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Synthesis and Chiral Separation of Fibratol, Isopropyl 2-(4-((4-chlorophenyl)(hydroxyl) methyl)-phenoxy)-2-methylpropanoate
Synthesis and Chiral Separation of Fibratol, Isopropyl 2-(4-((4-chlorophenyl)(hydroxyl) methyl)-phenoxy)-2-methylpropanoate
Amanda E. Kotheimer1, Wahajul Haq2, Ganesaratnam K. Balendiran1*
1Department of Chemistry, Youngstown State University, One University Plaza, Youngstown, OH, USA
2Medicinal Chemistry Division, Central Drug Research Institute, Lucknow, India
Practical synthetic route for the formation of enantiomeric mixture of Isopropyl 2-(4-((4-chlorophenyl)(hydroxyl)methyl)phenoxy)-2-methylpropanoate (Fibratol 2a/b) from isopropyl 2-(4-(4-chlorobenzoyl)phenoxy)-2-methylpropanoate (Fenofibrate 1) has been developed. Method has also been established for the chiral separation of enantiomers of Fibratol 2a/b that is synthesized using the route mentioned above. The optical activity determined for enantiomerically separated Fibratol (2a) and Fibratol (2b) are −5.2˚ and 8.0˚ which reflect their ability to rotate plane polarized light counterclockwise (levo) and clockwise (dextro), respectively.
Reduction, Chirality, Optical Activity, Fibrate
Rentsch [1] reported that about 56% of the synthetic drugs currently in use are chiral compounds. Though 88% of these chiral synthetic drugs are used therapeutically as racemates the recent trend in industry is to market the drugs in a pure enantiomeric form to give new life to old drugs for variety of reasons [1] . The leading single enantiomer blockbuster drugs (along with their corresponding medical/clinical application) are: Atorvastatin calcium (cardiovascular); Simvastatin (cardiovascular); Pravastatin sodium (cardiovascular); Paroxetine hydrochloride (CNS); Clopidogrel bisulfate (hematology); Sertraline hydrochloride (CNS); Fluticasone propionate and Salmeterol xinafoate (respiratory); Esomeprazole magnesium (gastrointestinal); Amoxicillin and Potassium clavulanate (antibiotic); and Valsartan (cardiovascular) [2] to mention a few.
Fenofibrate has shown inhibition properties towards Aldo-Keto reductase protein family members, Aldose Reductase and AKR1B10 recently [3] [4] . However, fibrates were previously believed to be ligands for the nuclear receptor, PPARα (peroxisome proliferator-activated receptor α) and are consequently used as therapeutic agents in the treatment of hyperlipidemia, heart disease and diabetic complications [5] [6] [7] . We describe herein methods to generate alcoholic/hydroxy derivatives of the above compounds, the enantiomeric separation and determine their chirality.
General Chemicals, Procedures and Instruments. The reaction was conducted at room temperature, unless otherwise noted [8] . Purification was accomplished by column chromatography, or high performance liquid chromatography. The 1H and 13C NMR were recorded on a Bruker Advance II 400 MHz NMR spectrometer with an indirect detection probe. Chemical shifts were reported in parts per million (ppm) from a standard of tetramethylsilane (TMS) in CDCl3 (0.1% w/v TMS). Signals in NMR spectra are defined as follows: s (singlet), d (doublet), t (triplet), m (multiplet), dd (doublet of doublets), pd (pseudo doublet), and all coupling constants (J) are labeled in Hertz. The mass spectra (MS) reported were obtained on a Bruker Esquire-HP LC/MS spectrometer in ESI+ detection mode. Samples were dissolved in methanol at a concentration of 1 mg/mL. Thin layer chromatography (TLC) was performed on oven dried Whatman aluminum-backed plates with varying eluent systems. Flash column chromatography was performed using oven dried 32 - 60 mesh 60-Å silica gel with varying eluent systems. A Perkin-Elmer 343 polarimeter was used to measure the optical rotation of all homogenous compounds. Infrared spectra were taken on a Thermo Electron Corporation IR 200 spectrophotometer and analyzed using EZ-OMNIC software.
2.1. Synthesis of Isopropyl 2-(4-((4-chlorophenyl)(hydroxyl) methyl)phenoxy)-2-methylpropanoate (Fibratol 2a/b)
To an oven-dried, 100 mL round bottomed flask, fitted with magnetic stir-bar, 0.425 g (1.18 mmol) of Fenofibrate (1) was added and then dissolved in 16 mL of methanol. When partially dissolved, an ice water bath was placed under the reaction flask and 0.093 g (2.46 mmol) of NaBH4 was added slowly in small portions. The ice water bath was removed and the solution was allowed to stir at room temperature (24˚C) for 3 hours. The reaction was monitored by TLC (7:3 ethyl acetate/hexane) and it resulted in two spots with corresponding Rf values of 0.73 and 0.77. The reaction mixture was placed in an ice water bath and 15 mL of chilled 5% HCl was added very slowly. The crude reaction mixture was poured onto 10 mL water, separated, extracted with ethyl acetate (2 × 15 mL) and dried over MgSO4. Using a Whatman #1 filter pad, the reaction mixture was filtered and the solvent was removed in vacuo resulting in a clear oil with 66.7% yield. Fibratol (2a/2b) has the following spectroscopic properties: 1H NMR: δ 1.21 (d, 6H, 3J= 5.76 Hz), 1.57 (s, 6H), 2.16 (s, 2H), 5.07 (septet, 1H, 3J = 6.31 Hz), 5.76 (s, 2H), 6.79 (pd (FF’ part of FF’HH’ pattern) 3J = 9.12 Hz, 2H), 7.19 (pd (GG’ part of GG’II’ pattern) 3J = 8.84 Hz, 2H), 7.29 (pd (HH’ part of HH’FF’ pattern and (pd (II’ part of II’GG’ pattern) 3J = 8.79 Hz, 4H); 13C NMR: δ 21.53, 25.39, 69.35, 79.47, 117.31, 128.55, 130.28, 131.16, 131.94, 136.47, 138.37, 159.77, 173.10, 194.25; ESI+ m/z (calculated by ACDLab2014): 362.8; m/z (experimental/found): 361.2 + 23 (385.2); IR spectrum (cm−1): 3466, 2985, 2939, 2876, 1904, 1728, 1606, 1505, 1465, 1381, 1096, 1012, 972, 827, 718, 684, 618.
2.2. Chiral Separation of Enantiomeric Products
High Performance liquid chromatography (HPLC) analyses were performed on a Waters 1525 Binary HPLC pump coupled to a Waters 2487 dual λ absorbance detector using Breeze software. All solvents were degassed for 1 hr under helium. A Lux Amylose-2 chiral column (5 µm, 4.6 × 250 mm) was selected for the separation of chiral compounds that had a unique selector amylose tris(5-chloro- 2-methylphenylcarbamate). Reduced racemic fenofibrate was separated by chiral separations techniques. A 5 mg of reaction product was dissolved in 1.5 mL of HPLC grade methanol. A 20 µL of this solution was injected by a 250 µL syringe at a flow rate of 1 mL/min and analyzed using HPLC by monitoring the flow through at 275 nm.
Specific rotation [α], reflects the magnitude of the rotation of each enantiomer. In addition the optical rotation is dependent on the wavelength of light used, temperature, solvent, concentration and the length of the polarimeter tube/cell. When monochromatic light from a sodium lamp D-line at 589 nm is used as the source, α can be defined by (Equation (1)) as
{\left[\alpha \right]}_{D}^{25}=\left\{\text{observedrotation}\left(\text{degrees}\right)/\text{lengthofsampletube}\left(dm\right)\times \text{concentration}\left(\frac{g}{ml}\right)\right\}
where T (=25˚C) is the measurement temperature, λ is the wavelength of light employed, α is the observed rotation, l is the path length and c is the concentration in grams per mL (the density of the pure substances) or grams per 100 mL. Specific rotation may also be expressed as degrees per mole of the substance where the conditions of measurement (i.e. solvent used, light source and path length) are also specified [9] .
Primary and secondary alcohols can be synthesized by reduction of the corresponding carbonyl compounds using a great variety of reagents [10] . Though ketones can be reduced by a wide variety of reagents, sodium borohydride (NaBH4) is considered to be the most useful one. The current reaction conditions (Scheme 1) were chosen because: 1) the reducing reagent, NaBH4, is inexpensive and readily available from commercial sources; 2) NaBH4 is more selective than any other hydride source because it reduces ketones but not carboxylic acids/esters; 3) the reaction can be maintained (tightly controlled) since it is conducted at room temperature (RT) and (iv) it is a one step reaction.
3.1. Synthesis of Fibratol (2a/2b)
The reduction (Scheme 1) of Fenofibrate (1), to Fibratol (2a/2b) was monitored via TLC (7:3 ethyl acetate/hexane v:v) until complete conversion of (1) to the final product (2a/2b) occur. Completion of the reaction after 3 hours is supported by 1) the shift of the -OH signal at 5.76 ppm in 1H NMR; 2) presence of an experimental m/z peak at 361.2, corresponding to the calculated m/z at 362.8 in MS; 3) appearance of an absorbance band at 3466 cm−1 in the IR spectra and 4) the disappearance of the signal at 194.0 ppm and appearance of 74.9 ppm peak in 13C NMR and further indicate the absence of the ketone carbonyl and the presence of OH group in the product.
3.2. Chiral Separation of Fibratol (2a/2b)
HPLC technique was selected for the enantiomeric separation of Fibratol (2a/b). The mobile phase 1 consisted of 500 mL of acetonitrile and 0.5 mL of diethylamine and mobile phase 2 comprised of 500 mL of isopropanol and 0.5 mL of diethylamine. For 20 µL sample size, the run time of 20 minutes at 275 nm with constant flow rate at 1 mL/min was used. Corresponding retention times for separation of each enantiomer are shown in Table 1 and Figure 1.
The difference in the retention time between peaks 2a and 2b increases from trial 1 to 4. Moreover trial 4 gave better resolved peaks for 2a and 2b compared to trial 1. Although there are minor differences in their retention times when varying the solvent systems, 60% of mobile phase 1 and 40% of mobile phase 2 yielded the best separation of the peaks.
Scheme 1. Formation of Fibratol (2a/b) from Fenofibrate (1).
Table 1. HPLC separation of racemic Fibratol (2a/2b) without any gradient but with varying amounts of % A (Pump A-mobile phase 1) and % B (Pump B-mobile phase 2).
Figure 1. Chiral separation of racemic Fibratol (2a/2b) without the use of any gradient method. Plot of Absorbance (AU) against Retention parameter (time) in min of the enantiomers are shown. Mobile phase 1 and 2 are delivered through Pump A and B, respectively.
The angle of rotation, α, of the plane of polarized light by an optically active forms of Fibratol (2a/2b) were measured by polarimetry technique. A 100 mg of Fibratol (2a) and Fibratol (2b) that were separated by chiral column were dissolved individually in 2 mL of methanol. The optical activity of Fibratol (2a) and Fibratol (2b) were −5.2˚ reflecting counterclockwise (levo) and 8.0˚ suggesting clockwise (dextro) rotation of plane polarized light, respectively. The trend demonstrated in resolving the chiral isomers of fibratol under otherwise identical condition (method, column, rate, pressure) reflects the amount of mobile phase 1 and 2 used in trials.
In conclusion, we have developed practical synthetic route to generate reduced derivatives of fenofibrate. This method is expected to be useful in converting ketone moiety of fenofibrate to its alcohol. This alcohol of fenofibrate will have chiral C atom. Molecular chirality is a fundamental consideration in drug discovery as it plays a critical role in therapeutics. Living organisms often display different biological responses to drug enantiomers when they are treated separately. Systematically it is customary for one enantiomer of a molecule to be biologically active while the other enantiomer to exist chemically toxic to the living systems. As a result, many pharmaceutical creations have taken advantage of this phenomenon and today the majority of drugs in the market are of a single-enantiomer.
This work is supported by National Institutes of Health Grant. We thank Ray Hoff for all the technical assistance.
Kotheimer, A.E., Haq, W. and Balendiran, G.K. (2018) Synthesis and Chiral Separation of Fibratol, Isopropyl 2-(4-((4-chlorophenyl)(hydroxyl) methyl)-phenoxy)-2-methylpropanoate. In- ternational Journal of Organic Chemistry, 8, 201-206. https://doi.org/10.4236/ijoc.2018.82015
1. Rentsch, K.M. (2002) The Importance of Stereoselective Determination of Drugs in the Clinical Laboratory. Journal of Biochemical and Biophysical Methods, 54, 1-9. https://doi.org/10.1016/S0165-022X(02)00124-0
2. Rouhi, A.M. (2003) Chiral Business. Chemical & Engineering News, 81, 45-55. https://doi.org/10.1021/cen-v081n018.p045
3. Balendiran, G.K. and Rajkumar, B. (2005) Fibrates Inhibit Aldose Reductase Activity in the Forward and Reverse Reactions. Biochemical Pharmacology, 70, 1653-1660. https://doi.org/10.1016/j.bcp.2005.06.029
4. Verma, M., Martin, H.-J., Haq, W., O’Connor, T.R., Maser, E. and Balendiran, G.K. (2008) Inhibiting Wild-Type and C299S Mutant AKR1B10; A Homologue of Aldose Reductase Upregulated in Cancers. European Journal of Pharmacology, 584, 213-221. https://doi.org/10.1016/j.ejphar.2008.01.036
5. Throp, J.M. (1962) Experimental Evaluation of an Orally Active Combination of Androsterone with Ethyl Chlorophenoxy-Isobutyrate. Lancet, 1, 1323-1326. https://doi.org/10.1016/S0140-6736(62)92423-6
6. Miller, D.B. and Spence, J.D. (1998) Clinical Pharmacokinetics of Fibric Acid Derivatives (Fibrates). Clinical Pharmacokinetics, 34, 155-162. https://doi.org/10.2165/00003088-199834020-00003
7. Forcheron, F., Cachefo, A., Thevenon, S., Pinteur, C. and Beylot, M. (2002) Mechanisms of the Triglyceride- and Cholesterol-Lowering Effect of Fenofibrate in Hyperlipidemic Type 2 Diabetic Patients. Diabetes, 51, 3486-3491. https://doi.org/10.2337/diabetes.51.12.3486
8. Kotheimer, A. (2010) Design, Synthesis and Characterization of Novel Chiral Inhibitors for Aldose Reductase. MSc Thesis, Youngstown State University, Youngstown.
9. Sheldon, R.A. (1993) Chirotechnology: Industrial Synthesis of Optically Active Compounds. Marcel Dekker, New York.
10. Norman, R.O.C. (1978) Principles of Organic Synthesis. Chapman and Hall Ltd., London. https://doi.org/10.1007/978-1-4899-3021-7
Central nervous system (CNS),
Coupling constants (J),
Doublet (d),
Doublet of doublets (dd),
Mass spectra (MS),
Multiplet (m),
Parts per million (ppm),
Pseudo doublet (pd),
Room temperature (RT),
Singlet (s),
Thin layer chromatography (TLC),
Tertramethylsilane (TMS),
Triplet (t).
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§ Mnemonics For Symmetric Polynomials
§ Some notation for partitions
\lambda \equiv (\lambda_1, \lambda_2, \dots \lambda_l)
of a partition of
N
L_0
norm of the partition will be
1 + 1 + \dots 1
l
times), which is equal to
N
|\lambda|_0 = l
L_0
norm of a partition is the number of parts of the partition.
L_1
|\lambda_1| + |\lambda_2| + \dots + |\lambda_l|
N
L_1
norm of a partition is the number it is partitoining. Thus,
|\lambda|_1 = N
§ Elementary Symmetric Polynomials (integer)
We need to define
e_k(\vec r)
k \in \mathbb N
r \in X^d
a sequence of variables (
r
for "roots").
These were elementary for Newton/Galois, and so has to do with the structure of roots.
e_k(\vec r)
is the coefficients of the "root polynomial"
(x - \vec r)
\begin{aligned} &(x+r_1)(x+r_2)(x+r_3) = 1x^3 + (r_1 + r_2 + r_3) x^2 + (r_1r_2 + r_2r_3 + r_1r_3) x + r_1r_2r_3 \cdot x^0 \\ &e_0 = 1 \\ &e_1 = r_1 + r_2 + r_3 \\ &e_2 = r_1 r_2 + r_2 r_3 + r_1 r_3 \\ &e_3 = r_1 r_2 r_3 \\ \end{aligned}
Formally, we define
e_k(\vec r)
to be the product of all terms
(r_a r_b\dots, r_k)
for distinct numbers
(a, b, \dots, k) \in [1, n]
\begin{aligned} e_k(\vec r) \equiv \sum_{1 \leq a < b < \dots k \leq n} r_a r_b \dots r_k \end{aligned}
§ Elementary Symmetric Polynomials (partition)
\vec \lambda \equiv (\lambda_1, \lambda_2, \dots, \lambda_l)
, the elementary symmetric polynomial
e_\lambda
is the product of the elementary symmetric polynomial
e_{\lambda_1} \cdot e_{\lambda_2} \dots e_{\lambda_l}
§ Monomial Symmetric Polynomials (partition)
We symmetrize the monomial dictated by the partition. To calculate
m_\lambda(\vec r)
\vec r^\lambda \equiv r_1^{\lambda_1} r_2^{\lambda_2} \dots r_l^{\lambda_l}
, and then symmetrize the above monomial.
m_{(3, 1, 1)}(r_1, r_2, r_3)
is given by symmetrizing
r_1^3 r_2^1 r_3^1
. So we must add the terms
r_1 r_2^3 r_3
r_1 r_2 r_3^3
m_{(3, 1, 1)}(r_1, r_2, r_3) \equiv r_1^3 r_2 r_3 + r_1 r_2^3 r_3 + r_1 r_2 r_3^3
§ Power Sum Symmetric Polynomials (number)
It's all in the name: take a sum of powers.
Alternatively, take a power and symmetrize it.
P_k(\vec r) \equiv r_1^k + r_2^k + \dots + r_n^k
§ Power Sum Symmetric Polynomials (partition)
Extend to partitions by taking product of power sets of numbers.
P_\lambda(\vec r) \equiv P_{\lambda_1}(\vec r) + P_{\lambda 2}(\vec r) + \dots + P_{\lambda_l}(\vec r)
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Determine the corresponding outputs for the given inputs of the following functions. If there is no solution, explain why not. Be careful: In some cases, there may be no solution or more than one possible solution.
Substitute 8 into the equation for
x
f(8) = \left|8\right| = ?
|
Multiple Choice: Which of the following expressions is the product of
\left(4y-3x\right)\left(2y+x\right)
8y^2-2xy-3x^2
6y^2-2xy-2x^2
8y^2+10xy-3x^2
6y^2-2x
Use a generic rectangle to multiply the expression together. Draw a rectangle and divide it into four equal parts.
Multiply each row by each column. Add the products together to get the final product. Added to the rectangle: Exterior left edge top is negative 3, X. Exterior left edge bottom is 4, y. Exterior bottom edge left is 2, y. Exterior bottom edge right is x.
(4y−3x)(2y+x)=8y^2−2xy−3x^2
Added to the rectangle: Interior upper left box is negative 6, x, y. Interior upper right box is negative 3, X squared. Interior lower left box is negative 8, y squared. Interior lower right box is 4, X, Y.
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Lrank - Maple Help
Home : Support : Online Help : Mathematics : Differential Equations : DEtools : Lie Symmetry Method : liesymm : Lrank
the Lie Rank of a set of forms
Lrank(form)
It removes forms which are redundant with respect to the generation of the determining equations.
\mathrm{with}\left(\mathrm{liesymm}\right):
\mathrm{setup}\left(\right)
[]
\mathrm{eq}≔\mathrm{Diff}\left(u\left(x,t\right),x,t\right)+\mathrm{Diff}\left(u\left(x,t\right),x\right)+{u\left(x,t\right)}^{2}=0
\textcolor[rgb]{0,0,1}{\mathrm{eq}}\textcolor[rgb]{0,0,1}{≔}\frac{{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}}^{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}\textcolor[rgb]{0,0,1}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{t}\right)\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}}{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}\textcolor[rgb]{0,0,1}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{t}\right)\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{t}\right)}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}
\mathrm{forms}≔\mathrm{makeforms}\left(\mathrm{eq},u\left(x,t\right),w\right)
\textcolor[rgb]{0,0,1}{\mathrm{forms}}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{u}\right)\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{w1}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{w2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{w2}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0,0,1}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)\textcolor[rgb]{0,0,1}{+}\left({\textcolor[rgb]{0,0,1}{u}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{w1}}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0,0,1}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)]
\mathrm{forms}≔\mathrm{close}\left(\mathrm{forms}\right)
\textcolor[rgb]{0,0,1}{\mathrm{forms}}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{u}\right)\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{w1}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{w2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{w2}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0,0,1}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)\textcolor[rgb]{0,0,1}{+}\left({\textcolor[rgb]{0,0,1}{u}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{w1}}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0,0,1}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{w1}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0,0,1}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{w2}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0,0,1}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)]
\mathrm{Lrank}\left(\mathrm{forms}\right)
[\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{u}\right)\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{w1}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{w2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{w2}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0,0,1}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)\textcolor[rgb]{0,0,1}{+}\left({\textcolor[rgb]{0,0,1}{u}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{w1}}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0,0,1}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)]
liesymm[wedge]
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The geometry of right-angled Artin subgroups of mapping class groups | EMS Press
We describe sufficient conditions which guarantee that a finite set of mapping classes generate a right-angled Artin group quasi-isometrically embedded in the mapping class group. Moreover, under these conditions, the orbit map to Teichmüller space is a quasi-isometric embedding for both of the standard metrics. As a consequence, we produce infinitely many genus
h
surfaces (for any
h
at least 2) in the moduli space of genus
g
g
at least 3) for which the universal covers are quasi-isometrically embedded in the Teichmüller space.
Matt T. Clay, Christopher J. Leininger, Johanna Mangahas, The geometry of right-angled Artin subgroups of mapping class groups. Groups Geom. Dyn. 6 (2012), no. 2, pp. 249–278
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Tags: machine learning notes all
Notes on some machine learning related topics
Getting started in machine learning sometimes feels like drinking water from a firehose (pardon my cliché). The topic has so many academic roots in a lot of different disciplines (bayesian statistics, optimization, and information theory - oh my!) so I decided to keep a personal glossary of various machine learning concepts (namely relating to neural networks and natural language processing) for my own benefit. These notes might not be complete or accurate - if you would like to see an idea to be written about here feel free to shoot me an email!
Variational Inference consists of framing inference as an optimization process. For example when we are working with an intractable probability distribution
p
, variational inference has significant gains over Markov-Chain Monte-Carlo estimation.
For data
x
and latent variable
z
p(z)
Likelihood:
p(x | z)
Posterior:
p(z | x)
As a result, we approximate the conditional density of latent variables given observed variables by using optimization methods. For a distribution
p
, we can approximate it with our own distribution
q
such that we minimize the KL-Divergence between the two distributions:
KL(q || p) = \sum_x {~ q(x) \cdot \operatorname{log} ~ \frac{q(x)}{p(x)}}
We can re-write the KL-Divergence between
q
p(z | x)
KL[q(z) ~||~ p(z | x)] = \mathop{\mathbb{E}}~[\operatorname{log} q(z)] - \mathop{\mathbb{E}}~[\operatorname{log} p(z | x)]
Then, we can expand the conditional term (and apply the logarithm identities) to get:
KL[q(z) ~||~ p(z | x)] = \mathop{\mathbb{E}}~[\operatorname{log} q(z)] - \mathop{\mathbb{E}}~[\operatorname{log} p(z, x)] + \operatorname{log} p(x)
\operatorname{log} p(x)
term makes computing the KL-Divergence intractable, since we assumed
p(x)
to be intractable. However, we can give a lower bound on this quantity since the KL-Divergence is always at least 0:
\operatorname{log} p(x) \geq \mathop{\mathbb{E}}~[\operatorname{log} p(z, x)] - \mathop{\mathbb{E}}~[\operatorname{log} q(z)]
We define the L.H.S to be ELBO: evidence lower bound. This is equivalent to the negative KL-Divergence plus
\operatorname{log} p(x)
. The nice thing about this is that
\operatorname{log} p(x)
is a constant with respect to distribution
q
, so we can minimize the KL-Divergence by maximizing ELBO, without calculating
p(x)
Autoencoders are models that consist of an encoder-model architecture where the encoder takes data and encodes it into a latent representation and the decoder takes a latent representation and approximates/re-generates the original data. The goal is to learn latent representations (posterior inference over
z
), as well as learn generation from latent spaces (marginal inference over
x
Autoencoders can be modelled using neural networks for both the encoder and decoder mechanisms. However, this can give us a lack of regularity in the latent space (i.e. non-continuous latent space) that makes generation hard for the decoder. We solve this using a variational autoencoder, which is an autoencoder that we regularize training for, not only so that we don't overfit but mainly so that the latent space is suitable for generation. We do this by encoding the autoencoder's input as a probability distribution.
In order to train our VAE, we must use backpropogation to compute the gradient of ELBO. However, since the network's nodes represent a stochastic process, we instead model stochastic neurons as having parameters
\sigma
\mu
that allows us propogate errors meaningfully throughout the network. This is know as the re-parameterization trick.
The Maximum Likelihood Estimation for an n-gram can be given by the formula:
P(w_t | w_{t - 1}) = \frac{c(w_{t - 1}, w_t)}{c(w_{t - 1})}
c(w)
is the frequency of
w
in the corpus. Sequence generation can be performed by sampling from this distribution.
Perplexity is an intrinsic evaluation method for language models that captures information about the entropy within the test set. Perplexity for a given test can can be computed as:
PP(W) = P(w_1, w_2, ... w_n)^{-\frac{1}{n}} = \sqrt[n]{\frac{1}{\Pi_{i = 1}^n P(w_i|w_1, w_2 ... w_{i - 1})}}
This estimate can be approximated using the Markov Assumption (for a bi-gram model):
PP(W) = \sqrt[n]{\frac{1}{\Pi_{i = 1}^n P(w_i|w_{i - 1})}}
Perplexity can be thought of as the weighted average branching factor of a language, and generally lower perplexity is better.
Word Embeddings are vectors in some space
\mathbb{R}^{d}
such that they encode lexical semantics. For example, the vectors of cat and kitten will have a small vector distance whereas the vectors of cat and chair will be far apart.
u
v
\operatorname{cos}(u, v) = \frac{u \cdot v}{|u| \cdot |v|} = \frac{\sum_{i = 1}^{n} u_i v_i}{\sqrt{\sum_{i = 1}^{n} u^2_i} \cdot \sqrt{\sum_{i = 1}^{n} v^2_i}}
Where the range is
[0, 1]
and we can define distance as the complement
1 - \operatorname{cos}(u, v)
Recurrent Neural Networks are like vanilla feed-forward networks, except they contain cycles, which allow the network to process sequential data. RNNs do this by mantaining a hidden state
h \in \mathbb{R}^{d}
that is updated at time-step
t
, and is later fed back into the network along with the network's previous output at time-step
t + 1
. The hidden state
h
lets the network maintain context while processing the sequence.
Sometimes too much context can be a burden for the network, and results in the vanishing gradient problem where errors are propogated too far and tend to zero. This problem is resolved with models that manage context better, namely selectively remembering and forgetting parts of the context. Examples of these models are LSTMs (Long Short Term Memory) and GRUs (Gated Recurrent Units).
Given some peice of text, certain words are more important than others and we want our neural network to understand their relative importances accordingly.
Markov Processes are systems that are rooted in the Markov Assumption which states that given sequential events
X_{n - 1}
X_{n - 2}
X_0
P(X_n | X_{n - 1}, X_{n - 2} ... X_0) = P(X_n | X_{n - 1})
In other words, our process only depends on the previous state and is memory-less.
A Markov Matrix
A
is a stochastic matrix, which means that the columns of
A
are probability vectors that model some distribution. In other words, the columns of
A
sum to 1 and obey the axiom of probability that each entry is non-negative. The reason that these stochastic matrices are called Markov Matrices is because
A
doesn't change with respect to time. In other words, we have that at time
t
, the probability distribution (across the states represented by the vector) is
A^tu_0
u_0
is the distribution at time
0
. This is nice because we can easily compute the exponentiation of matrices using diagonalization.
The steady state distribution
u_\infty
A
A^tu_0
tends to
\infty
Au_\infty = u_\infty
u_\infty
A
that corresponds to an eigenvalue of 1. In fact, the largest eigenvalue
\lambda
A
can have is 1.
A
be a square matrix with all non-negative values, with an eigenvalue
\lambda
|\lambda|
is maximized. Then, 1) we have that
|\lambda|
A
with a positive eigenvector and 2) the algebric and geometric multiplicity of
|\lambda|
A
be a real, symmetric matrix such that
A^T = A
. Then, we have that 1) all the eigenvalues of
A
are real and 2) there exists an orthonormal basis of eigenvectors for
A
https://people.scs.carleton.ca/~maheshwa/courses/3801/Projects17/PF-thm-report.pdf
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Maximal_ideal Knowpia
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all proper ideals.[1][2] In other words, I is a maximal ideal of a ring R if there are no other ideals contained between I and R.
Maximal ideals are important because the quotients of rings by maximal ideals are simple rings, and in the special case of unital commutative rings they are also fields.
In noncommutative ring theory, a maximal right ideal is defined analogously as being a maximal element in the poset of proper right ideals, and similarly, a maximal left ideal is defined to be a maximal element of the poset of proper left ideals. Since a one sided maximal ideal A is not necessarily two-sided, the quotient R/A is not necessarily a ring, but it is a simple module over R. If R has a unique maximal right ideal, then R is known as a local ring, and the maximal right ideal is also the unique maximal left and unique maximal two-sided ideal of the ring, and is in fact the Jacobson radical J(R).
It is possible for a ring to have a unique maximal two-sided ideal and yet lack unique maximal one sided ideals: for example, in the ring of 2 by 2 square matrices over a field, the zero ideal is a maximal two-sided ideal, but there are many maximal right ideals.
There are other equivalent ways of expressing the definition of maximal one-sided and maximal two-sided ideals. Given a ring R and a proper ideal I of R (that is I ≠ R), I is a maximal ideal of R if any of the following equivalent conditions hold:
There exists no other proper ideal J of R so that I ⊊ J.
For any ideal J with I ⊆ J, either J = I or J = R.
The quotient ring R/I is a simple ring.
There is an analogous list for one-sided ideals, for which only the right-hand versions will be given. For a right ideal A of a ring R, the following conditions are equivalent to A being a maximal right ideal of R:
There exists no other proper right ideal B of R so that A ⊊ B.
For any right ideal B with A ⊆ B, either B = A or B = R.
The quotient module R/A is a simple right R-module.
Maximal right/left/two-sided ideals are the dual notion to that of minimal ideals.
If F is a field, then the only maximal ideal is {0}.
In the ring Z of integers, the maximal ideals are the principal ideals generated by a prime number.
More generally, all nonzero prime ideals are maximal in a principal ideal domain.
{\displaystyle (2,x)}
is a maximal ideal in ring
{\displaystyle \mathbb {Z} [x]}
. Generally, the maximal ideals of
{\displaystyle \mathbb {Z} [x]}
are of the form
{\displaystyle (p,f(x))}
{\displaystyle p}
{\displaystyle f(x)}
{\displaystyle \mathbb {Z} [x]}
which is irreducible modulo
{\displaystyle p}
Every prime ideal is a maximal ideal in a Boolean ring, i.e., a ring consisting of only idempotent elements. In fact, every prime ideal is maximal in a commutative ring
{\displaystyle R}
whenever there exists an integer
{\displaystyle n>1}
{\displaystyle x^{n}=x}
{\displaystyle x\in R}
The maximal ideals of the polynomial ring
{\displaystyle \mathbb {C} [x]}
are principal ideals generated by
{\displaystyle x-c}
{\displaystyle c\in \mathbb {C} }
More generally, the maximal ideals of the polynomial ring K[x1, ..., xn] over an algebraically closed field K are the ideals of the form (x1 − a1, ..., xn − an). This result is known as the weak Nullstellensatz.
An important ideal of the ring called the Jacobson radical can be defined using maximal right (or maximal left) ideals.
If R is a unital commutative ring with an ideal m, then k = R/m is a field if and only if m is a maximal ideal. In that case, R/m is known as the residue field. This fact can fail in non-unital rings. For example,
{\displaystyle 4\mathbb {Z} }
is a maximal ideal in
{\displaystyle 2\mathbb {Z} }
{\displaystyle 2\mathbb {Z} /4\mathbb {Z} }
is not a field.
If L is a maximal left ideal, then R/L is a simple left R-module. Conversely in rings with unity, any simple left R-module arises this way. Incidentally this shows that a collection of representatives of simple left R-modules is actually a set since it can be put into correspondence with part of the set of maximal left ideals of R.
Krull's theorem (1929): Every nonzero unital ring has a maximal ideal. The result is also true if "ideal" is replaced with "right ideal" or "left ideal". More generally, it is true that every nonzero finitely generated module has a maximal submodule. Suppose I is an ideal which is not R (respectively, A is a right ideal which is not R). Then R/I is a ring with unity (respectively, R/A is a finitely generated module), and so the above theorems can be applied to the quotient to conclude that there is a maximal ideal (respectively, maximal right ideal) of R containing I (respectively, A).
Krull's theorem can fail for rings without unity. A radical ring, i.e. a ring in which the Jacobson radical is the entire ring, has no simple modules and hence has no maximal right or left ideals. See regular ideals for possible ways to circumvent this problem.
In a commutative ring with unity, every maximal ideal is a prime ideal. The converse is not always true: for example, in any nonfield integral domain the zero ideal is a prime ideal which is not maximal. Commutative rings in which prime ideals are maximal are known as zero-dimensional rings, where the dimension used is the Krull dimension.
A maximal ideal of a noncommutative ring might not be prime in the commutative sense. For example, let
{\displaystyle M_{n\times n}(\mathbb {Z} )}
be the ring of all
{\displaystyle n\times n}
{\displaystyle \mathbb {Z} }
. This ring has a maximal ideal
{\displaystyle M_{n\times n}(p\mathbb {Z} )}
{\displaystyle p}
, but this is not a prime ideal since (in the case
{\displaystyle n=2}
{\displaystyle A={\text{diag}}(1,p)}
{\displaystyle B={\text{diag}}(p,1)}
{\displaystyle M_{n\times n}(p\mathbb {Z} )}
{\displaystyle AB=pI_{2}\in M_{n\times n}(p\mathbb {Z} )}
. However, maximal ideals of noncommutative rings are prime in the generalized sense below.
For an R-module A, a maximal submodule M of A is a submodule M ≠ A satisfying the property that for any other submodule N, M ⊆ N ⊆ A implies N = M or N = A. Equivalently, M is a maximal submodule if and only if the quotient module A/M is a simple module. The maximal right ideals of a ring R are exactly the maximal submodules of the module RR.
Unlike rings with unity, a nonzero module does not necessarily have maximal submodules. However, as noted above, finitely generated nonzero modules have maximal submodules, and also projective modules have maximal submodules.
As with rings, one can define the radical of a module using maximal submodules. Furthermore, maximal ideals can be generalized by defining a maximal sub-bimodule M of a bimodule B to be a proper sub-bimodule of M which is contained in no other proper sub-bimodule of M. The maximal ideals of R are then exactly the maximal sub-bimodules of the bimodule RRR.
Anderson, Frank W.; Fuller, Kent R. (1992), Rings and categories of modules, Graduate Texts in Mathematics, vol. 13 (2 ed.), New York: Springer-Verlag, pp. x+376, doi:10.1007/978-1-4612-4418-9, ISBN 0-387-97845-3, MR 1245487
Lam, T. Y. (2001), A first course in noncommutative rings, Graduate Texts in Mathematics, vol. 131 (2 ed.), New York: Springer-Verlag, pp. xx+385, doi:10.1007/978-1-4419-8616-0, ISBN 0-387-95183-0, MR 1838439
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Bézier surface - Wikipedia
(Redirected from Bezier surface)
It has been suggested that Biharmonic Bézier surface be merged into this article. (Discuss) Proposed since November 2021.
Bézier surfaces are a species of mathematical spline used in computer graphics, computer-aided design, and finite element modeling. As with Bézier curves, a Bézier surface is defined by a set of control points. Similar to interpolation in many respects, a key difference is that the surface does not, in general, pass through the central control points; rather, it is "stretched" toward them as though each were an attractive force. They are visually intuitive, and for many applications, mathematically convenient.
3 Bézier surfaces in computer graphics
Sample Bézier surface; red – control points, blue – control grid, black – surface approximation
A given Bézier surface of degree (n, m) is defined by a set of (n + 1)(m + 1) control points ki,j where i = 0, ..., n and j = 0, ..., m. It maps the unit square into a smooth-continuous surface embedded within the space containing the ki,j s – for example, if the ki,j s are all points in a four-dimensional space, then the surface will be within a four-dimensional space.
A two-dimensional Bézier surface can be defined as a parametric surface where the position of a point p as a function of the parametric coordinates u, v is given by:[1]
{\displaystyle \mathbf {p} (u,v)=\sum _{i=0}^{n}\sum _{j=0}^{m}B_{i}^{n}(u)\,B_{j}^{m}(v)\,\mathbf {k} _{i,j}}
evaluated over the unit square, where
{\displaystyle B_{i}^{n}(u)={n \choose i}u^{i}(1-u)^{n-i}}
is a basis Bernstein polynomial, and
{\displaystyle {n \choose i}={\frac {n!}{i!(n-i)!}}}
Some properties of Bézier surfaces:
A Bézier surface will transform in the same way as its control points under all linear transformations and translations.
All u = constant and v = constant lines in the (u, v) space, and – in particular – all four edges of the deformed (u, v) unit square are Bézier curves.
A Bézier surface will lie completely within the convex hull of its control points, and therefore also completely within the bounding box of its control points in any given Cartesian coordinate system.
The points in the patch corresponding to the corners of the deformed unit square coincide with four of the control points.
However, a Bézier surface does not generally pass through its other control points.
Generally, the most common use of Bézier surfaces is as nets of bicubic patches (where m = n = 3). The geometry of a single bicubic patch is thus completely defined by a set of 16 control points. These are typically linked up to form a B-spline surface in a similar way as Bézier curves are linked up to form a B-spline curve.
Simpler Bézier surfaces are formed from biquadratic patches (m = n = 2), or Bézier triangles.
Bézier surfaces in computer graphics[edit]
Ed Catmull's "Gumbo" model, composed from patches
Bézier patch meshes are superior to triangle meshes as a representation of smooth surfaces. They require fewer points (and thus less memory) to represent curved surfaces, are easier to manipulate, and have much better continuity properties. In addition, other common parametric surfaces such as spheres and cylinders can be well approximated by relatively small numbers of cubic Bézier patches.
However, Bézier patch meshes are difficult to render directly. One problem with Bézier patches is that calculating their intersections with lines is difficult, making them awkward for pure ray tracing or other direct geometric techniques which do not use subdivision or successive approximation techniques. They are also difficult to combine directly with perspective projection algorithms.
For this reason, Bézier patch meshes are in general eventually decomposed into meshes of flat triangles by 3D rendering pipelines. In high-quality rendering, the subdivision is adjusted to be so fine that the individual triangle boundaries cannot be seen. To avoid a "blobby" look, fine detail is usually applied to Bézier surfaces at this stage using texture maps, bump maps and other pixel shader techniques.
A Bézier patch of degree (m, n) may be constructed out of two Bézier triangles of degree m + n, or out of a single Bézier triangle of degree m + n, with the input domain as a square instead of a triangle.
A Bézier triangle of degree m may also be constructed out of a Bézier surface of degree (m, m), with the control points so that one edge is squashed to a point, or with the input domain as a triangle instead of a square.
Bézier triangle
Visualisation of Bezier Surface with code
^ Farin, Gerald (2002). Curves and Surfaces for CAGD (5th ed.). Academic Press. ISBN 1-55860-737-4.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Bézier_surface&oldid=1059250603"
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1
{a}_{1}^{{a}_{2}^{{\phantom{\rule[-0.0ex]{0.5ex}{0.0ex}}}^{{⋰}^{{a}_{n}}}}}
as an iterated power of ordinals and non-negative integers
{a}_{1},{a}_{2},\dots ,{a}_{n}
a=1\phantom{\rule[-0.0ex]{0.5ex}{0.0ex}}⇔\phantom{\rule[-0.0ex]{0.5ex}{0.0ex}}n=0
a=0
n=1
{a}_{1}=0
a\ge 2
{a}_{k}
\ge 2
{a}_{k}\ge 2
{b}^{c}
b,c\ge 2
{a}_{k}
k=n=1
a={a}_{k}=\mathbf{\omega }
{a}_{k}
a
i∈\left\{1,\dots ,n\right\}
{a}_{i}
{a}_{i+1},\dots ,{a}_{n}
\ge 2
i\ge 2
{a}_{1}=\phantom{\rule[-0.0ex]{0.5ex}{0.0ex}}⋯\phantom{\rule[-0.0ex]{0.5ex}{0.0ex}}={a}_{i-2}=2
\mathrm{degree}\left({a}_{i}\right)\ge 1>0=\mathrm{tdegree}\left({a}_{i}\right)
{a}_{i-1}\ge 2
\mathrm{value}\left(\mathrm{Decompose}\left(a,\mathrm{output}=\mathrm{inert}\right)\right)=a
a
\mathrm{with}\left(\mathrm{Ordinals}\right)
[\textcolor[rgb]{0,0,1}{\mathrm{`+`}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{`.`}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{`<`}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{<=}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Add}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Base}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Dec}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Decompose}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Div}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Eval}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Factor}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Gcd}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Lcm}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{LessThan}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Log}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Max}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Min}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Mult}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Ordinal}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Power}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Split}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Sub}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{`^`}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{degree}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{lcoeff}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{log}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{lterm}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{\omega }}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{quo}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{rem}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{tcoeff}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{tdegree}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{tterm}}]
\mathrm{Decompose}\left({\mathrm{\omega }}^{\mathrm{\omega }}\right)
[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}]
\mathrm{Decompose}\left({\mathrm{\omega }}^{\mathrm{\omega }},\mathrm{output}=\mathrm{inert}\right)
{\left(\textcolor[rgb]{0,0,1}{2}\right)}^{{\left(\textcolor[rgb]{0,0,1}{2}\right)}^{{\left(\textcolor[rgb]{0,0,1}{2}\right)}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}}}}
\mathrm{value}\left(\right)
{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}
\mathrm{Decompose}\left({\mathrm{\omega }}^{\mathrm{\omega }+1}\right)
[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}]
\ne \mathbf{\omega }
\mathrm{Decompose}\left(\mathrm{\omega }·3\right)
[\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}]
\mathrm{Decompose}\left({\mathrm{\omega }}^{3}\right)
[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}]
\mathrm{Decompose}\left({\mathrm{\omega }}^{2}·3\right)
[\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}]
\mathrm{Decompose}\left({\mathrm{\omega }}^{\mathrm{\omega }+1}·2,\mathrm{output}=\mathrm{inert}\right)
{\left(\textcolor[rgb]{0,0,1}{2}\right)}^{{\left(\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)}^{\textcolor[rgb]{0,0,1}{2}}}
\mathrm{value}\left(\right)
{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}
\mathbf{\omega }
2\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&ˆ\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{\omega }={2}^{\mathrm{\omega }}
{\left(\textcolor[rgb]{0,0,1}{2}\right)}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}
\mathrm{Decompose}\left(\mathrm{\omega }\right)
[\textcolor[rgb]{0,0,1}{\mathbf{\omega }}]
b≔{\mathrm{\omega }}^{2}+\mathrm{\omega }
\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}
a≔{\mathrm{\omega }}^{b+2}+{\mathrm{\omega }}^{b+1}
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}}
\mathrm{Decompose}\left(a,\mathrm{output}=\mathrm{inert}\right)
{\left({\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\right)}^{{\left(\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)}^{\textcolor[rgb]{0,0,1}{2}}}
c≔a+{\mathrm{\omega }}^{b}·3
\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}
\mathrm{Decompose}\left(c,\mathrm{output}=\mathrm{inert}\right)
{\left({\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}\right)}^{{\left(\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)}^{\textcolor[rgb]{0,0,1}{2}}}
p≔{\left(\mathrm{\omega }+2\right)}^{2}
\textcolor[rgb]{0,0,1}{p}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}
\mathrm{Decompose}\left(p,\mathrm{output}=\mathrm{inert}\right)
{\left(\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\right)}^{\textcolor[rgb]{0,0,1}{2}}
q≔{\left(\mathrm{\omega }+3\right)}^{p}
\textcolor[rgb]{0,0,1}{q}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}
\mathrm{Decompose}\left(q,\mathrm{output}=\mathrm{inert}\right)
{\left(\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}\right)}^{{\left(\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\right)}^{\textcolor[rgb]{0,0,1}{2}}}
r≔{\left(\mathrm{\omega }+5\right)}^{q}
\textcolor[rgb]{0,0,1}{r}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}}
\mathrm{Decompose}\left(r,\mathrm{output}=\mathrm{inert}\right)
{\left(\textcolor[rgb]{0,0,1}{2}\right)}^{{\left(\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}\right)}^{{\left(\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\right)}^{\textcolor[rgb]{0,0,1}{2}}}}
f≔\mathrm{Ordinal}\left([[8,1],[7,2],[6,3],[5,2],[4,3],[3,2],[2,3],[1,2],[0,3]]\right)
\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{8}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{7}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{6}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}
\mathrm{Decompose}\left(f,\mathrm{output}=\mathrm{inert}\right)
{\left({\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}\right)}^{{\left(\textcolor[rgb]{0,0,1}{2}\right)}^{\textcolor[rgb]{0,0,1}{2}}}
\mathrm{Decompose}\left(2417851639229258349412352,\mathrm{output}=\mathrm{inert}\right)
{\left(\textcolor[rgb]{0,0,1}{2}\right)}^{{\left(\textcolor[rgb]{0,0,1}{3}\right)}^{{\left(\textcolor[rgb]{0,0,1}{2}\right)}^{\textcolor[rgb]{0,0,1}{2}}}}
u≔\mathrm{Ordinal}\left([[2,x],[1,3x],[0,3]]\right)
\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\left(\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}
\mathrm{Decompose}\left(u\right)
\mathrm{Decompose}\left(\mathrm{Eval}\left(u,x=x+1\right),\mathrm{output}=\mathrm{inert}\right)
{\left(\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\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}{3}\right)}^{\textcolor[rgb]{0,0,1}{2}}
v≔\mathrm{Ordinal}\left([[\mathrm{\omega }+3,1],[\mathrm{\omega }+2,x],[\mathrm{\omega }+1,1]]\right)
\textcolor[rgb]{0,0,1}{v}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}}
\mathrm{Decompose}\left(v,\mathrm{output}=\mathrm{inert}\right)
{\left({\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\right)}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}}
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Random Numbers Within a Sphere - MATLAB & Simulink - MathWorks Switzerland
This example shows how to create random points within the volume of a sphere, as described by Knuth [1]. The sphere in this example is centered at the origin and has a radius of 3.
One way to create points inside a sphere is to specify them in spherical coordinates. Then you can convert them to Cartesian coordinates to plot them.
Calculate an elevation angle for each point in the sphere. These values are in the open interval,
\left(-\pi /2,\pi /2\right)
, but are not uniformly distributed.
Create an azimuth angle for each point in the sphere. These values are uniformly distributed in the open interval,
\left(0,2\pi \right)
Create a radius value for each point in the sphere. These values are in the open interval,
\left(0,3\right)
Convert to Cartesian coordinates and plot the result.
If you want to place random numbers on the surface of the sphere, then specify a constant radius value to be the last input argument to sph2cart. In this case, the value is 3.
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Imitate Nonlinear MPC Controller for Flying Robot - MATLAB & Simulink - MathWorks India
Design Nonlinear MPC Controller
Behavior Cloning Approach
Data Aggregation Approach
Compare Trained DAgger Network with NLMPC Controller
Animate the Flying Robot with Trained DAgger Network
This example shows how to train, validate, and test a deep neural network (DNN) that imitates the behavior of a nonlinear model predictive controller for a flying robot. It then compares the behavior of the deep neural network with that of the original controller. To train the deep neural network, this example uses the data aggregation (DAgger) approach as in [1].
Nonlinear model predictive control (NLMPC) solves a constrained nonlinear optimization problem in real time based on the current state of the plant. Since NLMPC solves its optimization problem in an open-loop fashion, there is the potential to replace the controller with a trained DNN. Doing so is an appealing option, since evaluating a DNN can be more computationally efficient than solving a nonlinear optimization problem in real-time.
If the trained DNN reasonably approximates the controller behavior, you can then deploy the network for your control application. You can also use the network as a warm starting point for training the actor network of a reinforcement learning agent. For an example that does so with a DNN trained for an MPC application, see Train DDPG Agent with Pretrained Actor Network.
Design a nonlinear MPC controller for a flying robot. The dynamics for the flying robot are the same as in Trajectory Optimization and Control of Flying Robot Using Nonlinear MPC (Model Predictive Control Toolbox) example. First, define the limit for the control variables, which are the robot thrust levels.
umax = 3;
Create the nonlinear MPC controller object nlobj. To reduce command-window output, disable the MPC update messages.
nlobj = createMPCobjImFlyingRobot(umax);
Load the input data from DAggerInputDataFileImFlyingRobot.mat. The columns of the data set contain:
\mathit{x}
\mathit{y}
\theta
\stackrel{˙}{\mathit{x}}
is the velocity of the robot along the x-axis.
\stackrel{˙}{\mathit{y}}
is the velocity of the robot along the y-axis.
\underset{}{\overset{˙}{\theta }}
is the angular velocity of the robot.
{\mathit{u}}_{\mathit{l}}
is the thrust on the left side of the flying robot
{\mathit{u}}_{\mathit{r}}
is the thrust on the right side of the flying robot
{\mathit{u}}_{\mathit{l}}^{*}
is the thrust on the left side computed by NLMPC
{\mathit{u}}_{\mathit{r}}^{*}
is the thrust on the right side computed by NLMPC
The data in DAggerInputDataFileImFlyingRobot.mat is created by computing the NLMPC control action for randomly generated states (
\mathit{x}
\mathit{y}
\theta
\stackrel{˙}{\mathit{x}}
\stackrel{˙}{\mathit{y}}
\underset{}{\overset{˙}{\theta }}
), and previous control actions (
{\mathit{u}}_{\mathit{l}}
{\mathit{u}}_{\mathit{r}}
). To generate your own training data, use the collectDataImFlyingRobot function.
fileName = 'DAggerInputDataFileImFlyingRobot.mat';
DAggerData = load(fileName);
data = DAggerData.data;
existingData = data;
numCol = size(data,2);
The deep neural network architecture uses the following types of layers.
scalingLayer scales the value to the range to [-3,3].
Create the deep neural network that will imitate the NLMPC controller after training.
numObservations = numCol-2;
fullyConnectedLayer(numActions,'Name','fcLast')
tanhLayer('Name','tanhLast')
scalingLayer('Name','ActorScaling','Scale',umax)
regressionLayer('Name','routput')];
One approach to learning an expert policy using supervised learning is the behavior cloning method. This method divides the expert demonstrations (NLMPC control actions in response to observations) into state-action pairs and applies supervised learning to train the network.
% intialize validation cell array
validationCellArray = {0,0};
'ValidationData', validationCellArray, ...
'GradientThreshold', 10, ...
'MaxEpochs', 40 ...
You can train the behavior cloning neural network by following below steps
Collect data using the collectDataImFlyingRobot function.
Train the behavior cloning network using the behaviorCloningTrainNetwork function.
Training a DNN is a computationally intensive process. To save time, load a pretrained neural network object.
load('behaviorCloningMPCImDNNObject.mat');
imitateMPCNetBehaviorCloningObj = behaviorCloningNNObj.imitateMPCNetObj;
The training of the DNN using behavior cloning reduces the gap between the DNN and NLMPC performance. However, the behavior cloning neural network fails to imitate the behavior of the NLMPC controller correctly on some randomly generated data.
To improve the performance of the DNN, you can learn the policy using an interactive demonstrator method. DAgger is an iterative method where the DNN is run in the closed-loop environment. The expert, in this case the NLMPC controller, outputs actions based on the states visited by the DNN. In this manner, more training data is aggregated and the DNN is retrained for improved performance. For more information, see [1].
Train the deep neural network using the DAggerTrainNetwork function. It creates DAggerImFlyingRobotDNNObj.mat file that contains the following information.
DatasetPath: path where the dataset corresponding to each iteration is stored
policyObjs: policies that were trained in each iteration
finalData: total training data collected till final iteration
finalPolicy: best policy among all the collected policies
First, create and initialize the parameters for training. Use the network trained using behavior cloning (imitateMPCNetBehaviorCloningObj) as the starting point for the DAgger training.
[dataStruct,nlmpcStruct,tuningParamsStruct,neuralNetStruct] = loadDAggerParameters(existingData, ...
numCol,nlobj,umax,options,imitateMPCNetBehaviorCloningObj);
To save time, load a pretrained neural network by setting doTraining to false. To train the DAgger yourself, set doTraining to true.
DAgger = DAggerTrainNetwork(nlmpcStruct,dataStruct,neuralNetStruct,tuningParamsStruct);
load('DAggerImFlyingRobotDNNObj.mat');
DNN = DAgger.finalPolicy;
As an alternative, you can train the neural network with a modified policy update rule using the DAggerModifiedTrainNetwork function. In this function, after every 20 training iterations, the DNN is set to the most optimal configuration from the previous 20 iterations. To run this example with a neural network object with the modified DAgger approach, use the DAggerModifiedImFlyingRobotDNNObj.mat file.
To compare the performance of the NLMPC controller and the trained DNN, run closed-loop simulations with the flying robot model.
Set initial condition for the states of the flying robot (
\mathit{x}
\mathit{y}
\theta
\stackrel{˙}{\mathit{x}}
\stackrel{˙}{\mathit{y}}
\underset{}{\overset{˙}{\theta }}
) and the control variables of flying robot (
{\mathit{u}}_{\mathit{l}}
{\mathit{u}}_{\mathit{r}}
x0 = [-1.8200 0.5300 -2.3500 1.1700 -1.0400 0.3100]';
u0 = [-2.1800 -2.6200]';
Run a closed-loop simulation of the NLMPC controller.
