the Exponential Distribution by Properties of the Difference
X_{k+s:n} – X_{k:n}
of Order Statistics. Z. Anal. Anwend. 16 (1997), no. 1, pp. 191–200
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6 02 * 1020molecules of a substance weigh 44mg what is the molar mass of the substance - Chemistry - Some Basic Concepts of Chemistry - 9076271 | Meritnation.com
6.02 * 1020molecules of a substance weigh 44mg. what is the molar mass of the substance?
×
1020 molecules weighs = 44 mg = 44
×
×
1023 molecules weigh =
\frac{44×{10}^{-3}}{6.02×{10}^{20}}×6.022×{10}^{23 }
Hence , the molecular mass of the substance is 44 g.
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Order statistic - Wikipedia
(Redirected from Order statistics)
In statistics, the kth order statistic of a statistical sample is equal to its kth-smallest value.[1] Together with rank statistics, order statistics are among the most fundamental tools in non-parametric statistics and inference.
2.1 Cumulative distribution function of order statistics
2.2 Probability distributions of order statistics
2.2.1 Order statistics sampled from an uniform distribution
2.2.2 The joint distribution of the order statistics of the uniform distribution
2.2.3 Order statistics sampled from an exponential distribution
2.2.4 Order statistics sampled from an Erlang distribution
2.2.5 The joint distribution of the order statistics of an absolutely continuous distribution
3 Application: confidence intervals for quantiles
3.1 A small-sample-size example
3.2 Large sample sizes
4 Application: Non-parametric density estimation
5 Dealing with discrete variables
6 Computing order statistics
7.1 Examples of order statistics
Notation and examples[edit]
{\displaystyle x_{(1)}=3,\ \ x_{(2)}=6,\ \ x_{(3)}=8,\ \ x_{(4)}=9,\,}
{\displaystyle X_{(1)}=\min\{\,X_{1},\ldots ,X_{n}\,\}}
{\displaystyle X_{(n)}=\max\{\,X_{1},\ldots ,X_{n}\,\}.}
{\displaystyle {\rm {Range}}\{\,X_{1},\ldots ,X_{n}\,\}=X_{(n)}-X_{(1)}.}
{\displaystyle X_{(m+1)}}
{\displaystyle X_{(m)}}
{\displaystyle X_{(m+1)}}
Probabilistic analysis[edit]
Cumulative distribution function of order statistics[edit]
{\displaystyle F_{X}(x)}
, the order statistics for that sample have cumulative distributions as follows[2] (where r specifies which order statistic):
{\displaystyle F_{X_{(r)}}(x)=\sum _{j=r}^{n}{\binom {n}{j}}[F_{X}(x)]^{j}[1-F_{X}(x)]^{n-j}}
{\displaystyle f_{X_{(r)}}(x)={\frac {n!}{(r-1)!(n-r)!}}f_{X}(x)[F_{X}(x)]^{r-1}[1-F_{X}(x)]^{n-r}.}
{\displaystyle F_{X_{(n)}}(x)=\operatorname {Prob} (\max\{\,X_{1},\ldots ,X_{n}\,\}\leq x)=[F_{X}(x)]^{n}}
{\displaystyle F_{X_{(1)}}(x)=\operatorname {Prob} (\min\{\,X_{1},\ldots ,X_{n}\,\}\leq x)=1-[1-F_{X}(x)]^{n}}
Probability distributions of order statistics[edit]
Order statistics sampled from an uniform distribution[edit]
{\displaystyle X_{1},X_{2},\ldots ,X_{n}}
{\displaystyle F_{X}}
{\displaystyle U_{i}=F_{X}(X_{i})}
{\displaystyle U_{1},\ldots ,U_{n}}
{\displaystyle U_{(i)}=F_{X}(X_{(i)})}
{\displaystyle U_{(k)}}
is equal to[3]
{\displaystyle f_{U_{(k)}}(u)={n! \over (k-1)!(n-k)!}u^{k-1}(1-u)^{n-k}}
that is, the kth order statistic of the uniform distribution is a beta-distributed random variable.[3][4]
{\displaystyle U_{(k)}\sim \operatorname {Beta} (k,n+1\mathbf {-} k).}
{\displaystyle U_{(k)}}
{\displaystyle O(du^{2})}
{\displaystyle (0,u)}
{\displaystyle (u,u+du)}
{\displaystyle (u+du,1)}
{\displaystyle {n! \over (k-1)!(n-k)!}u^{k-1}\cdot du\cdot (1-u-du)^{n-k}}
The joint distribution of the order statistics of the uniform distribution[edit]
{\displaystyle f_{U_{(i)},U_{(j)}}(u,v)=n!{u^{i-1} \over (i-1)!}{(v-u)^{j-i-1} \over (j-i-1)!}{(1-v)^{n-j} \over (n-j)!}}
{\displaystyle O(du\,dv)}
{\displaystyle (0,u)}
{\displaystyle (u,u+du)}
{\displaystyle (u+du,v)}
{\displaystyle (v,v+dv)}
{\displaystyle (v+dv,1)}
{\displaystyle f_{U_{(1)},U_{(2)},\ldots ,U_{(n)}}(u_{1},u_{2},\ldots ,u_{n})=n!.}
{\displaystyle 0<u_{1}<\cdots <u_{n}<1}
{\displaystyle 0<s<n/2}
{\displaystyle {\sqrt {2sn}}}
{\displaystyle s,n}
{\displaystyle sn}
{\displaystyle U_{(n)}-U_{(1)}}
{\displaystyle n\geq k>j\geq 1}
{\displaystyle U_{(k)}-U_{(j)}}
{\displaystyle U_{(k)}-U_{(j)}\sim \operatorname {Beta} (k-j,n-(k-j)+1)}
{\displaystyle \operatorname {Cov} (U_{(k)},U_{(j)})={\frac {j(n-k+1)}{(n+1)^{2}(n+2)}}}
{\displaystyle \operatorname {Var} (U_{(k)}-U_{(j)})=\operatorname {Var} (U_{(k)})+\operatorname {Var} (U_{(j)})-2\cdot \operatorname {Cov} (U_{(k)},U_{(j)})={\frac {k(n-k+1)}{(n+1)^{2}(n+2)}}+{\frac {j(n-j+1)}{(n+1)^{2}(n+2)}}-2\cdot \operatorname {Cov} (U_{(k)},U_{(j)})}
{\displaystyle \operatorname {Var} (U)={\frac {(k-j)(n-(k-j)+1)}{(n+1)^{2}(n+2)}}}
{\displaystyle U\sim \operatorname {Beta} (k-j,n-(k-j)+1)}
Order statistics sampled from an exponential distribution[edit]
{\displaystyle X_{1},X_{2},..,X_{n}}
{\displaystyle X_{(i)}{\stackrel {d}{=}}{\frac {1}{\lambda }}\left(\sum _{j=1}^{i}{\frac {Z_{j}}{n-j+1}}\right)}
where the Zj are iid standard exponential random variables (i.e. with rate parameter 1). This result was first published by Alfréd Rényi.[5][6]
Order statistics sampled from an Erlang distribution[edit]
The Laplace transform of order statistics may be sampled from an Erlang distribution via a path counting method[clarification needed].[7]
The joint distribution of the order statistics of an absolutely continuous distribution[edit]
{\displaystyle dF_{X}(x)=f_{X}(x)\,dx}
{\displaystyle u=F_{X}(x)}
{\displaystyle du=f_{X}(x)\,dx}
{\displaystyle f_{X_{(k)}}(x)={\frac {n!}{(k-1)!(n-k)!}}[F_{X}(x)]^{k-1}[1-F_{X}(x)]^{n-k}f_{X}(x)}
{\displaystyle f_{X_{(j)},X_{(k)}}(x,y)={\frac {n!}{(j-1)!(k-j-1)!(n-k)!}}[F_{X}(x)]^{j-1}[F_{X}(y)-F_{X}(x)]^{k-1-j}[1-F_{X}(y)]^{n-k}f_{X}(x)f_{X}(y)}
{\displaystyle x\leq y}
{\displaystyle f_{X_{(1)},\ldots ,X_{(n)}}(x_{1},\ldots ,x_{n})=n!f_{X}(x_{1})\cdots f_{X}(x_{n})}
{\displaystyle x_{1}\leq x_{2}\leq \dots \leq x_{n}.}
Application: confidence intervals for quantiles[edit]
A small-sample-size example[edit]
As an example, consider a random sample of size 6. In that case, the sample median is usually defined as the midpoint of the interval delimited by the 3rd and 4th order statistics. However, we know from the preceding discussion that the probability that this interval actually contains the population median is[clarification needed]
{\displaystyle {6 \choose 3}(1/2)^{6}={5 \over 16}\approx 31\%.}
{\displaystyle \left[{6 \choose 2}+{6 \choose 3}+{6 \choose 4}\right](1/2)^{6}={25 \over 32}\approx 78\%.}
Large sample sizes[edit]
{\displaystyle U_{(\lceil np\rceil )}\sim AN\left(p,{\frac {p(1-p)}{n}}\right).}
{\displaystyle X_{(\lceil np\rceil )}\sim AN\left(F^{-1}(p),{\frac {p(1-p)}{n[f(F^{-1}(p))]^{2}}}\right)}
where f is the density function, and F −1 is the quantile function associated with F. One of the first people to mention and prove this result was Frederick Mosteller in his seminal paper in 1946.[8] Further research led in the 1960s to the Bahadur representation which provides information about the errorbounds.
