text:
# Tag Info

## Hot answers tagged bugs

16

Preface: For everyone: be aware that this behaviour is very likely to change soon (the name of the hook variable flag, etc). In fact, I did change this for 10.2, where Export would fall back to the standard Export in case when the specialised hook fails. Leonid Shifrin I get the same behaviour using Mathematica version 10.1 on Windows 7/64-bit. ...

6

There are a few things to be said here. First, it should not crash and a future version will behave better in this scenario. What I am not quite sure abuot is why the call ToElementMesh[DiscretizeRegion[...]] in the first place. Note that both mesh = ToElementMesh[\[CapitalOmega]]; and DiscretizeRegion[\[CapitalOmega]]; work fine. In your example you ...

5

Just to summarize my understanding: First of all, the documentation states that Simplify assumes that variables are real when they occur algebraically in inequalities. Clearly, there are no inequalities in the logical expression y == 0 && x^2 == -1, and therefore x and y should be assumed to be general complex numbers. If they were real, then the ...

5

Apparently in version 9 listability of the CDF for TransformedDistribution, at least in this particular case, was broken. As of version 10 this is no longer the case. However, there does appear to be a bug here. Named tests like PearsonChiSquareTest actually call DistributionFitTest under the hood so I will use the former to help point to the problem. The ...

4

I will expand on Szabolcs's answer. Starting with the same triangulation, we can see that the problematic cells consist of "triangle" whose vertices are collinear. These have areas that are either negative (wrong orientation), zero, or nearly zero (below 2.*^-15). They arise no doubt from round-off error. I could not find any means of improving the ...

4

I can confirm the above incorrect results are due to a bug in the Intel MKL library shipping with Mathematica 10.1 which affects FFT convolution. The problem is only known to be triggered on some processors (for example AMD chips, or virtual machine emulated CPUs). The following workaround will use an alternative implementation that does not rely on MKL ...

3

WDX is not a good format. We are likely to deprecate it or entirely replace it with a different implementation that is not backward-compatible (which is obviously problematic). Dataset will never directly support it. There are candidates for a possible native format for dataset, like Cap'n Proto, HDF5, and a couple others. Or XML. No, just kidding :-)

3

Extended comment: $Version "10.1.0 for Mac OS X x86 (64-bit) (March 24, 2015)" eq = 0.0055356626 + 0.000029 x + 0.000034 y <= 1000.; eq // Simplify x + 1.17241 y <= 3.44826*10^7 Use of assumptions reproduces the problem: Simplify[eq, {x > 0, y > 0}] False However, use of Rationalize avoids the problem with using ... 2 No problem in version 10:$Version (* Out[3]= "10.1.0 for Microsoft Windows (64-bit) (March 24, 2015)" *) RSolve[{(k + 2) c[k + 2] q^(k + 1) - c[k] == 0}, c[k], k] \left\{\left\{c(k)\to \frac{c_1 2^{-\frac{k}{2}} \left(\frac{1}{q^2}\right)^{\frac{1}{8} k (k+2)} q^{k/2}}{\Gamma \left(\frac{k}{2}+1\right)}+\frac{c_2 (-1)^k 2^{-\frac{k}{2}} ...

1

The present state of affairs The kind of manual manipulation demonstrated by kglr should theoretically not be necessary. There may have been some problems in the initial version 10 release but in either 10.0.2 or 10.1.0 many Themes already combine correctly. Emulating his penultimate example: Plot[Evaluate@Table[BesselJ[n, x], {n, 5}], {x, 0, 10}, ...

1

I received an answer from support that it's a Manipulate bug. The following code is stable(it's working on Mathematica 10, Windows 7): co = 2.0*^8; ω = 2 Pi \[ScriptF] 10^6; τ = \[ScriptCapitalT] 10^-9; t = ts*10^-9; sol = Solve[{a + b == 1, (a E^(-I ω τ) + b E^(I ω τ)) == (a E^(-I ω τ) - b E^(I ω τ))*RL/Z0}, {a, b}]; Vi[ts_, x_, ...

1

This is in response to Chris's request in comments, but not an answer to his question. I tried running the code you asked: {ev, if, mesh} = helmholzSolve3D[\[CapitalOmega], 4, MaxCellMeasure -> 0.25] with the definition of Omega that you have in the current question, and the definition of helmholzSolve3D from ...

1

After removing the superfluous θ[0] == 0 from bc, the key to the solution is a subtle change in the use of Piecewise in the definition of Cd: Cd[x_] := Piecewise[{{Cj/Sqrt[1 - x/ϕ], x < 0}}, Cs*Exp[x/ϕ]] eqn = {V'[t] == (j[t] - Id[V[t]])/Cd[V[t]], j'[t] == (V0[t] - Rt*j[t] - V[t])/L}; bc = {V[0] == A, j[0] == 0 (*, \[Theta][0] == 0*)}; pl = ...

1

Even with Rubi's help, the result returned by Mathematica is an ugly looking elliptic integral expression with complex parameter. (Actually, even Maple's result is far from optimal as well.) Using methods similar to what I did here, here is a closed form for the OP's integral: N[(12 Sqrt[7] + 3^(3/4) InverseJacobiCN[(8 Sqrt[3] - 19)/13, (2 + Sqrt[3])/4])/5, ...

Only top voted, non community-wiki answers of a minimum length are eligible


text:
## Multiplicity and concentration results for a fractional Choquard equation via penalization method.(English)Zbl 1408.35001

The paper is focused on the nonlinear fractional Choquard equation $\varepsilon^{2s}(-\Delta)^su+V(x)u=\varepsilon^{\mu-N}\left(\frac{1}{|x|^{\mu}} * F(u)\right)f(u),\text{ in }\mathbb{R}^N,\tag{1}$ where $$\varepsilon$$ is a positive parameter, $$s\in (0,1)$$, $$N>2s$$, $$0<\mu<2s$$, the potential $$V:\mathbb{R}^N\to\mathbb{R}$$ is a positive continuous function, the nonlinearity $$f:\mathbb{R}\to\mathbb{R}$$ is a continuous function with subcritical growth, $$F(t)=\int_0^tf(\tau)$$ $$d\tau$$, and the nonlocal operator $$(-\Delta)^s$$ is the fractional Laplacian. Under some assumptions on $$V$$ and $$f$$, the authors study the multiplicity and the concentration of positive solutions of equation $$(1)$$ by using the Ljusternik-Schnirelmann theory and a Moser iteration scheme.

### MSC:

 35A15 Variational methods applied to PDEs 35B09 Positive solutions to PDEs 35R11 Fractional partial differential equations 45G05 Singular nonlinear integral equations
Full Text:

### References:

 [1] Ackermann, N., On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z., 248, 423-443, (2004) · Zbl 1059.35037 [2] Alves, C.O., Miyagaki, O.H.: Existence and concentration of solution for a class of fractional elliptic equation in $$\mathbb {R}^{N}$$ via penalization method. Calc. Var. Partial Diff. Equa. 55(3), Art. 47, 19 pp. (2016) · Zbl 1366.35212 [3] Alves, CA; Yang, M., Existence of semiclassical ground state solutions for a generalized Choquard equation, J. Differ. Equ., 257, 4133-4164, (2014) · Zbl 1309.35036 [4] Alves, CA; Yang, M., Investigating the multiplicity and concentration behaviour of solutions for a quasilinear Choquard equation via penalization method, Proc. Roy. Soc. Edinburgh Sect. A, 146, 23-58, (2016) · Zbl 1366.35050 [5] Ambrosetti, A.; Rabinowitz, PH, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14, 349-381, (1973) · Zbl 0273.49063 [6] Ambrosio, V., Multiplicity of positive solutions for a class of fractional Schrödinger equations via penalization method, Ann. Mat. Pura Appl. (4) 196, 6, 2043-2062, (2017) · Zbl 06812662 [7] Ambrosio, V., Ground states for superlinear fractional Schrödinger equations in $$\mathbb {R}^{N}$$ℝN, Ann. Acad. Sci. Fenn. Math. 41, 2, 745-756, (2016) · Zbl 1365.35196 [8] Belchior, P.; Bueno, H.; Miyagaki, OH; Pereira, GA, Remarks about a fractional Choquard equation: Ground state, regularity and polynomial decay, Nonlinear Anal., 164, 38-53, (2017) · Zbl 1373.35111 [9] Benci, V.; Cerami, G., Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology, Calc. Var. Partial Diff. Equa., 2, 29-48, (1994) · Zbl 0822.35046 [10] Caffarelli, LA; Silvestre, L., An extension problem related to the fractional Laplacian, Comm Partial Diff. Equa., 32, 1245-1260, (2007) · Zbl 1143.26002 [11] Chen, YH; Liu, C., Ground state solutions for non-autonomous fractional Choquard equations, Nonlinearity, 29, 1827-1842, (2016) · Zbl 1381.35213 [12] Coti Zelati, V.; Nolasco, M., Ground states for pseudo-relativistic Hartree equations of critical type, Rev. Mat. Iberoam., 29, 1421-1436, (2013) · Zbl 1283.35121 [13] d’Avenia, P.; Siciliano, G.; Squassina, M., On fractional Choquard equations, Math. Models Methods Appl. Sci., 25, 1447-1476, (2015) · Zbl 1323.35205 [14] Dávila, J.; del Pino, M.; Wei, J., Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Diff. Equa. 256, 2, 858-892, (2014) · Zbl 1322.35162 [15] Del Pino, M.; Felmer, PL, Local Mountain Pass for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Diff. Equa., 4, 121-137, (1996) · Zbl 0844.35032 [16] Di Nezza, E.; Palatucci, G.; Valdinoci, E., Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. math., 136, 521-573, (2012) · Zbl 1252.46023 [17] Ekeland, I., On the variational principle, J. Math. Anal. Appl., 47, 324-353, (1974) · Zbl 0286.49015 [18] Fall, MM; Fethi, M.; Valdinoci, E., Ground states and concentration phenomena for the fractional Schrödinger equation, Nonlinearity, 28, 1937-1961, (2015) · Zbl 1320.35320 [19] Felmer, P.; Quaas, A.; Tan, J., Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142, 1237-1262, (2012) · Zbl 1290.35308 [20] Frank, R.L, Lenzmann, E.: On ground states for the $$L$$2,-critical boson star equation, preprint arXiv:0910.2721 [21] He, X.; Zou, W., Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities, Calc. Var. Partial Diff. Equa., 55, paper no. 91, 39 pp, (2016) · Zbl 1395.35193 [22] Laskin, N.: Fractional Schrödinger equation, vol. 66 (2002) [23] Lieb, EH, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Stud. Appl Math., 57, 93-105, (1977) · Zbl 0369.35022 [24] Lieb, E.H., Loss, M.: Analysis, Graduate Studies in Mathematics, vol. 14. American Mathematical Society, Providence (1997). xviii+ 278 pp. [25] Lions, PL, The Choquard equation and related questions, Nonlinear Anal., 4, 1063-1072, (1980) · Zbl 0453.47042 [26] Ma, L.; Zhao, L., Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195, 455-467, (2010) · Zbl 1185.35260 [27] Mawhin, J., Willen, M.: Critical Point Theory and Hamiltonian Systems. Springer-Verlag (1989) [28] Molica Bisci, G., Rădulescu, V., Servadei, R.: Variational Methods for Nonlocal Fractional Problems, p. 162. Cambridge University Press, Cambridge (2016) · Zbl 1356.49003 [29] Moser, J., A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math., 13, 457-468, (1960) · Zbl 0111.09301 [30] Moroz, V.; Schaftingen, J., Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265, 153-84, (2013) · Zbl 1285.35048 [31] Moroz, V.; Schaftingen, J., A guide to the Choquard equation, J. Fixed Point Theory Appl., 19, 773-813, (2017) · Zbl 1360.35252 [32] Palatucci, G.; Pisante, A., Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calc. Var. Partial Diff. Equa., 50, 799-829, (2014) · Zbl 1296.35064 [33] Pekar, S.: Untersuchung uber die Elektronentheorie der Kristalle. Akademie Verlag (1954) · Zbl 0058.45503 [34] Penrose, R., Quantum computation, entanglement and state reduction, Philos. Trans. Roy. Soc., 356, 1-13, (1998) · Zbl 1152.81659 [35] Secchi, S., A note on Schrödinger-Newton systems with decaying electric potential, Nonlinear Anal., 72, 3842-3856, (2010) · Zbl 1187.35254 [36] Secchi, S., Ground state solutions for nonlinear fractional Schrödinger equations in $$\mathbb {R}^{N}$$ℝN, J. Math. Phys., 54, 031501, (2013) · Zbl 1281.81034 [37] Shen, Z.; Gao, F.; Yang, M., Ground states for nonlinear fractional Choquard equations with general nonlinearities, Math. Methods Appl. Sci., 39, 4082-4098, (2016) · Zbl 1344.35168 [38] Silvestre, L., Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60, 67-112, (2007) · Zbl 1141.49035 [39] Szulkin, A., Weth, T.: The method of Nehari manifold. In: Gao, D. Y., Montreanu, D. (eds.) Handbook of Nonconvex Analysis and Applications, pp. 597-632. International Press, Boston (2010) · Zbl 1218.58010 [40] Yang, M.; Zhang, J.; Zhang, Y., Multi-peak solutions for nonlinear Choquard equation with a general nonlinearity, Commun. Pure Appl. Anal., 16, 493-512, (2017) · Zbl 1364.35027 [41] Wei, J., Winter, M.: Strongly interacting bumps for the Schrödinger-Newton equation. J. Math. Phys., 50 (2009) [42] Willem, M.: Minimax theorems, Progress in Nonlinear Differential Equations and their Applications, vol. 24. Birkhäuser Boston, Inc., Boston (1996). x + 162 pp · Zbl 0856.49001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.