Ts = nlobj.Ts;
% Simulation steps
Tsteps = Tf/Ts+1;
% Run NLMPC in closed loop.
[xHistoryMPC,uHistoryMPC] = simModelMPCImFlyingRobot(x0,u0,nlobj,Tf);
Run a closed-loop simulation of the trained DAgger network.
[xHistoryDNN,uHistoryDNN] = simModelDAggerImFlyingRobot(x0,u0,DNN,Ts,Tf);
Plot the results, and compare the NLMPC and trained DNN trajectories.
plotSimResultsImFlyingRobot(nlobj,xHistoryMPC,uHistoryMPC,xHistoryDNN,uHistoryDNN,umax,Tf)
The DAgger neural network successfully imitates the behavior of the NLMPC controller. The flying robot states and control action trajectories for the controller and the DAgger deep neural network closely align. The closed-loop simulation time for the DNN is significantly less than that of the NLMPC controller.
To validate the performance of the trained DNN, animate the flying robot with data from the DNN closed-loop simulation. The flying robot lands at the origin successfully.
x = xHistoryDNN(ct,1);
y = xHistoryDNN(ct,2);
theta = xHistoryDNN(ct,3);
tL = uHistoryDNN(ct,1);
tR = uHistoryDNN(ct,2);
rl.env.viz.plotFlyingRobot(x,y,theta,tL,tR,Lx,Ly);
% Turn on MPC messages
mpcverbosity on;
[1] Osa, Takayuki, Joni Pajarinen, Gerhard Neumann, J. Andrew Bagnell, Pieter Abbeel, and Jan Peters. ‘An Algorithmic Perspective on Imitation Learning’. Foundations and Trends in Robotics 7, no. 1–2 (2018): 1–179. https://doi.org/10.1561/2300000053.
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Bis(ammonium) Zoledronate Dihydrate
Małgorzata Sikorska, Jarosław Chojnacki, "Bis(ammonium) Zoledronate Dihydrate", Journal of Crystallography, vol. 2013, Article ID 741483, 5 pages, 2013. https://doi.org/10.1155/2013/741483
Małgorzata Sikorska1 and Jarosław Chojnacki 1
1Department of Chemistry, Gdańsk University of Technology, G. Narutowicza 11/12, 80233 Gdańsk, Poland
Academic Editor: P. Macchi
Neutralization of 2-(1-imidazole)-1-hydroxyl-1,1′-ethylidenediphosphonic acid (zoledronic acid) by an excess of ammonia yielded bis(ammonium) zoledronate dihydrate, { , 2(H4N+), 2(H2O)}. The product is readily soluble in water and forms monocrystals for which the X-ray structural analysis was carried out. The zoledronic anion is of double negative charge due to deprotonation of three P–OH groups and protonation of the nitrogen in the imidazole ring. The structure is stabilized by extensive network of N–H⋯O and O–H⋯O hydrogen bonds expanding through the crystal in plane (002). The imidazole ring is involved in - stacking interactions with its symmetry equivalents related by inversion centers at (1 0 0) and (1 1/2 0), with distances between centroids (Cg–Cg) of 3.819 (2) and 3.881 (2) Å, respectively.
Bisphosphonates have been subject of intensive research mainly due to their application in medicine as bone resorption inhibitors [1]. 2-(1-imidazole)-1-hydroxyl-1,1′-ethylidenediphosphonic acid, known as zoledronic acid, is regarded as the third generation of drugs for osteoporosis. Due to its structure, imidazolyl –CH2–C(OH){O=P(OH)2}2, the acid may be deprotonated at all four P–OH groups and protonated at the imidazolic N atom. The kind of the ion formed should influence not only coordination mode and intermolecular interactions but also the water solubility of the product, which may alter bioavailability.
Salts of zoledronic acid in the solid state usually contain monovalent anion resulting from deprotonation of two OH groups and protonation of the imidazole nitrogen. Examples include salts of sodium Na+ C5H9N2O7 4H2O [3] and a series of compounds of general formula (C5H9N2O7 )2M(H2O)2 M=Mg [4], Zn [5], Mn [6], Cu [7], Co, and Ni [8]. In some cases (for Cu, Co, Mn, and Ni), polynuclear complexes or coordination polymers are formed, with potentially interesting magnetic properties (e.g., [9]). Doubly negative anion was encountered in the catena((μ3-zoledronato)aqua calcium) [10]. Triply negative anion was found in a tris(dicyclohexylammonium) zoledronate ethanol solvate monohydrate [11]. It is one of the few structures reported with nonprotonated imidazole ring. The other two are the cobalt derivative with coordination of the ring nitrogen to a metal atom Co N [9] and the mentioned above calcium salt [10]. Structure of another organic ammonium salt of the acid was recently reported (albeit with mononegative anion)—cytosinium zoledronate trihydrate [12]. In this paper, we carried out neutralization of zoledronic acid with excess of ammonia to reveal whether mono-, bis-, or trisammonium salt will be formed in the solid state and to study the basic properties and the structure of the obtained product.
Starting material 2-(1-imidazole)-1-hydroxyl-1,1′-ethylidenediphosphonic acid monohydrate (zoledronic acid monohydrate) was supplied by Polpharma SA (Starogard Gdanski, Poland) and used as such (m.p. 239°C). Other reagents were of analytical grade and were used without further workup.
Preparation of C5H20N4O9P2 is given as follows. Solution of 3 mol/dm3 ammonia (eight-fold excess, 0.86 mL, 2.6 mmol) and solid zoledronic acid (88 mg, 0.325 mmol) were added to a mixture of water (8 mL) and ethanol (4 mL). The contents were stirred and heated to boiling. The solution was concentrated under vacuum to ca 2 mL and left to crystallize by slow evaporation, and after, a few days, colorless crystals (m.p. 260-261°C) suitable for X-ray diffraction were obtained.
Elemental analysis, CHNS, found the following (calculated for C5H20N4O9P2): C 17.73% (17.55), H 5.75% (5.89), N 16.01% (16.37), and S 0% (0). Apparatus used is Vario El Cube CHNS (Elementar).
Diffraction data were collected using KM4CCD (Kuma), Sapphire2, large Be window, 4-axis kappa diffractometer with graphite monochromator, and Mo radiation, at ambient temperature. More details are given in Table 1.
Empirical formula C5H20N4O9P2
Unit cell dimensions a = 6.8415 (4) Å
b = 7.5070 (5) Å
c = 13.9398 (10) Å
Volume 698.84 (8) Å3
Density (calcd.) 1.626 Mg m−3
Absorption coeff. µ = 0.36 mm−1
Crystal color and shape Colorless, block
θ range data collection θ = 2.8–25.1°
Limiting indices h = −5→8
k = −8→8
Independent reflections 2481 ( = 0.023)
Refinement method Full-matrix least-square on
Data/restraints/parameters 2481/14/235
Final indices = 0.0481
indices (all data) = 0.0586
( ) = 0.1279
Residual el. density = 0.49 e Å−3
= −0.34 e Å−3
Crystal data, data collection, and refinement details for C5H8N2O7 ·2(H2O)·2(H4N+).
Structure was refined with all heavy atoms treated as anisotropic and H-atoms as isotropic. All C–H and C–OH hydrogen atoms were refined as riding on their bonded counterpart atoms with the usual constrains. Hydrogen atoms belonging to water molecules were found in the Fourier map and refined with O–H bond length constrained to 0.82 Å. The same location method was applied to the H atoms bound to ammonium nitrogen atoms, whose N–H distances were constrained to 0.86 Å. Hydrogen atom at O6 was found in the electron density map and refined as riding on the oxygen atom.
Programs used are the following: for data collection, CrysAlis PRO (Agilent Technologies) [13]; for cell refinement and data reduction, CrysAlis PRO [13]; for solving structure, SUPERFLIP [14]; for refineing structure, SHELXL97 [15] and WinGX [16]; for molecular graphics, Mercury [17] and ORTEP-3 [18]; software used for preparing material for publication, PLATON [2], CSD [19], and publCIF [20].
Supplementary crystallographic data for C5H20N4O9P2, CCDC 935170, can be obtained free of charge via http://www.ccdc.cam.ac.uk/conts/retrieving.html or from the Cambridge Crystallographic Data Centre, Cambridge, UK.
Reaction of zoledronic acid with an excess of ammonia yielded the title compound: C5H8N2O7 2(H4N+) 2(H2O).
Its molecular structure is shown in Scheme 1 and Figure 1, and geometric parameters are given in Table 2. The bisphosphonate anion is of double negative charge due to deprotonation of three P–OH groups and protonation of the nitrogen in the imidazole ring. The longest P–O bond is noted for the only P–OH group (at O6 atom). Contrary to tris(dicyclohexylammonium) zoledronate [11], no intramolecular P–OH O hydrogen bond was formed. Protonation at the ring nitrogen N2 is confirmed by location of H2N from the electron density peak in the Fourier map. Presence of a strong hydrogen bond N2–H2N O (charge assisted, see Table 3) is an additional evidence of the protonation.
P1–O1 1.516 (2) O7–C5 1.443 (3)
P1–O3 1.526 (2) N2–C3 1.325 (4)
P1–C5 1.879 (3) N1–C3 1.337 (4)
P2–O6 1.578 (2) C1–C2 1.353 (5)
P2–C5 1.861 (3) C5–C4 1.543 (4)
O1–P1–O3 112.51 (12) C3–N1–C1 108.4 (3)
O1–P1–C5 109.45 (12) C2–C1–N1 106.5 (3)
O2–P1–C5 105.88 (12) O7–C5–C4 104.3 (2)
O4–P2–O5 116.33 (13) O7–C5–P2 109.87 (18)
O4–P2–O6 109.22 (13) C4–C5–P2 106.06 (18)
O4–P2–C5 107.62 (12) C4–C5–P1 112.52 (19)
O5–P2–C5 109.04 (13) P2–C5–P1 111.85 (14)
O6–P2–C5 103.87 (13) N2–C3–N1 108.5 (3)
C3–N1–C1–C2 −0.2 (3) O2–P1–C5–O7 34.8 (2)
C4–N1–C1–C2 178.6 (3) O1–P1–C5–C4 157.95 (19)
N1–C1–C2–N2 0.0 (4) O3–P1–C5–C4 36.7 (2)
C3–N2–C2–C1 0.2 (4) O2–P1–C5–C4 −82.2 (2)
O4–P2–C5–O7 173.78 (18) O1–P1–C5–P2 38.68 (18)
O5–P2–C5–O7 −59.2 (2) O3–P1–C5–P2 −82.56 (16)
O6–P2–C5–O7 58.1 (2) O2–P1–C5–P2 158.53 (14)
O4–P2–C5–C4 −74.1 (2) C2–N2–C3–N1 −0.3 (4)
O5–P2–C5–C4 52.9 (2) C1–N1–C3–N2 0.3 (3)
O6–P2–C5–C4 170.17 (18) C4–N1–C3–N2 −178.6 (3)
O4–P2–C5–P1 48.92 (17) C3–N1–C4–C5 84.1 (3)
O5–P2–C5–P1 175.92 (13) C1–N1–C4–C5 −94.5 (3)
O6–P2–C5–P1 −66.81 (16) O7–C5–C4–N1 −60.4 (3)
O1–P1–C5–O7 −85.1 (2) P2–C5–C4–N1 −176.36 (19)
O3–P1–C5–O7 153.69 (18) P1–C5–C4–N1 61.1 (3)
Selected geometric parameters (Å, °).
D–H H⋯A D⋯A D–H⋯A
O7–H7⋯O8ii 0.82 1.95 2.699 (3) 151
O8–H8A⋯O5 0.81 (2) 1.95 (2) 2.751 (4) 168 (4)
O8–H8B⋯O2iii 0.80 (2) 2.01 (2) 2.806 (4) 171 (5)
O9–H9A⋯O1iii 0.83 (2) 1.96 (2) 2.769 (3) 166 (4)
O9–H9B⋯O3iv 0.84 (2) 2.00 (2) 2.841 (3) 179 (4)
N2–H2N⋯O2v 0.86 (2) 1.77 (2) 2.613 (3) 165 (4)
N3–H3A⋯O1vi 0.88 (2) 1.89 (2) 2.761 (4) 168 (4)
N3–H3C⋯O4 0.88 (2) 2.14 (2) 3.003 (4) 169 (5)
N3–H3D⋯O9 0.86 (2) 2.16 (3) 2.909 (4) 146 (4)
N4–H4A⋯O9vii 0.88 (2) 2.02 (3) 2.812 (4) 149 (3)
N4–H4B⋯O2vi 0.87 (2) 2.01 (2) 2.857 (4) 165 (3)
N4–H4D⋯O5vii 0.88 (2) 2.16 (2) 3.000 (4) 159 (4)
Symmetry codes: i ; ii ; iii ; iv ; v ; vi ; vii .
Hydrogen-bond geometry (Å, °).
Molecular structure of C5H20N4O9P2 showing atom labeling scheme. All hanging hydrogen bonds are not shown; displacement ellipsoids 50%.
It is worthy noting that P1–O2 bond is longer than other P1–O bonds probably due to the mentioned hydrogen bond. Analysis of Table 3 reveals that it is a bond with the second smallest H A distance. Even stronger is the interaction O6–H6 O , of the only phosphorus hydroxyl group with the negatively charged oxygen, forming a first-level chain C(6) motif passing in parallel to the -axis. The hydrogen bond network is complex with most of the first-level motifs being discrete. Among the second-level graph motifs, notable is a centrosymmetric ring R2,4(8) motif with four N3–H donors and two O9 acceptors placed on the inversion center at ( ). Packing of molecules is organized in hydrophilic and hydrophobic layers parallel to the (i.e., (001)) plane, Figure 2. The hydrogen-bonding network is present in vicinity of the planes at ( , the set of all integers), while close to the planes at dominating are the stacking interactions. The imidazole groups, involved in - ring stacking, interact with their symmetry equivalents related by inversion centers at (1 0 0) and (1 0), with Cg–Cg distances 3.819 (2) and 3.881 (2) Å, respectively; for details, see Table 4. The title compound is very readily soluble in water.
Ring(I) Ring(II) Cg(I)–Cg(II) CgI_perp Slippage
Imid Imida 3.819 (2) 34.20 34.20 −3.1585 (16) 2.147
Imid Imidb 3.881 (2) 27.64 27.64 3.4376 (16) 1.800
Symmetry codes: a ; b .
Ring stacking interaction parameters [Å, °], calculated with PLATON program [2]. Imid = (N1–C1–C2–N2–C3); all dihedral angles between rings due to symmetry.
Layers of hydrogen bonding, about plane at (blue) and ring stacking in vicinity of the plane at (pink) in the crystals of bis(ammonium) zoledronate dehydrate.
Neutralization of 2-(1-imidazole)-1-hydroxyl-1,1′-ethylidenediphosphonic acid (zoledronic acid) by an excess of ammonia gives bis(ammonium) zoledronate dehydrate: {C5H8N2O7 , 2(H4N+), 2(H2O)}. Crystals are readily soluble in water. The anion is of double negative charge due to deprotonation of three OH groups and protonation of the nitrogen atom in the imidazole ring. The structure is stabilized by extensive network of N–H O and O–H O hydrogen bonds and - stacking interactions between the imidazole rings.
The authors have no conflict of interests with the mentioned commercial entity.
The authors thank the Polpharma SA Company (Starogard Gdanski, Poland) for the donation of samples of 2-(1-imidazole)-1-hydroxyl-1, -ethylidenediphosphonic acid monohydrate (zoledronic acid monohydrate).
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Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation | EMS Press
We address the problem of the existence of finite energy solitary waves for nonlinear Klein-Gordon or Schrödinger type equations
\Delta u - u + f(u) =0
\mathbb R^N
u \in H^1 (\mathbb R^N)
N\geq 2
. Under natural conditions on the nonlinearity
f
, we prove the existence of infinitely many nonradial solutions in any dimension
N \geq 2
. Our result complements earlier works of Bartsch and Willem (
N=4
N \geq 6
) and Lorca-Ubilla (
N=5
) where solutions invariant under the action of
O(2) \times O(N-2)
are constructed. In contrast, the solutions we construct are invariant under the action of
D_k \times O(N-2)
D_k \subset O(2)
denotes the dihedral group of rotations and reflexions leaving a regular planar polygon with
k
sides invariant, for some integer
k\geq 7
, but they are not invariant under the action of
O(2) \times O(N-2)
Juncheng Wei, Monica Musso, Frank Pacard, Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation. J. Eur. Math. Soc. 14 (2012), no. 6, pp. 1923–1953
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It is much easier to work with context-free grammars if the given CFG is its normal form. In this article we discuss the Chomsky Normal Form.
Normally it is much easier to work with a context-free grammar if the given context-free grammar is in a normal form.
Normal forms include:
Chomsky Normal Form, whereby productions are in the form of A → BC or A → a, where A, B and C are variables and a is a terminal symbol.
Greibach Normal Form, where productions are in the form of A → aα, where α ∈ V* and A ∈ V.
Rules in a context free grammar G = (V,Σ, R, S) are of the form:
A → w where A is a variable and w is a string over the alphabet V ∪ Σ.
We will show that every context-free grammar G can be converted into a context-free grammar G′ such that L(G) =L(G′) and rules of G′ are of restricted form as defined below.
Definition 1: A context-free grammar G = (V,Σ, R, S) is said to be in chomsky normal form if every rule in R has one of the forms:
A → BC where A, B and C are elements of V, B ≠ S, and C ≠ S.
A → a where A is an element of V and a is also an element of Σ.
S → ϵ where S is the start variable.
Now, we should convince ourselves that for such a grammar G, R contains the rule number 3 iff ϵ ∈ L(G)
Theorem 1: Let Σ be an alphabet and L ⊆ Σ* be a context-free language. Then there exists a context-free grammar in chomsky normal for whose language is L.
Proof: L is a context-free language and there exists a grammar G = (V,Σ, R, S), such that L(G) = L. Now we transform G to a grammar in chomsky normal form in five steps:
We eliminate the start variable from the right side of the rules by defining G1= (V1,Σ, R1, S1), where S1 is the start variable, V1 = V ∪ {S1} and R1 = R ∪ {S1 → S}.
This grammar will have the following property:
The start variable S1 doesn't occur on the right side of any rule in R1
L(G1) =L(G)
An ϵ-rule is a rule in the form of A → ϵ where A is a variable that is not equal to the start variable.
In this step we eliminate all ϵ-rules from G1
Now we consider rules one by one. Let A → ϵ be an example of a rule where A ∈ V1 and A ≠ S1. We change G1 as follows:
First we remove the rule A → ≠ from the current set R1.
Then, for each rule in the current set R1 in the form of:
(a). B → A, add rule B → ≠, unless the rule has been deleted from R1. In this way we replace the two-step derivation B ⇒ A ⇒ ≠ by a one-step derivation B ⇒ ≠
(b). B → uAv, where u and v are strings, both of which are not empty. We add the rule B →uv to R1 now observe that we have replaced a two-step derivation B ⇒ uAv ⇒ uv with a single-step derivation B ⇒ uv.
(c). B → uAvAw, where u, v and w are all strings. We add the rules B → uvw, B → uAvw, and B → uvAw to R1. If u = v = w = ϵ and the rule B → ϵ has been deleted from R1, we don't add the rule B → ϵ
(d). We shall treat all rules where A occurs multiple times on the right-hand side in a similar way.
This process is repeated until all ϵ-rules are eliminated. Let R2 be the set of rules after all ϵ-rules are eliminated. We define G2 = (V2,Σ, R2, S2) where V2 = V1 and S2 = S1.
The properties of this grammar are:
It's start variable S2 doesn't occur on the right-hand side of any rule in R2.
R2 doesn't contain any ϵ-rule but may contain S2 → ϵ rule.
L(G2) = L(G1) = L(G)
A unit-rule is in the form A → B where A and B are variables. In this step we eliminate all unit rules from G2. To do this we consider all rules .
Let A → B be an example of a rule where A and B are elements of V2. Knowing that B ≠ S2 we change G2 as follows:
First we remove rule A → B from the current R2 set.
For each rule in R2 set in the form of B → u where u ∈(V2 ∪ Σ)*, we add the rule A → u to the current set R2 unless it is a unit-rule that has been eliminated. This way, we replace a two-step derivation A ⇒ B ⇒ u with a one-step derivation A ⇒ u.
The process is repeated until all unit-rules are eliminated.
Let R3 be thes set of rules after all the rules have been eliminated. We therefore define G3 = (V3,Σ, R3, S3), where V3 = V2 and S3 = S2.
The above grammar has the following properties:
The start variable S3 doesn't occur on the right side of any rule in R3.
R3 doesn't contain any unit-rule
L(G3) = L(G2) =L(G1) = L(G)
This step involves eliminating all rules with more than two symbols on the right side. For each rule in the set R3 in the form of A → u1u2, ..., uk,, where k ≥ 3 and each
{\mathrm{u}}_{\mathrm{i}}
is an element of V3 ∪ Σ we change G3 as follows:
First we remove the rule A → u1 u2, ...,
{\mathrm{u}}_{\mathrm{i}}
from the current set R3.
Next we add the following rules to set R3.
A → u1A1
A1 → u2A2
Ak−3 → uk−2Ak−2
Ak−2 → uk−1uk
{\mathrm{A}}_{1}
{\mathrm{A}}_{2}
{\mathrm{A}}_{k-2}
are new variables added to the current set V3.
In this way we replace the one-step derivation A ⇒ u1 u2, ...,
{\mathrm{u}}_{\mathrm{i}}
by the (k-1)-step derivation.
A⇒{u}_{1}{A}_{1}⇒{u}_{1}{u}_{2}{A}_{2}⇒...⇒{u}_{1}{u}_{2}...{u}_{k-2}{A}_{k-2}⇒{u}_{1}{u}_{2}...{u}_{k}
Let R4 and V4 be the set of rules and variables respectively, when we eliminate all rules with more than two symbols on the right. We define G4 = (V4,Σ, R4, S4), where S4 = S3.
R4 doesn't contain any ϵ-rule but mat contain S3 → ϵ rule.
R4 doesn't contain any unit-rule.
R4 doesn't contain any rule with more than two symbols on the right hand side.
L(G4) = L(G3) = L(G2) = L(G1) = L(G)
Now we eliminate all rules with the form A →
{u}_{1}{u}_{2}
where u1 and u2 are not both variables.
For each rule in the current set R4 in the form of A →
{u}_{1}{u}_{2}
where u1 and u2 are not both contained in V4 we modify G3 as follows:
If u1 ∈ Σ and u2 ∈ V4, we replace the rule A →
{u}_{1}{u}_{2}
in the current R4 set by the two rules A →
{U}_{1}{u}_{2}
{U}_{1}\to {u}_{1}
{U}_{1}
is a new variable added to the current set V4. In this way we replace the one-step derivation A ⇒ u1u2 with a two-step derivation A ⇒ U1u2 ⇒ u1u2.
If u1 ∈ V4 and u2 ∈ Σ, we replace the rule A → u1u2 in the current set R4 by the two rules A →
{u}_{1}{U}_{2}
{u}_{2}\to {u}_{2}
{U}_{2}
is a new variable added to the current V4 set. This way we replace a one-step derivation A ⇒ u1u2 with a two-step derivation A ⇒ u1U2 ⇒ u1u2
If u1 ∈ Σ, u2 ∈ Σ, and u1 ≠ u2, we replace A →
{u}_{1}{u}_{2}
in the current R4 set by the three rules, A →
{U}_{1}
{U}_{2}
{U}_{1}
{u}_{2}
{U}_{2}
{u}_{2}
where U1 and U2 are new variables added to the current V4 set. In this way we replace the one-step derivation A ⇒ u1u2 by a three-step derivation A ⇒ U1U2 ⇒ u1U2 ⇒ u1u2.
If u1 ∈ Σ, u2 ∈ Σ, and u1 = u2, we replace the rule A → u1u2 = u1u1 in the current R4 set by the two rules A →
{U}_{1}
{U}_{1}
{U}_{1}
{u}_{1}
{U}_{1}
is a new variable added to the current V4 set. In this way we replace a one-step derivation A ⇒ u1u2 = u1u1 with a three-step derivation A ⇒ U1U1 ⇒ u1U1 ⇒ u1u1.
Let R5 be the set of rules and V5 the set of variables after the above step is completed. We define G5 = (V5,Σ, R5, S5) where S5 = S4.
R5 doesn't have any ϵ-rule but may have S5 → ϵ rule.
R5 doesn't have any unit-rule.
R5 doesn't have any rule with more than two symbols on the right side.
R5 doesn't have any rule in the form of
A\to {u}_{1}{u}_{2}
{u}_{1}
{u}_{2}
are not both V5 variables.
L(G5) = L(G4) = L(G3) = L(G2) = L(G1) = L(G).
And since the G5 grammar is in Chomsky normal form we have completed our proof.
It is easier to work with CFGs if a given CFG is in its normal form, following this we have discussed the Chomsky normal form.
Chomsky Normal Form is a form whereby productions are in the form of A → BC or A → a, where A, B and C are variables and a is a terminal symbol.
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§ Discrete schild's ladder
If one is given a finite graph
(V, E)
, which we are guaranteed came from discretizing a grid, can we recover a global sense of orientation?
More formally, assume the grid was of dimensions
W \times H
. So we have the vertex set as
V \equiv \{ (w, h) 1 \leq w \leq W, 1 \leq h \leq H \}
. We have in the edge set all elements of the form
((w, h), (w \pm 1, h \pm 1))
as long as respective elements
(w, h)
(w \pm 1, h \pm 1)
V
We lose the information about the grid as comprising of elements of the form
(w, h)
. That is, we are no longer allowed to "look inside". All we have is a pile of vertices and edges
V, E
Can we somehow "re-label" the edges
e \in E
as "north", "south", "east", and "west" to regain a sense of orientation?
Yes. Start with some corner. Such a corner vertex will have degree 2. Now, walk "along the edge", by going from a vertex of degree 2 to an neighbour of degree 2. If we finally reach a vertex that has unexplored neighbours of degree 3 or 4, pick the neighbour of degree 3. This will give us "some edge" of the original rectangle.
We now arbitrary declare this edge to be the North-South edge. We now need to build the perpendicular East-West edge.
This entire construction is very reminisecent of Schild's Ladder
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Microwaves101 | Noise Conversion Calculator
Enter Noise Figure (dB), Noise Temperature (K), or Noise Factor to calculate other equivalent parameters
Noise Figure, Noise Factor and Noise Temperature are all figures of merit to evaluate the sensitivity of a given system to random and uncorrelated fluctuations adding to the signal of interest. There are two mechanisms that contribute to the total noise of a system: charge carrier velocity fluctuations (thermal noise) and fluctuations in the number of charge carriers (shot noise). Both thermal and shot noise are purely random and have a Gaussian amplitude distribution. Generally, noise power per unit bandwidth is uniform with frequency but excess (or flicker or 1/f) noise does not have a uniform power spectral distribution. Shot noise can become dependent on frequency when the period of the signal approaches the charge carrier transit time within the device. Thanks to Hadrien Theveneau for improving on my original version of this calculator.
The Noise Factor is the ratio of the signal-to-noise ratio at the input to the signal-to-noise ratio at the output
\frac{{\text{SNR}}_{\text{in}}}{{\text{SNR}}_{\text{out}}}
\frac{\text{SNR}_\text{in}}{\text{SNR}_\text{out}}
The Noise Figure (dB) is
10\cdot {\mathrm{log}}_{10}\left(\text{Noise Factor}\right)
10 \cdot \log_{10}(\text{Noise Factor})
The Noise Temperature (K) is
290\cdot \left(\text{Noise Factor}-1\right)
290 \cdot (\text{Noise Factor} - 1)
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Remote Dynamic Triggering of Earthquakes in Three Unconventional Canadian Hydrocarbon Regions Based on a Multiple‐Station Matched‐Filter ApproachRemote Dynamic Triggering of Earthquakes in Three Unconventional Canadian Hydrocarbon Regions | Bulletin of the Seismological Society of America | GeoScienceWorld
Institute of Geology, Mineralogy, and Geophysics, Ruhr University Bochum, 44801 Bochum, Germany, bei.wang@mail.mcgill.ca
Also at Department of Earth and Planetary Sciences, McGill University, 3450 University Street, Montreal, Quebec, Canada, H3A 0E8.
Department of Earth and Planetary Sciences, McGill University, 3450 University Street, Montreal, Quebec, Canada H3A 0E8
Geological Survey of Canada, Pacific Geoscience Centre, 9860 West Saanich Road, Sidney, British Columbia, Canada V8L 1B9
Bei Wang, Rebecca M. Harrington, Yajing Liu, Honn Kao, Hongyu Yu; Remote Dynamic Triggering of Earthquakes in Three Unconventional Canadian Hydrocarbon Regions Based on a Multiple‐Station Matched‐Filter Approach. Bulletin of the Seismological Society of America 2018;; 109 (1): 372–386. doi: https://doi.org/10.1785/0120180164
We investigate the occurrence of remote dynamic triggering in three Canadian unconventional hydrocarbon regions where recent fluid injection activity is correlated with increasing numbers of earthquakes. We select mainshocks with an estimated local peak ground velocity exceeding
0.01 cm/s
occurring between 2013 and 2015, when station coverage was increased to monitor injection activity. A twofold approach, using continuous waveform data and an enhanced earthquake catalog created using a multiple‐station matched‐filter detection algorithm, suggests that remote dynamic triggering occurs at all three regions. The waveform‐based approach shows evidence for direct triggering in the surface wavetrain of the mainshock, as well as directly afterward in the coda. The enhanced catalog approach shows qualitative increases in earthquake rates at all three regions that are both immediate, and in some cases, sustained over 10‐day time windows and are corroborated with two types of statistical tests: a
p
‐value test to quantify the statistical significance of earthquake rate change following a stressing event and an interevent time test that provides a statistical measure of changes in seismicity rates. The occurrence of both direct and delayed triggering following transient stress perturbations of
<10 kPa
in all three regions suggests that local faults may remain critically stressed over periods similar to the time frame of our study (
∼2
yrs) or longer, potentially due to high pore pressures maintained in tight shale formations following injection. The results interpreted in the context of injection history and recent poroelastic modeling results may have implications for the mechanisms of remote triggering. Namely, triggering via poroelastic stresses may provide a unifying mechanism that can explain both delayed and immediate triggering observations.
M
Effects of sedimentary facies on structural styles in the Canadian Rocky Mountain Fold and Thrust Belt
Reservoir Characterization and Distribution in Rift and Synrift Basin Fill—Examples from the Triassic Fundy Basin and Orpheus Graben of the Scotian Margin
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Using Randomness Instead of Coordination?
Another Real Example: Load Shedding
My rule for when to write a simulator:
Simulate anything that involves more than one probability, probabilities over time, or queues.
Anything involving probability and/or queues you will need to approach with humility and care, as they are often deceivingly difficult: How many people with their random, erratic behaviour can you let into the checkout at once to make sure it doesn’t topple over? How many connections should you allow open to a database when it’s overloaded? What is the best algorithm to prioritize asynchronous jobs to uphold our SLOs as much as possible?
If you’re in a meeting discussing whether to do algorithm X or Y with this nature of problem without a simulator (or amazing data), you’re wasting your time. Unless maybe one of you has a PhD in queuing theory or probability theory. Probably even then. Don’t trust your intuition for anything the rule above applies to.
My favourite illustration of how bad your intuition is for these types of problems is the Monty Hall problem:
— Wikipedia Entry for the Monty Hall problem
Against your intuition, it is to your advantage to switch your choice. You will win the car twice as much if you do! This completely stumped me. Take a moment to think about it.
I frantically read the explanation on Wikipedia several times: still didn’t get it. Watched videos, now I think that.. maybe… I get it? According to Wikipedia, Erdős, one of the most renowned mathematicians in history also wasn’t convinced until he was shown a simulation!
After writing my simulation, however, I finally feel like I get it. Writing a simulation not only gives you a result you can trust more than your intuition but also develops your understanding of the problem dramatically. I won’t try to offer an in-depth explanation here, click the video link above, or try to implement a simulation — and you’ll see!
# https://gist.github.com/sirupsen/87ae5e79064354b0e4f81c8e1315f89b
$ ruby monty_hall.rb
Switch strategy wins: 666226 (66.62%)
No Switch strategy wins: 333774 (33.38%)
The short of it is that the host always opens the non-winning door, and not your door, which reveals information about the doors! Your first choice retains the 1/3 odds, but switching at this point, incorporating ‘the new information’ of the host opening a non-winning door, you improve your odds to 2/3 if you always switch.
This is a good example of a deceptively difficult problem. We should simulate it because it involves probabilities over time. If someone framed the Monty Hall problem to you you’d intuitively just say ‘no’ or ‘1/3’. Any problem involving probabilities over time should humble you. Walk away and quietly go write a simulation.
Now imagine when you add scale, queues, … as most of the systems you work on likely have. Thinking you can reason about this off the top of your head might constitute a case of good ol’ Dunning-Kruger. If Bob’s offering a perfect algorithm off the top of his head, call bullshit (unless he carefully frames it as a hypothesis to test in a simulator, thank you, Bob).
When I used to do informatics competitions in high school, I was never confident in my correctness of the more math-heavy tasks — so I would often write simulations for various things to make sure some condition held in a bunch of scenarios (often using binary search). Same principle at work: I’m much more confident most day-to-day developers would be able to write a good simulation than a closed-form mathematical solution. I once read something about a mathematician that spent a long time figuring out the optimal strategy in Monopoly. A computer scientist came along and wrote a simulator in a fraction of the time.
A few years ago, we were revisiting old systems as part of moving to Kubernetes. One system we had to adapt was a process spun up for every shard to do some book-keeping. We were discussing how we’d make sure we’d have at least ~2-3 replicas per shard in the K8s setup (for high availability). Previously, we had a messy static configuration in Chef to ensure we had a service for each shard and that the replicas spread out among different servers, not something that easily translated itself to K8s.
Below, the green dots denote the active replica for each shard. The red dots are the inactive replicas for each shard:
We discussed a couple of options: each process consulting some shared service to coordinate having enough replicas per shard, or creating a K8s deployment per shard with the 2-3 replicas. Both sounded a bit awkward and error-prone, and we didn’t love either of them.
As a quick, curious semi-jokingly thought-experiment I asked:
“What if each process chooses a shard at random when booting, and we boot enough that we are near certain every shard has at least 2 replicas?”
To rephrase the problem in a ‘mathy way’, with n being the number of shards:
“How many times do you have to roll an n-sided die to ensure you’ve seen each side at least m times?”
This successfully nerd-sniped everyone in the office pod. It didn’t take long before some were pulling out complicated Wikipedia entries on probability theory, trawling their email for old student MatLab licenses, and formulas soon appeared on the whiteboard I had no idea how to parse.
Insecure that I’ve only ever done high school math, I surreptitiously started writing a simple simulator. After 10 minutes I was done, and they were still arguing about this and that probability formula. Once I showed them the simulation the response was: “oh yeah, you could do that too… in fact that’s probably simpler…” We all had a laugh and referenced that hour endearingly for years after. (If you know a closed-form mathematical solution, I’d be very curious! Email me.)
# https://gist.github.com/sirupsen/8cc99a0d4290c9aa3e6c009fdce1ffec
$ ruby die.rb
P999: 1842
P9999: 2147
It followed from running the simulation that we’d need to boot 2000+ processes to ensure we’d have at least 2 replicas per shard with a 99.99% probability with this strategy. Compare this with the ~400 we’d need if we did some light coordination. As you can imagine, we then did the napkin cost of 1600 excess dedicated CPUs to run these book-keepers at [
10/month][costs]. Was this strategy worth ~
16,000 a month? Probably not.
Throughout my career I remember countless times complicated Wikipedia entries have been pulled out as a possible solution. I can’t remember a single time that was actually implemented over something simpler. Intimidating Wikipedia entries might be another sign it’s time to write a simulator, if nothing else, to prove that something simpler might work. For example, you don’t need to know that traffic probably arrives in a Poisson distribution and how to do further analysis on that. That will just happen in a simulation, even if you don’t know the name. Not important!
At Shopify, a good chunk of my time there I worked on teams that worked on reliability of the platform. Years ago, we started working on a ‘load shedder.’ The idea was that when the platform was overloaded we’d prioritize traffic. For example, if a shop got inundated with traffic (typically bots), how could we make sure we’d prioritize ‘shedding’ (red arrow below) the lowest value traffic? Failing that, only degrade that single store? Failing that, only impact that shard?
Hormoz Kheradmand led most of this effort, and has written this post about it in more detail. When Hormoz started working on the first load shedder, we were uncertain about what algorithms might work for shedding traffic fairly. It was a big topic of discussion in the lively office pod, just like the dice-problem. Hormoz started writing simulations to develop a much better grasp on how various controls might behave. This worked out wonderfully, and also served to convince the team that a very simple algorithm for prioritizing traffic could work which Hormoz describes in his post.
Of course, before the simulations, we all started talking about Wikipedia entries of the complicated, cool stuff we could do. The simple simulations showed that none of that was necessary — perfect! There’s tremendous value in exploratory simulation for nebulous tasks that ooze of complexity. It gives a feedback loop, and typically a justification to keep V1 simple.
Do you need to bin-pack tenants on n shards that are being filled up randomly? Sounds like probabilities over time, a lot of randomness, and smells of NP-completion. It won’t be long before someone points out deep learning is perfect, or some resemblance to protein folding or whatever… Write a simple simulation with a few different sizes and see if you can beat random by even a little bit. Probably random is fine.
You need to plan for retirement and want to stress-test your portfolio? The state of the art for this is using Monte Carlo analysis which, for the sake of this post, we can say is a fancy way to say “simulate lots of random scenarios.”
I hope you see the value in simulations for getting a handle on these types of problems. I think you’ll also find that writing simulators is some of the most fun programming there is. Enjoy!
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Why the division of a galvonometer scale are equally spaced - Physics - Magnetism And Matter - 10960603 | Meritnation.com
Why the division of a galvonometer scale are equally spaced?
Deflection in the pointer is proportional to the current passed .The number of divisions in the deflection will be proportional to the current passed , i.e
I\propto \varphi
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String theory - Uncyclopedia, the content-free encyclopedia
Schrödinger's cat twists reality with a coherent superstring vertex operator, otherwise known as a cosmic superstring.
String theory is the theory that matter, energy and women are made up of tiny strings. It states that whenever you put a set of perfectly arranged strings in any container, they will come out completely tangled, no matter what the arrangement or the container. The aforementioned three ingredients (plus lard that acts as the glue) give rise to various elaborate, sophisticated and highly complicated yet subtly simple and non-functioning existences, such as: iPod headphones, Christmas tree lights, garden hoses, electric cords, string panties, shoelaces, your Blu-Ray player; although surprisingly beautiful and functioning constructions have also appeared, such as horse intestines, beetle legs, belly-button fluff, the area behind your computer desk and smurfs.
Physicists now think that various dualities, e.g. AdS/CFT or gauge/string duality, link the various perceived realities. For instance, gauge/string duality states that belly-button hair is (when the embedding manifold is sufficiently and appropriately curved) dual to a conformal gauge theory of sweat living on the surface or boundary of the embedding manifold — in this case a pink porous surface known by the layman as skin. The gauge theory of sweat is a CFT (conformal field theory) in the sense that the underlying physics is independent of the size of the stinky bearer. On the string theory side, the bulk theory is correctly described by a qunatum (otherwise known as quantum, from the Greek κούνα τον) superstring theory in a fixed ten-dimensional space time background. It has been experimentally difficult to probe this region, primarily due to the un-probabilistic quantum (or deterministic) nature of non-existence of topological universes containing ten-dimensional arthropods and women. Namely, the ten-dimensional arthropod erectus a la carbonara cannot grasp that they are much more than merely two-handed and two-legged four-dimensional existences with a big and small bump in between the former and latter regions, otherwise known as humans. It is widely believed that the theory of string was first envisioned during a curious orgasmic array of lightning strikes in one of the offices of QWUL (Queen Mary University of London).
String theory has since proven a number of fascinating and glorious facts: that nonsense universes don't exist, that gravity exists (at least approximately), that the universes are typically multidimensional, that (using gauge/string duality) quantum field theories are indistinguishable from string theories (unless they are wrong), that Riemann surfaces are very interesting indeed, and finally that there is much more to universes that meets the Iris and Bob as well as the ten-dimensional arthropod erectus a la carbonara.
2 Famous knots
5 Cheeseology
7.2 Fate of the Universe
8 What it's used for
The first hints that everything is made of string began with the math equation, C + 2x = M (C being cat, and M being matter). The scientists found that X, no matter what M and C equaled, always was equal to string. This therefore meant that all matter was made up of strings. Soon the greatest scientist of the time, Paris Hilton, suggested the radical new idea that string theory was no longer string theory, but rather rope theory. This was because 2s = R, meaning that C + R = M, simplifying the equation, now causing 2 * string (2S) to become rope (R). Not long after that, she claimed that if this was true, than the universe was merely a giant rope which we all lived on. However, for this to be true, stated rival scientist Miley Cyrus, there needed to be more than the 4 dimensions that scientists had believed were the only dimensions out there. This was because, for the universe to be a giant rope, it cannot merely move within the 4 dimensions. This is because if the equation C + R = M and U/D * V = M are both true (U being universe, V being velocity and D being dimension) than D must equal more than 4 and less than 12. The equation U/D * V = M was proven to be true many times before, so that means either C + R = M is completely wrong, or there are more than four dimensions.
Famous knots[edit]
An edible knot
The most complicated knot tied to date is the famous Tibetan master knot, which is tied in the hair of a monk living off yeti flesh in the Great Mountains of Durka-durka-stan.
Another famous knot is the Gordian, which was misspelled by a clerk with a bad kidney in the fifteenth century. The Gordian knot was officially renamed the Accordion knot by President Maelin Seed's daughter, supermodel Appel Microsoft Seed. A specific theory about this knot was developed by Alexander the Great, a Shaolin monk who reached the Tao and therefore gained supernatural powers that allowed him to control a vast empire, but he did not build enough coliseums and theatres. Mao Tse Tung built the William Shakespeare's Theatre wonder before him, and therefore his empire suffered many rebellions.
A knot used by fierce, ancient warriors of the Agahapula tribe of southern Lirpaland in the continent of Urkulekela was the crazy-monkey fist knot. This knot was infused with poison dart frog poison and the thorn of the juju bush and swung around on a rope, hitting any opponent within a two-foot radius, and making them hallucinate for up to two hours. "I saw many pink unicorns and flying elephants on my journey," said a victim of this weapon.
After initiation of the Large Hadron Collider, a bizarre item instantaneously appeared and disappeared. Baffled by the contradiction to standard physics, a Swedish physicist defined this anomalous form of matter as "dark panties" or "the anti-panty", which coincidentally is the nickname of a Norwegian pimp who lives in South Africa.
The effects of String theory can be seen in every day life after doing the laundry. This is known as The Missing Sock Anomaly which is dependent upon the energy put into not losing your clothing. This energy includes a contribution from the "cashmere effect", to wit fluctuations in the quality of the clothing, e.g. more expensive socks. Clothing lost through the Missing Sock Anomaly end up in another universe where garden gnomes use them as oil rags.
String theory is a very, very vast and complex subject, and only monks who spend their whole life on it can reach the mysteries of knots, which lies beyond the Tao, or that guy's apartment building, whichever one you're willing to find first.
There are seven initiation steps.
The Shoelace Knot step, being the first one, is the easiest. Yet it requires amazing concentration, and monks have to spend years studying the Force to properly master the stunning Bunny-ears manuever. Masters of the Shoelace Knot are believed to be the most dangerous men in the world. Seriously, if you even think of them they WILL kill the rat that lives under your floorboards. YOU will be next.
The second step, the boating knots step, is far more difficult than the first one, and young apprentices (known as padawans) have to spend several years sailing until they can walk on water, to master this complex and subtle subject.
The third step lies in the deep, black and silent pools of the cavernous Quantum Physics : monks have to spend up to three years in a special room, called the University, to reach a level of interior peace high enough to perceive the mysteries of the Tao. Most are driven completely insane, and attend college parties and drink endless amounts of beer for the rest of their lives.
The other steps are mind elevation steps through the contemplation of the Tao and mastering of the Force.
Once these steps are finished, the Knots Masters can master their body and their mind, and gain supernatural powers, like ubiquity, through a space-time continuum rupture or levitation through a shifting of their Karma in the 27th dimension and a displacement of their Ka in the 13th along the lines of force of the Spiritual Whole God, or cast powerful spells, like cowation, or even reach the nirvana and bring back powerful artifacts, like a Blade +3 of Roses, or even a monk-only Carshemir + 5 which can invoke magical Guinea Pigs.
In fact, the ultimate goal of the String Theory is to deviate photons using the dimensions 4 to 37.17. (Space-time having a fractal shape, this is not a problem.) Once this trouser deviation is perfected, one of the monks studying the field hopes to unknot a panty or two.
Masterful use of string theory can lead to the mastery of all master uses!
Cheeseology[edit]
Created by bored physicists at some university somewhere, String Theory sought to resolve some of the unresolvable issues of Einsteinian Physics in accordance to Quantum Physics. The theory postulates that the sum of all matter (particles) in the universe is made up of really, really, really, really small Cheese Strings and governed by the laws of Quantum cheddardynamics. These Cheese Strings were created in an extremely large explosion called the String Cheese Incident. String Theory is not, I repeat: String Theory is not to be confused with String theory, the other white meat.
Current data show signs that the type of matter particles that the Cheese Strings create are dependent on their type of cheese. For example, Bosons are made of Soft cheeses, whereas fermions consist of more solid cheeses such as parmesan. Up quarks are Cheddar, down quarks are wensleydale, Muons are made of Swiss, Bozos are made of Le Roule, Gravitons, which may or may not exist, are tiny vibrating loops of ricotta, Gluons are made of mascarpone, which is why they are sticky, Electrons are made of stilton, Photons are made of Cream cheese, Morons are made of Dubliner, Neutrinos are made of Chunky Cheese, and Taus are made of Darth Feta. These Cheese Strings are collectively known as the 12 Fundamental Cheeses, however not all have been observed in a scientific fashion (ie, whilst drinking port).
The unaccounted-for dark matter of the universe consists of anti-cheese. For every fundamental cheese there is an anti-cheese equivalent. Usually these anti-cheeses are referred to as "crackers". For each fundamental cheese there is an associated cracker. For example, anti-bosons (bosons being soft cheese, remember) are made of Ryvita. When anti-cheese comes into contact with cheese a large quantity of wind, energy is released — hence the name "letting wind". It is thought that our universe was created from a Big Bang caused by the collision of astronomical quantities of Cheddar, Philadelphia and some soggy Jacobs Cream crackers.
How long is a piece of string?[edit]
A kitten scientist unravels the mystery.
The greatest theory of all is often quoted by idiots who have no idea about what they're doing or what they are about to do, and in extreme cases what they just did. A piece of string is 18.29 metres long (or wide, depending on how you look at it) (or deep, depending on how you look at it) but only when you concentrate really hard when measuring it; lack of concentration can make the "length" appear to be 18.2899999 metres. This has only ever happened once hence why people are always asking the great question.
Perhaps the most striking point about the above theory is that the string always appears to be 18.29 metres long, even if you are also 18.29 metres long, and moving at the exact same velocity as the string. This perplexing fact drove many scientists quite mad, including Barbra Streisand, Ellen Degenerate, and Fred Flintstone.
A string is always (almost) twice as long as half a piece of string. Except if the string is made of cheese, then results may differ depending on the Chunk Factor.
M-Theory[edit]
M-Theory? So there must be W-Theory somewhere.
M-Theory is the most sophisticated of all string theories. M-Theory name is due to cat's name, which sound Meow (foreign). It claims that the Universe is, in fact, a ball of wool stranded with a lot of threads, "a lot" being either 11 or 26; stars, planets, comets ad nauseam are but dust particles sticking to those threads. At least one of the threads is thought to be temporal, to wit time, and is being pulled by Cosmic Cat away from where Schrödinger shat. Nevertheless some scientists view things a bit differently and see the role of Cosmic Mouse as being pursued, which may explain the cat's need to pull in only one direction. Detractors claim incompatibility with Occam's Razor, which both prohibits multiplying without necessity and mandates shaving one's beard to the bare skin.
The string is being pulled at a constant speed of 299,998 km/s^2 in a constant direction. The strength of this pull is the biggest in the universe and nothing can move into the opposite direction. Also, light is being produced due to the friction between the threads and that's why time is related to the speed of light. Furthermore, energy being produced at the pull equals to the product of object's mass and the square of c, thus:
{\displaystyle E=mc^{2}}
The role of the Cosmic Cat has been recognized long before the M-Theory has been created; for instance, the famous physicist Albert Einstein who came up with the mentioned equation named the speed of light with a sign of c, the first letter in the word cat.
Black holes in the M-Theory are the extremely tight knots on one or many threads; they're so tight that nothing can escape from them, and the place of the tightest knotting is called a singularity; it's so tight that it disappears from the ball altogether, as does anything touching it.
Fate of the Universe[edit]
The Universe ends when the string begins fraying. The wool is being unfolded by many forces, including the Cosmic Cat's pull of the Time String. This is why galaxies seem to move farther and farther away, what in turn is being called Expansion of the Universe. Nobody knows for sure, what will happen to the world after the ball of wool will unfold completely. Some physicists claim that what will ensue will be Big Rip, where there is no more a ball and the matter is thrown into total chaos. Another possibility is that the cat dies or becomes bored with pulling; in such a case, the time will stop. Others think as the Time Rope gets longer the past begins to fray, diverging history in a new direction. Anyway, our fate won't be nice.
What it's used for[edit]
Fascinating cats (see also: kitten huffing)
Tying Shoelaces (or even larger more complex knots like Stephen Hawking does)
Very long and expensive books
Confusing the blood of the innocent
Something ... something pope
Yaught Crisenings
By European nudists who defeated Kublai Kahn at the great siege of the forest moon of Endor
Some stores accept pieces of string as payment
Defeating vast empires
Cutting things in half, such as cheese and potatoes
Silly String Theory, a special form of string theory (much like special relativity)
Quantum Murphydynamics, a competing (and far superior) theory that unites quantum mechanics and general relativity in an elegant explanatory framework without being completely ridiculous
Puppet String Theory, the theory that there is someone with a hand up your ass
Ring Theory, the theory purported by popular scientist Elrond Hubbard
Bling Theory, first hypothesised by Mr. T and Snoop Dogg as an excuse for their gold fetish
For those without comedic tastes, the so-called experts at Wikipedia have an article about String theory.
G-String Theory
The 12 Fundamental Cheeses
*Not to be confused with "Holey" Cheese
The 3 Noble Cheeses
*Also known as "Negative Cheese" or "Dark Dematta"
Retrieved from "https://uncyclopedia.com/w/index.php?title=String_theory&oldid=6167981"
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On explicit L2-convergence rate estimate for piecewise deterministic Markov processes in MCMC algorithms
April 2022 On explicit
{\mathit{L}}^{2}
-convergence rate estimate for piecewise deterministic Markov processes in MCMC algorithms
Jianfeng Lu, Lihan Wang
Jianfeng Lu,1 Lihan Wang2
1Department of Mathematics, Department of Physics, and Department of Chemistry, Duke University
We establish
{\mathit{L}}^{2}
-exponential convergence rate for three popular piecewise deterministic Markov processes for sampling: the randomized Hamiltonian Monte Carlo method, the zigzag process and the bouncy particle sampler. Our analysis is based on a variational framework for hypocoercivity, which combines a Poincaré-type inequality in time-augmented state space and a standard
{\mathit{L}}^{2}
energy estimate. Our analysis provides explicit convergence rate estimates, which are more quantitative than existing results.
This work is supported in part by National Science Foundation via grants CCF-1910571 and DMS-2012286.
Jianfeng Lu. Lihan Wang. "On explicit
{\mathit{L}}^{2}
-convergence rate estimate for piecewise deterministic Markov processes in MCMC algorithms." Ann. Appl. Probab. 32 (2) 1333 - 1361, April 2022. https://doi.org/10.1214/21-AAP1710
Received: 1 August 2020; Revised: 1 March 2021; Published: April 2022
Primary: 60J22 , 60J25 , 65C40
Keywords: convergence rate , hypocoercivity , Piecewise deterministic Markov process , Poincaré-type inequality
Jianfeng Lu, Lihan Wang "On explicit
{\mathit{L}}^{2}
-convergence rate estimate for piecewise deterministic Markov processes in MCMC algorithms," The Annals of Applied Probability, Ann. Appl. Probab. 32(2), 1333-1361, (April 2022)
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The Political Economy Research Institute Definition
The Political Economy Research Institute
What Is the Political Economy Research Institute?
The Political Economy Research Institute (PERI) is a progressive, left-leaning economic think tank at the University of Massachusetts Amherst, which conducts economic research intended to influence the public debate and be put into practical policy proposals to improve the quality of human life. Although PERI's research spans many fields, from environmentalism to social causes, one of its most well-known ventures is determining which companies make the Toxic 100 list—the list of the top 100 air polluters in the United States.
The Political Economy Research Institute (PERI) is an independent research unit of the University of Massachusetts Amherst.
PERI sponsors economic research, public policy studies, and conferences that focus on various progressive causes, especially the intersection of economics and environmental policy.
PERI produces an annual list of the top 100 air-polluting companies in the U.S. and has published several studies promoting the economic benefits of climate change policy.
Understanding the Political Economy Research Institute
Established in 1998, PERI works to conduct research that can be implemented into policy for the greater good. The economist Robert Heilbroner, known for his belief that economics should help improve the well-being of people at work and of the society they work in, once said that PERI "strive[s] to make a workable science out of morality." PERI collaborates with university faculty and students as well as other think tanks and researchers from around the globe, and it is closely linked to the UMass at Amherst's Department of Economics, though it is technically an independent unit of UMass.
