user10478

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$Mathematics$ Mathematics

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user10478

Nov 11, 2024 19:58

Okay, I'll look up those terms, thanks.

user10478

Nov 11, 2024 19:52

@copper.hat Hmmm, not sure in which way exactly. For example, if I have a function z of x and y, and I just take a partial wrt x and set it to the constant 0, I'll get some relation between x and y which I can substitute into the objective function to reduce it to either z of x or z of y.

user10478

Nov 11, 2024 18:02

In an (unconstrained?) optimization problem, is there an interpretation of the function you get if you set only some of the variables' partials to zero and plug the result back into the objective function, as opposed to the entire gradient?

user10478

Oct 3, 2024 01:37

@Jakobian fair point

user10478

Oct 2, 2024 18:59

Would be be fair to say that the law of identity (for all a, a = a) is not likely to be considered mathematically beautiful? Or will people want to fight me on that? Is there an example of some math too mundane to be beautiful without much controversy?

user10478

Sep 13, 2024 18:52

Say I solve a problem, maybe with some branching computations as opposed to each step following linearly from the previous step, and in the end I choose a set of steps within the computations which seem important. So I have some steps I'll label A, B, C, D, E, F..., and I want to understand which pairs of these steps are "bijection-like."
For example, I would want to represent whether knowing B lets you compute D and knowing D lets you compute B. If you can't, I would want to represent the direction of "information loss." I guess for a linear computation, this could be represented for al…

user10478

Aug 23, 2024 22:38

Does anyone see why none of my four attempts at writing the Lagrangian form of this program yield the same answer as the non-Lagrangian form? i.imgur.com/8tYQrcJ.png

user10478

Aug 16, 2024 16:10

(that is, generalizing countable additivity further to include both countable and uncountable)

user10478

Aug 16, 2024 16:10

Is there an intuitive explanation of why uncountable additivity in probability theory in impossible, without needing to know measure theory?

user10478

Jul 30, 2024 21:18

Is this expression (i.imgur.com/6usYwhW.png) supposed to be the law of total probability, or something else? It looks mistaken to me, like x and D should both be in both probability "denominators" in the integrand or something.

user10478

Jul 24, 2024 20:16

Is there a good intuition for why a primal constrained minimization problem is equivalent to the max over the dual variables of the min over the primal variables of the Lagrangian?

user10478

Jul 12, 2024 23:10

The solution to $y' = a(t)y$ is $y = Ce^{\int a(t)dt}$, so if I start with a general solution and assume it's from a linear first order ODE with no forcing term, I can solve for $a(t)$ to recover the ODE. I'm having trouble doing this with recurrences. The only solution I can find to $z[n + 1] = b[n]z[n]$ is $z[n] = D\Pi_{i = 1}^{n - 1} b[i]$. Is there a way I can solve this for $b[n]$?

user10478

Jul 10, 2024 18:34

Can someone explain what this procedure (reference.wolfram.com/language/ref/FindLinearRecurrence.html) is outputting? I'm not sure what the 10 in the input means (or what a minimum linear recurrence is, or what they mean by kernel here). I am trying to find the recurrence relation which has the solution $a_n = a_0(1 + \frac{.06t}{n})^n$. Is the output somehow encoding coefficients to such a recurrence, or is it doing something totally different?

user10478

Jun 28, 2024 17:03

Is this a general fact of differentiable objective functions (perhaps related to the fact that the steepest uphill/downhill direction is always perpendicular to the contour lines), or does it hinge on some special fact about regularization?

user10478

Jun 28, 2024 17:03

I'm watching a lecture on regularization, and the professor seems to be making the assumption that solving a constrained optimization problem will always yield the feasible point which has the shortest distance to the minimum (or maximum as appropriate) of the objective function. This isn't entirely obvious to me.

user10478

Jun 19, 2024 03:13

@Jakobian Sorry, I didn't see latest reply right away. Yeah it might be better if I make a picture in Mathematica at a later time and show what I have in mind.

user10478

Jun 19, 2024 02:03

Can I treat a $\theta$ obtained in that way as a random variable?