{\displaystyle B(k,n+1-k)\ {\stackrel {\mathrm {d} }{=}}\ {\frac {X}{X+Y}},}
{\displaystyle X=\sum _{i=1}^{k}Z_{i},\quad Y=\sum _{i=k+1}^{n+1}Z_{i},}
Application: Non-parametric density estimation[edit]
Moments of the distribution for the first order statistic can be used to develop a non-parametric density estimator.[9] Suppose, we want to estimate the density
{\displaystyle f_{X}}
{\displaystyle x^{*}}
{\displaystyle Y_{i}=|X_{i}-x^{*}|}
{\displaystyle g_{Y}(y)=f_{X}(y+x^{*})+f_{X}(x^{*}-y)}
{\displaystyle f_{X}(x^{*})={\frac {g_{Y}(0)}{2}}}
{\displaystyle Y_{(1)}}
{\displaystyle N}
{\displaystyle E(Y_{(1)})={\frac {1}{(N+1)g(0)}}+{\frac {1}{(N+1)(N+2)}}\int _{0}^{1}Q''(z)\delta _{N+1}(z)\,dz}
{\displaystyle Q}
{\displaystyle g_{Y}}
{\displaystyle \delta _{N}(z)=(N+1)(1-z)^{N}}
{\displaystyle N}
{\displaystyle \{x_{\ell }\}_{\ell =1}^{M}}
{\displaystyle a\in (0,1)}
(usually 1/3).
{\displaystyle \{{\hat {f}}_{\ell }\}_{\ell =1}^{M}}
{\displaystyle m_{N}=\operatorname {round} (N^{1-a})}
{\displaystyle s_{N}={\frac {N}{m_{N}}}}
{\displaystyle s_{N}\times m_{N}}
{\displaystyle M_{ij}}
{\displaystyle m_{N}}
{\displaystyle s_{N}}
samples each.
4: Create a vector
{\displaystyle {\hat {f}}}
to hold the density evaluations.
{\displaystyle \ell =1\to M}
6: for
{\displaystyle k=1\to m_{N}}
7: Find the nearest distance
{\displaystyle d_{\ell k}}
{\displaystyle x_{\ell }}
{\displaystyle k}
8: end for
9: Compute the subset average of distances to
{\displaystyle x_{\ell }:d_{\ell }=\sum _{k=1}^{m_{N}}{\frac {d_{\ell k}}{m_{N}}}}
{\displaystyle x_{\ell }:{\hat {f}}_{\ell }={\frac {1}{2(1+s_{N})d_{\ell }}}}
11: end for
{\displaystyle {\hat {f}}}
In contrast to the bandwidth/length based tuning parameters for histogram and kernel based approaches, the tuning parameter for the order statistic based density estimator is the size of sample subsets. Such an estimator is more robust than histogram and kernel based approaches, for example densities like the Cauchy distribution (which lack finite moments) can be inferred without the need for specialized modifications such as IQR based bandwidths. This is because the first moment of the order statistic always exists if the expected value of the underlying distribution does, but the converse is not necessarily true.[10]
Dealing with discrete variables[edit]
{\displaystyle X_{1},X_{2},\ldots ,X_{n}}
{\displaystyle F(x)}
{\displaystyle f(x)}
{\displaystyle k^{\text{th}}}
{\displaystyle p_{1}=P(X<x)=F(x)-f(x),\ p_{2}=P(X=x)=f(x),{\text{ and }}p_{3}=P(X>x)=1-F(x).}
{\displaystyle k^{\text{th}}}
{\displaystyle {\begin{aligned}P(X_{(k)}\leq x)&=P({\text{there are at least }}k{\text{ observations less than or equal to }}x),\\&=P({\text{there are at most }}n-k{\text{ observations greater than }}x),\\&=\sum _{j=0}^{n-k}{n \choose j}p_{3}^{j}(p_{1}+p_{2})^{n-j}.\end{aligned}}}
{\displaystyle P(X_{(k)}<x)}
{\displaystyle {\begin{aligned}P(X_{(k)}<x)&=P({\text{there are at least }}k{\text{ observations less than }}x),\\&=P({\text{there are at most }}n-k{\text{ observations greater than or equal to }}x),\\&=\sum _{j=0}^{n-k}{n \choose j}(p_{2}+p_{3})^{j}(p_{1})^{n-j}.\end{aligned}}}
{\displaystyle X_{(k)}}
{\displaystyle {\begin{aligned}P(X_{(k)}=x)&=P(X_{(k)}\leq x)-P(X_{(k)}<x),\\&=\sum _{j=0}^{n-k}{n \choose j}\left(p_{3}^{j}(p_{1}+p_{2})^{n-j}-(p_{2}+p_{3})^{j}(p_{1})^{n-j}\right),\\&=\sum _{j=0}^{n-k}{n \choose j}\left((1-F(x))^{j}(F(x))^{n-j}-(1-F(x)+f(x))^{j}(F(x)-f(x))^{n-j}\right).\end{aligned}}}
Computing order statistics[edit]
Examples of order statistics[edit]
^ David, H. A.; Nagaraja, H. N. (2003). Order Statistics. Wiley Series in Probability and Statistics. doi:10.1002/0471722162. ISBN 9780471722168.
^ Casella, George; Berger, Roger. Statistical Inference (2nd ed.). Cengage Learning. p. 229. ISBN 9788131503942.
^ a b Gentle, James E. (2009), Computational Statistics, Springer, p. 63, ISBN 9780387981444 .
^ Jones, M. C. (2009), "Kumaraswamy's distribution: A beta-type distribution with some tractability advantages", Statistical Methodology, 6 (1): 70–81, doi:10.1016/j.stamet.2008.04.001, As is well known, the beta distribution is the distribution of the m’th order statistic from a random sample of size n from the uniform distribution (on (0,1)).
^ David, H. A.; Nagaraja, H. N. (2003), "Chapter 2. Basic Distribution Theory", Order Statistics, Wiley Series in Probability and Statistics, p. 9, doi:10.1002/0471722162.ch2, ISBN 9780471722168
^ Rényi, Alfréd (1953). "On the theory of order statistics". Acta Mathematica Hungarica. 4 (3): 191–231. doi:10.1007/BF02127580.
^ Hlynka, M.; Brill, P. H.; Horn, W. (2010). "A method for obtaining Laplace transforms of order statistics of Erlang random variables". Statistics & Probability Letters. 80: 9–18. doi:10.1016/j.spl.2009.09.006.
^ Mosteller, Frederick (1946). "On Some Useful "Inefficient" Statistics". Annals of Mathematical Statistics. 17 (4): 377–408. doi:10.1214/aoms/1177730881. Retrieved February 26, 2015.
^ Garg, Vikram V.; Tenorio, Luis; Willcox, Karen (2017). "Minimum local distance density estimation". Communications in Statistics - Theory and Methods. 46 (1): 148–164. arXiv:1412.2851. doi:10.1080/03610926.2014.988260.
^ David, H. A.; Nagaraja, H. N. (2003), "Chapter 3. Expected Values and Moments", Order Statistics, Wiley Series in Probability and Statistics, p. 34, doi:10.1002/0471722162.ch3, ISBN 9780471722168
Order statistics at PlanetMath. Retrieved Feb 02,2005
Weisstein, Eric W. "Order Statistic". MathWorld. Retrieved Feb 02,2005
Retrieved from "https://en.wikipedia.org/w/index.php?title=Order_statistic&oldid=1088665050"
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Home : Support : Online Help : Mathematics : Evaluation : evala : Expand
expansion over algebraic extension fields
evala(Expand(a))
This function expands product and powers of rational functions with algebraic coefficients. Powers of algebraic numbers and functions are reduced and denominators are rationalized.