PERI's goals include:
Raising awareness of issues that affect human and ecological well-being, such as globalization, income inequality, and environmentalism
Growing its network of research collaborators
PERI's research spans a variety of specialties, but it tends to focus on economic costs, benefits, and solutions and finding ways to implement policy changes that have positive impacts on the ecological system and the economy.
PERI's research is divided into many categories:
Finance, Jobs, and Macroeconomics: Research focuses on the relationships between financial institutions and economic inequality and instability.
Environmental and Energy Economics: Research focuses on economic solutions to environmental issues.
Economics for the Developing World: Research focuses on economic issues faced by developing countries.
Health Policy: Research focuses on economic and social factors that affect health, particularly addressing income support, social policies, and health disparities.
These are just a few of PERI's areas of research, all of which aim to raise awareness, guide public policy discussions, and offer solutions to the problems our world faces. It has also collaborated with the left-wing Center for American Progress in producing and publishing a series of studies on promoting economic growth through government policy to combat climate change.
PERI is perhaps most well-known for its research into the Toxic 100, or the top 100 air polluting companies in the United States. In order to score and rank each company, PERI pulls data from the Environmental Protection Agency (EPA), which provides insight into each company's emissions and toxic waste. Companies must report their chemical emissions data to the EPA's Toxic Release Inventory (TRI). Data from the TRI is then used by the EPA's Risk Screening Environmental Indicators (RSEI) system to determine weighted toxicity levels and the risk to human health.
Toxic scores are determined using the following equation:
\text{Emissions}\times\text{Toxicity}\times\text{Population Exposure}
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§ Mean value theorem and Taylor's theorem. (TODO)
I realise that there are many theorems that I learnt during my preparation for JEE that I simply don't know how to prove. This is one of them. Here I exhibit the proof of Taylor's theorem from Tu's introduction to smooth manifolds.
Taylor's theorem: Let
f: \mathbb R \rightarrow \mathbb R
be a smooth function, and let
n \in \mathbb N
be an "approximation cutoff". Then there exists for all
x_0 \in \mathbb R
a smooth function
r \in C^{\infty} \mathbb R
such that: f(x) = f(x 0) + (x - x 0) f'(x 0)/1! + (x - x 0)^2 f'(x 0)/2! + \dots + (x - x 0)^n f^{(n)'}(x 0)/n! + (x - x 0)^{n+1} r
We prove this by induction on
n = 0
, we need to show that there exists an
r
f(x) = f(x_0) + r
. We begin by parametrising the path from
x_0
x
p(t) \equiv (1 - t) x_0 + tx
. Then we consider
(f \circ p)'
\begin{aligned} &\frac{f(p(t))}{dt} = \frac{df((1 - t) x_0) + tx)}{dt} \\ &= (x - x_0) \frac{df((1 - t)x_0) + tx)}{dx} \end{aligned}
Integrating on both sides with limits
t=0, t=1
\begin{aligned} &\int_0^1 \frac{df(p(t))}{dt} dt = \int_0^1 (x - x_0) \frac{df((1 - t)x_0) + tx)}{dx} dt \\ f(p(1)) - f(p(0)) = (x - x_0) \int_0^1 \frac{df((1 - t)x_0) + tx)}{dx} dt \\ f(x) - f(x_0) = (x - x_0) g[1](x) \\ \end{aligned}
g[1](x) \equiv \int_0^1 \frac{df((1 - t)x_0) + tx)}{dx} dt
g[1](x)
witnesses that we have the first derivative of
in its expression. By rearranging, we get:
\begin{aligned} f(x) - f(x_0) = (x - x_0) g[1](x) \\ f(x) = f(x_0) + (x - x_0) g[1](x) \\ \end{aligned}
If we want higher derivatives, then we simply notice that
g[1](x)
\begin{aligned} g[1](x) \equiv \int_0^1 f'((1 - t)x_0) + tx) dt \\ g[1](x) \equiv \int_0^1 f'((1 - t)x_0) + tx) dt \\ \end{aligned}
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List of open-source software for mathematics - Wikipedia
Find sources: "List of open-source software for mathematics" – news · newspapers · books · scholar · JSTOR (June 2013) (Learn how and when to remove this template message)
Mathics[edit]
Geogebra[edit]
Geogebra (Geometry and Algebra) - combines geometric objects like circles and graphs of functions with its algebraic representation e.g.
{\displaystyle x^{2}+y^{2}=r^{2}}
respresenting a circle with the radius
{\displaystyle r}
. Designed for use in schools and educational settings.
Theorem provers[edit]
This section is an excerpt from Automated theorem proving § Free software.[edit]
Recreational mathematics software[edit]
Retrieved from "https://en.wikipedia.org/w/index.php?title=List_of_open-source_software_for_mathematics&oldid=1075250684"
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Converse (logic) - Simple English Wikipedia, the free encyclopedia
In mathematics and logic, a converse is a variant of an implication. More specifically, given an implication of the form
{\displaystyle P\to Q}
, the converse is the statement
{\displaystyle Q\to P}
While a converse is similar to its originating implication, they are not logically equivalent.[2] This means that the truth of an implication does not guarantee the truth of its converse (and vice versa).[1]
As a logical connective, the converse of
{\displaystyle P}
{\displaystyle Q}
can be represented by the symbol
{\displaystyle \leftarrow }
(as in
{\displaystyle P\leftarrow Q}
↑ 1.0 1.1 "The Definitive Glossary of Higher Mathematical Jargon". Math Vault. 2019-08-01. Retrieved 2020-10-09.
↑ Taylor, Courtney. "What Are the Converse, Contrapositive, and Inverse?". ThoughtCo. Retrieved 2020-10-09. {{cite web}}: CS1 maint: url-status (link)
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Converse_(logic)&oldid=7139921"
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Demodulate using QPSK method - MATLAB - MathWorks Switzerland
CustomDerotateFactorDataType
Specific to comm.QPSKDemodulator
BER Estimate of QPSK Signal
Hard-Decision QPSK Demodulation
Soft-Decision QPSK Demodulation
Demodulate using QPSK method
The comm.QPSKDemodulator object demodulates a signal that was modulated using the quadrature phase shift keying (QPSK) method. The input is a baseband representation of the modulated signal.
To demodulate a signal that was modulated using the QPSK method:
Create the comm.QPSKDemodulator object and set its properties.
qpskdemod = comm.QPSKDemodulator
qpskdemod = comm.QPSKDemodulator(Name=Value)
qpskdemod = comm.QPSKDemodulator(phase=Name,Value)
qpskdemod = comm.QPSKDemodulator creates a System object™ to demodulate input QPSK signals.
qpskdemod = comm.QPSKDemodulator(Name=Value) sets properties using one or more name-value arguments. For example, DecisionMethod="Hard decision" specifies demodulation using the hard-decision method.
qpskdemod = comm.QPSKDemodulator(phase=Name,Value) sets the PhaseOffset property to phase, and optional name-value arguments. Specify phase in radians.
PhaseOffset — Phase of zeroth point in constellation
Phase of the zeroth point in the constellation in radians, specified as a scalar.
Example: PhaseOffset=0 aligns the QPSK signal constellation points on the axes {(1,0), (0,j), (-1,0), (0,-j)}.
Set this property to false to output symbols as integer values in the range [0, 3] with length equal to the input data vector length.
Set this property to true to output a column vector of bit values with length equal to twice the input data vector length.
Demodulation decision method, specified as 'Hard decision', 'Log-likelihood ratio', or 'Approximate log-likelihood ratio'. When you set the BitOutput property to false, the object always performs hard-decision demodulation.
To enable this property, set the BitOutput property to true and the DecisionMethod property to 'Log-likelihood ratio' or 'Approximate log-likelihood ratio'.
Noise variance, specified as a positive scalar.
To enable this property, set the BitOutput property to true, the DecisionMethod property to 'Log-likelihood ratio' or 'Approximate log-likelihood ratio', and the VarianceSource property to 'Property'.
'Full precision' (default) | 'Smallest unsigned integer' | 'double' | ...
Data type of the output, specified as 'Full precision', 'Smallest unsigned integer', 'double', 'single', 'int8', 'uint8', 'int16', 'uint16', 'int32', or 'uint32','logical'.
When the input data type is single or double precision and you set the BitOutput property to true, the DecisionMethod property to 'Hard decision', and the OutputDataType property to 'Full precision', the output has the same data type as that of the input.
When the input data is of a fixed-point type, the output data type behaves as if you had set the OutputDataType property to 'Smallest unsigned integer'.
When you set BitOutput to true and the DecisionMethod property to 'Hard Decision', then 'logical' data type is a valid option.
When you set the BitOutput property to true and the DecisionMethod property to 'Log-likelihood ratio' or 'Approximate log-likelihood ratio', the output data type is the same as that of the input and the input data type must be single or double precision.
To enable this property, set the BitOutput property to false or set the BitOutput property to true and the DecisionMethod property to 'Hard decision'.
Data type of the derotate factor, specified as 'Same word length as input' or 'Custom'. The object uses the derotate factor in the computations only when the input signal is a fixed-point type and the PhaseOffset property has a value that is not an even multiple of π/4.
CustomDerotateFactorDataType — Fixed-point data type of derotate factor
numerictype([],16) (default) | unscaled numerictype object
Fixed-point data type of the derotate factor, specified as an unscaled numerictype (Fixed-Point Designer) object with a Signedness of Auto.
To enable this property, set the DerotateFactorDataType property to 'Custom'.
Data Types: numerictype object
y = qpskdemod(x)
y = qpskdemod(x,var)
y = qpskdemod(x) applies QPSK demodulation to the input signal and returns the demodulated signal.
y = qpskdemod(x,var) uses soft decision demodulation and noise variance var. This syntax applies when you set the BitOutput property to true, the DecisionMethod property to 'Approximate log-likelihood ratio' or 'Log-likelihood ratio', and the VarianceSource property to 'Input port'.
x — QPSK-modulated signal
QPSK-modulated signal, specified as a scalar or column vector.
The object accepts inputs with a signed integer data type or signed fixed point (sfi (Fixed-Point Designer)) objects when you set the BitOutput property to false or you set the DecisionMethod property to 'Hard decision' and the BitOutput property to true.
Data Types: double | single | int | fi
To enable this argument, set the VarianceSource property to 'Input port', the BitOutput property to true, and the DecisionMethod property to 'Approximate log-likelihood ratio' or 'Log-likelihood ratio'.
Output signal, returned as a scalar or column vector. To specify whether the object outputs values as integers or bits, use the BitOutput property. To specify the output data type, use the OutputDataType property.
Create a QPSK modulator and demodulator pair that operate on bits.
qpskModulator = comm.QPSKModulator('BitInput',true);
qpskDemodulator = comm.QPSKDemodulator('BitOutput',true);
Create an AWGN channel object and an error rate counter.
Generate random binary data and apply QPSK modulation.
txSig = qpskModulator(data);
Pass the signal through the AWGN channel and demodulate it.
rxData = qpskDemodulator(rxSig);
Calculate the error statistics. Display the BER.
errorStats = errorRate(data,rxData);
errorStats(1)
SNR
{E}_{b}/{N}_{o}
The signal preprocessing required for QPSK demodulation depends on the configuration.
This figure shows the hard-decision QPSK demodulation signal diagram for the trivial phase offset (odd multiple of π/4) configuration.
This figure shows the hard-decision QPSK demodulation floating point signal diagram for the nontrivial phase offset configuration.
This figure shows the hard-decision QPSK demodulation fixed-point signal diagram for the nontrivial phase offset configuration.
comm.QPSKModulator | comm.PSKDemodulator | comm.PSKModulator | comm.DPSKDemodulator | comm.OQPSKDemodulator
QPSK Demodulator Baseband | M-PSK Demodulator Baseband
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Beta decay - New World Encyclopedia
Previous (Bessie Smith)
Next (Beta movement)
Henri Becquerel · Marie Curie · Pierre Curie
In nuclear physics, beta decay is a type of radioactive decay involving the emission of beta particles. Beta particles are high-energy, high-speed electrons or positrons emitted by certain types of radioactive atomic nuclei such as potassium-40. These particles, designated by the Greek letter beta (β), are a form of ionizing radiation and are also known as beta rays.
There are two forms of beta decay: "beta minus" (β−), involving the release of electrons; and "beta plus" (β+), involving the emission of positrons (which are antiparticles of electrons). In beta minus decay, a neutron is converted into a proton, an electron, and an electron antineutrino. In beta plus decay, a proton is converted into a neutron, a positron, and an electron neutrino (a type of neutrino associated with the electron). In either case, the number of nucleons (neutrons plus protons) in the nucleus remains the same, while the number of protons in the nucleus changes.
2 β− decay (electron emission)
3 β+ decay (positron emission)
4 Electron capture
5 Effects of beta decay
Beta-minus (β-) decay. The intermediate emission of a W- boson is omitted.
Alpha radiation consists of helium nuclei and is readily stopped by a sheet of paper. Beta radiation, consisting of electrons, is halted by an aluminum plate. Gamma radiation is eventually absorbed as it penetrates a dense material.
The Feynman diagram for beta decay of a neutron into a proton, electron, and electron antineutrino via an intermediate heavy W- boson.
If the atomic nuclei of a chemical element undergo beta decay, this process leads to the transmutation of that element into another. It is one way by which unstable atomic nuclei acquire greater stability. Beta minus decay is a common process in the neutron-rich fission by-products produced in nuclear reactors, accounting for the large numbers of electron antineutrinos produced by these reactors. Free neutrons also decay by this process.
Historically, the study of beta decay provided the first physical evidence of the neutrino. In 1911, Lise Meitner and Otto Hahn performed an experiment that showed that the energies of electrons emitted by beta decay had a continuous rather than discrete spectrum. This was in apparent contradiction to the law of conservation of energy, as it appeared that energy was lost in the beta decay process. A second problem was that the spin of the Nitrogen-14 atom was 1, in contradiction to the Rutherford prediction of ½.
In 1920-1927, Charles Drummond Ellis (along with James Chadwick and colleagues) established clearly that the beta decay spectrum really is continuous, ending all controversies.
In a famous letter written in 1930, Wolfgang Pauli suggested that in addition to electrons and protons atoms also contained an extremely light neutral particle which he called the neutron. He suggested that this "neutron" was also emitted during beta decay and had simply not yet been observed. In 1931, Enrico Fermi renamed Pauli's "neutron" to neutrino, and in 1934 Fermi published a very successful model of beta decay in which neutrinos were produced.
An unstable atomic nucleus with an excess of neutrons may undergo β− decay. In this process, a neutron is converted into a proton, an electron, and an electron-type antineutrino (the antiparticle of the neutrino):
{\displaystyle n^{0}\rightarrow p^{+}+e^{-}+{\bar {\nu }}_{e}}
At the fundamental level (depicted in the Feynman diagram below), this process is mediated by the weak interaction. A neutron (one up quark and two down quarks) turns into a proton (two up quarks and one down quark) by the conversion of a down quark to an up quark, with the emission of a W- boson. The W- boson subsequently decays into an electron and an antineutrino.
Beta decay commonly occurs among the neutron-rich fission byproducts produced in nuclear reactors. This process is the source of the large numbers of electron antineutrinos produced by fission reactors. Free neutrons also decay via this process.
Unstable atomic nuclei with an excess of protons may undergo β+ decay, or inverse beta decay. In this case, energy is used to convert a proton into a neutron, a positron (e+), and an electron-type neutrino (
{\displaystyle \nu _{e}}
{\displaystyle \mathrm {energy} +p^{+}\rightarrow n^{0}+e^{+}+{\nu }_{e}}
On a fundamental level, an up quark is converted into a down quark, emitting a W+ boson that then decays into a positron and a neutrino.
Unlike beta minus decay, beta plus decay cannot occur in isolation, because it requires energy — the mass of the neutron being greater than the mass of the proton. Beta plus decay can only happen inside nuclei when the absolute value of the binding energy of the daughter nucleus is higher than that of the mother nucleus. The difference between these energies goes into the reaction of converting a proton into a neutron, a positron and, a neutrino and into the kinetic energy of these particles.
(See main article on Electron capture.)
In all cases where β+ decay is allowed energetically (and the proton is part of an atomic nucleus surrounded by electron shells), it is accompanied by the "electron capture" process, also known as inverse beta decay. In this process, a proton in the atomic nucleus captures an atomic electron (from an inner orbital), with the emission of a neutrino. The proton is converted into a neutron. The process may be written as follows:
{\displaystyle \mathrm {energy} +p^{+}+e^{-}\rightarrow n^{0}+{\nu }_{e}}
If, however, the energy difference between initial and final states is low (less than 2mec2), then β+ decay is not energetically possible, and electron capture is the sole decay mode.
Effects of beta decay
Beta decay does not change the number of nucleons A in the nucleus, but changes only its charge Z. Thus, during beta decay, the parent nuclide and daughter nuclide share the same A value.
The beta decay of atomic nuclei results in the transmutation of one chemical element into another. For example:
Beta minus:
{\displaystyle \mathrm {{}^{1}{}_{55}^{37}Cs} \rightarrow \mathrm {{}^{1}{}_{56}^{37}Ba} +e^{-}+{\bar {\nu }}_{e}}
Beta plus:
{\displaystyle \mathrm {~_{11}^{22}Na} \rightarrow \mathrm {~_{10}^{22}Ne} +e^{+}+{\nu }_{e}}
For comparison, the electron capture process may be written as follows:
Electron capture:
{\displaystyle \mathrm {~_{11}^{22}Na} +e^{-}\rightarrow \mathrm {~_{10}^{22}Ne} +{\nu }_{e}}
In nature, most isotopes are beta stable, but a few exceptions exist with half-lives so long that they have not had enough time to decay since the moment of their nucleosynthesis. One example is 40K, which undergoes beta minus and beta plus decay and electron capture, with a half-life of 1.277×109 years.
It should be noted that a beta-stable nucleus may undergo other kinds of radioactive decay, such as alpha decay.
Some nuclei can undergo double beta decay (ββ decay), where the charge of the nucleus changes by two units. In most practically interesting cases, single beta decay is energetically forbidden for such nuclei, because when β and ββ decays are both allowed, the probability of β decay is (usually) much higher, preventing investigations of very rare ββ decays. Thus, ββ decay is usually studied only for beta stable nuclei. Like single beta decay, double beta decay does not change the value of A. Thus, at least one of the nuclides with a given A value has to be stable, for both single and double beta decay.
Krane, Kenneth S., and David Halliday. 1988. Introductory Nuclear Physics. New York: Wiley. ISBN 047180553X.
Martin, Brian. 2006. Nuclear and Particle Physics: An Introduction. Hoboken, NJ: Wiley. ISBN 0470025328.
Poenaru, D. N. 1996. Nuclear Decay Modes. Fundamental and Applied Nuclear Physics Series. Philadelphia: Institute of Physics. ISBN 0750303387.
Tipler, Paul, and Ralph Llewellyn. 2002. Modern Physics. 4th ed. New York, NY: W.H. Freeman. ISBN 0716743450.
Turner, James E. 1995. Atoms, Radiation, and Radiation Protection. 2nd ed. New York: Wiley. ISBN 0471595810.
Beta decay. Jefferson Lab.
Beta_decay history
Beta_particle history
History of "Beta decay"
Retrieved from https://www.newworldencyclopedia.org/p/index.php?title=Beta_decay&oldid=1063323
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Napkin friends, from near and far, it’s time for napkin problem number four! If you are wondering why you’re receiving this email, you likely watched my talk on napkin math and decided to sign up for some monthly training.
Since last, there has been some smaller updates to the napkin-math repository and the accompanying program. I’ve been brushing up on x86 to ensure that the base-rates truly represent the upper-bound, which will require some smaller changes. The numbers are unlikely to change by an order of magnitude, but I am dedicated to make sure they are optimum. If you’d like to help with providing some napkin calculations, I’d love contributions around serialization (JSON, YAML, …) and compression (Gzip, Snappy, …). I am also working on turning all my notes from the above talk into a long, long blog post.
With that out of the way, we’ll do a slightly easier problem than last week this week! As always, consult sirupsen/napkin-math for resources and help to solve today’s problem.
Today, as you were preparing you organic, high-mountain Taiwanese oolong in the kitchennette, one of your lovely co-workers mentioned that they were looking at adding more Redises because it was maxing out at 10,000 commands per second which they were trending aggressively towards. You asked them how they were using it (were they running some obscure O(n) command?). They’d BPF-probes to determine that it was all GET <key> and SET <key> <value>. They also confirmed all the values were about or less than 64 bytes. For those unfamiliar with Redis, it’s a single-threaded in-memory key-value store written in C.
Unphased after this encounter, you walk to the window. You look out and sip your high-mountain Taiwanese oolong. As you stare at yet another condominium building being built—it hits you. 10,000 commands per second. 10,000. Isn’t that abysmally low? Shouldn’t something that’s fundamentally ‘just’ doing random memory reads and writes over an established TCP session be able to do more?
What kind of throughput might we be able to expect for a single-thread, as an absolute upper-bound if we disregard I/O? What if we include I/O (and assume it’s blocking each command), so it’s akin to a simple TCP server? Based on that result, would you say that they have more investigation to do before adding more servers?
You can read the problem in the archive, here.
We have 4 bitmaps (one per condition) of 10^6 product ids, each of 64 bits. That’s 4 * 10^6 * 64 bits = 32 Mb. Would this be in memory or on SSDs? Well, let’s assume the largest merchants have 10^6 products and 10^3 attributes, that means a total of 10^6 * 10^3 * 64 bits = 8Gb. That’d cost us about
8 in memory, or about
1 to store on disk. In terms of performance, this is nicely sequential access. For memory, 32 mb * 100us/mb = 3.2 ms. For SSD (about 10x cheaper, and 10x slower than memory), 30 ms. 30 ms is a bit high, but 3 ms is acceptable. $8 is not crazy, given that this would be the absolute largest merchant we have. If cost becomes an issue, we could likely employ good caching.
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Floating-Point Numbers - MATLAB & Simulink - MathWorks Benelux
IEEE 754 Standard for Floating-Point Numbers
The Sign Bit
The Fraction Field
The Exponent Field
Double-Precision Format
Single-Precision Format
Half-Precision Format
Floating-Point Data Type Parameters
Exceptional Arithmetic
Fixed-point numbers are limited in that they cannot simultaneously represent very large or very small numbers using a reasonable word size. This limitation can be overcome by using scientific notation. With scientific notation, you can dynamically place the binary point at a convenient location and use powers of the binary to keep track of that location. Thus, you can represent a range of very large and very small numbers with only a few digits.
You can represent any binary floating-point number in scientific notation form as f2e, where f is the fraction (or mantissa), 2 is the radix or base (binary in this case), and e is the exponent of the radix. The radix is always a positive number, while f and e can be positive or negative.
When performing arithmetic operations, floating-point hardware must take into account that the sign, exponent, and fraction are all encoded within the same binary word. This results in complex logic circuits when compared with the circuits for binary fixed-point operations.
The Fixed-Point Designer™ software supports half-precision, single-precision, and double-precision floating-point numbers as defined by the IEEE® Standard 754.
A direct analogy exists between scientific notation and radix point notation. For example, scientific notation using five decimal digits for the fraction would take the form
±d.dddd×{10}^{p}=±ddddd.0×{10}^{p-4}=±0.ddddd×{10}^{p+1},
d=0,...,9
and p is an integer of unrestricted range.
Radix point notation using five bits for the fraction is the same except for the number base
±b.bbbb×{2}^{q}=±bbbbb.0×{2}^{q-4}=±0.bbbbb×{2}^{q+1},
b=0,1
and q is an integer of unrestricted range.
For fixed-point numbers, the exponent is fixed but there is no reason why the binary point must be contiguous with the fraction. For more information, see Binary Point Interpretation.
The IEEE Standard 754 has been widely adopted, and is used with virtually all floating-point processors and arithmetic coprocessors, with the notable exception of many DSP floating-point processors.
This standard specifies several floating-point number formats, of which singles and doubles are the most widely used. Each format contains three components: a sign bit, a fraction field, and an exponent field.
IEEE floating-point numbers use sign/magnitude representation, where the sign bit is explicitly included in the word. Using sign/magnitude representation, a sign bit of 0 represents a positive number and a sign bit of 1 represents a negative number. This is in contrast to the two's complement representation preferred for signed fixed-point numbers.
Floating-point numbers can be represented in many different ways by shifting the number to the left or right of the binary point and decreasing or increasing the exponent of the binary by a corresponding amount.
To simplify operations on floating-point numbers, they are normalized in the IEEE format. A normalized binary number has a fraction of the form 1.f, where f has a fixed size for a given data type. Since the leftmost fraction bit is always a 1, it is unnecessary to store this bit and it is therefore implicit (or hidden). Thus, an n-bit fraction stores an n+1-bit number. The IEEE format also supports Denormalized Numbers, which have a fraction of the form 0.f.
In the IEEE format, exponent representations are biased. This means a fixed value, the bias, is subtracted from the exponent field to get the true exponent value. For example, if the exponent field is 8 bits, then the numbers 0 through 255 are represented, and there is a bias of 127. Note that some values of the exponent are reserved for flagging Inf (infinity), NaN (not-a-number), and denormalized numbers, so the true exponent values range from -126 to 127. See Inf and NaN for more information.
The IEEE double-precision floating-point format is a 64-bit word divided into a 1-bit sign indicator s, an 11-bit biased exponent e, and a 52-bit fraction f.
The relationship between double-precision format and the representation of real numbers is given by
value=\left\{\begin{array}{ll}{\left(-1\right)}^{s}\left({2}^{e-1023}\right)\left(1.f\right)\hfill & \text{normalized, }0<e<2047,\hfill \\ {\left(-1\right)}^{s}\left({2}^{e-1022}\right)\left(0.f\right)\hfill & \text{denormalized, }e=0,\text{ }f>0\hfill \\ \text{exceptional value }\hfill & \text{otherwise}\text{.}\hfill \end{array},
See Exceptional Arithmetic for more information.
The IEEE single-precision floating-point format is a 32-bit word divided into a 1-bit sign indicator s, an 8-bit biased exponent e, and a 23-bit fraction f.
The relationship between single-precision format and the representation of real numbers is given by
value=\left\{\begin{array}{ll}{\left(-1\right)}^{s}\left({2}^{e-127}\right)\left(1.f\right)\hfill & \text{normalized, }0<e<255,\hfill \\ {\left(-1\right)}^{s}\left({2}^{e-126}\right)\left(0.f\right)\hfill & \text{denormalized, }e=0,\text{ }f>0,\hfill \\ \text{exceptional value }\hfill & \text{otherwise}\text{.}\hfill \end{array}
The IEEE half-precision floating-point format is a 16-bit word divided into a 1-bit sign indicator s, a 5-bit biased exponent e, and a 10-bit fraction f.
Half-precision numbers are supported in MATLAB® and Simulink®. For more information, see half and The Half-Precision Data Type in Simulink.
The range of a number gives the limits of the representation. The precision gives the distance between successive numbers in the representation. The range and precision of an IEEE floating-point number depends on the specific format.
The range of representable numbers for an IEEE floating-point number with f bits allocated for the fraction, e bits allocated for the exponent, and the bias of e given by bias = 2(e−1)−1 is given below.
Normalized positive numbers are defined within the range 2(1−bias) to (2−2−f)2bias.
Normalized negative numbers are defined within the range −2(1−bias) to −(2−2−f)2bias.
Positive numbers greater than (2−2−f)2bias and negative numbers less than −(2−2−f)2bias are overflows.
Positive numbers less than 2(1−bias) and negative numbers greater than −2(1−bias) are either underflows or denormalized numbers.
Zero is given by a special bit pattern, where e = 0 and f = 0.
Overflows and underflows result from exceptional arithmetic conditions. Floating-point numbers outside the defined range are always mapped to ±Inf.
You can use the MATLAB commands realmin and realmax to determine the dynamic range of double-precision floating-point values for your computer.
A floating-point number is only an approximation of the “true” value because of a finite word size. Therefore, it is important to have an understanding of the precision (or accuracy) of a floating-point result. A value v with an accuracy q is specified by v ± q. For IEEE floating-point numbers,
v = (−1)s(2e–bias)(1.f)
q = 2–f × 2e–bias
Thus, the precision is associated with the number of bits in the fraction field.
In the MATLAB software, floating-point relative accuracy is given by the command eps, which returns the distance from 1.0 to the next larger floating-point number. For a computer that supports the IEEE Standard 754, eps = 2−52 or 2.22045 · 10-16.
Because floating-point numbers are represented using sign/magnitude, there are two representations of zero, one positive and one negative. For both representations e = 0 and f.0 = 0.0.
The IEEE Standard 754 specifies practices and procedures so that predictable results are produced independently of the hardware platform. Denormalized numbers, Inf, and NaN are defined to deal with exceptional arithmetic (underflow and overflow).
If an underflow or overflow is handled as Inf or NaN, then significant processor overhead is required to deal with this exception. Although the IEEE Standard 754 specifies practices and procedures to deal with exceptional arithmetic conditions in a consistent manner, microprocessor manufacturers might handle these conditions in ways that depart from the standard.
Denormalized numbers are used to handle cases of exponent underflow. When the exponent of the result is too small (i.e., a negative exponent with too large a magnitude), the result is denormalized by right-shifting the fraction and leaving the exponent at its minimum value. The use of denormalized numbers is also referred to as gradual underflow. Without denormalized numbers, the gap between the smallest representable nonzero number and zero is much wider than the gap between the smallest representable nonzero number and the next larger number. Gradual underflow fills that gap and reduces the impact of exponent underflow to a level comparable with round-off among the normalized numbers. Denormalized numbers provide extended range for small numbers at the expense of precision.
Arithmetic involving Inf (infinity) is treated as the limiting case of real arithmetic, with infinite values defined as those outside the range of representable numbers, or −∞ ≤ (representable numbers) < ∞. With the exception of the special cases discussed below (NaN), any arithmetic operation involving Inf yields Inf. Inf is represented by the largest biased exponent allowed by the format and a fraction of zero.
A NaN (not-a-number) is a symbolic entity encoded in floating-point format. There are two types of NaN: signaling and quiet. A signaling NaN signals an invalid operation exception. A quiet NaN propagates through almost every arithmetic operation without signaling an exception. The following operations result in a NaN: ∞–∞, –∞+∞, 0×∞, 0/0, and ∞/∞.
Both signaling NaN and quiet NaN are represented by the largest biased exponent allowed by the format and a nonzero fraction. The bit pattern for a quiet NaN is given by 0.f, where the most significant bit in f must be a one. The bit pattern for a signaling NaN is given by 0.f, where the most significant bit in f must be zero and at least one of the remaining bits must be nonzero.
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Fellow computer-napkin-mathers, it’s time for napkin problem #2. The last problem’s solution you’ll find at the end! I’ve updated sirupsen/napkin-math with last week’s tips and tricks—consult that repo if you need a refresher. My goal for that repo is to become a great resource for napkin calculations in the domain of computers. My talk from SRECON’s video was published this week, you can see it here.
Problem #2: Your SSD-backed database has a usage-pattern that rewards you with a 80% page-cache hit-rate (i.e. 80% of disk reads are served directly out of memory instead of going to the SSD). The median is 50 distinct disk pages for a query to gather its query results (e.g. InnoDB pages in MySQL). What is the expected average query time from your database?
Reply to this email with your answer, happy to provide you mine ahead of time if you’re curious.
Last Problem’s Solution
Question: How much will the storage of logs cost for a standard, monolithic 100,000 RPS web application?
Answer: First I jotted down the basics and convert them to scientific notation for easy calculation ~1 *10^3 bytes/request (1 KB), 9 * 10^4 seconds/day, and 10^5 requests/second. Then multiplied these numbers into storage per day: 10^3 bytes/request * 9 * 10^4 seconds/day * 10^5 requests/second = 9 * 10^12 bytes/day = 9 Tb/day. Then we need to use the monthly cost for disk storage from sirupsen/napkin-math (or your cloud’s pricing calculator) — $0.01 GB/month. So we have 9 Tb/day * $0.01 GB/month. We do some unit conversions (you could do this by hand to practise, or on Wolframalpha) and get to $3 * 10^3 per month, or
3,000 per month. Most of those who replied got somewhere between
1,000 and $10,000 — well within an order of magnitude!
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evalapply - Maple Help
Home : Support : Online Help : Programming : Evaluation : evalapply
user definable control over function application
`evalapply/V`(f, t))
When nontrivial Maple objects such as unevaluated function calls, lists, or sets are applied to arguments as if they were functions, the evalapply command determines the outcome. For example, an object of type function can be applied to arguments, as in V(f, g)(a, b, c).
The effect of applying the result of most built-in constructors to a sequence of arguments is determined by an internal evalapply function. The evalapply command also implements the application semantics for compositions and compositional powers.
The effect of applying a function call to arguments can be specified by defining an optional procedure of the name `evalapply/V`, where V is the name of the function.
When present, the procedure of the name `evalapply/V` is automatically invoked with f set to V(f, g), and t set to [a, b, c] in response to the function invocation V(f, g)(a, b, c).
V\left(f,g\right)\left(a,b,c\right)
\textcolor[rgb]{0,0,1}{V}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{g}\right)\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{c}\right)
`evalapply/V` := proc(f,t) local i;
V(seq(op(i,f)(op(t)),i=1..nops(f)));
V\left(f,g\right)\left(a,b,c\right)
\textcolor[rgb]{0,0,1}{V}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{c}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{c}\right)\right)
[a,b]\left(x,y\right)
[\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)]
{a,b}\left(x,y\right)
{\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)}
\left(a=b\right)\left(x,y\right)
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)
\left(a+b\right)\left(x,y\right)
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)
\left(a-b\right)\left(x,y\right)
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)
\left(ab\right)\left(x,y\right)
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)
\left(\frac{a}{b}\right)\left(x,y\right)
\frac{\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)}{\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)}
\left({a}^{b}\right)\left(x,y\right)
{\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)}^{\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)}
\left(a·b\right)\left(x,y\right)
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{·}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)
\left(a,b\right)\left(x,y\right)
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)
\left(a..b\right)\left(x,y\right)
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)
\left(a\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{and}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}b\right)\left(x,y\right)
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0,0,1}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)
\left(a\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{or}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}b\right)\left(x,y\right)
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0,0,1}{\mathbf{or}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)
\left(a\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{xor}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}b\right)\left(x,y\right)
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0,0,1}{\mathbf{xor}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)
\left(a⇒b\right)\left(x,y\right)
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{⇒}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)
\left(\mathbf{not}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}a\right)\left(x,y\right)
\textcolor[rgb]{0,0,1}{\mathbf{not}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)
\left(a@b\right)\left(x,y\right)
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)\right)
{a}^{\left(n\right)}\left(a\left(x,y\right)\right)
{\textcolor[rgb]{0,0,1}{a}}^{\left(\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)
\mathrm{combinat}['\mathrm{choose}']\left(4,3\right)
[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\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}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]]
\mathrm{map}\left({\mathrm{op}},\mathrm{combinat}['\mathrm{choose}']\left(4,3\right)\right)
[{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\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}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}}]
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The Relation Between the Porous Medium and the Eikonal Equations in Several Space Dimensions | EMS Press
We study the relation between the porous medium equation
u_t = \Delta(u^m), \; m>1
, and the eikonal equation
\nu_t = |D\nu|^2
. Under quite general assumptions, we prove that the pressure and the interface of the solution of the Cauchy problem for the porous medium equation converge as
m \downarrow 1
to the viscosity solution and the interface of the Cauchy problem for the eikonal equation. We also address the same questions for the case of the Dirichlet boundary value problem.
Pierre-Louis Lions, Panagiotis E. Souganidis, Juan Luis Vázquez, The Relation Between the Porous Medium and the Eikonal Equations in Several Space Dimensions. Rev. Mat. Iberoam. 3 (1987), no. 3, pp. 275–310
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Noncommutative algebraic geometry | EMS Press
The need for a noncommutative algebraic geometry is apparent in classical invariant and moduli theory. It is, in general, impossible to find commuting parameters parametrizing all orbits of a Lie group acting on a scheme. When one orbit is contained in the closure of another, the orbit space cannot, in a natural way, be given a scheme structure. In this paper we shall show that one may overcome these difficulties by introducing a noncommutative algebraic geometry, where affine "schemes" are modeled on associative algebras. The points of such an affine scheme are the simple modules of the algebra, and the local structure of the scheme at a finite family of points, is expressed in terms of a noncommutative deformation theory proposed by the author in \cite{Laudal2002}. More generally, the geometry in the theory is represented by a {\it swarm}, i.e. a diagram (finite or infinite) of objects (and if one wants, arrows) in a given
k
-linear Abelian category (
k
a field), satisfying some reasonable conditions. The noncommutative deformation theory refered to above, permits the construction of a presheaf of associative
k
-algebras, locally {\it parametrizing} the diagram. It is shown that this theory, in a natural way, generalizes the classical scheme theory. Moreover it provides a promising framework for treating problems of invariant theory and moduli problems. In particular it is shown that many moduli spaces in classical algebraic geometry are commutativizations of noncommutative schemes containing additional information.
Olav Arnfinn Laudal, Noncommutative algebraic geometry. Rev. Mat. Iberoam. 19 (2003), no. 2, pp. 509–580
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Multiphase Reactive Power Sensor - MapleSim Help
Home : Support : Online Help : MapleSim : MapleSim Component Library : Electrical : Multiphase Systems : Sensors : Multiphase Reactive Power Sensor
Reactive Power Sensor
Multiphase reactive power sensor
The Reactive Power Sensor component contains three single-phase power meters to measure total reactive power in a three-phase system.
{P}_{R}={i}_{1}\left({v}_{2}-{v}_{3}\right)+{i}_{2}\left({v}_{3}-{v}_{1}\right)+{i}_{3}\left({v}_{1}-{v}_{2}\right)
{\mathrm{plug}}_{p}
{\mathrm{plug}}_{n}
\mathrm{reactivePower}
Real output signal; reactive power
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Steiner Ellipse, Minimal Area Through Three Points | Brilliant Math & Science Wiki
The Steiner ellipse has the minimal area surrounding a triangle. It is characterized by having its center coincident with the triangle's centroid.
The Steiner ellipse can be extended to higher dimensions with one more point than the dimension. It is useful in a number of fields, such as statistics for determining which data points are outliers.
Proving That It Is Minimal
A Bit About Affine Transformations
Two Different Methods Are Presented
Getting the Ellipse's Conjugate Semidiameters (There's a Reason for This)
Getting the Ellipse's Equation
Second Method Follows
A 3-Dimensional Example
For a given chord or triangle base, the maximal area triangle occurs when the third vertex of the triangle is at the intersection of the chord's perpendicular bisector. Since that needs to be true for all three sides of the triangle, the triangle must be equilateral.
Using a calculus-based approach,
\text{Maximize}\left[\left\{\text{Area}\left[\text{Triangle}\left[\left( \begin{array}{cc} 1 & 0 \\ \cos (2 \pi \theta ) & \sin (2 \pi \theta ) \\ \cos (2 \pi \phi ) & \sin (2 \pi \phi ) \\ \end{array} \right)\right]\right],0\leq \theta \leq 1\land 0\leq \phi \leq 1\right\},\{\theta ,\phi \}\right] \Rightarrow
\left\{\frac{3 \sqrt{3}}{4},\left\{\theta \to \frac{2}{3},\phi \to \frac{1}{3}\right\}\right\}.
The area within the circle is
\pi
. The area within the triangle is
\frac{3 \sqrt{3}}{4}
. The ratio of the areas is
\frac{4 \pi }{3 \sqrt{3}}
2.41839915231229
Affine transformations preserve length ratios and linearities. This means they also preserve area ratios as those are just lengths squared, volume ratios in the 3-dimensional case as those are just lengths cubed, etc.
Affine transformations can be described by linear equations and therefore by matrix algebra. These matrix algebra operations can be eased by augmenting the matrices. Using
n
for new,
o
for old, and
a
for the augmented affine transformation matrix,
o
n
resemble
\left( \begin{array}{ccc} x_1 & x_2 & x_3 \\ y_1 & y_2 & y_3 \\ 1 & 1 & 1 \\ \end{array} \right).
The augmented affine transformation matrix
a
\left( \begin{array}{ccc} a_{1,1} & a_{1,2}& b_1 \\ a_{2,1} & a_{2,2} & b_2 \\ 0 & 0 & 1 \\ \end{array} \right).
The augmented affine transformation matrix equation is
n=a\cdot o
, where the dot
\cdot
is matrix multiplication.
Using this problem Is it a Circle? as an example, solve the following problem:
Find the forward augmented affine transformation matrix
\bm{a}
\bm{o}
\bm{n}
, from the problem's triangle to the equilateral triangle within the unit circle.
\bm{a}
matrix is as above.
\bm{o}
matrix is
\left( \begin{array}{ccc} -2 & 0 & 2 \\ 2 & -1 & 0 \\ 1 & 1 & 1 \\ \end{array} \right).
\bm{n}
\left( \begin{array}{ccc} 1 & -\frac{1}{2} & -\frac{1}{2} \\ 0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2} \\ 1 & 1 & 1 \\ \end{array} \right).
The matrix equation is
\bm{a}\cdot\bm{o}=\bm{n}.
Invert the
\bm{o}
matrix. If the matrix is singular, i.e. is not invertible, then the triangle is collinear and has a zero area.
Right matrix multiply both sides of the matrix equation above by the inverse of
\bm{o}
\bm{a}\cdot\bm{o}\cdot\bm{o}^{-1}=\bm{n}\cdot\bm{o}^{-1} \implies \bm{a} = \bm{n}\cdot\bm{o}^{-1},
\bm{a}s follows:
\left( \begin{array}{ccc} -2 & -\frac{2}{\sqrt{3}} & 0 \\ \frac{5}{3} & -\frac{1}{\sqrt{3}} & \frac{1}{3} \\ 0 & 0 & 1 \\ \end{array} \right).
By inverting that transformation matrix, one may go in the opposite direction.
\bm{a}
\left( \begin{array}{ccc} -\frac{3}{16} & \frac{3}{8} & -\frac{1}{8} \\ -\frac{1}{16} \left(5 \sqrt{3}\right) & -\frac{1}{8} \left(3 \sqrt{3}\right) & \frac{\sqrt{3}}{8} \\ 0 & 0 & 1 \\ \end{array} \right).
One method follows immediately. The other follows after the first method.
Unfortunately, which diameters of the unit circle map to the major and minor axes of the ellipse are still unknown.
What can be availed are a pair of conjugate semidiameters. One end of the semidiameters is the shared center of the triangle and of the Steiner ellipse. The centroid of the triangle is easily computed as it is the average or mean of the triangle's vertices' coordinates. A pair of conjugate semidiameters are orthogonal radii of the unit circle which when reverse affine transformed to the original environment become a set of conjugate semidiameters. One member of a set of conjugate semidiameters is readily at hand as one of the triangle's vertices was mapped to
x
-axis unit vector position,
(1,0)
. The other member of that set of conjugate semidiameters is not much harder as the base of the equilateral triangle is parallel to the
y
-axis and its length is
\sqrt{3}
in the unit circle environment. Therefore, another member of a conjugate semidiameter pair is one half of the Euclidean distance between the base vertices divided by
\sqrt{3}
in the original environment projected from the triangle's centroid parallel to the triangle's base, in either direction from the centroid.
One method for getting the major and minor axes and then the equation of the Steiner ellipse is a method called Rytz's construction.
Here is the original triangle:
Here is the same triangle decorated with the circle projected into the original environment, the solution ellipse which is the orange ellipse which surrounds the projected circle (they are the same), the two conjugate semidiameters in red and the parallelogram which is tangent to the ellipse where the conjugate semidiameters touch the ellipse, the corners of which are projections of the
(1,1),\,(1,-1),\,(-1,-1),\,(-1,1)
points in the unit circle environment.
The computations are done more easily with the ellipse center and triangle centroid at the origin.
Sorting the conjugate semidiameters by their distance from the origin gives
\left(\frac{2}{\sqrt{3}},\frac{1}{\sqrt{3}}\right)
\left(-2,\frac{5}{3}\right)
Rotate the shorter conjugate semidiameter by
90^o
in the direction that puts the end closer to the end of the longer conjugate semidiameter to get
\left(-\frac{1}{\sqrt{3}},\frac{2}{\sqrt{3}}\right)
and call that point
P'.
Call the point midway between
P'
and the end of the longer conjugate semidiameter
D:
\bigg(\frac{1}{2} \left(-2-\frac{1}{\sqrt{3}}\right),\frac{1}{2} \left(\frac{5}{3}+\frac{2}{\sqrt{3}}\right)\bigg).
Intersect an infinite line passing through
P'
D
and a circle centered at
D
and radius such that it passes through the origin (the green circle).
a=\Big(\frac{2}{39} \left(4 \sqrt{3}+3\right),\frac{1}{13} \left(4 \sqrt{3}+3\right)\Big)
b=\Big(\frac{1}{13} (-7) \left(\sqrt{3}+4\right),\frac{14}{39} \left(\sqrt{3}+4\right)\Big).
Intersect an infinite half-line starting at the origin and in the direction of point
a
immediately above with a (red) circle centered at the origin and of a radius of the Euclidean distance between the end of the longer conjugate semidiameter and point
b
immediately above. That is the end of the ellipse's semi-minor axis:
\left(\frac{4}{\sqrt{39}},2 \sqrt{\frac{3}{13}}\right).
b
immediately above with a (blue) circle centered at the origin and of a radius of the Euclidean distance between the end of the longer conjugate semidiameter and point
a
\left(-\frac{8}{\sqrt{13}},\frac{16}{3 \sqrt{13}}\right).
Therefore, the ellipse will be tangent to the red and blue circles mentioned in the previous two paragraphs.
The general equation for an ellipse centered at the origin is
\text{ga}\,\text{xs}^2+\text{gb}\,\text{xs}\,\text{ys}+\text{gc}\,\text{ys}^2=1
, using two character variable names.
Using the coordinates of the ends of the semi-major and semi-minor axes and one of the triangles' vertices, three equations in three unknowns can be written:
\frac{64 \text{ga}}{13}-\frac{128 \text{gb}}{39}+\frac{256 \text{gc}}{117}=1\land \frac{16 \text{ga}}{39}+\frac{8 \text{gb}}{13}+\frac{12 \text{gc}}{13}=1\land \frac{4 \text{ga}}{3}+\frac{2 \text{gb}}{3}+\frac{\text{gc}}{3}=1
. Solving those equations gives
\text{ga}\to \frac{21}{64},\text{gb}\to \frac{9}{16},\text{gc}\to \frac{9}{16}
. This results in the equation:
21\, x^2+36\, x\, y+36 y^2=64
. Moving the center from the origin to its proper location gives the final ellipse equation:
7\,x^2+12\, x\,y+12 y^2=4 (x+2 y+5)
A copy of the inverse of the transformation matrix
\bm{a}
, which is called
\bm{b},
\left( \begin{array}{cc|c} -2 & -\frac{2}{\sqrt{3}} & 0 \\ \frac{5}{3} & -\frac{1}{\sqrt{3}} & \frac{1}{3} \\ \hline 0 & 0 & 1 \\ \end{array} \right).
The first two columns and first two rows are affine matrix (the return rotation matrix).
The third column first two rows of
\bm{b}
is the translation from the origin of the center of the ellipse.
The parametric form of the ellipse equation is
\left\{-\frac{2 \sin (\theta )}{\sqrt{3}}-2 \cos (\theta ),-\frac{\sin (\theta )}{\sqrt{3}}+\frac{5 \cos (\theta )}{3}+\frac{1}{3}\right\}
The sIngular-value decomposition, also on Wikipedia and Wolfram MathWorld, of the first two columns and the first two rows, a square 2
\times
2 matrix, is
\begin{aligned} \bm{u}&=\left( \begin{array}{cc} -\frac{3}{\sqrt{13}} & -\frac{2}{\sqrt{13}} \\ \frac{2}{\sqrt{13}} & -\frac{3}{\sqrt{13}} \\ \end{array} \right)\\ \bm{\sigma}&=\left( \begin{array}{cc} \frac{8}{3} & 0 \\ 0 & \frac{2}{\sqrt{3}} \\ \end{array} \right)\\ \bm{v}&=\left( \begin{array}{cc} \frac{7}{2 \sqrt{13}} & -\frac{\sqrt{\frac{3}{13}}}{2} \\ \frac{\sqrt{\frac{3}{13}}}{2} & \frac{7}{2 \sqrt{13}} \\ \end{array} \right). \end{aligned}
These three matrices are
\bm{u}
, the rotation of the ellipse into its final orientation
\bm{\sigma}
, the scaling that establishes the lengths of the semi-major and semi-minor axes, respectively
\bm{v}
, the rotation of what will become the semi-major and semi-minor axes onto the
x
y
x
y
onto the unit circle coordinate system using the
\bm{a}ffine transform:
\left\{-\frac{3 x}{16}+\frac{3 y}{8}-\frac{1}{8},-\frac{5 \sqrt{3} x}{16}-\frac{1}{8} 3 \sqrt{3} y+\frac{\sqrt{3}}{8}\right\}.
Substitute those transformed
x
y
definitions into the equation of the unit circle
x^2+y^2=1
7 x^2+12 x y+12 y^2=20+4 x+8 y.
Create an affine transformation from
\bm{v}
used to transform the unit vectors and their negatives into the unit circle coordinate system to locate the ends of the ellipse's axes:
\left(\bm{t}= \begin{array}{cc|c} \frac{7}{2 \sqrt{13}} & -\frac{\sqrt{\frac{3}{13}}}{2} & 0 \\ \frac{\sqrt{\frac{3}{13}}}{2} & \frac{7}{2 \sqrt{13}} & 0 \\ \hline 0 & 0 & 1 \\ \end{array} \right).
\bm{b}
\bm{t}
. Note, the application order is
\bm{t}
\bm{b}
\left( \begin{array}{cc|c} -\frac{8}{\sqrt{13}} & -\frac{4}{\sqrt{39}} & 0 \\ \frac{16}{3 \sqrt{13}} & -2 \sqrt{\frac{3}{13}} & \frac{1}{3} \\ \hline 0 & 0 & 1 \\ \end{array} \right)
Transforming {1, 0}, {-1, 0}, {0, 1}, {0, -1} using the transformation immediately above gives
\left\{-\frac{8}{\sqrt{13}},\frac{1}{3}+\frac{16}{3 \sqrt{13}}\right\},\left\{\frac{8}{\sqrt{13}},\frac{1}{3}-\frac{16}{3 \sqrt{13}}\right\},\left\{-\frac{4}{\sqrt{39}},\frac{1}{3}-2 \sqrt{\frac{3}{13}}\right\},\left\{\frac{4}{\sqrt{39}},2 \sqrt{\frac{3}{13}}+\frac{1}{3}\right\}.
The black dots are the original points, the ellipse is plotted from the equation provided above in upper and lower sections, and the red lines are the semi-major and semi-minor axes of the ellipse:
This method readily can be extended to higher dimensions.
The following two animated GIFs illustrate the same ellipsoid. The difference is what is kept constant.
In this clip, the visual image size is kept constant and the axes are allowed to change their length to do that:
In this clip, the axes are kept constant size and the visual image size changes to do that:
Note well that the ratio of the ellipsoid volume divided by the tetrahedron volume remains constant:
\frac{3 \sqrt{3} \pi }{2} \approx 8.16209713905398.