user10478

Jun 19, 2024 01:38

I was just wondering if those probabilities over $\theta$ automatically sum to $1$ and satisfy sigma additivity, so as to constitute an actual probability distribution.

user10478

Jun 19, 2024 01:19

@Jakobian Yeah, so if for example $\theta$ happened to be discrete, your new probability distribution be $Pr(\theta = 1)$, $Pr(\theta = 2), ... and the old variable $f$ used to range over would now be a parameter.

user10478

Jun 19, 2024 00:01

Would this distribution in $\lambda$ adhere to the probability axioms in that case (for each fixed $k$)? Is this basically what Bayesian statistics is doing under the hood by positing a distribution of belief over a parameter?

user10478

Jun 19, 2024 00:01

Is it possible to, if not always then at least for most famous probability distributions, construct a (discrete if appropriate) surface by gluing the PDFs with neighboring parameter values next to each other, and reslice the surface in an orthogonal direction to get another probability distribution? For example, the surface could be the glued Poisson PDFs for each value of $\lambda$, which would then be resliced treating $k$ as the parameter and $\lambda$ as the support variable.

user10478

Jun 2, 2024 18:03

okie

user10478

Jun 2, 2024 18:02

Okay, I may not be familiar with some of the notation and concepts here, but essentially think of Minkowski distances as lengths you could get by trying different inner products for your vector space?

user10478

Jun 2, 2024 17:50

Yeah that should be Euclidean.

user10478

Jun 2, 2024 17:44

Should I think of Minkowski distance as a generalization of Euclidean distance completely unrelated to the generalization of Euclidean distance found in Linear Algebra by generalizing dot products to inner products and/or ordinary geometric vectors to other vector spaces such as functions, or do these generalizations overlap?

user10478

May 7, 2024 05:28

Does anyone know the name of the logical form P, ¬ Q, therefore ¬ (P -> Q)?

user10478

Apr 26, 2024 02:28

Can someone explain if this (stackoverflow.com/a/64974659/2364796) makes any sense at all Linear Algebraically? I don't get "The literal position of that vector maybe garbage because by checking it in the euclidean space, you will anchor it on the origin." I get vector addition geometry, but surely the only way Queen - Woman ≈ King - Man and Queen - King ≈ Woman - Man is if the vector Queen - Women is near Royal or some similar words and Woman - Man is near some gender toggling words?

user10478

Mar 24, 2024 01:59

I think this might even be fine for nonlinear programs (aside from the matrix vector notation, I don't really know how to notate nonlinear programs but the linearity here appears incidental to the form of $f$, $g$, $h$, and the constraints).

user10478

Mar 24, 2024 01:58

Put another way, $(min_{\vec x}\ (-3\ f(\vec x))$, $A\vec x \leq \vec b$, $\vec x \geq \vec 0) = (-3\ max_{\vec x}\ f(\vec x)$, $A\vec x \leq \vec b$, $\vec x \geq \vec 0)$; you have to toggle between mins and maxes to pull out negative constants. Similarity for additivity with negatives.

user10478

Mar 24, 2024 01:58

and $(extr_{\vec x}\ g(\vec x)$, $A\vec x \leq \vec b$, $\vec x \geq \vec 0) + (extr_{\vec x}\ h(\vec x)$, $A\vec x \leq \vec b$, $\vec x \geq \vec 0) = (extr_{\vec x}\ (g(\vec x) + h(\vec x))$, $A\vec x \leq \vec b$, $\vec x \geq \vec 0)$, where $extr()$ picks out whichever of $min()$ or $max()$ is needed to make the linearity conditions true.

user10478

Mar 24, 2024 01:57

@Jakobian Okay, I was thinking about this more for some examples in $A = I, \mathbb{R^n} = \mathbb{R}$. I believe LPs have a property I'm tentatively calling "vague linearity." If I'm correct, then $(extr_{\vec x}\ (k\ f(\vec x))$, $A\vec x \leq \vec b$, $\vec x \geq \vec 0) = (k\ extr_{\vec x}\ f(\vec x)$, $A\vec x \leq \vec b$, $\vec x \geq \vec 0)$

user10478

Mar 23, 2024 19:58

Okie, thanks

user10478

Mar 23, 2024 19:56

@Jakobian Okay, I think I see the problem. So is there any notion of adding two LPs (either in objective function, constraints, or both) if you play with the details (change $max g$ to $min -g$, switch back and forth between primal and dual, and so forth), or can LPs only be scaled by constants and not added?