More precisely, the output satisfies the following properties:
Positive powers, products of sums and products of positive powers are expanded. The first operand of negative powers is expanded recursively, but negative powers are not expanded. Products of negative powers are not expanded either.
Algebraic numbers and functions have been reduced modulo the minimal polynomials. See Normal for a more precise definition.
Denominators have been rationalized. In other words, RootOfs and radicals defining algebraic numbers and functions have been removed from the denominator of rational functions.
Arguments of functions have been expanded recursively. Note that, unlike expand, Expand has no effect on mathematical functions such as sin or exp.
If a is a set, a list, a range, a relation, or a series, then Expand is mapped over the object.
This function can be used to normalize polynomials over algebraic number fields. If the coefficients are algebraic functions or if a is not a polynomial, Expand cannot be used to decide whether a is mathematically equal to zero. See Normal in this case.
This function does not check that the algebraic quantities are independent.
\mathrm{s1}≔\frac{x+\mathrm{sqrt}\left(2\right)}{\left(x-\mathrm{sqrt}\left(2\right)\right)x}
\textcolor[rgb]{0,0,1}{\mathrm{s1}}\textcolor[rgb]{0,0,1}{≔}\frac{\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\sqrt{\textcolor[rgb]{0,0,1}{2}}}{\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\sqrt{\textcolor[rgb]{0,0,1}{2}}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}}
\mathrm{evala}\left(\mathrm{Expand}\left(\mathrm{s1}\right)\right)
\frac{\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{2}}}{{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{x}}{{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{2}}{\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\right)}
\mathrm{alias}\left(\mathrm{\alpha }=\mathrm{RootOf}\left({y}^{2}-y+x,y\right)\right):
\mathrm{s2}≔{\left(\mathrm{\alpha }-y\right)}^{2}
\textcolor[rgb]{0,0,1}{\mathrm{s2}}\textcolor[rgb]{0,0,1}{≔}{\left(\textcolor[rgb]{0,0,1}{\mathrm{\alpha }}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{y}\right)}^{\textcolor[rgb]{0,0,1}{2}}
\mathrm{evala}\left(\mathrm{Expand}\left(\mathrm{s2}\right)\right)
\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{\alpha }}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{\alpha }}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}
\mathrm{s3}≔{\left(x-\mathrm{RootOf}\left({x}^{2}-4\right)\right)}^{2}
\textcolor[rgb]{0,0,1}{\mathrm{s3}}\textcolor[rgb]{0,0,1}{≔}{\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{RootOf}}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{\mathrm{_Z}}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{4}\right)\right)}^{\textcolor[rgb]{0,0,1}{2}}
\mathrm{evala}\left(\mathrm{Expand}\left(\mathrm{s3}\right)\right)
\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{RootOf}}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{\mathrm{_Z}}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{4}\right)\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}
\mathrm{s4}≔-\frac{1}{x-1}+\frac{x}{x-1}-1
\textcolor[rgb]{0,0,1}{\mathrm{s4}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{x}}{\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}
\mathrm{evala}\left(\mathrm{Expand}\left(\mathrm{s4}\right)\right)
\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{x}}{\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}
\mathrm{normal}\left(\mathrm{s4}\right)
\textcolor[rgb]{0,0,1}{0}
\mathrm{s5}≔\mathrm{sin}\left({\mathrm{\alpha }}^{2}-\mathrm{\alpha }+2x+y\right)
\textcolor[rgb]{0,0,1}{\mathrm{s5}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{\mathrm{\alpha }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{\alpha }}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\right)
\mathrm{evala}\left(\mathrm{Expand}\left(\mathrm{s5}\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\right)
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Recall that the vector projection of a vector
\stackrel{⇀}{u }
onto another vector
\stackrel{⇀}{v }
{\mathrm{proj}}_{\stackrel{⇀}{v }}\left(\stackrel{⇀}{u }\right) = \frac{\stackrel{⇀}{u }·\stackrel{⇀}{v }}{{∥\stackrel{⇀}{v }∥}^{2}}\stackrel{⇀}{v }
\stackrel{\mathbf{⇀}}{\mathbit{u}\mathbf{ }}
onto a plane can be calculated by subtracting the component of
\stackrel{⇀}{u }
that is orthogonal to the plane from
\stackrel{⇀}{u }
. If you think of the plane as being horizontal, this means computing
\stackrel{⇀}{u }
minus the vertical component of
\stackrel{⇀}{u }
, leaving the horizontal component. This "vertical" component is calculated as the projection of
\stackrel{⇀}{u }
onto the plane normal vector
\stackrel{⇀}{n }
{\mathrm{proj}}_{\mathrm{Plane}}\left(\stackrel{⇀}{u }\right) = \stackrel{⇀}{u } - {\mathrm{proj}}_{\stackrel{⇀}{n }}\left(\stackrel{⇀}{u }\right) = \stackrel{⇀}{u } - \frac{\stackrel{⇀}{u }·\stackrel{⇀}{n }}{{∥\stackrel{⇀}{n }∥}^{2}}\stackrel{⇀}{n}
Choose the coordinates of a plane normal vector
\stackrel{\mathit{⇀}}{n}
\mathit{ }\stackrel{\mathit{⇀}}{u\mathit{ }}
and notice how the perpendicular of the vector projection of
\stackrel{\mathit{⇀}}{u\mathit{ }}
\stackrel{\mathit{⇀}}{n}
\stackrel{\mathit{⇀}}{u\mathit{ }}
onto the plane.
\stackrel{\textcolor[rgb]{1,1,1}{\mathbf{⇀}}}{\textcolor[rgb]{1,1,1}{\mathbit{n}}}
\stackrel{\textcolor[rgb]{1,1,1}{\mathbf{⇀}}}{\textcolor[rgb]{1,1,1}{\mathbit{u}}}
{x}_{n} =
{x}_{u}=
{y}_{n}=
{y}_{u}=
{z}_{n}=
{z}_{u}=
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Variable Ratio Write Definition
What Is a Variable Ratio Write?
A variable ratio write is a strategy in options investing that requires holding a long position in the underlying asset while simultaneously writing multiple call options at varying strike prices. It is essentially a ratio buy-write strategy.
The trader's goal is to capture the premiums paid for the call options. Variable ratio writes have limited profit potential. The strategy is best used on stocks with little expected volatility, particularly in the near term.
A variable ratio write is an options strategy used by traders who seek a side source of income for a stock that they own.
The strategy is used when a trader thinks the stock will remain static in price for some period of time.
The trader invests in multiple call options at varying strike prices.
The potential profit is in the premiums paid for the call options.
Understanding Variable Ratio Writes
In ratio call writing, the word "ratio" represents the number of options sold for every 100 shares owned in the underlying stock.
For example, in a 2:1 variable ratio write, the trader might own 100 shares of the underlying stock and sell 200 options.
Two calls are written: One is "out of the money." That is, the strike price is higher than the current value of the underlying stock. On the other, the strike price is "in the money," or lower than the price of the underlying stock.
The payoff in a variable ratio write resembles that of a reverse strangle. In the options trade, any strangle strategy involves buying both a call and a put on the same underlying asset.
The variable ratio write is aptly described as having limited profit potential and unlimited risk.
Variable ratio writes have limited upside and unlimited downside.
When the Variable Ratio Write Is Used
As an investment strategy, the variable ratio write should be avoided by inexperienced options traders as it is a strategy with unlimited risk potential.
The losses begin if the stock's price makes a strong move to the upside or downside beyond the upper and lower breakeven points set by the trader.
There is no limit to the maximum possible loss on a variable ratio write position. Despite its significant risks, the variable ratio write technique can bring the experienced trader a fair amount of flexibility with managed market risk while providing attractive income.
There are two breakeven points for a variable ratio write position. These breakeven points can be found as follows:
\begin{aligned} &\text{Upper Breakeven Point} = SPH+PMP\\ &\text{Lower Breakeven Point} = SPL-PMP\\ &\textbf{where:}\\ &SPH=\text{Strike price of higher strike short call}\\ &PMP=\text{Points of maximum profit}\\ &SPL=\text{Strike price of lower strike short call} \end{aligned}
Upper Breakeven Point=SPH+PMPLower Breakeven Point=SPL−PMPwhere:SPH=Strike price of higher strike short callPMP=Points of maximum profitSPL=Strike price of lower strike short call
Real-World Example of a Variable Ratio Write
Consider an investor who owns 1,000 shares of the company XYZ, currently trading at $100 per share. The investor believes that the stock is unlikely to move much over the next two months.