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## Mimi_x3 Group Title Projectile Motion: Two particles are projected simultaneously from $$A$$ with the same velocity $$V$$ at the angles of elevation $$\alpha$$, $$\beta$$ on a horizontal plane respectively. If they both pass through $$B$$, one after time $$t_1$$ and the other after time $$t_2$$. (a) show that $$\alpha+ \beta\=\frac{\pi}{2}$$ (b) Prove that $$4V^2 = g^2 (t_{1}^{2} + t_{2}^{2}$$ Will draw diagram below one year ago one year ago

1. Mimi_x3 Group Title

|dw:1344001551890:dw|

2. Mimi_x3 Group Title

well how do i start? anyone able to give me a hint?

3. Vaidehi09 Group Title

proving part b is simple. use the formula for time of flight of a projectile: T = 2vsinx/g......for both the projectiles. so u'll have t1 = 2vsinA/g.........[A = alpha]............(i) t2 = 2vsinB/g.........[B = beta]..............(ii) now, the RHS of ur question has t1^2 + t2^2..........so do exactly that. first square both (i) and (ii) and then add those. and then simplify and arrange the terms as required in the question.

4. Mimi_x3 Group Title

ok i got it. how about a?

5. Mimi_x3 Group Title

for b its the touchdown velocity right?

6. Vaidehi09 Group Title

still working on that one..i just remember that if the range is same for 2 projectiles then their angles of projection are complementary. i don't remember the proof though. so trying to derive that. and for b, its the initial velocity....given as V.

7. Ishaan94 Group Title

Range is same. $\frac{u^2 \sin2\alpha}{g} = \frac{u^2\sin 2\beta }{g}\implies \sin 2\alpha = \sin2\beta$ $\implies 2\alpha = 2\beta\;or\; \pi - 2\alpha = 2\beta \implies \frac{\pi}2 = \alpha +\beta$

8. Mimi_x3 Group Title

thank you!


text:
# AR Controller¶

Picture

The AR Controller is used to control your AR Goggles wirelessly. You can draw anything you wish into an overlay that will be visible as long as you're wearing the goggles.

Hint

To link your goggles to an AR Controller, right click it with them in your hand. Multiple Goggles can be linked to one Controller.

Bug

The AR Goggles are currently slightly buggy. Sometimes, they just don't do what they should do. We will rework the AR System in 0.8r/1.0r

Peripheral Name Interfaces with Has events Introduced in
arController AR Goggles No 0.5b

While the Controller is in relative mode, it interprets all coordinates as if they were on a virtual screen of the size you specified, and then scales them according to your screen size.

All color values are hexadecimal color codes (for example 0xff00ff)

Everything that is painted onto the canvas remains there until clear() or if the element is removed using clearElement(id), however, it might be repositioned if relative mode is toggled on or off.

## Functions¶

clear() -> void

Clears the entire canvas.

### clearElement¶

clearElement(id: string) -> void

Clears the element with the given id.

### horizontalLine¶

horizontalLine(minX: number, maxX: number, y: number, color: number) -> void

Draws a horizontal line in the given color from minX to maxX at vertical y.

### horizontalLineWithId¶

horizontalLineWithId(id: string, minX: number, maxX: number, y: number, color: number) -> void

The same as horizontalLine(), but has an id so it can be overridden later on or can be completely cleared.

### verticalLine¶

verticalLine(x: number, minY: number, maxY: number, color: number) -> void

Draws a vertical line in the given color from minY to maxY at horizontal x.

### verticalLineWithId¶

verticalLineWithId(id: string, x: number, minY: number, maxY: number, color: number) -> void

The same as verticalLine(), but has an id so it can be overridden later on or can be completely cleared.

### drawString¶

drawString(text: string, x: number, y: number, color: number) -> void

Draws the given string to the specified position with the specified text color.

### drawStringWithId¶

drawStringWithId(id: string, text: string, x: number, y: number, color: number) -> void

The same as drawString(), but has an id so it can be overridden later on or can be completely cleared.

### drawCenteredString¶

drawCenteredString(text: string, x: number, y: number, color: number) -> void

The same as drawString(), but centers the string horizontally around the specified position.

### drawCenteredStringWithId¶

drawCenteredStringWithId(id: string, text: string, x: number, y: number, color: number) -> void

The same as drawCenteredString(), but has an id so it can be overridden later on or can be completely cleared.

### drawRightboundString¶

drawRightboundString(text: string, x: number, y: number, color: number) -> void

The same as drawString(), but the string is positioned with its right end at the specified position.

### drawRightboundStringWithId¶

drawRightboundStringWithId(id: string, text: string, x: number, y: number, color: number) -> void

The same as drawRightboundString(), but has an id so it can be overridden later on or can be completely cleared.

### drawItemIcon¶

drawItemIcon(itemId: string, x: number, y: number) -> void

Draws the given item to the specified position.

### drawItemIconWithId¶

drawItemIconWithId(id: string, itemId: string, x: number, y: number) -> void

The same as drawItemIcon(), but has an id so it can be overridden later on or can be completely cleared.

### drawCircle¶

drawCircle(x: number, y: number, radius: number, color: number) -> void

Draws a circle without filling it.

### drawCircleWithId¶

drawCircleWithId(id: string, x: number, y: number, radius: number, color: number) -> void

The same as drawCircle(), but has an id so it can be overridden later on or can be completely cleared.

### fill¶

fill(minX: number, minY: number, maxX: number, maxY: number, color: number)

Fills a rectangle with the given color from the corner minX, minY to maxX, maxY.

### fillWithId¶

fill(id: string, minX: number, minY: number, maxX: number, maxY: number, color: number)

The same as fill(), but has an id so it can be overridden later on or can be completely cleared.

### fillCircle¶

fillCircle(x: number, y: number, radius: number, color: number) -> void

Draws a full circle.

### fillCircleWithId¶

fillCircleWithId(id: string, x: number, y: number, radius: number, color: number) -> void

The same as fillCircle(), but has an id so it can be overridden later on or can be completely cleared.

fillGradient(minX: number, minY: number, maxX: number, maxY: number, colorFrom: number, colorTo: number) -> void

Draws a rectangular gradient from colorFrom to colorTo with the given corners.

fillGradientWithId(id: string, minX: number, minY: number, maxX: number, maxY: number, colorFrom: number, colorTo: number) -> void

The same as fillGradient(), but has an id so it can be overridden later on or can be completely cleared.

### isRelativeMode¶

isRelativeMode(): true, number, number | false

Returns true and the size of the virtual screen if relative mode is active, or just false if it isn't.

### setRelativeMode¶

setRelativeMode(enabled: boolean, virtualScreenWidth?: number, virtualScreenHeight?: number) -> void

Activates or deactivates relative mode. Requires virtual screen width and height if it is being enabled.

Hint

Use negative coordinates to specify an x value from the right end of the screen or a y value from the bottom!

## Examples¶

### Example 1¶

Olfi01 made a simple script that shows the current date and time in the top right corner of the screen and updates every second.

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 local controller = peripheral.find("arController") -- Finds the peripheral if one is connected if controller == nil then error("arController not found") end controller.setRelativeMode(true, 1600, 900) -- Convenient Aspect ratio for most screens while true do local timer = os.startTimer(1) local event, id repeat event, id = os.pullEvent("timer") until id == timer controller.clear() -- If you don't do this, the texts will draw over each other controller.drawRightboundString(os.date(), -10, 10, 0xffffff) end

### Example 2¶

Olfi01 made another script to draw .nfp files, which you can draw with the paint program, in cc in your HUD.

First, we have a script with more adaptability. You can define x, y, width and height.

Script 1: Github

And we have another script which depends on the script above, but is simpler to use.

Script 2: Github

## Changelog/Trivia¶

0.5.2b
Added fillCircle, drawCircle and drawItemIcon.

0.5b


text:
It is currently 20 Apr 2019, 22:50

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# If the average (arithmetic mean) of two numbers is 35 and o

Author Message
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Kudos [?]: 1199 [0], given: 5709

If the average (arithmetic mean) of two numbers is 35 and o [#permalink]  19 Jun 2017, 12:01
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Question Stats:

100% (00:26) correct 0% (00:00) wrong based on 13 sessions

If the average (arithmetic mean) of two numbers is 35 and one of the numbers is k, then what is the other number in terms of k ?

A) 35 + k

B) 35 – k

C) 35 – 2k

D) 70 – k

E) 70 – 2k
[Reveal] Spoiler: OA

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Kudos [?]: 355 [1] , given: 66

Re: If the average (arithmetic mean) of two numbers is 35 and o [#permalink]  20 Sep 2017, 05:14
1
KUDOS
The average between k and an unknown number x is $$frac{k+x}{2}=35$$. Thus rearranging is straightforward to find that $$x=70-k$$. Answer D!
Re: If the average (arithmetic mean) of two numbers is 35 and o   [#permalink] 20 Sep 2017, 05:14
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# If the average (arithmetic mean) of two numbers is 35 and o

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text:
Journal of Symbolic Logic

The Deduction Rule and Linear and Near-Linear Proof Simulations

Abstract

We introduce new proof systems for propositional logic, simple deduction Frege systems, general deduction Frege systems, and nested deduction Frege systems, which augment Frege systems with variants of the deduction rule. We give upper bounds on the lengths of proofs in Frege proof systems compared to lengths in these new systems. As applications we give near-linear simulations of the propositional Gentzen sequent calculus and the natural deduction calculus by Frege proofs. The length of a proof is the number of lines (or formulas) in the proof. A general deduction Frege proof system provides at most quadratic speedup over Frege proof systems. A nested deduction Frege proof system provides at most a nearly linear speedup over Frege system where by "nearly linear" is meant the ratio of proof lengths is $O(\alpha(n))$ where $\alpha$ is the inverse Ackermann function. A nested deduction Frege system can linearly simulate the propositional sequent calculus, the tree-like general deduction Frege calculus, and the natural deduction calculus. Hence a Frege proof system can simulate all those proof systems with proof lengths bounded by $O(n \cdot \alpha(n))$. Also we show that a Frege proof of $n$ lines can be transformed into a tree-like Frege proof of $O(n \log n)$ lines and of height $O(\log n)$. As a corollary of this fact we can prove that natural deduction and sequent calculus tree-like systems simulate Frege systems with proof lengths bounded by $O(n \log n)$.

Article information

Source
J. Symbolic Logic, Volume 58, Issue 2 (1993), 688-709.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183744256

Mathematical Reviews number (MathSciNet)
MR1233933

Zentralblatt MATH identifier
0848.03028

JSTOR


text:
# How can I make this graph look better?

Here's what my graph looks like:

I have several issues with this:

1. I expected the external edges to have similar sizes. As you can see, the ones on the sides are much longer than the ones on the top/bottom. Is there a way to make them even, and the graph look more "circular"?
2. The nodes are of different sizes. The one in the middle is way too big. Is there a way to make them the same size (possibly make all the labels smaller, so that the nodes aren't really big)?
3. The labels on the edges are placed almost randomly. Is there a way to make this more consistent?