user10478

Mar 23, 2024 19:48

@Jakobian So, if $g = -h$, then every $\vec x$ in the input space maximizes $g + h$ with objective function value $0$.

user10478

Mar 23, 2024 19:41

Hmm, maybe not actually.

user10478

Mar 23, 2024 19:38

Yeah

user10478

Mar 23, 2024 19:37

@Jakobian Doesn't any non-trivial instantiation of that violate non-negativity constraints?

user10478

Mar 23, 2024 19:34

...$(max_x g(\vec x)$, $A\vec x \leq \vec b$, $\vec x \geq \vec 0) = 4$ and $(max_x h(\vec x)$, $A\vec x \leq \vec b$, $\vec x \geq \vec 0) = 3$ jointly entail $(max_x (g(\vec x) + h(\vec x))$, $A\vec x \leq \vec b$, $\vec x \geq \vec 0) = 7$ when $g$, $h$, and the constraints are linear?

user10478

Mar 23, 2024 19:33

I'm surprised this doesn't appear to be on Wiki. Is there a way to exploit linearity to shortcut solving linear programs? If I'm thinking about the geometry correctly, scaling by constants is unproblematic ($(max_x f(\vec x)$, $A\vec x \leq \vec b$, $\vec x \geq \vec 0) = 5$ entails $(max_x 2f(\vec x)$, $A\vec x \leq \vec b$, $\vec x \geq \vec 0) = 10$ whether or not $f$ is linear, or even if the constraints are made nonlinear), but do you also get...

user10478

Mar 21, 2024 01:44

It's kind of revolutionizing math on Youtube. It's almost hard not to come across them searching for arbitrary topics.

user10478

Mar 21, 2024 01:44

Like, 3B1B has similar videos on other math subjects (Differential Equations, Probability, Neural Networks, etc.), and the channel owner Grant Sanderson did a great series for Khan Academy on the differential half of Multivariate/Vector Calculus. He also hosts a yearly math video competition which has spawned tons of other content creators making similar style videos.

user10478

Mar 21, 2024 01:37

What are you trying to learn? More linear algebra or something else?

user10478

Mar 20, 2024 22:30

Does Von Neumann's Minimax Theorem apply to zero-sum games only if they consist of a single move made by each player, or also in zero-sum games with several alternating moves? If the latter, then why does it not imply that in Chess neither white nor black has an edge? If the former, then why does it appear when players use multiplicative weights to play zero-sum games over a sequence of days (Example: youtube.com/watch?v=-_pIUD5Jzl0)?

user10478

Mar 16, 2024 16:49

I think I want an "interesting" maximum as the parameter varies (i.e., left and right neighbors along the dimension of varying the parameter are both strictly less), not just an invariant.

user10478

Mar 16, 2024 16:46

I might have asked the wrong question.

user10478

Mar 16, 2024 16:44

hmm

user10478

Mar 16, 2024 16:33

This appears to be false of the Gaussian and Poisson, according to an eyeball test.

Wolfram Mathematica

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user10478

Aug 26, 2024 21:25

user10478

Aug 26, 2024 21:25

I'm doing something wrong with the Lagrangian but can't figure out what. This is my correct (verified on Desmos) attempt to enter the constrained optimization form of a problem followed by four incorrect attempts to enter a Lagrangian form (adding VS subtracting the substantive constraints, and keeping VS dropping the non-negativity constraints). What do I have wrong?

The Ivory Tower

General discussion for academia.stackexchange.com

user10478

Mar 23, 2024 17:09

Are questions about how specific subjects are classified into broader disciplines, what (if any) subject or dialog lies at the intersection of multiple other specified subjects (possibly from different disciplines), etc., considered to be on topic?