The investor can hold onto the stock and still earn a positive return on it while it remains static in price. This is achieved by initiating a variable ratio write position, selling 30 of the 110 strike calls on XYZ that are due to expire in two month's time. The options premium on the 110 calls is $0.25, so our investor will collect $750 from selling the options.
That is, if the investor is correct in predicting that the stock's price will remain flat.
After two months, if XYZ shares remain below $110, the investor will book the entire $750 premium as profit, since the calls will be worthless when they expire.
If the shares rise above the breakeven $110.25, however, the gains on the long stock position will be more than offset by losses by the short calls. The options represented 3,000 shares of XYZ, triple the number that the trader owns.
The Options Guide. "Variable Ratio Write." Accessed July 23, 2021.
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Unitary Representations of the Group of Diffeomorphisms via Restricted Product Measures with Infinite Mass II | EMS Press
This paper concerns the problem of irreducibly decomposing unitary representations of the group
\text{Diff}_0(M)
of diffeomorphisms with compact support on the smooth manifold
M
. As was shown in [19], these representations are decomposable under a fairly mild condition. In this paper, we consider a specific example of unitary representations
(T,\text{Diff}_0(M))
that has been considered by [4].
(T,\text{Diff}_0(M))
is already a factor representation of type
\text{II}_\infty
; in addition, it may be decomposed into irreducible components through the left regular representatation of the group
{\frak{S}}_\infty
of the finite permutations. We describe the concrete realization of these irreducible components. The results obtained herein have some resemblance to the finite-dimensional case of [20] with the exception of the factor representatation. In addition, we will give another proofs of the irreducibility and equivalence that were obtained by [4].
Hiroaki Shimomura, Unitary Representations of the Group of Diffeomorphisms via Restricted Product Measures with Infinite Mass II. Publ. Res. Inst. Math. Sci. 48 (2012), no. 1, pp. 183–213
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\mathrm{content}\left(3-3x,x\right)
\textcolor[rgb]{0,0,1}{3}
\mathrm{content}\left(3xy+6{y}^{2},x\right)
\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}
\mathrm{content}\left(3xy+6{y}^{2},[x,y]\right)
\textcolor[rgb]{0,0,1}{3}
\mathrm{content}\left(-4xy+6{y}^{2},x,'\mathrm{pp}'\right)
\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}
\mathrm{pp}
\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}
In this example, you can see the effect of calling normal, which happens because the polynomial doesn't have purely numeric coefficients (the coefficient of x is
\frac{1}{a}
\mathrm{content}\left(\frac{x}{a}-\frac{1}{2},x,'\mathrm{pp}'\right)
\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{a}}
\mathrm{pp}
\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{a}
\mathrm{normal}\left(\frac{x}{a}-\frac{1}{2}\right)
\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{a}}{\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{a}}
\mathrm{content}\left(2.ux-2.v,x,'\mathrm{pp}'\right)
\textcolor[rgb]{0,0,1}{1}
\mathrm{pp}
\textcolor[rgb]{0,0,1}{2.}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2.}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{v}
\mathrm{content}\left(2.ux-2.u,x,'\mathrm{pp}'\right)
\textcolor[rgb]{0,0,1}{u}
\mathrm{pp}
\textcolor[rgb]{0,0,1}{2.}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2.}
Non-numeric, nonpolynomial coefficients are also considered indivisible with respect to each other. For example, you could consider
\sqrt{2}
to be a common divisor between the two coefficients
\sqrt{10}
\sqrt{6}
, but they are considered indivisible with respect to each other for this command and the content is considered to be 1.
\mathrm{content}\left(\mathrm{sqrt}\left(10\right)x+\mathrm{sqrt}\left(6\right),x\right)
\textcolor[rgb]{0,0,1}{1}
\mathrm{primpart}\left(-4xy+6{y}^{2},x\right)
\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}
\mathrm{primpart}\left(\frac{x}{a}-\frac{1}{2},x\right)
\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{a}
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Eating Disorders - Uncyclopedia, the content-free encyclopedia
“BLAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAGH HACK-HACK-HACK!”
~ Oscar Wilde on Bulimia
“My wife has one of these, I think...”
~ Evil Pizza on eating disorders
“People with these have no troubles at all, and don't go crazy if I throw a chicken leg at them.”
~ Captain Sarcasm on eating disorders
“I tried to kill myself with an eating disorder once. It didn't work...”
~ Emo Man on eating disorders
For those without comedic tastes, the so-called experts at Wikipedia have an article very remotely related to Eating Disorders.
Eating disorders are a source of hilarity for cool, non-hideous people. As of recent history, the medical community has been dealing with this problem, trying to answer the unsolvable question of why worthless fatties even bother trying to puke out their soullessness. Eating disorders are most common in America, as it has the least amount of worthy human flesh per capita. Scientists recommend that beautiful people should badger the fatties, making the world a more comfortable and humorous place for those of us who are totally awesome and worthwhile. National Eating Disorders Awareness Month is held annually in February to make people more aware of the advantages and benefits of eating disorders. Additionally, the public is educated on the most popular methods of exercise, caloric restriction, and purging, in case any aspiring anorexics or bulimics are short on ideas or inspiration.
2 Shorted Mouth-Nerve
3 Feedeetitus
4 OscarWildeSyndrome
Anorexia[edit]
When stupid people run away screaming from anything that isn't fat-sugar-carbs-taste free.
Shorted Mouth-Nerve[edit]
This disorder is the only one that is present from birth. Sure, people can have morbidly obese babies or stick-thin babies; but that doesn't mean they instantly have anorexia or feedeetitus. It's actually caused by a malfuction in the developing of the nerve control center of the brain. The nerves that help the brain recongnize the mouth just aren't there. Basically, people have no bloody idea that their mouths exist on their faces. They also believe that they talk through their eyes, breathe through their noses, and that moving hole on their chins is a useless part of the face. As such, they often cram food into other external body parts. Quite obviously, most are undernurished because their food has not started the first step in digestion. People afflicted with this disease must be spoon-fed by male nurses or family members. If you happen to see a girl walking down the street with a cuccumber in her ear; spaghetti in her hair; a Twinkie up her nose; a drumstick commin' out of her butt; an ice cream cone in her clevage; and/or a Subway Sub duct-taped to her back, please be a kind soul and place all of the food items into her mouth.
Feedeetitus[edit]
Feedeetitus is something all BBWs, potential BBWs, feedees, and any overeating girl has. They either love the idea of being fat or just can't stop eating. Girls in this category have large appitites.
{\displaystyle appitite=A}
That means they love to eat (duh!). Their style of eating can be represented by this:
{\displaystyle A^{2}+113(5A/H)=B}
{\displaystyle hight=H}
{\displaystyle bellybulge=B}
As you can see, girls with this disorder eat until they are stuffed full and then eat some more. This leads to the exspansion of their bellies (or boobs) and the gainage of weight. Often times this leaves their bellies stretched and painfuly sore. To calculate belly soreness, use the following formula;
{\displaystyle BH^{3}=*S}
{\displaystyle bellysoreness=S}
As you can see above, this formula can only approximate belly soreness after eating. To really understand the pain girls with Feedeetitus go through, you would have to swap bodies with them and feel the pain. (this is not suggested because you'll probably get stuck in that body for all eternity and have idiots like me study you and create mathimatical formulas to explain your exsistance) However, we must remember something very important.
{\displaystyle S<A}
So we are left with a weird question: If this expression exsists, then are girls able to continue eating with belly soreness? This ? has plagued doctors for years. Doctor Mario developed the following expression:
{\displaystyle ST=A}
{\displaystyle time=T}
He was trying to suggest that, after a long period of time, belly soreness would go away and this would allow the girls to eat again. This seemed credible for 23 years beause girls who stuffed themselves to the limit often would take naps. Dr. Luke Skywalker followed Doctor Mario's work and submitted a formula that explained naps.
{\displaystyle (S-A)T=2N}
{\displaystyle naps=N}
The 2N represented the fact that fat girls sleep twice as long as regular girls. The works of Doctor Mario and Luke seemed to have finally explained Feedeetitus. Unfortanently, scientific experiments tested these equations and found them to be void 38% of the time. Many of the feedee girls in the study diden't take naps after stuffing themselves and eventually continued to eat. How then were the girls getting rid of their belly soreness? It was a mysterious figure in the medical community who, after studying his Feedeetitus-affected wife for three years, discovered the the secret.
