Positive Definition and 987 Threads

Prove that $\dfrac{y^2z}{x}+y^2+z\ge\dfrac{9y^2z}{x+y^2+z}$ for all positive real numbers $x,\,y$ and $z$.

For a pair of positive integers $(a,\,b)$, $Q(a,\,b)$ is defined by $Q(a,\,b)=\dfrac{a^2b+2ab^2-5}{ab+1}$. Let $(a_1,\,b_1),\,(a_2,\,b_2),\,\cdots, (a_n,\,b_n)$ be all pairs of positive integers such that $Q(a,\,b)$ is an integer. Calculate $\displaystyle \sum_{i=1}^n a_i$,

So this is more of an intuitive question rather than a mathematical one. I present the problem. Assume I have 2 charges of charge +q at a distance r from each other on the z axis. Position of two charges is (0,0,r/2) and (0,0,-r/2). Assume now that I want to calculate the force these two...

So I'm confused on a few things about this circuit. I know my math is right (since I got the absolute values correct), but the signs are off since I can't tell when the voltage, current or if the general power equation is positive or negative. A) So my first question is, where is the current...

If $x$ and $y$ are positive real numbers, prove that $4x^4+4y^3+5x^2+y+1\ge 12xy$.

Let $a$ and $b$ be positive real numbers such that $a+b=1$. Prove that $a^ab^b+a^bb^a\le 1$.

Show that the equation $x(x+1)(x+2)\cdots(x+2020)-1=0$ has exactly one positive solution $x_0$ and prove that this solution $x_0$ satisfies $\dfrac{1}{2020!+10}<x_0<\dfrac{1}{2020!+6}$.

Let $a,\,b,\,c,\,d,\,e,\,f$ be positive integers and $S=a+b+c+d+e+f$. Suppose that the number $S$ divides $abc+def$ and $ab+bc+ca-de-ef-df$, prove that $S$ is composite.

$a,\,b$ and $c$ are positive integers that satisfy the inequality $ab+3b+2c>a^2+b^2+c^2+3$. Evaluate $a+b+c$.

X^3 + y^2 - z = z^3 - x^2 + y what are all the positive integers for x,y, z that satisfy this equation?

I need some help resolving the follow problem. I really don't know where to put the "twice as large as the resultant force on Q3" in order to build an equation. Thank you !

Proof: Let a be a even positive integer of the form a=2m & b of the form b=2n (This is where b is a even positive integer) ab = 2m*2n = 2(mn) = Let k = mn = 2k Therefore, ab is even. Let a be a even positive integer a=2m & b be a odd positive integer b = 2n+1 ab = (2m)*(2n+1)...

Hi all, I have a positive displacement (diaphragm) pump that I am looking to use to supply spray nozzles. The pump has a max pressure rating of 69 bar at 30.6 L/min (1450 rpm). I however want my nozzles to spray at 15 bar pressure and from the nozzle performance data sheet that equates to 0.68...

Since distances increase, their first derivative which is velocity (Hubble constant) should be positive if not increasing too. Accelerated expansion needs the velocity to increase. What about the third derivative which is acceleration? An accelerated universe could have third derivative (called...

I think we can say that (u,v,u^v) is a positive basis, so as (w^v,v,w) and (u,w^u,w). (1) So u^v = βw v^w = γu w^u = λv where λ, β, and γ > 0 (*) (u^v, v^w,w^u) = (βw,γu,λv) \begin{vmatrix} 0 & 0 & β \\ γ & 0 & 0 \\ 0 & λ & 0 \\ \end{vmatrix} This determinant is positive by (*) What you...

How many of the first 1000 positive integers can be expressed in the form $\lfloor 2x \rfloor+\lfloor 4x \rfloor+\lfloor 6x \rfloor+\lfloor 8x \rfloor$, where $x$ is a real number?

Determine all possible integers $n$ for which $\dfrac{n^2+1}{\lfloor{\sqrt{n}}\rfloor^2+2}$ is an integer.

Hi all! I am currently working on a problem that involves calculation of Green functions. I am having serious doubts on some part of the calculation, as I am getting a GF that has positive values of its imaginary part. According to Lehman's rule, the retarded GF can be decomposed by the spectrum...

Can we predict the positive parity, and zero spin of the Higgs boson from the decay mode: 𝐻 → 𝛾𝛾?

Hey! 😊 Let $A=L^TDL$ be the Cholesky decomposition of a symmetric matrix, at which the left upper triangular $L$ hat only $1$ on the diagonal and $D$ is a diagonal matrix with positiv elements on the diagonal. I want to show that such a decomposition exists if and only if $A$ is positive...

I know how to find integral solutions of linear equations like x+y=C or x+y+z=C where C is a constant. But I don't have any idea how to solve these type of questions.I am only able to predict that both x and y will be greater than 243554.Please help.

3.7.4. The sum of two positive numbers is 16. What is the smallest possible value of the sum of their squares? $x+y=16\implies y=16-x$ Then $x^2+(16-x)^2=2 x^2 - 32x + 256$ So far ... Hopefully

Not another Corona thread surely?...yes but this time it's something positive we can take forward, when it's over. Several posters @Tom O [USER=283516]@davenn and others have made reference to reduced seismic activity due to lock down. That's great for Earth sciences and reduction in RTAs but...