My code:

\begin{tikzpicture}[->,node distance=3cm,auto,
main node/.style={circle,draw}]

\node[main node] (0) {$$f1, f2, f3$$};
\node[main node] (1) [above of=0] {$$f1, f2$$};
\node[main node] (2) [above right of=0] {$$f2$$};
\node[main node] (3) [below right of=0] {$$f2, f3$$};
\node[main node] (4) [below of=0] {$$f3$$};
\node[main node] (5) [below left of=0] {$$f1, f3$$};
\node[main node] (6) [above left of=0] {$$f1$$};

\path[every node/.style={font=\sffamily\small}]
(0) edge node {$$f3$$} (1)
(0) edge node {$$f1$$} (3)
(0) edge node {$$f2$$} (5)
(1) edge node {$$f1$$} (2)
(1) edge node {$$f2$$} (6)
(3) edge node {$$f3$$} (2)
(3) edge node {$$f2$$} (4)
(5) edge node {$$f1$$} (4)
(5) edge node {$$f3$$} (6);

\end{tikzpicture}
• Welcome to TeX SX! Did you try with tikz-cd for the more circular look? Oct 11 '19 at 12:42
• There are graph drawing libraries that have circular layouts, e.g. the examples on p. 266 of the pgfmanual.
– user194703
Oct 11 '19 at 12:47
• Thanks. I'm (very) new to LaTeX so I'm not familiar with any of those libraries. Can you point me to a relevant example?
– Mr F
Oct 11 '19 at 12:59

\documentclass[tikz,margin=3mm]{standalone}
\usetikzlibrary{arrows.meta,
quotes}

\begin{document}
\begin{tikzpicture}[
MN/.style = {%Main Node
circle, draw, minimum size=15mm, font=\footnotesize,
inner sep = 0pt, outer sep=0pt},
every edge/.style = {draw, -Stealth}
]
\node[MN] (mn0)             {$f1, f2, f3$};
\node[MN] (mn1) at ( 30:3)  {$f1, f2$};
\node[MN] (mn2) at ( 90:3)  {$f2$};
\node[MN] (mn3) at (150:3)  {$f2, f3$};
\node[MN] (mn4) at (210:3)  {$f3$};
\node[MN] (mn5) at (270:3)  {$f1, f3$};
\node[MN] (mn6) at (330:3)  {$f1$};

\foreach \i in {1,3,5}
\path   (mn0) edge["$f3$"]    (mn\i);
\path   (mn1) edge["$f1$" ']  (mn2)
(mn1) edge["$f2$"]    (mn6)
(mn3) edge["$f3$"]    (mn2)
(mn3) edge["$f2$" ']  (mn4)
(mn5) edge["$f1$"]    (mn4)
(mn5) edge["$f3$" ']  (mn6);
\end{tikzpicture}
\end{document}

or with added style for edge labels:

every edge quotes/.style = {font=\footnotesize, inner sep=2pt, sloped, auto}

• Beautiful, thank you!
– Mr F
Oct 11 '19 at 13:07

Just for fun: using TikZ modulo arithmetic. Helps to avoid mislabeling (if I understand the logic, of course).

\documentclass[tikz,border=3mm]{standalone}
\begin{document}
\begin{tikzpicture}[circ/.style={circle,draw,minimum size=4em}]
\node[circ] (n0) {$f_1,f_2,f_3$};
\foreach \X [evaluate=\X as \NextX using {int(1+mod(\X,3))},
evaluate=\X as \PrevX using {int(1+Mod(\X-2,3))}] in {1,2,3}
{
\path (-150+\X*120:3) node[circ] (n\the\numexpr2*\X-1)
{$\ifnum\PrevX<\NextX f_\PrevX,f_\NextX \else f_\NextX,f_\PrevX \fi$}
(-90+\X*120:3) node[circ] (n\the\numexpr2*\X) {$f_\PrevX$}
;
\draw[-stealth] (n0) -- (n\the\numexpr2*\X-1) node[midway,auto]{$f_\X$};
}
\foreach \X [evaluate=\X as \NextX using {int(1+mod(\X,3))},
evaluate=\X as \PrevX using {int(1+Mod(\X-2,3))}] in {1,2,3}
{
\draw[-stealth]  (n\the\numexpr2*\X-1) -- (n\the\numexpr2*\X)
node[midway,auto]{$f_\NextX$};
\ifnum\X=1
\draw[-stealth]  (n\the\numexpr2*\X-1) -- (n6)
node[midway,auto]{$f_\PrevX$};
\else
\draw[-stealth]  (n\the\numexpr2*\X-1) -- (n\the\numexpr2*\X-2)
node[midway,auto]{$f_\PrevX$};
\fi
}
\end{tikzpicture}
\end{document}


text:
# Calculate the density of silver which crystallizes in the face centred cubic structure. The distance between the nearest silver atoms in this structure is 287 pm. (molar mass of silver = $108.87 g mol^{−1}$).

$10.73gcm^{-3}$


text:
# Math Help - D.e mixing problem

1. ## D.e mixing problem

300 gallon tank contains 100 gallons of brine with a concentration of 1 pound of salt per gallon.a brine containing one half point of salt per gallon runs into the tank at rate of four gallons per minute nad the well stirred mixture drains from the tank at the same rate .when is the concentration in the tank 0.7 pound per gallon .

2. Let c be the concentration. Then c changes according to:

$\frac{dc}{dt}=\frac{\text{salt entering}-\text{salt leaving}}{100\text{ gallons}}$

The amount of salt entering the tank is $\frac{1}{2}*4$ pounds per minute.

The amount of salt leaving the tank is c*4 pounds per minute.

So you solve the differential equation and apply the initial condition c(0)=1 pound per gallon to evaluate the constant of integration, then set c(t)=0.7 and solve for t.

Post again in this thread if you are still having trouble.

- Hollywood


text:
# Python Basics¶

## Unit 3, Lecture 2¶

Numerical Methods and Statistics

# Floating Point Representation¶

Floating point numbers are represented with a mantissa and exponent: $$\underbrace{0.3423}_{\textrm{mantissa}}\times{}\underbrace{10^{-2}}_{\textrm{exponent}}$$

We are working with computers though, so the mantissa is a binary and the exponent is in powers of 2. Let's look at them as decimals to begin though.

In [1]:
from IPython.display import Math
from math import frexp, pi
import math

#Convert a float into its mantissa and exponent and print as LaTeX
def fprint(x):
m,e = frexp(x)
return Math('{:4} \\times 2^{{{:}}}'.format(m, int(e)))

#Convert a mantissa from decimal to binary and print as LaTeX
def ffrac_ltx(x, terms=10):
bits = []
exp = 1
latex = ''
while x > 0 and exp < terms:
bits.append(int(x / 2.0 ** -exp))
x -= bits[-1] * 2.0 ** -exp
latex += '\\frac{{ {} }}{{ 2^{{ {} }} }} + '.format(bits[-1], exp)
exp += 1
return Math(latex[:-3] + ' = {}'.format(sum([b * 2**-i for b,i in zip(bits,range(1,exp))])))

def ffrac(x, terms=10):
bits = []
exp = 1
while x > 0 and exp < terms:
bits.append(int(x / 2.0 ** -exp))
x -= bits[-1] * 2.0 ** -exp
exp += 1
return ''.join(['{}'.format(x) for x in bits])
In [2]:
fprint(4.3)
Out[2]:
$\displaystyle 0.5375 \times 2^{3}$
In [3]:
fprint(0.1)
Out[3]:
$\displaystyle 0.8 \times 2^{-3}$
In [4]:
fprint(pi)
Out[4]:
$\displaystyle 0.7853981633974483 \times 2^{2}$

A 'decimal' in binary is challenging to think about, because each integer location is a $1 / 2^{n}$, where $n$ is the location. That means to represent a binary-mantissa exacly, its denominator must be a power of $2$. For example:

In [5]:
ffrac_ltx(0.5)
Out[5]:
$\displaystyle \frac{ 1 }{ 2^{ 1 } } = 0.5$
In [6]:
ffrac_ltx(0.75)
Out[6]:
$\displaystyle \frac{ 1 }{ 2^{ 1 } } + \frac{ 1 }{ 2^{ 2 } } = 0.75$
In [7]:
ffrac_ltx(0.5232)
Out[7]:
$\displaystyle \frac{ 1 }{ 2^{ 1 } } + \frac{ 0 }{ 2^{ 2 } } + \frac{ 0 }{ 2^{ 3 } } + \frac{ 0 }{ 2^{ 4 } } + \frac{ 0 }{ 2^{ 5 } } + \frac{ 1 }{ 2^{ 6 } } + \frac{ 0 }{ 2^{ 7 } } + \frac{ 1 }{ 2^{ 8 } } + \frac{ 1 }{ 2^{ 9 } } = 0.521484375$
In [8]:
#This is the maximum number of bits used in the 0.1 Mantissa as a string.
print(ffrac(0.1, 100))
0001100110011001100110011001100110011001100110011001101
In [9]:
print('{:0.28f}'.format(0.1))
0.1000000000000000055511151231
In [10]:
0.1 + 0.2 == (1.0 + 2.0) / 10.0
Out[10]:
False

In [11]:
print(abs(-43))
43
In [12]:
x = -5
x = abs(x)
print(x)
5

# Imports and Modules¶

Try typing math.fa and hit TAB in ipython notebook. You should see suggestions. Next, type math.fabs( and hit TAB and ipython notebook will tell you what the function does.

In [13]:
from math import fabs
print(fabs(-4.34322))
4.34322
In [14]:
from math import fabs, sin, cos
print(fabs(sin(4) * cos(43.)))
0.4201111317694318
In [15]:
from math import *
print(e)
2.718281828459045

# Boolean Logic¶

In [16]:
print(2 == 5)
False
In [17]:
a = 3
print(a == 3)
True
In [18]:
print(a <= 0)
False
In [19]:
print(a > 0 and a < 4)
True

Boolean Logic uses Short-Circuiting - only as much as necessary is evaluated.

In [20]:
print(True or 1 / 0.)
True
In [21]:
print(True and 1 / 0.)
---------------------------------------------------------------------------
ZeroDivisionError                         Traceback (most recent call last)
----> 1 print(True and 1 / 0.)

ZeroDivisionError: float division by zero

# Flow with Boolean¶

You can make code only accessible if a condition is true with if. You can also use else and elif for alternatives.

To indicate where your if code is located, press tab (which adds 4 spaces).

In [22]:
x = 65
if x > 43:
print('Surprise! {} is greater than 43'.format(x))
Surprise! 65 is greater than 43
In [23]:
x = 54
if x < 0:
print('{} is negative'.format(x))
elif x > 0:
print('{} is positive'.format(x))
else:
print('{} is 0'.format(x))
54 is positive
In [24]:
x = -32
if x < 0:
if abs(x) > 5:
print('{} is less than 0 and has magnitude greater than 5'.format(x))
else:
print('{} is not interesting'.format(x))

#or you can do:
if x < 0 and abs(x) > 5:
print('{} is less than 0 and has magnitude greater than 5'.format(x))
if x < 0 and not abs(x) > 5:
print('{} is not interesting'.format(x))
-32 is less than 0 and has magnitude greater than 5
-32 is less than 0 and has magnitude greater than 5

# Full list of boolean operators¶

Here are the boolean operators: >, <, >=, <=, ==, !=, is

Recall our keywords from Lecture 1: and, or, and not

is is used to see if a variables is None or if two variables point to the same memory address.

In [25]:
a = 28238
b = 28238
print(a is b, a == b)
False True
In [26]:
a = b
print(a is b)
True

is None is special case. None is used as a "sentinel" value to indicate if something is not ready or invalid.

In [27]:
var = None
print(var is None)
True
In [28]:
a = 43
b = 23
print(a != b)
True

# Default Boolean - Don't Rely on this!¶

Just to be aware of, Python has a way of checking if an object is False based on its value (0 or None).

In [29]:
x = 1
if(x):
print('True')
else:
print('False')
True
In [30]:
x = None
if(x):
print('True')
else:
print('False')
False
In [31]:
x = 'Hello'
if(x):
print('True')
else:
print('False')
True
In [32]:
x = ''
if(x):
print('True')
else:
print('False')
False
In [33]:
x = math.sin(0.0)
if(x):
print('True')
else:
print('False')
False

It is better to just be explicit and give a full boolean expression. You'll most often see these used by accident

In [34]:
x = 0.15 + 0.15
y = 0.1 + 0.2
print(x == y)
False

# The solution?¶

Only use <, >. Never use == with boolean

In [35]:
x = 0.15 + 0.15
y = 0.1 + 0.2
abs(x - y) < 10**-8
Out[35]:
True

# Lists¶

In python, groups of values can be stored together into lists, which are like groups of variables.