Cite as: Steiner Ellipse, Minimal Area Through Three Points. Brilliant.org. Retrieved from https://brilliant.org/wiki/steiner-ellipse-minimal-area-through-three-points/
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Option price by Heston model using FFT and FRFT - MATLAB optByHestonFFT - MathWorks India
\mathrm{max}\left(St-K,0\right)
\mathrm{max}\left(K-St,0\right)
\begin{array}{l}d{S}_{t}=\left(r-q\right){S}_{t}dt+\sqrt{{v}_{t}}{S}_{t}d{W}_{t}\\ d{v}_{t}=\kappa \left(\theta -{v}_{t}\right)dt+{\sigma }_{v}\sqrt{{v}_{t}}d{W}_{t}^{v}\\ \text{E}\left[d{W}_{t}d{W}_{t}^{v}\right]=pdt\end{array}
{f}_{Hesto{n}_{j}}\left(\varphi \right)
\begin{array}{l}{f}_{Hesto{n}_{j}}\left(\varphi \right)=\mathrm{exp}\left({C}_{j}+{D}_{j}{v}_{0}+i\varphi \mathrm{ln}{S}_{t}\right)\\ {C}_{j}=\left(r-q\right)i\varphi \tau +\frac{\kappa \theta }{{\sigma }_{v}^{2}}\left[\left({b}_{j}-p{\sigma }_{v}i\varphi +{d}_{j}\right)\tau -2\mathrm{ln}\left(\frac{1-{g}_{j}{e}^{{d}_{j}\tau }}{1-{g}_{j}}\right)\right]\\ {D}_{j}=\frac{{b}_{j}-p{\sigma }_{v}i\varphi +{d}_{j}}{{\sigma }_{v}^{2}}\left(\frac{1-{e}^{{d}_{j}\tau }}{1-{g}_{j}{e}^{{d}_{j}\tau }}\right)\\ {g}_{j}=\frac{{b}_{j}-p{\sigma }_{v}i\varphi +{d}_{j}}{{b}_{j}-p{\sigma }_{v}i\varphi -{d}_{j}}\\ {d}_{j}=\sqrt{{\left({b}_{j}-p{\sigma }_{v}i\varphi \right)}^{2}-{\sigma }_{v}^{2}\left(2{u}_{j}i\varphi -{\varphi }^{2}\right)}\\ \text{where for }j=1,2:\\ {u}_{1}=\frac{1}{2},{u}_{2}=-\frac{1}{2},{b}_{1}=\kappa +{\lambda }_{VolRisk}-p{\sigma }_{v},{b}_{2}=\kappa +{\lambda }_{VolRisk}\end{array}
\begin{array}{l}{C}_{j}=\left(r-q\right)i\varphi \tau +\frac{\kappa \theta }{{\sigma }_{v}{}^{2}}\left[\left({b}_{j}-p{\sigma }_{v}i\varphi -{d}_{j}\right)\tau -2\mathrm{ln}\left(\frac{1-{\epsilon }_{j}{e}^{-{d}_{j}\tau }}{1-{\epsilon }_{j}}\right)\right]\\ Dj=\frac{{b}_{j}-p{\sigma }_{v}i\varphi -{d}_{j}}{{\sigma }_{v}^{2}}\left(\frac{1-{e}^{-{d}_{j}\tau }}{1-{\epsilon }_{j}{e}^{-{d}_{j}\tau }}\right)\\ {\epsilon }_{j}=\frac{{b}_{j}-p{\sigma }_{v}i\varphi -{d}_{j}}{{b}_{j}-p{\sigma }_{v}i\varphi +{d}_{j}}\end{array}
\begin{array}{l}Call\left(k\right)=\frac{{e}^{-\alpha k}}{\pi }{\int }_{0}^{\infty }\mathrm{Re}\left[{e}^{-iuk}\psi \left(u\right)\right]du\\ \psi \left(u\right)=\frac{{e}^{-r\tau }{f}_{2}\left(\varphi =\left(u-\left(\alpha +1\right)i\right)\right)}{{\alpha }^{2}+\alpha -{u}^{2}+iu\left(2\alpha +1\right)}\\ Put\left(K\right)=Call\left(K\right)+K{e}^{-r\tau }-{S}_{t}{e}^{-q\tau }\end{array}
\mathrm{ln}\left({S}_{t}\right)-\frac{N}{2}\Delta k
\mathrm{ln}\left({S}_{t}\right)+\left(\frac{N}{2}-1\right)\Delta k
{S}_{t}\mathrm{exp}\left(-\frac{N}{2}\Delta k\right)
{S}_{t}\mathrm{exp}\left[\left(\frac{N}{2}-1\right)\Delta k\right]
Call\left({k}_{n}\right)=\Delta u\frac{{e}^{-\alpha {k}_{n}}}{\pi }\sum _{j=1}^{N}\mathrm{Re}\left[{e}^{-i\Delta k\Delta u\left(j-1\right)\left(n-1\right){e}^{i{u}_{j}}\left[\frac{N\Delta k}{2}-\mathrm{ln}\left({S}_{t}\right)\right]}\psi \left({u}_{j}\right)\right]{w}_{j}
\Delta k\Delta u=\left(\frac{2\pi }{N}\right)
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Bipul Sarma, "Some Double Sequence Spaces of Fuzzy Real Numbers of Paranormed Type", Journal of Mathematics, vol. 2013, Article ID 627047, 4 pages, 2013. https://doi.org/10.1155/2013/627047
Bipul Sarma1
1Department of Mathematics, Madhab Choudhury College-Gauhati University, Barpeta, Assam 781301, India
Academic Editor: Pierpaolo D'Urso
We study different properties of convergent, null, and bounded double sequence spaces of fuzzy real numbers like completeness, solidness, sequence algebra, symmetricity, convergence-free, and so forth. We prove some inclusion results too.
Throughout the paper, a double sequence is denoted by , a double infinite array of elements , where each is a fuzzy real number.
The initial work on double sequences is found in Bromwich [1]. Later on, it was studied by Hardy [2], Móricz [3], Tripathy [4], Basarir and Sonalcan [5], and many others. Hardy [2] introduced the notion of regular convergence for double sequences.
The concept of paranormed sequences was studied by Nakano [6] and Simons [7] at the initial stage. Later on, it was studied by many others.
After the introduction of fuzzy real numbers, different classes of sequences of fuzzy real numbers were introduced and studied by Tripathy and Nanda [8], Choudhary and Tripathy [9], Tripathy et al. [10–13], Tripathy and Dutta [14–16], Tripathy and Borgogain [17], Tripathy and Das [18], and many others.
Let denote the set of all closed and bounded intervals on , the real line. For , , we define where and . It is known that is a complete metric space.
A fuzzy real number is a fuzzy set on , that is, a mapping associating each real number with its grade of membership .
The -level set of the fuzzy real number , for , is defined as .
The set of all upper semicontinuous, normal, and convex fuzzy real numbers is denoted by , and throughout the paper, by a fuzzy real number, we mean that the number belongs to .
Let , and let the -level sets be ; the product of and is defined by
A fuzzy real number is called convex if , where .
If there exists such that , then the fuzzy real number is called normal.
A fuzzy real number is said to be upper semicontinuous if, for each , , for all , is open in the usual topology of .
The set of all real numbers can be embedded in . For , is defined by
The absolute value, of , is defined by (see, e.g., [19])
A fuzzy real number is called nonnegative if , for all . The set of all nonnegative fuzzy real numbers is denoted by .
Then defines a metric on .
The additive identity and multiplicative identity in are denoted by , respectively.
A sequence of fuzzy real numbers is said to be convergent to the fuzzy real number if, for every , there exists such that , for all .
A sequence of fuzzy numbers converges to a fuzzy number if both and hold for every [20].
A sequence of generalized fuzzy numbers converges weakly to a generalized fuzzy number (and we write ) if distribution functions converge weakly to and converge weakly to [21].
A double sequence of fuzzy real numbers is said to be convergent in Pringsheim’s sense to the fuzzy real number if, for every , there exists , such that , for all , .
A double sequence of fuzzy real numbers is said to be regularly convergent if it converges in Pringsheim’s sense, and the following limits exist:
A fuzzy real number sequence is said to be bounded if , for some .
For and , we define
Throughout the paper , , , , , and denote the classes of all, bounded, convergent in Pringsheim’s sense, null in Pringsheim’s sense, regularly convergent, and regularly null fuzzy real number sequences, respectively.
A double sequence space is said to be solid (or normal) if , whenever , for all , , for some .
Let , and let be a double sequence space. A K-step space of is a sequence space
A canonical preimage of a sequence is a sequence defined as follows:
A canonical preimage of a step space is a set of canonical preimages of all elements in .
A double sequence space is said to be monotone if contains the canonical preimage of all its step spaces.
From the above definitions, we have the following remark.
Remark 1. A sequence space is solid is monotone.
A double sequence space is said to be symmetric if , whenever , where is a permutation of .
A double sequence space is said to be sequence algebra if , whenever , .
A double sequence space is said to be convergence-free if , whenever , and implies that .
Sequences of fuzzy real numbers relative to the paranormed sequence spaces were studied by Choudhary and Tripathy [9].
In this paper, we introduce the following sequence spaces of fuzzy real numbers.
Let be a sequence of positive real numbers
For , we get the class .
Also a fuzzy sequence if , and the following limits exist:
For the class of sequences , .
We define , .
Theorem 2. Let be bounded. Then, the classes of sequences , , , , and are complete metric spaces with respect to the metric defined by
Proof. We prove the result for . Let be a Cauchy sequence in . Then, for a given , there exists such that
Since is complete, there exist fuzzy numbers such that , for each , .
Taking in (13), we have
Using the triangular inequality we have . Hence, is complete.
Property 1. The space is symmetric, but the spaces , , , , , and are not symmetric.
Proof. Obviously the space is symmetric. For the other spaces, consider the following example.
Example 3. Consider the sequence space . Let , for all and , otherwise. Let the sequence be defined by and for ,
Let be a rearrangement of defined by and for ,
Then, , but . Hence, is not symmetric. Similarly, it can be established that the other spaces are also not symmetric.
Theorem 4. The spaces , , , and are solid.
Proof. Consider the sequence space . Let , and let be such that .
The result follows from the inequality
Hence, the space is solid. Similarly, the other spaces are also solid.
Property 2. The spaces , , and are not monotone and hence are not solid.
Proof. The result follows from the following example.
Example 5. Consider the sequence space . Let for even and , otherwise. Let . Let be defined by the following:
for all , , Then, . Let be the canonical preimage of for the subsequence of . Then, Then, . Thus, is not monotone. Similarly, the other spaces are also not monotone. Hence, the spaces , , and are not solid.
Property 3. The spaces , , , , , , and are not convergence-free.
The result follows from the following example.
Example 6. Consider the sequence space . Let , for all , , otherwise. Consider the sequence defined by and for other values,
Let the sequence be defined by and for other values, Then, , but . Hence, the space is not convergence-free. Similarly, the other spaces are also not convergence-free.
Theorem 7. , for , , , . The inclusions are strict.
Proof. Since convergent sequences are bounded, the proof is clear.
Theorem 8. Let , for all . Then, for , , , , , .
Proof. Consider the sequence spaces and . Let .
Then, , for all .
The result follows from the inequality .
Theorem 9. The spaces , , , , , , and are sequence algebras.
Proof. Consider the sequence space . Let , . Then, the result follows immediately from the inequality
The author’s work is supported by UGC Project no. F. 5-294/2009-10 (MRP/NERO).
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Copyright © 2013 Bipul Sarma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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David Young 2021-12-16 Answered
y={x}^{3},\text{ }x+y=30,\text{ }y=0
y={x}^{3},\text{ }x+y=30,\text{ }y=0
A={\int }_{0}^{27}\left(30-y\right)-{y}^{\frac{1}{3}}dy
A={\left(30y-\frac{{y}^{2}}{2}-\frac{3}{4}{y}^{\frac{4}{3}}\right)}_{0}^{27}
A=\left(30\left(27\right)-\frac{{27}^{2}}{2}-\frac{3}{4}{\left(27\right)}^{\frac{4}{3}}\right)
A=384.75
{M}_{x}={\int }_{0}^{27}y\left(30-y-{y}^{\frac{1}{3}}\right)dy
{M}_{x}={\left(15{y}^{2}-\frac{{y}^{3}}{3}-\frac{3}{7}{y}^{\frac{7}{3}}\right)}_{0}^{23}
{M}_{x}=\frac{24057}{7}
{M}_{y}=\frac{1}{2}{\int }_{0}^{27}{\left(30-y\right)}^{2}-{y}^{\frac{2}{3}}dy
{M}_{y}=\frac{22113}{5}
In a truck-loading station at a post office, a small 0.200-kg package is released from rest a point A on a track that is one quarter of a circle with radius 1.60 m.The size of the packageis much less than 1.60m, so the package can be treated at aparticle. It slides down the track and reaches point B with a speed of 4.80 m/s. From point B, it slides on a level surface a distanceof 3.00 m to point C, where is comes to rest.
(a) What is the coefficient of kinetic friction on the horizontal surface?
(b) How much work is done on the package by friction as it slides down the circular arc from A to B?
\left\{x\in {R}^{\prime }\mid 0<x<1\right\}
\left\{x\in R\mid x\le 0\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}x⇒1\right\}
\left\{n\in Z\mid n\text{ }is\text{ }a\text{ }factor\text{ }of\text{ }6\right\}
\left\{n\in Z\cdot \mid n\text{ }is\text{ }a\text{ }factor\text{ }of\text{ }6\right\}
Suppose, household color TVs are replaced at an average age of
\mu =7.4years
after purchase, and the (95% of data) range was from 5.4 to 9.4 years. Thus, the range was
9.4-5.4=4.0
years. Let x be the age (in years) at which a color TV is replaced. Assume that x has a distribution that is approximately normal.
(a) The empirical rule indicates that for a symmetric and bell-shaped distribution, approximately 95% of the data lies within two standard deviations of the mean. Therefore, a 95% range of data values extending from
\mu -2\sigma \text{ }\to \text{ }\mu +2\sigma
is often used for "commonly occurring" data values. Note that the interval from
\mu -2\sigma \text{ }\to \text{ }\mu +2\sigma \text{ }is\text{ }4\sigma
in length. This leads to a "rule of thumb" for estimating the standard deviation from a 95% range of data values.
Estimating the standard deviation
For a symmetric, bell-shaped distribution,
\text{standard deviation}\approx range\text{ }4\approx high\text{ }value\text{ }4-low\text{ }value
where it is estimated that about 95% of the commonly occurring data values fall into this range.
Use this "rule of thumb" to approximate the standard deviation of x values, where x is the age (in years) at which a color TV is replaced. (Round your answer to one decimal place.)
(b) What is the probability that someone will keep a color TV more than 5 years before replacement? (Round your answer to four decimal places.)
(c) What is the probability that someone will keep a color TV fewer than 10 years before replacement? (Round your answer to four decimal places.)
Find the margin of error for the given values of
c,\sigma ,
c=0.90,\sigma =2.9,n=81
A person stands on a scale inside an elevator at rest. The Scale reads 800N.
a) What is the persons
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Precise Position Adjustment of Automotive Electrohydraulic Coupling System With Parameter Perturbations | J. Dyn. Sys., Meas., Control. | ASME Digital Collection
Mingming Mei,
Mingming Mei
School of Vehicle and Mobility, Tsinghua University
Haidian District, Beijing 100084,
e-mail: mmm19@mails.tsinghua.edu.cn
Shuo Cheng,
1Corresponding author. e-mail: shuocheng9@yeah.net
e-mail: liangl@tsinghua.edu.cn
e-mail: yanbj0219@126.com
Liang Li Professor
Mei, M., Cheng, S., Li, L., and Yan, B. (March 7, 2022). "Precise Position Adjustment of Automotive Electrohydraulic Coupling System With Parameter Perturbations." ASME. J. Dyn. Sys., Meas., Control. May 2022; 144(5): 051008. https://doi.org/10.1115/1.4053430
Based on the guaranteed cost theory, this paper proposes a robust controller for the automotive electrohydraulic coupling system. However, parameter perturbation caused by the model linearization is a critical challenge for the nonlinear electrohydraulic coupling system. Generally, the electrical brake booster system (E-booster) can be separated into three parts, a permanent magnet synchronous motor (PMSM), a hydraulic model of the master cylinder, and the transmission mechanism. In this paper, the robust guaranteed cost controller (RGCC) is adopted to achieve an accurate regulation of the pushrod position of the E-booster and has strong robustness against internal uncertainties, and the linear extended state observer (LESO) is utilized to optimize E-booster's dynamic performance. Specifically, the tracking differentiator (TD) and LESO are used to improve the dynamic precision and reduce the hysteresis effect. The overshoot is suppressed by TD, and the disturbance caused by nonlinear uncertainty is restrained by LESO. The experimental results show that RGCC sacrifices a 6% phase lag in the low-frequency domain for a 10% and 40% reduction in first and second-order, respectively, compared with the proportion integration differentiation (PID). Results demonstrate that RGCC has higher precision and stronger robustness in dynamic behavior.
Control equipment, Cylinders, Design, Engines, Friction, Motors, State estimation, Dynamic models, Permanent magnets, Displacement, Signals, Brakes, Uncertainty, Pressure, Simulation results, Simulation
A Novel Regenerative Electrohydraulic Brake System: Development and Hardware-in-Loop Tests
A Novel Pressure Control Strategy of an Electro-Hydraulic Brake System Via Fusion of Control Signals
Proc. Inst. Mech. Eng., Part D J. Autom. Eng.
Precise Active Brake-Pressure Control for a Novel Electro-Booster Brake System
, and Songyun, X.,
Hydraulic Pressure Control System of Integrated-⋅ Electro–Hydraulic Brake System Based on Byrnes-Isidori Normalized Form
Adaptive Cascade Control of a Brake-by-Wire Actuator for Sport Motorcycles
Nonlinear Pressure Control for BBW Systems Via Dead-Zone and Antiwindup Compensation
Vimala Starbino
Improved Position Tracking Performance of Electro Hydraulic Actuator Using PID and Sliding Mode Controller
Quantitative Fault Tolerant Control Design for a Leaking Hydraulic Actuator
Position Tracking Control of Electro-Hydraulic Single-Rod Actuator Based on an Extended Disturbance Observer
G. T.-C.
Robust H
∞
Position Control Synthesis of an Electro-Hydraulic Servo System
∞
Tracking Control for Water Level in the u-Tube Steam Generator
,” M.D. thesis, Harbin Engineering University.
High Accuracy Adaptive Robust Control for Permanent Magnet Synchronous Motor Systems
Decentralized Robust Adaptive Control of Nonlinear Systems With Unmodeled Dynamics
Adaptive Output Feedback Control of Nonlinear Systems Represented by Input-Output Models
Robust Adaptive Observer for Nonlinear Systems With Unmodeled Dynamics
Delay Compensation Position Tracking Control of Electro-Hydraulic Servo Systems Based on a Delay Observer
Proc. Inst. Mech. Eng., Part I J. Syst. Control Eng.
Development of an Electrically Driven Intelligent Brake System
SAE Int. J. Passenger Cars-Mech. Syst.
Adaptive Guaranteed Cost Control of Systems With Uncertain Parameters
An LMI Approach to Guaranteed Cost Control of Linear Uncertain Time-Delay Systems
Quadratic Guaranteed Cost Control With Robust Pole Placement in a Disk
.10.1049/ip-cta:19960058
Output Feedback H
∞
Control of Systems With Parameter Uncertainty
Nonlinear Tracking-Differentiator
J. Syst. Math.
Scaling and Bandwidth-Parameterization Based Controller Tuning
, Denver, CO, June 4–6, pp.
Position and Current Control of an Interior Permanent-Magnet Synchronous Motor by Using Loop-Shaping Methodology: Blending of H ∞ Mixed-Sensitivity Problem and T–S Fuzzy Model Scheme
State Estimation Using Randomly Delayed Measurements
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Discrete Optimization Modelling (with MiniZinc) | Brilliant Math & Science Wiki
Discrete Optimization Modelling (with MiniZinc)
Combinatorial Problems are extremely difficult. In the industry, we are more interested in tasks like "Maximize Profit", "Minimize Costs" than questions like "in how many ways..". Some of this problems are proven to be so hard (NP-Complete) that a general solving time shoots up exponentially unless P = NP.
In the real world, we need an answer to these problems anyway. There are many sophisticated solvers such as gecode, flatzinc, Google or-tools and many more. (You might think of Prolog too.) They try to exploit substructures and simplify the problems with sophisticated techniques gathered from several PhD thesies and research papers.
These solvers can be used to solve a variety of problems ranging from several logic puzzles, solving a minesweeper board to serious problems like linear programming, bin packing or scheduling problems.
Dissecting the first example
Models vs. Instances
Structure of a MiniZinc Model
Modelling is in no way a form procedural programming. It is a form of declarative programming. To clarify, when modelling we make a very specific and more importantly, non-ambiguous description of the problem.
This description can be handed over to the solver, who decides how to solve the problem. That is not to say that all equivalent models of the same problem are equally as powerful or we have absolutely no control over the solving mechanism. Just like the human brain, the solver have varying difficulties solving a problem even though they are technically equivalent. Also, to some extent, it is possible to specify the search technique that the solver is going to use.
Here is a simple example of a MiniZinc model of a Linear Programming Problem
We know how to make two sorts of cakes.
A banana cake which takes 250g of self-raising flour, 2 mashed bananas, 75g sugar and 100g of butter, and a chocolate cake which takes 200g of self-raising flour, 75g of cocoa, 150g sugar and 150g of butter. We can sell a chocolate cake for $4.50 and a banana cake for $4.00. And we have 4kg self-raising flour, 6 bananas, 2kg of sugar, 500g of butter and 500g of cocoa.
How many cakes of each sort should we bake to maximize profit?
output ["no. of banana cakes = ", show(b), "\n",
"no. of chocolate cakes = ", show(c), "\n"];
MiniZinc is a constraint modelling language that offers a higher level modelling interface for specifying problems to a number of lower level solvers like Flatzinc or gecode.
One can easily download the MiniZinc packages from here
One can just download the command line executables or the IDE if he pleases.
Imagine for a second that this were a Math class. How would you go about formulating the problem?
Here is a possible formulation:
be the number of banana cakes,
c
the number of chocolate cakes.
p(b,c) = 400b + 500c
250b + 200c \leq 4000 \\ 2b \leq 6 \\ 75b + 150c \leq 2000 \\ 100b + 150c \leq 500 \\ 75c \leq 500
Notice that our model is exactly the same as that.
First, we begin with declaring our decision variables, i.e, the things we wish for the solver to solve.
var int: b; % no. of banana cakes
var int: c; % no. of chocolate cakes
Then, we specify the constraints:
And then comes what we want the solver to do:
If you're using the IDE, this section will not help much but it is still recommended you learn the tricks
The next most important thing you should do is to download the model and run it. Note that minizinc files always end with a .mzn extension.
Go to the appropriate directory, and type minizinc cakes.mzn.
On my computer, I see something like this.
A couple of remarks to make:
The ---------- marks the end of a solution.
The ========== tells you that the solver has proven this solution to be the optimal solution (according to the criteria given).
You might get an ====UNSATISFIABLE==== implying that your model has no feasible solutions.
If you are interested in seeing all feasible solutions, run minizinc --all-solutions cakes.mzn.
Sometimes, we want to run models with additional data files. (See the next section).
The generic bash command to supply a model is minizinc <model.mzn> <data.dzn>
The data files are required to end with a .dzn extension.
Usually, we want to write models that are very general. In contrast, an instance is a combination of a model with some data that is passed on to a solver.
Here is an example of how we could split up the above problem into a generalized model and a data file:
cakes2.mzn
"Amount of flour is non-negative");
"Amount of banana is non-negative");
"Amount of sugar is non-negative");
"Amount of butter is non-negative");
"Amount of cocoa is non-negative");
pantry.dzn
Cite as: Discrete Optimization Modelling (with MiniZinc). Brilliant.org. Retrieved from https://brilliant.org/wiki/discrete-optimization-modelling-with-minizinc/
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Network security equipment evaluation based on attack tree with risk fusion
CHENG Ran,LU Yue-ming
School of Information and Communication Engineering,Beijing University of Posts and Communications,Beijing 100876,China
作者简介:CHENG Ran (1994-),born in Anhui. She is working on her master degree at Beijing University of Posts and Telecommunications. Her research interests include distributed computation and information security.|LU Yueming (1969-),born in Jiangsu. He received his Ph.D degree of computer architecture from Xi’an Jiaotong University in 2000. He is a professor in Beijing University of Posts and Telecommunications. His research interests include network simulation,network security and distributed computing.
The Research of Key Technology and Application of Information Security Certification Project(2016YFF0204001)
Network security equipment is crucial to information systems,and a proper evaluation model can ensure the quality of network security equipment.However,there is only a few models of comprehensive models nowadays.An index system for network security equipment was established and a model based on attack tree with risk fusion was proposed to obtain the score of qualitative indices.The proposed model implements attack tree model and controlled interval and memory (CIM) model to solve the problem of quantifying qualitative indices,and thus improves the accuracy of the evaluation.
关键è¯: attack tree, evaluation, network security equipment, risk fusion
CHENG Ran,LU Yue-ming. Network security equipment evaluation based on attack tree with risk fusion[J]. 网络与信æ¯å®‰å…¨å¦æŠ¥, 2017, 3(7): 70-77.
Ran CHENG,Yue-ming LU. [J]. Chinese Journal of Network and Information Security, 2017, 3(7): 70-77.
Network security equipment evaluation model based on attack tree with risk fusion"
The index system of network security equipment"
A simple attack tree"
The attack tree corresponding to data response index"
The risk distributions of atom events"
Event Risk interval
0.05～0.15 0.15～0.25 0.25～0.35 0.35～0.45 0.45～0.55
\frac{1}{10}
\frac{2}{10}
\frac{4}{10}
\frac{2}{10}
\frac{1}{10}
\frac{1}{10}
\frac{1}{10}
\frac{2}{10}
\frac{3}{10}
\frac{3}{10}
\frac{3}{10}
\frac{4}{10}
\frac{2}{10}
\frac{1}{10}
\frac{1}{10}
\frac{1}{10}
\frac{5}{10}
\frac{3}{10}
Fusion of the risk of C1 and C2 using parallel model"
Risk Interval Risk Distribution
0.05～0.15
\frac{1}{10}×\frac{1}{10}=0.01
\frac{2}{10}×\left(\frac{1}{10}+\frac{1}{10}\right)+\frac{1}{10}×\frac{1}{10}=0.05
\frac{4}{10}×\left(\frac{1}{10}+\frac{1}{10}+\frac{2}{10}\right)+\frac{2}{10}×\left(\frac{1}{10}+\frac{2}{10}\right)=0.22
\frac{2}{10}×\left(\frac{1}{10}+\frac{1}{10}+\frac{2}{10}+\frac{3}{10}\right)+\frac{3}{10}×\left(\frac{1}{10}+\frac{2}{10}+\frac{4}{10}\right)=0.35
\frac{1}{10}×\left(\frac{1}{10}+\frac{1}{10}+\frac{2}{10}+\frac{4}{10}+\frac{3}{10}\right)+\frac{3}{10}×\left(\frac{1}{10}+\frac{2}{10}+\frac{4}{10}+\frac{2}{10}\right)=0.37
The risk distribution of event B3"
0.05～0.15 0.000 3
0.25～0.35 0.168
0.45～0.55 0.37
The risk distribution of event A"
Interval’s midpoint Interval’s probability
[1] LIPPMANN R P , FRIED D J , GRAF I ,et al. Evaluating intrusion detection systems:the 1998 DARPA off-line intrusion detection evaluation[C]// DARPA Information Survivability Conference and Exposition(DISCEX'00). 2000: 12-26.
[2] HERRMANN D S . Using the common criteria for IT security evaluation[M]. Florida: CRC PressPress, 2002.
[3] HAN L M Q . Analysis and study on AHP-fuzzy comprehensive evaluation[J]. China Safety Science Journal, 1983.
[4] FUNG C , CHEN Y L , WANG X ,et al. Survivability analysis of distributed systems using attack tree methodology[C]// Military Communications Conference. 2005: 583-589.
[5] CHAPMAN C B , COOPER D F . Risk engineering:basic controlled interval and memory models[J]. Journal of the Operational Research Society, 1983,34(1): 51-60.
[6] SAATY L . How to make a decision:the analytic hierarchy process[J]. European Journal of Operational Research, 1990,48(1): 9-26.
[7] PETTERSSON J . A study on software management approaches:proposing a project support tool[J]. University West Library, 2003.
[8] VAN-HOLSTEIJN F A . The motivation of attackers in attack tree analysis[J]. TU Delft Library, 2015.
[9] ZHANG X , BAI Y , LV L . Application of the controlled interval and memory model in the risk assessment of city gas transmission and distribution networks[C]// The International Conference on Pipelines and Trenchless Technology. 2012.
[10] GONG Y , MABU S , CHEN C ,et al. Intrusion detection system combining misuse detection and anomaly detection using genetic network programming[C]// Iccas-Sice. 2009: 3463-3467.
[1] Huanruo LI,Yunfei GUO,Shumin HUO,Guozhen CHENG,Wenyan LIU. Survey on quantitative evaluations of moving target defense[J]. 网络与信æ¯å®‰å…¨å¦æŠ¥, 2018, 4(9): 66-76.
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Fractals/Iterations in the complex plane/atomdomains - Wikibooks, open books for an open world
Fractals/Iterations in the complex plane/atomdomains
4 Properities
6.1 whole parameter plane
6.3 bof61
7 Size of the atom domain
atom domain.[1][2]
BOF61( see book Beauty Of Fractals page 61 )[3][4]
orbit trap at (0,0) [5]
mclosetime
fractalforums : cellular-coloring-of-mandelbrot-insides
3D view of bof61
2D Mandelbrot zoom (4K, 60fps) ayoungblood
You tube video : Mandelbrot, 1080p 25fps by Warren Garabrandt
Atom domains in case of the Mandelbrot set ( parameter plane) are parts of parameter plane with the same the index p.
it is positive integer
{\displaystyle p\geq 1}
for p=1 domain is a whole plane because in the algorithm value of complex modulus is compared to infinity
the period of hyperbolic component of the Mandelbrot set which is inside domain
iteration at which modulus of z is minimized during iteration of critical point
ProperitiesEdit
atom domains are overlapping
"Atom domains surround hyperbolic components of the same period, and are generally much larger than the components themselves"[6]
"These domains completely enclose the hyperbolic components of the same period"
atom domain contain :
component of mandelbrot set with period n mv
exterior of this component
fast finding ( aprioximating) of period n components of Mandelbrot set and it's centers,
whole parameter planeEdit
// code from :
// http://mathr.co.uk/blog/2014-11-02_practical_interior_distance_rendering.html
// C program by Claude Heiland-Allen
complex double z = 0;
double minimum_z2 = infinity; // atom domain
for (int n = 1; n <= maxiters; ++n) {
double z2 = cabs2(z);
if (z2 < minimum_z2) {
minimum_z2 = z2;
period = n;}}
Shadertoy bof61
Filtered atom domains
modified atom domains - which makes smaller domains more visible.
bof61Edit
This is the method described in the book "The Beauty of Fractals" on page 63, but the image in on page 61.
Color of point is proportional to :
the time it takes z to reach its smallest value
iterate of the critical point makes the closest approach
Index (c) is the iteration when point of the orbit was closest to the origin. Since there may be more than one, index(c) is the least such.
This algorithms shows borders of domains with the same index(c)[7] .[8]
Fragment of code : fractint.cfrm from Gnofract4d [9]
bof61 {
int current_index = -1 ; -1 to match Fractint's notion of iter count
int index_of_closest_point = -1
float mag_of_closest_point = 1e100
float zmag = |z|
if zmag < mag_of_closest_point
index_of_closest_point = current_index
mag_of_closest_point = zmag
#index = index_of_closest_point /256.0
Cpp function
// function is based on function mclosetime
// from mbrot.cpp
// from program mandel by Wolf Jung
// http:www.iram.rwth-aachen.de/~jung/indexp.html
////8 = iterate = bof61
// bailout2=4
int mclosetime(std::complex<double> C , int iter_max, int bailout2)
{ int j, cln = 0;
double x = C.real(), y = C.imag(), u, cld = 4;
for (j = 0; j <= iter_max; j++)
{ u = x*x + y*y;
if (u > bailout2) return j;
if (u < cld) {cld = u;cln = j; }
u = x*x - y*y + C.real();
y = 2*x*y + C.imag();
return iter_max + cln % 15; //iterate, bof61
// compute escape time
int last_iter= mclosetime(C,iter_max,bailout2);
// drawing code */
if (last_iter>=iter_max) { putpixel(ix,iy,last_iter - iter_max);} // interior
else putpixel(ix,iy,WHITE); // exterior
Atom domains : note that atom domains overlapp and "completely enclose the hyperbolic components of the same period"[10]
Note that this method can be applied to both exterior and interior. It is called atom domain.[11] It can also be modified [12]
Size of the atom domainEdit
estimation of size [13]
Function for computing size estimation of atom domain from nucleus c and its period p :
// code by Claude Heiland-Allen
// from http://mathr.co.uk/blog/2013-12-10_atom_domain_size_estimation.html
real_t atom_domain_size_estimate(complex_t c, int_t p) {
complex_t z = c;
complex_t dc = 1;
real_t abszq = cabs(z);
for (int_t q = 2; q <= p; ++q) {
real_t abszp = cabs(z);
if (abszp < abszq && q < p) {
abszq = abszp;
return abszq / cabs(dc);
↑ Atom Domain by Robert P. Munafo
↑ Practical interior distance rendering by Claude Heiland-Allen
↑ The_Beauty_of_Fractals in english wikipedia
↑ Bof61 algorithm in wikibooks
↑ An orbit trap at (0,0) by hobold
↑ misiurewicz_domains by Claude
↑ Fractint doc by Noel Giffin
↑ A Series of spiral bays in the Mandelbrot set by Patrick Hahn
↑ gnofract4d
↑ Atom Domain From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo
↑ Modified Atom Domains by Claude Heiland-Allen
↑ Atom domain size estimation by Claude Heiland-Allen
Retrieved from "https://en.wikibooks.org/w/index.php?title=Fractals/Iterations_in_the_complex_plane/atomdomains&oldid=3677982"
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Hamed H. Alsulami, Erdal Karapınar, Marwan A. Kutbi, Antonio-Francisco Roldán-López-de-Hierro, "An Illusion: “A Suzuki Type Coupled Fixed Point Theorem”", Abstract and Applied Analysis, vol. 2014, Article ID 235731, 8 pages, 2014. https://doi.org/10.1155/2014/235731
Hamed H. Alsulami,1 Erdal Karapınar,1,2 Marwan A. Kutbi,3 and Antonio-Francisco Roldán-López-de-Hierro 4
2Department of Mathematics, Atilim University, İncek 06836, Ankara, Turkey
4University of Jaén, Campus las Lagunillas s/n, 23071 Jaén, Spain
We admonish to be careful on studying coupled fixed point theorems since most of the reported fixed point results can be easily derived from the existing corresponding theorems in the literature. In particular, we notice that the recent paper [Semwal and Dimri (2014)] has gaps and the announced result is false. The authors claimed that their result generalized the main result in [Ðoric and Lazović (2011)] but, in fact, the contrary case is true. Finally, we present a fixed point theorem for Suzuki type (, r)-admissible contractions.
Throughout this note, we follow the notions and notations given in [1, 2]. Let be the mapping defined, for all , by Let be a metric space. We denote by (or by when it is convenient to clarify the involved metric) the class of all nonempty closed and bounded subsets of . For every , , let where for all and all . It is well known that is a metric on .
Very recently, Semwal and Dimri [1] announced the following result.
Theorem 1 (Semwal and Dimri [1], Theorem 2.1). Let be a complete metric space and let be mapping from into . Assume that there exists such that implies for all , , , . Then there exist , such that and .
Semwal and Dimri [1] claimed that it was a generalization of Đoric and Lazović’s recent result (see [2]), which is the following theorem.
Theorem 2 (Đoric and Lazović [2], Theorem 2.1). Let be a complete metric space and let be mapping from into . Assume that there exists such that implies for all . Then, there exists such that .
This note is devoted to the following three aims. We will show that the proof of the main result of Semwal and Dimri [1] is incorrect; in fact, it is possible to fix the glitch of the given proof in [1]. By modifying its contractivity condition, we obtain a correct version of Theorem 1 but, in such a case, we realize that the obtained result is a simple consequence of Theorem 2. Finally, we present a generalization of Theorem 2 for Suzuki type -admissible contractions.
2. Main Gaps
Let us review the lines of their proof. First at all, notice that, in general, if is a metric space, we know that, for all , , , all and all : The contrary inequalities can be false.
The authors took , arbitrarily and, later, they chose Taking into account that and using (7), the authors wrote the following (see [1], page 3, line 17): However, this inequality is not strong enough to apply the contractivity condition given in Theorem 1 because in the antecedent condition cannot appear the points and in the second member if they are not in the first member. Therefore, the contractivity condition that we found in Theorem 1 is not applicable.
Furthermore, assume that we would have been able to apply the mentioned contractivity condition. In this case, the authors wrote the following (see [1], page 3, lines 19–22): Immediately, they deduced that However, this inequality is based on which, in general, is false. Then, this argument is not correct.
The same mistake occurred when the authors tried to upper bound the terms and (see [1], page 4).
3. A Correct Version of Theorem 1
If we want to modify the contractivity condition given in Theorem 1 in order that (9) can be applied, then antecedent condition (3) must be replaced by the following one: for all , , , . In such a case, we obtain the following result.
Theorem 3. Let be a complete metric space and let be a mapping. Assume that there exists such that implies for all , , , . Then there exist , such that and .
However, we claim that this result is not a proper generalization of Theorem 2, but it is an immediate consequence of such theorem. To prove it, we need some preliminaries.
Lemma 4 (see, e.g., [3, 4]). Given a metric on , define , for all , by Then is a metric on . In addition to this, if is complete and then is also complete.
Remark 5. Notice that .
Given a mapping , denote by the mapping If there exists a point such that , then there exist two points such that and . This is precisely the thesis of Theorem 3. Therefore, we only have to prove that has a fixed point .
Notice that antecedent condition (14) can be written as Moreover, the second member in (15) is Notice that, associated to the metric , we can also consider given, for all , , by In such a case,
Theorem 6. Theorem 3 is a consequence of Theorem 2.
Proof. It is evident from (17) that (14) and (15) are equivalent to (5) and (6). Regarding Lemma 4, Remark 5, and the observations above, we conclude that, under the conditions of Theorem 3, all hypotheses of Theorem 2 are satisfied.
4. A Fixed Point Theorem for Suzuki Type -Admissible Contractions
In this section, we introduce a generalization of Theorem 2 using a slightly different kind of contractivity condition. We use the following preliminaries. Let be a metric space. Given a mapping , let be the mapping for all , . We say that the mapping is transitive if and implies that .
Definition 7 (see [5]). Let be a metric space and let be a mapping. We say that a mapping is -admissible if for all , such that .
We say that the metric space is -regular if for all provided that is a sequence such that and for all .
Remark 8. If for all , then is transitive, any metric space is -regular and any mapping is -admissible.
Definition 9. Let be a metric space, let be a multivalued mapping, let be a mapping, and let . One says that is a Suzuki type -admissible contraction if implies for all , .
In the following theorem, we will use the following condition, which can be verified for .
: if is a sequence such that verifying for all and , then there exist and such that
We must clarify that this condition is always satisfied when .
Lemma 10. If is a metric space and verifies for all , then condition holds for all and all .
Proof. Since and is closed, then . If , there is nothing to prove. Assume that and let be any positive real number in the interval As is an infimum, there exists such that . Therefore
Theorem 11. Let be a complete metric space and let be a Suzuki type -admissible multi-valued contraction from into . Suppose also that(i)is -admissible;(ii)there exist and such that ;(iii)at least, one of the following properties holds:(iii.1) is continuous, or(iii.2) is -regular and
Then has, at least, a fixed point; that is, there exists such that .
Taking into account Remark 8, this result admits Theorem 2 as a particularization to the case in which for all . Notice that the following proof is a slightly modified version of the proof of Theorem 2.1 in [2] using .
Proof. We follow the lines of the proof of Theorem 1.2 in [2], doing slight changes due to mapping . Let be arbitrary.
Step 1. There exists a sequence such that, for all , Starting from and such that , we notice that because is -admissible. If , then , so is a fixed point of and the proof is finished. On the contrary, assume that . As we can apply contractivity condition (24) and we deduce that As , and . Moreover, Hence, If , then the maximum is , and we have . As , we deduce . Therefore, , and as is closed, we conclude and the proof is finished. Suppose, on the contrary, that . In such a case, (33) means that Since is an infimum and , there exists such that Furthermore, Repeating this argument, either there exists such that (in this case, and the proof is finished) or there exists a sequence verifying (29).
Step 2. There exists such that . This fact is a consequence of being . Following a classical argument, it is easy to prove that is Cauchy in and, therefore, by the completeness, it is convergent.
Step 3. Assume that is continuous. In such a case, we have that ; that is, . By (7), it follows that for all (because ), and, taking limit as , we deduce that . Therefore, , and as is closed, we conclude and the proof is finished.
Step 4. Assume that is -regular and condition (28) holds.
In this case, using that is -regular, we have that and taking into account that is -admissible, we also have that If , the proof is also finished in this case. Therefore, we assume that and we will get a contradiction.
Next, we are going to show the following claim: Let be such that for all . As is -admissible, As and , there exists such that Therefore, for all , As we can apply contractivity condition (24), we obtain that, for all , Letting , we deduce that Assume that the maximum value is the last term. In that case, This proves that if the maximum in (46) is , then it is also true that . Therefore, in any case, we have that and it follows that (41) holds.
Next we distinguish between the cases and .
Case 4.1. Assume that and condition holds. Then, there exists and such that If , the proof is finished. On the contrary, assume that ; that is, . As and for all , property (41) guarantees that Since , we have that , and we can use contractivity condition (24). Notice that, as , , and therefore As and , If we suppose that , then , and the previous inequality leads to , so , which is false. Then and (50) implies that However, in this case, using the last two inequalities, (7) and (49), which is a contradiction. Then, we must admit that and the proof is finished in this case.
Case 4.2. Assume that and (or ) is transitive. We claim that, in this case, for all such that , we have that If , there is nothing to prove. Assume that . As is -admissible, Using (38) and the transitivity of , and, as is -admissible, then (The same conclusion is valid if is transitive.) By (41) and (58), we have that
Given , as and is an infimum, there exists a sequence such that Therefore, by (60) and (61), Letting , we deduce that
Two cases can be considered. If , then the previous inequality means that Therefore Hence, we can use contractivity condition (24), which means, using (57), that which guarantees that (56) holds.
On the contrary, assume that . Hence, inequality (63) yields that is, As we can also apply contractivity condition (24), reasoning as in (66), we deduce that, in this case, (56) holds.
In any case, using (56), we deduce that As , it follows that , so , which is the desired contradiction.
If we take for all , then we have the following result (using Remark 8).
Corollary 12. Theorem 2 follows from Theorem 11.
In the next example we show that Theorem 11 improves Theorem 2.
Example 13. Let be provided with the Euclidean metric for all . Define and by Clearly, is continuous and, if we identify , then Let and . Then Any number verifies . However, condition is false. Therefore, Theorem 2 cannot be applied to deduce that has a fixed point because is not contractive in the sense of Theorem 2. However, let . We will show that Theorem 11 is applicable.
Indeed, let , be such that . If , then there is nothing to prove. Assume that . In this case, , which means that , , and . Taking into account that then condition is obvious. Therefore, Theorem 11 guarantees that has a fixed point.
This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, under Grant no. 55-130-35-HiCi. The authors, therefore, acknowledge technical and financial support of KAU. The fourth author has been partially supported by Junta de Andaluca by Project FQM-268 of the Andalusian CICYE.
P. Semwal and R. C. Dimri, “A Suzuki type coupled fixed point theorem for generalized multivalued mapping,” Abstract and Applied Analysis, vol. 2014, Article ID 820482, 8 pages, 2014. View at: Publisher Site | Google Scholar | MathSciNet
D. Đorić and R. Lazović, “Some Suzuki-type fixed point theorems for generalized multivalued mappings and applications,” Fixed Point Theory and Applications, vol. 2011, article 40, 2011. View at: Publisher Site | Google Scholar
B. Samet, E. Karapinar, H. Aydi, and V. Ć. Rajić, “Discussion on some coupled fixed point theorems,” Fixed Point Theory and Applications, vol. 2013, article 50, 2013. View at: Publisher Site | Google Scholar
J. H. Asl, S. Rezapour, and N. Shahzad, “On fixed points of α-
\mathrm{\psi }
|
(cerate article from part of main page)
This wiki has the visual editor installed, which makes creating and editing articles very similar to editing a document in a simple word processor.
Each wiki page has an '''edit''' tab at the top of the page. This lets you edit the entire page, including such elements as categories.
You can find the full [https://www.mediawiki.org/wiki/Help:VisualEditor/User_guide '''manual for the visual editor here'''] if you need further guidance.
Many pages also have '''edit''' links at the right of the page, opposite section headings. These allow you to edit just that section.
===Editing Articles===
Each wiki page has '''edit''' and '''edit source''' tabs at the top of the page. These lets you edit the entire page. The edit link will normally load the visual editor, and the edit source link will take you to the traditional wiki markup editor.
Many pages also have '''edit''' links next to each section heading. These allow you to jump in and edit just that section.
is probably the canonical reference on editing this 'make' of wiki (mediawiki) although this particular installation may differ from the examples given at wikimedia.org, depending on how this installation has been configured.
The visual editor makes editing simple. Although it you want to dabble with the older style wiki markup editor you will need to know a bit about wiki markup. http://meta.wikimedia.org/wiki/Help:Editing is probably the canonical reference on editing this 'brand' of wiki (mediawiki).
One of the nice things about the wiki is that we can upload pictures to include in articles. However the wiki seems to keep all uploaded images in one pool ([[Special:Imagelist]]), so if we're to be able to manage this we probably need to pay a bit of attention to the names we give to the image files. So instead of, say, '''boiler.jpg''' followed by '''boiler2.jpg''' ... '''boiler1001.jpg''' maybe we could name them like '''Boiler_Worcester-Bosch_Greenstar_Junior_24i_front_outside_view.jpg'''. That's a bit of a mouth (or keyboard)-full but we'll probably all be grateful by the time we get to Wikipedia size :-) Supplying a meaningful description when we upload images should also help.
One of the nice things about the wiki is that we can upload pictures to include in articles (or just so that you can link to them from usenet posts). However the wiki seems to keep all uploaded images in one pool ([[Special:Imagelist]]), so if we're to be able to manage this we probably need to pay a bit of attention to the names we give to the image files. So instead of, say, '''boiler.jpg''' followed by '''boiler2.jpg''' ... '''boiler1001.jpg''' maybe we could name them like '''Boiler_Worcester-Bosch_Greenstar_Junior_24i_front_outside_view.jpg'''. That's a bit of a mouth (or keyboard)-full but we'll probably all be grateful by the time we get to Wikipedia size :-) Supplying a meaningful description when we upload images should also help.
=== Creating Articles ===
====Equations and other maths related stuff====
This wiki includes the "math" module, that enables you to include equations and other things with a mathematical style of presentation. So if you feel the need to include stuff like:
You can. Also the visual editor has lots of math related assistance built in to help you get what you want.
===Creating Articles===
You can create an article by searching for an article with the name you want to use. If it doesn't exist, you are given the chance to create it.
So say you want to create an article about plastering. Put the word "plastering" in the search box and click go. If that article doesn't exist, you get a link to create it. (You may want to consider re-submitting the query for the exact title of the article you want. So search again for 'Skim Plastering' rather than 'plastering' to create the article with the full title Capitalised).
====Naming====
When referring to one article from another it helps to have a consistent naming convention. Unlike Wikipedia's naming convention [[http://en.wikipedia.org/wiki/Wikipedia:Naming_conventions]] this Wiki seems to have evolved its own ''de facto'' convention:
* Use plural form for article names about classes of things e.g. '''Round Tuits'''
* Capitalise first word and main words in article names e.g. '''Where to get Round Tuits'''
*Use plural form for article names about classes of things e.g. '''Round Tuits'''
*Capitalise first word and main words in article names e.g. '''Where to get Round Tuits'''
Here is an [[Example_article|example article]]. It contains an example of the structure and many of the formatting markups you may want in an article, which you can use as a template.
All contributions to this Wiki are considered to be released under the GNU Free Documentation License. For more information see [[Project:Copyrights]].
Some of [http://en.wikipedia.org Wikipedia]'s help pages are more-or-less applicable and/or helpful for this Wiki:
*[http://en.wikipedia.org/wiki/Wikipedia:Five_pillars The five pillars of Wikipedia] (Note DIY Wiki has a different agenda!)
*[http://en.wikipedia.org/wiki/Wikipedia:How_to_edit_a_page How to edit a page]
*[http://en.wikipedia.org/wiki/Help:Contents Help pages]
*[http://en.wikipedia.org/wiki/Wikipedia:Tutorial Tutorial]
*[http://en.wikipedia.org/wiki/Wikipedia:How_to_write_a_great_article How to write a great article]
*[http://en.wikipedia.org/wiki/Wikipedia:Manual_of_Style Manual of Style]
*[http://meta.wikimedia.org/wiki/Help:Editor Metawiki Guide to Article Formatting]
You can find the full manual for the visual editor here if you need further guidance.
1.2 Equations and other maths related stuff
Each wiki page has edit and edit source tabs at the top of the page. These lets you edit the entire page. The edit link will normally load the visual editor, and the edit source link will take you to the traditional wiki markup editor.
Many pages also have edit links next to each section heading. These allow you to jump in and edit just that section.
One of the nice things about the wiki is that we can upload pictures to include in articles (or just so that you can link to them from usenet posts). However the wiki seems to keep all uploaded images in one pool (Special:Imagelist), so if we're to be able to manage this we probably need to pay a bit of attention to the names we give to the image files. So instead of, say, boiler.jpg followed by boiler2.jpg ... boiler1001.jpg maybe we could name them like Boiler_Worcester-Bosch_Greenstar_Junior_24i_front_outside_view.jpg. That's a bit of a mouth (or keyboard)-full but we'll probably all be grateful by the time we get to Wikipedia size :-) Supplying a meaningful description when we upload images should also help.
Equations and other maths related stuff
{\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}}
The five pillars of Wikipedia (Note DIY Wiki has a different agenda!)
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Naive Bayes classification for multiclass classification - MATLAB - MathWorks Switzerland
f\left(x\right)=0.5I\left\{|x|\le 1\right\}
f\left(x\right)=0.75\left(1-{x}^{2}\right)I\left\{|x|\le 1\right\}
f\left(x\right)=\frac{1}{\sqrt{2\pi }}\mathrm{exp}\left(-0.5{x}^{2}\right)
f\left(x\right)=\left(1-|x|\right)I\left\{|x|\le 1\right\}
\stackrel{^}{P}\left(Y=k|{X}_{1},..,{X}_{P}\right)=\frac{\pi \left(Y=k\right)\prod _{j=1}^{P}P\left({X}_{j}|Y=k\right)}{\sum _{k=1}^{K}\pi \left(Y=k\right)\prod _{j=1}^{P}P\left({X}_{j}|Y=k\right)},
\pi \left(Y=k\right)
\stackrel{^}{P}\left(Y=k|{X}_{1},..,{X}_{P}\right)\propto \pi \left(Y=k\right){P}_{mn}\left({X}_{1},...,{X}_{P}|Y=k\right),
{P}_{mn}\left({X}_{1},...,{X}_{P}|Y=k\right)
{\overline{x}}_{j|k}=\frac{\sum _{\left\{i:{y}_{i}=k\right\}}{w}_{i}{x}_{ij}}{\sum _{\left\{i:{y}_{i}=k\right\}}{w}_{i}},
{s}_{j|k}={\left[\frac{\sum _{\left\{i:{y}_{i}=k\right\}}{w}_{i}{\left({x}_{ij}-{\overline{x}}_{j|k}\right)}^{2}}{{z}_{1|k}-\frac{{z}_{2|k}}{{z}_{1|k}}}\right]}^{1/2},
P\left(\text{token }j|\text{class }k\right)=\frac{1+{c}_{j|k}}{P+{c}_{k}},
{c}_{j|k}={n}_{k}\frac{\sum _{\left\{i:{y}_{i}=k\right\}}^{}{x}_{ij}{w}_{i}^{}}{\sum _{\left\{i:{y}_{i}=k\right\}}^{}{w}_{i}},
{w}_{i}^{}
{c}_{k}=\sum _{j=1}^{P}{c}_{j|k},
P\left(\text{predictor }j=L|\text{class }k\right)=\frac{1+{m}_{j|k}\left(L\right)}{{m}_{j}+{m}_{k}},
{m}_{j|k}\left(L\right)={n}_{k}\frac{\sum _{\left\{i:{y}_{i}=k\right\}}^{}I\left\{{x}_{ij}=L\right\}{w}_{i}^{}}{\sum _{\left\{i:{y}_{i}=k\right\}}^{}{w}_{i}^{}},
I\left\{{x}_{ij}=L\right\}=1
{w}_{i}^{}
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Architecture and Mathematics(in progress) | Brilliant Math & Science Wiki
Architecture and Mathematics(in progress)
B Jones contributed
An architect is a person who designs buildings, prepares structures (blueprints), and manages construction.
How math apply's to architecture: - Geometry: Conversions(unit of measurement) when buying supply from the US for example. Optimizing space(volume) - Finance: budgets and costs (products from other places).Exchange rate and net income (revenue-expenses). - Algebra: ax^2+bx+c parabola (how high a building can be) - 1-variable stat: Inflation of costs (labor or material)
You are an architect hired by a family to design a small 1 story house with 2 bedrooms, 2 baths, kitchen, dining room, and living room. The size of the house will be 34ft by 24ft. They also would like an estimate on cost of material for shingles on the roof and vinyl siding. All materials must be bought in the US and converted to Canadian dollars.
1)What is the area and perimeter of the house as a whole?
perimeter of house: 2l+2w 2(24)+2(36) 48+72 =120ft
Area of house: lxw 36x24 =864ft^2
2)What is the area of each room on the plan and the hallway (as one space)?
Living room: 18x12=216ft^2 Bathroom+laundry: 9x9=81ft^2 Bedroom: 9x9=81ft^2 Kitchen: 12x7.5=90ft^2 Dining room: 10.5x9=94.5ft^2 Ensuite bathroom: 9x6=54ft^2 Master bedroom: 9x12=108ft^2
Hallway: 81+31.5+27= 139.5ft^2
3)If the average cost of shingles is $5.11/per square foot, how much will it cost to put shingles on the roof?
roof 3: a^2+b^2=c^2 18^2+b^2=19.21^2 324+b^2=369.02 b^2=369.02-324 (square root) b=6.70
\frac{18x6.7}{2}
=120.6ft^2
roof 1: 9^2+12^2=C^2 81+144=C^2 C^2=225 (square root) c=15ft
15^2+12^2=c^2 c^2=225+144 C^2=369 (square foot) c=19.21 area:
\frac{12x13 }{2}
=180ft^2
What is the smallest positive integer with four different prime factors?
Cite as: Architecture and Mathematics(in progress). Brilliant.org. Retrieved from https://brilliant.org/wiki/architecture-and-mathematicsin-progress/
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Composite Figures Warmup Practice Problems Online | Brilliant
A, B, C, D
are equally spaced points on a line. Semicircles are constructed on both sides, and colored in, as in the above image.
Which region is the smallest?
Green region Yellow region Red region The regions have equal area
To draw an alien's head, Mary drew a large circle, and then added 2 equal black circles for the eyes. She colored the smaller circles black, and colored the rest of the figure grey.
Which region is larger, the grey region or the black region?
Black region Grey region The regions have equal area
In the above square of side length 2, we draw in 2 quarter circles and shade them blue and red respectively.
What is the area of the purple oval region, where these quarter circles overlap?
2 \pi - 4
\frac{\pi}{2}
\pi - 1
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Reaction Rates - Course Hero
General Chemistry/Chemical Kinetics/Reaction Rates
Some chemical reactions occur very quickly. For example, an explosion is a chemical reaction that happens in less than one second. Some chemical reactions occur much more slowly. Iron will rust when exposed to air. This reaction occurs slowly, over years in dry weather. Iron will rust more quickly in the presence of water. The field that studies the rates of chemical reactions is called chemical kinetics. All industrial processes that produce a specific material use chemical kinetics to optimize its production.
The speed at which a reaction occurs is called the reaction rate. During a chemical reaction, reactants are converted into products. Therefore, the amounts of reactants and products change over time. Reaction rates are studied by quantifying the changes in the amount of reactants and products.