{\displaystyle (SH/M)^{B}=R}
{\displaystyle handmotion=M}
{\displaystyle bellyrub=R}
Yes, he had discovered the belly rub! His wife, after stuffing herself with food, would rub her belly to soothe belly soreness. The equation showed that belly rubbing also stimulated digestion, which allowed for more things to be put in the stomach. This figure then applaied the belly rub to the Feedeetitus problem and got the following result:
{\displaystyle R(B^{2}+S^{3})=A}
The medical community rejoyced with this news. When the formula was tested, scientists found that feedee-girls would pat, slap, massage, rub, and play with their belly after gorging. In fact, of the girls studied, 78% would rub their belly and 23% would nap. Scientists also discovered that it didn't even matter who submitted the belly rub. Wearing radioactive suits, scientists enetered the testing site and began to rub the bloated bellies of the girls. Much to their surprise, this incouraged the girls to eat more because they then didn't have to stop eating to soothe their bellies. The scientists also found out that it was fun to play with a feedee-girl's belly (some of the scientists were never seen again). Celebrate the anniversary of this discovery (which is the May 21th) by playing with a feedee-girl's gut today!
OscarWildeSyndrome[edit]
This is a very rare eating disorder, but on of the simplest to explain. Basically, paitents with this disorder strive themselves to weigh exactly the same as Oscar Wilde. First known patient? Oscar Wilde (or he's the source of the disease. I'm not entirely sure; I always get the two mixed up). But does anyone know how much pounds or kilograms Oscar Wilde weighs?
HowTo:Make a Fat Girl
Retrieved from "https://uncyclopedia.com/w/index.php?title=Eating_Disorders&oldid=6165031"
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Latin Squares and Finite Groups
Last semester, for an algebra homework, I was trying to prove that there exist only 2 groups of order 6 (namely
\mathbb{Z}_6
S_3
). The usual argument uses the classification of groups with order
pq
p
q
prime), which itself uses Sylow theorems, but I wondered if I could prove it computationally. Here's my attempt.
Definition: a magma is a pair
(G,\ast)
G
is a non-empty set and
\ast
G
. A quasigroup is a magma in which the equations
gx=h
yg=h
g,h\in G
, always have a unique solution for
x
y
G
. A group is a magma in which
\ast
is associative, modulative and invertive.
Another way of defining a quasigroup is as a magma in which the left and right cancellation laws hold. Note that we're not asking
\ast
to be associative in a magma (let alone in a quasigroup).
All there is to know about a finite magma
(G,\ast)
is encoded in its Cayley table: a matrix that shows how the elements in
M
operate among themselves. In general, if
G = \{a_1, \dots, a_n\}
, the Cayley table looks like this:
\begin{matrix} \ast&a_1&\dots &a_n\\ \hline a_1&a_1a_1&\dots &a_1a_n\\ \vdots&\vdots&\dots&\vdots\\ a_n&a_na_1&\dots &a_na_n\\ \end{matrix}
That is, the element in the
(i,j)
-th position is the result of multiplying
a_i
a_j
in that order.
You can tell a lot about a magma and its operation just by looking at its Cayley table. For example, consider Klein's 4 group
V
\begin{array}{c|cccc} *&e&a&b&c\\ \hline e&e&a&b&c\\ a&a&e&c&b\\ b&b&c&e&a\\ c&c&b&a&e\\ \end{array}
You can immediately tell its commutative (because the matrix is symmetric), you can tell there is an identity (namely
e
) and that each element is its own inverse. You could also tell if the binary operation is associative from its Cayley table using Light's test, although it isn't any better, computationally speaking, that just verifying every case by hand.
We will use Cayley tables as the bridge between algebra and combinatorics.
Latin squares and quasigroups
Definition: a
n\times n
matrix of
n
different entries is called a latin square if no element appears more than once in any row or column. This property is called the latin square property.
We will deal with latin squares of size
n
whose entries are the integers from
0
n-1
\begin{bmatrix} 0&1&2\\ 1&2&0\\ 2&0&1\\ \end{bmatrix}
is a latin square of size
3
. We will also start indexing by 0.
(G, \ast)
is a quasigroup, then its Cayley table is a latin square.
G = \{a_1, \dots, a_n\}
and that an element
b\in G
appears twice in row
l
(say, in columns
j
k
), by the definition of the Cayley table, this means that
a_la_j = b = a_la_k
G
is a quasigroup, the left cancellation law implies that
a_j = a_k
, which is absurd because we assumed that
j
k
were different. Analogously, the right cancellation law implies that no element appears twice in any column. Q.E.D.
This theorem has a reciprocal of some sort:
Theorem 2: Given a latin square
L = (l_{ij})
, one can construct a quasigroup whose Cayley table is
L
G = \{l_{11}, \dots, l_{1n}\}
g_i := l_{1i}
\ast
g_i * g_j = g_{l_{ij}}
\ast
is well defined (that is, it is closed in the set). We need to check that the equations
gx =h
yg = h
have unique solutions. Consider
g_lx = g_k
L
is a latin square,
g_k = l_{1k}
appears somewhere in row
l
, call the column it appears in
m
x = g_m
g_lx = g_k
. It is unique, because if there existed
g_{\widetilde{m}}
g_lg_{\widetilde{m}} = g_k = g_lg_m
g_k
would appear twice in row
l
L
is a latin square. Analogously, now arguing with columns,
yg = h
G
So now we're set!, we only need to find all latin squares of size
n
and to verify if they represent a valid binary operation for a group. Moreover, we could force the existence of an identity by focusing on finding normalized (or reduced) latin squares (that is, latin squares where the first row and column are
0, 1, \dots, n-1
The algorithm for finding normalized latin squares of size n.
I use a depth-first-search style algorithm, starting with a normalized
n\times n
A = \begin{bmatrix} 0&1&\cdots&n-1\\ 1&-1&\cdots&-1\\ \vdots&\vdots&\ddots&\vdots\\ n-1&-1&\cdots&-1\\ \end{bmatrix}
where the unvisited locations are labeled with a
-1
. We also start with an empty stack
S
. The algorithm goes as follows:
Put matrix
A
in the stack
S
If the stack
S
is empty, stop; if it isn't, pop a matrix
B
from it.
Find the first unvisited position
(i,j)
B
(i.e. the first
-1
), if there isn't any, it is finished, put it in a special list of finished latin squares and go to step 2.
Push into the stack the result of replacing this
-1
with every number from
0
n-1
that isn't already on its row or column.
Here's the algorithm implemented in python:
def dfs_in_matrix(A): # First we create an empty stack and we put the initial matrix # in it. list_of_solutions = [] stack = LifoQueue() stack.put(A) while not stack.empty(): # We pop a matrix from the stack B = stack.get() # We check if it's finished. if is_finished(B): list_of_solutions.append(B) continue # We find an unvisited position position = find_first_unvisited_position(B) if position == None: continue span = span_of_position(position, B) for k in range(len(B)): if k not in span: C = B.copy() C[position] = k stack.put(C) return list_of_solutions def find_normalized_latin_squares(number): A = np.zeros((number, number)) for k in range(number): A[0, k] = k A[k, 0] = k for i in range(1, number): for j in range(1, number): A[i, j] = -1 list_of_solutions = dfs_in_matrix(A) return list_of_solutions
(the functions is_finished, find_first_unvisited_position and span_of_position are auxiliary, check this jupyter notebook for all the code discussed in this post). It checks out with the literature on the topic1, saying that there are 9408 normalized latin squares of size 6.
The Magma class
Once we have all the normalized latin squares, we can build up a Magma class in python and we can write a verification function to find which of these correspond to associative operations (and thus to groups).
class Magma: def __init__(self, _matrix): self.cayley_table = _matrix self.set = set(range(0, len(_matrix[0,:]))) def mult(self, a, b): return int(self.cayley_table[a, b]) def is_magma_associative(mag): ''' This function verifies if magma `mag` is associative by brute force. ''' n = len(mag.cayley_table) _flag = True for a in range(n): for b in range(n): for c in range(n): _flag = _flag and (mag.mult(a, mag.mult(b,c)) == mag.mult(mag.mult(a,b),c)) return _flag def find_groups(number): latin_squares = find_normalized_latin_squares(number) associative_magmas = [sol for sol in latin_squares if is_magma_associative(Magma(sol))] return associative_magmas
After running this is_magma_associative in all 9408 reduced latin squares of order 6 we're left with 80 reduced latin squares such that, when interpreted as quasigroups, are associative. That is, only 80 of the original 9408 reduced latin squares of size 6 can be interpreted as Cayley tables for groups.
