Positive Definition and 987 Threads

I need an idea. Thank you.

Let ##\mathbb{R}^\omega_{+}## be the set of all sequences ##(a_n)_{n=1}^\infty## of positive real numbers. Determine the infimum $$\inf\left\{\limsup_{n \to \infty} \left(\frac{1 + a_{n+1}}{a_n}\right)^n : (a_n) \in \mathbb{R}^\omega_{+}\right\}$$

For this theorem, I'm trying to prove why ## a > 0## and ##4ac - b^2 > 0## for the function to be positive definite. My working is, ##V(x,y) = x(ax + by) + cy^2## (I try on write the function in alternative forms) ##V(x,y) = ax^2 + y(bx + cy)## However, does anybody please know where to...

why test charge is always positive ?

Show that for all positive integers ##n##, $$\binom{n}{1} - \frac{1}{2}\binom{n}{2} + \cdots + (-1)^{n-1}\frac{1}{n}\binom{n}{n} = 1 + \frac{1}{2} + \cdots + \frac{1}{n}$$

I know to break it down into its x and y components and then use Pythagorean: Acceleration in the x direction is Fx/m ---> (7.50 x 10^6*cos55) / (4.50 x 10^5 kg) = 9.56 m/s^2 Acceleration in the y direction is: (Fy - mg)/m ---> ((7.50 x 10^6*sin55) - (4.5 x 10^5* 9.8 m/s^2)) / (4.5 x 10^5 kg)...

My attempt: $$ \begin{vmatrix} 1-\lambda & b\\ b & a-\lambda \end{vmatrix} =0$$ $$(1-\lambda)(a-\lambda)-b^2=0$$ $$a-\lambda-a\lambda+\lambda^2-b^2=0$$ $$\lambda^2+(-1-a)\lambda +a-b^2=0$$ The value of ##\lambda## will be positive if D < 0, so $$(-1-a)^2-4(a-b^2)<0$$ $$1+2a+a^2-4a+4b^2<0$$...

Based on the fact of observed cosmological redshift, scientists have proposed different ideas to explain. One interesting question is whether gravity does negative or positive work now: According to universe expanding in Big Bang theory (Lemaitre, 1927), obviously gravity does overall negative...

So I have this question. I get all the working out, but then I feel like the answer should be; -2/(b-a). Then I thought -2/(-a+b) must just be the same thing... all good so far.. Then they somehow just to 2/(a-b) as the final answer.. I'm lost there. How does that conversion happen...

For part (a) of this problem, The solution is However, my solution is Am I correct? In the solutions that don't appear to plot the electric potential as units of ## \frac {k_eQ} {a} ## like I have which the problem statement said to do. Many thanks!

I am thinking of powder coating at home. I know the part to be coated is negatively charged because it is grounded. I assume the powder is positively charged by the gun. I wonder if there is a way to add more positive charge cheaply to an inexpensive gun like the one at harbor freight? I think...

Let ##x,\,y>0## and ##x+y \leq 1##. Prove that ##(1-\frac{1}{x^2})(1-\frac{1}{y^2})\geq 9##.

I wonder how it is possible that a positive charge can exert el. field beyond negative charge? Shouldn't they "connect" and therefore positive charge should stop to have el. field beyond neg. charge? I mean, I am obviously wrong about that, but can someone please explain why/how el. field from...

I am trying to reproduce the results from this paper. On page 10 of the paper, they have an equation: $$ \frac{S}{T}=\int dt\sum _{n=0,1} (\dot{c_n}{}^2-c_n^2 \omega _n^2)+11.3 c_0^3+21.5 c_0 c_1^2+10.7 c_0 \dot{c_0}{}^2+3.32 c_0 \dot{c_1}{}^2+6.64 \dot{c_0} c_1 \dot{c_1} \tag{B12} $$ where they...

hello i would like to understand something, i found the right answer but there is still something i don't understand. here is the figure and here is my correct solution what i don't understand is why F(3,Q) is 3kQ/r^2 i mean why is the 3? i only calculat the force between q3 and Q so why...

Im wondering if plasma is possible to be separated into a positive nucleus and negative electrons and contained within a magnetic bottle ? If possible, what is the most efficient method of achieving it ?

Let a and b be positive real numbers such that ##a^5+b^3 \le a^2+b^2##. Prove that ##a(a+b) \le 2##.