The amounts of both reactants and products can be expressed as concentrations in the reaction mixture. The average reaction rate can be defined based on concentrations of reactants and products over a period of time:
\text{average reaction rate}=\frac{\text{change in concentration}}{\text{change in time}}
A chemical reaction occurs when there is an interaction between multiple molecules or atoms. The chance of a collision is greater when there are more molecules or atoms present. Concentration is a factor that affects reaction rate. For a given reaction, reaction rates are higher at higher concentrations of reactants. Reaction rates are lower at lower concentrations of reactants. Thus, as a reaction proceeds and the concentration of reactants changes, the reaction rate also changes. Average reaction rate indicates the average rate over a selected period of time. The actual rate is likely to be different than the average reaction. The instantaneous rate is the rate of a chemical reaction at a particular moment. If the change in one of the reactants is graphed over time for a chemical reaction, the graph will form a curve for which the absolute value of the slope decreases over time. A graph of reactant concentration versus time can be used to calculate instantaneous rates.
The instantaneous rate can be found using the slope of the line tangent to the curve at any time. The slope is the change in the reactant concentration divided by the change in time.
The rate of reaction between nitrogen oxide and carbon monoxide can be defined by any of the four compounds involved in it:
{\rm{NO_2}}(g)+{\rm{CO}}(g)\rightarrow{\rm{NO}}(g)+{\rm{CO_2}}(g)
For products, the rate is equal to the concentration increase over time:
{\rm{rate}}=\frac{\Delta\lbrack\rm{NO}\rbrack}{\Delta t}=\frac{\Delta\lbrack{\rm{CO}}_2\rbrack}{\Delta t}
For reactants, the change in concentration is negative. Therefore, the rate is equal to their concentration change over time multiplied by –1.
{\rm{rate}}=-\frac{\Delta\lbrack{\rm{NO}}_2\rbrack}{\Delta t}=-\frac{\Delta\lbrack\rm{CO}\rbrack}{\Delta t}=\frac{\Delta\lbrack\rm{NO}\rbrack}{\Delta t}=\frac{\Delta\lbrack{\rm{CO}}_2\rbrack}{\Delta t}
Note that the unit for concentration is M, which is mol/L. The unit for time is seconds (s). The unit for reaction rate is therefore M/s or mol/L·s. In the decomposition reaction of hydrogen iodide (HI), the rate of consumption of HI is twice the rate of formation of hydrogen (H2) and iodine (I2) gases. This is shown by the coefficients of each species in the reaction equation:
2\rm{HI}(\mathit g)\rightarrow{\rm H}_2(\mathit g)+{\rm I}_2(\mathit g)
The rate in terms of products is straightforward.
{\rm{rate}}=\frac{\Delta\lbrack{\rm{H}}_2\rbrack}{\Delta t}=\frac{\Delta\lbrack{\rm{I}}_2\rbrack}{\Delta t}
The rates of formation of hydrogen gas and iodine gas are half the rate that hydrogen iodide (HI) is consumed. The coefficient of two causes the hydrogen iodide (HI) term to have a multiplier of
\frac{1}{2}
. Also note the negative sign, because hydrogen iodide (HI) is a reactant.
{\rm{rate}}=-\frac{1}2\frac{\Delta\lbrack{\rm H}_2\rbrack}{\Delta t}
Hence, the rate in terms of hydrogen iodide consumed (HI) can be written in terms of hydrogen gas formation.
\frac{\Delta[{\rm H}_2]}{\Delta t}=-\frac{1}{2}\frac{\Delta[{\rm {HI}}]}{\Delta t}
In general, the rate for the reaction
a{\rm A}+b{\rm B}\rightarrow c{\rm C}+d{\rm D}
{\rm{rate}}=-\frac1a\frac{\Delta\lbrack\rm A\rbrack}{\Delta t}=-\frac1b\frac{\Delta\lbrack\rm B\rbrack}{\Delta t}=\frac1c\frac{\Delta\lbrack\rm C\rbrack}{\Delta t}=\frac1d\frac{\Delta\lbrack\rm D\rbrack}{\Delta t}
Reaction Rate Calculation for the Decomposition of Dinitrogen Pentoxide
2{\rm N}{_2}{\rm O}{_5}(\mathit g)\rightarrow4{\rm{NO}}{_2}(\mathit g)+{\rm O}{_2}(\mathit g)
The instantaneous rate of formation of oxygen gas (O2) is
2.0\times10^{-7}\;\rm M/\rm s
. What is the rate of formation of nitrogen dioxide (NO2) gas and rate of consumption of dinitrogen pentoxide (N2O5)?
Write the rate in terms of each product and reactant.
{\rm{rate}}=-\frac12\frac{\Delta\lbrack{\rm N}{_2}{\rm O}{_5}\rbrack}{\Delta t}=\frac14\frac{\Delta\lbrack{\rm{NO}}_2\rbrack}{\Delta t}=\frac11\frac{\Delta\lbrack{\rm O}{_2}\rbrack}{\Delta t}
The rate of formation of oxygen is given. This can be used to solve for the rate of formation of nitrogen dioxide (NO2) gas and the rate of consumption of dinitrogen pentoxide (N2O5).
\begin{aligned}-\frac12\frac{\Delta\lbrack{\rm N}_2{\rm O}_5\rbrack}{\Delta t}&=2.0\times10^{-7}\;\rm M/\rm s\\\frac{\Delta\lbrack{\rm N}_2{\rm O}_5\rbrack}{\Delta t}&=(-2)(2.0\times10^{-7}\;\rm M/\rm s)\\&=-4.0\times10^{-7}\;\rm M/\rm s\end{aligned}
Note the negative sign, indicating dinitrogen pentoxide is consumed.
\begin{aligned}\frac14\frac{\Delta[{\rm{NO}}_2]}{\Delta t}&=2.0\times10^{-7}\;\rm M/\rm s\\\frac{\Delta[{\rm{NO}}_2]}{\Delta t}&=(4)(2.0\times10^{-7}\;\rm M/\rm s)\\&=8.0\times10^{-7}\;\rm M/\rm s\end{aligned}
<Vocabulary>Collision Theory
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Transformation of Mouse C3H/10T½ Cells by Single and Fractionated Doses of X-Rays and Fission-Spectrum Neutrons1 | Cancer Research | American Association for Cancer Research
Transformation of Mouse C3H/10T½ Cells by Single and Fractionated Doses of X-Rays and Fission-Spectrum Neutrons1
Antun Han;
Antun Han
Antun Han, M. M. Elkind; Transformation of Mouse C3H/10T½ Cells by Single and Fractionated Doses of X-Rays and Fission-Spectrum Neutrons1. Cancer Res 1 January 1979; 39 (1): 123–130.
A mouse embryo-derived cell line, C3H/10T
1\over2
, was used to measure the frequency of in vitro neoplastic transformation induced by 50-kVp X-rays and by fission-spectrum neutrons from the JANUS reactor at the Argonne National Laboratory. The transformation frequency after single X-ray doses rises exponentially, reaching a plateau at about 3 × 10-3 transformant/survivor. The induction curve following single neutron doses, while qualitatively similar, initially rises more steeply and levels off at a maximum of about 6 × 10-3 transformant/survivor. For both radiations, transformation frequency varies with changes in the number of viable cells per dish, showing about a 10-fold decrease in transformation frequency when the number of viable cells per 90-mm dish was increased from about 300 to about 1000. Fractionation of a total X-ray dose of 700 rads results in an approximately 6-fold increase in survival and reduction in transformation frequency over a 16-hr interval. Fractionation of the total neutron dose of 378 rads has no effect upon cell survival, and transformation frequency declines by a factor of only about 1.7 at most over a 24-hr period. Cells derived from transformed foci formed fibrosarcomas when injected into appropriately treated mice.
This research was supported by the United States Department of Energy.
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Muirhead Inequality | Brilliant Math & Science Wiki
Patrick Corn, Worranat Pakornrat, and Jimin Khim contributed
Muirhead's inequality is a generalization of the AM-GM inequality. Like the AM-GM inequality, it involves a comparison of symmetric sums of monomials involving several variables. It is often useful in proofs involving inequalities.
a_1,\ldots,a_n
be nonnegative real numbers. Let
x_1,\ldots,x_n
be variables. Then
\sum_{\text{sym}} x_1^{a_1} x_2^{a_2} \ldots x_n^{a_n}
n!
terms of the form
x_{\sigma(1)}^{a_1} x_{\sigma(2)}^{a_2} \ldots x_{\sigma(n)}^{a_n},
\sigma
runs over the permutations of
\{1, \ldots,n\}.
\sum_{\text{sym}} xy^3z^2 = xy^3z^2+xy^2z^3 + x^2y^3z + x^2yz^3 + x^3y^2z + x^3yz^2
a_1,\ldots, a_n
b_1,\ldots,b_n
are two sequences of nonnegative real numbers satisfying the following conditions:
a_1\ge a_2\ge \cdots \ge a_n
b_1\ge b_2\ge \cdots \ge b_n
a_1 \ge b_1,
a_1+a_2 \ge b_1+b_2,
\ldots,
a_1+a_2+\cdots+a_{n-1} \ge b_1+b_2+\cdots+b_{n-1}
a_1+a_2+\cdots+a_n = b_1+b_2+\cdots + b_n.
(a_i)
is said to majorize the sequence
(b_i).
7,3,1
majorizes
5,4,2,
7 \ge 5,
7+3 \ge 5+4,
7+3+1=5+4+2.
Muirhead's Inequality:
(a_i)
(b_i).
Then, for all nonnegative
x_i,
\sum_{\text{sym}} x_1^{a_1}\ldots x_n^{a_n} \ge \sum_{\text{sym}} x_1^{b_1}\ldots x_n^{b_n}.
From the example in the previous section, if
x,y,z
are nonnegative real numbers, then
\begin{aligned} x^7y^3z + x^7yz^3 + x^3y^7z + x^3yz^7 + xy^7z^3 +xy^3z^7 \ge x^5y^4z^2 + x^5y^2z^4 + x^4y^5z^2 + x^4y^2z^5 + x^2y^5z^4 + x^2y^4z^5.\end{aligned}
In the interest of more compact notation, write
M(a_1,a_2,\ldots,a_n) = \sum_{\text{sym}} x_1^{a_1}\ldots x_n^{a_n}.
Then Muirhead's inequality says that if
(a_i)
(b_i),
M(a_1,\ldots,a_n) \ge M(b_1,\ldots,b_n).
(1,0,0,\ldots,0)
\left( \frac1{n}, \frac1{n}, \ldots, \frac1{n}\right),
M(1,0,0,\ldots,0) \ge M\left( \frac1{n},\frac1{n},\ldots,\frac1{n}\right)
by Muirhead's inequality. In the symmetric sum for
M(1,0,0,\ldots,0),
(n-1)!
permutations of the variables that give each term. For instance, the term
x_1^1 x_2^0 x_3^0 \ldots x_n^0
is preserved by all the permutations that keep
1
fixed and move
2\ldots n
around. In the symmetric sum for
M\left( \frac1{n},\frac1{n},\ldots,\frac1{n}\right),
all of the
n!
permutations give the same monomial.
So Muirhead's inequality becomes
\begin{aligned} (n-1)!(x_1 + x_2 + \cdots + x_n) &\ge n!\left(x_1^{1/n} x_2^{1/n} \ldots x_n^{1/n}\right) \\ \frac{x_1+x_2+\cdots+x_n}{n} &\ge (x_1x_2\ldots x_n)^{1/n}, \end{aligned}
which is the AM-GM inequality. So this is a special case of Muirhead's inequality.
\frac{(a+b)(b+c)(c+a)}{abc}
a,b,c
range over positive real numbers.
Multiplying out the numerator gives
2abc + M(2,1,0).
M(1,1,1) = 6abc,
so the expression is
2+\frac{M(2,1,0)}{abc} = 2+\frac{6 M(2,1,0)}{M(1,1,1)},
M(2,1,0) \ge M(1,1,1)
by Muirhead's inequality, so the expression is
\ge 2+6 = 8.
But note that if
a=b=c,
then the expression evaluates to
8,
8.
_\square
x \ge y
y \ge x
always They are always equal It depends on
a,b,c
x = a+b+c, \qquad y = \frac{a^3}{bc} + \frac{b^3}{ca} + \frac{c^3}{ab}
a,b,c
are positive real numbers, which is bigger,
x
y?
x^6+5x^5y+10x^4y^2+kx^3y^3+10x^2y^4+5xy^5+y^6\ge 0
Find the absolute value of the smallest possible
k
such that the inequality above is true for all non-negative reals
x
y
Note: You may use the algebraic identities below.
(x+y)^5=x^5+5x^4y+10x^3y^2+10x^2y^3+5xy^4+y^5
(x+y)^6=x^6+6 x^5 y+15 x^4 y^2+20 x^3 y^3+15 x^2 y^4+6 x y^5+y^6
xyz
-plane above, let
O=(0,0,0)
be the point of origin,
P=(a,b,c)
Q=(a+b, b+c, c+a)
a,b,c
What is the maximum value of the ratio
\frac{OQ}{OP}?
S = \left \{\left(\frac{x+y-z}{x+y+z}\right)^2 + \left(\frac{x-y+z}{x+y+z}\right)^2 + \left(\frac{-x+y+z}{x+y+z}\right)^2: x,y,z \in \mathbb R^+ \right \}
S
be a set defined as above, where
x, y, z
are positive real numbers.
a = \text{sup } S
(
supremum of
S)
b = \text{inf } S
(
infimum of
S),
\frac{a}{b}
Muirhead's inequality can sometimes be extended to prove non-homogeneous inequalities:
x,y,z
xyz=1.
x^{10}+y^{10}+z^{10} \ge x^9+y^9+z^9.
The trick is to multiply
x^9+y^9+z^9
(xyz)^{1/3}=1,
which makes the degrees of both sides equal to each other. That is,
\begin{aligned} M(10,0,0) &\ge M\left(9\!\frac13,\frac13,\frac13\right) \\ 2!\left(x^{10}+y^{10}+z^{10}\right) &\ge 2!\left(x^9(xyz)^{1/3} + y^9(xyz)^{1/3} + z^9(xyz)^{1/3}\right) \\ x^{10}+y^{10}+z^{10} &\ge x^9+y^9+z^9.\ _\square \end{aligned}
This last example is quite a bit harder, but illustrates the power of the technique.
[IMO 1995] For
a,b,c>0
abc=1,
\frac1{a^3(b+c)} + \frac1{b^3(c+a)} + \frac1{c^3(a+b)} \ge \frac32.
Multiply by the product of denominators:
b^3c^3(c+a)(a+b) + a^3c^3(b+c)(a+b) + a^3b^3(b+c)(c+a) \ge \frac32 a^3b^3c^3(b+c)(c+a)(a+b).
a^3b^3c^3=1
on the right side, this turns into
M(4,3,1) + \frac12 M(4,4,0) + \frac12 M(3,3,2) \ge \frac32\big(2abc+M(2,1,0)\big) = 3abc + \frac32 M(2,1,0).
Now the left side has degree
8,
and the right side has degree
3.
To get the degrees on both sides to match up, we can multiply the right side by
(abc)^{5/3}=1.
(abc)^{8/3} = \frac16 M\left(\frac83,\frac83,\frac83\right).
M(4,3,1) + \frac12 M(4,4,0) + \frac12 M(3,3,2) \ge \frac12 M\left(\frac83,\frac83,\frac83\right)+\frac32 M\left(\frac{11}3,\frac83,\frac53\right).
4,3,1
4,4,0
both majorize
\frac{11}3,\frac83,\frac53,
M(4,3,1)+\frac12M(4,4,0) \ge M\left(\frac{11}3,\frac83,\frac53\right)+\frac12 M\left(\frac{11}3,\frac83,\frac53\right) = \frac32 M\left(\frac{11}3,\frac83,\frac53\right),
3,3,2
\frac83,\frac83,\frac83,
\frac12 M(3,3,2) \ge \frac12 M\left(\frac83,\frac83,\frac83\right),
and adding these inequalities gives the result.
_\square
Cite as: Muirhead Inequality. Brilliant.org. Retrieved from https://brilliant.org/wiki/muirhead-inequality/
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§ Shrinking wedge of circles / Hawaiian earring (TODO)
I've been trying to make peace with the fact that countably infinite wedge of circles is so different from the hawaiian earring. Here are some thoughts:
Topology on the hawaiian earring is very different. Eg: small opens around the center of infinite wedge is contractible, while no small open around the origin of the hawaiian earring is contractible.
We can take infinite products of group elements in the hawaiian earring, since the radii decrease. For example, we can take a loop at each circle of radius
1/n
by making it traverse the circle of radius
1/n
t \in [(n-1)/n, n/(n+1)]
, and stay at
(0, 0)
t=1
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The interesting and non-obvious bit is thatthere's only one bandlimited signal that passes exactly through each sample point; it's a unique solution. If you sample a bandlimited signal and then convert it back, the original input is also the only possible output.
Before you say, "I can draw a different signal that passes through those points", if it differs even minutely from the original, it includes frequency content at or beyond Nyquist, breaks the bandlimiting requirement and isn't a valid solution.
{\displaystyle \ squarewave(t)={\begin{cases}1,&|t|<T_{1}\\0,&T_{1}<|t|\leq {1 \over 2}T\end{cases}}}
{\displaystyle {\begin{aligned}\ squarewave(t)={\frac {4}{\pi }}\sin(\omega t)+{\frac {4}{3\pi }}\sin(3\omega t)+{\frac {4}{5\pi }}\sin(5\omega t)+\\{\frac {4}{7\pi }}\sin(7\omega t)+{\frac {4}{9\pi }}\sin(9\omega t)+{\frac {4}{11\pi }}\sin(11\omega t)+\\{\frac {4}{13\pi }}\sin(13\omega t)+{\frac {4}{15\pi }}\sin(15\omega t)+{\frac {4}{17\pi }}\sin(17\omega t)+\\{\frac {4}{19\pi }}\sin(19\omega t)+{\frac {4}{21\pi }}\sin(21\omega t)+{\frac {4}{23\pi }}\sin(23\omega t)+\\{\frac {4}{25\pi }}\sin(25\omega t)+{\frac {4}{27\pi }}\sin(27\omega t)+{\frac {4}{29\pi }}\sin(29\omega t)+\\{\frac {4}{31\pi }}\sin(31\omega t)+{\frac {4}{33\pi }}\sin(33\omega t)+\cdots \end{aligned}}}
|
Q 9 Find the value of a so that 1 - 7x is a factor of -14x3-47x2-14x+a - Maths - Factorisation - 12382843 | Meritnation.com
Q.9. Find the value of a so that 1 - 7x is a factor of
-14{x}^{3}-47{x}^{2}-14x+a
\mathbf{9}\mathbf{.} Let f\left(x\right) = -14{x}^{3}-47{x}^{2}-14x+a\phantom{\rule{0ex}{0ex}}\mathbit{G}\mathbit{i}\mathbit{v}\mathbit{e}\mathbit{n}\mathbf{:} 1-7x is a factor of f\left(x\right). i.e., f\left(\frac{1}{7}\right) = 0\phantom{\rule{0ex}{0ex}}Substituting x=\frac{1}{7}in f\left(x\right),\phantom{\rule{0ex}{0ex}}-14{\left(\frac{1}{7}\right)}^{3}-47{\left(\frac{1}{7}\right)}^{2}-14\left(\frac{1}{7}\right)+a=0\phantom{\rule{0ex}{0ex}}-\frac{2}{49}-\frac{47}{49}-2+a=0\phantom{\rule{0ex}{0ex}}\frac{-2-47-98+49a}{49}=0\phantom{\rule{0ex}{0ex}}-147+49a=0\phantom{\rule{0ex}{0ex}}a=\frac{147}{49}= 3
|
Cérou, Frédéric ; Guyader, Arnaud
X
be a random element in a metric space
\left(ℱ,d\right)
Y
be a random variable with value
0
1
Y
is called the class, or the label, of
X
{\left({X}_{i},{Y}_{i}\right)}_{1\le i\le n}
be an observed i.i.d. sample having the same law as
\left(X,Y\right)
. The problem of classification is to predict the label of a new random element
X
k
-nearest neighbor classifier is the simple following rule: look at the
k
nearest neighbors of
X
in the trial sample and choose
0
1
for its label according to the majority vote. When
\left(ℱ,d\right)=\left({ℝ}^{d},||.||\right)
, Stone (1977) proved in 1977 the universal consistency of this classifier: its probability of error converges to the Bayes error, whatever the distribution of
\left(X,Y\right)
. We show in this paper that this result is no longer valid in general metric spaces. However, if
\left(ℱ,d\right)
is separable and if some regularity condition is assumed, then the
k
-nearest neighbor classifier is weakly consistent.
Mots clés : classification, consistency, non parametric statistics
author = {C\'erou, Fr\'ed\'eric and Guyader, Arnaud},
title = {Nearest neighbor classification in infinite dimension},
AU - Cérou, Frédéric
AU - Guyader, Arnaud
TI - Nearest neighbor classification in infinite dimension
Cérou, Frédéric; Guyader, Arnaud. Nearest neighbor classification in infinite dimension. ESAIM: Probability and Statistics, Tome 10 (2006), pp. 340-355. doi : 10.1051/ps:2006014. http://www.numdam.org/articles/10.1051/ps:2006014/
[1] C. Abraham, G. Biau and B. Cadre, On the kernel rule for function classification. submitted (2003). | Zbl 1100.62066
[2] G. Biau, F. Bunea and M.H. Wegkamp, On the kernel rule for function classification. IEEE Trans. Inform. Theory, to appear (2005). | MR 2235289
[3] T.M. Cover and P.E. Hart, Nearest neighbor pattern classification. IEEE Trans. Inform. Theory IT-13 (1967) 21-27. | Zbl 0154.44505
[4] S. Dabo-Niang and N. Rhomari, Nonparametric regression estimation when the regressor takes its values in a metric space, submitted (2001). | Zbl 1020.62034
[5] L. Devroye, On the almost everywhere convergence of nonparametric regression function estimates. Ann. Statist. 9 (1981) 1310-1319. | Zbl 0477.62025
[6] L. Devroye, L. Györfi, A. Krzyżak and G. Lugosi, On the strong universal consistency of nearest neighbor regression function estimates. Ann. Statist. 22 (1994) 1371-1385. | Zbl 0817.62038
[7] L. Devroye, L. Györfi and G. Lugosi, A probabilistic theory of pattern recognition 31, Applications of Mathematics (New York). Springer-Verlag, New York (1996). | MR 1383093 | Zbl 0853.68150
[8] L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992). | MR 1158660 | Zbl 0804.28001
[9] H. Federer, Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York (1969). | MR 257325 | Zbl 0176.00801
[10] D. Preiss, Gaussian measures and the density theorem. Comment. Math. Univ. Carolin. 22 (1981) 181-193. | Zbl 0459.28015
[11] D. Preiss, Dimension of metrics and differentiation of measures, in General topology and its relations to modern analysis and algebra, V (Prague, 1981), Sigma Ser. Pure Math., Heldermann, Berlin 3 (1983) 565-568. | Zbl 0502.28002
[12] D. Preiss and J. Tišer, Differentiation of measures on Hilbert spaces, in Measure theory, Oberwolfach 1981 (Oberwolfach, 1981), Springer, Berlin. Lect. Notes Math. 945 (1982) 194-207. | Zbl 0495.28010
[13] C.J. Stone, Consistent nonparametric regression. Ann. Statist. 5 (1977) 595-645. With discussion and a reply by the author. | Zbl 0366.62051
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Solve this : Q State whether the following statements are true or false Give proof for your - Maths - Playing with Numbers - 12596125 | Meritnation.com
Q. State whether the following statements are true or false. Give proof for your answers.
(i) For any real number x, x2
\ge
(ii) The sum of two even numbers is even.
\mathrm{Hi}, \phantom{\rule{0ex}{0ex}}\left(\mathrm{i}\right)\phantom{\rule{0ex}{0ex}}\mathrm{real} \mathrm{number} \mathrm{can} \mathrm{be} \mathrm{positive} , \mathrm{negative} \mathrm{or} \mathrm{zero} \phantom{\rule{0ex}{0ex}}\mathrm{so} \mathrm{when} \mathrm{x} \mathrm{is} \mathrm{positive} {\mathrm{x}}^{2}= +{\mathrm{x}}^{2}\phantom{\rule{0ex}{0ex}}\mathrm{for} \mathrm{example} \mathrm{x} = 1,2,3,4,...\phantom{\rule{0ex}{0ex}}{\mathrm{x}}^{2}= 1,4,9,16,.... \mathrm{so} \mathrm{basically} \mathrm{all} \mathrm{are} \mathrm{greater} \mathrm{than} \mathrm{zero} \phantom{\rule{0ex}{0ex}}\mathrm{now} \mathrm{when} \mathrm{x} \mathrm{is} 0 \mathrm{then} \phantom{\rule{0ex}{0ex}}{\mathrm{x}}^{2}= {0}^{2}= 0 \phantom{\rule{0ex}{0ex}}\mathrm{so} \mathrm{it} \mathrm{is} \mathrm{zero} \phantom{\rule{0ex}{0ex}}\mathrm{now} \mathrm{when} \mathrm{x} \mathrm{is} \mathrm{negative} \phantom{\rule{0ex}{0ex}}{\mathrm{x}}^{2}= \left(-\mathrm{x}\right)×\left(-\mathrm{x}\right)= {\mathrm{x}}^{2} \mathrm{again} \mathrm{positive} \phantom{\rule{0ex}{0ex}}\mathrm{for} \mathrm{example} \mathrm{x} = -1,-2,-3,-4,...\phantom{\rule{0ex}{0ex}}{\mathrm{x}}^{2}= 1,4,9,16,.... \mathrm{so} \mathrm{basically} \mathrm{all} \mathrm{are} \mathrm{greater} \mathrm{than} \mathrm{zero} \phantom{\rule{0ex}{0ex}}\mathrm{so} \mathrm{it} \mathrm{is} \mathrm{true}. \phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\left(\mathrm{ii}\right)\phantom{\rule{0ex}{0ex}}\mathrm{two} \mathrm{even} \mathrm{numbers} \mathrm{can} \mathrm{be} \mathrm{of} \mathrm{the} \mathrm{form} 2\mathrm{x}, 2\mathrm{y} \phantom{\rule{0ex}{0ex}}\mathrm{sum} = 2\mathrm{x}+2\mathrm{y} =2\left(\mathrm{x}+\mathrm{y}\right) \mathrm{it} \mathrm{is} \mathrm{also} \mathrm{even} \mathrm{as} \mathrm{it} \mathrm{has} 2 \mathrm{as} \mathrm{its} \mathrm{factor} \phantom{\rule{0ex}{0ex}}\mathrm{so} \mathrm{it} \mathrm{is} \mathrm{also} \mathrm{true}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}
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Double-slit Experiment | Brilliant Math & Science Wiki
Matt DeCross, Adam Strandberg, and Jimin Khim contributed
Plane wave representing a particle passing through two slits, resulting in an interference pattern on a screen some distance away from the slits. [1].
The double-slit experiment is an experiment in quantum mechanics and optics demonstrating the wave-particle duality of electrons, photons, and other fundamental objects in physics. When streams of particles such as electrons or photons pass through two narrow adjacent slits to hit a detector screen on the other side, they don't form clusters based on whether they passed through one slit or the other. Instead, they interfere: simultaneously passing through both slits, and producing a pattern of interference bands on the screen. This phenomenon occurs even if the particles are fired one at a time, showing that the particles demonstrate some wave behavior by interfering with themselves as if they were a wave passing through both slits.
Niels Bohr proposed the idea of wave-particle duality to explain the results of the double-slit experiment. The idea is that all fundamental particles behave in some ways like waves and in other ways like particles, depending on what properties are being observed. These insights led to the development of quantum mechanics and quantum field theory, the current basis behind the Standard Model of particle physics, which is our most accurate understanding of how particles work.
The original double-slit experiment was performed using light/photons around the turn of the nineteenth century by Thomas Young, so the original experiment is often called Young's double-slit experiment. The idea of using particles other than photons in the experiment did not come until after the ideas of de Broglie and the advent of quantum mechanics, when it was proposed that fundamental particles might also behave as waves with characteristic wavelengths depending on their momenta. The single-electron version of the experiment was in fact not performed until 1974. A more recent version of the experiment successfully demonstrating wave-particle duality used buckminsterfullerene or buckyballs, the
C_{60}
allotrope of carbon.
Modeling the Double-slit Experiment
To understand why the double-slit experiment is important, it is useful to understand the strong distinctions between wave and particles that make wave-particle duality so intriguing.
Waves describe oscillating values of a physical quantity that obey the wave equation. They are usually described by sums of sine and cosine functions, since any periodic (oscillating) function may be decomposed into a Fourier series. When two waves pass through each other, the resulting wave is the sum of the two original waves. This is called a superposition since the waves are placed ("-position") on top of each other ("super-"). Superposition is one of the most fundamental principles of quantum mechanics. A general quantum system need not be in one state or another but can reside in a superposition of two where there is some probability of measuring the quantum wavefunction in one state or another.
Left: example of superposed waves constructively interfering. Right: superposed waves destructively interfering.[2]
If one wave is
A(x) = \sin (2x)
B(x) = \sin (2x)
, then they add together to make
A + B = 2 \sin (2x)
. The addition of two waves to form a wave of larger amplitude is in general known as constructive interference since the interference results in a larger wave.
A(x) = \sin (2x)
B(x) = \sin (2x + \pi)
A + B = 0
\big(
\sin (2x + \pi) = - \sin (2x)\big).
This is known as destructive interference in general, when adding two waves results in a wave of smaller amplitude. See the figure above for examples of both constructive and destructive interference.
This wave behavior is quite unlike the behavior of particles. Classically, particles are objects with a single definite position and a single definite momentum. Particles do not make interference patterns with other particles in detectors whether or not they pass through slits. They only interact by colliding elastically, i.e., via electromagnetic forces at short distances. Before the discovery of quantum mechanics, it was assumed that waves and particles were two distinct models for objects, and that any real physical thing could only be described as a particle or as a wave, but not both.
In the more modern version of the double slit experiment using electrons, electrons with the same momentum are shot from an "electron gun" like the ones inside CRT televisions towards a screen with two slits in it. After each electron goes through one of the slits, it is observed hitting a single point on a detecting screen at an apparently random location. As more and more electrons pass through, one at a time, they form an overall pattern of light and dark interference bands. If each electron was truly just a point particle, then there would only be two clusters of observations: one for the electrons passing through the left slit, and one for the right. However, if electrons are made of waves, they interfere with themselves and pass through both slits simultaneously. Indeed, this is what is observed when the double-slit experiment is performed using electrons. It must therefore be true that the electron is interfering with itself since each electron was only sent through one at a time—there were no other electrons to interfere with it!
When the double-slit experiment is performed using electrons instead of photons, the relevant wavelength is the de Broglie wavelength
\lambda:
\lambda = \frac{h}{p},
h
p
is the electron's momentum.
1.5 \times 10^{-11}\text{ m}
7.3 \times 10^{-11}\text{ m}
1.5 \times 10^{-10}\text{ m}
7.3 \times 10^{-10}\text{ m}
1.0 \times 10^{7} \text{ m/s}.
r_{B} = 5.29 \times 10^{-11} \text{m}
4 \times 10^{-5} r_{B}
-5.
While the de Broglie relation was postulated for massive matter, the equation applies equally well to light. Given light of a certain wavelength, the momentum and energy of that light can be found using de Broglie's formula. This generalizes the naive formula
p = m v
, which can't be applied to light since light has no mass and always moves at a constant velocity of
c
regardless of wavelength.
The below is reproduced from the Amplitude, Frequency, Wave Number, Phase Shift wiki.
In Young's double-slit experiment, photons corresponding to light of wavelength
\lambda
d,
as shown in the diagram below. After passing through the slits, they hit a screen at a distance of
D
away with
D \gg d,
and the point of impact is measured. Remarkably, both the experiment and theory of quantum mechanics predict that the number of photons measured at each point along the screen follows a complicated series of peaks and troughs called an interference pattern as below. The photons must exhibit the wave behavior of a relative phase shift somehow to be responsible for this phenomenon. Below, the condition for which maxima of the interference pattern occur on the screen is derived.
Left: actual experimental two-slit interference pattern of photons, exhibiting many small peaks and troughs. Right: schematic diagram of the experiment as described above. [3]
D \gg d
\theta
y
is the vertical displacement to an interference peak from the midpoint between the slits, it is therefore true that
D\tan \theta \approx D\sin \theta \approx D\theta = y.
\Delta L
\Delta L
\Delta L
number of wavelengths is exactly
2\pi n
, which is the same as no phase shift and therefore constructive interference. From the above diagram and basic trigonometry, one can write
\Delta L = d\sin \theta \approx d\theta = n\lambda.
\theta = \frac{y}{D}
, one can see that the condition for maxima of the interference pattern, corresponding to constructive interference, is
n\lambda = \frac{dy}{D},
i.e. the maxima occur at the vertical displacements of
y = \frac{n\lambda D}{d}.
The analogous experimental setup and mathematical modeling using electrons instead of photons is identical except that the de Broglie wavelength of the electrons
\lambda = \frac{h}{p}
is used instead of the literal wavelength of light.
Lookang, . CC-3.0 Licensing. Retrieved from https://commons.wikimedia.org/w/index.php?curid=17014507
Haade, . CC-3.0 Licensing. Retrieved from https://commons.wikimedia.org/w/index.php?curid=10073387
Cite as: Double-slit Experiment. Brilliant.org. Retrieved from https://brilliant.org/wiki/double-slit-experiment/
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BartlettHannWindow - Maple Help
Home : Support : Online Help : Science and Engineering : Signal Processing : Windowing Functions : BartlettHannWindow
multiply an array of samples by a Bartlett-Hann windowing function
BartlettHannWindow(A)
The BartlettHannWindow(A) command multiplies the Array A by the Bartlett-Hann windowing function and returns the result in an Array having the same length.
The Bartlett-Hann windowing function
w\left(k\right)
N
w\left(k\right)=0.62-0.48|\frac{2k-N}{2N}|-0.38\mathrm{cos}\left(\frac{2k\mathrm{\pi }}{N}\right)
The SignalProcessing[BartlettHannWindow] command is thread-safe as of Maple 18.
\mathrm{with}\left(\mathrm{SignalProcessing}\right):
N≔1024:
a≔\mathrm{GenerateUniform}\left(N,-1,1\right)
{\textcolor[rgb]{0,0,1}{\mathrm{_rtable}}}_{\textcolor[rgb]{0,0,1}{36893627978157125988}}
\mathrm{BartlettHannWindow}\left(a\right)
{\textcolor[rgb]{0,0,1}{\mathrm{_rtable}}}_{\textcolor[rgb]{0,0,1}{36893627978008285180}}
c≔\mathrm{Array}\left(1..N,'\mathrm{datatype}'='\mathrm{float}'[8],'\mathrm{order}'='\mathrm{C_order}'\right):
\mathrm{BartlettHannWindow}\left(\mathrm{Array}\left(1..N,'\mathrm{fill}'=1,'\mathrm{datatype}'='\mathrm{float}'[8],'\mathrm{order}'='\mathrm{C_order}'\right),'\mathrm{container}'=c\right)
{\textcolor[rgb]{0,0,1}{\mathrm{_rtable}}}_{\textcolor[rgb]{0,0,1}{36893627978008261084}}
u≔\mathrm{`~`}[\mathrm{log}]\left(\mathrm{FFT}\left(c\right)\right):
\mathbf{use}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{plots}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{display}\left(\mathrm{Array}\left(\left[\mathrm{listplot}\left(\mathrm{ℜ}\left(u\right)\right),\mathrm{listplot}\left(\mathrm{ℑ}\left(u\right)\right)\right]\right)\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end use}
The SignalProcessing[BartlettHannWindow] command was introduced in Maple 18.
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Sensors | Special Issue : Compressed Sensing for ECG Data Acquisition and Processing
Compressed Sensing for ECG Data Acquisition and Processing
Special Issue "Compressed Sensing for ECG Data Acquisition and Processing"
Prof. Dr. Luca De Vito
Department of Engineering, Università degli Studi del Sannio, Benevento, Italy
Interests: instrumentation and measurement; data converters; data acquisition systems; Internet of Things (IoT); biomedical instrumentation
Dr. Francesco Picariello
Interests: electrical and electronic instrumentation; data acquisition systems (DAQ) based on compressive sampling (CS); biomedical instrumentation; distributed measurement systems, including wireless sensor networks (WSNs); Internet of Things (IoT); unmanned aerial systems (UASs); aerial photogrammetry
Special Issue in Sensors: New Sensors for Metrology for Aerospace
Special Issue in Sensors: Selected Papers from the 2018 IEEE International Workshop on Metrology for the Sea
Topical Collection in Sensors: Sensors and Measurements for Unmanned Systems
Dr. Ioan Tudosa
Department of Engineering, University of Sannio, Piazza Roma, 21, 82100 Benevento, Italy
Interests: time and frequency measurements; measurement methods for signal/instrument characterization/calibration; data acquisition systems (DAQ); hardware design for instrumentation; applied Compressed Sensing (CS) and compressive sampling techniques
Special Issue in Sensors: Selected Papers from 24th IMEKO TC4 International Symposium and 22nd International Workshop on ADC and DAC Modelling and Testing
Compressed sensing (CS) has recently been applied to ECG monitoring systems with the aim of either compressing the acquired data rate, reducing the noise, or even processing the ECG signal to discover anomalies.
This Special Issue seeks innovative contributions on the application of recent CS results to the acquisition and processing of ECG signals, related but not restricted to the following topics:
Signal acquisition schemes based on CS;
Signal dictionaries and methods for dictionary optimization, learning, and adaptation;
Reconstruction algorithms;
Characterization and assessment of CS ECG monitoring systems;
Hardware implementations of ECG monitoring systems based on CS;
Analog-to-information converters for ECG monitoring;
Processing of ECG samples acquired by CS;
CS-based ECG signal denoising;
Anomaly detection from compressed samples;
CS-based heartrate and heartrate variation measurements;
CS-based Internet of Things and Internet of Medical Things systems;
Machine learning for CS;
Energy-efficient CS systems;
CS-based ECG segmentation and feature extraction.
ECG Monitoring Based on Dynamic Compressed Sensing of Multi-Lead Signals
This paper presents an innovative method for multiple lead electrocardiogram (ECG) monitoring based on Compressed Sensing (CS). The proposed method extends to multiple leads signals, a dynamic Compressed Sensing method, that were previously developed on a single lead. The dynamic sensing method makes [...] Read more.
This paper presents an innovative method for multiple lead electrocardiogram (ECG) monitoring based on Compressed Sensing (CS). The proposed method extends to multiple leads signals, a dynamic Compressed Sensing method, that were previously developed on a single lead. The dynamic sensing method makes use of a sensing matrix in which its elements are dynamically obtained from the signal to be compressed. In this method, for the application to multiple leads, it is proposed to use a single sensing matrix for which its elements are obtained from a combination of multiple leads. The proposed method is evaluated on a wide set of signals and acquired on healthy subjects and on subjects affected by different pathologies, such as myocardial infarction, cardiomyopathy, and bundle branch block. The experimental results demonstrated that the proposed method can be adopted for a Compression Ratio (
CR
) up to 10, without compromising signal quality. In particular, for
CR=
10, it exhibits a percentage of root-mean-squared difference average among a wide set of ECG signals lower than 3%. Full article
(This article belongs to the Special Issue Compressed Sensing for ECG Data Acquisition and Processing)
This paper presents a new approach for the optimization of a dictionary used in ECG signal compression and reconstruction systems, based on Compressed Sensing (CS). Alternatively to fully data driven methods, which learn the dictionary from the training data, the proposed approach uses [...] Read more.
This paper presents a new approach for the optimization of a dictionary used in ECG signal compression and reconstruction systems, based on Compressed Sensing (CS). Alternatively to fully data driven methods, which learn the dictionary from the training data, the proposed approach uses an over complete wavelet dictionary, which is then reduced by means of a training phase. Moreover, the alignment of the frames according to the position of the R-peak is proposed, such that the dictionary optimization can exploit the different scaling features of the ECG waves. Therefore, at first, a training phase is performed in order to optimize the overcomplete dictionary matrix by reducing its number of columns. Then, the optimized matrix is used in combination with a dynamic sensing matrix to compress and reconstruct the ECG waveform. In this paper, the mathematical formulation of the patient-specific optimization is presented and three optimization algorithms have been evaluated. For each of them, an experimental tuning of the convergence parameter is carried out, in order to ensure that the algorithm can work in its most suitable conditions. The performance of each considered algorithm is evaluated by assessing the Percentage of Root-mean-squared Difference (PRD) and compared with the state of the art techniques. The obtained experimental results demonstrate that: (i) the utilization of an optimized dictionary matrix allows a better performance to be reached in the reconstruction quality of the ECG signals when compared with other methods, (ii) the regularization parameters of the optimization algorithms should be properly tuned to achieve the best reconstruction results, and (iii) the Multiple Orthogonal Matching Pursuit (M-OMP) algorithm is the better suited algorithm among those examined. Full article
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Boiling Heat Transfer Enhancement Using a Submerged, Vibration-Induced Jet | J. Electron. Packag. | ASME Digital Collection
Steven W. Tillery,
Steven W. Tillery
Samuel N. Heffington,
Samuel N. Heffington
Marc K. Smith,
Tillery, S. W., Heffington, S. N., Smith, M. K., and Glezer, A. (February 1, 2006). "Boiling Heat Transfer Enhancement Using a Submerged, Vibration-Induced Jet." ASME. J. Electron. Packag. June 2006; 128(2): 145–149. https://doi.org/10.1115/1.2188954
In this paper we describe a new two-phase cooling cell based on channel boiling and a vibration-induced liquid jet whose collective purpose is to delay the onset of critical heat flux by forcibly dislodging the small vapor bubbles that form on the heated surface during nucleate boiling and propelling them into the cooler bulk liquid within the cell. The submerged turbulent vibration-induced jet is generated by a vibrating piezoelectric diaphragm operating at resonance. The piezoelectric driver induces pressure oscillations in the liquid near the surface of the diaphragm, resulting in the time-periodic formation and collapse of cavitation bubbles that entrain surrounding liquid and generate a strong liquid jet. The resultant jet is directed at the heated surface in the channel. The jet enhances boiling heat transfer by removing attached vapor bubbles that insulate the surface and provides additional forced convection heat transfer on the surface. A small cross flow maintained within the cell increases heat transfer even further by sweeping the bubbles downstream, where they condense. In addition, the cross flow keeps the temperature of the liquid within the cell regulated. In the present experiments, the cell dimensions were
51×25×76mm
and water was the working liquid. Heat fluxes above
300W∕cm2
were obtained at surface temperatures near
150°C
for a horizontal cell.
electronics packaging, heat transfer, jets, bubbles
Boiling, Bubbles, Heat transfer, Temperature, Vibration, Vapors, Water, Diaphragms (Mechanical devices), Diaphragms (Structural), Heat flux
Final Report on Next Generation Thermal Management (NGTM) for Power Electronics
,” NSWCCD Technical Report TR-82-2002012.
An Introduction to Heat Pipes
A Small Scale Thermosyphon for Immersion Cooling of a Disc Heat Source
,” Heat Transfer in Electronic Equipment, ASME Symposium HTD-Vol.
Microelectronic Cooling by Fluorocarbon Liquid Films
Proc. Int. Symposium of Cooling Technology for Electronic Equipment
Immersion Cooling of High Heat Flux Microelectronics with Dielectrid Liquids
4th Int. Symposium and Exhibition on Advanced Packaging-Materials—Processes-Properties
Thermal Management of Electronic Components With Dielectric Liquids
Jet Impingement Heat Transfer — A Literature Survey
,” ASME Paper, 87-HT-35.
Heat Transfer From Impinging Jets — A Literature Review
,” Technical Report AFWAL-TR-81-3504,
, Wright Patterson AFB, Ohio.
Experimental Investigation on Two-Phase Two-Component Jet Impingement Heat Transfer from Simulated Microelectronic Heat Sources
Boiling Jet Impingement Cooling of Simulated Microelectronic Chips
,” ASME, HTD-Vol.
Study of Mechanism of Burn-Out in a High Heat-Flux Boiling System With an Impinging Jet
Int. Heat Transfer Conference
A Round Turbulent Jet Produced by an Oscillating Diaphragm
Rajaratnan
Enhanced Boiling Heat Transfer by Submerged, Vibration Induced Jets
,” 9th Therminic Conference, Aix-en-Provence, France, Sept.
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Complete the two-way table, please.
Using data from the 2000 census, a random sample of 348 U.S. residents aged 18 and older was selected. The two-way table summarizes the relationship between marital status and housing status for these residents.
\begin{array}{lccc}& Married& Notmarried& Total\\ Own& 172& 82& 254\\ Rent& 40& 54& 94\\ Total& 212& 136& 348\end{array}
The two-way table shows the results from a survey of dog and cat owners about whether their pet prefers dry food or wet food.
\begin{array}{ccc}& \text{ Dry }& \text{ Wet }\\ \text{ Cats }& 10& 30\\ \text{ Dogs}& 20& 20\end{array}
Does the two-way table show any difference in preferences between dogs and cats? Explain.
What is the second name for a two-way frequency table of categorical data?
a. a two-way by two-way table
b. a multiple-treatment table
c. an equal probability table
d. a contingency table
Suppose that each student in a sample had been categorized with respect to political views, marijuana usage, and religious preference, with the categories of this latter factor being Protestant, Catholic, and other. The data could be displayed in three different two-way tables, one corresponding to each category of the third factor.
The ___________ relative frequencies are the sums of each row and column in a two-way table. (joint, marginal, or conditional)
Diabetes and unemployment. A Gallup poll surveyed Americans about their employment status and whether or not they have diabetes. The survey results indicate that 1.5% of the 47,774 employed (full or part time) and 2.5% of the 5,855 unemployed 18-29 year olds have diabetes. (Gallup, 2012)
a. Create a two-way table presenting the results of this study.
b. State appropriate hypotheses to test for difference in proportions of diabetes between employed and unemployed Americans.
c. The sample difference is about 1%. If we completed the hypothesis test, we would find that the p-value is very small (about 0), meaning the difference is statistically significant. Use this result to explain the difference between statistically significant and practically significant findings.
Use a two-way table to explore the probabilities related to each group or category in the table.
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Applied Science AQA/Energy and Efficiency - Wikibooks, open books for an open world
Applied Science AQA/Energy and Efficiency
1.2.1 The meaning of efficiency
It is useful for energy consultants to be able to compare the efficiency of different devices in our homes and workplaces. Energy is transferred by different devices, and the rate at which energy is transferred is called ‘power’.
Architects and energy consultants use U values to measure how effective different materials used in buildings are as insulators. That is, how effective they are at preventing heat energy from transmitting between the inside and the outside of a building.
• the meaning of ‘efficiency’:
• why efficiency is important and why a device can never be 100% efficient
• methods of improving the efficiency of a system or device
• the formula
{\displaystyle efficiency={\frac {usefulenergy(orpower)output}{usefulenergy(orpower)input}}}
• the importance of efficiency in making the best use of available energy
• ways in which efficiency can be increased in mechanical and thermal systems
Work out the efficiency.
why is efficiency useful?
• examples of situations where thermal transfer needs to be maximised and situations where it needs to be minimised
• the meaning of U values
{\displaystyle U={\frac {Q}{At{\vartriangle }T}}}
The meaning of efficiencyEdit
Efficiency is a measure of how much work or energy is kept and not lost in a process. in many processes, work or the energy is lost, this can be through heat or sound. the efficiency is the energy output, this is divided by the energy input and is expressed as a percentage. A perfect process would of an efficiency of 100%.
Retrieved from "https://en.wikibooks.org/w/index.php?title=Applied_Science_AQA/Energy_and_Efficiency&oldid=3393204"
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Pairwise distance between two sets of observations - MATLAB pdist2 - MathWorks América Latina
{d}_{st}^{2}=\left({x}_{s}-{y}_{t}\right)\left({x}_{s}-{y}_{t}{\right)}^{\prime }.
{d}_{st}^{2}=\left({x}_{s}-{y}_{t}\right){V}^{-1}\left({x}_{s}-{y}_{t}{\right)}^{\prime },
{d}_{st}^{2}=\left({x}_{s}-{y}_{t}\right){C}^{-1}\left({x}_{s}-{y}_{t}{\right)}^{\prime },
{d}_{st}=\sum _{j=1}^{n}|{x}_{sj}-{y}_{tj}|.
{d}_{st}=\sqrt[p]{\sum _{j=1}^{n}{|{x}_{sj}-{y}_{tj}|}^{p}}.
{d}_{st}={\mathrm{max}}_{j}\left\{|{x}_{sj}-{y}_{tj}|\right\}.
{d}_{st}=\left(1-\frac{{x}_{s}{{y}^{\prime }}_{t}}{\sqrt{\left({x}_{s}{{x}^{\prime }}_{s}\right)\left({y}_{t}{{y}^{\prime }}_{t}\right)}}\right).
{d}_{st}=1-\frac{\left({x}_{s}-{\overline{x}}_{s}\right){\left({y}_{t}-{\overline{y}}_{t}\right)}^{\prime }}{\sqrt{\left({x}_{s}-{\overline{x}}_{s}\right){\left({x}_{s}-{\overline{x}}_{s}\right)}^{\prime }}\sqrt{\left({y}_{t}-{\overline{y}}_{t}\right){\left({y}_{t}-{\overline{y}}_{t}\right)}^{\prime }}},
{\overline{x}}_{s}=\frac{1}{n}\sum _{j}{x}_{sj}
{\overline{y}}_{t}=\frac{1}{n}\sum _{j}{y}_{tj}.
{d}_{st}=\left(#\left({x}_{sj}\ne {y}_{tj}\right)/n\right).
{d}_{st}=\frac{#\left[\left({x}_{sj}\ne {y}_{tj}\right)\cap \left(\left({x}_{sj}\ne 0\right)\cup \left({y}_{tj}\ne 0\right)\right)\right]}{#\left[\left({x}_{sj}\ne 0\right)\cup \left({y}_{tj}\ne 0\right)\right]}.
{d}_{st}=1-\frac{\left({r}_{s}-{\overline{r}}_{s}\right){\left({r}_{t}-{\overline{r}}_{t}\right)}^{\prime }}{\sqrt{\left({r}_{s}-{\overline{r}}_{s}\right){\left({r}_{s}-{\overline{r}}_{s}\right)}^{\prime }}\sqrt{\left({r}_{t}-{\overline{r}}_{t}\right){\left({r}_{t}-{\overline{r}}_{t}\right)}^{\prime }}},
{\overline{r}}_{s}=\frac{1}{n}\sum _{j}{r}_{sj}=\frac{\left(n+1\right)}{2}
{\overline{r}}_{t}=\frac{1}{n}\sum _{j}{r}_{tj}=\frac{\left(n+1\right)}{2}
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Cichoń's diagram - Wikipedia
Cichoń's diagram
(Redirected from Cichon's diagram)
In set theory, Cichoń's diagram or Cichon's diagram is a table of 10 infinite cardinal numbers related to the set theory of the reals displaying the provable relations between these cardinal characteristics of the continuum. All these cardinals are greater than or equal to
{\displaystyle \aleph _{1}}
, the smallest uncountable cardinal, and they are bounded above by
{\displaystyle 2^{\aleph _{0}}}
, the cardinality of the continuum. Four cardinals describe properties of the ideal of sets of measure zero; four more describe the corresponding properties of the ideal of meager sets (first category sets).
Let I be an ideal of a fixed infinite set X, containing all finite subsets of X. We define the following "cardinal coefficients" of I:
{\displaystyle \operatorname {add} (I)=\min\{|{\mathcal {A}}|:{\mathcal {A}}\subseteq I\wedge \bigcup {\mathcal {A}}\notin I{\big \}}.}
The "additivity" of I is the smallest number of sets from I whose union is not in I any more. As any ideal is closed under finite unions, this number is always at least
{\displaystyle \aleph _{0}}
; if I is a σ-ideal, then add(I) ≥
{\displaystyle \aleph _{1}}
{\displaystyle \operatorname {cov} (I)=\min\{|{\mathcal {A}}|:{\mathcal {A}}\subseteq I\wedge \bigcup {\mathcal {A}}=X{\big \}}.}
The "covering number" of I is the smallest number of sets from I whose union is all of X. As X itself is not in I, we must have add(I) ≤ cov(I).
{\displaystyle \operatorname {non} (I)=\min\{|{\mathcal {A}}|:{\mathcal {A}}\subseteq X\ \wedge \ {\mathcal {A}}\notin I{\big \}},}
The "uniformity number" of I (sometimes also written
{\displaystyle \operatorname {unif} (I)}
) is the size of the smallest set not in I. By our assumption on I, add(I) ≤ non(I).
{\displaystyle \operatorname {cof} (I)=\min\{|{\mathcal {A}}|:{\mathcal {A}}\subseteq I\wedge (\forall B\in I)(\exists A\in {\mathcal {A}})(B\subseteq A){\big \}}.}
The "cofinality" of I is the cofinality of the partial order (I, ⊆). It is easy to see that we must have non(I) ≤ cof(I) and cov(I) ≤ cof(I).
Furthermore, the "bounding number" or "unboundedness number"
{\displaystyle {\mathfrak {b}}}
and the "dominating number"
{\displaystyle {\mathfrak {d}}}
{\displaystyle {\mathfrak {b}}=\min {\big \{}|F|:F\subseteq {\mathbb {N} }^{\mathbb {N} }\ \wedge \ (\forall g\in {\mathbb {N} }^{\mathbb {N} })(\exists f\in F)(\exists ^{\infty }n\in {\mathbb {N} })(g(n)<f(n)){\big \}},}
{\displaystyle {\mathfrak {d}}=\min {\big \{}|F|:F\subseteq {\mathbb {N} }^{\mathbb {N} }\ \wedge \ (\forall g\in {\mathbb {N} }^{\mathbb {N} })(\exists f\in F)(\forall ^{\infty }n\in {\mathbb {N} })(g(n)<f(n)){\big \}},}
{\displaystyle \exists ^{\infty }n\in {\mathbb {N} }}
" means: "there are infinitely many natural numbers n such that …", and "
{\displaystyle \forall ^{\infty }n\in {\mathbb {N} }}
" means "for all except finitely many natural numbers n we have …".