We're trying to prove the following theorem:
Theorem 3: There are only 2 groups of order 6, namely
S_3
\mathbb{Z}_6
It is useful, then, to cleary state what we interpret as
S_3
\mathbb{Z}_6
\mathbb{Z}_6
are the usual integers modulo 6 with sum modulo 6, but note that
\mathbb{Z}_6
can also be interpreted in the following way: its a group of six elements
\{a_1, a_2, a_3, a_4, a_5, 0\}
a_1
a_5
have order 6.
a_2
a_4
a_3
a_2^2 = a_4
a_1a_2 = a_3
(note that we just changed
i
a_i
). We can use this information to find an isomorphism between a latin-square-generated group and
\mathbb{Z}_6
S_3 = \{\sigma_1, \sigma_2, \sigma_3, \rho_1, \rho_2, \text{id}\}
is usually interpreted as the group of symmetries of a triangle (where
\sigma_i
is the reflection that fixes vertex
i
\rho_j
120*j
degrees, but we prefer the following presentation:
S_3 = \langle \sigma, \rho\,\vert\,\sigma^2 = \rho^3 = \text{id},\, \sigma\rho = \rho^2\sigma \rangle
In this presentation, the 6 different elements are
\text{id}, \sigma, \rho, \rho\sigma, \rho^2\sigma
\rho^2
So, to prove theorem 3, we will follow this strategy: we will give an isomorphism from either
S_3
\mathbb{Z}_6
to each of the 80 groups found using latin squares:
Proof (of Theorem 3): Theorem 1 and 2 show that all possible groups of a certain order are restricted by the amount of normalized latin squares of said order. After filtering the normalized latin squares of size 6 by verifying which represent an associative binary operation, we're left with 80 Cayley tables for groups. In this jupyter notebook we show an explicit isomorphism between each of these 80 latin square generated groups and either
\mathbb{Z}_6
S_3
, but for the sake of completeness we show how these isomorphisms were constructed with explicit examples for
\mathbb{Z}_6
S_3
. Consider the following normalized latin square:
\begin{bmatrix} 0 & 1 & 2 & 3 & 4 & 5\\ 1 & 5 & 4 & 2 & 3 & 0\\ 2 & 3 & 0 & 1 & 5 & 4\\ 3 & 4 & 5 & 0 & 1 & 2\\ 4 & 2 & 1 & 5 & 0 & 3\\ 5 & 0 & 3 & 4 & 2 & 1\end{bmatrix}
and call the group it induces
G
. After inspecting it, we can tell that the orders of their elements are either
2
3
, so it is a candidate for being isomorphic to
S_3
4\mapsto \sigma
5\mapsto \rho
(5*5)*4 = 1*4 = 3 = 4*5,
that is, this group obeys the presentation given for
S_3
. The isomorphism would then be given by
\begin{matrix} S_3 & & G\\ \hline \text{id}&\mapsto&0\\ \sigma_1&\mapsto&4\\ \sigma_2&\mapsto&3\\ \sigma_3&\mapsto&2\\ \rho&\mapsto&5\\ \rho^2&\mapsto&1 \end{matrix}
Now consider the group
H
given by the following reduced latin square:
\begin{bmatrix} 0 & 1 & 2 & 3 & 4 & 5\\ 1 & 5 & 4 & 2 & 3 & 0\\ 2 & 4 & 5 & 1 & 0 & 3\\ 3 & 2 & 1 & 0 & 5 & 4\\ 4 & 3 & 0 & 5 & 1 & 2\\ 5 & 0 & 3 & 4 & 2 & 1\end{bmatrix}
because the elements of
H
have order either 2, 3 or 6, we will construct an isomorphism between
H
\mathbb{Z}_6
using the identification
\mathbb{Z}_6 = \{a_1, a_2, a_3, a_4, a_5, 0\}
stated before. First note that
3\in H
is an element of order 2,
2, 4\in H
have order 6 and
1, 5\in H
have order 3. Because
1\ast1 = 5
4\ast1 = 3
, we construct the following isomorphism
\begin{matrix} \mathbb{Z}_6 & & H\\ \hline 0&\mapsto&0\\ 1&\mapsto&4\\ 2&\mapsto&1\\ 3&\mapsto&3\\ 4&\mapsto&5\\ 5&\mapsto&2\\ \end{matrix}
The results the algorithm gave were in par with what's said in Small Latin Squares, Quasigroups and Loops, an article by Brendan D. Mackay, Alison Meynert and Wendy Myrvold. Check Table 1 of their article for more details. ↩
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Use the sum or difference of cubes and what you already know about factoring to factor the following expressions as completely as possible.
Refer to problem 12-73 for the patterns of sums and differences of cubes.
x^5 + 8x^2y^3
x^2
is a common factor for both terms.
Factor it out. Then use the sum of cubes to factor it completely.
8 = 2^3
8y^6 - 125x^3
This is already in the form you need.
Use the difference of cubes to factor it.
What are the cubes?
x^6 - y^6
(Note: This is tricky. If you start it as the difference of two cubes, you will not be able to factor it completely. Think of it as the difference of two squares and then factor the factors as the sum and difference of two cubes.)
\left(x^3\right)^2-\left(y^3\right)^2
(x + y)(x^2 - 2xy + y^2)(x - y)(x^2 + 2xy + y^2)
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Three Positive Periodic Solutions to Nonlinear Neutral Functional Differential Equations with Parameters on Variable Time Scales
2012 Three Positive Periodic Solutions to Nonlinear Neutral Functional Differential Equations with Parameters on Variable Time Scales
Using two successive reductions: B-equivalence of the system on a variable time scale to a system on a time scale and a reduction to an impulsive differential equation and by Leggett-Williams fixed point theorem, we investigate the existence of three positive periodic solutions to the nonlinear neutral functional differential equation on variable time scales with a transition condition between two consecutive parts of the scale
\left(d/dt\right)\left(x\left(t\right)+c\left(t\right)x\left(t-\alpha \right)\right)=a\left(t\right)g\left(x\left(t\right)\right)x\left(t\right)-{\sum }_{j=1}^{n}{\lambda }_{j}{f}_{j}\left(t,x\left(t-{v}_{j}\left(t\right)\right)\right)
\left(t,x\right)\in {\mathbb{T}}_{0}\left(x\right)
\Delta t{|}_{\left(t,x\right)\in {\mathcal{S}}_{2i}}={\Pi }_{i}^{1}\left(t,x\right)-t
\Delta x{|}_{\left(t,x\right)\in {\mathcal{S}}_{2i}}={\Pi }_{i}^{2}\left(t,x\right)-x
{\Pi }_{i}^{1}\left(t,x\right)={t}_{2i+1}+{\tau }_{2i+1}\left({\Pi }_{i}^{2}\left(t,x\right)\right)
{\Pi }_{i}^{2}\left(t,x\right)={B}_{i}x+{J}_{i}\left(x\right)+x,i=1,2,\dots .{\lambda }_{j}\left(j=1,2,\dots ,n\right)
{\mathbb{T}}_{0}\left(x\right)
is a variable time scale with
\left(\omega ,p\right)
c\left(t\right),a\left(t\right)
{v}_{j}\left(t\right),
{f}_{j}\left(t,x\right)\left(j=1,2,\dots ,n\right)
\omega
-periodic functions of
t
{B}_{i+p}={B}_{i},{J}_{i+p}\left(x\right)={J}_{i}\left(x\right)
i\in \mathbb{Z}
Yongkun Li. Chao Wang. "Three Positive Periodic Solutions to Nonlinear Neutral Functional Differential Equations with Parameters on Variable Time Scales." J. Appl. Math. 2012 1 - 28, 2012. https://doi.org/10.1155/2012/516476
Yongkun Li, Chao Wang "Three Positive Periodic Solutions to Nonlinear Neutral Functional Differential Equations with Parameters on Variable Time Scales," Journal of Applied Mathematics, J. Appl. Math. 2012(none), 1-28, (2012)
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Thorax Model for the Studies of Hemodynamic Monitoring by Implanted Devices | J. Med. Devices | ASME Digital Collection
Jingying Lin,
Jingying Lin
Tadashi Allen,
David Benditt,
Hsiang Hsiung,
Lin, J., Allen, T., Patterson, R., Benditt, D., Hsiung, H., and Zhang, J. (August 12, 2010). "Thorax Model for the Studies of Hemodynamic Monitoring by Implanted Devices." ASME. J. Med. Devices. June 2010; 4(2): 027539. https://doi.org/10.1115/1.3443783
Impedance incorporated implanted device provides a unique approach to monitor hemodynamics. The challenge of the use of this method is the optimization of electrode configurations. To alleviate this issue, a 3D thorax model is presented in this study. The model was developed from CT images of a patient, covering from the neck to the lower abdomen. A MATLAB-based program was developed and used to delineate different tissues/organs. The model contains 467 layers and 37 different types of tissues. Each layer had 262,144 pixels with a resolution of
1.0×1.0 mm2
, approximately
122×106 pixels
(voxels) in total. This high-resolution model can be used as a virtual phantom to optimize electrode configuration for the monitoring of hemodynamics by an implanted device.
biological organs, biological tissues, biomedical electrodes, computerised tomography, haemodynamics, medical computing, optimisation, patient monitoring, phantoms, physiological models, prosthetics
Biological tissues, Electrodes, Hemodynamics, Optimization, Phantoms
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Einstein's Malicious Theories - Uncyclopedia, the content-free encyclopedia
Albert Einstein, best known for his work inventing Gravity and Light, devising new theories on the cooking time of goose eggs, the braking speed of the Ford Cortina in an oil slick and the natural propensity for objects, when released in midair, to hit the ground, was also responsible for the propagation throughout the physics world of theories designed to anger fellow physicists. The 'Malicious Theories' are seen by some as aberrations, by some others as 'Easter Eggs' in his otherwise dull work.