In [36]:
numbers = [4,7,24,11,2]
print(numbers)
[4, 7, 24, 11, 2]

Individual elements can be accessed with []. The thing that goes inside the brackets is called the index. Recall that python starts counting from 0

In [37]:
print(numbers[0], numbers[1])
4 7

## Slicing Lists¶

You can access multiple elements with the : symbol, which is called slicing.

In [38]:
print(numbers[0:3])
[4, 7, 24]

The last element to the slice operator is not inclusive, so 0:3 means 0, 1, 2. Why? Becuase you expect $3 - 0$ elements, which is three, starting from the $0$ index.

In [39]:
numbers[2:5]
Out[39]:
[24, 11, 2]

If it's more convienent, you can count backwards from the end of the list by using negative indices. The trick is that -0 is not really a thing, so the last element in the list is -1.

In [40]:
print(numbers)
print(numbers[-1], numbers[-2])
[4, 7, 24, 11, 2]
2 11

The index arguments to a slice are optional.

In [41]:
print(numbers[:4])
[4, 7, 24, 11]
In [42]:
print(numbers)
print(numbers[:-1])
[4, 7, 24, 11, 2]
[4, 7, 24, 11]
In [43]:
print(numbers[3:])
[11, 2]
In [44]:
print(numbers[-3:])
[24, 11, 2]

You can add a third argument by adding another : It is the step-size.

In [45]:
print(numbers)
print(numbers[0:-1:2])
[4, 7, 24, 11, 2]
[4, 24]

The other slicing indices are optional.

In [46]:
print(numbers[0::2])
[4, 24, 2]
In [47]:
print(numbers)
print(numbers[::2])
[4, 7, 24, 11, 2]
[4, 24, 2]
In [48]:
print(numbers[2::2])
[24, 2]

Using a negative stepsize allows you to count downwards

In [49]:
print(numbers[4:0:-1])
[2, 11, 24, 7]

Note that we still include the first slice index (4), but not the second one (0).

A common idiom is to reverse the order of list this way:

In [50]:
print(numbers, numbers[::-1])
[4, 7, 24, 11, 2] [2, 11, 24, 7, 4]

Notice that Python was clever and knew with a step-size of -1, we actually wanted to start at the far end and count downwards.

# List Functions¶

Generally, most functions (like cos) do not take a list. One common function that we need is len, which returns the length of a list.

In [51]:
x = range(100)
len(x)
Out[51]:
100
In [52]:
x = range(10)
print(max(x))
9
In [53]:
import math
math.cos(x)
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-53-87cdf88e21ce> in <module>
1 import math
----> 2 math.cos(x)

TypeError: must be real number, not range


text:
]]>
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# Sequences and Series

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Have a doubt? Clear it now.
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format_list_bulleted Take this Lesson Test

#### Sequences and Series - Lesson Summary

When a collection of objects is listed in a sequential manner or a definite order such that every member has a definite position, that is, it comes either before or after, every other member, it is called a sequence.

A collection of numbers arranged in a definite order according to some definite rule is called a sequence.

The members or numbers that are listed in the sequence are called its terms.

The terms of a sequence are denoted by a1, a2, a3,…,an,…

The nth term of a sequence is denoted by an. an is also known as the general term of the sequence.

Finite or infinite sequence:

If the number of terms in a sequence is finite or countable, then it is called a finite sequence.

If the sequence goes on forever or has an uncountable number of terms, then it is called an infinite sequence.

Ex: 1, 2, 3, 4, ... is an infinite sequence. 2, 4, 6, 8, 10 is a finite sequence.

Terms of a sequence can be expressed by an algebraic formula.

Consider the sequence 1/2, 1/4, 1/8, 1/16,....

The first term can be written as a1 = 1/2 = 1/21;

The second term can be written as a2 = 1/4 = 1/22;

The third term can be written as a3 = 1/8 = 1/23 ;

The fourth term can be written as a4 = 1/16 = 1/24, and so on.

nth term of this sequence can be expressed as an = 1/2n, where n is a natural number.

Any term of the sequence can be found by substituting the term value in place of n.

e.g. a8 = 1/28 = 1/256

In some cases, an arrangement of numbers in a sequence like {1,1, 3, 7, 17, 41, ………} has no apparently visible pattern, but the terms of this sequence are generated by the relation given by
a1 = a2 =1

a3 = a1 + 2a2

a4 = a2 + 2a3, and so on.

The nth term of this sequence is an = an-2 + 2an-1, n>2.

Sequence {1, 1, 2, 3, 5, 8 …} can be described as:

a1 = a2 =1;

a3 = a1 + a2;

a4 = a2 + a3 and an = an-2 + an-1, n >2.

This sequence is known as the Fibonacci sequence.

Sometimes, it is practically impossible to express the terms of a sequence by means of a particular algebraic formula. Such sequences can only be described verbally, like the sequence of prime numbers {2,3,5,7,………..}.

A sequence is a collection of all images of a function, x, from N or any subset of N to X, where N is the set of all natural numbers and X is any set, may be of objects, events or numbers.

Domain of function x = N

{ x(1), x(2), x(3),  x(4), …} Or {x1,x2,x3,x4,...}, where  x(1)= x1, x(2)= x2, x(3)= x3, x(4) = x4 ,

Sequences are also called progressions because the terms in a sequence progress in a definite manner obeying some specific algebraic rule.

Series

Let {a1, a2, a3, a4,...} be a sequence.

Then the expression a1 + a2 + a+ a4 + ... + an + ... is called the series corresponding to the given sequence.

The expression obtained on adding the terms of a sequence is called a series.

Here, a1 is called the first term, a2 the second term… and an the nth term of the series.

A series is finite if the given sequence is finite, and infinite if the given sequence is infinite.

In summation notation, the infinite series a1 + a2 + a+ a4 + ... + an + ... can be abbreviated as ∑n=1an .

Similarly, in summation notation, the finite series a1 + a2 + a+ a4 + ... + an + ... can be written as ∑k=1ak

n=1an or ∑k=1ak are known as the compact forms of a series.


text:
# Homework Help: Finding the capcitance of this capacitator

1. Apr 24, 2009

### Dell

given the capacitator C1 filled with dialectric subtances, which change linearly from ε1 to ε2, what is the capacitance of C1?

since i know that ε changes linerly, according to the distance between the plates, and i know that at x=0 ε=ε1 and x=d ε=ε2

εr=ax+b
εr1=a*0+b
b=ε1
εr2=a*d+ε1
a=(ε21)/d

εr=(ε21)x/d+ε1
εr=((ε21)x+ε1d)/d

know this capacitator C1 is like millions of tiny little capacitators, dC, each with dialectric substance changing like εr=((ε21)x+ε1d)/d, all connected in a column,

1/Cp=$$\int$$d/(Aε0εr)d(εr) from (ε1 to ε2)

=d/(Aε0)$$\int$$d(d/((ε21)x+ε1d))

=d2/(Aε0)$$\int$$dx/((ε21)x+ε1d) (from 0 to d)

=d2/(Aε0)*1/(ε21)* ln((ε21)x+ε1d)|from 0 to d

=[d2ln(ε21)]/ε021)Aε0

but the correct answer is meant to be 1/Cp=[dln(ε21)]/ε021)Aε0

can anyone see whre i went wrong?

2. Apr 25, 2009

### Redbelly98

Staff Emeritus
While I don't exactly follow what you are doing, it appears that your extra factor of d first appears here.

Just as a general rule, I wouldn't name a variable d if integration will be involved in solving the problem. Too easy to lose track of which is the variable d, and which is used to denote the variable of integration.

3. Apr 25, 2009

### Dell

thats exactly right, the problem comes from the fact that i already have one 'd' and my εr is also dependant on 'd' εr=((ε2-ε1)x+ε1d)/d and thats where the second factor comes from, but i cannot see why this should be incorrect mathematically.

4. Apr 25, 2009

### Redbelly98

Staff Emeritus
I would have set up the integral differently to begin with. What is 1/C for a slab of thickness dx? (d→differential, not capacitor thickness here) Then add them by doing the appropriate integral, as you are aware. It's not clear to me how you got the first integral you wrote.

p.s. there is a problem with the "correct" answer you quoted,

1/Cp=[d ln(ε21)] / [ε021) A ε0],

because the units are inconsistent.

5. Apr 25, 2009

### Dell

and the units of my answer??

6. Apr 25, 2009

### Redbelly98

Staff Emeritus
It would be beneficial for you to work that out. They should come out to be 1/F.

7. Apr 25, 2009

### Redbelly98

Staff Emeritus
I'll offer this hint:

The quantity d / (A ε0) has the correct units of 1/Farads. So any remaining terms should, overall, be unitless.

8. Apr 25, 2009

### Dell

d / (A ε0) = 1/F

d2ln(ε2/ε1)]/(ε2-ε1)Aε0=d/(A ε0)* (d*ln(ε2/ε1)/(ε2/ε1))

what would the units of (d*ln(ε2/ε1)/(ε2/ε1)) be

9. Apr 25, 2009

### Redbelly98

Staff Emeritus
The ε's are all unitless, leaving ... ?

10. Apr 25, 2009

### Dell

so i get m/F,

but you said that there was a problem with the "correct " answers units, as far as i see they come to 1/F

1/Cp=[dln(ε2/ε1)]/(ε2-ε1)Aε0

i just saw that i wrote
1/Cp=[dln(ε2/ε1)]/ε0(ε2-ε1)Aε0
and there is an extra ε0, but you said that they are unitless

11. Apr 25, 2009

### Redbelly98

Staff Emeritus
e0 does have units, e1 and e2 do not.

So, the "correct" answer has the correct units afterall.

Last edited: Apr 25, 2009
12. Apr 26, 2009

### Dell

so how would you have solved it?
the capacitance you spoke of would be

1/C=dx/(ε0εrA)
=dx/(ε0((ε2-ε1)x+ε1D)/D)A)

i used D for the distance

= D/(ε0A)∫dx/((ε21)x+ε1D)
=D/(ε0A(ε21)) * [ln(ε21)]
-----------------------------------------------------------------------------------

if i would not have taken 1/C but instead C, why would i not reach the correct answer,

13. Apr 26, 2009

### Redbelly98

Staff Emeritus
The slabs of thickness dx act as many capacitors in series.

For capacitors in series, we add the reciprocals of capacitance to get total capacitance.

14. Apr 26, 2009

### Redbelly98

Staff Emeritus
Oh, I forgot to say in my last post ... I would do exactly what you did here:


text:
# brutus_ppxf module¶

brutus: a set of Python modules to process datacubes from integral field spectrographs.

This file contains functions to call ppxf to fit the continuum in a given spectra. It is based on the examples provided by M. Cappellari within ppxf istelf (e.g. ppxf_kinematics_example_sauron.py & ppxf_population_example_sdss.py), but has been modified to account for a wavelength dependant spectral resolution of the data.

Basically, this is designed with MUSE in mind.

Created February 2016, F.P.A. Vogt - frederic.vogt@alumni.anu.edu.au

brutus_ppxf.ppxf_population(specerr, templates=None, velscale=None, start=None, goodpixels=None, plot=False, moments=4, degree=-1, vsyst=None, clean=False, mdegree=10, regul=None)[source]

Function that calls ppxf, designed to be called from multiprocessing (i.e. only require 1 argument). Everything else is taken care of outside of this.

brutus_ppxf.setup_spectral_library(velscale, inst, sl_name, fit_lims, z)[source]

This prepares the set of stellar templates to be fed to ppxf.

Param: velscale - the velocity scale for the log-rebinning inst - the instrument that took the data, e.g. ‘MUSE’ sl_name - the name of the stellar library to use fit_lims - the spectral range for the fit (data & templates overlap) z - redshift of target - needed for correct reshaping of fwhm_target. ...lots of stuf...
brutus_ppxf.sl_resample(fn, z=0, inst='MUSE', sl_name='MILES_ppxf_default', do_cap=True)[source]

This function resamples a given spectrum (i.e. from a stellar library) to match theresolution of the instrument that took the data.

