Applied Mathematics

Hi. A 2 metre long solar panel has its frame supported with a post(s) into soil. The lower end is held anchored to flat soil. What is an optimal position "?" to get less convex bending or concave sagging ? Its ~20Kg weight distribution is very uniform, there is nothing more massive at any place.

If not, when it is ? Can the make it a function to understand the nature of the formula ?

Can anyone give me another example of outrageous large numbers in the real universe? Here is the best I can do for now. The number of tiny Planck Volumes in the observable universe is about 10 to the power of 185. The number of Planck Volumes in a level teaspoon is about a googol (10 to the power of 100). But here is what surprised me some more. The number of Planck Volumes in a level teaspoon is a quadrillion times the number of teaspoons in the observable universe, and if you subtract 10 to the power of 85 from a googol, the remainder is very, very nearly a googol. Now please surprise me with some big numbers. Graham's number is incredibly large but if you subtract Gra…

Good day! I would like to present an idea (hopefully, a fresh one) in the field of logic called "Context-Gradual Referential Logic." The theory is described in the author's paper by Maxim Petrov-Verin. I have attached the paper. Context-Graded Referential Logic — Max. Petrov-Verov. — Self-published, 2025, 4 p.pdf The author does not claim scientific novelty, academic recognition, or educational value — these are his personal reflections (see disclaimer). Possible value of the work: It presents a very simple and intuitive system (accessible even at the high school level) based on well-known principles from modal logic, fuzzy logic, and referential approaches. Combined, the…

The "Twistable Truth" Framework Math Is a Language, and We’re Just Getting Started Picture this: English, Japanese, Russian, Spanish—they’re all languages, right? Each one’s got its own rules, its own flavor, shaped by the people who speak it. They’re different, but they all carry meaning, and you can translate between them. Now, what if math’s the same? We’ve been speaking one dialect—let’s call it "Standard Math"—with its tidy equations and single answers, like 1 + 1 = 2. But what if that’s just one way to talk? What if math is a whole family of languages, each with its own syntax, its own truths, waiting to be explored? We’re not here to say we’ve …

When you have to multiply two quantities with different units together, how do you interpret the meaning of the combined units? Take the formula for momentum, p=mv, for example - you have (say) kg times m/s. I know how to interpret m/s - for every second that passes by, so many meters are traversed. But what does kgxm mean? For every second, there are so many kilogram-meters. But what is a "kilogram-meter"?

Airy stress function in boundary conditions in 2D elasticity has a requirement that its x and y derivatives be related to the x and y boundary tractions Rx and Ry, i.e.: dAiry/dy=Rx, dAiry/dx=-Ry. Then dRx/dx=-dRy/dy? How to understand this identity?

Where can one find warping and bending function formulae of common cross sections of a beam in St Venant problems in elasticity?

Hello, I am currently dealing with a mathematical problem I am still not able to resolve: Imagine there is a sphere with radius 1 and the center is just in the middle of the coordinate system, so the equation is (x - 0)^2 + (y - 0)^2 + (z - 0)^2 = r^2 And there is a point with the coordinates (5, 0, 0) which is the tip of a cone that is directed directly towards the center of the sphere till the end of the sphere. What I now want to calculate is the exact angle this cone needs so that the volume it occupies within the sphere is exactly half of the sphere. And I need to figure out how long the line between the first intersection of the cone when hitting…

Hi. I don't know much about interpolation or curve fitting and need help, please. Fit/approximate a function through the points (0,0), (7,11), (13,33). The type of function is of the form y=tan(exp(ax^2+bx+c)). Any help is appreciated. I hope I have given enough information. Thanks.

This manuscript is submitted solely for feedback and is not intended for publication in any editorial journal. To streamline the review process, only the proving section is included, with non-technical components such as the Introduction, Abstract, and References removed. Readers are invited to provide constructive feedback on both the solution and the writing style. Suggestions, such as merging subcases, eliminating the Methodology section, or simplifying the essay for conciseness, are particularly welcome. Your insights will greatly enhance the clarity and rigor of this work. This proof has been available as a preprint on ResearchGate, where readers can review…

First of all I apologise for the bad English, I am Austrian so my mother tongue is German and I used Google Translate for most of the translations. I also have not mathematical education; this is why this document don´t fulfil the standards for a scientific publication. But I will do my best to present my Theory and give the readers an appropriate understanding of it. In this document I will describe my Theory of the distribution of prime numbers. This includes a attempt to proof the infinity of twin primes as well as an explanation of the emergence of prime gap´s and prime k-tuple´s. Known Facts: "A prime number is a number that is only divisible b…

Hello, In short, we can transform the SAT problem into AX=1 withX a vector and 1 the vector of 1, but the determinant A is often zero, so A^-1 does not exist. However, assuming that since I have the form of the A determinant, I could construct a polynomial-time algorithm where I add more variables to avoid having a zero determinant, I could then find 𝐴′^-1, and the solution would be in polynomial time to find X': X in X′.

I have a function y (x) defined over a range from x0 to x1 I want to know the probability that y will equal some number within a given interval if I choose a random value for x.