This asshole nearly lost us the fucking WAR
There are several well documented 'Malicious Theories', which were responsible for the loss of 103,000 man-hours of American nuclear physicists during the height of World War II, according to the National Science Foundation. Several have been posited as further malicious theories, but so far most have been plausible. Whilst this article is not exhaustive, it details some of the most obvious and important of the Malicious Theories.
The discovery of the Malicious Theories resulted in the blocking of Einstein's position as guest host of the Johnny Carson show some years later.
Other Malicious theories have been posited since, by other physicists, and, more recently, biologists bored after dissections.
1 Energy-Matzoh Equivalence
2 The Weeble Theory
3 The Momentum Force Theory
4 The Parallax Conundrum
5 The Theory of Accelerated Light
6 The Special Theory of Hyperbolic Reduction
7 The Theory of Numerical Equivalence
8 The Speed of Thought
9 The Vader Theory
Energy-Matzoh Equivalence[edit]
Einstein, had established long before 1905 that
{\displaystyle e=mbcs}
usually read "energy is matzoh balls in chicken soup". Ernest Rutherford, who had little tolerance any sort of B.S., reduced it to
{\displaystyle e=mc}
Einstein concurred and, feeling that this was a square meal, recast the equation in its final form:
{\displaystyle e=mc^{2}}
The Weeble Theory[edit]
{\displaystyle C=n(fO)*(fO^{-}1)}
This states that the Speed of Light may 'wobble' itself at sufficiently high speeds, but will never 'fall over'. For some years, this was taken rather seriously by the Physics establishment, as it seemed to suggest a flux in acceleration for light. Despite the fact that Einstein had accepted a 'defined' lightspeed as a constant in a vacuum, some scientists got a little caught up in hero worship.
The Momentum Force Theory[edit]
This non-expressed theory was a fake thought experiment, released to the Royal Society of the United Kingdom. It concluded that if a previously stable substance was bombarded with high speed protons, a vortex and momentum would build, imploding the source material and making the accumulation of material accelerate to an infinite degree. It seemed to suggest that bombarding a substance would lead inevitably to an immediate runaway accumulation of mass, until, 24 seconds after the creation of the vortex, a sphere of material of infinite mass and gravity would be generated, most likely absorbing the universe.
He would have gotten away with this if it hadn't been for those meddling kids.
The Parallax Conundrum[edit]
{\displaystyle gD=M(g)/u(A)!}
Deliberately misnamed, this was a theory Einstein had come up with to help him remember how many drinks he needed to buy Bessie Kirchenstein before she'd get paralytic enough to give him a foot job.
The Theory of Accelerated light would suggest the spoon would appear less bent if there was a brighter light source.
The Theory of Accelerated Light[edit]
{\displaystyle C=(nS/e)}
The first theory was the easiest to work out as a malicious theory. The theory essentially states that the speed of light is determined by the total number of light sources available, or, in simple terms, that very bright lights travel faster than dimmer ones.
The Special Theory of Hyperbolic Reduction[edit]
{\displaystyle nE=nE-1/C(PiNK)}
Einstein pretended for years that this line was an integral part of his working out for the Special Theory of Relativity. in reality, the equation suggests that the number of energies (sic) is equal to the number of energies minus one speed of light, times 3.14159 gravities. Clearly, this theory has no significance at all.
The Theory of Numerical Equivalence[edit]
{\displaystyle a=b=c=d=e=f=g=h=i=j=k=l=m=n=o=p=q=r=s=t=u=v=w=x=y=z}
Written the day after Einstein's 45th Birthday party (under the influence of liquor and some child tv shows), this theory shows clearly how bored Einstein was with mathematics as a discipline.
The Speed of Thought[edit]
{\displaystyle T(vel)var/=n(Cf)}
At first considered an error in notation, this appears to be Einstein's assertion that an increase in cups of coffee would not result in an increased speed of thought.
The Vader Theory[edit]
{\displaystyle f(lS)=(aS)=(Dv)!}
This was Einstein's most haunting theory. It predicts that Darth Vader (Dv) is Anakin Skywalker, and therefore Luke Skywalker's father. (f(lS))
Bizarrely, this formula is derived from a calculation of the speed at which light decelerates relative to a massive object.
Retrieved from "https://uncyclopedia.com/w/index.php?title=Einstein%27s_Malicious_Theories&oldid=6090877"
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Parts (a) through (d) of problem 12-72 represent a general pattern known as the sum and difference of cubes. Use this pattern to factor each of the following polynomials.
The sum and difference of cubes formulas are:
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
x^3 + y^3
Factor using the sum of cubes formula.
(x + y)(x^2 - xy + y^2)
x^3 - 27
\text{27 is } 3^3.
(x - 3)(x^2 + 3x + 9)
8x^3 − y^3
\text{Rewrite the polynomial as}\ (2x)^3 - y^3.
Now factor using the difference of cubes formula.
x^3 + 1
\text{Rewrite the polynomial as } x^3 + 1^3.
Now factor using the sum of cubes formula.
Make up another problem involving the sum or difference of cubes and show how to factor it.
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Correlation - Simple English Wikipedia, the free encyclopedia
In statistics and probability theory, correlation is a way to indicate how closely related two sets of data are.
Correlation does not always mean that one causes the other. In fact, it is very possible that there is a third factor involved.
Correlation usually has one of two directions. These are positive or negative. If it is positive, then the two sets go up together. If it is negative, then one goes up while the other goes down.
Lots of different measurements of correlation are used for different situations. For example, on a scatter graph, people draw a line of best fit to show the direction of the correlation.
This scatter graph has positive correlation. You can tell because the trend is up and right. The red line is a line of best fit.
1 Explaining correlation
2 Correlation vs causation
Explaining correlation[change | change source]
Strong and weak are words used to describe the strength of correlation. If there is strong correlation, then the points are all close together. If there is weak correlation, then the points are all spread apart. There are ways of making numbers show how strong the correlation is. These measurements are called correlation coefficients. The best known is the Pearson product-moment correlation coefficient, sometimes denoted by
{\displaystyle r}
or its Greek equivalent
{\displaystyle \rho }
.[1][2] You put in data into a formula, and it gives you a number between -1 and 1.[3] If the number is 1 or −1, then there is strong correlation. If the answer is 0, then there is no correlation. Another kind of correlation coefficient is Spearman's rank correlation coefficient.
Correlation vs causation[change | change source]
Correlation does not always mean that one thing causes the other (causation), because there might be something else that is at play.
For example, on hot days people buy ice cream, and people also go to the beach where some are eaten by sharks. There is a correlation between ice cream sales and shark attacks (they both go up as the temperature goes up in this case). But just because ice cream sales go up does not mean ice cream sales cause (causation) more shark attacks or vice versa.[4]
Because correlation does not imply causation, scientists, economists, etc. will test their theories by creating isolated environments where only one factor is changed (where this is possible). However, politicians, salesmen, news outlets and others often suggest that a particular correlation implies causation. This may be due to ignorance or a wish to persuade. Thus, a news report may attract attention by saying that people who consume a particular product more often have a particular health problem, implying a causation that could be actually due to something else.
↑ Even though it is called 'Pearson', it was first made by Francis Galton.
↑ Weisstein, Eric W. "Statistical Correlation". mathworld.wolfram.com. Retrieved 2020-08-22.
↑ "Ice cream and shark attacks". Big Think. 2019-02-21. Archived from the original on 2020-09-28. Retrieved 2020-08-22.
