calculus – Differentiating from – something that is not present in the equation

$ MRS = frac {u_1} {u_2} $

Differentiating from $ x_1 $:

the book shows that you get $ u_2 ((u_ {11} + u_ {12}) dx_2 / dx_1) – u_1 (u_ {21} + u_ {22} dx_2 / dx_1) / u_2 ^ 2 $

To give some context, this is the marginal rate of substitution in economics and we are making the second order condition.

What is the complexity of the E-KRHyper test (E-hyper tableau calculus)?

Before the question, let me explain better what E-KRHyper is:

E-KRHyper is a system for generating models and proof of theorems for first-order logic with equality. It is an implementation of the E-hyper tableau calculation, which integrates an equality management based on the superposition in the hyper-tableau calculation (source: System description: E-KRHyper).

I am interested in the complexity of the E-KRHyper system because it is used in the Log-Answer question and answer system (LogAnswer: a question based on deduction
Answering system (system description)

I found a partial answer:

our calculation is a non-trivial decision procedure for this
fragment (with equality), which captures the complexity class NEXT (Source: Hyper Tableaux with Equality).

I don't understand much about complexity theory, so my question is:

What is the complexity of a theorem to be tested in terms of the number of axioms in the database and in terms of some parameter of the question to answer?

Stochastic calculus – Distances between crosses up and down in Gaussian processes

Given a Gaussian process $ g: = mathcal {GP} left ( mu, Sigma right) $,
where $ mu $ is the average and $ Sigma $ is the covariance function, I'm interested in estimating the average value $ L_m $ Of the distances between up and down with a constant level. $ u $, that is, these distances:

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In this plot, I use $ u = 0 $, but ideally I would like $ u $ be generic I suspect this is related to Rice's formula, which estimates the number of ascending crosses for a given Gaussian process and a given length domain, but I do not know how

Variational calculus … Prove that a linear functional φ[h] can not have an end unless φ[h] ≡ 0

From the definitions in the variation calculation book – Gelfand and fomin

I try to prove that a linear functional. $ varphi[h]$ can not have an end unless $ varphi[h] ≡ 0 $.

I tried the following
1- prove that $ varphi[h]$ is differentiable and uses that theorem 2 page 13 in the book

2- I tried the use that a $ J[y]$ has an end in $ t $ Yes $ J[y] -J

Any help is good.

Thank you.

The following is the mathematical derivation for Deep Learning. Could someone explain to me the differential calculus?

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Stochastic calculus – integral Ito and true martingale.

Consider a function twice differentiable. $ F $ in $ R $ with bounded
first derivative $ F & # 39; $ and a Brownian movement $ W $. Show that $ F (W_t) – frac {1} {2} int_ {0} ^ {t} F & # 39; & # 39; (W_s) ds $ It is a true martingale.

I tried to show it using this, but I just got confused and did not find any solution.

  1. Yes $ M $ it's a local martingale with continuous trajectories, so it's a real martingale and $ E (M_t ^ 2) < infty $ for all $ t geq 0 $

  2. Yes $ M $ it's a local martingale with continuous trajectories, so it's a real martingale and $ E ([M]_t) < infty $ for all t.

Multivariable calculus – How to get to the Jacobian formula in the change of variables

According to the variable change formula for the multivariate calculation,
$$ d vec {v} = left | det (D varphi) ( vec {u}) right | d vec {u} $$
where $ vec {v} = varphi vec {u} $ Y $ det (D varphi) ( vec {u}) $ It is the Jacobian matrix of the partial derivatives of $ varphi $ on the point $ vec {u} $.

How to get to this relationship (preferably conceptually)?

textbooks of calculus, linear algebra and probability and statistics for nonmathematical careers

What are the most popular textbooks of calculus, linear algebra and probability and statistics for majors that are not mathematics in the United States?
Also, what about other common math courses for undergraduate non-mathematics students?

Stochastic calculus: How do the increments of the integral of Ito come together?

Given a Brownian movement $ {W_t } _ {t in[0;T]$ and a continuous process, adapted and square-integrable (bounded if desired) $ { sigma_t } _ {t in[0;T]$ Y $ varepsilon> 0 $, I want to prove that there is a $ delta> 0 $ such that

For all $ s in [0;T]$ and all $ M en mathcal F_s $, is

mathbb E bigg (1_M max_ {s le t le (s + delta) wedge T} bigg | int_s ^ t sigma_u mathrm dW_u bigg | bigg)
le varepsilon.

by $ sigma equiv 1 $, this is easy because we only consider
mathbb E Big (1_M max_ {s le t le (s + delta) wedge T} | W_t – W_s | Big)

for which we have a limit due to the distribution of the maximum $ W $ and the increases in Brownian motion are independent of the past.

Is there anything similar for arbitrary Ito integrals (or those that satisfy some assumptions)?

Propositional calculus – Enderton logic integrity test theorem, $ Gamma coupon Theta coupon Lambda $ gratiability

I am reading the proof of the Integrity Theorem of "A Mathematical Introduction to Logic" by Enderton. I have problems to see how the following highlighted sentence is fulfilled (excerpt on page 137).

Leave $ Lambda $ Be the set of logical axioms for expanded language. As $ Gamma cup Theta $ is consistent, there is no formula $ beta $ such that $ Gamma cup Theta coupon Lambda $ tautologically implies both $ beta $ Y $ neg beta $. (This is by Theorem 24B, here the compactness theorem of sentential logic is used). Hence, there is a real task. $ v $ For the set of all the main formulas that satisfy. $ Gamma cup Theta coupon Lambda $.

I have tried to reason "by contrapositive". That is, suppose a set of formulas (sentential) $ Sigma $ It is unsatisfiable. Then, vacuously, each task really satisfies. $ Sigma $ It will also satisfy any formula at all. Thus $ Sigma $ Tautologically implies any formula. In particular, for any given formula. $ beta $, $ Sigma $ tautologically implies both $ beta $ Y $ neg beta $.

Is my reasoning correct?