{\displaystyle {\mathcal {B}}}
be the σ-ideal of those subsets of the real line that are meager (or "of the first category") in the euclidean topology, and let
{\displaystyle {\mathcal {L}}}
be the σ-ideal of those subsets of the real line that are of Lebesgue measure zero. Then the following inequalities hold:
{\displaystyle \operatorname {cov} ({\mathcal {L}})}
{\displaystyle \longrightarrow }
{\displaystyle \operatorname {non} ({\mathcal {B}})}
{\displaystyle \longrightarrow }
{\displaystyle \operatorname {cof} ({\mathcal {B}})}
{\displaystyle \longrightarrow }
{\displaystyle \operatorname {cof} ({\mathcal {L}})}
{\displaystyle \longrightarrow }
{\displaystyle 2^{\aleph _{0}}}
{\displaystyle \uparrow }
{\displaystyle \uparrow }
{\displaystyle {\Bigg \uparrow }}
{\displaystyle {\mathfrak {b}}}
{\displaystyle \longrightarrow }
{\displaystyle {\mathfrak {d}}}
{\displaystyle {\Bigg \uparrow }}
{\displaystyle \uparrow }
{\displaystyle \uparrow }
{\displaystyle \aleph _{1}}
{\displaystyle \longrightarrow }
{\displaystyle \operatorname {add} {({\mathcal {L}})}}
{\displaystyle \longrightarrow }
{\displaystyle \operatorname {add} {({\mathcal {B}})}}
{\displaystyle \longrightarrow }
{\displaystyle \operatorname {cov} {({\mathcal {B}})}}
{\displaystyle \longrightarrow }
{\displaystyle \operatorname {non} {({\mathcal {L}})}}
Where an arrow from
{\displaystyle x}
{\displaystyle y}
is to mean that
{\displaystyle x\leq y}
. In addition, the following relations hold:
{\displaystyle \operatorname {add} ({\mathcal {B}})=\min\{\operatorname {cov} ({\mathcal {B}}),{\mathfrak {b}}\}}
{\displaystyle \operatorname {cof} ({\mathcal {B}})=\max\{\operatorname {non} ({\mathcal {B}}),{\mathfrak {d}}\}.}
It turns out that the inequalities described by the diagram, together with the relations mentioned above, are all the relations between these cardinals that are provable in ZFC, in the following limited sense. Let A be any assignment of the cardinals
{\displaystyle \aleph _{1}}
{\displaystyle \aleph _{2}}
to the 10 cardinals in Cichoń's diagram. Then if A is consistent with the diagram's relations, and if A also satisfies the two additional relations, then A can be realized in some model of ZFC.
For larger continuum sizes, the situation is less clear. It is consistent with ZFC that all of the Cichoń's diagram cardinals are simultaneously different apart from
{\displaystyle \operatorname {add} ({\mathcal {B}})}
{\displaystyle \operatorname {cof} ({\mathcal {B}})}
(which are equal to other entries),[2][3][4] but (as of 2019) it remains open whether all combinations of the cardinal orderings consistent with the diagram are consistent.
Some inequalities in the diagram (such as "add ≤ cov") follow immediately from the definitions. The inequalities
{\displaystyle \operatorname {cov} ({\mathcal {B}})\leq \operatorname {non} ({\mathcal {L}})}
{\displaystyle \operatorname {cov} ({\mathcal {L}})\leq \operatorname {non} ({\mathcal {B}})}
are classical theorems and follow from the fact that the real line can be partitioned into a meager set and a set of measure zero.
The British mathematician David Fremlin named the diagram after the Polish mathematician from Wrocław, Jacek Cichoń [pl].[5]
The continuum hypothesis, of
{\displaystyle 2^{\aleph _{0}}}
being equal to
{\displaystyle \aleph _{1}}
, would make all of these relations equalities.
Martin's axiom, a weakening of the continuum hypothesis, implies that all cardinals in the diagram (except perhaps
{\displaystyle \aleph _{1}}
) are equal to
{\displaystyle 2^{\aleph _{0}}}
^ Bartoszyński, Tomek (2009), "Invariants of Measure and Category", in Foreman, Matthew (ed.), Handbook of Set Theory, Springer-Verlag, pp. 491–555, arXiv:math/9910015, doi:10.1007/978-1-4020-5764-9_8, ISBN 978-1-4020-4843-2, S2CID 15079978
^ Martin Goldstern, Jakob Kellner, Saharon Shelah (2019), "Cichoń's maximum", Annals of Mathematics, 190 (1): 113–143, arXiv:1708.03691, doi:10.4007/annals.2019.190.1.2, S2CID 119654292 {{citation}}: CS1 maint: uses authors parameter (link)
^ Martin Goldstern, Jakob Kellner, Diego A. Mejía, Saharon Shelah (2019), Cichoń's maximum without large cardinals, arXiv:1906.06608 {{citation}}: CS1 maint: uses authors parameter (link)
^ Martin Goldstern, Jakob Kellner, "A Deep Math Dive into Why Some Infinities Are Bigger Than Others", Scientific American, retrieved 2021-08-23 {{citation}}: CS1 maint: uses authors parameter (link)
^ Fremlin, David H. (1984), "Cichon's diagram", Sémin. Initiation Anal. 23ème Année-1983/84, Publ. Math. Pierre and Marie Curie University, vol. 66, Zbl 0559.03029, Exp. No.5, 13 p. .
Retrieved from "https://en.wikipedia.org/w/index.php?title=Cichoń%27s_diagram&oldid=1078847247"
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Multi-parameter paraproducts | EMS Press
We prove that the classical Coifman-Meyer theorem holds on any polydisc
\mathbf{T}^d
of arbitrary dimension
d\geq 1
Camil Muscalu, Jill Pipher, Terence Tao, Christoph Thiele, Multi-parameter paraproducts. Rev. Mat. Iberoam. 22 (2006), no. 3, pp. 963–976
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Revision as of 13:52, 30 July 2013 by NikosA (talk | contribs) (→Deriving Physical Quantities: added Band-Averaged Solar Spectral Irradiance as constants)
{\displaystyle {\frac {W}{m^{2}*sr*nm}}}
{\displaystyle L(\lambda ,Band)={\frac {K*DN\lambda }{Bandwidth\lambda }}}
{\displaystyle \rho _{p}={\frac {\pi *L\lambda *d^{2}}{ESUN\lambda *cos(\Theta _{S})}}}
{\displaystyle \rho }
{\displaystyle \pi }
{\displaystyle L\lambda }
{\displaystyle d}
{\displaystyle Esun}
{\displaystyle cos(\theta _{s})}
{\displaystyle {\frac {W}{m^{2}*\mu m}}}
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Electric Dipoles | Brilliant Math & Science Wiki
Two charges are brought closer together. In the limit where the distance between them goes to zero, the become an ideal electric dipole and are represented simply with their dipole vector. [1]
An electric dipole is composed of two electric charges with opposite signs brought very close together. An ideal electric dipole is one in which the two charges are only infinitesimally separated. Equations defined particularly for electric dipoles are defined in terms of the electric dipole moment
(\vec{p}).
The electric dipole moment for two charges
+q
-q
separated by a displacement vector
\vec{d}
\vec{p} = q\vec{d}
-8 \hat{x} + 6 \hat{y}
8 \hat{x} - 6 \hat{y}
4 \hat{x} - 3 \hat{y}
-4 \hat{x} + 3 \hat{y}
2 \text{ C}
charge is placed at
(0\text{ m},3\text{ m})
-2\text{ C}
charge is at
(4\text{ m},0\text{ m}).
What is the electric dipole moment?
Energy stored in a dipole
If an electric dipole is placed in an external electric field the charges will feel forces in opposite directions by virtue of their opposite signs, causing a rotation if the two charges remain bound to one another. Hence, the forces actually manifest themselves as a torque.
The black external electric field causes red forces in opposite directions on the charges, resulting in rotation of the blue dipole moment in the direction indicated by the yellow arrow.
Since the torque is largest when
\vec{p}
\vec{E}
and decreases as the two vectors become more parallel, it is best modeled as the cross product between the two vectors.
The torque on a dipole
\vec{p}
placed in an external field
\vec{E}
\tau = \vec{p} \times \vec{E}
+Q
-Q
are placed at
x = 1\text{ m}
x = -1\text{ m}
respectively in an external field
\vec{E} = 3 \hat{x} + 4 \hat{y}.
What is the torque on the dipole?
In order to evaluate the cross product, write down the dipole.
\vec{p} = q\vec{d} = (Q) (2 \hat{x}) = 2Q\hat{x}
Now evaluate the cross product.
\tau = \vec{p} \times \vec{E} =( 2Q\hat{x}) \times (3 \hat{x} + 4 \hat{y}) = (8Q \text{ Nm})\hat{z}
-AB None of these choices AB 0
What is the torque on an ideal electric dipole with moment
\vec{p} = -A \hat{z}
placed in an external electric field
\vec{E} = B \hat{z}?
The dipole wants to rotate until its moment points in the same direction as the external electric field. This orientation thus represents the potential minimum, and the potential energy increases as the angle between the dipole moment and the external field increases.
U=-\vec{p}\bullet\vec{E}
How much work is required to flip an electric dipole from its potential minimum to an orientation antiparallel to the external field?
Recall that work is defined in terms of potential energy according to the work-energy theorem.
W=-\Delta U = - (U_f -U_0) = -[pE\cos(180^\circ) - pE\cos(0^\circ) ] = 2pE
Geek3, . VFPt dipole animation electric. Retrieved June 8, 2016, from https://commons.wikimedia.org/wiki/File:VFPt_dipole_animation_electric.gif
Cite as: Electric Dipoles. Brilliant.org. Retrieved from https://brilliant.org/wiki/electric-dipoles/
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Zhouling Wang, Wenwu Wang, Ya Yang, Wei Li, Lianghuan Feng, Jingquan Zhang, Lili Wu, Guanggen Zeng, "The Structure and Stability of Molybdenum Ditelluride Thin Films", International Journal of Photoenergy, vol. 2014, Article ID 956083, 6 pages, 2014. https://doi.org/10.1155/2014/956083
Zhouling Wang,1 Wenwu Wang,1 Ya Yang,1 Wei Li ,1 Lianghuan Feng,1 Jingquan Zhang,1 Lili Wu,1 and Guanggen Zeng1
1College of Materials Science and Engineering, Sichuan University, Chengdu 610064, China
Molybdenum-tellurium alloy thin films were fabricated by electron beam evaporation and the films were annealed in different conditions in N2 ambient. The hexagonal molybdenum ditelluride thin films with well crystallization annealed at 470°C or higher were obtained by solid state reactions. Thermal stability measurements indicate the formation of MoTe2 took place at about 350°C, and a subtle weight-loss was in the range between 30°C and 500°C. The evolution of the chemistry for Mo-Te thin films was performed to investigate the growth of the MoTe2 thin films free of any secondary phase. And the effect of other postdeposition treatments on the film characteristics was also investigated.
Molybdenum ditelluride (MoTe2) belongs to the large family of layered transition metal dichalcogenides, which is bound by weak van der Waals interactions along the -axis [1]. The electronic, optical, magnetic, and catalytic properties of the transition metal dichalcogenides have been extensively studied [2–6]. MoTe2 can act as an efficient absorbing layer in solar cells only if the crystallites of the films are textured with the -axis perpendicular to the plane of the substrate [7]. Because of the layered structure of MoTe2, various metal atoms can be doped between the layers to change its optical and electrical properties [8]. It has been found that H absorption on MoTe2 monolayers results in large spatial extensions of spin density and weak AFM coupling between local magnetic moments even at the distance of above 12.74 Å [4]. MoTe2 has a bandgap of around 1.1 eV and a high work function of ~4.7 eV, and the valence band offset of CdTe/MoTe2 is only 0.03 eV [1]; these are advantageous to the hole transport between cadmium telluride and molybdenum ditelluride. Therefore, MoTe2 is a potential candidate for a stable Cu-free back contact to CdS/CdTe solar cells.
Tellurium pressure or vapor is often indispensable for the preparation of the MoTe2 thin films [9, 10], and the adhesion or reproducibility of the films is very poor. In this work, MoTe2 thin films were synthesized by solid state reactions between Mo and Te thin films followed by an anneal in N2 ambient. The structural property and stability of Mo-Te thin films were investigated under different postdeposition treatment conditions.
Molybdenum-tellurium alloy thin films were deposited by electron beam evaporation at room temperature in the pressure of ~10−4 Pa. The multilayer Mo/Te films with a stacking sequence Te-Mo-Te-Mo were deposited independently and alternately using high purity molybdenum (99.999% purity, Alfa Aesar) and tellurium (99.999% purity, Alfa Aesar). The deposition rates of molybdenum and tellurium were monitored by a thickness monitor. The total thickness of the molybdenum-tellurium multilayer thin films was 200~450 nm. After deposition, a posttreatment was performed at different temperatures in N2 ambient. The temperatures reproducible to ±1 K were obtained from repeated runs on the same sample.
The structure and the surface morphology of the samples were done by X-ray diffraction (XRD) (DX-2600, Dandong, China) and atomic force microscope (AFM) (MFP-3D-BIO, Asylum Research, USA). The film thickness was surveyed using a stylus profiler (XP-2, Ambios Technology Inc., USA). To study the effect of annealing on the MoTe2 thin films, thermogravimetry and differential scanning calorimetry (TG/DSC) (STA 449C, NETZSCH, Germany) analysis were carried out. X-ray photoelectron spectra (XPS) (ESCALAB 250, Thermo Fisher SCIENTIFIC, UK) were performed to determine the atom chemical states.
Figure 1 shows the XRD patterns of the Mo-Te thin films with 450 nm thickness annealed at different temperatures in N2 ambient. The peaks marked by rhombus were indexed to the phase of Te (JCPDS NO. 65-3370), and the others were the phase of MoTe2 (JCPDS number 15-0658) (see Figure 1). The reflection positions in the XRD patterns of as-deposited layers were at the angles of 23.029°, 27.560°, 40.454°, and 49.650°, which correspond to Te (100), (011), (110), and (021). In order to form the crystalline compound MoTe2 thin films with a simple hexagonal Bravais lattice, thermal postdeposition treatment was performed. The as-deposited thin films were annealed at 300°C, 356°C, 450°C, 470°C, and 475°C in N2 ambient, respectively. After annealing at 300°C in N2 ambient, many more diffraction peaks of Te, such as (012), (111), and (200), emerged at 38.263°, 43.353°, and 47.060°. When the annealing temperature reached 356°C, the patterns of the films were very different from those of the films annealed at 300°C; the peaks corresponding to MoTe2 (002) and (004) were revealed by XRD patterns as shown in Figure 1. It was noticed that two diffraction peaks of Te (011) and (112) were also observed in the XRD patterns. The results show that a considerable amount of Te and Mo diffuses into the Mo and the Te films, respectively. The chemical reaction that took place in the thin elemental layers can be described as 2Te + Mo → MoTe2. The thin films with well crystallization were achieved when annealing was performed at 470°C or higher. The peaks of MoTe2 (006) could be observed at 38.905°, and there were no peaks of Te. These results indicate the disappearance of Te but the presence of MoTe2 with increasing the annealing temperature. Therefore, annealing promotes the formation of MoTe2 and annealing at 470°C leads to the single phase MoTe2 thin films.
XRD patterns of Mo-Te thin films annealed at different temperatures for 15 min in N2 ambient.
To study the stability of the molybdenum-tellurium alloy thin films, TG and DSC analysis were performed. The thin films as-deposited were cleaved from the substrates; approximately 2.952 mg of sample was used in this work. The gas rate was 30 mL/min, and the heating rate was 10 K/min. Figure 2 shows the TG-DSC curves of as-deposited films. The peaks of as-deposited MoTe2 thin films were extended from 30°C to 600°C. The peaks in DSC values meant endothermic reactions happened while heating. The first endothermic peak was located at 68.5°C and it was the endothermic peak of water. Then with the increase of temperature, there were endothermic peaks appearing at about 350°C and 448.5°C due to the energy consumption, which results from Te atoms and Mo atoms moving into lattice sites so as to form MoTe2. This behavior is consistent with the results of XRD (Figure 1); that is, polycrystalline MoTe2 was gradually formed when the annealing temperature increased up to 350°C. From the TG values also shown in Figure 2, one can see that there was almost no weight loss at the temperatures lower than 500°C, whereas the TG curve dropped sharply at the temperatures higher than 500°C. The severe weight-loss might be the reevaporation of tellurium for MoTe2, the results of which will be discussed later.
TG and DSC analysis of the as-deposited films.
With the increase of the annealing temperature, the intensities of the peaks for the thin films became significantly strong and the positions of the peaks were not changed (Figure 3). These indicate the improved crystallinity and stable structure of MoTe2 thin films due to the interdiffusion. As the annealing temperature increased up to 500°C, although the weight-loss was about 2% (Figure 2), an amorphous baseline distinctly appeared. We attribute the poor crystallinity of the films to the severe reevaporation of tellurium in the process of the postdeposition treatments. When the annealing temperature was further increased, the adhesion of the films was poor. Grain growth of MoTe2 thin films may introduce stress at the glass/MoTe2 interface, resulting in film blistering or peeling.
XRD patterns of Mo-Te thin films with a thickness of 450 nm annealed at 485°C, 495°C, and 500°C for 15 minutes in N2 ambient.
Based upon the investigations of XRD and TG/DSC analysis, the evolution of the morphology and chemistry for MoTe2 thin films annealed at temperatures lower than 500°C was studied by AFM and XPS (Figures 4-5). Figure 4 shows the atomic force microscopy of Mo-Te thin films as-deposited and annealed at 475°C in N2 ambient. The surface of the as-deposited MoTe2 thin film was smoother than the one annealed at 475°C. The root mean square of the as-deposited film was 1.179 nm while it became 19.803 nm after annealing. That means annealing could promote the growth of the grains. The well-distributed atoms moved to lattice sites so the valleys and peaks on the surface were detected by AFM, and this is consistent with the XRD results.
AFM images of Mo-Te thin films as-deposited (a) and (b) annealed at 475°C for 15 min in N2 ambient.
XPS spectra of (a) Mo 3d, (b) Te 3d, and (c) O 1s for Mo-Te thin films annealed at different temperatures.
Figure 5 shows XPS spectra of Mo-Te thin films at different temperatures. When the annealing temperature was below 450°C, it was noticed that there was only one emission peak in XPS spectra of Mo 3d3/2, which drew core level lines at about 232.75 eV, and it was a double peak. The emission peak with lower binding energies was determined to be the Mo 3d5/2 in MoTe2 which occurred at 229.2 eV when the annealing temperature reached 450°C or higher. That means MoTe2 was formed when the annealing temperature rose to 450°C and it has been confirmed by XRD in Figure 1. In Figure 5, the doublet of Te 3d was doubled (the two Te 3d5/2 peaks being at about 572.9 eV and 576.5 eV), while the Te 3d5/2 position of TeO2 occurred at 576 eV; maybe there was a small amount of tellurium oxidation at the surface of the films. It was obvious that the ratio of ionized Te/elemental Te increased dramatically after annealing since the peak area was proportional to the chemical composition. The results show that a large amount of Te was not alloyed in the as-deposited films. After annealing, most of Te was alloyed with Mo in the form of MoTe2 due to the interdiffusion. This could explain why the peak of Te was firstly detected but disappeared after annealing in XRD patterns (see Figure 1).
To further explore the effect of annealing on the structure of Mo-Te thin films, the other thermal postdeposition treatments such as annealing time and thickness were carried out. Figure 6 shows XRD of Mo-Te thin films with thickness of 450 nm annealed at 470°C for different annealing time. It was noticed that when the annealing time was 10 minutes, peaks of MoTe2 (002), (004), (006), and (008) were revealed with Te (100) at 23.029°, while the annealing time reached 15 minutes and the Te phase disappeared. With the increase of the annealing time (e.g., 20 minutes), the crystallization of the films was not better than that of the films annealed for 15 minutes and the intensities of the peaks became weak. The poor crystallization can be also attributed in part to the reevaporation of tellurium.
XRD patterns of Mo-Te thin films with thickness of 450 nm annealed at 470°C for different times in N2 ambient.
From Figure 7, peaks of Te (100), (110), (111), (200), (021), and (210) for 300 nm thick Mo-Te thin films were detected with MoTe2 (002). As the film thickness increased to 400 nm, the number of diffraction peaks of Te was suppressed and the peak of MoTe2 (004) was firstly observed. With the increase of the thickness for the thin films, more peaks of single phase MoTe2, such as (006) and (008), were detected in the 450 nm thick Mo-Te thin films.
XRD patterns of Mo-Te thin films annealed at 470°C in N2 ambient with different thicknesses.
Mo-Te thin films were deposited at room temperature by electron beam evaporation, and then the films were annealed in N2 ambient under different conditions. The formation of MoTe2 thin films took place at about 350°C and the structure of the thin films was stable from room temperature to 500°C. As the temperature increased, the growth of the MoTe2 phase was predominated and the Te secondary phase was suppressed. The thin films were single phase MoTe2 and well crystallized in the hexagonal structure annealed at 470°C or higher. At a temperature of 500°C or higher, or even for a long annealing time, the thin films were poor due to the reevaporation of tellurium and the adhesion-loss problems.
This work was supported by the National Basic Research Program of China (Grant no. 2011CBA007008), National Natural Science Foundation of China (Grant no. 61076058), and the Science and Technology Program of Sichuan Province, China (Grant no. 13ZC2185).
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Research on Initialization on EM Algorithm Based on Gaussian Mixture Model
Research on Initialization on EM Algorithm Based on Gaussian Mixture Model
Ye Li, Yiyan Chen*
UK Systems Science Association, London, UK
The EM algorithm is a very popular maximum likelihood estimation method, the iterative algorithm for solving the maximum likelihood estimator when the observation data is the incomplete data, but also is very effective algorithm to estimate the finite mixture model parameters. However, EM algorithm can not guarantee to find the global optimal solution, and often easy to fall into local optimal solution, so it is sensitive to the determination of initial value to iteration. Traditional EM algorithm select the initial value at random, we propose an improved method of selection of initial value. First, we use the k-nearest-neighbor method to delete outliers. Second, use the k-means to initialize the EM algorithm. Compare this method with the original random initial value method, numerical experiments show that the parameter estimation effect of the initialization of the EM algorithm is significantly better than the effect of the original EM algorithm.
EM Algorithm, Gaussian Mixture Model, K-Nearest Neighbor, K-Means Algorithm, Initialization
Assessment of this performance of an algorithm generally relates to its efficiency, ease of operation and operation results. We are concerned about the efficiency of the iteration, and one of the factors that affect the efficiency of the iteration is the selection of the initial value. EM algorithm has a very important application in the Gaussian mixture model (GMM). In simple terms, if we don’t know neither the parameters of the mixed model nor the classification of the observed data, EM algorithm is a popular algorithm for estimating the parameters of finite mixture model. However, sometimes its performance is poor. This algorithm has an obvious shortcoming: it is very sensitive to the initial value. Therefore, in order to get the parameter estimation of the closest to the true value, we have to find a method to initialize the EM algorithm. We can list several usual methods with initialization: random center, hierarchical clustering, k-means algorithm and so on [1]. As a result of the k-means clustering algorithm is also a kind of dynamic iterative algorithm and decides the classification number by subjective factors. Further, it is accordant with EM algorithm for parameter estimation of finite mixture model. Hence we can use outlier detection based on proximity to remove the outliers in order to reduce the influence of noise for the parameter estimation. Then, a rough grouping of the rest of the mixed data is given by k-means clustering. Finally, a rough estimate of parameters is given based on packet data.
2. Gaussian Mixture Modeling
The mixture model is a useful tool for density estimation, and can be viewed as a kind of kernel method [2]. If the d-dimensional random vector has a finite mixture normal distribution, its probability density function is as follows:
p\left(x|\theta \right)=\sum _{i=1}^{k}\text{ }\text{ }{\alpha }_{i}{p}_{i}\left(x|{\theta }_{i}\right)=\sum _{i=1}^{k}\text{ }\text{ }{\alpha }_{i}{\left(2\pi \right)}^{-\frac{d}{2}}{|{\sum }_{i}|}^{-\frac{1}{2}}\mathrm{exp}\left(-\frac{1}{2}{\left(x-{\mu }_{i}\right)}^{T}{\sum }_{i}^{-1}\left(x-{\mu }_{i}\right)\right)
{p}_{i}\left(x|{\theta }_{i}\right)={\left(2\pi \right)}^{-\frac{d}{2}}{|{\sum }_{i}|}^{-\frac{1}{2}}\mathrm{exp}\left(-\frac{1}{2}{\left(x-{\mu }_{i}\right)}^{T}{\sum }_{i}^{-1}\left(x-{\mu }_{i}\right),i=1,2,\cdots ,k
This is the probability density function of the i-th branch, which has a mean
{\mu }_{i}
, a covariance
{\sum }_{i}
, and mixing proportion
{\alpha }_{i}
\sum _{i=1}^{k}{\alpha }_{i}=1
{\theta }_{i}={\left({\mu }_{i}^{T},{\sum }_{i}\right)}^{T}
\theta ={\left({\theta }_{1}^{T},{\theta }_{2}^{T},\cdots ,{\theta }_{k}^{T},{\alpha }_{1},{\alpha }_{2},\cdots ,{\alpha }_{k}\right)}^{T}
is a vector corresponding to all unknown parameters.
2.1. The EM Algorithm
The classical and natural method for computing the maximum-likelihood estimates (MLEs) for mixture distributions is the EM algorithm (Dempster et al., 1977), which is known to possess good convergence properties. In other words, the EM algorithm is a kind of the iterative algorithm to solve the maximum likelihood estimate when the observation data is incomplete data, which has good application value as well. The process of parameter estimation of the EM algorithm is given in reference [3]: Y is generally assumed to be the observed data, Z is potential data,
\theta
is unknown parameters, a posteriori distribution density function based on the observed data Y of parameter
\theta
p\left(\theta |Y\right)
, called observed posterior distribution,
p\left(\theta |Y,Z\right)
represents that the posterior distribution density function of parameter
\theta
is obtained after the addition of data Z, called addition posterior distribution,
p\left(Z|\theta ,Y\right)
represents the conditional distribution density function of the potential data Z in given
\theta
and observation data Y. The goal is to estimate the parameter values of the observed posterior distribution
{\theta }^{\left(t\right)}
, hence the EM algorithm is carried out as follows: Let
{\theta }^{\left(t\right)}
be the estimate values of the posterior mode at the beginning of the t + 1 iteration. The two step of the t + 1 iteration is:
Expectation Step: calculate mathematical expectation of
p\left(\theta |Y,Z\right)
\mathrm{log}p\left(\theta |Y,Z\right)
on the conditional distribution of Z, namely:
Maximization Step: maximize the
Q\left(\theta |{\theta }^{\left(t\right)},Y\right)
, that is to find a point
{\theta }^{\left(t+1\right)}
Q\left({\theta }^{\left(t+1\right)}|{\theta }^{\left(t\right)},Y\right)=\underset{\theta }{\mathrm{max}}Q\left(\theta |{\theta }^{\left(t\right)},Y\right)
after getting
{\theta }^{\left(t+1\right)}
, this forms an iteration
{\theta }^{\left(t\right)}\to {\theta }^{\left(t+1\right)}
, iterate Expectation step and Maximization step until
‖{\theta }^{\left(t+1\right)}-{\theta }^{\left(t\right)}‖
‖Q\left({\theta }^{\left(t+1\right)}|{\theta }^{\left(t\right)},Y\right)-Q\left({\theta }^{\left(t\right)}|{\theta }^{\left(t\right)},Y\right)‖
2.2. Outlier Detection Based on Proximity
Outliers are data objects that are inconsistent with behavior or model of most of the data in the entire data set [4]. An object is abnormal if it is far away from most points. This method is more general and easier to use than the statistical method, because it is easier to determine meaningful proximity metrics than statistical distribution for data sets.
One of the easiest ways to measure whether an object is far away from most points is using the distance to the k-nearest neighbor. The outlier score of an object is given by the distance between the object and its the k-nearest neighbor. The minimum outlier score is 0, and the maximum value is the maximum value of the distance function: it is usually infinite.
Outlier score is highly sensitive to the value of k. If the k is too small, then a small amount of adjacent outliers can lead to a low outlier score. If k is too large, all objects in the clusters with points less than k are likely to be outliers. In order to make the scheme more robust for the selection of k, we can use the average distance of the first k nearest neighbors.
The K-means algorithm is one of the most popular iterative descent clustering methods [5]. This algorithm takes
k
as the parameter, then
objects are divided into
k
clusters, in the same cluster objects in a high similarity, while objects in different clusters in the greater dissimilarity. First, selecting
k
objects at random, each object represents a cluster center. For the rest of each object, which assigned to the most similar class according to the distance between the object and the cluster centers. And then calculate the new center of each cluster. This process is iterated until convergence. The method is based upon the following steps:
Step 1: Choose samples
m
randomly:
{\overline{x}}_{1}^{\left(k\right)},{\overline{x}}_{2}^{\left(k\right)},\cdots ,{\overline{x}}_{m}^{\left(k\right)}
as the cluster centers of mixed data.
Step 2: For the remaining
n-m
data, noted:
{x}_{1},{x}_{2},\cdots ,{x}_{j},\cdots ,{x}_{n-m},\text{\hspace{0.17em}}{d}_{ji}^{\left(k\right)}=\left|{x}_{j}-{\overline{x}}_{i}^{\left(k\right)}\right|
This function represents the distance from
{x}_{j}
{\overline{x}}_{i}^{\left(k\right)}
j=1,2,\cdots ,n-m
i=1,2,\cdots ,m
. For fixed
{x}_{j}
\left|{x}_{j}-{\overline{x}}_{i}^{\left(k\right)}\right|=\mathrm{min}{d}_{ji}^{\left(k\right)},i=1,2,\cdots ,m
{x}_{j}
is classified into the category
i
. Thus, we divide the data into
m
classes:
{C}_{1}^{\left(k\right)},{C}_{2}^{\left(k\right)},\cdots ,{C}_{m}^{\left(k\right)}
Step 3: Use formula:
{\overline{x}}_{i}^{k+1}=\frac{1}{{n}_{i}^{\left(k\right)}}{\displaystyle \sum _{j=1}^{{n}_{i}^{\left(k\right)}}{x}_{ji}},{x}_{ji}\in {C}_{i}^{\left(k\right)}
{n}_{1}^{\left(k\right)}+{n}_{2}^{\left(k\right)}+\cdots +{n}_{i}^{\left(k\right)}=n,i=1,2,\cdots ,m
Find out the sample mean of the data of each class in the second step as a new cluster center.
Step 4: For a given number
\epsilon >0
{\displaystyle \sum _{i=1}^{m}\left|{\overline{x}}_{i}^{\left(k+1\right)}-{\overline{x}}_{i}^{\left(k\right)}\right|}<\epsilon
, then stop iteration,
Otherwise, the new clustering center in the third step into the first step to replace the old clustering center, repeat the second step and third step operation.
2.4. The K-Means Algorithm Is Used to Initialize EM Algorithm after Deleting Outliers
First, we compute the distance between each observation data and its the k-nearest neighbor (this distance is called the k-NN distance), deleting those points which have relatively large the k-NN distances, namely abnormal observations that is far from most points. Second, we give a rough grouping of mixed data using k-means cluster method for the rest of data. Then according to the packet data, a rough estimate of parameters, as the initial value of the iterative algorithm, are given. Finally, execute EM algorithm until convergence to estimate the parameters of gaussian mixture modeling.
We compare the effect of parameter estimation about the improved initial value method and original method on the simulation data in this section.
First, we generate a one-dimensional data set; its histogram is shown in Figure 1. Simulated example with two classes, the data in first class are generated from one-dimensional normal population
N\left(3,1\right)
, and the data in second class are generated from one-dimensional normal population
N\left(-2,{2}^{2}\right)
. The numbers of the data points of the 2 classes are 2000 and 3000. After iteration of the EM algorithm, the parameter estimation results are shown in later.
The original random initial value method of EM algorithm is very unstable, sometimes it need to be iterated nearly 100 times, and usually can not be very
Figure 1. Histogram and density curve of the original data.
close the true value of the parameters. However, number of iterations of EM algorithm is greatly reduced by the improved initial value method which is pretty stable. And the final parameter estimates are closer to the true value compared to original method. Here, we select a test result to draw a line graph (see Figure 2 in next page).
The average iteration times of the original EM algorithm is approximately 33, and the final iteration result is also highly unstable. Then we compare the parameter estimation results of the original method and the improved method on some realization. We can draw some conclusion from Figure 2, after 22 iterations, the final output parameter estimates of the original method are:
\begin{array}{l}{\stackrel{^}{\mu }}_{1}=2.829,{\stackrel{^}{\mu }}_{2}=-2.233\\ {\stackrel{^}{\sigma }}_{1}=1.112,{\stackrel{^}{\sigma }}_{2}=1.795\\ {\stackrel{^}{\alpha }}_{1}=0.443,{\stackrel{^}{\alpha }}_{2}=0.556\end{array}
The outliers are removed by the distances between each sample and its 10-nearest neighbor in improved initial value method. Then use the k-means method to classify the remaining points. The class center of each class is used as the initial value of the mean, the sample variance of each class is used as the initial value of the variance. The proportion of each class is used as the initial value of the coefficient. After 7 iterations, the change of the parameter is less than 10−2. The results of parameter estimation are as follows
\begin{array}{l}{\stackrel{^}{\mu }}_{1}=2.949,{\stackrel{^}{\mu }}_{2}=-2.034\\ {\stackrel{^}{\sigma }}_{1}=1.011,{\stackrel{^}{\sigma }}_{2}=1.992\\ {\stackrel{^}{\alpha }}_{1}=0.404,{\stackrel{^}{\alpha }}_{2}=0.596\end{array}
Obviously, the performance of the improved method outperforms that of the original method [6]. The improved method has less number of iterations and
Figure 2. Effect comparison between improved initial value method and original method for parameter estimation.
better parameter estimation results. What is more, its statistical meaning is easy to understand. Then, the improved initial value method can be used not only for one-dimensional Gaussian mixture model, but also for multidimensional Gaussian mixture model.
However, it is need to further study how to choose the k when using the k-NN distance to remove outliers. At the same time, if we can further optimize the process of initial value selection (reducing complexity of deleting outliers and k-means clustering), it will bring greater use value.
Li, Y. and Chen, Y.Y. (2018) Research on Initialization on EM Algorithm Based on Gaussian Mixture Model. Journal of Applied Mathematics and Physics, 6, 11-17. https://doi.org/10.4236/jamp.2018.61002
1. Wang, X. (2012) Gaussian Mixture Model Based k-Means to Initialize the EM Algorithm. Journal of Shangqiu Normal University, 28, 11-14.
2. Wang, J.K. and Gai, J.Y. (1995) Mixture Distribution and Its Application. Journal of Biomathematics, 10, 87-92.
3. Trevor, H., Robert, T. and Jerome, F. (2001) The Elements of Statistical Learning, Springer-Verlag, New York.
4. Zhang, Z.P., Xu, X.Y. and Wang, P. (2011) Spatial Outlier Mining Algorithm Based on KNN Graph. Computer Engineering, 3737-3739.
5. Zhu, J.Y. (2013) Research and Application of K-Means Algorithm. Dalian University of Technology.
6. Zhai, S.D. (2009) Research on Clustering Algorithm Based on Mixtured Model. Northwest University.
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PDEs in a Nutshell Practice Problems Online | Brilliant
Besides being nonlinear, last quiz's equation systems had one other thing in common: their only independent variable was time
t.
The other part of our course looks at problems where the situation is reversed:
\begin{aligned} \textcolor{#D61F06}{\text{nonlinear section}} & \implies \textcolor{#D61F06}{\begin{cases} \text{several equations} \\ \text{single independent variable} \end{cases}} \\ \textcolor{#3D99F6}{\text{pde section}} & \implies \textcolor{#3D99F6}{\begin{cases} \text{single equation} \\ \text{several independent variables}. \end{cases}}\end{aligned}
When the unknown function depends on several variables, it necessarily involves partial derivatives, hence the name partial differential equation, or PDE for short.
Let's take a look at some of the PDEs we'll encounter and how we'll go about solving them!
A partial derivative of a multivariable function
f
is the rate of change with respect to a single variable with all others fixed.
\frac{\partial f}{\partial x_{i}} = f_{x_{i}} = \lim\limits_{h\to 0} \frac{f(x_{1}, \dots, x_{i}+h, \dots , x_{n})-f(x_{1}, \dots, x_{i}, \dots, x_{n})}{h}.
PDEs in a Nutshell
You'll probably recognize our first PDE not from the equation itself but from its solutions: Tie a rope to a post and wiggle it up and down. The wiggles reflect back when they hit the post; the reflections and incoming wiggles combine and eventually form a standing wave.
The amount by which the rope is displaced depends on where we look and when, i.e.
u(x,t)
(see figure) depends on
x
t.
For reasons we'll get into later, the rope's wave equation is given by
u_{tt} = v^2 u_{xx},
v
is the constant wave speed and
u_{tt} \ (u_{xx})
is the second partial derivative with respect to time (space).
Select all options solving this wave equation. Since we don't know how to solve a PDE from scratch yet, take the second partials of each and find those where
u_{tt}
v^2 u_{xx}.
f
\frac{\partial f}{\partial x_{i}} = f_{x_{i}} = \lim\limits_{h\to 0} \frac{f(x_{1}, \dots, x_{i}+h, \dots , x_{n})-f(x_{1}, \dots, x_{i}, \dots, x_{n})}{h}.
\cos( v t) \sin(x)
e^{- v t} \sin(x)
(vt)^2 - x^2
\sin( v t) \cos(x)
The last problem not only shows us that a PDE can have more than one solution, but it also gives us a clue about how we can start making our own from scratch.
Notice that both solutions
\cos(v t) \sin(x)
\sin(v t) \cos(x)
“split” into
t
x
parts. Let's see if we can make a similar split
u(x,y,t) = X(x) Y(y) T(t)
for the 2D wave equation
\frac{\partial^2 u}{\partial t^2 } = v^2 \left[ \frac{\partial^2 u}{\partial x^2 }+\frac{\partial^2 u}{\partial y^2 } \right],
which describes the vibrations of the rectangular drumhead (blue) graphed below:
(The graph is touch interactive, so be sure to practice changing the perspective and zooming; we'll see many more graphs like this in the future!)
X(x) Y(y) T(t)
into the 2D wave equation, and then divide both sides by
X(x) Y(y) T(t)
after you're done taking derivatives. What can you say about the result?
x, y,
t
all split without mixing. Variable
x
splits off from
y
t,
which are mixed together. Variable
y
x
t,
t
x
y,
which are mixed together.
This method of variable separation is our first line of attack when it comes to PDEs. It won't always work, but when it does, it reduces a difficult PDE problem to something easier.
For example, the 2D wave equation splits as
\frac{T''(t)}{T(t)} = v^2 \left[ \frac{X''(x)}{X(x)} + \frac{Y''(y)}{Y(y)} \right];
x,y,
t
are all independent variables, this equation can only be true if each individual piece is constant:
\frac{X''(x)}{X(x)} = - 4 \pi^2,\ \ \frac{Y''(y)}{Y(y)} = - 16 \pi^2,\ \ \frac{T''(t)}{T(t)} = -20 \pi^2 v^2
are the choices that make the example defined on the rectangle
[0,1] \times \left[ 0, \frac{1}{2} \right]
Each of these equations can be solved separately; what option solves the
X
equation?
X(x) = \sin( \pi x)
X(x) = \sin(2 \pi x)
X(x) = \sin^2(2 \pi x)
X(x) = \sin(2 \pi x)\cos(2\pi x)
Waves on bounded domains like our drum are all infinite sums (called Fourier series) of “split” solutions like the ones we just found in the 2D case.
We'll also encounter another kind of infinite sum: power series. They're needed in a wide variety of real-world engineering and physics problems. Besides describing vibrations on a circular drumhead, they come up in fluid problems and even quantum theory.
In the finale of our course, we'll use power series together with separation of variables to solve the hydrogen atom, one of the most important scientific achievements of the 20
^\text{th}
We'll even use our solutions to sketch an iconic image from basic chemistry: an electron orbital!
Separating the variables can be a useful thing to try when the domain has a simple shape like a rectangle or a circle, but even then we might need some advanced technique like power series.
Things are different for infinite domains, like that of the 3D compression wave we'll study later:
This wave obeys the 3D wave equation
u_{tt} = v^2 [ u_{xx}+ u_{yy}+u_{zz} ]
u_{tt} = v^2 \nabla^2 u
\big(
\nabla^2
will be a recurring feature of our course!
\big)
u(x,y,z,t)
measures the compression/expansion of air at position
(x,y,z)
t.
A 2D slice is shown below; to solve for
u,
we'll need a new tool, the Fourier transform.
The Fourier transform turns a PDE into an easier problem, like an ordinary differential equation. It works best when the domain is all of space, or
\mathbb{R}^{n},
and the unknown
u
For example, consider the classic Drunkard's walk problem in probability.
Jack's trying to make his way home after a night at the pub, but in his current state he moves left or right in a random way.
x = 0
is his starting point, then
u(x,t)
measures the probability of locating Jack on the
x
-axis running along the sidewalk at time
t.
It obeys the diffusion equation
\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}\ \ \ \text{and} \ \int\limits_{x = - \infty}^{x=\infty} u(x,t) dx = 1,
where the integral means we're certain to find Jack somewhere on the sidewalk at any given
t.
Assuming the sidewalk is really really long, what “boundary conditions” do we need for this diffusion equation?
u(x,t) \to 1
\mid x \mid \to \infty
t.
u(x,t) \to 0
\mid x \mid \to \infty
t.
u(x,t) \to \infty
\mid x \mid \to \infty
t.
u \to 0
|x| \to \infty,
the diffusion equation is a perfect candidate for a Fourier transform.
The details will have to wait until later, but for now we can treat the transform
\mathcal{F}
like a magic wand that turns a “
\frac{\partial}{\partial x}
” into a constant
i \omega = \sqrt{-1} \omega
\mathcal{F}\left[ \frac{\partial f}{\partial x} \right] = i \omega \mathcal{F}[f].
\mathcal{F}[f]
is called the Fourier transform of
f,
\omega
now and not
x.
For example, an important Fourier transform we'll prove later and use in the next problem is
\mathcal{F}\left[ A e^{-\frac{a x^2}{2}} \right] = \sqrt{\frac{1}{2 \pi a } } A e^{-\frac{\omega^2}{2a}}.
If the Fourier transform doesn't affect
at all, what differential equation does
\mathcal{F}[u]
obey if
u
satisfies the diffusion equation
u_{t} = u_{xx} ?
\frac{d}{dt} \mathcal{F}[u] = -i \omega^3 \mathcal{F}[u]
\frac{d}{dt} \mathcal{F}[u] =i \omega \mathcal{F}[u]
\frac{d}{dt} \mathcal{F}[u] = - \omega^2 \mathcal{F}[u]
The Fourier transform really is magical: it turned the difficult diffusion problem
u_{t} = u_{xx}
into the simpler first-order ordinary differential equation
\frac{d}{dt} \mathcal{F}[u] = - \omega^2 \mathcal{F}[u] \implies \mathcal{F}[u] =\text{(constant)} \cdot e^{-\omega^2 t}.
But with all magic, there's a price: reversing the Fourier transform to get
u
is normally tough.
Jack's random walk is an exception, though. Since we're certain he sets off from
x=0,
\mathcal{F}[u] = \frac{1}{2\pi} e^{-\omega^2 t};
we'll work out the
\frac{1}{2 \pi}
later. In the last problem, we quoted a specific Fourier transform:
\mathcal{F}\left[ A e^{-\frac{a x^2}{2}} \right] = \sqrt{\frac{1}{2 \pi a } } A e^{-\frac{\omega^2}{2a}}.
Use this to find
u(x,t).
\frac{1}{\sqrt{4\pi t}}e^{-\frac{ x^2}{4t}}
\sqrt{4\pi t}e^{-\frac{ x^2}{4t}}
\frac{1}{\sqrt{4\pi t}}e^{-4 t x^2}
\frac{1}{\sqrt{\pi t}}e^{-\frac{ x^2}{2t}}
Jack's drunken stagger is a fun way to introduce the diffusion equation, one of the most important PDEs for many random processes appearing in physics, chemistry, and finance.
For example, atoms and molecules in a gas are jostled about unpredictably due to thermal motion, but a particle's trajectory is very much a 3D random walk with probabilities given by
u_t = \nabla^2u;
u(\vec{x},t)
measures the likelihood it has diffused from its starting point to
\vec{x}
t.
The Fourier transform can help us solve this higher-dimensional problem, too, but we first need to take the time to develop it and its inverse transform as triple integrals. More on this later!
In a nutshell, a partial differential equation (PDE) has several independent variables.
There are many methods for approaching such an equation: separation of variables, power series, and the Fourier transform are just a few we touched on in this intro quiz.
Full mastery of PDEs and nonlinear equations takes time to develop, so let's begin!
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Yearly\phantom{\rule{0.5em}{0ex}}Historical\phantom{\rule{0.5em}{0ex}}Surfac{e}_{j}=Surfac{{e}_{x,y}}^{×}\left[\frac{{\Sigma }_{k=1}^{{N}_{naps}}\left(\frac{1}{\left({d}_{x,y,k}\right)}x\frac{\stackrel{̄}{\mathit{\text{NAP}}{\mathit{\text{S}}}_{k}^{J}}}{Surfac{e}_{k}}\right)}{{\Sigma }_{k=1}^{{}_{naps}}\frac{1}{\left({d}_{x,y,k}\right)}}\right]
Where for each year between 1975 and 1994 the annual historical surface for pollutant j is equal to the current spatial surface of pollutant j (Surface x,y ) at coordinates x,y multiplied by the IDW interpolation of the ratio's of spatial co-located historical NAPS and surface estimates. d x,y,k is the distance (km) from NAPS monitoring station k to location x,y.
\stackrel{̄}{NAP{S}_{K}^{J}}
and Surface k are coincidently sampled pollutant concentrations of j at station k. A smooth interpolation option (smooth factor = 0.2) was included in the IDW interpolation (not shown in equation 1 for simplicity), which uses three ellipses in the interpolation method: points that fall outside the smaller ellipse but inside the largest ellipse are weighted using a sigmoid function [28]. The smoothed IDW function was used to reduce abrupt changes in the yearly calibration surfaces as these do not reflect spatial patterns of pollution change.
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Revision as of 09:59, 31 July 2013 by NikosA (talk | contribs) (→Automatising Conversions: bash script for DN > Radiance > ToAR)
{\displaystyle {\frac {W}{m^{2}*sr*nm}}}
{\displaystyle L(\lambda ,Band)={\frac {K*DN\lambda }{Bandwidth\lambda }}}
{\displaystyle \rho _{p}={\frac {\pi *L\lambda *d^{2}}{ESUN\lambda *cos(\Theta _{S})}}}
{\displaystyle \rho }
{\displaystyle \pi }
{\displaystyle L\lambda }
{\displaystyle d}
{\displaystyle Esun}
{\displaystyle cos(\theta _{s})}
{\displaystyle {\frac {W}{m^{2}*\mu m}}}
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P.Oxy.LXIV 4404
{\displaystyle {\mathfrak {P}}}
Matthew 21v34-37; 43 and 45(?)
Matt. 21:44 omitted
Papyrus 104 (in the Gregory-Aland numbering), designated by the symbol
{\displaystyle {\mathfrak {P}}}
104, is a fragment that is part of a leaf from a papyrus codex, it measures 2.5 by 3.75 inches (6.35 by 9.5 cm) at its widest. It is conserved in the Papyrology Rooms at Sackler Library, Oxford, UK. The front (recto) contains lines from the Gospel of Matthew 21:34-37, in Greek, the back (verso) contains tentative traces of lines from verses 43 and 45.[2]
This papyrus ranks among the earliest surviving texts of Matthew. It consists of six verses from the Gospel of Matthew, in a fragmentary condition, and is dated late 2nd century.[3][4] The text of the manuscript concurs with the NA27/UBS4 (Greek New Testaments) completely, with the exception that it does not include Matthew 21:44. This verse is also omitted in manuscripts: Codex Bezae, Minuscule 33, some Old-Latin manuscripts, Syriac Sinaiticus (syrs), Diatessaron. However, it is included in Sinaiticus, Vaticanus, Ephraemi, Regius, Washingtonianus, and Dublinensis. This verse thus belongs to the so-called Western non-interpolations, making
{\displaystyle {\mathfrak {P}}}
104 the earliest witness to the interpolated nature of this verse.
Greek text[edit]
The papyrus is written on both sides, and the surviving portion also includes part of the top and outer margins of the page. Since the text for the verso is nearly illegible, only the text for the recto is given. The characters that are in bold style are the ones that can be seen in Papyrus
{\displaystyle {\mathfrak {P}}}
TOYΣ ΓEΩPΓOYΣ ΛABEIN TOYΣ KAP-
ΠOYΣ AYTOY KAI ΛABONTEΣ OI ΓEΩP-
ΔE EΛIΘOBOΛHΣAN ΠAΛIN AΠE-
ΣTEIΛEN AΛΛOYΣ ΔOYΛOYΣ ΠΛEIO-
AYTOIΣ ΩΣAYTΩΣ YΣTEPON ΔE AΠE-
tous geōrgous labein tous kar-
pous autou kai labontes oi geōr-
de elithobolēsan palin ape-
steilen allous doulous pleio-
autois ōsautōs usteron de ape-
...he sent his servants to
unto them likewise. But last of all he sent...
A total of 110 legible letters are visible on the recto side of the fragment, representing 18 out of the 24 letters of the Greek Alphabet; zeta, theta, xi, phi, chi, and psi being missing. "The scribe uses rough breathings, but no other lectional feature or punctuation is found".[5] The hand is 'early', i.e., before c. 250. It is very carefully written, with extensive use of serifs.
^ Jongkind, Dirk (January 13, 2015), "What is the Oldest Manuscript of the New Testament?", Evangelical Textual Criticism, retrieved March 21, 2017
^ P.Oxy.LXIV 4404
^ Gregory-Aland numbers register
^ P. M. Head, "Some recently published NT Papyri from Oxyrhynchus: An Overview and Preliminary Assessment Archived 2011-07-06 at the Wayback Machine," Tyndale Bulletin 51 (2000), pp. 1-16.
Thomas, J. David. The Oxyrhynchus Papyri LXIV (London: 1997), pp. 7–9.
Oxyrhynchus Online, P.Oxy.LXIV 4404
University of Münster,New Testament Transcripts Prototype. Select P104 from 'Manuscript descriptions' box
„Fortsetzung der Liste der Handschriften“ Institut für Neutestamentliche Textforschung, Universität Münster. (PDF-Datei; 147 kB)
2nd-century biblical manuscripts
Gospel of Matthew papyri
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Young’s modulus is a quantitative measure of stiffness of an
Young’s modulus is a quantitative measure of stiffness of an elastic material. Suppose that for aluminum alloy sheets of a particular type
\mu ˉx=70GPa
qˉx=0.4GPa
Show from first principles, i.e., by using the definition of linear independence,
are linearly independent solutions of ˙X = AX.
Use (a) to solve the system (see image)
Using "Proof by Contraposition",sho that: if n is any odd integer and m is any even integer, then,
3{m}^{3}+2{m}^{2}
A bipolar alkaline water electrolyzer stack module comprises 160 electrolytic cells that have an effective cell area of
2{m}^{2}
. At nominal operation, the current density for a single cell of the electrolyzer stack is 0.40
\frac{A}{c}{m}^{2}
. The nominal operating temperature of the water electrolyzer stack is
{70}^{\circ }
C and pressure 1 bar. The voltage over a single electrolytic cell is 1.96 V at nominal load and 1.78 V at 50% of nominal load. The Faraday efficiency of the water electrolyzer stack is 95% at nominal current density, but at 50% of nominal load, the Faraday efficiency decreases to 80%.
(Give your answer to at least three significant digits.)
Calculate the nominal stack voltage:
Answer in V
Calculate the nominal stack current:
Answer in A
Calculate the nominal power on the water electrolyzer stack:
Answer in kW
Express the following propositions in symbolic form, and identify the principal connective:
i.Either Karen is studying computing and Ketsi is not studying mathematics, or Ketsi is studying mathematics.
ii.It is not the case case that if it is sunny then i will carry an umbrella.