Param: fn: filename of stellar template z: redshift of the target observations inst: which instrument are we using sl_name: which templates do we use ? do the convolution using M.Cappellari in-built function

Returns the convolved spectrum


text:
# zbMATH — the first resource for mathematics

## Ioffe, Alexander Davidovich

Compute Distance To:
 Author ID: ioffe.alexander-d Published as: Ioffe, A.; Ioffe, A. D.; Ioffe, Aleksandr D.; Ioffe, Alexander; Ioffe, Alexander D.; Ioffe, Alexandre D. Homepage: http://www2.math.technion.ac.il/~ioffe/ External Links: MGP · Math-Net.Ru · Wikidata · dblp · GND
 Documents Indexed: 167 Publications since 1966, including 9 Books Biographic References: 3 Publications
all top 5

#### Co-Authors

 104 single-authored 10 Tikhomirov, Vladimir Mikhaĭlovich 7 Drusvyatskiy, Dmitriy 7 Lewis, Adrian S. 7 Schwartzman, Efim 6 Roberto, Lucchetti E. 4 Bonnans, Joseph Frédéric 4 Revalski, Julian Petrov 3 Fabian, Marián J. 3 Levin, Vladimir L’vovich 3 Penot, Jean-Paul 3 Reich, Simeon 2 Malanowski, Kazimierz D. 2 Sekiguchi, Yoshiyuki 2 Shafrir, Itai 2 Shapiro, Alexander 2 Tröltzsch, Fredi 2 Yudin, David Borisovich 1 Bodneva, Valeriya L. 1 Boltyanskij, Vladimir Grigor’evich 1 Borwein, Jonathan Michael 1 Cibulka, Radek 1 Dikusar, V. V. 1 Dmitruk, Andrei V. 1 Fabian, Martin 1 Freddi, Lorenzo 1 Gel’fand, Israil’ Moiseevich 1 Ginsburg, Boris 1 Kazhdan, Ya. M. 1 Khenkin, Gennadiĭ Markovich 1 Marcus, Moshe M. 1 Miłosz, T. 1 Osmolovskiĭ, Nikolaĭ Pavlovich 1 Outrata, Jiří V. 1 Rockafellar, Ralph Tyrrell 1 Rubinov, Alexander M. 1 Tihomirov, V. M. 1 Yudin, B. D. 1 Zaslavski, Alexander Yakovlevich
all top 5

#### Serials

 10 SIAM Journal on Control and Optimization 9 Journal of Convex Analysis 8 Mathematical Programming. Series A. Series B 6 Control and Cybernetics 6 Soviet Mathematics. Doklady 6 Transactions of the American Mathematical Society 5 SIAM Journal on Optimization 4 Mathematics of Operations Research 4 Proceedings of the American Mathematical Society 4 Comptes Rendus de l’Académie des Sciences. Série I 4 Set-Valued Analysis 4 Journal of Fixed Point Theory and Applications 3 Russian Mathematical Surveys 3 Nonlinear Analysis. Theory, Methods & Applications 2 Journal of Mathematical Analysis and Applications 2 Uspekhi Matematicheskikh Nauk [N. S.] 2 Differentsial’nye Uravneniya 2 Differential Equations 2 Journal of Optimization Theory and Applications 2 Mathematika 2 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 2 Transactions of the Moscow Mathematical Society 2 Trudy Moskovskogo Matematicheskogo Obshchestva 2 Optimization 2 Calculus of Variations and Partial Differential Equations 2 Journal of the Australian Mathematical Society 2 Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences, Série A 2 Chapman & Hall/CRC Research Notes in Mathematics 2 Set-Valued and Variational Analysis 2 Optimizacija 2 Advances in Nonlinear Analysis 1 Mathematical Notes 1 Zhurnal Vychislitel’noĭ Matematiki i Matematicheskoĭ Fiziki 1 Fundamenta Mathematicae 1 Funktsional’nyĭ Analiz i ego Prilozheniya 1 Izvestiya Akademii Nauk SSSR. Tekhnicheskaya Kibernetika 1 Journal of the London Mathematical Society. Second Series 1 Rendiconti del Seminario Matemàtico e Fisico di Milano 1 Applied Mathematics Letters 1 Doklady Bolgarskoĭ Akademii Nauk 1 Journal de Mathématiques Pures et Appliquées. Neuvième Série 1 Mathematics of the USSR, Sbornik 1 U.S.S.R. Computational Mathematics and Mathematical Physics 1 Bulletin of the American Mathematical Society. New Series 1 Cybernetics and Systems Analysis 1 Journal of Mathematical Sciences (New York) 1 Top 1 Serdica Mathematical Journal 1 Vietnam Journal of Mathematics 1 Abstract and Applied Analysis 1 Foundations of Computational Mathematics 1 Pacific Journal of Optimization 1 Bulletin of the American Mathematical Society 1 Springer Monographs in Mathematics
all top 5

#### Fields

 123 Calculus of variations and optimal control; optimization (49-XX) 53 Operations research, mathematical programming (90-XX) 41 Global analysis, analysis on manifolds (58-XX) 36 Functional analysis (46-XX) 21 Real functions (26-XX) 18 Operator theory (47-XX) 17 General topology (54-XX) 11 Measure and integration (28-XX) 8 Systems theory; control (93-XX) 5 General and overarching topics; collections (00-XX) 5 Numerical analysis (65-XX) 4 Algebraic geometry (14-XX) 4 Several complex variables and analytic spaces (32-XX) 4 Ordinary differential equations (34-XX) 3 Convex and discrete geometry (52-XX) 3 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 2 History and biography (01-XX) 1 Partial differential equations (35-XX) 1 Dynamical systems and ergodic theory (37-XX) 1 Sequences, series, summability (40-XX) 1 Manifolds and cell complexes (57-XX) 1 Mechanics of particles and systems (70-XX)