Suppose we play the game where I draw a N(0, 1) random number A, show it to you, then draw another independent N(0, 1) one B, but before showing it to you ask you to guess whether B will be larger than A or not. If you are right, you win if you are wrong you lose. What is your strategy, and what will be your long term success rate if we keep playing? Author's Solution: Since the numbers are zero-mean, the strategy is to guess the sign of the difference B − A as positive if A < 0, and negative otherwise. The long-term success rate will be \(\frac12 P(B-A) > 0| A < 0) + \frac12 P(B-A) \leq 0 | A \geq 0)\) Due to the symmetr…

I have a right angle, the 90 degrees angle is at the top right corner. A force is applied along AB. AB is the horizontal line from the 90 degree corner marked A to the left marked B, with a length 10.5 cm. The force along AB is 102.31 N. The force AC is the vertical line of action measured at a length of 13.5 cm. The hypotenuse final length BC is 17 cm. Hope your with me to this point! According to a engineering book I've read that includes the methods for calculating the Traingle of Forces, where two or more forces are known, the side lengths of the triangle are measured and then multiplied by the acting force to record the results, however, either I'm missing …

I believe it was in 1995 that Andrew Wiles proved Fermat's Conjecture, that no integer greater than n=2 could possibly satisfy the equation an + bn = cn Not knowing anything about Maths beyond GCSE I hope I've got that right. Does Andrew Wiles' proof have any practical application? Eg does it allow space engineers or architects or anyone else to do something they could not do previously? Or is this purely knowledge for its own sake? Not that there's anything wrong with that of course! Either way it's still beautiful and fascinating cheerz GIAN🙂XXX

Greetings. There is a scheme the electric utility companies credit / not pay! for solar generation in dwellings, at 75% of the electric power rate. That is; if my solar panels on the roof deliver/inject electric energy into the grid which is priced at -say $0.10 per KWh- If I inject/deliver into the utility electric grid 100KWh @ $0.10/each KWh = $10 What does it mean... The utility company gives me credit of $7.50 but received $2.50 from my neighbors... So the utility company cost of the $7.50 credited to me costs them $2.50 less, being actually $5.00 ? Am I seeing it wrong ?

The robot arm dilemma is a mathematical riddle about probability that I devised. It’s called this because I use the example of a robotic arm to siege it. But there are a thousand ways to explain this riddle, which makes us think it’s a lot more around us than we think. I apologize for any errors of English but I am Italian and I translated the text with the translator Anyway, the riddle says, "consider a list of n elements, which contains the numbers from 1 to n (it is not necessary to place them in ascending order, but it is more convenient to imagine it. potentially they could be in random positions). This list is literally written on paper as a shopping l…

I don't understand the author's solution given to the following question. Would any member of this forum take some efforts to explain this solution?

For the two attached pictures below, let [math]{P_n}[/math] denote the proportion of the big circle covered by the small circles as function of the number of the small circle in the outer layer. Find the expression for [math]P_n[/math] and compute [math]\lim_{n\to \infty} P_n [/math]. Would you answer this question in details? **Author's answer to this question as follows:** Let R be the radius of the big circle, [math]r_n[/math] be the radius of the small circles in the outer layer, and \(R_n\) be the radius of the circle encompassing all small circles except those in the outer layer. Then in both cases \(P_n R^2\pi = n r^2_n \pi + P_n R^2_n \pi (\star)\) and…

If one knows the physics He can do things with graphical coordinates that can tool the microrobotics that control the program-user interface to do the right moves. I've demonstrated that motion can be just as accurate as a UR3 if I'd had more time to disassemble the device "If I have C=1 and A=<1 anything I do has a small probability of falling onto an exact degree. .99^2 + B^2 = 1, B=sqrt(1-.9801)=0.14106735970. sin(1deg)=.01745240643, for what I just got 1/x=.01745/0.14107=0.1237 so 1=0.1237x, so x=1/0.1237=8.08407437348deg". Keeping me at a disadvantage won't do you any favors in the long run.

I believe to have found a pattern for the goldbach conjecture, however I need help formalizing it into an equation and proof.

This is just an attempt to associate sums or differences of prime numbers with points lying on an ellipse or hyperbola. Certain pairs of prime numbers can be represented as radius-distances from the focuses to points lying either on the ellipse or on the hyperbola. 1. The ellipse equation can be written in the following form: |p(k)| + |p(t)| = 2n 2. The hyperbola equation can be written in the following form: ||p(k)| - |p(t)|| = 2n where p(k) and p(t) are prime numbers (p(1) = 2, p(2) = 3, p(3) = 5, p(4) = 7,...), k and t are indices of prime numbers, 2n is a given even number, k, t, n ∈ N. If we construct ellipses a…

Hello, I have π(x)~x/ln(x), the larger the value of x, the more true this relationship becomes. Suppose I have two very large twin prime numbers, then I would have n/ln(n)~N so E(n/ln(n))=N and (n+2)/ln(n+2)~N+1 so E((n+2)/ln(n+2))=N+1 with Using E to denote the floor function. So, if there exists an infinity of twin primes, the equation E((n+2)/ln(n+2))=1+E(n/ln(n)) would have a solution where n is very large. If this equation does not have a solution with a sufficiently large n, then there are a finite number of twin primes that are less than a specific value of n. This equation is true only if I have infinity = 1 + infinity,So, twin prime numbers are i…