{\left({x}^{4}{y}^{5}\right)}^{\frac{1}{4}}{\left({x}^{8}{y}^{5}\right)}^{\frac{1}{5}}={x}^{\frac{j}{5}}{y}^{\frac{k}{4}}
j-k
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Structural Health Monitoring of Glass/Epoxy Composite Plates Using PZT and PMN-PT Transducers | J. Eng. Mater. Technol. | ASME Digital Collection
Valeria La Saponara,
e-mail: vlasaponara@ucdavis
David A. Horsley,
, Prescott, AZ 86301
La Saponara, V., Horsley, D. A., and Lestari, W. (December 2, 2010). "Structural Health Monitoring of Glass/Epoxy Composite Plates Using PZT and PMN-PT Transducers." ASME. J. Eng. Mater. Technol. January 2011; 133(1): 011011. https://doi.org/10.1115/1.4002644
The structural health monitoring of composite structures presents many challenges, ranging from sensors’ reliability and sensitivity to signal processing and a robust assessment of life to failure. In this research project, sensors constructed with both PZT-4 ceramic and single-crystal PMN-PT, i.e.,
Pb(Mg1/3Nb2/3)O3−PbTiO3
, were investigated for structural health monitoring of composite plates. Fiberglass/epoxy specimens were manufactured with a delamination starter located in the middle of the plate, and were subjected to axial tensile fatigue at a high stress ratio. A surface-mounted PMN-PT pair and a surface-mounted PZT-4 pair were positioned on each side of the delamination starter and excited in turns at set intervals during fatigue loading. This project had two goals: (1) assess the performance of the two piezoelectric materials and (2) develop a signal processing technique based on wavelet transforms capable of detecting damage features that are independent of the transducers (being damaged concurrently to the host composite specimens) and thus can estimate life to failure of the composite specimens. Results indicate that the PMN-PT transducers may be more resilient to fatigue damage of the host structure and possibly generate less dispersive Lamb waves. However, these aspects are compounded with higher costs and manufacturing difficulties. Moreover, the proposed signal processing method shows promise in estimating impending failure in composites: It could, in principle, capture and quantify the complex wave propagation problem of dispersion, scattering, and mode conversion across a delamination front, and it will be further investigated.
condition monitoring, delamination, disperse systems, fatigue, glass fibre reinforced composites, lead compounds, piezoelectric materials, piezoelectric transducers, signal processing, structural engineering, surface acoustic wave sensors, surface acoustic waves, transducers, wave propagation, wavelet transforms, composite, structural health monitoring, wavelet, fatigue
Composite materials, Damage, Delamination, Epoxy adhesives, Epoxy resins, Fatigue, Plates (structures), Signal processing, Signals, Structural health monitoring, Transducers, Wavelet transforms, Wavelets, Wave propagation, Sensors, Glass, Cycles, Resonance, Manufacturing, Waves, Failure, Fatigue damage, Stress, Piezoelectric materials
Plate Mode Velocities in Graphite/Epoxy Plates
Takatsubo
Lamb Wave Method for Quick Inspection of Impact-Induced Delamination in Composite Laminates
The Interaction of Lamb Waves With Delaminations in Composite Laminates
Damage Detection in Composite Materials Using Lamb Wave Methods
Damage Assessment in Composites by Lamb Waves and Wavelet Coefficients
Díaz Valdés
Real-Time Nondestructive Evaluation of Fiber Composite Laminate Using Low-Frequency Lamb Waves
Investigation of the Dielectric and Piezoelectric Properties of PMN and PMN-PT Materials for Ultrasonic Transducer Applications
MEMS Deformable Mirrors for Adaptive Optics Using Single Crystal PMN-PT
, OPT MEMS, pp.
Tire Tread Deformation Sensor and Energy Harvester Development for ‘Smart Tire’ Applications
PMN-PT Single-Crystal Transducer for Non-Destructive Evaluation
La Saponara
Embedded Sensors for Composite Structural Health Monitoring
, Long Beach, CA.
Influence of Embedded Structural Health Monitoring Sensors on the Mechanical Performance of Glass/Epoxy Composites
Damage Detection of Structures by Wavelet Analysis
Wavelet-Based Active Sensing for Delamination Detection in Composite Structures
Structural Damage Detection in Beams by Wavelet Transforms
Wavelet-Based Outlier Analysis for Guided Wave Structural Health Monitoring: Application to Multi-Wire Strands
Introduction to Wavelets and Wavelet Transforms–A Primer
Adaptive Thresholding of Wavelet Coefficients
WAVELAB 850, developed by professors of the Department of Statistics of Stanford University and their collaborators, and available under http://www-stat.stanford.edu/~wavelab/http://www-stat.stanford.edu/~wavelab/
The Effect of Dispersion on Long-Range Inspection Using Ultrasonic Guided Waves
E. -H.
Effectiveness of the Continuous Wavelet Transform in the Analysis of Some Dispersive Elastic Waves
Practical Issues in the Detection of Damage in Beams Using Wavelets
The Transforms and Applications Handbook: Second Edition
Poularikas
Delamination Detection in Smart Composite Beams Using Lamb Waves
Assessing Damage in Corroded Reinforced Concrete Using Acoustic Emission
CONTOUR2AREA.M, developed by Per Sundqvist and posted on 2010-01-26, available under http://www.mathworks.com/matlabcentral/fileexchange/26480-contour2areahttp://www.mathworks.com/matlabcentral/fileexchange/26480-contour2area. This code was verified against a well-known geometric problem (areas of concentric circles).
Composite Structural Health Monitoring Through Use of Embedded PZT Sensors
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Answer the following: Q 1 Find counter-examples to disprove the following statements: (i) If the corresponding angles in two - Maths - Playing with Numbers - 12599247 | Meritnation.com
Q.1. Find counter-examples to disprove the following statements:
(i) If the corresponding angles in two triangles are equal, then the triangles are congruent.
2{n}^{2}+11
is a prime for all whole numbers n.
{n}^{2}-n+41
is a prime for all positive integers n.
\left(i\right) Let there be two equilateral triangles, one of side 6 cm and the other of side 4 cm. Their corresponding angles are all equal - as each angle is 60°. However, the two triangles are not congruent, as they don\text{'}t have equal sides.\phantom{\rule{0ex}{0ex}}\left(ii\right) When n = 11, 2{\left(11\right)}^{2} + 11 = 2\left(121\right) + 11 = 242 + 11 = 253 = 23 × 11, which is not prime\phantom{\rule{0ex}{0ex}}\left(iii\right) When n = 41, {41}^{2} - \overline{)41} + \overline{)41} = 1681 = 41 × 41, which is not prime
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Automate Signal Labeling with Custom Functions - MATLAB & Simulink - MathWorks Nordic
Load, Resample, and Import Data into Signal Labeler
Label QRS Complexes and R Peaks
Export Labeled Signals and Compute Heart-Rate Variability
findQRS Function: Find QRS Complexes
computeFSST Function: Reshape Input and Compute Fourier Synchrosqueezed Transforms
findRpeaks Function: Find R Peaks
This example shows how to use custom autolabeling functions in Signal Labeler to label QRS complexes and R peaks of electrocardiogram (ECG) signals. One custom function uses a previously trained recurrent deep learning network to identify and locate the QRS complexes. Another custom function uses a simple peak finder to locate the R peaks. In this example, the network labels the QRS complexes of two signals that are completely independent of the network training and testing process.
The QRS complex, which consists of three deflections in the ECG waveform, reflects the depolarization of the right and left ventricles of the heart. The QRS is also the highest-amplitude segment of the human heartbeat. Study of the QRS complex can help assess the overall health of a person's heart and the presence of abnormalities [1]. In particular, by locating R peaks within the QRS complexes and looking at the time intervals between consecutive peaks, a diagnostician can compute the heart-rate variability of a patient and detect cardiac arrhythmia.
The deep learning network in this example was introduced in Waveform Segmentation Using Deep Learning, where it was trained using ECG signals from the publicly available QT Database [2] [3]. The data consists of roughly 15 minutes of ECG recordings from a total of 105 patients, sampled at 250 Hz. To obtain each recording, the examiners placed two electrodes on different locations on a patient's chest, which resulted in a two-channel signal. The database provides signal region labels generated by an automated expert system [1]. The added labels make it possible to use the data to train a deep learning network.
The signals labeled in this example are from the MIT-BIH Arrhythmia Database [4]. Each signal in the database was irregularly sampled at a mean rate of 360 Hz and was annotated by two cardiologists, allowing for verification of the results.
Load two MIT database signals, corresponding to records 200 and 203. Resample the signals to a uniform grid with a sample time of 1/250 second, which corresponds to the nominal sample rate of the QT Database data.
y200 = resample(ecgsig,tm,250);
Open Signal Labeler. On the Labeler tab, click Import and select From Workspace in the Members list. In the dialog box, select the signals y200 and y203. Add time information: Select Time from the drop-down list and specify a sample rate of 250 Hz. Click Import and Close. The signals appear in the Labeled Signal Set Browser. Plot the signals by selecting the check boxes next to their names.
Define labels to attach to the signals.
Define a categorical region-of-interest (ROI) label for the QRS complexes. Click Add Definition on the Labeler tab. Specify the Label Name as QRSregions, select a Label Type of ROI, enter the Data Type as categorical, and add two Categories, QRS and n/a, each on its own line.
Define a sublabel of QRSregions as a numerical point label for the R peaks. Click QRSregions in the Label Definitions browser to select it. Click Add Definition and select Add sublabel definition. Specify the Label Name as Rpeaks, select a LabelType of Point, and enter the Data Type as numeric.
Create two Custom Labeling Functions, one to locate and label the QRS complexes and another to locate and label the R peak within each QRS complex. (Code for the findQRS and findRpeaks functions appears later in the example.) To create each function, in the Labeler tab, expand the Automate Value gallery and select Add Custom Function. Signal Labeler shows a dialog box asking for the name, description, and label type of the function.
For the function that locates the QRS complexes, enter findQRS in the Name field and select ROI as the Label Type. You can leave the Description field empty or you can enter your own description.
For the function that locates the R peaks, enter findRpeaks in the Name field and select Point as the Label Type. You can leave the Description field empty or you can enter your own description.
Find and label the QRS complexes of the input signals.
In the Labeled Signal Set Browser, select the check box next to y200.
Select QRSregions in the Label Definitions browser.
In the Automate Value gallery, select findQRS.
Click Auto-Label and select Auto-Label All Signals. In the dialog box that appears, enter the 250 Hz sample rate in the Arguments field and click OK.
Signal Labeler locates and labels the QRS complexes for all signals, but displays labels only for the signals whose check boxes are selected. The QRS complexes appear as shaded regions in the plot and in the label viewer axes. Activate the panner by clicking Panner on the Display tab and zoom in on a region of the labeled signal.
Find and label the R peaks corresponding to the QRS complexes.
Select Rpeaks in the Label Definitions browser.
Go back to the Labeler tab. In the Automate Value gallery, select findRpeaks.
Click Auto-Label and select Auto-Label All Signals. Click OK in the dialog box that appears.
The labels and their numeric values appear in the plot and in the label viewer axes.
Export the labeled signals to compare the heart-rate variability for each patient. On the Labeler tab, click Export and select To File in the Labeled Signal Set list. In the dialog box that appears, give the name HeartRates.mat to the labeled signal set and add an optional short description. Click Export.
Go back to the MATLAB® Command Window. Load the labeled signal set. For each signal in the set, compute the heart-rate variability as the standard deviation of the time differences between consecutive heartbeats. Plot a histogram of the differences and display the heart-rate variability.
load HeartRates
nms = getMemberNames(heartrates);
for k = 1:heartrates.NumMembers
v = getLabelValues(heartrates,k,["QRSregions" "Rpeaks"]);
hr = diff(cellfun(@(x) x.Location,v));
histogram(hr,0.5:0.025:1.5)
legend("hrv = " + std(hr))
ylabel(nms(k))
The findQRS function finds and labels the QRS complexes of the input signals.
The function uses an auxiliary function, computeFSST, to reshape the input data and compute the Fourier synchrosqueezed transform (FSST). You can either store computeFSST in a separate file in the same directory or nest it inside findQRS by inserting it before the final end statement.
findQRS uses the classify (Deep Learning Toolbox) function and the trained deep learning network net to identify the QRS regions. The deep learning network outputs a categorical array that labels every point of the input signal as belonging to a P region, a QRS complex, a T region, or to none of these. This function converts the point labels corresponding to a QRS complex to QRS region-of-interest labels using signalMask and discards the rest. The df parameter selects as regions of interest only those QRS complexes whose duration is greater than 20 samples. If you do not specify a sample rate, the function uses a default value of 250 Hz.
function [labelVals,labelLocs] = findQRS(x,t,parentLabelVal,parentLabelLoc,varargin)
% This is a template for creating a custom function for automated labeling
% x is a matrix where each column contains data corresponding to a
% channel. If the channels have different lengths, then x is a cell array
% of column vectors.
% t is a matrix where each column contains time corresponding to a
% channel. If the channels have different lengths, then t is a cell array
% parentLabelVal is the parent label value associated with the output
% sublabel or empty when output is not a sublabel.
% parentLabelLoc contains an empty vector when the parent label is an
% attribute, a vector of ROI limits when parent label is an ROI or a point
% location when parent label is a point.
% labelVals must be a column vector with numeric, logical or string output
% values.
% labelLocs must be an empty vector when output labels are attributes, a
% two column matrix of ROI limits when output labels are ROIs, or a column
% vector of point locations when output labels are points.
labelVals = cell(2,1);
labelLocs = cell(2,1);
load("trainedQTSegmentationNetwork","net")
for kj = 1:size(x,2)
sig = x(:,kj);
% Reshape input and compute Fourier synchrosqueezed transforms
mitFSST = computeFSST(sig,Fs);
% Use trained network to predict which points belong to QRS regions
netPreds = classify(net,mitFSST,MiniBatchSize=50);
% Create a signal mask for QRS regions and specify minimum sequence length
QRS = categorical([netPreds{1} netPreds{2}]',"QRS");
msk = signalMask(QRS,MinLength=df,SampleRate=Fs);
r = roimask(msk);
% Label QRS complexes as regions of interest
labelVals{kj} = r.Value;
labelLocs{kj} = r.ROILimits;
labelVals = vertcat(labelVals{:});
% Insert computeFSST here if you want to nest it inside findQRS.
This function uses the fsst function to compute the Fourier synchrosqueezed transform (FSST) of the input. As discussed in Waveform Segmentation Using Deep Learning, the network performs best when given as input a time-frequency map of each training or testing signal. The FSST results in a set of features particularly useful for recurrent networks because the transform has the same time resolution as the original input. The function:
Pads the input data with random numbers and reshapes it into the 2-by-5000 cell array stack expected by net.
Specifies a Kaiser window of length 128 and default shape factor
\beta =0.5
to provide adequate frequency resolution.
Extracts data over the frequency range from 0.5 Hz to 40 Hz.
Treats the real and imaginary parts of the FSST as separate features.
Normalizes the data by subtracting the mean and dividing by the standard deviation.
function signalsFsst = computeFSST(xd,Fs)
xd = reshape([xd;randn(10000-length(xd),1)/100],5000,2);
signalsFsst = cell(1,2);
[ss,ff] = fsst(xd(:,k),Fs,kaiser(128));
sp = ss(ff>0.5 & ff<40,:);
signalsFsst{k} = normalize([real(sp);imag(sp)],2);
This function locates the most prominent peak of the QRS regions of interest found by findQRS. The function applies the MATLAB® islocalmax function to the absolute value of the signal in the intervals located by findQRS.
function [labelVals,labelLocs] = findRpeaks(x,t,parentLabelVal,parentLabelLoc,varargin)
labelVals = zeros(size(parentLabelLoc,1),1);
labelLocs = zeros(size(parentLabelLoc,1),1);
for kj = 1:size(parentLabelLoc,1)
tvals = t>=parentLabelLoc(kj,1) & t<=parentLabelLoc(kj,2);
ti = t(tvals);
xi = x(tvals);
lc = islocalmax(abs(xi),MaxNumExtrema=1);
labelVals(kj) = xi(lc);
labelLocs(kj) = ti(lc);
[2] Goldberger, Ary L., Luis A. N. Amaral, Leon Glass, Jeffery M. Hausdorff, Plamen Ch. Ivanov, Roger G. Mark, Joseph E. Mietus, George B. Moody, Chung-Kang Peng, and H. Eugene Stanley. "PhysioBank, PhysioToolkit, and PhysioNet: Components of a New Research Resource for Complex Physiologic Signals." Circulation. Vol. 101, No. 23, 2000, pp. e215–e220. [Circulation Electronic Pages: http://circ.ahajournals.org/content/101/23/e215.full].
[3] Laguna, Pablo, Roger G. Mark, Ary L. Goldberger, and George B. Moody. "A Database for Evaluation of Algorithms for Measurement of QT and Other Waveform Intervals in the ECG." Computers in Cardiology. Vol. 24, 1997, pp. 673–676.
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Dev Kant Shandilya1*, Rekha Israni1, Peter Edward Joseph2, Venkata Siva Satyanarayana Kantamreddi3
3 Centre for Chemical Analysis, Central Research Laboratory, GIT, GITAM University, Visakhapatnam, India.
Abstract: Introduction: A possible fragmentation pathway of atorvastatin was proposed based on rational interpretation of high resolution collision induced dissociation (CID) fragmentation spectral data. Method: The mass spectral (MS and MS/MS) data of atorvastatin was obtained by using a flow injection analysis, LC coupled with high resolution mass analyzer system with Q-TOF technology. Results: The elemental composition for each major fragment was proposed with a calculated mass error in parts per million (ppm). The mass error found in this study is from 0.3 to 5.7 ppm; strongly supports all the proposed elemental composition of fragments. Based on the fragments, possible fragmentation pathway was proposed. Conclusion: The workflow followed for interpretation can also address the structural possibilities of similar type of small organic molecules.
Keywords: Atorvastatin, Small Drug Molecules, High Resolution Mass Spectrometry, Fragmentation, Rational Workflow, Interpretation
A variety of spectroscopic techniques are available for the identification and characterization of drug molecules. One of the most advanced analytical tools is high resolution mass spectrometry (HRMS) that has become the method of choice in structural characterization of small molecules, drug products, proteins and metabolites in different biological matrix [1] [2] [3] [4] . With advancements in ionization methods and instrumentation, high performance liquid chromatography coupled with high resolution mass spectrometry (LC-HRMS) has become a powerful tool for the research [5] - [11] . Interpretation of the fragmentation spectra and establishment of fragments structures are most time taking and essential tasks. The workflow [12] for the interpretation of the full scan atmospheric pressure ionization mass spectra (MS) and collision induced dissociation fragmentation spectra (MS/MS) of small organic molecules was efficiently applied for structural conformational studies; small nitrogenous organic molecule with hydroxyl and carboxylic acid functional group; atorvastatin was selected for this study.
Atorvastatin [13] [14] [15] is an inhibitor of 3-hydroxy-3-methylglutaryl- coenzyme A (HMG-CoA) reductase. This enzyme catalyzes the conversion of HMG-CoA to mevalonate, an early and rate-limiting step in cholesterol biosynthesis. It is used primarily as a lipid-lowering agent and for prevention of events associated with cardiovascular disease especially people with type 2 diabetes, coronary heart disease, or other risk factors. It is an off-white crystalline powder and chemically known as Calcium-(3R,5R)-7-[2-(4-fluorphenyl)-5-isopropyl- 3-phenyl-4-(phenylcarbamoyl)-1H-pyrrol-1-yl]-3,5-dihydroxyheptanoathydrat(1:2:3). The empirical formula of atorvastatin calcium trihydrate is (C66H68CaF2 N4O10)3H2O and its molecular weight is 1209.4. Atorvastatin free form empirical formula is C33H35FN2O5; monoisotopic molecular weight is 558.2530 and molecular structure of atorvastatin free form is presented in Figure 1.
During this study, high resolution mass spectral data of atorvastatin was generated using electrospray ionization and collision induced dissociation, followed by interpretation using basic interpretation rules and rational workflow [12] approach.
Figure 1. HRMS spectrum of atorvastatin using ESI+ ion source.
2.1. Drug Sample
Atorvastatin was extracted from generic dosage form. A final concentration was about 10 μg/ml in water and methanol for mass spectrometric studies.
The ultrapure water (18.2 MΩ) was obtained using MilliQ apparatus from Millipore (Milford, USA) and the HPLC grade methanol was purchased from J.T. Baker.
Shimadzu Prominence 20 AD HPLC (Kyto Japan) was coupled with Triple TOF (AB SCIEX) high resolution mass spectrometer system equipped with dual ionization (electrospray ionization and atmospheric pressure chemical ionization) source was used for this analysis. External calibrant delivery system (CDS) was used to calibrate the TOF mass analyzer with APCI small molecules calibration solution provided by instrument vendor.
2.4. Chromatographic and Mass Spectrometric Conditions
The extracted drug sample of atorvastatin was subjected to high resolution MS and MS/MS analysis via flow injection analysis (FIA) mode; liquid chromatography system was used to introduce the sample to mass spectrometer ion source. Liquid chromatography system set to isocratic flow 0.1 ml/min of mobile phase water and methanol in a ratio of 1:1 and injection volume was 10 μl. Electrospray ion source was selected to achieve the intense protonated parent ion and high resolution mass analyzer selected to get the accurate mass; which supports to predict the right elemental composition. Spray voltage and collision energy optimized by using direct infusion mode to get optimum quality to spectral data. Experiments were acquired using optimized parameters; spray voltage of +5.0 kV for MS and collision energy (CE) setting of 35 V with a spread of ±15 V applied to all parent ions for CID (collision induced dissociation) for MS/MS.
The LC coupled with HRMS technique is a modern alternative method for the identification of small molecules and their impurities. The experimental data (MS and MS-MS spectral data) of atorvastatin was generated using a high performance liquid chromatography (HPLC) coupled with high resolution mass spectrometer system (HRMS) with Time of Flight (TOF) mass analyzer via Flow Injection Analysis (FIA) mode and Electro-spray Ionization (ESI+) ion source.
The full scan accurate mass and product ion spectrum of atorvastatin was obtained from IDA (information dependent acquisition) experiments. The workflow [12] was applied for the interpretation of high resolution MS and MS/MS spectral data; molecular ion peak was displayed [M + H]+ at m/z 559.2635 Da. The most typical adduct [M + Na]+ at m/z 581.2442 with a less abundant adduct [M + K]+ at 597.2183 in mass spectrum of atorvastatin is further confirming the molecular ion peak at m/z 559.2635 with empirical formula
{\text{C}}_{33}{\text{H}}_{36}{\text{FN}}_{2}{\text{O}}_{5}^{\text{+}}
The TOF MS/MS spectrum of atorvastatin also exhibited molecular ion peak at m/z 559.2635 Da as [M + H]+ (calculated for
{\text{C}}_{33}{\text{H}}_{36}{\text{FN}}_{2}{\text{O}}_{5}^{\text{+}}
, exact mass 559.2603, difference 5.7 ppm, Figure 2 and Table 1). It was further fragmented in collision cell (Q2) into eleven major fragments at m/z 250.1038, 276.1194, 292.1508, 318.1654, 362.1561, 380.1675, 404.2032, 422.2139, 440.2239, 448.1936 and 466.2046. The elemental composition, exact mass, assigned structures and mass error of these major fragments were illustrated in Figure 2 and Table 1. Predicted fragmentation pathway also presented in Figure 2; the molecular ion peak (m/z 559.2635 Da) was initially fragmented into m/z 466.2046 (calculated formula
{\text{C}}_{27}{\text{H}}_{29}{\text{FNO}}_{5}^{\text{+}}
, exact mass 466.2024, mass error 4.7 ppm) by the loss of C6H7N, fragment ion 466.2046 yielded two ions at m/z 448.1936 (calculated formula
{\text{C}}_{27}{\text{H}}_{27}{\text{FNO}}_{4}^{\text{+}}
, exact mass 448.1919, mass error 3.8 ppm) and 440.2239 (calculated formula
{\text{C}}_{26}{\text{H}}_{31}{\text{FNO}}_{4}^{\text{+}}
, exact mass 440.2232, mass error 1.6 ppm) by loss of water and by loss of CO, respectively. Fragment m/z 440.2239 was further fragmented into two ions at m/z 422.2139 (calculated formula
{\text{C}}_{26}{\text{H}}_{29}{\text{FNO}}_{3}^{\text{+}}
, exact mass 422.2126, mass error 3.1 ppm) and 292.1508 (calculated formula C20H19FN+, exact mass 292.1496, mass error 4.1 ppm) by loss of a water molecule and C4H12O4, respectively. Further, a loss of water and C3H6 to the ion with m/z 422.2139 lead fragment ions m/z 404.2032 (calculated formula
{\text{C}}_{26}{\text{H}}_{27}{\text{FNO}}_{2}^{\text{+}}
Table 1. Interpretation of MS/MS spectra of atorvastatin based on rational work flow [12] .
EE: even electron; EN: even nitrogen; ON: odd nitrogen; *Mass error = difference between measured accurate mass and calculated accurate mass/calculated accurate mass × 106.
Figure 2. CID mass spectra (HR-MS/MS), proposed fragments and probable fragmentation pathway.
exact mass 404.2020, mass error 3.0 ppm) and 380.1675 (calculated formula
{\text{C}}_{23}{\text{H}}_{23}{\text{FNO}}_{3}^{\text{+}}
, exact mass 380.1656, mass error 5.0 ppm) was observed, respectively. The fragment ion with m/z 362.1561 (calculated formula
{\text{C}}_{23}{\text{H}}_{21}{\text{FNO}}_{2}^{\text{+}}
, exact mass 362.1551, mass error 2.8 ppm) shows a loss of C3H6 to a fragment ion with m/z 404.2032, which was further fragmented at m/z 318.1654 (calculated formula C22H21FN+, exact mass 318.1653, mass error 0.3 ppm) corresponding to the loss of COOH.
The fragment at m/z 292.1508 (calculated formula C20H19FN+, exact mass 292.1496, mass error 4.1 ppm) corresponding to the loss of C4H12O4 from the ion m/z 440.2239. A subsequent fragment of ion m/z 292.1508 resulting peak at m/z 276.1194 (calculated formula C19H15FN+, exact mass 276.1183, mass error 4.0) by losing methane, which further resulted an ion at m/z 250.1038 (calculated formula C17H13FN+, exact mass 250.1027, mass error 4.4 ppm) corresponding to the loss of C2H2. Fragmentation pathway described above and presented in figure no. 2 is a probable pathway; formation of fragments can be simultaneous because of multiple transitions can occur simultaneously.
The molecular ion peak as [M + H]+ in MS/MS spectra of atorvastatin appeared at m/z 559.2635 Da. Further, the CID fragmentation of protonated [M + H]+ ion generated and interpretation was carried followed by the proposal of fragmentation pathway (based on neutral losses, single and multiple bond cleavages). Fragmentation pathway is only a predication. Pathway can be different and there are some other probabilities also considering the occurrence of multiple transitions simultaneously. The mass error found in this study is 0.3 to 5.7 ppm, which strongly supports the fragment ions molecular structures and elemental composition. Nowadays various software tools are available for the interpretation of mass spectrometry data; during the study no software tool was used for interpretation, predication of the fragments structure and pathway of formation. In addition to above, the study also provides the insights about workflow based interpretation of parent and product ion spectra in combination with the basic rules. The workflow [12] applied in this study was found efficient and can be applied for structure verification studies of small organic molecules and for identification of similar type of drug molecules or small organic molecules and their impurities.
This paper is part of Ph.D thesis of Dev Kant Shandilya. Author expresses his gratitude to the Dean, Department of Research, Bhagwant University, Ajmer, Rajasthan, India for extending his constant support.
LC: Liquid chromatography;
MS: Mass spectrometry;
HRMS: High resolution mass spectrometer;
MS/MS: Tandem mass spectrum;
m/z : mass-to-charge ratio;
API: Atmospheric pressure ionization;
APCI: Atmospheric pressure chemical ionization;
ESI: Electrospray ionization;
CID: Collision induced dissociation;
Q-TOF: Quadrupole-Time of Flight;
IDA: Information dependent acquisition;
FIA: Flow Injection Analysis
Cite this paper: Shandilya, D. , Israni, R. , Joseph, P. and Kantamreddi, V. (2017) Prediction of the Fragmentation Pathway of Atorvastatin by Using High Resolution Collision Induced Dissociation (HR-MS/MS) Spectral Information. Open Access Library Journal, 4, 1-8. doi: 10.4236/oalib.1103473.
[1] Gorog, S. (2005) The Sacred Cow: The Questionable Role of Assay Methods in Characterising the Quality of Bulk Pharmaceuticals. Journal of Pharmaceutical and Biomedical Analysis, 36, 931-937. https://doi.org/10.1016/j.jpba.2004.06.025
[2] Domon, B. and Aebersold, R. (2006) Mass Spectrometry and Protein Analysis. Science, 312, 212-217. https://doi.org/10.1126/science.1124619
[4] Nicolas, C.E. and Schoolz, T.H. (1998) Active Drug Substances Impurity Profiling Part II. LC/MS/MS Fingerprinting. Journal of Pharmaceutical and Biomedical Analysis, 16, 825-836. https://doi.org/10.1016/S0731-7085(97)00132-5
[5] Cooks, R.G., Chen, G., Wong, P. and Wollnik, H. (1997) Mass Spectrometers. In: Trigg, G.L., Ed., Encyclopedia of Applied Physics, Vol. 19, VCH Publishers, New York, 289.
[6] Chen, G., Pramanik, B.N., Liu, Y.-H. and Mirza, U.A. (2007) Applications of LC/MS in Structure Identifications of Small Molecules and Proteins in Drug Discovery. Journal of Mass Spectrometry, 42, 279-287. https://doi.org/10.1002/jms.1184
[7] Kumar, Y.R., Babu, J.M., Sarma, M.S.P., Seshidhar, B., Reddy, S.S., Reddy, G.S. and Vyas, K. (2003) Application of LC-MS/MS for the Identification of a Polar Impurity in Mosapride, a Gastroprokinetic Drug. Journal of Pharmaceutical and Biomedical Analysis, 32, 361-368. https://doi.org/10.1016/S0731-7085(03)00076-1
[8] Ermer, J. (1998) The Use of Hyphenated LC-MS Technique for Characterisation of Impurity Profiles during Drug Development. Journal of Pharmaceutical and Biomedical Analysis, 18, 707-714. https://doi.org/10.1016/S0731-7085(98)00267-2
[9] Angelika, G., Harrison, M.W., Herniman, J.M., Skylaris, C.-K. and Langely, G.J. (2013) A Predictive Science Approach to Aid Understanding of Electrospray Ionization Trandem Mass Spectrometric Fragmentation Pathway of Small Molecules Using Density Functional Calculations. Rapid Communications in Mass Spectrometry, 27, 964-970. https://doi.org/10.1002/rcm.6536
[10] Kumar, A., Darekar, G., Ramagiri, S., Bhasin, N., Pillai, M. and Shandilya, D.K. (2015) Generic Workflow Using Advanced Analysis and Data Interpretation Tools for Identification of Irbesartan Degradation Products by Liquid Chromatography to High Resolution Mass Spectrometry. ACAIJ, 15, 352-363.
[11] Pillai, M.G., Kumar, A., Sharma, R. and Bhasin, N. (2014) LC-MS Based Workflows for Qualitative and Quantitative Analysis for Homeopathic Preparation of Hydrastis Canadensis. Chromatographia, 7, 119-131. https://doi.org/10.1007/s10337-013-2577-5
[12] Shandilya, D.K., Joseph, P.E. and Kantamreddi, V.S.S. (2017) Interpretation of Full Scan Atmospheric Pressure Ionization Mass Spectra (MS) and Collision Induced Dissociation Fragmentation Spectra (MS/MS) of Small Organic Molecules—A Mini Review. Systematic Reviews in Pharmacy, 8, 23-25. https://doi.org/10.5530/srp.2017.1.9
[13] https://en.wikipedia.org/wiki/Atorvastatin
[14] http://www.drugs.com/atorvastatin.html
[15] http://www.rxlist.com/lipitor-drug.htm
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Generate colored noise signal - MATLAB - MathWorks Switzerland
10{\mathrm{log}}_{10}
1/|f|
S\left(f\right)=\frac{L\left(f\right)}{|f{|}^{\alpha }}
L\left(f\right)
\mathrm{ln}S\left(f\right)=-\alpha \mathrm{ln}|f|+\mathrm{ln}L\left(f\right).
10\mathrm{log}S\left(f\right)=-10\alpha \frac{\mathrm{ln}\left(2\right){\mathrm{log}}_{2}\left(f\right)}{\mathrm{ln}\left(10\right)}+10\frac{\mathrm{ln}\left(L\left(f\right)\right)}{\mathrm{ln}\left(10\right)}
-10\alpha \frac{\mathrm{ln}\left(2\right){\mathrm{log}}_{2}\left(f\right)}{\mathrm{ln}\left(10\right)}
\frac{1}{|f{|}^{\alpha }}
\frac{1}{|f{|}^{2}}
S\left(f\right)=\frac{L\left(f\right)}{|f{|}^{\alpha }}
\begin{array}{l}{a}_{0}=1,\\ {a}_{k}=\left(k-1-\frac{\alpha }{2}\right)\frac{{a}_{k-1}}{k},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}k=1,2,\dots ,63\end{array}
\begin{array}{l}{b}_{0}=1,\\ {b}_{k}=\left(k-1+\frac{\alpha }{2}\right)\frac{{b}_{k-1}}{k},\text{\hspace{0.17em}}\text{\hspace{0.17em}}k=1,2,\cdots ,255\end{array}
|
wsEXOD - EXODIA
wsEXOD is the abbreviation for wrapped staked EXOD, which is the wrapped version of sEXOD.
A wrapped token's value is pegged to that of another token. In our case wsEXOD derives its value from the following calculation:
Current market value of EXOD * Current Index = wsEXOD value
As the current index increases with each rebase, so does the value of wsEXOD relative to the current price of EXOD.
In some tax jurisdictions a rebase reward is considered a taxable income. Holding wsEXOD removes this taxable event. Effectively, you are still collecting rebase rewards through the current index multiplier as your wsEXOD increases in value through the above equation.
You may also deposit your wsEXOD into ‘The Monolith’ LP on Beethoven-x and gain LP tokens. We recommend balancing your deposit to the pool by using 50% EXOD and 50% wsEXOD. These can then be used to bond or you can hold them to earn the APR from the pool.
At this time, we would not recommend holding LP tokens unless you plan on using them to bond.
Future use cases for wsEXOD
wsEXOD could be used as collateral.
wsEXOD could be used cross chain.
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Sophia starts a job at a restaurant. She deposits $40 from each paycheck into her savings
Sophia starts a job at a restaurant. She deposits $40 from each paycheck into her savings account. There was no money in the account prior to her firs
Sophia starts a job at a restaurant. She deposits $40 from each paycheck into her savings account. There was no money in the account prior to her first deposit. Represent the amount of money in the savings account after Sophia receives each of her first 6 paychecks with a numeric sequence.
Since her savings account initially had $0 in it, the amount of money in her savings account after her first deposit of $40 will be $40. Since she deposits $40 from each paycheck into her savings account, then the amount of money in the account will increase by 40 after each deposit. The amount of money in her account after her first 6 deposits is then represented by the numeric sequence of 40,80,120,160,200,240.
Use what you know about bonded, monotonic sequences to show that the following sequences converge
{a}_{n}=\frac{1}{3}\left(1-\frac{1}{{3}^{n}}\right)
Which of them are bounded?
\frac{n}{14n+1}
13n
{10}^{n}
{\left(-11\right)}^{n}
{\left\{{x}_{n}\right\}}_{n=1}^{\mathrm{\infty }}
{\left\{{y}_{n}\right\}}_{n=1}^{\mathrm{\infty }}
be two sequences of positive numbers. Assume that
{x}_{n}⟨{y}_{n}
For each one of the following claims, decide whether they are always true, always false, or sometimes true and sometimes false (depending on the specific sequences) and prove it.
{x}_{n}⟨\frac{{x}_{n}+{y}_{n}}{2}
\frac{{x}_{n}+{y}_{n}}{2}⟨{y}_{n}
Determine whether each statement makes sense or does not make sense, and explain your reasoning. By modeling attitudes of college freshmen from 1969 through 2010, I can make precise predictions about the attitudes of the freshman class of 2020.
What is the next number in this sequence: 1,2,6,24,120 ...?
Substitute n=1, 2, 3, 4, 5 and find the first five sequences in sequence
\left\{\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots +\frac{1}{{2}^{n}}\right\}
The graph shows the number of teaspoons of water, y, that have dripped from a leaky faucet at the end of x minutes.Leaky Faucet
Which equation represents the relationship between x and y shown in the
A. Y= 3х
B. Y =X - 3
C. Y= 1/3x
D. Y= x + 3
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Find the planes tangent to the following surfaces at the
Find the planes tangent to the following surfaces at the indicated points:
{x}^{2}+2{y}^{2}+3xz=10
, at the point (
1,2,\frac{1}{3}
{y}^{2}-{x}^{2}=3
, at the point (1,2,8)
xyz=1
f:{\mathbb{R}}^{n}\to \mathbb{R}
be a differentiable function. Recal that the tangent plane of the surface consisting of points such that f(x)=k for some constant k at the point
{x}_{0}
is given with:
▽f\left({x}_{0}\right)\cdot \left(x-{x}_{0}\right)=0
a) Here we set
f\left(x,y,z\right)={x}^{2}+2{y}^{2}+3xz,{x}_{0}=\left(1,2,\frac{1}{3}\right)
and k=10. Note that
f\left({x}_{0}\right)=k
▽f\left(x\right)=\left(2x+3z,4y,3x\right)⇒▽f\left(1,2,\frac{1}{3}\right)=\left(3,8,3\right)
Using (1) we easily get:
0=\left(3,8,3\right)\cdot \left(x-1,y-2,z-\frac{1}{3}\right)=3\left(x-1\right)+8\left(y-2\right)+3\left(z-\frac{1}{3}\right)
Thus, the tangent plane equation is:
3x+8y+3z=20
b) Here we set
f\left(x,y,z\right)={y}^{2}-{x}^{2},{x}_{0}=\left(1,2,8\right)
and k=3. Note that
f\left({x}_{0}\right)=k
▽f\left(x\right)=\left(-2x,2y,0\right)⇒▽f\left(1,2,8\right)=\left(-2,4,0\right)
0=\left(-2,4,0\right)\cdot \left(x-1,y-2,z-8\right)=-2\left(x-1\right)+4\left(y-2\right)+0\left(z-8\right)
c) Here we set
f\left(x,y,z\right)=xyz,{x}_{0}=\left(1,1,1\right)
f\left({x}_{0}\right)=k
. Now we can calculate:
▽f\left(x\right)=\left(yz,xz,xy\right)⇒▽f\left(1,1,1\right)=\left(1,1,1\right)
0=\left(1,1,1\right)\cdot \left(x-1,y-1,z-1\right)=1\left(x-1\right)+1\left(y-1\right)+1\left(z-1\right)
f\left(x\right)=\sqrt{1-x}
a=0
\sqrt{0.9}
\sqrt{0.99}
f\left(x\right)=-3x+4
f\left(x\right)=-3{x}^{2}+7
f\left(x\right)=\frac{x+1}{x+2}
d\right)f\left(x\right)={x}^{5}+1
Use factor formula to show that
\mathrm{sin}5a+\mathrm{sin}2a-\mathrm{sin}a=\mathrm{sin}2a\left(2\mathrm{cos}3a+1\right)
Delia purchased a new car for $25,350. This make and model straight line depreciated to zero after 13 years. Determine the slope of the depreciation equation.
The joint density function of X and Y is given by
f\left(x,y\right)=x{e}^{-x\left(y+1\right)}x>0,y>0
Find the conditional density of Z, given Y=y, and that of Y, given X=x.
f\left(x\right)=3{x}^{3}+12{x}^{2}-36x
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Assessing the Floristic Biodiversity and Carbon Stock in a Republic of Congo’s Forest Ecosystem ()
1Ecole Nationale Supérieure d’Agronomie et de Foresterie, Université Marien Ngouabi, Brazzaville, Republic of Congo.
2Beijing Key Laboratory of Forest Resources and Ecosystem Process, Beijing Forestry University, Beijing, China.
Reforestation management goes through the knowledge of the tired out evolution after a forestry rest. The aim of this study was to assess biodi-versity concerning Terminalia superba Engl. & Diels and its undergrowth, and then quantify sequestered carbon stocks to appreciate the impact of reforestation on forest recovery and the enhancement of carbon stocks. The study was conducted at Bilala artificial forest in southeastern Republic of Congo, in Kouilou Department (Mayombe), close to Mvouti District with an altitude of 30 m. The floristic inventory was carried out in 9 rectangular sub-plots of 20 × 25 m each, installed in three blocks of Terminalia superba Engl. & Diels, for a total area of 0.5 ha. These blocks consisted of the 64, 31 and 20 years old plantations. Within ten sub-plots censuses, all trees with a DBH ≥5 cm were identified and measured. 51 trees of Terminalia superba Engl. & Diels and around 3007 trees in its undergrowth have been recorded belonging to 33 botanical families and 52 species. The results showed that the biomass recorded in this forest has been 275 t·ha﹣1 and the carbon stock was 129.6 t·ha﹣1. Terminalia superba Engl. Diels had for itself a biomass of 181.42 t·ha﹣1 out of the 275 t·ha﹣1 quantified. The amount of CO2 captured in the atmosphere by the recorded floristic procession deducted from this carbon stock was 475.89 tons of CO2, with an economic value equal to US$ 2379.45 or XAF 1,413,278. This study demonstrated that the forestry method of reforestation has a positive impact on biodiversity recovery and carbon sequestration.
Nzala, D. , Ekoungoulou, R. and Ngoumba, B. (2019) Assessing the Floristic Biodiversity and Carbon Stock in a Republic of Congo’s Forest Ecosystem. Open Access Library Journal, 6, 1-20. doi: 10.4236/oalib.1105638.
{P}_{i}=\frac{{N}_{i}}{N}
{N}_{i}
{H}^{\prime }=-{\displaystyle {\sum }_{i=1}^{n}\left({P}_{i}\times \mathrm{log}2{P}_{i}\right)}
E=\frac{{H}^{\prime }}{\mathrm{log}2\times S}
\text{Survivalrate}\left(\%\right)=\frac{\text{Numberofalivetrees}}{\text{Totalnumberoftrees}}\times 100
\text{AGB}=0.067\times {\left(\rho {D}^{2}H\right)}^{0.976}
\rho
\begin{array}{l}y=0.205\times \text{AGB}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}\text{AGB}\le 125\text{\hspace{0.17em}}\text{t}\cdot {\text{ha}}^{-1}\\ y=0.235\times \text{AGB}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}\text{AGB}\ge 125\text{\hspace{0.17em}}\text{t}\cdot {\text{ha}}^{-1}\end{array}
[1] CDB (2010) Third Edition of Global Perspective on Biodiversity. Convention on Biological Diversity. Montreal.
[2] Pan, Y., Birdsey, R.A., Fang, J., Houghton, R., Kauppi, P.E., Kurz, W.A. and Phillips, O.L. (2011) A Large and Persistent Carbon Sink in the World. Science, 333, 988-993. https://doi.org/10.1126/science.1201609
[3] PEFC (2017) European Program for Forest Certification.
[4] FAO (2005) Press Communication: Deforestation Is Growing with an Alarming Rate. Food and Agriculture Organization for United Nations, Rome.
[5] Robert, B. (2006) From National Natural History Museum, the Elephant in the Game of the Skittle. Seuil, Ardennes.
[6] Van Der Werf, G.R., Morton, D.C., Defries, R.S., Olivier, J.G.J., Kasibhatla, P.S. and Jackson (2009) CO2 Emissions from Forest Loss. Nature Geoscience, 2, 737-738.
[7] Laurent, D. and Maitre, H.F. (1992) Destruction of Tropical Forest Resources: Logging Is a Cause? CTFT/FAO, Rome.
[8] Ngoumba, B.Y. (2019) Biodiversity, Estimation of Carbon Stock and Forest Management for Terminalia superba Engl. & Diels Reforestation in Bilala (Kouilou Department). Master Memory. National Superior School for Agronomy and Forestry. Marien Ngouabi University, Brazzaville.
[9] Saya, A., Marien, J.N. and Mallet, B. (2016) Planted Forests for the Area Management: Case of Plantations in Republic of Congo. CRDPI, CIRAD.
[10] Ekoungoulou, R., Liu, X.D., Loumeto, J.J., Ifo, S.A., Bocko, Y.E., Koula, F.E. and Niu, S.K. (2014) Tree Allometry in Tropical Forest of Congo for Carbon Stocks Estimation in Above-Ground Biomass. Open Journal of Forestry, 4, 481-491.
[11] Ekoungoulou, R., Niu, S.K., Folega, F., Nzala, D. and Liu, X.D. (2018) Carbon Stocks of Coarse Woody Debris in Central African Tropical Forests. Sustainability in Environment, 3, 142-160. https://doi.org/10.22158/se.v3n2p142
[12] Jonard, M., Colmant, R. and Heylen, C. (2014) Impact of Coniferous Forestation on Carbon Sequestration of Peruvian Andes: Case of Pinus patula Plantations in the Cooperative of Granja Porcon (Cajamarca), Perou. Bois et Forêts des Tropiques, 322, 17-27. https://doi.org/10.19182/bft2014.322.a31226
[13] Boundzanga, G.C. (2018) Dendrometrical Work to a Better Evaluation of Rough and Net Comercial Volumes of Trees in Republic of Congo’s Forest Concession. CN-REDD, Brazzaville.
[14] Ekoungoulou, R., Liu X.D., Loumeto, J.J. and Ifo, S.A. (2014) Tree Above-And Below-Ground Biomass Allometries for Carbon Stocks Estimation in Secondary Forest of Congo. Journal of Environmental Science, Toxicology and Food Technology, 8, 9-20. https://doi.org/10.9790/2402-08420920
[15] Ekoungoulou, R., Liu, X.D., Ifo, S.A., Loumeto, J.J. and Folega, F. (2014) Carbon Stock Estimation in Secondary Forest and Gallery Forest of Congo Using Allometric Equations. International Journal of Scientific and Technology Research, 3, 465-474.
[16] Kimpouni, V., Loumeto, J.L. and Mizingou, J. (2008) Floristic Diversity of Forest Facies of Aucoumea klaineana (Okoume) in the Congolese Littoral. Acta Botanica Gallic: Botany Letters, 153, 323-334.
[17] Boukono, D.R.V. (2018) Estimation of Carbon Stock in a Block of Okoumea klaineana from the Reforestation National Service. Bachelor Memory. National Superior School for Agronomy and Forestry, Brazzaville.
[18] Ekoungoulou, R., Niu, S.K., Loumeto, J.J., Ifo, S.A., Bocko, Y.E., Mikieleko, F.E.K., Guiekisse, E.D.M., Senou, H. and Liu, X.D. (2015) Evaluating the Carbon Stock in Above-And Below-Ground Biomass in a Moist Central African Forest. Applied Ecology and Environmental Sciences, 3, 51-59.
[19] Moutsambote, J.M., Nzala, D. and Ngondo, J.C. (2000) Evolution of Forest Exhausted after Cassava Production in Mayombe. Cahiers Agricultures, 9, 141-144.
[20] Goma-Tchimbakala, J., Ndondou-Hockemba, M., Kokolo, A. and Mboussou-Kimbangou, A.N.S. (2005) Variations on Litter Bringing and Mineral Component in the Limba (Terminalia superba) Plantation in Congo. Tropicultura, 23, 53-59.
[21] ANAC (2018) Annual Report of National Agency of Civil Aviation. National Agency of Civil Aviation (ANAC), Brazzaville.
[22] Pearson, T.S. and Brown, S. (2005) Guide for Measuring and Monitoring Carbon in Forests and Grasslands. Winrock International, Arlington.
[23] ATIBT (2006) Botanic-Inventory Biodiversity. Volume VIII, Association of International Technicians on Tropical Timbers, Libreville.
[24] CNIAF (2016) List of Inventoried Flora in Southern Congo. National Center of Inventory and Management of Wildlife and Forest Resources. Brazzaville.
[25] Pedel, L. and Fabri, M.C. (2012) Art State on Existing Index Concerning Ecological State of Benthic Habitats of Deep Domain. Laboratory of Environment Resources (LER). MEDDTL Convention, Ifremer for DCSMM, BEE.
[26] Ekoungoulou, R., Nzala, D., Liu, X.D. and Niu, S.K. (2017) Ecological and Structural Analyses of Trees in an Evergreen Lowland Congo Basin Forest. International Journal of Biology, 10, 31-43. https://doi.org/10.5539/ijb.v10n1p31
[27] Ekoungoulou, R., Folega, F., Mukete, B., Ifo, S.A., Loumeto, J.J., Liu, X.D. and Niu, S.K. (2018) Assessing the Effectiveness of Protected Areas on Floristic Diversity in Tropical Forests. Applied Ecology and Environmental Research, 16, 837-853.
[28] Ekoungoulou, R. (2014) Carbon Stocks Evaluation in Tropical Forest, Congo. Carbon Stocks in Forest Ecosystems. Lambert Academic Publishing, Saarbrucken.
[29] Ekoungoulou, R. (2018) Managing Tropical Forest Ecosystems. Tropical Trees. Lambert Academic Publishing, Saarbrucken.
[30] Chave, J., Rejou-Mechain, M., Burquez, A., Chidumayo, E., Colgan, M.S., Delitti, W.B.C., Duque, A., Eid, T., Fearnside, P.M., Goodman, R.C., Henry, M., Martinez-Yrizar, A., Mugasha, W.A., Muller-Landau, H.C., Mencuccini, M., Nelson, B.W., Ngomanda, A., Nogueira, E.M., Ortiz-Malavassi, E., Pelissier, R., Ploton, P., Ryan, C.M., Saldarriaga, J.G. and Vieilledent, G. (2014) Improved Allometric Models to Estimate the Aboveground Biomass of Tropical Trees. Global Change Biology, 20, 3177-3190. https://doi.org/10.1111/gcb.12629
[31] Djomo, A.N., Picard, N., Fayolle, A., Henry, M., Ngomanda, A., Ploton, P., McLellan, J., Saborowski, J., Adamou, I. and Lejeune, P. (2016) Tree Allometry for Estimation of Carbon Stocks in African Tropical Forests. Forestry, 89, 446-455.
[32] Ekoungoulou, R., Nzala, D., Liu, X.D. and Niu, S.K. (2018) Tree Biomass Estimation in Central African Forests Using Allometric Models. Open Journal of Ecology, 8, 209-237. https://doi.org/10.4236/oje.2018.83014
[33] Fayolle, A., Loubota, P.G.J., Drouet, T., Swaine, M.D., Bauwens, S., Vleminckx, J., Biwole, A., Lejeune, P. and Doucet, J.L. (2016) Taller Trees, Denser Stands and Greater Biomass in Semi-Deciduous than in Evergreen Lowland Central African Forests. Forest Ecology and Management, 374, 42-50.
[34] Mokany, K., Raison, R.J. and Prokushkin, A.S. (2006) Critical Analysis of Root: Shoot Ratios in Terrestrial Biomes. Global Change Biology, 12, 84-96.
[35] IPCC (2006) The 2006 Directrix Line of International Panel on Climate Change (IPCC) for the Nationals Inventories of Greenhouse Gas. Agriculture, Forestry and Others Earth Affections. Chapter 3: Earth Coherent Representation.
[36] EMP (2018) State of the Voluntary Carbon Market. Ecosystem Services. Ecosystems Marketplace. http://www.ecosystemmarketplace.com
[37] Awe, D.V. (2016) Diversity Floristic and Carbon Stock in the Plantation of Anacardium. Master Memory in Plant Organisms Biology. University of Ngaoundere, Ngaoundere.
[38] Tareil, J. and Groulez, J. (1958) Limba Plantations of Moyen Congo. Bois et Forêts des Tropiques, 61, 9-25. https://doi.org/10.19182/bft1958.61.a18715
[39] Nzala, D., Nongamani, A., Moutsambote, J. and Mapangui, A. (1997) Floristic Diversity in the Monocultures of Eucalyptus Pine in Congo. Cahiers Agricultures, 6, 169-174.
[40] Adamou, M.B. (2010) Biodiversity and Carbon Sequestration in Cocoa Plantation at the West Periphery of Lobeke National Park. Master Memory. University of Yaounde, Yaounde.
[41] Tsoumou, B.R., Lumande, K.J., Kampe, J.P. and Nzila, J.D. (2016) Estimation of Carbon Stock Sequestred by Dimonika Reference Forest (Southwestern Republic of Congo). Revue Scientifique et Technique Foret et Environnement du Bassin du Congo, 6, 39-45.
[42] Temgoua, L.F., Dongmo, W., Nguimdo, V. and Nguena, C. (2018) Ligneous Diversity and Carbon Stock of Agroforestry System Based on Cocoa Plantations at East of Cameroon. Journal of Applied Biosciences, 122, 12274-12286.
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Calculating momentum of an object | Brilliant Math & Science Wiki
Calculating momentum of an object
Hari prasad Varadarajan, Sravanth C., Tom Verhoeff, and
Everybody knows that it is dangerous to drive in front of a big truck on the highway because of how long it takes the big truck to slow down, even though it is going the same speed as all the small cars. Likewise, if a little kid going very quickly crashes into a slow moving adult on an ice skating rink, it is a very different outcome than if a fast adult crashes into a slow kid. The reason for these things is the connection between force and momentum. In fact, we can see this connection from Newton's second law.
Calculating the momentum of an object
The momentum of an object is defined as its mass (the kind defined by
m = F/a
) times its velocity. Like the velocity, it has a magnitude as well as a direction. Practically, momentum can be thought of as the tendency for an object to stay along its current path through space. The more momentum an object has, the bigger the force required to change its velocity.
Writing it out in the form
F=ma
, we see that force is equal to the mass times the acceleration. Acceleration is, by definition, the rate at which the velocity changes. If we assume that the mass of an object is constant, then we can reinterpret the second law in a useful way.
Suppose that before and after the application of a force
F
, the velocity of an object is
v_i
v_f
Because the mass
m
is the same, before and after, the momentum before and after is
p_i = mv_i
p_f = mv_f
The change in momentum is then
\Delta p = p_f - p_i = m(v_f-v_i) = m\Delta v
The rate at which the momentum changed is then
\displaystyle\Delta p / \Delta t = m\Delta V/ \Delta t
. But,
\Delta v/\Delta t
is just the acceleration, therefore, the rate of change in momentum is
\displaystyle ma = F
So, we can see clearly that force is an exact measure of how quickly the momentum of an object will change. Whereas with velocity, we have to divide by
m
to find how much
v
will be changed by
F
, with momentum the relationship is direct. This suggests that momentum is the more natural quantity to work with in classical mechanics. In fact, as you progress through mechanics, quantum physics, and on to the frontiers of theoretical physics, one hardly talks about velocities, but of momenta. But that's for later.
As we showed, we can reinterpret the second law
F=ma
F = \Delta p / \Delta t
, meaning that force is equal to the rate of change of momentum. Therefore, if we have a known force that acts on an object for some known amount of time
\Delta t
, we can use the second law to calculate the change in momentum:
\Delta p = F\Delta t
For example, if a backcountry skier is pulled by a rope with a constant force of tension
T=
10 N, for
t=
10 s, their momentum must increase by
F\Delta t
= 100 kg m/s.