#### Citations contained in zbMATH Open

127 Publications have been cited 2,294 times in 1,466 Documents Cited by Year
Theory of extremal problems. Translated from the Russian by K. Makowski. Zbl 0407.90051
Ioffe, A. D.; Tihomirov, V. M.
1979
Metric regularity and subdifferential calculus. Zbl 0979.49017
Ioffe, A. D.
2000
Approximate subdifferentials and applications. I: The finite dimensional theory. Zbl 0531.49014
Ioffe, A. D.
1984
On lower semicontinuity of integral functionals. I. Zbl 0361.46037
Ioffe, A. D.
1977
Nonsmooth analysis: differential calculus of nondifferentiable mappings. Zbl 0651.58007
Ioffe, A. D.
1981
Approximate subdifferentials and applications. III: The metric theory. Zbl 0713.49022
Ioffe, A. D.
1989
On metric and calmness qualification conditions in subdifferential calculus. Zbl 1156.49013
Ioffe, Alexander D.; Outrata, Jiří V.
2008
Regular points of Lipschitz functions. Zbl 0427.58008
Ioffe, Alexander D.
1979
Necessary and sufficient conditions for a local minimum. 1: A reduction theorem and first order conditions. Zbl 0417.49027
Ioffe, A. D.
1979
Necessary and sufficient conditions for a local minimum. 3: Second order conditions and augmented duality. Zbl 0417.49029
Ioffe, A. D.
1979
Proximal analysis and approximate subdifferentials. Zbl 0725.46045
Ioffe, A. D.
1990
Duality of convex functions and extremum problems. Zbl 0191.13101
Ioffe, A. D.; Tikhomirov, V. M.
1968
Extension of variational problems. Zbl 0194.13902
Ioffe, A. D.; Tikhomirov, V. M.
1968
Approximate subdifferentials and applications. II. Zbl 0601.49008
Ioffe, A. D.
1986
Subdifferentials of convex functions. Zbl 0281.46039
Ioffe, A. D.; Levin, V. L.
1974
On the local surjection property. Zbl 0642.49010
Ioffe, A. D.
1987
Variational analysis of regular mappings. Theory and applications. Zbl 1381.49001
Ioffe, Alexander D.
2017
Metric critical point theory. I: Morse regularity and homotopic stability of a minimum. Zbl 0852.58018
Ioffe, A.; Schwartzman, E.
1996
Transversality and alternating projections for nonconvex sets. Zbl 1338.49057
Drusvyatskiy, D.; Ioffe, A. D.; Lewis, A. S.
2015
An invitation to tame optimization. Zbl 1182.90083
Ioffe, A. D.
2009
Necessary conditions in nonsmooth optimization. Zbl 0548.90088
Ioffe, A. D.
1984
Codirectional compactness, metric regularity and subdifferential calculus. Zbl 0966.49014
Ioffe, Alexander
2000
Euler-Lagrange and Hamiltonian formalisms in dynamic optimization. Zbl 0876.49024
Ioffe, Alexander
1997
The Euler and Weierstrass conditions for nonsmooth variational problems. Zbl 0838.49015
Ioffe, A. D.; Rockafellar, R. T.
1996
On the theory of subdifferentials. Zbl 1277.49019
Ioffe, Alexander D.
2012
On lower semicontinuity of integral functionals. II. Zbl 0379.46022
Ioffe, A. D.
1977
Variational principles and well-posedness in optimization and calculus of variations. Zbl 0997.49023
Ioffe, Alexander D.; Zaslavski, Alexander J.
2000
On subdifferentiability spaces. Zbl 0599.46065
Ioffe, Aleksandr D.
1983
Theorie der Extremalaufgaben. Übersetzung aus dem Russischen von Bernd Luderer. Zbl 0403.49001
Ioffe, A. D.; Tikhomirov, V. M.
1979
Metric regularity – a survey. I: Theory. Zbl 1369.49001
Ioffe, A. D.
2016
Nonlinear regularity models. Zbl 1275.47106
Ioffe, Alexander D.
2013
A Lagrange multiplier rule with small convex-valued subdifferentials for nonsmooth problems of mathematical programming involving equality and nonfunctional constraints. Zbl 0782.90095
Ioffe, Alexander
1993
Necessary and sufficient conditions for a local minimum. 2: Conditions of Levitin-Miljutin-Osmolovskii type. Zbl 0417.49028
Ioffe, A. D.
1979
Regularity on a fixed set. Zbl 1275.49028
Ioffe, A. D.
2011
Proximal analysis in smooth spaces. Zbl 0858.49015
Borwein, Jonathan M.; Ioffe, Alexander
1996
Towards variational analysis in metric spaces: metric regularity and fixed points. Zbl 1190.49026
Ioffe, A. D.
2010
Subdifferentiability spaces and nonsmooth analysis. Zbl 0554.58008
Ioffe, A. D.
1984
Existence and relaxation theorems for unbounded differential inclusions. Zbl 1104.34006
Ioffe, A.
2006
Quadratic growth and critical point stability of semi-algebraic functions. Zbl 1325.49021
Drusvyatskiy, D.; Ioffe, A. D.
2015
Second-order sufficiency and quadratic growth for nonisolated minima. Zbl 0846.90095
Bonnans, Joseph Frédéric; Ioffe, Alexander
1995
Sous-différentielles approchees de fonctions numériques. Zbl 0482.46029
Ioffe, Alexandre D.
1981
Metric regularity – a survey. II: Applications. Zbl 1369.49002
Ioffe, A. D.
2016
On regularity concepts in variational analysis. Zbl 1205.49023
Ioffe, A. D.
2010
Critical values of set-valued maps with stratifiable graphs. Extensions of Sard and Smale-Sard theorems. Zbl 1191.49015
Ioffe, A. D.
2008
Fuzzy principles and characterization of trustworthiness. Zbl 0927.49009
Ioffe, Alexander
1998
Subdifferentials of performance functions and calculus of coderivatives of set-valued mappings. Zbl 0874.49018
Ioffe, Alexander D.; Penot, Jean-Paul
1996
Variational analysis of a composite function: A formula for the lower second order epi-derivative. Zbl 0778.49021
Ioffe, Alexander
1991
Regularity estimates for convex multifunctions. Zbl 1157.49026
Ioffe, A. D.; Sekiguchi, Y.
2009
On perturbation stability of metric regularity. Zbl 1005.47045
Ioffe, Alexander D.
2001
Limiting subhessians, limiting subjets and their calculus. Zbl 0884.49012
Ioffe, Alexander D.; Penot, Jean-Paul
1997
On primal regularity estimates for single-valued mappings. Zbl 1327.49030
Cibulka, Radek; Fabian, Marián; Ioffe, Alexander D.
2015
Almost every convex or quadratic programming problem is well posed. Zbl 1082.49031
Ioffe, A. D.; Lucchetti, R. E.; Revalski, J. P.
2004
On robustness of the regularity property of maps. Zbl 1127.49023
Ioffe, Alexander D.
2003
Différentielles généralisées d’applications localement lipschitziennes d’un espace de Banach dans un autre. Zbl 0421.46039
Ioffe, Alexandre D.
1979
Curves of descent. Zbl 1345.26027
Drusvyatskiy, D.; Ioffe, A. D.; Lewis, A. S.
2015
A note on subdifferentials of pointwise suprema. Zbl 1258.49016
Ioffe, A. D.
2012
The dimension of semialgebraic subdifferential graphs. Zbl 1236.49039
Drusvyatskiy, D.; Ioffe, A. D.; Lewis, A. S.
2012
A Sard theorem for tame set-valued mappings. Zbl 1121.49016
Ioffe, A.
2007
Towards metric theory of metric regularity. Zbl 0986.54039
Ioffe, Alexander
2001
Some remarks on variational principles. Zbl 0928.49013
Ioffe, A. D.; Tikhomirov, V. M.
1997
On sensitivity analysis of nonlinear programs in Banach spaces: The approach via composite unconstrained optimization. Zbl 0820.90105
Ioffe, Alexander
1994
Global surjection and global inverse mapping theorems in Banach spaces. Zbl 0709.47054
Ioffe, A. D.
1987
Metric regularity, fixed points and some associated problems of variational analysis. Zbl 1311.49042
Ioffe, Alexander D.
2014
On regularity estimates for mappings between embedded manifolds. Zbl 1168.49036
Ioffe, Alexander
2007
Typical convex program is very well posed. Zbl 1082.49030
Ioffe, A.; Lucchetti, R. E.
2005
On stability estimates for the regularity property of maps. Zbl 1054.58006
Ioffe, A. D.
2003
A variational principle for problems with functional constraints. Zbl 1023.49021
Ioffe, A. D.; Lucchetti, R. E.; Revalski, J. P.
2001
On the theory of subdifferential. Zbl 0604.49010
Ioffe, A. D.
1986
Single-valued representation of set-valued mappings. II: Application to differential inclusions. Zbl 0539.49009
Ioffe, A. D.
1983
A new proof of the equivalence of the Hahn-Banach extension and the least upper bound properties. Zbl 0469.46005
Ioffe, A. D.
1981
An existence theorem for a general Bolza problem. Zbl 0341.49004
Ioffe, A. D.
1976
Theorie der Extremalprobleme. (Teorija ekstremal’nyh zadac.). Zbl 0292.90042
Ioffe, A. D.; Tikhomirov, V. M.
1974
Separable reductions and rich families in the theory of Fréchet subdifferentials. Zbl 1350.49012
Fabian, Marián; Ioffe, Alexander
2016
Generic minimizing behavior in semialgebraic optimization. Zbl 1334.49057
Drusvyatskiy, D.; Ioffe, A. D.; Lewis, Adrian S.
2016
Three theorems on subdifferentiation of convex integral functionals. Zbl 1139.26010
Ioffe, A.
2006
Optimality alternative: a non-variational approach to necessary conditions. Zbl 1113.49027
Ioffe, A. D.
2005
Abstract convexity and non-smooth analysis. Zbl 1063.49016
Ioffe, A. D.
2001
The maximum principle in optimal control of systems governed by semilinear equations. Zbl 0887.49022
Ginsburg, Boris; Ioffe, Alexander
1996
Quadratic growth and stability in convex programming problems with multiple solutions. Zbl 0839.90090
Bonnans, J. Frédéric; Ioffe, Alexander D.
1995
Survey of measurable selection theorems: Russian literature supplement. Zbl 0382.28004
Ioffe, A. D.
1978
Transversality in variational analysis. Zbl 1382.49014
Ioffe, Alexander D.
2017
Clarke subgradients for directionally Lipschitzian stratifiable functions. Zbl 1315.49002
Drusvyatskiy, Dmitriy; Ioffe, Alexander D.; Lewis, Adrian S.
2015
Variational analysis and mathematical economics. II: Nonsmooth regular economies. Zbl 1216.91008
Ioffe, A. D.
2011
Exact formulae for regularity estimates. Zbl 1193.49014
Ioffe, Alexander D.; Sekiguchi, Yoshiyuki
2007
Abstract convexity and nonsmooth analysis: Global aspects. Zbl 1045.49023
Ioffe, A. D.; Rubinov, A. M.
2002
Variational methods in local and global non-smooth analysis. Notes by Igor Zelenko. Zbl 0940.49018
Ioffe, Alexander
1999
An existence theorem for problems of the calculus of variations. Zbl 0275.49009
Ioffe, A. D.
1972
On minimization of integral functionals. Zbl 0193.07603
Ioffe, A. D.; Tikhomirov, V. M.
1969
On generalized Bolza problem and its application to dynamic optimization. Zbl 1420.49027
Ioffe, Alexander D.
2019
Separable reduction in the theory of Fréchet subdifferentials. Zbl 1284.49016
Fabian, Marián; Ioffe, Alexander
2013
Variational analysis and mathematical economics. I: Subdifferential calculus and the second theorem of welfare economics. Zbl 1230.91047
Ioffe, A. D.
2009
Variational problem associated with a model of welfare economics with a measure spaces of agents. Zbl 1194.91115
Ioffe, Alexander
2006
Generic well-posedness in minimization problems. Zbl 1099.49023
Ioffe, A.; Lucchetti, R. E.
2005
Metric critical point theory. II. Deformation techniques. Zbl 0897.58008
Ioffe, A.; Schwartzman, E.
1997
Directional compactness, scalarization and nonsmooth semi-Fredholm mappings. Zbl 0912.46004
Ioffe, A. D.
1997
Expansion of exact and approximate solutions in nonlinear programming. Zbl 0769.90070
Bonnans, J. Frédéric; Ioffe, Alexander D.; Shapiro, Alexander
1992
Composite optimization: Second order conditions, value functions and sensitivity. Zbl 0708.90080
Ioffe, Alexander
1990
Absolutely continuous subgradients of nonconvex integral functionals. Zbl 0612.49013
Ioffe, A. D.
1987
Single-valued representation of set-valued mappings. Zbl 0417.28006
Ioffe, A. D.
1979
Die Dualität von konvexen Funktionen und Extremalprobleme. Zbl 0167.42202
Ioffe, A. D.; Tikhomirov, V. M.
1968
Nonsmooth optimization using Taylor-like models: error bounds, convergence, and termination criteria. Zbl 1459.65083
Drusvyatskiy, D.; Ioffe, A. D.; Lewis, A. S.
2021
Towards the theory of strong minimum in calculus of variations and optimal control: a view from variational analysis. Zbl 1436.49026
Ioffe, Alexander D.
2020
On generalized Bolza problem and its application to dynamic optimization. Zbl 1420.49027
Ioffe, Alexander D.
2019
On variational inequalities over polyhedral sets. Zbl 1391.49014
Ioffe, Alexander D.
2018
Separable reduction of local metric regularity. Zbl 1457.58004
Fabian, M.; Ioffe, A. D.; Revalski, J.
2018
Variational analysis of regular mappings. Theory and applications. Zbl 1381.49001
Ioffe, Alexander D.
2017
Transversality in variational analysis. Zbl 1382.49014
Ioffe, Alexander D.
2017
Metric regularity – a survey. I: Theory. Zbl 1369.49001
Ioffe, A. D.
2016
Metric regularity – a survey. II: Applications. Zbl 1369.49002
Ioffe, A. D.
2016
Separable reductions and rich families in the theory of Fréchet subdifferentials. Zbl 1350.49012
Fabian, Marián; Ioffe, Alexander
2016
Generic minimizing behavior in semialgebraic optimization. Zbl 1334.49057
Drusvyatskiy, D.; Ioffe, A. D.; Lewis, Adrian S.
2016
Transversality and alternating projections for nonconvex sets. Zbl 1338.49057
Drusvyatskiy, D.; Ioffe, A. D.; Lewis, A. S.
2015
Quadratic growth and critical point stability of semi-algebraic functions. Zbl 1325.49021
Drusvyatskiy, D.; Ioffe, A. D.
2015
On primal regularity estimates for single-valued mappings. Zbl 1327.49030
Cibulka, Radek; Fabian, Marián; Ioffe, Alexander D.
2015
Curves of descent. Zbl 1345.26027
Drusvyatskiy, D.; Ioffe, A. D.; Lewis, A. S.
2015
Clarke subgradients for directionally Lipschitzian stratifiable functions. Zbl 1315.49002
Drusvyatskiy, Dmitriy; Ioffe, Alexander D.; Lewis, Adrian S.
2015
Metric regularity, fixed points and some associated problems of variational analysis. Zbl 1311.49042
Ioffe, Alexander D.
2014
Separable reduction of metric regularity properties. Zbl 1282.49012
Ioffe, A. D.
2014
Nonlinear regularity models. Zbl 1275.47106
Ioffe, Alexander D.
2013
Separable reduction in the theory of Fréchet subdifferentials. Zbl 1284.49016
Fabian, Marián; Ioffe, Alexander
2013
Convexity and variational analysis. Zbl 1305.90399
Ioffe, A. D.
2013
On the theory of subdifferentials. Zbl 1277.49019
Ioffe, Alexander D.
2012
A note on subdifferentials of pointwise suprema. Zbl 1258.49016
Ioffe, A. D.
2012
The dimension of semialgebraic subdifferential graphs. Zbl 1236.49039
Drusvyatskiy, D.; Ioffe, A. D.; Lewis, A. S.
2012
Regularity on a fixed set. Zbl 1275.49028
Ioffe, A. D.
2011
Variational analysis and mathematical economics. II: Nonsmooth regular economies. Zbl 1216.91008
Ioffe, A. D.
2011
Separable reduction revisited. Zbl 1216.49017
Ioffe, A. D.
2011
Towards variational analysis in metric spaces: metric regularity and fixed points. Zbl 1190.49026
Ioffe, A. D.
2010
On regularity concepts in variational analysis. Zbl 1205.49023
Ioffe, A. D.
2010
Typical convexity (concavity) of Dini-Hadamard upper (lower) directional derivatives of functions on separable Banach spaces. Zbl 1208.46043
Ioffe, A.
2010
The principle of Lagrange and critical values of optimization problems. Zbl 1357.49077
Ioffe, A. D.
2010
An invitation to tame optimization. Zbl 1182.90083
Ioffe, A. D.
2009
Regularity estimates for convex multifunctions. Zbl 1157.49026
Ioffe, A. D.; Sekiguchi, Y.
2009
Variational analysis and mathematical economics. I: Subdifferential calculus and the second theorem of welfare economics. Zbl 1230.91047
Ioffe, A. D.
2009
On metric and calmness qualification conditions in subdifferential calculus. Zbl 1156.49013
Ioffe, Alexander D.; Outrata, Jiří V.
2008
Critical values of set-valued maps with stratifiable graphs. Extensions of Sard and Smale-Sard theorems. Zbl 1191.49015
Ioffe, A. D.
2008
On regularity of convex multifunctions. Zbl 1146.49024
Ioffe, A. D.
2008
Critical points of simple functions. Zbl 1220.49009
Ioffe, A. D.; Lewis, A. S.
2008
A Sard theorem for tame set-valued mappings. Zbl 1121.49016
Ioffe, A.
2007
On regularity estimates for mappings between embedded manifolds. Zbl 1168.49036
Ioffe, Alexander
2007
Exact formulae for regularity estimates. Zbl 1193.49014
Ioffe, Alexander D.; Sekiguchi, Yoshiyuki
2007
Existence and relaxation theorems for unbounded differential inclusions. Zbl 1104.34006
Ioffe, A.
2006
Three theorems on subdifferentiation of convex integral functionals. Zbl 1139.26010
Ioffe, A.
2006
Variational problem associated with a model of welfare economics with a measure spaces of agents. Zbl 1194.91115
Ioffe, Alexander
2006
Typical convex program is very well posed. Zbl 1082.49030
Ioffe, A.; Lucchetti, R. E.
2005
Optimality alternative: a non-variational approach to necessary conditions. Zbl 1113.49027
Ioffe, A. D.
2005
Generic well-posedness in minimization problems. Zbl 1099.49023
Ioffe, A.; Lucchetti, R. E.
2005
On necessary conditions in variable end-time optimal control problems. Zbl 1167.49315
Ioffe, A.
2005
Almost every convex or quadratic programming problem is well posed. Zbl 1082.49031
Ioffe, A. D.; Lucchetti, R. E.; Revalski, J. P.
2004
On robustness of the regularity property of maps. Zbl 1127.49023
Ioffe, Alexander D.
2003
On stability estimates for the regularity property of maps. Zbl 1054.58006
Ioffe, A. D.
2003
Abstract convexity and nonsmooth analysis: Global aspects. Zbl 1045.49023
Ioffe, A. D.; Rubinov, A. M.
2002
Convergence and extensions of variational problems with non-coercive functionals. Zbl 1125.49303
Ioffe, Alexander D.
2002
On a characterization of $$C^{1,1}$$ functions. Zbl 1029.49017
Ioffe, A.; Miłosz, T.
2002
On perturbation stability of metric regularity. Zbl 1005.47045
Ioffe, Alexander D.
2001
Towards metric theory of metric regularity. Zbl 0986.54039
Ioffe, Alexander
2001
A variational principle for problems with functional constraints. Zbl 1023.49021
Ioffe, A. D.; Lucchetti, R. E.; Revalski, J. P.
2001
Abstract convexity and non-smooth analysis. Zbl 1063.49016
Ioffe, A. D.
2001
Metric regularity and subdifferential calculus. Zbl 0979.49017
Ioffe, A. D.
2000
Codirectional compactness, metric regularity and subdifferential calculus. Zbl 0966.49014
Ioffe, Alexander
2000
Variational principles and well-posedness in optimization and calculus of variations. Zbl 0997.49023
Ioffe, Alexander D.; Zaslavski, Alexander J.
2000
Generic existence, uniqueness and stability in optimization problems. Zbl 0972.90090
Ioffe, Alexander; Lucchetti, Roberto
2000
Variational methods in local and global non-smooth analysis. Notes by Igor Zelenko. Zbl 0940.49018
Ioffe, Alexander
1999
Fuzzy principles and characterization of trustworthiness. Zbl 0927.49009
Ioffe, Alexander
1998
Euler-Lagrange and Hamiltonian formalisms in dynamic optimization. Zbl 0876.49024
Ioffe, Alexander
1997
Limiting subhessians, limiting subjets and their calculus. Zbl 0884.49012
Ioffe, Alexander D.; Penot, Jean-Paul
1997
Some remarks on variational principles. Zbl 0928.49013
Ioffe, A. D.; Tikhomirov, V. M.
1997
Metric critical point theory. II. Deformation techniques. Zbl 0897.58008
Ioffe, A.; Schwartzman, E.
1997
Directional compactness, scalarization and nonsmooth semi-Fredholm mappings. Zbl 0912.46004
Ioffe, A. D.
1997
An extension of the Rabinowitz bifurcation theorem to Lipschitz potential operators in Hilbert spaces. Zbl 0885.58012
Ioffe, Alexander; Schwartzman, Efim
1997
Metric critical point theory. I: Morse regularity and homotopic stability of a minimum. Zbl 0852.58018
Ioffe, A.; Schwartzman, E.
1996
The Euler and Weierstrass conditions for nonsmooth variational problems. Zbl 0838.49015
Ioffe, A. D.; Rockafellar, R. T.
1996
Proximal analysis in smooth spaces. Zbl 0858.49015
Borwein, Jonathan M.; Ioffe, Alexander
1996
Subdifferentials of performance functions and calculus of coderivatives of set-valued mappings. Zbl 0874.49018
Ioffe, Alexander D.; Penot, Jean-Paul
1996
The maximum principle in optimal control of systems governed by semilinear equations. Zbl 0887.49022
Ginsburg, Boris; Ioffe, Alexander
1996
Separable reduction theorem for approximate subdifferentials. Zbl 0856.49014
Ioffe, Alexander
1996
Non-smooth subdifferentials: Their calculus and applications. Zbl 0854.49016
Ioffe, A. D.
1996
Second-order sufficiency and quadratic growth for nonisolated minima. Zbl 0846.90095
Bonnans, Joseph Frédéric; Ioffe, Alexander
1995
Quadratic growth and stability in convex programming problems with multiple solutions. Zbl 0839.90090
Bonnans, J. Frédéric; Ioffe, Alexander D.
1995
On sensitivity analysis of nonlinear programs in Banach spaces: The approach via composite unconstrained optimization. Zbl 0820.90105
Ioffe, Alexander
1994
A Lagrange multiplier rule with small convex-valued subdifferentials for nonsmooth problems of mathematical programming involving equality and nonfunctional constraints. Zbl 0782.90095
Ioffe, Alexander
1993
Expansion of exact and approximate solutions in nonlinear programming. Zbl 0769.90070
Bonnans, J. Frédéric; Ioffe, Alexander D.; Shapiro, Alexander
1992
Optimization and nonlinear analysis. Proceedings of a binational workshop, held at Technion City, Haifa, Israel, 21-27 March 1990. Zbl 0745.00050
Ioffe, A. (ed.); Marcus, M. (ed.); Reich, S. (ed.)
1992
Variational analysis of a composite function: A formula for the lower second order epi-derivative. Zbl 0778.49021
Ioffe, Alexander
1991
Proximal analysis and approximate subdifferentials. Zbl 0725.46045
Ioffe, A. D.
1990
Composite optimization: Second order conditions, value functions and sensitivity. Zbl 0708.90080
Ioffe, Alexander
1990
Approximate subdifferentials and applications. III: The metric theory. Zbl 0713.49022
Ioffe, A. D.
1989
On some recent developments in the theory of second order optimality conditions. Zbl 0713.90094
Ioffe, A.
1989
On the local surjection property. Zbl 0642.49010
Ioffe, A. D.
1987
Global surjection and global inverse mapping theorems in Banach spaces. Zbl 0709.47054
Ioffe, A. D.
1987
Absolutely continuous subgradients of nonconvex integral functionals. Zbl 0612.49013
Ioffe, A. D.
1987
Approximate subdifferentials and applications. II. Zbl 0601.49008
Ioffe, A. D.
1986
On the theory of subdifferential. Zbl 0604.49010
Ioffe, A. D.
1986
Approximate subdifferentials and applications. I: The finite dimensional theory. Zbl 0531.49014
Ioffe, A. D.
1984
Necessary conditions in nonsmooth optimization. Zbl 0548.90088
Ioffe, A. D.
1984
Subdifferentiability spaces and nonsmooth analysis. Zbl 0554.58008
Ioffe, A. D.
1984
On subdifferentiability spaces. Zbl 0599.46065
Ioffe, Aleksandr D.
1983
Single-valued representation of set-valued mappings. II: Application to differential inclusions. Zbl 0539.49009
Ioffe, A. D.
1983
Second order conditions in nonlinear nonsmooth problems of semi-infinite programming. Zbl 0522.90080
Ioffe, A. D.
1983
Nonsmooth analysis: differential calculus of nondifferentiable mappings. Zbl 0651.58007
Ioffe, A. D.
1981
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#### Cited by 1,243 Authors