Let ## 5k+4=p_{1}^{k_{1}}p_{2}^{k_{2}}\dotsb p_{s}^{k_{s}} ##. Then ## 5\equiv 0\pmod {5} ## and ## 5k+4\equiv 4\pmod {5} ##. Thus ## p_{i}^{k_{i}}\not \equiv 0\pmod {5} ## for ## i=1, 2,..., s ##. Suppose all ## p_{i}^{k_{i}}\equiv 1\pmod {5} ##. Then ## p_{1}^{k_{1}}p_{2}^{k_{2}}\dotsb...

Find all triples (a, b, c) of positive integers such that ##a^3+b^3+c^3=(abc)^2##.

Hi In QM the inner product satisfies < a | a > ≥ 0 with equality if and only if a = 0. Is this positive definite or positive semidefinite because i have seen it described as both Thanks

As I was looking for an example for a metric tensor that isn't among the usual suspects, I observed that the Cartan matrix I wanted to use is positive definite (I assume all are), but not symmetric. Are the symmetry breaks in quantum physics related to this fact?

I do not mean neutral electrical charge, but a forth kind (if exists) I am in 9th grade, and someone asked the teacher if there is an electrical charge that is not positive, not negative and not neutral, maybe something in the middle of them. The teacher said that there is a charge like that...

Hello, I would like to know, if there's a closed form solution to the following problem: Given a sum of say, 3 sines, with the form y = sin(a.2.PI.t) + sin(b.2.PI.t) + sin(c.2.PI.t) where a,b,c are constants and PI = 3.141592654 and the periods in the expression are multiplication signs, what...

Hi! So, I've actually already solved this problem.. for the most part. I have split up the work into two sections, floor 0 to 10, and floor 10 to 15. From floor 0 to 10, I did ## F_{elevator} = w_{pass.} + w_{elev.} ## ## F = (70)(20 (num. of pass.))(9.8) + (800)(9.8) ## ## F_{elev.} = 21560N...

This is the code that i wrote Clear["Global`*"] Z = 500; W = 100000; G = 250; H = 100; K = 0.5; T = 30; L = 4000; P = 5; S = 2.5; Y = 1; A = 0.1; V = 2.5; J = 8000; f[x_] := 1/ x {(J*Z*x*(2*Y - x))/( 2*Y) - ((W + T*G) + ((L + T*P)*2*Z*Y*(1 - ((Y - x)/Y)^1.5))/ 3 + (H + T*S +...

In MOSCAP, why does the band stop bending as soon as the Si Fermi level touches either the conduction band (inversion) or the valance band (accumulation)?

I need to prove the following: A symmetric real matrix ##A## with positive elements ##a_{i,j}\geq 0## can’t be definite positive matrix (i.e. with only positive eigenvalues) if the following condition holds: $$\sum_{i=1}^{N-1}a_{i,i+1}>\frac{1}{2}\sum_{i=1}^{N}a_{i,i}=\frac{1}{2}\text{Tr}(A)$$...

Greetings the solution is the following which I understand I do understand why the current orientation of the Path is positive regarding to stocks (the surface should remain to the left) but I don´t understand why the current N vector of the surface is positive regarding stockes theorem...

$$\frac{1}{x}+\frac{1}{y}+\frac{1}{xy}=\frac{1}{2021}$$ $$\frac{y+x+1}{xy}=\frac{1}{2021}$$ $$xy = 2021y + 2021 x + 2021$$ Then I am stuck. How to continue? Thanks

Positive displacement pumps are used a lot in the industry for pumping water or other liquids. So why is it not used as turbines in hydropower plants?

In this problem, it doesn't tell you where is the focal length. So how do you know if it is in front or behind the object?

When I was 18 years old, I worked at an automobile repair shop doing oil changes and other light mechanical work. The automobile repair shop hired me for this work to free up their real auto mechanics to do more advanced automobile repair work. One time I was assigned the task of replacing a...

I am confused with the solution. It says ##\vec E = \frac{\sigma}{\epsilon_0}##. Shouldn't E = ##2*\vec E = 2*\frac{\sigma}{\epsilon_0}##? Electric field of the positive plate and electric field of the negative plate.