This is fine is we have a known force. However, in many cases momentum is relatively easy to measure by experiment, while force is more difficult to measure. For example, if we throw a ball at a wall, it bounces off in the opposite direction, a change in momentum. It is clear that the momentum has changed and we can measure it by timing the motion of the ball, but how can we possibly measure the force between the wall and the ball?
Happily, we can turn the second law on its head and use the change in momentum to find the unknown force.
Suppose we throw a ball of mass
m
at a wall, and it bounces off elastically meaning that it bounces back at the same speed. If we record the compression of the ball against the wall, we can figure out the approximate amount of time it spent interacting with the wall,
\Delta t
. Using the second law, we can then infer the strength (on average) of the force with which the wall pushed on the ball!
It states that = Net external impulse = change in momentum in the direction of F. This is very useful when you have to get the net external impulse of the object. This is widely used in collision mechanics, generally where friction comes into play. This theorem is always valid where as the Conservation of momentum is not valid when there is a net external force.
Cite as: Calculating momentum of an object. Brilliant.org. Retrieved from https://brilliant.org/wiki/calculating_momentum_object/
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Modified Duration | Brilliant Math & Science Wiki
The modified duration of a bond is the price sensitivity of a bond. It measures the percentage change in price with respect to yield. As such, it gives us a (first order) approximation for the change in price of a bond, as the yield changes.
When continuously compounded, the modified duration is equal to the Macaulay duration.
Calculating Modified Duration from Prices
Effect of yield change on bond prices
The theoretical calculation of the Modified Duration is
ModD = - \frac{1}{P } \cdot \frac{ \partial P } { \partial y } = - \frac{ \partial \ln P } { \partial y } ,
P
is the price of the bond.
To account for the fact that bond prices are negative This was defined historically, and we do not want to change the convention Negative duration implies that an increase in yield causes a decrease in price When you differentiate
\frac{1}{ x}
- \frac{1}{x^2 }
The formula for the modified duration is
Mod \, D(y) = - \frac{1}{P} \frac{ \partial P } { \partial y }
What is the reason for the negative sign?
Let's take this theoretical definition, and apply it to determine a calculation for the modified duration. If the yield is compounded annually, then the price of the bond is
P = \sum_{t=1 } ^ T \frac{ C_t } { ( 1 + y ) ^ t } .
\frac{ \partial P } { \partial y } = \sum_{t=1 } ^ T (- t) \times \frac{ C_t } { ( 1 + y ) ^ {t + 1 } } = \frac{ MacD \times P } { ( 1 + y ) } .
Mod D = \frac{ MacD } { (1 + y ) }.
What is the Modified Duration for a 10 year bond with fixed coupon payments of 5% and a face value of $1000, if the current price is $1100?
Continuing from the example in Macaulay Duration, we know that the YTM is
2.82 \%
and the MacD is
4.571
ModD = \frac{ Mac D } { 1 + y } = \frac{ 4.571 } { 1 + 2.82 \% } = 4.445.
What is the modified duration (in %) of a ten-year 5% par bond?
Note: Your answer should be positive.
More generally, if the yield is compounded
k
times a year, then
Mod D = \frac{ Mac D } { ( 1 + \frac{ y}{k} ) } .
Thus, when the yield is compounded continuously, we have
k \rightarrow \infty
Mod D = Mac D
From Calculus, we know that
\frac{ \partial P } { \partial y }
can be approximated by using
\frac{ P ( y + \delta y ) - P ( y - \delta y ) } { 2 \delta y }
. As such, this gives us:
If we are given the bond prices across different yield rates, then we can estimate the modified duration by
Mod D(y) \approx - \frac{ P ( y + \Delta y ) - P ( y - \Delta y ) } { 2 P \Delta y }.
This offers us a way to approximate the modified duration when we have a list of the price of the bond at different yields.
A bond has the following prices at different yields
Yield (in %) Price (in $)
What is the modified of the bond at an 8 % yield?
From the definition of Modified duration, we can use it to estimate the change in price of a bond as interest rate changes.
Consider a bond currently priced at $1100 with a modified duration of 4.445. What would be the bond price as yields increase by 1%?
By substituting in the formula for Modified Duration, we get that
4.445 = - \frac{1}{1100} \times \frac{ \Delta P } { 1 \% }.
\Delta P = - 4.445 \times 1100 \times 1 \% = - \$48.895
. Thus, the new price would be
P + \Delta P = \$1100 - \$48.895 = \$1051.105.
This example shows how knowing the modified duration allows us to make a simple calculation to determine the (approximate) price of the bond. Of course, we could recalculate the price of the bond by accounting for the yield changes, but that is more complicated then the above approach.
Cite as: Modified Duration. Brilliant.org. Retrieved from https://brilliant.org/wiki/modified-duration/
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A Brief History of Currency Practice Problems Online | Brilliant
A sad reality of life is that, in all likelihood, you don’t have all the things you want. What's worse, other people have them.
For thousands of years, we've devised ways to remedy this situation, exchanging stuff we need less for stuff we’d like more. Since you’re starting this course, you’ve probably heard of a relatively new method of exchanging goods and services called cryptocurrency.
To understand cryptocurrency, and the truth behind the claims of the problems it will solve and the dangers it will create, it helps to make sure we understand some basics about money in general.
Before we consider money, we can look at a system for exchanging goods and services that's easier to understand: bartering. Bartering is a direct exchange, trading something you have that the other person wants for something they have that you want.
You’ve probably bartered in the past yourself, like that time you traded your apple for a cookie in the school cafeteria.
The advantage of bartering is in its simplicity: you exchange directly for whatever it is that you want. However, a challenge with bartering arises when you want something from someone who wants something you don’t have. For example, you have an apple, and you’d really like to trade it for your friend’s cookie, but your friend is craving a salty snack.
Is it possible for both you and your friend to get what you want by bartering?
No, it's not possible to give up your apple to receive the cookie. Only if your friend is willing to compromise. Yes, but it requires a third person with different snacks and preferences.
Bartering can be simple and direct, but getting what you want in exchange for what you have might require coordination between more than just you and the person who has what you want.
A solution to the issue of coordination is money: something we can use as a medium of exchange.
With currency (which is what we call money that's in circulation, such as dollar bills), we can make a trade with someone even if they don’t want the goods that we have: we pay using currency rather than trading them goods directly.
If your friend thinks that trading you her cookie in exchange for one dollar is a good deal for her, what is she assuming?
The cookie is worth exactly one dollar to her — no more, no less. She'll be able to use the dollar to buy something else she wants later. The paper bill can directly meet her wants and needs. All of the above
Early examples of money helped to meet basic needs independent of their currency status and therefore had inherent value. For instance, furs have been used to facilitate trading since they can keep you warm, as have barley or salt, which can be eaten.
Using furs, barley, or salt as currency is not that different from bartering since the currency is something you can actually use.
These are called commodity monies, since their value comes from the fact that they're made out of a commodity — an economically useful good whose units are interchangeable. Sulfuric acid is an example of a commodity since any batch is essentially indistinguishable from any other batch.
Tennis shoes, on the other hand, are not a commodity since shoppers are willing to pay much more for certain brands of tennis shoes.
What are the downsides of using bushels of barley as currency? Select all answers that apply.
Barley is bulky and inconvenient to carry around. Barley can degrade if not used or traded away quickly. People who don’t like to eat barley wouldn’t accept it as payment. None of the above
Commodities that meet basic needs are nice because we can trust that (almost) everyone will be able to make use of them and they're unlikely to lose value, but they’re not always the most convenient things to carry around or store.
Some solutions are to use a commodity that's more convenient (without worrying if it can meet a basic need) or to trade things that represent the commodities. This makes the money system more convenient, but it requires a lot more trust.
Today, this concept has been fully embraced in most parts of the world and only a small fraction of exchange happens via bartering or commodity monies.
Which of the following would be the most convenient material to use as a currency? That is, one that is easy to carry around and doesn’t degrade over time.
Helium Iron Gold Mercury Uranium
For thousands of years, precious metals like gold and silver were used to mint coins for currency because they have a number of convenient characteristics:
They don’t degrade over time.
They’re rare enough that it takes a lot of work to find more.
They’re common enough that it's possible to find more.
The amounts that most people would have are easy to carry around.
The fact that precious metals don’t meet a basic need (unless you consider jewelry and computer chips as basic needs) didn’t dissuade people from assigning them value. The collective agreement to value precious metals made for a much smoother system of exchange.
The money we use today doesn’t represent anything that meets basic needs, and it isn't even backed by a convenient monetary commodity like gold. But people will still exchange it for these things.
That's because it's backed by governments. It’s easy to trust that the monetary system will keep working (and that others will continue to honor the value of money) because large, powerful entities make claims that they’ll enforce this system.
You don’t even need to trade gold coins or pieces of fancy paper (i.e. cash) for goods, you also have the option to trade a promise that you’ll pay the person back later.
This system is called credit.
Why might someone be hesitant to accept credit as payment rather than cash?
Credit doesn’t have any inherent value. Credit can be stolen more easily than cash. They have to trust you’ll fulfill your promise to pay later. There’s no reason someone should accept cash and not credit.
In theory, merchants take on risk when they accept credit as payment: the risk that you don’t pay what you promised. However, for the way most people use credit on a day-to-day basis, this is not a concern.
When you use a credit card to pay for something, the merchant receives the money immediately. In the moment, you're essentially making a promise to the credit card company that you'll pay them later for the money they're disbursing on your behalf. For most everyday uses of credit, risk has been transferred from the merchant to the credit card company.
All you need to make use of the credit system is a piece of information: the numbers associated with your credit card account. This system is especially convenient for shopping online, where we can exchange information but can’t directly exchange physical currency.
What is one downside to using credit cards when shopping online?
You need to have a lot of cash on hand to use a credit card. You have to share your credit card number. It takes a long time for your payment info to go through. There are no downsides.
We've learned to live with the security risks associated with sharing credit card info online, largely thanks to the fraud protection offered by credit card companies. But this requires putting our trust in credit card companies, which isn't strictly necessary for making purchases online.
Cryptocurrencies offer another way to exchange money in a way that's fundamentally different from credit cards, bank transfers, and other common methods of exchange. Cryptocurrencies accomplish a monetary system that doesn't depend on trust in a central authority or the people you're transacting with. It requires trust in mathematics and certain algorithms.
Because of how the rules of cryptocurrencies were designed, it's difficult to control them unilaterally. For example, a cryptocurrency's rules couldn't be changed by a government acting in the interest of aggressive bankers at the expense of everyone else
(
which some people would argue is what happened to traditional currencies in the wake of the
2008
).
A cryptocurrency, powered by its blockchain, can help to protect your savings from unscrupulous financial "professionals."
Since cryptocurrencies are new and different from traditional money, you may have heard all sorts of opinions about them, including from people who don't understand how they work.
Proponents argue that cryptocurrencies will wrest power out of the hands of the few, that they'll lead to lower transaction fees, and that they would have prevented the devastating effects of hyperinflation experienced by Zimbabwe in 2008 or by Venezuela in recent years. On the other hand, detractors argue that cryptocurrencies are a scam and that their valuation is a bubble waiting to burst.
In this course, we'll cut through the noise in order to teach you the underlying mechanics that cryptocurrencies are built on, so that you can decide for yourself whether they're the next big thing or just a mathematician's pipe dream.
Money is built on trust, and if you understand how cryptocurrencies work, you'll be much better equipped to determine if they're trustworthy.
We’ll start this course following the fable of Cryptonia. The story of the people of Cryptonia will provide an understanding of the systems inside cryptocurrencies, giving you the foundation you need to dive into the deeper details through the rest of this course.
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Lithium gold boride - EverybodyWiki Bios & Wiki
Lithium gold boride
Lithium gold boride (chemical formula LiAu3B) is a material with a honeycomb-like structure.[1] that can be used to make superconducting batteries.
2.1 Lattice structure
4.1.1 Diffusion barrier
In 2018, LiAu3B metal was discovered.[2] Just like the other lithium boride materials, this material also is predicted to have a critical temperature where superconductivity is achieved at 5.8K.
LiAu3B is a ternary metal, consisting of the three elements lithium, gold and boron. The material is stable under extraction of its lithium atoms, during which it experiences a volume deviation of 0.42%.[2]
Lattice structure[edit]
The unit cell consists of three formula units of LiAu3B. The gold atoms are placed in a hexagonal structure. Together with the boron atoms, that are placed at the centre of trigonal prisms, they form the main skeleton of the material. The lithium atoms are placed at the centre of the hexagonal structure of the gold atoms.
Electronic structure[edit]
The electronic structure of the material makes it useful for battery applications. The lithium atoms at the centre of the structure behave as donors, while the Au3B units surrounding the lithium behave as acceptors. Therefore the lithium atoms have a positive charge, whereas the host structure of gold and boron atoms has a negative charge. This follows from Mulliken charge analysis.
Mechanical stability[edit]
Under an increase of pressure, the B-Au bonds are less affected than the Au-Au bonds. Under an increase from 0 to 50 GPa, all bond lengths become smaller except those of the B-Au bonds. This is a result of the strong hybridization between the gold and boron atoms mentioned earlier. Furthermore, under increasing pressure, the electronegativity of the gold and boron atoms rises, while the electronegativity of the lithium atoms decreases. This can be explained by the fact that the length of the Au-Au bonds decreases under higher pressure. Because of the hexagonal structure of the gold atoms, the gold atoms will become closer to the lithium atoms and will be better able to 'steal' its electron. This means that the acceptation capacities of the gold and boron atoms increases under higher pressure.[1]
Band structure[edit]
This is the band structure of the material as calculated.[1] The total density of states can be separated into 4 regions up to the Fermi energy. The first group is a sharp peak at approximately -48 eV. This one is due to the contribution of the 1s2 orbitals of the lithium atoms. The second peak is the result of the s-states of gold and boron, and is located between -11 and -9 eV. The third peak is caused by the d-states of gold and p-states of boron, located between -7 to -3 eV. The densities of both orbitals are relatively high in this region. Because the d- and p-states have approximately the same amount of energy in this region, it is easy for the electrons to hop between these two different orbitals. This, combined with the relatively high density of states in this region, will create a strong hybridization between the d-states and p-states of gold and boron respectively. The last peak is from -3 eV to the Fermi level. It is due to the p-states of gold and there is hybridization between p-states of gold and boron atoms. The hybridizations are responsible for higher covalency in Au-B bonds.
Calculations have showed that ternary metal borides promise a higher critical temperature Tc for superconductivity then[clarification needed].[2]
The flow of lithium ions from the cathode to the anode and vice versa, while facilitated by an Li+ conducting electrolyte, is analogous to the flow of current in ordinary batteries. It allows for simple charging and discharging of LiAu3B batteries. Furthermore, the lattice structure is preserved even when all lithium ions are removed from the structure, to a (temporary) host structure, and only a small volumetric deviation is observed when all lithium ions are removed. The deviation is only 0.42% of the total volume when there are no lithium atoms in the Au3B. This is because the lithium atoms are located at the centre of the hexagonal gold structures. Adding or removing the lithium atoms will not drastically change the structure of the gold and therefore also will not affect the volume too much. This is an important result for batteries because it is not desired to have batteries that change their volume while being charged or discharged. The average open-circuit voltage of 1.30V also makes the material promising to use in batteries.[2]
Diffusion barrier[edit]
The diffusion barrier is defined as the energy difference of the system when the lithium atoms are placed inside the host structure (inside the cathode) and when the lithium atoms are outside the structure (inside the anode). So the diffusion barrier is important for batteries, as it says something about how much energy is needed to charge or discharge the system. It also depends highly on pressure, as the Au-Au bond lengthes decreases under high pressures, and thus there will be less free space for the lithium atom to be placed in. Less free space means a higher diffusion barrier. For 0 GPA the diffusion barrier is approximately 0.30 eV and increases to 0.51 eV at 15 GPa. Boltzmann's constant kB can then be used to find a temperature dependent expression for the diffusion mobility. At room temperature at 15 GPa, the diffusion mobility is in the order of
{\displaystyle 3.4\cdot 10^{-}3}
times smaller than at 0 GPa.
Like most other lithium boride metals, superconductivity is also a property of LiAu3B. This is because the boron atoms create metal bonds. The superconductive properties can be derived from the phonon spectra of LiAu3B at 0 hydrostatic pressure. The critical temperature where superconductivity occurs can be calculated using McMillan's equation:
{\displaystyle T_{c}={\frac {w_{log}}{1.2}}\exp \left({\frac {-1.04(1+\lambda )}{\lambda (1-0.62\mu ^{*})-\mu ^{*}}}\right)}
Here, λ is given by the relation
{\displaystyle \lambda =\sum _{q,v}\lambda _{q,v}=2\int {\frac {\alpha ^{2}F(w)}{w}}dw}
where the individual
{\displaystyle \lambda _{q,v}}
for a mode v at wave vector q are the electron-phonon coupling constants, and given by
{\displaystyle \lambda _{q,v}={\frac {\gamma _{q,v}}{\pi \hbar N(E_{F})w_{qv}^{2}}}}
{\displaystyle \gamma _{q,v}}
the phonon line width, and
{\displaystyle N(E_{F})}
the density of states at the Fermi level. Using this method, the critical temperature Tc is calculated to be 5.8K. This result is of the same order as other already known superconducting materials.[2]
One problem of LiAu3B is that it is mostly made out of gold atoms (3 out of 5 atoms per unit cell). As gold is a relatively expensive material, these batteries will be rather costly as well. Therefore, it is not expected that LiAu3B batteries will be used for the foreseeable future.
↑ 1.0 1.1 1.2 Aydin, Sezgin; Şimşek, Mehmet (2018). "Stability and Pressure Dependent Properties of Ternary Lithium Borides of Gold and Silver". Physica Status Solidi B. 255 (6): 1700666. Bibcode:2018PSSBR.25500666A. doi:10.1002/pssb.201700666.
↑ 2.0 2.1 2.2 2.3 2.4 Aydin, Sezgin; Şimşek, Mehmet (2018). "A superconducting battery material: Lithium gold boride (LiAu3B)". Solid State Communications. 272: 8–11. Bibcode:2018SSCom.272....8A. doi:10.1016/j.ssc.2018.01.007.
This article "Lithium gold boride" is from Wikipedia. The list of its authors can be seen in its historical and/or the page Edithistory:Lithium gold boride. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.
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G
e
G
M
\mathrm{μ} :G→.M
\mathrm{μ}\left(e, x\right) = x
\mathrm{μ}\left(a*b,x\right) = \mathrm{μ}\left(a, \mathrm{μ}\left(b, x\right)\right)
a, b ∈G
x ∈ M
\mathrm{μ},
{\mathrm{μ}}_{1,a}:M→M
{\mathrm{\mu }}_{1,a}\left(x\right) = \mathrm{μ}\left(a, x\right)
{\mathrm{μ}}_{2,x}:G→M
{\mathrm{\mu }}_{2,x}\left(a\right) = \mathrm{μ}\left(a, x\right)
\mathrm{μ}
{\mathrm{Γ}}_{\mathrm{μ}}
M
{\mathrm{\mu }}_{2,x}
G
{\mathrm{Γ}}_{\mathrm{μ} }
{\mathrm{\mu }}_{2,x}
and evaluating the results at the identity.
\mathrm{μ}
{\mathrm{Γ}}_{\mathrm{μ}} = \mathrm{Γ}
{\mathrm{\mu }}_{1,a}
{\mathrm{\mu }}_{2,x}
\mathrm{μ}
\mathrm{𝔤}
\mathrm{Γ}
\mathrm{𝔤}\mathit{ }
\mathrm{𝔤}
\mathrm{Γ}
\mathrm{𝔤}
\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):
\mathrm{with}\left(\mathrm{GroupActions}\right):
\mathrm{with}\left(\mathrm{LieAlgebras}\right):
\mathrm{with}\left(\mathrm{Library}\right):
\left[x,y\right].
\mathrm{DGsetup}\left([x,y],M\right):
\mathrm{Γ}
\mathrm{Gamma}≔\mathrm{evalDG}\left([\mathrm{D_x},\mathrm{D_y},y\mathrm{D_x}]\right)
\textcolor[rgb]{0,0,1}{\mathrm{Γ}}\textcolor[rgb]{0,0,1}{:=}\left[\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\right]
\mathrm{Γ}
\mathrm{LieAlgebraData}\left(\mathrm{Gamma}\right)
\left[\left[\textcolor[rgb]{0,0,1}{\mathrm{e2}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e3}}\right]\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{e1}}\right]
\mathrm{DGsetup}\left([\mathrm{z1},\mathrm{z2},\mathrm{z3}],G\right):
\mathrm{μ1}≔\mathrm{Action}\left(\mathrm{Gamma},G\right)
\textcolor[rgb]{0,0,1}{\mathrm{μ1}}\textcolor[rgb]{0,0,1}{:=}\left[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{z3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{z2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{z3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{z1}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{z2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\right]
\mathrm{newGamma}≔\mathrm{InfinitesimalTransformation}\left(\mathrm{μ1},[\mathrm{z1},\mathrm{z2},\mathrm{z3}]\right)
\textcolor[rgb]{0,0,1}{\mathrm{newGamma}}\textcolor[rgb]{0,0,1}{:=}\left[\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\right]
\mathrm{Γ2}≔\mathrm{evalDG}\left([y\mathrm{D_x},\mathrm{D_x},\mathrm{D_y}]\right)
\textcolor[rgb]{0,0,1}{\mathrm{Γ2}}\textcolor[rgb]{0,0,1}{:=}\left[\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\right]
\mathrm{L2}≔\mathrm{LieAlgebraData}\left(\mathrm{Γ2},\mathrm{Alg2}\right)
\textcolor[rgb]{0,0,1}{\mathrm{L2}}\textcolor[rgb]{0,0,1}{:=}\left[\left[\textcolor[rgb]{0,0,1}{\mathrm{e1}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e3}}\right]\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{e2}}\right]
\mathrm{DGsetup}\left(\mathrm{L2}\right)
\textcolor[rgb]{0,0,1}{\mathrm{Lie algebra: Alg2}}
\mathrm{Adjoint}\left(\right)
\left[\left[\begin{array}{rrr}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\end{array}\right]\textcolor[rgb]{0,0,1}{,}\left[\begin{array}{rrr}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\end{array}\right]\textcolor[rgb]{0,0,1}{,}\left[\begin{array}{rrr}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\end{array}\right]\right]
\mathrm{μ1},B≔\mathrm{Action}\left(\mathrm{Γ2},G,\mathrm{output}=["ManifoldToManifold","Basis"]\right)
\textcolor[rgb]{0,0,1}{\mathrm{μ1}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{B}\textcolor[rgb]{0,0,1}{:=}\left[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{z2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{z1}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{z3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\right]\textcolor[rgb]{0,0,1}{,}\left[\left[\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}\right]\textcolor[rgb]{0,0,1}{,}\left[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}\right]\textcolor[rgb]{0,0,1}{,}\left[\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}\right]\right]
\mathrm{newGamma}≔\mathrm{InfinitesimalTransformation}\left(\mathrm{μ1},[\mathrm{z1},\mathrm{z2},\mathrm{z3}]\right)
\textcolor[rgb]{0,0,1}{\mathrm{newGamma}}\textcolor[rgb]{0,0,1}{:=}\left[\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\right]
\mathrm{map}\left(\mathrm{DGzip},B,\mathrm{Γ2},"plus"\right)
\left[\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\right]
\mathrm{DGsetup}\left([x,y],M\right):
\mathrm{Γ3}≔\mathrm{Retrieve}\left("Gonzalez-Lopez",1,[22,17],\mathrm{manifold}=M\right)
\textcolor[rgb]{0,0,1}{\mathrm{Γ3}}\textcolor[rgb]{0,0,1}{:=}\left[\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{,}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\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}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\right]
\mathrm{DGsetup}\left([\mathrm{z1},\mathrm{z2},\mathrm{z3},\mathrm{z4},\mathrm{z5}],\mathrm{G3}\right)
\textcolor[rgb]{0,0,1}{\mathrm{frame name: G3}}
\mathrm{\mu }≔\mathrm{Action}\left(\mathrm{Γ3},\mathrm{G3}\right)
\textcolor[rgb]{0,0,1}{\mathrm{μ}}\textcolor[rgb]{0,0,1}{:=}\left[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{z5}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{z5}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{z1}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{z2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{z3}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\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}{\mathrm{z4}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\right]
\mathrm{InfinitesimalTransformation}\left(\mathrm{\mu },[\mathrm{z1},\mathrm{z2},\mathrm{z3},\mathrm{z4},\mathrm{z5}]\right)
\left[{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{,}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\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}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\right]
\mathrm{DGsetup}\left([x,y,u,v],\mathrm{M4}\right):
\mathrm{Γ4}≔\mathrm{Retrieve}\left("Petrov",1,[32,6],\mathrm{manifold}=\mathrm{M4}\right)
\textcolor[rgb]{0,0,1}{\mathrm{Γ4}}\textcolor[rgb]{0,0,1}{:=}\left[\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_u}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_u}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_u}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\right]
\mathrm{DGsetup}\left([\mathrm{z1},\mathrm{z2},\mathrm{z3},\mathrm{z4}],\mathrm{G4}\right)
\textcolor[rgb]{0,0,1}{\mathrm{frame name: G4}}
\mathrm{\mu }≔\mathrm{Action}\left(\mathrm{Γ4},\mathrm{G4}\right)
\textcolor[rgb]{0,0,1}{\mathrm{μ}}\textcolor[rgb]{0,0,1}{:=}\left[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{z3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{z4}}\right)\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{z3}}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{cos}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{z4}}\right)\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{z3}}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{z1}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{z4}}\right)\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{z3}}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{cos}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{z4}}\right)\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{z3}}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{z2}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{v}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{v}\right]
\mathrm{InfinitesimalTransformation}\left(\mathrm{\mu },[\mathrm{z1},\mathrm{z2},\mathrm{z3},\mathrm{z4}]\right)
\left[\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_u}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_u}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_u}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\right]
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Cotangent of angle in radians - MATLAB cot - MathWorks España
Cotangent of Vector of Complex Angles
Cotangent of angle in radians
Y = cot(X) returns the cotangent of elements of X. The cot function operates element-wise on arrays. The function accepts both real and complex inputs.
For real values of X, cot(X) returns real values in the interval [-∞, ∞].
For complex values of X, cot(X) returns complex values.
Plot the cotangent function over the domain
-\pi <x<0
0<x<\pi
plot(x1,cot(x1),x2,cot(x2)), grid on
Calculate the cotangent of the complex angles in vector x.
0.0000 + 1.3130i -0.0000 - 1.0903i -0.0006 - 0.9997i
Y — Cotangent of input angle
Cotangent of input angle, returned as a real-valued or complex-valued scalar, vector, matrix or multidimensional array.
\text{cot}\left(\alpha \right)=\frac{1}{\mathrm{tan}\left(\alpha \right)}=\frac{\text{adjacent side}}{\text{opposite side}}=\frac{b}{a}\text{\hspace{0.17em}}.
\text{cot}\left(\alpha \right)=\frac{i\left({e}^{i\alpha }+{e}^{-i\alpha }\right)}{\left({e}^{i\alpha }-{e}^{-i\alpha }\right)}\text{\hspace{0.17em}}.
In floating-point arithmetic, cot is a bounded function. That is, cot does not return values of Inf or -Inf at points of divergence that are multiples of pi, but a large magnitude number instead. This stems from the inaccuracy of the floating-point representation of π.
cotd | coth | acot | acotd | acoth
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Paradoxes in Probability | Brilliant Math & Science Wiki
Raghav Vaidyanathan, Mohammad Farhat, Calvin Lin, and
Paradoxes in probability often arise because people have an incorrect connotation of probability or because the phrasing is ambiguous, which leads to multiple interpretations. More specifically, the intuitive way of viewing some problems makes it seem as though an incomplete enumeration of the possible outcomes for a problem is actually a complete one.
Two Envelopes Problem (Expected Value)
See the famous Monty Hall problem.
Introduced by French mathematician Joseph Bertrand, this paradox remains highly debated till date. It was intended as an example to show that the mechanism of production of a random variable may affect the probability of certain events.
Consider an equilateral triangle inscribed in a circle. When a chord of the circle is chosen at random, what's the probability that the chord is longer than a side of the triangle?
Bertrand followed this up with three different but seemingly correct solutions. The paradox lies in the fact that each of these solutions gives a different answer to the problem.
We generate the random chord by randomly choosing its end on the circle.
By symmetry, we can fix one of the ends. The moving end generates a chord with a length greater than the square root of 3 when it's on the portion away from the fixed end, with length one-third of the circle length.
So, the probability that the random chord has a length greater than the square root of 3 is
\frac13.
By symmetry, we can reduce the problem to examining only the chord with a chosen direction.
Then, the factor which decides whether the chord is smaller or greater than the square root of 3 is the distance to the center of the circle. The chord is greater than the square root of 3 if and only if its distance to the center of the circle is smaller than
\frac12.
So, the probability is
\frac12.
Choose a point anywhere within the circle and construct a chord with the chosen point as its midpoint.
The chord is longer than a side of the inscribed triangle if the chosen point falls within a concentric circle of
\frac12
the radius of the larger circle.
The area of the smaller circle is one fourth the area of the larger circle, so the probability that a random chord is longer than a side of the inscribed triangle is
\frac14.
Classical solution:
The classical solution to this paradox depends on the mode of selection of chords. The problem has a well-defined answer only when the selection procedure is predefined. Since there is no restriction on the selection procedure, there isn't any reason to choose one method over another. Further, if one decides to resolve this using physical experiments, one is met with the paradox again as different experiments can be designed, each of which gives one of the different answers mentioned above.
See the two envelopes problem.
See the mind-bending Simpson's paradox.
Cite as: Paradoxes in Probability. Brilliant.org. Retrieved from https://brilliant.org/wiki/paradoxes-in-probability/
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LMIs in Control/pages/NI-Lemma - Wikibooks, open books for an open world
LMIs in Control/pages/NI-Lemma
LMIs in Control/pages/NI-Lemma Positive real systems are often related to systems involving energy dissipation. But the standard Positive real theory will not be helpful in establishing closed-loop stability. However transfer functions of systems with a degree more than one can be satisfied with the negative imaginary conditions for all frequency values and such systems are called "systems with negative imaginary frequency response" or "negative imaginary systems".
3 The LMI: LMI for Negative Imaginary Lemma
{\displaystyle (A,B,C,D)}
{\displaystyle {\begin{aligned}{\dot {x}}(t)=Ax(t)+Bu(t)\\y=Cx(t)+Du(t)\end{aligned}}}
{\displaystyle A\in \mathbb {R} ^{n\times n},B\in \mathbb {R} ^{n\times m},C\in \mathbb {R} ^{m\times n},D\in \mathbb {S} ^{m}}
The LMI: LMI for Negative Imaginary LemmaEdit
According to the Lemma, The aforementioned system is negative imaginary under either of the following equivalent necessary and sufficient conditions
{\displaystyle P}
{\displaystyle {\begin{aligned}\mathbb {S} \end{aligned}}}
n,where
{\displaystyle P\geq 0}
{\displaystyle {\begin{bmatrix}A^{T}P+PA&PB-A^{T}C^{T}\\B^{T}P-CA&-(CB+B^{T}C^{T})\end{bmatrix}}}
{\displaystyle {\begin{aligned}\leq 0\end{aligned}}}
{\displaystyle Q}
{\displaystyle {\begin{aligned}\mathbb {S} \end{aligned}}}
{\displaystyle Q\geq 0}
{\displaystyle {\begin{bmatrix}QA^{T}+AQ&B-QA^{T}C^{T}\\B^{T}-CAQ&-(CB+B^{T}C^{T})\end{bmatrix}}}
{\displaystyle {\begin{aligned}\leq 0\end{aligned}}}
The system is strictly negative if det(
{\displaystyle A}
{\displaystyle \neq }
0 and either of the above LMIs are feasible with resulting
{\displaystyle P>0}
{\displaystyle Q>0}
Xiong, Junlin & Petersen, Ian & Lanzon, Alexander. (2010). A Negative Imaginary Lemma and the Stability of Interconnections of Linear Negative Imaginary Systems.
Retrieved from "https://en.wikibooks.org/w/index.php?title=LMIs_in_Control/pages/NI-Lemma&oldid=4052236"
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Normalized number - Wikipedia
In applied mathematics, a number is normalized when it is written in scientific notation with one non-zero decimal digit before the decimal point.[1] Thus, a real number, when written out in normalized scientific notation, is as follows:
{\displaystyle \pm d_{0}.d_{1}d_{2}d_{3}\dots \times 10^{n}}
where n is an integer,
{\textstyle d_{0},d_{1},d_{2},d_{3},\ldots ,}
are the digits of the number in base 10, and
{\displaystyle d_{0}}
is not zero. That is, its leading digit (i.e., leftmost) is not zero and is followed by the decimal point. Simply speaking, a number is normalized when it is written in the form of a × 10n where 1 ≤ a < 10 without leading zeros in a. This is the standard form of scientific notation. An alternative style is to have the first non-zero digit after the decimal point.
As examples, the number 918.082 in normalized form is
{\displaystyle 9.18082\times 10^{2},}
while the number −0.00574012 in normalized form is
{\displaystyle -5.74012\times 10^{-3}.}
Clearly, any non-zero real number can be normalized.
The same definition holds if the number is represented in another radix (that is, base of enumeration), rather than base 10.
In base b a normalized number will have the form
{\displaystyle \pm d_{0}.d_{1}d_{2}d_{3}\dots \times b^{n},}
{\textstyle d_{0}\neq 0,}
and the digits,
{\textstyle d_{0},d_{1},d_{2},d_{3},\ldots ,}
are integers between
{\displaystyle 0}
{\displaystyle b-1}
In many computer systems, binary floating-point numbers are represented internally using this normalized form for their representations; for details, see normal number (computing). Although the point is described as floating, for a normalized floating-point number, its position is fixed, the movement being reflected in the different values of the power.
^ Fleisch, Daniel; Kregenow, Julia (2013), A Student's Guide to the Mathematics of Astronomy, Cambridge University Press, p. 35, ISBN 9781107292550 .
Retrieved from "https://en.wikipedia.org/w/index.php?title=Normalized_number&oldid=1016985403"
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Relation (mathematics) - Simple English Wikipedia, the free encyclopedia
This article uses too much jargon, which needs explaining or simplifying. Please help improve the page to make it understandable for everybody, without removing the technical details.
A (binary)relation R among sets A and B is a subset of the Cartesian product of the set of b
{\displaystyle R\subseteq A\times B}
. The above picture shows an example of two sets A={a,b,c} and B={x,y,z}, their Cartesian product is a complete relation among A and B, and any other subset of AxB is a relation too. The Venn diagram shows a function. A function is a relation that "maps" elements of one set to another set. That means that each element in the first set can appear at most in one pair in the first entry. The example shows two elements of the first set can be mapped to the same element of the second set. Both elements b and c of the first set map to element z of the second set.
In mathematics, an n-ary relation on n sets, is any subset of Cartesian product of the n sets (i.e., a collection of n-tuples),[1] with the most common one being a binary relation, a collection of order pairs from two sets containing an object from each set.[2] The relation is homogeneous when it is formed with one set.
For example, any curve in the Cartesian plane is a subset of the Cartesian product of real numbers, RxR. The homogeneous binary relations are studied for properties like reflexiveness, symmetry, and transitivity, which determine different kinds of orderings on the set.[3] Heterogeneous n-ary relations are used in the semantics of predicate calculus, and in relational databases.
In relational databases jargon, the relations are called tables. There is a relational algebra consisting in the operations on sets, because relations are sets, extended with operators like projection, which forms a new relation selecting a subset of the columns (tuple entries) in a table, the selection operator, which selects just the rows (tuples),according to some condition, and join which works like a composition operator.
The use of the term "relation" is often used as shorthand to refer to binary relations, where the set of all the starting points is called the domain and the set of the ending points is the codomain.[4]
Different types of relationship[change | change source]
An example for such a relation might be a function. Functions associate each key with one value. The set of all functions is a subset of the set of all relations - a function is a relation where the first value of every tuple is unique through the set.
Other well-known relations are the equivalence relation and the order relation. That way, sets of things can be ordered: Take the first element of a set, it is either equal to the element looked for, or there is an order relation that can be used to classify it. That way, the whole set can be classified (i.e., compared to some arbitrarily chosen element).
Relations can be transitive.One example of a transitive relation is the "smaller-than" relation. If X "is smaller than" Y,and Y is "smaller than" Z,then X "is smaller than" Z. In general, a transitive relation is a relation such that if relations (a,b) and (b,c) both belong to R, then (a,c) must also belongs to R.
Relations can be symmetric. One example of a symmetric relation is the relation "is equal to". If X "is equal to" Y, then Y "is equal to" X. In general, a symmetric relation is a relation such that if (a,b) belongs to R, then (b,a) must belong to R as well.
Relations can be asymmetric, such as the relation " is smaller than". In general, a relation is asymmetric if whether (a,b) belongs to R, (b,a) does not belong to R.
Relations can be reflexive. One example of a reflexive relation is the relation "is equal to" (e.g., for all X, X "is equal to" X). In general, a reflexive relation is a relation such that for all a in A, (a,a) belongs to R.
By definition, every subset of AxB is a relation from A to B.
In category theory, relations play an important role in the Cartesian closed categories, which transform morphisms from tuples to morphisms of single elements. That corresponds to Currying in the Lambda calculus.
Databases and Relations[change | change source]
In the relational database theory, a database is a set of relations. To model a real world, the relations should be in a canonical form called normalized form in the data base argot. That transformation ensure no loss of information, nor the insertion of spurious tuples with no corresponding meaning in the world represented in the database. The normalization process takes into account properties of relations like functional dependencies among their entries, keys and foreign keys, transitive and join dependencies.
↑ "The Definitive Glossary of Higher Mathematical Jargon — Relation". Math Vault. 2019-08-01. Retrieved 2019-12-11.
↑ "Relation definition - Math Insight". mathinsight.org. Retrieved 2019-12-11.
↑ "Relations | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2019-12-11.
↑ Stapel, Elizabeth. "Functions versus Relations". Purplemath. Retrieved 2019-12-11.
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Relation_(mathematics)&oldid=8179643"
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Tie in selecting row and column (Vogel's Approximation Method - VAM) | Numerical | Solving Transportation Problem | Transportation Model | Education Lessons
Solve the following problem for BFS using VAM method (Tie in selecting row and column)
Note: Important point to remember for VAM - Case of Tie
If the "smallest cost" in a row or column are repeating, then difference for that row or column is "0".
In VAM, we have to select the row or column which is having higher difference
But if there is a tie in selection, then we have to select the row or column which contains minimum cost.
In case there's a tie in minimum cost too, select the cell in which maximum allocation can be done.
Step 1: Balance the problem
The given problem is already balanced.
Step 2: Find row difference and column difference
Step 3: Find highest row/column difference and start allocation
→ Select row/column with highest difference. In the same row/column, select the cell with minimum cost, then allocate smallest value of demand or supply in that cell.
→ Here, we have [3] as the highest difference. Selecting column with [3] as column difference and finding cell with minimum cost.
→ As we can see here "1" in the second row is the minimum cost in this last column with highest difference of [3].
→ So, allocating 10 to "1"(min. cost) with highest column difference.
Step 4: Remove the row/column as per fulfillment of supply or demand
→ Remove the row/column whose supply or demand is fulfilled and prepare new matrix as shown below.
→ Check that here, we have multiple highest difference as "[2]".
→ [Tie] We have to select the row/column which has minimum cost included. (check note above for this step).
→ So, selecting first column with highest column difference as [2] and minimum cost as "1" and allocating same as we have done in third step above.
Step 5: Repetition of above steps
→ Repeat the procedure until all allocations are done.
→ You may get [Tie] again in this and further steps. Just repeat step 4 & 3 as above shown to get the answer.
Step 6: After all allocations are done, write the allocations and find transportation cost
\begin{aligned} &Transportation \ Cost\\ &= (1 \times 20) + (2 \times 30) + (1 \times 10) + (2 \times 20) + (3 \times 20)\\ &= 190 \end{aligned}
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Generate sine wave, using simulation time as time source - Simulink - MathWorks France
y=amplitude×\mathrm{sin}\left(frequency×time+phase\right)+bias.
\begin{array}{l}\mathrm{sin}\left(t+\Delta t\right)=\mathrm{sin}\left(t\right)\mathrm{cos}\left(\Delta t\right)+\mathrm{sin}\left(\Delta t\right)\mathrm{cos}\left(t\right)\\ \mathrm{cos}\left(t+\Delta t\right)=\mathrm{cos}\left(t\right)\mathrm{cos}\left(\Delta t\right)-\mathrm{sin}\left(t\right)\mathrm{sin}\left(\Delta t\right)\end{array}
\left[\begin{array}{c}\mathrm{sin}\left(t+\Delta t\right)\\ \mathrm{cos}\left(t+\Delta t\right)\end{array}\right]=\left[\begin{array}{cc}\mathrm{cos}\left(\Delta t\right)& \mathrm{sin}\left(\Delta t\right)\\ -\mathrm{sin}\left(\Delta t\right)& \mathrm{cos}\left(\Delta t\right)\end{array}\right]\left[\begin{array}{c}\mathrm{sin}\left(t\right)\\ \mathrm{cos}\left(t\right)\end{array}\right]
\left[\begin{array}{cc}\mathrm{cos}\left(\Delta t\right)& \mathrm{sin}\left(\Delta t\right)\\ -\mathrm{sin}\left(\Delta t\right)& \mathrm{cos}\left(\Delta t\right)\end{array}\right]
\mathrm{sin}\left(t\right)
\mathrm{sin}\left(t+\Delta t\right)
y=A\mathrm{sin}\left(2\pi \left(k+o\right)/p\right)+b
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4.1 The Tangle | IOTA Wiki
This specification describes the data structure used in the IOTA protocol, and introduces its main terminology.
4.1.1 Description
The Tangle represents a growing partially-ordered set of messages, linked with each other through cryptographic primitives, and replicated to all nodes in the peer-to-peer network. The Tangle enables the possibility to store data and to keep a ledger, the latter being based on UTXO-DAG formed by transactions contained in messages.
In mathematical terms, the Tangle is a Directed Acyclic (multi)Graph with messages as vertices and directed edges as references to existing messages. Directed edges are labelled:
0
represents direct references flagged as weak, and
1
represents direct references flagged as strong (see Section 6.4 - Finalization. Messages are linked with each other through cryptographic hashes. The acyclicity condition means that there is no directed cycle composed of weak or strong edges.
In this section we provide some useful terminology which is useful to understand the basic elements of the protocol.
Here we present the set of rules, called approval switch, which allow nodes to alternatively approve single messages or the entire past cone of a message.
Parent: The protocol requires that each message contains a field parents in order to guarantee cryptographic references among messages in the Tangle. These references can be of two types, strong and weak, and are defined by the field parents type. Intuitively, we say that
y
is a strong (resp. weak) parent of a message
x
x
has a directed strong (resp. weak) edge to
y
. Each message has a possible number of parents from 2 to 8 with repetitions (with 2 as a default value), where at least one is a strong parent. More information about parents can be found in Section 2.2 - Message layout.
Approvers: A message
x
directly approves
y
y
x
x
is a strong (resp. weak) approver of
y
y
is a strong (resp. weak) parent of
x
. More generally, we say that a message
x
strongly approves
y
if there is a directed path of strong approvals from
x
y
x
weakly approves
y
if there is a directed path of approvals of any type from
x
y
Past cone: We say that the (strong) past cone of
x
is the set of all messages strongly approved by
x
, and the weak past cone of
x
is the set of all messages weakly or strongly approved by
x
Future cone: We define the future cone of a message
x
as the set of messages that weakly or strongly approve
x
. Please note that, unlike its past cone, the future cone of a message changes over time.
Genesis: The genesis is the message that creates the entire token supply. Note that this implies that no other tokens will ever be created or, equivalently, no mining occurs. This message has no outgoing edges and is in the weak past cone of every other message.
In short, strong approvals are equivalent to the approvals in the legacy IOTA: if
x
y
, it approves also
y
's past cone. Moreover, weak approvals emulate the White Flag approach from Chrysalis: approving a message does not necessarily approve its past cone. This feature allows, for instance, two conflicting messages to be part of the Tangle without creating unmergeable branches.
4.1.2.2 Validity
This subsection introduces the definitions of validity for transactions and messages.
(Transaction) Validity: A transaction is valid if it passes the syntactical filter and its references are valid (see Section 2.3 - Payloads Layout for information):
It is syntactically correct.
Unblock conditions are met (see Section 5.1 - UTXO for further information).
Balances are zero, i.e., the sum of the inputs in the transaction's payload is equal to the sum of the outputs spent.
No conflicts in the past cone on the UTXO DAG (two transactions conflict if they consume the same UTXO output).
(Message) Individual Validity: A message is considered individually valid if it passes all the objective filters, i.e. the ones included in the Message Parser (see Section 2.4 - Data Flow):
Its signature is valid.
Its Proof of Work is correct.
(Message) Weak Validity: A message is weakly valid if:
Its Individually Valid.
Its parents are weakly valid.
Its transaction is valid.
It passes the Parent Age Check.
(Message) Strong Validity: A message is strongly valid if:
It is weakly valid.
Its strong parents do not have a conflicting past.
Its strong parents are strongly valid.
4.1.2.3 Branches
In the IOTA protocol, multiple version of the ledger state can temporarily coexist. These multiple versions of the ledger state are called branches. As soon as conflicts are detected, new branches are generated where the outputs created by conflicting transactions and those created by transactions that spend these outputs are tracked. If all past conflicts are resolved and no new conflicts are detected, then only one branch will exist. More rigorously, we can refer to a branch as a non-conflicting collection of past cones of outputs in the UTXO DAG. Additional information, as well as the distinction between conflict and aggregated branches, is given in Section 5.2 - Ledger state.
4.1.2.2 Solidification
Due to the asynchronicity of the network, we may receive messages for which their past cone has missing elements. We refer to these messages as unsolid messages. It is not possible neither to approve nor to gossip unsolid messages. The actions required to obtain such missing messages is called solidification procedure, and are described in detail in Section 4.4 - Solidification.
4.1.3 Example
Image 4.1.1 shows an example of the Tangle (strong edges are with a continuous line, weak edges are with a dotted line). Message
D
contains a transaction that has been rejected. Thus, in the legacy IOTA implementation, its future cone would be orphaned due to the monotonicity rule. In particular, both messages
E
(data) and
F
(transaction) directly reference
D
. In IOTA 2.0, the introduction of the approval switch allows that these messages can be picked up via a weak approval, as messages
G
H
exemplify.
Image 4.1.1: Example of the IOTA Tangle.
4.1.2.2 Solidification
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m (→Importing data: looping r.in.gdal over GeoTIFF bands)
Once inside a Location that is defined by the spatial reference system in which the bands of interest are projected, they can be imported with the {{cdm|r.in.gdal}} module.
For example, GeoTIFF files can be imported by looping {{cdm|r.in.gdal}} over all of them
{\displaystyle {\frac {W}{m^{2}*sr*nm}}}
{\displaystyle L\lambda ={\frac {10^{4}*DN\lambda }{CalCoef\lambda *Bandwidth\lambda }}}
{\displaystyle \rho _{p}={\frac {\pi *L\lambda *d^{2}}{ESUN\lambda *cos(\Theta _{S})}}}
{\displaystyle \rho }
{\displaystyle \pi }
{\displaystyle L\lambda }
{\displaystyle d}
{\displaystyle Esun}
{\displaystyle cos(\theta _{s})}
Once inside a Location that is defined by the spatial reference system in which the bands of interest are projected, they can be imported with the Template:Cdm module.
For example, GeoTIFF files can be imported by looping Template:Cdm over all of them
{\displaystyle {\frac {W}{m^{2}*\mu m}}}
{\displaystyle [0,255]}
{\displaystyle [0,2047]}
|
in a solvent 50% of benzoic acid dimerises while the rest ionises determine molar mass of acid which is - Chemistry - Solutions - 7578809 | Meritnation.com
in a solvent 50% of benzoic acid dimerises while the rest ionises. determine molar mass of acid which is observed and also its van't hoff factor.
Formula to calculate i ( Van't haff factor) using
\alpha
(degree of ionizaion) is as follows:
i = (1-
\alpha
\frac{\alpha }{n}
\alpha
= 50/100 = 0.5
n = 2 (dimerisation)
i = (1.0-0.5) + 0.5/2
\frac{\mathrm{Observed}\quad \mathrm{molar}\quad mass}{\text{Actual Molar mass}}
\frac{\mathrm{Observed}\quad \mathrm{molar}\quad mass}{122\quad \text{g/mol}}
Observed Molar mass = 0.75 x 122 g/mol
Najeeb Choudhury answered this
alpha = 50/100 = 0.5 put in formula i = 1 - 0.5 + 0.5/2 here 2 is the dimerization or dissociation as its dissociated to 2 ions so i = 0.5 + 0.25 = 0.75 now for mass put in the equation i = observed mass/normal mass = 0.75 = x/122 = x = 91.5 so mass is 91.5 grams observed
|
Sobol quasirandom point set - MATLAB - MathWorks í•œêµ
\left\{\begin{array}{cc}0,& i=1\text{â}\\ {\mathrm{γ}}_{i}\left(1\right){v}_{j}\left(1\right)â{\mathrm{γ}}_{i}\left(2\right){v}_{j}\left(2\right)â...,& i>1.\end{array}
{\mathrm{γ}}_{i}\left(n\right)
iâ1=\underset{n=1}{â}{\mathrm{γ}}_{i}\left(n\right){2}^{nâ1}.
{\mathrm{γ}}_{i}\left(n\right)
The ⊕ operator is the bitwise exclusive-or operator. For two numbers expressed in binary, the ⊕ operator compares the digits in each position. For a given digit position, the ⊕ operator returns a 1 if the digits in that position differ and returns a 0 if the digits in that position are the same.
19â24={\left(10011\right)}_{2}â{\left(11000\right)}_{2}={\left(01011\right)}_{2}=11.
\frac{1}{2}â\frac{3}{4}={\left(0.1\right)}_{2}â{\left(0.11\right)}_{2}={\left(0.01\right)}_{2}=\frac{1}{4}.
{v}_{j}\left(n\right):=\frac{{m}_{j}\left(n\right)}{{2}^{n}}.
{\mathrm{â¤}}_{2}
{x}^{{s}_{j}}+{a}_{j}\left(1\right){x}^{{s}_{j}â1}+{a}_{j}\left(2\right){x}^{{s}_{j}â2}+...+{a}_{j}\left({s}_{j}â1\right)x+1.
The remaining direction numbers are determined by the following recurrence relation, which uses the coefficients of the primitive polynomial, the previous direction numbers, and the ⊕ bitwise exclusive-or operator.
{m}_{j}\left(n\right):=2{a}_{j}\left(1\right){m}_{j}\left(nâ1\right)â{2}^{2}{a}_{j}\left(2\right){m}_{j}\left(nâ2\right)â...â{2}^{{s}_{j}â1}{a}_{j}\left({s}_{j}â1\right){m}_{j}\left(nâ{s}_{j}+1\right)â{2}^{{s}_{j}}{m}_{j}\left(nâ{s}_{j}\right)â{m}_{j}\left(nâ{s}_{j}\right).
[1] Bratley, P., and B. L. Fox. “Algorithm 659 Implementing Sobol's Quasirandom Sequence Generator.†ACM Transactions on Mathematical Software. Vol. 14, No. 1, 1988, pp. 88–100.
[2] Hong, H. S., and F. J. Hickernell. “Algorithm 823: Implementing Scrambled Digital Sequences.†ACM Transactions on Mathematical Software. Vol. 29, No. 2, 2003, pp. 95–109.
[3] Joe, S., and F. Y. Kuo. “Remark on Algorithm 659: Implementing Sobol's Quasirandom Sequence Generator.†ACM Transactions on Mathematical Software. Vol. 29, No. 1, 2003, pp. 49–57.
[4] Kocis, L., and W. J. Whiten. “Computational Investigations of Low-Discrepancy Sequences.†ACM Transactions on Mathematical Software. Vol. 23, No. 2, 1997, pp. 266–294.
[5] Matousek, J. “On the L2-Discrepancy for Anchored Boxes.†Journal of Complexity. Vol. 14, No. 4, 1998, pp. 527–556.
|
Revision as of 03:30, 26 February 2013 by Xiphmont (talk | contribs) (→Stairsteps)
1 kHz is still a fairly low frequency, so perhaps the stairsteps are just hard to see or they're being smoothed away. Next, set the signal generator to 15kHz, which is much closer to Nyquist. Now the sine wave is represented by less than three samples per cycle, and the digital waveform appears rather poor! Yet the analog output is still a perfect sine wave, exactly like the original.
{\displaystyle \ squarewave(t)={\begin{cases}1,&|t|<T_{1}\\0,&T_{1}<|t|\leq {1 \over 2}T\end{cases}}}
{\displaystyle {\begin{aligned}\ squarewave(t)={\frac {4}{\pi }}\sin(\omega t)+{\frac {4}{3\pi }}\sin(3\omega t)+{\frac {4}{5\pi }}\sin(5\omega t)+\\{\frac {4}{7\pi }}\sin(7\omega t)+{\frac {4}{9\pi }}\sin(9\omega t)+{\frac {4}{11\pi }}\sin(11\omega t)+\\{\frac {4}{13\pi }}\sin(13\omega t)+{\frac {4}{15\pi }}\sin(15\omega t)+{\frac {4}{17\pi }}\sin(17\omega t)+\\{\frac {4}{19\pi }}\sin(19\omega t)+{\frac {4}{21\pi }}\sin(21\omega t)+{\frac {4}{23\pi }}\sin(23\omega t)+\\{\frac {4}{25\pi }}\sin(25\omega t)+{\frac {4}{27\pi }}\sin(27\omega t)+{\frac {4}{29\pi }}\sin(29\omega t)+\\{\frac {4}{31\pi }}\sin(31\omega t)+{\frac {4}{33\pi }}\sin(33\omega t)+\cdots \end{aligned}}}
|
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