 48 Ioffe, Alexander Davidovich 46 Mordukhovich, Boris S. 36 Penot, Jean-Paul 35 Kruger, Alexander Ya. 25 Yao, Jen-Chih 23 López-Cerdá, Marco Antonio 19 Outrata, Jiří V. 19 Thibault, Lionel 18 Huynh Van Ngai 18 Yang, Xiaoqi 17 Zaslavski, Alexander Yakovlevich 16 Borwein, Jonathan Michael 16 Dontchev, Asen L. 16 Drusvyatskiy, Dmitriy 16 Phan Quôc Khánh 15 Jourani, Abdeljalil 15 Théra, Michel A. 14 Cánovas, María Josefa 14 Parra, Juan 14 Zheng, Xiyin 13 Cibulka, Radek 13 Geoffroy, Michel H. 13 Jourani, Abderrahim 12 Degiovanni, Marco 12 Lewis, Adrian S. 12 Rossi, Riccarda 11 Arutyunov, Aram Vladimirovic 11 Hantoute, Abderrahim 11 Ng, Kung-Fu 11 Thao, Nguyen Hieu 10 Durea, Marius 10 Lassonde, Marc 10 Páles, Zsolt 10 Phạm Tiên So’n 10 Thibault, Louis-Philippe 10 Vinter, Richard B. 10 Wang, Bingwu 9 Balder, Erik J. 9 Chikriĭ, Arkadiĭ Alekseevich 9 Fabian, Marián J. 9 Gfrerer, Helmut 9 Shapiro, Alexander 9 Strugariu, Radu 9 Tret’yakov, Alexey A. 9 Uderzo, Amos 9 Ye, Jane J. 8 Bounkhel, Messaoud 8 Chuong, Thai Doan 8 Correa, Rafael 8 Eberhard, Andrew C. 8 Frankowska, Hélène 8 Gaydu, Michaël 8 Henrion, René 8 Liu, Jiaquan 8 Nam, Nguyen Mau 8 Nguyen Thi Toan 8 Rockafellar, Ralph Tyrrell 8 Tolstonogov, Aleksandr Aleksandrovich 8 Tröltzsch, Fredi 8 Wang, Shawn Xianfu 8 Nguyen Dong Yen 7 Azé, Dominique 7 Klatte, Diethard 7 Li, Chong 7 Papageorgiou, Nikolaos S. 7 Pickenhain, Sabine 7 Seeger, Alberto 7 Shao, Yongheng 6 Adam, Lukáš 6 Buttazzo, Giuseppe 6 Clarke, Francis H. 6 Corvellec, Jean-Noël 6 Daniilidis, Aris 6 Demyanov, Vladimir Fedorovich 6 Dinh The Luc 6 Dolgopolik, Maxim Vladimirovich 6 Hoàng Xuân Phù 6 Jeyakumar, Vaithilingam 6 Kien, Bui Trong 6 Ledyaev, Yuri Semenovich 6 Ledzewicz, Urszula 6 Lee, Gue Myung 6 Li, Shengjie 6 Roshchina, Vera 6 Schimperna, Giulio 6 Treiman, Jay S. 6 Visintin, Augusto 6 Wagner, Marcus M. 6 Wang, Zhi-Qiang 6 Zeidan, Vera Michel 6 Zhang, Binbin 6 Zhu, Qiji Jim 5 Adly, Samir 5 Amahroq, Tijani 5 Anh, Lam Quoc 5 Bonnans, Joseph Frédéric 5 Bouchitté, Guy 5 Burke, James V. 5 Casas, Eduardo 5 Červinka, Michal ...and 1,143 more Authors
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text:
# Arc

In geometry, an arc is defined as the portion or part of a circle’s circumference. Moreover, you can find this arc in other geometrical shapes like ellipse, but it refers to circles in most scenarios.

To neglect this mistake, it is also called a circular arc. For example, if a small curve line is drawn from point B to C, it is called an arc, and the distance between both curve points is known as arc length.

For a more precise understanding, let see the diagram listed below:

Note:

To calculate arc length, you can use our Arc Length Calculator.


text:
+0

# HELP PLZ

0
59
1

Find all values of z such that {3-z}/{z+1} ≥ 1. Answer in interval notation.

Oct 14, 2019

#1
+961
+1

$$\frac{3-z}{z+1} \geq 1, 3-z \geq z+1$$

Now we can add z and subtract 1 to both sides, then solve and get the interval notation. Can you solve that?

You are very welcome!

:P

Oct 14, 2019


text:
# The Caves of Clojure: Part 7.1

Posted on October 15, 2012.

This post is part of an ongoing series. If you haven’t already done so, you should probably start at the beginning.

This entry corresponds to the beginning of post seven in Trystan’s tutorial.

If you want to follow along, the code for the series is on Bitbucket and on GitHub. Update to the entry-07-1 tag to see the code as it stands after this post.

It’s been a while since the last post, but I’ve been taking care of things and hopefully should be able to write a bit more now.

This post is going to be short. It’ll cover a relatively self-contained but interesting bit of Trystan’s seventh post. The rest of it will be covered in the next entry.

## Summary

In Trystan’s seventh post he adds vertical levels and stairs connecting them. I’m going to cover the first part of that now: mapping out regions.

There’s been a bit of refactoring since my last post which I’m not going to cover. If you want to see what changed, diff the tags in your VCS of choice.

