To speak of "fitting" a function to a data set does not describe a specific mathematical process. Statistically, a person who speaks of "fitting" z = f(x,y) = Ax^2 + Bxy + Cy^2 + Dx + Ey + F to a dataset is most often seeking the function ##f## that minimizes
##g(A,B,C,...) = \sum_{i=1}^n...
Imagine we have 5 blocks with the symbol "|" printed on them. We also have 5 blocks with the symbol "P" written on them. We arrange the blocks in a row with the condition that the first and last block in the row must have a "|" on it.
This produces patterns like:
|PPP|P||P|
That pattern...
A vector in ##C^n({R})## is also vector in ##C({R})##.
I think what you mean to ask about is a situation where ##u(t)## is ##n## times continuously differentiable, but not ##n+1## times continuously differentiable. You are worried that the ##u^n(t)## term in ##F(u)## might cause ##F'(u) =...
Yes, your are correct that it does, if you are not thinking of "the second" boy as being the name of one particular boy and thinking of all prizes as being identical.
Your calculation is treating the boys as being distinct, but treating the prizes as being indistinct.
For example , the...
Where do you account for the First boy getting 4 prizes and the third boy getting 1 prize?
It's simpler to think of each prize chosing a boy who gets it.
Asking about the "proper" way to "combine" measurements is not a well defined mathematical question. If you don't want to tackle the sophisticated concepts involved in statistics, find some authority that has done similar work and copy what they did.
A slightly inferior version of that...
You have to be specific about the interpretation of the number you wish to publish. Your example of the "uncertainty" in measuring with a ruler doesn't involve probability. If the ruler can be read to ##\pm## a tenth of an inch, then a measurement with the ruler is guanteed to be within that...
We can be more bold about generalizing. A general viewpoint is to consider situations where decomposition of a function into sums of functions is equivalent to solving a set of simultaneous equations and a relation among values of component functions can be used to reduce the number of...
You aren't specific about the meaning of "error". (What is meant by "the mean" is also ambiguous.)
On one hand, a measuring instrument can have a calibration guarantee such as ##\pm##2%.
On the other hand, a particular person can have a weight that differs from the mean weight of the...
Good idea.
A well known technique in mathematics is treat certain sets of functions as infinite dimensional vector spaces. In your original post, you make an analogy to 2 dimensional vectors.
Look at the definition of a vector space - the sophisticated definition, not the secondary school...
That's what I did on my second try. Working at a bench, I soldered the brass adapater from 3/4 inch copper to 1 inch PEX onto a copper couping. Then I soldered the coupling on the copper line, working in a trench. So far, no leaks.
You needn't give up. You should exert yourself to define "dominant over the other" precisely. What do you want that to mean? If we are given two functions f(x), g(x), how do we decide if f(x) is "dominant over" g(x)?
Restrict the type of functions that are considered. You picked a...
That's one alternative technology to burying lines. The way I imagine the municipal water lines in my town, they wouldn't form loops, but I've never looked into the matter.
Circulation pumps are commonly used in large hotels and motels on hot water systems to provide "instant" hot water...
In the USA, the usual solution to keeping water lines from freezing is to bury them. In cold climates this implies digging deep trenches. (On another forum, a person from Minnesota said water lines are typically buried 7 ft deep.) In the days before modern excavating equipment, were...
I don't know what you mean specifically by "handled". Are you asking how to compute a sampling distribution?
To talk about a sampling distribution, you must first specified what statistic is being sampled. One can compute statistics from a sample in various ways. For example, from a sample...
A photo shows two plants that are known locally (southern NM, USA) as "pigweed". The shorter plant has burrs that develop along the main stem. The taller plant has no burrs. What is the scientific name for these plants? Are these plants distinct varieties? - or different genders of the same...
I don't detect a mathematical question in that. It sounds like a question about psychology or sociology.
Use notation that distinguishes between an outcome versus the probability of an outcome.
"##X_1 = 0##" is an outcome, not a probability.
Don't use the symbol "+" to mean the word "and"...
The flux applied to copper pipes and fittings before they are soldered together is said to suck the solder into the space between the fitting and the pipe. Empirically, the solder does flow into the narrow space. What physical phenomenon causes this? - an actual vacuum?
I like to hear from anyone who has tried "loose lay" vinyl plank flooring. (e.g. ).
I don't understand how it can possibly work. Local hardware stores don't stock it, so I haven't seen any samples.
The dead apparati do not interpret what they are measuring. Investigating probabilities experimentally involves repeated measurements of similar events. It's human's who judge that a series of actions repeats "the same" experiment. Two repetitions of literally "the same" experiment wouldn't...
When soldering a brass fitting (e.g. https://www.pexuniverse.com/1-pex-x-3-4-pex-brass-pex-copper-fitting-adapter-lead-free-brass ) to copper (e.g. a 3/4 inch copper coupling) I think there is a danger that copper gets hot enough to melt the solder, but the brass fitting (being more massive)...
Is there a term for the general methods archeologists use to date things? - more general that the specific method of radiocarbon dating. "Historical markers" doesn't seem right.
An example is pull tabs on soda cans. There was short time period (in the USA) where the tabs came completely off...
Apparently, what you mean by "using standard limits" is assuming things like:
If ##lim_{x \rightarrow a} g(x) = L## then ##lim_{x \rightarrow a} f(g(x)) = f(L)##
and
If ##lim_{x \rightarrow a} g(x) = L_1## and ##lim_{x \rightarrow a} h(x) = L_2## then ##\lim_{x \rightarrow a} f(g(x), h(x)) =...
That rule may be an old one, but when I bought my house in the 80's, the bank didn't investigate where I got the down payment. I have the impression that the gift affidavit is recent requirement. Is that correct?
Recently, I've encountered two examples of how the Federal Housing Adminsitration requires banks to scrutinize borrowers who acknowledge receiving gifts of money to help purchase a house. The banks require an "affidavit of gift" document from the giver and a copy of a bank statement showing...
Some forum members agree with the late John Boogle and are vocal advocates for investing in "index funds". Of course, I don't know if projecting this strategy back to years ago is realistic.