Proof: Suppose ## a ## is a positive integer and ## \sqrt[n]{a} ## is rational. Then we have ## \sqrt[n]{a}=\frac{b}{c} ## for some ## b,c\in\mathbb{Z} ## such that ## gcd(b, c)=1 ## where ## c\neq 0 ##. Thus ## \sqrt[n]{a}=\frac{b}{c} ## ## (\sqrt[n]{a})^{n}=(\frac{b}{c})^n ##...

Proof: Suppose a positive integer ## a>1 ## is a square. Then we have ## a=b^2 ## for some ## b\in\mathbb{Z} ##, where ## b=p_{1}^{n_{1}} p_{2}^{n_{2}} \dotsb p_{r}^{n_{r}} ## such that each ## n_{i} ## is a positive integer and ## p_{i}'s ## are prime for ## i=1,2,3,...,r ## with ##...

I have in the past been criricised for inappropriate postings that I could have resolved with research so this time I have done the research first. The best solution I have found is from wiki "that causes it to experience a force when placed in an electromagnetic field." What causes the force...

My attempt: $$P(\text{B is positive}|\text{A is positive})=\frac{P(\text{B is positive} \cap \text{A is positive})}{P(\text{A is positive})}$$ $$=\frac{P(\text{B is positive})\times P(\text{A is positive})}{P(\text{A is positive})}$$ $$=P(\text{B is positive})$$ $$=0.01 \times 0.99 + 0.99 \times...

Playing around with my calculator, I realized that if I do successive rooting operations on any positive non-zero number, I always get the number one. Can I conclude that the infinite root of any positive number will always be zero? If the statement is true, is there any synthesized formula to...

I got as far as simplifying the expression to $$\frac{4}{9}(n_x^4 n_y^4 + n_x^4 n_z^4 + n_y^4 + n_z^4 - n_x^4 n_y^2 n_z^2 - n_x^2 n_y^4 n_z^2 - n_x^2 n_y^2 n_z^4)$$ But that doesn't seem to be a form that is necessarily positive and satisfies the criteria of the homework statement. Little help...

The following Python 3 code is provided as the solution to this problem (https://leetcode.com/problems/subsets/solution/) that asks to find all subsets of a list of integers. For example, for the list below the output is [[], [1], [1, 2], [1, 2, 3], [1, 3], [2], [2, 3], [3]]. I am not familiar...

##m##, ##n##, ##a##, and ##b## are in ##\mathbb{Z}##. The set of natural numbers is an abelian group, so ##\{ma+nb|m,n\in\mathbb{Z}\}## is a subset of ##\mathbb{Z}##. ##a ## and ##b## are either relatively prime or not relatively prime. If ##a ## and ##b## are relatively prime, then there are...

in a cours of electrostatic when we have a positive charge and we bring another one (also postitive)we have to do work and apply a force that equals the force of repultion over the distance which seems weird because if we do that the net force will be equal to 0 and the charge will not move can...

Find the smallest positive integer N that satisfies all of the following conditions: • N is a square. • N is a cube. • N is an odd number. • N is divisible by twelve prime numbers. How many digits does this number N have?Please Explain your steps in detail.

If I connect a voltmeter with the positive terminal of a battery and leave the other wire hanging in the air, it won't measure anything since it's not a closed circuit. But if there was a way to measure the voltage between those two points, what would the voltage be? Or is this a meaningless...

Compute the number of positive integer divisors of 10!. By the fundamental theorem of arithmetic and the factorial expansion: 10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 2 x 5 x 3^2 x 7 x 2 x 3 x 5 x 2^2 x 3 x 2 x 1 = 2^8 x 3^4 x 5^2 x 7 Then there are 9 possibilities for 2, 5 for 3, 3 for...

An electron is shot horizontally. There is a proton located somewhere else, but not in the horizontal path of the electron. Is there a distance of closest approach, and how do you derive it? A physical explanation would be appreciated too.

I recently had an incident in which a person visited my office and a few days later informed me he had tested positive for Covid. He had been healthy but took the test as required for any passenger boarding an airline. He later went for another test (the next day in fact) and that was negative...

Hey! A function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is convex if for all $x,y\in \mathbb{R}^n$ the inequality $$f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)$$ holds for all $t\in [0,1]$. Show that a twice continuously differentiable funtion $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is convex iff the...

Find all positive integers $n$ for which the equation $(x^2+y^2)^n=(xy)^{2014}$ has positive integer solutions.