Cohen, J., Cohen P., West, S.G., & Aiken, L.S. (2003). Applied multiple regression/correlation analysis for the behavioral sciences. (3rd ed.) Hillsdale, NJ: Lawrence Erlbaum Associates.
Correlation Information – At StatisticalEngineering.com
Statsoft Electronic Textbook Archived 2009-02-27 at the Wayback Machine
Pearson's Correlation Coefficient – How to work it out it quickly
Learning by Simulations – The spread of the correlation coefficient
CorrMatr.c Archived 2007-12-04 at the Wayback Machine simple program for working out a correlation matrix
Understanding Correlation – More beginner's information by a Hawaii professor
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Correlation&oldid=7930084"
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Periodicities of T-systems and Y-systems, Dilogarithm Identities, and Cluster Algebras I: Type $B_r$ | EMS Press
Periodicities of T-systems and Y-systems, Dilogarithm Identities, and Cluster Algebras I: Type
B_r
We prove the periodicities of the restricted T-systems and Y-systems associated with the quantum ane algebra of type
B_r
at any level. We also prove the dilogarithm identities for the Y-systems of type
B_r
at any level. Our proof is based on the tropical Y-systems and the categorication of the cluster algebra associated with any skew-symmetric matrix by Plamondon. Using this new method, we also give an alternative and simplied proof of the periodicities of the T-systems and Y-systems associated with pairs of simply laced Dynkin diagrams.
Rei Inoue, Bernhard Keller, Osamu Iyama, Atsuo Kuniba, Tomoki Nakanishi, Periodicities of T-systems and Y-systems, Dilogarithm Identities, and Cluster Algebras I: Type
B_r
. Publ. Res. Inst. Math. Sci. 49 (2013), no. 1, pp. 1–42
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Cash Available for Distribution (CAD)?
What Does CAD Tell You?
CAD vs. FFO
Cash available for distribution (CAD) refers to a real estate investment trust's (REIT) cash-on-hand that is available to be distributed as shareholder dividends. The CAD value is calculated by taking the REIT's funds from operations (FFO) and subtracting its recurring capital expenditures (CAPEX).
CAD is the most liquid subset of funds available for distribution (FAD). The benefits of having a stockpile of CAD are that it provides a more complete picture of a REIT's adjusted cash flows and how much investors can expect to receive in the form of dividend distributions.
Cash Available for Distribution (CAD) is a REIT metric that subtracts recurring capital expenditures from funds from operations (FFO).
CAD is a non-GAAP measure that is used as a proxy for a REIT's cash flow for investors
CAD can be increased organically or through the acquisitions of new properties.
The computation of CAD does not follow a standardized formula in the REIT sector, so analysts and investors should be careful to note the methodology used.
Formula for Cash Available for Distribution (CAD)
\begin{aligned} &CAD = FFO - RCE\\ &\textbf{where:}\\ &CAD = \text{Cash available for distribution}\\ &FFO = \text{Funds from operations}\\ &RCE = \text{Recurring capital expenditures} \end{aligned}
CAD=FFO−RCEwhere:CAD=Cash available for distributionFFO=Funds from operations
How to Calculate CAD
Calculating cash available for distribution is done by subtracting recurring capital expenditures from funds from operations. The formula and calculation for FFO appear below.
What Does Cash Available for Distribution Tell You?
A real estate investment trust (REIT) is a pooled investment vehicle that holds a portfolio of income-producing properties and/or mortgages and is required to distribute nearly all its taxable net income to maintain REIT status. In fact, REITs are required to pay out 90% of taxable income earned to investors. While there is no standardized method for calculating funds available for distribution, many REITs calculate CAD in a similar way by adjusting the funds from operations value for straight-line rents, non-cash items, and any recurring real estate-related expenses.
To income-oriented investors, cash available for distribution is a key metric to assess a REIT's strength. REITs can increase it organically or through an acquisition.
For REITs, there is no hard and fast rule about CADs and how it’s calculated. Thus, when the metric is calculated by a REIT, the calculation could vary from company to company. As a result, it is a non-GAAP measure and should be treated as pro-forma.
Example of Cash Available for Distribution
Boston Properties (BXP) is a commercial property REIT that owns buildings in Boston, New York, San Francisco, Los Angeles, Washington D.C., and Reston, Virginia. In 2020, the REIT's CAD payout ratio was 96.4% compared with 86.7% in 2019.
Boston Properties' financial statements indicate that it calculates CAD by adding to FFO lease transaction the costs that qualify as rent inducements, non-real estate depreciation, non-cash losses from early extinguishment of debt, and stock-based compensation expense; then eliminating the effects of straight-line rent and straight-line ground rent expense adjustment; and finally, subtracting maintenance capital expenditures, hotel improvements, and equipment upgrades and replacements. This list of cash flow adjustment items is not exhaustive, but it shows how cash and non-cash items are handled to present a more accurate figure of actual funds available for distribution to investors.
The Difference Between CAD and FFO
Cash available for distribution calculations does not adhere to a standardized formula in the REIT sector, but it is generally defined as the difference between FFO and recurring expenses. Recurring capital expenses that are typically subtracted from the FFO to determine the CAD value include replacing building roofs, HVAC system repairs, resurfacing of parking lots, and other significant routine maintenance. Some REITs may choose to deduct tenant improvements, straight-lining of rents, or leasing commissions from FFO.
The National Association Real Estate Investment Trusts (NAREIT), a trade group for the industry, defines FFO as net income plus depreciation less the gain on property sale plus loss on the property sale.
An expanded formula for FFO is:
\begin{aligned} &FFO = NI + DA - II + IE - GP + LP - IV + LV\\ &\textbf{where:}\\ ∋ = \text{ Net income}\\ &DA = \text{ Depreciation and amortization}\\ &II = \text{ Interest income}\\ &IE = \text{ Interest expense}\\ &GP = \text{ Gain on property sale}\\ &LP = \text{ Loss on property sale}\\ &IV = \text{ Income from unconsolidated ventures}\\ &LV = \text{ Loss from unconsolidated ventures} \end{aligned}
FFO=NI+DA−II+IE−GP+LP−IV+LVwhere:NI= Net incomeDA= Depreciation and amortizationII= Interest incomeIE= Interest expenseGP= Gain on property saleLP= Loss on property saleIV= Income from unconsolidated venturesLV= Loss from unconsolidated ventures
Boston Properties. "BXP Quarterly Investor Overview - Q4 2020," Pages 8 and 20. Accessed May 14, 2021.
Boston Properties. "BXP Quarterly Investor Overview - Q4 2020," Page 56. Accessed May 14, 2021.
Funds from operations, or FFO, refers to the figure used by real estate investment trusts (REITs) to define the cash flow from their operations.
Adjusted Funds From Operations—AFFO
Adjusted funds from operations is a financial performance measure primarily used in the analysis of real-estate income trusts (REITs).
Funds available for distribution is an internal, non-GAAP measure of the amount of capital that is on hand for REITS to pay to investors.
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Quasi-geostrophic equations, nonlinear Bernstein inequalities and $\alpha$-stable processes | EMS Press
Quasi-geostrophic equations, nonlinear Bernstein inequalities and
\alpha
We prove some functional inequalities for the fractional differentiation operator
(-\Delta)^\alpha
through the formalism of semi-groups. This gives us an estimate of the regularity of Marchand’s weak solutions for the dissipative quasi-geostrophic equation.
Diego Chamorro, Pierre Gilles Lemarié-Rieusset, Quasi-geostrophic equations, nonlinear Bernstein inequalities and
\alpha
-stable processes. Rev. Mat. Iberoam. 28 (2012), no. 4, pp. 1109–1122
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Differential-Functional Inequalities for Bounded Vector-Valued Functions | EMS Press
Differential-Functional Inequalities for Bounded Vector-Valued Functions
\mathbb R^n
ordered by a cone and some functions
f : \mathbb R^{n+mn} \to \mathbb R^n
h_1, ..., h_m : \mathbb R \to \mathbb R
we consider differential-functional inequalities of the type
v'' + cv' + f v,v(_1), ..., v(h_m) ≤ u'' + cu' + f u, u(h_1), ..., u(h_m)
and conclude
u ≤ v
under suitable conditions on
u, v, h_k
f
. The result can be applied to obtain existence and uniqueness results for differential-functional boundary value problems of the form
u'' + cu' + f u, u(h_1), ..., u(h_m) = q
u \in C^2 (\mathbb R, \mathbb R^n
bounded.
Gerd Herzog, Differential-Functional Inequalities for Bounded Vector-Valued Functions. Z. Anal. Anwend. 20 (2001), no. 4, pp. 1055–1063
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