## Region Mapping

In order to decide where to place stairs, Trystan maps out “regions” of contiguous, walkable tiles after he generates and smooths the world. I’m going to do the same thing.

My goal is to create a “region map”, which is a map of coordinates to region numbers. For example, consider the following tiny world map:

..##..
..#...
..#.##
..#.#.

There are three distinct regions in this map:

11##22
11#222
11#2##
11#2#3

So the region map would look like:

; x y  region
{[0 0] 1
[1 0] 1
[4 0] 2
[5 0] 2
...
[5 3] 3}

This makes it easy to tell which region a particular tile is in (if any).

As usual, I’ll start with a few helper functions. These two functions are just for convenience and readability:

(def all-coords
(let [[cols rows] world-size]
(for [x (range cols)
y (range rows)]
[x y])))

(defn get-tile-from-level [level [x y]]
(get-in level [y x] (:bound tiles)))

The all-coords function simply returns a lazy sequence of [x y] coordinates representing every coordinate in a level.

get-tile-from-level encapsulates the act of pulling out a tile from a level given an [x y] coordinate. This is helpful because of the way I’m storing tiles (note the ugly [y x]).

Next up is a function that filters a set of coordinates to only contain those that are actually walkable in the given level (i.e.: those that don’t contain a wall tile):

(defn filter-walkable
"Filter the given coordinates to include only walkable ones."
[level coords]
(set (filter #(tile-walkable? (get-tile-from-level level %))
coords)))

This uses the get-tile-from-level function as well as tile-walkable? from world.core.

Next is a function to take a coordinate and return which of its neighboring coordinates are walkable:

(defn walkable-neighbors
"Return the neighboring coordinates walkable from the given coord."
[level coord]
(filter-walkable level (neighbors coord)))

This one is almost trivial, but I like building up functions in small steps like this because it’s easier for me to read.

Now we come to a function with a bit more meat. This is the core of the “flood fill” algorithm I’m going to use to fill in the region map.

(defn walkable-from
"Return all coordinates walkable from the given coord (including itself)."
[level coord]
(loop [walked #{}
to-walk #{coord}]
(if (empty? to-walk)
walked
(let [current (first to-walk)
walked (conj walked current)
to-walk (disj to-walk current)
candidates (walkable-neighbors level current)
to-walk (union to-walk (difference candidates walked))]
(recur walked to-walk)))))

In a nutshell, this function loops over two sets: walked and to-walk.

Each iteration it grabs a coordinate from to-walk and sticks it into the walked set. It then finds all the coordinates it can walk to from that coordinate using a helper function. It uses clojure.set/difference to determine which of those are new (i.e.: still need to be walked) and sticks them into the to-walk set. Then it recurs.

The code for this is surprisingly simple and easy to read. It’s mostly just shuffling things between sets. Eventually the to-walk set will be empty and walked will contain all the coordinates that we want.

Finally comes the function to create the region map for an entire level:

(defn get-region-map
[level]
(loop [remaining (filter-walkable level all-coords)
region-map {}
n 0]
(if (empty? remaining)
region-map
(let [next-coord (first remaining)
next-region-coords (walkable-from level next-coord)]
(recur (difference remaining next-region-coords)
(into region-map (map vector
next-region-coords
(repeat n)))
(inc n))))))

This function also uses Clojure sets to its advantage. Once again, I loop over a couple of variables.

remaining is a set containing all the coordinates whose regions has not yet been determined.

Each iteration it pulls off one of the remaining coordinates. Note that I’m using first to do this. remaining is a set, which is unordered, so first effectively could return any element in the set. For this loop that doesn’t matter, but it’s important to be aware of if you’re going to use the same strategy.

After pulling off a coordinate, it finds all coordinates walkable from that coordinate with the walkable-from flood-fill function. It removes all of those from the remaining set, shoves them into the region map, and increments the region number before recurring.

## Visualization

I’m going to save the rest of Trystan’s seventh post for another entry, but since this one ended up pretty short I’m also going to go over visualizing the region map I’ve just created.

First I need to generate the region map when I create the world, and attach it to the world itself so we can access it from other places:

(defn random-world []
(let [world (->World (random-tiles) {})
world (nth (iterate smooth-world world) 3)
world (populate-world world)
world (assoc world :regions (get-region-map (:tiles world)))]
world))

The last line in the let is where it gets generated. It’s pretty straightforward.

I’d like to be able to toggle the visualization of regions off and on, so I’m going to introduce a new concept to the game: “debug flags”.

I updated the Game record to include a slot for these flags:

(defrecord Game [world uis input debug-flags])

I then updated the new-game function to initialize them (currently there’s only one) to default values:

(defn new-game []
(map->Game {:world nil
:uis [(->UI :start)]
:input nil
:debug-flags {:show-regions false}}))

The user needs a way to toggle them. For now I’ll just bind it to a key. In the future I could make a debug UI with a nice menu.

(defmethod process-input :play [game input]
(case input
:enter     (assoc game :uis [(->UI :win)])
:backspace (assoc game :uis [(->UI :lose)])
\q         (assoc game :uis [])

\h (update-in game [:world] move-player :w)
\j (update-in game [:world] move-player :s)

; ...

\R (update-in game [:debug-flags :show-regions] not)

game))

Now when the user presses R (Shift and R) it will toggle the state of the :show-regions debug flag in the game.

All that’s left is to actually draw the regions somehow. First, we only want to do this if :show-regions is true. I edited the :play UI’s drawing function to do this:

(defmethod draw-ui :play [ui game screen]
(let [world (:world game)
{:keys [tiles entities regions]} world
player (:player entities)
[cols rows] (s/get-size screen)
vcols cols
vrows (dec rows)
origin (get-viewport-coords game (:location player) vcols vrows)]
(draw-world screen vrows vcols origin tiles)
; ******************
(when (get-in game [:debug-flags :show-regions])
(draw-regions screen regions vrows vcols origin))
; ******************
(doseq [entity (vals entities)]
(draw-entity screen origin vrows vcols entity))
(draw-hud screen game)
(draw-messages screen (:messages player))
(highlight-player screen origin player)))

The marked lines are the only new ones. I’m going to draw regions after/above the world tiles (so they’ll show up at all) but before/below the entities (so we can still see what’s going on).

Drawing the regions is fairly simple, if a bit tedious:

(defn draw-regions [screen region-map vrows vcols [ox oy]]
(letfn [(get-region-glyph [region-number]
(str
(nth
"0123456789abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ"
region-number)))]
(doseq [x (range ox (+ ox vcols))
y (range oy (+ oy vrows))]
(let [region-number (region-map [x y])]
(when region-number
(s/put-string screen (- x ox) (- y oy)
(get-region-glyph region-number)
{:fg :blue}))))))

For now, bad things will happen if we have more than 62 regions in a single level. In practice I usually end up with about 20 to 30, so it’s not a big deal.

To sum up this function: it iterates through every coordinate in the level that’s displayed in the viewport, looks up its region number in the region map, and draws the appropriate letter if it has a region number.

## Results

Now that I’ve got a way to visualize regions it becomes much easier to check whether they’re getting set correctly. Here’s an example of what it looks like when you toggle :show-regions with R:

As you can see, the small, closed off areas have their own numbers, while the larger regions sprawl across the map.

You can view the code on GitHub if you want to see the end result.

The next article will finish Trystan’s seventh post by adding multiple z-levels to the caves.


text:
Courses

# Circular Motion Class 11 Notes | EduRev

## Class 11 : Circular Motion Class 11 Notes | EduRev

The document Circular Motion Class 11 Notes | EduRev is a part of the Class 11 Course Physics For JEE.
All you need of Class 11 at this link: Class 11

Circular Motion

When a particle moves in a plane such that its distance from a fixed (or moving) point remains constant then its motion is called as the circular motion with respect to that fixed (or moving) point. That fixed point is called centre and the distance between fixed point and particle is called radius.

For Example:

The car is moving in a straight line with respect to the man A. But the man B continuously rotate his face to see the car. So with respect to man A

But with respect to man B

Therefore we conclude that with respect to A the motion of car is straight line but for man B it has some angular velocity.

Kinematics of Circular Motion

1. Angular Position

The angle made by the position vector with given line (reference line) is called angular position Circular motion is a two dimensional motion or motion in a plane.

Suppose a particle P is moving in a circle of radius r and centre O. The position of the particle P at a given instant may be described by the angle q between OP and OX. This angle θ is called the angular position of the particle. As the particle moves on the circle its angular position θ change. Suppose the point rotates an angle Δθ in time Δt.

2. Angular Displacement

➢ Definition

Angle rotated by a position vector of the moving particle in a given time interval with some reference line is called its angular displacement

Important Points

• It is dimensionless and has proper unit SI unit radian while other units are degree or revolution 2p rad = 360° = 1 rev
• Infinitely small angular displacement is a vector quantity but finite angular displacement is not because the addition of the small angular displacement is commutative while for large is not.
• Direction of small angular displacement is decided by right hand thumb rule. When the fingers are directed along the motion of the point then thumb will represents the direction of angular displacement.
• Angular displacement can be different for different observers

3. Angular velocity ω

➢ Average Angular Velocity

where θ1 and θ2 are angular position of the particle at time t1 and t2 respectively.

➢ Instantaneous Angular Velocity

The rate at which the position vector of a particle with respect to the centre rotates, is called as instantaneous angular velocity with respect to the centre.

Important points:

• It is an axial vector with dimensions [T-1] and SI unit rad/s.
• For a rigid body as all points will rotate through same angle in same time, angular velocity is a characteristic of the body as a whole, e.g., angular velocity of all points of earth about its own axis is (2π/24) rad/hr.
• If a body makes `n' rotations in `t' seconds then angular velocity in radian per second will be
If T is the period and ‘f’ the frequency of uniform circular motion.
• If θ = a – bt + ct2 then ω = dθ/dt = – b + 2ct

➢ Relation Between Speed and Angular Velocity

The rate of change of angular velocity is called the angular acceleration (α). Thus,

The linear distance PP' travelled by the particle in time Δt is

Here, v is the linear speeds of the particle
It is only valid for circular motion.

v = rω is a scalar quantity.

4. Relative Angular Velocity

Angular velocity is defined with respect to the point from which the position vector of the moving particle is drawn Here angular velocity of the particle w.r.t. `O' and `A' will be different.

➢ Definition

Relative angular velocity of a particle ‘A’ with respect to the other moving particle ‘B’ is the angular velocity of the position vector of ‘A’ with respect to ‘B’. That means it is the rate at which position vector of ‘A’ with respect to ‘B’ rotates at that instant

Important Points:

• If two particles are moving on the same circle or different coplanar concentric circles in same direction with different uniform angular speed ωA and ωB respectively, the rate of change of angle between OA and OB is

So the time taken by one to complete one revolution around O w.r.t. the other
• If two particles are moving on two different concentric circles with different velocities then angular velocity of B relative to A as observed by A will depend on their positions and velocities. consider the case when A and B are closest to each other moving in same direction as shown in figure. In this situation

5. Angular Acceleration α

➢ Average Angular Acceleration

Let ω1 and ω2 be the instantaneous angular speeds at times t1 and t2 respectively, then the average angular acceleration αav is defined as

➢ Instantaneous Angular Acceleration

It is the limit of average angular acceleration as ∆t approaches zero, i.e.,

Important points:

• It is also an axial vector with dimension [T-2] and unit rad/s2
• If α = 0, circular motion is said to be uniform.
•  As
i.e., second derivative of angular displacement w.r.t time gives angular acceleration.
• α is a axial vector and direction of α is along ω
if ω increases and opposite to ω if ω decreases.

Types of Circular Motion

For better understanding, we can classify the circular motion in following three

types:

1. Uniform circular motion in which v and ω are constant.

In this motion,

a is towards centre, v is tangential and according to the shown figure, ω is perpendicular to paper inwards or in ⊗ direction.

2. Accelerated circular motion in which v and ω are increasing. So, ais in the direction of v and α is in the direction of ω.

In the figure shown, α and ω both are perpendicular to paper inwards. Further,

and

3. Retarded circular motion in which v and ω are decreasing. So, at is in the opposite direction of v and α is in the opposite direction of ω.

In the figure shown, ω is perpendicular to paper inwards in ⊗direction and

α is perpendicular to paper outwards in O direction.

Note: In the above figures, θ is the angle between v and a.

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188 videos|511 docs|265 tests

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