Start with a closed surface of positive Gauss curvature embedded smoothly in ##R^3##. At each point, choose two independent eigenvectors of the shape operator whose lengths are the corresponding principal curvatures. By declaring them to be orthonormal one gets - I think - a new metric on the...

interesting: https://www.ncbi.nlm.nih.gov/pubmed/32133832?fbclid=IwAR3KtTRFTp9eb0aA-8ThVONEMGeGhecOnjMJtIblbClp8ufahx71HOFSWdg Objective: As the prevention and control of COVID-19continues to advance, the active nucleic acid test screening in the close contacts of the patients has been...

The term “negative mass” gets puts forth occasionally, and it’s definitions can sometimes be unclear. the topic I’m interested in is particles which have both positive inertial mass and negative gravitational mass. So far, what theories do physicists have of speculating on the existence of...

My attempt I calculated the partial derivatives of n wrt P and T. They are given below. ##\frac {\partial n}{\partial P} = \frac{nb -1}{\left(2an-Pb-3abn^2-kT\right )}## ##\frac {\partial n}{\partial T}= \frac {nk}{\left(2an-Pb-3abn^2-kT \right ) }## I know that if the partial derivative is...

ok I don't don't know de jure on this so ... is it just plug and play?? find factors of -48 $-1(48)=-48$ $-2(24)=-48$ $-3(16)=-48$ $-4(12)=-48$ $-6(8)=-48$ check sums for positive number $-1+48=47$ $-2+24=22$ $-3+16=13$ $-4+12=8$ $-6+8=2$it looks like c. 5

I am needing clarification for a concept. I understand that electrons carry a negative charge and that protons carry a positive charge. I also understand that a plastic rod picks up electrons when I rub it with a piece of wool. From the conservation of charge, the piece of wool must have a...

ik that i have to add the positive and negative ions but the cross section (surface) confuses me how do i resolve this?

In textbooks or in lectures it is said that : Work can be either positive or negative: if the force has a component in the same direction as the displacement of the object, the force is doing positive work. If the force has a component in the direction opposite to the displacement, the force...

Can anyone explain what the positive energy theorem is layman terms ?

I am working out an example problem from one of my textbooks and I am a bit confused on why a value is negative. The problem asks: Calculate the final speed of a free electron accelerated from rest through a potential difference of 100 V. This is a conservation of energy problem. Ultimately you...

As per (b) in the above image, or easily solved with t=rFsin(theta), the perpendicular force is 260N. When inputting that value into the equation for torque, the value for torque is 520Nm, as per t=2.00*300*sin60. Because the wedge prevents the door from moving, the torque on/at the doorknob...

Summary: Seeking ideas how to create a positively static charged open end container to attract negatively static charged plastic items. Hi, I have got a first world problem that I'm trying to overcome. We have a small home based business and sometimes pack a few hundred items at a time using...

Quote: "In mathematics, an exponential function is a function of the form f(x)=ab^{x} where b is a POSITIVE REAL number" Wait. Give me a reason, why exponent base must be positive and real? What happens if b<0?

What is the physical difference between negative potential energy and positive potential energy? PS- When I was reading about surface tension I came through the following line - ' The molecules at bulk inside the fluid has lesser negative potential energy than those on the surface.' In this...

Hi all, I read on "Intoduction to Elementary Particle Physics" (A. Bettini) that baryons with positive strangeness cannot exist. I don't know what to conclude from this sentence: sigma-baryons have negative strangeness, since there's a sigma as valence quark. But these baryons have, of course...

I am having trouble solving the following problem. I am given two positive charges on the x axis: I know that the electric field strength at point P is ##E=150 \frac{V}{m}##, ##d=1.8m## and ##a=2.5m##. I want to find the charge of ##Q##. As far as I know, the electric field on the y-axis...

Suppose , there is a body B1(-vely charged) and body B2(+vely charged). When both are on contact, electrons move from B1 to B2 as B2 is having electron deficit. But is reverse also possible that protons move from B2 to B1 as B1 is having proton deficit.

Minute physics made an interesting video for laymen describing the feasibility of muon catalyzed fusion, specifically stating that the current state of technology would lead to an endothermic process. I was wondering if it would be possible to use muons in different fusion configurations, for...

I decided to set up ##\LaTeX## in my PC, windows 7, 32 bit. The pc has a legal version of QuickHeal Internet Security antivirus. I downloaded TexStudio and MikTex. When I started installing TexStudio, QuickHeal said that the installer file was a potential keylogger, and put it into quarantine...

I'm stuck on a proof involving the Bolzano-Weierstrass theorem. Consider the following statement: $$f'(x)>0 \ \text{on} \ [a,b] \implies \forall x_1,x_2\in[a,b], \ f(x_1)<f(x_2) \ \text{for} \ x_1<x_2 $$ i.e. a positive derivative over an interval implies that the function is growing over the...

Prove the existence of a positive integer divisible by $2019$ the sum of whose decimal digits is $2019$.Source: Nordic Math. Contest

Homework Statement Do five positive integers exist such that the positive difference between any two is the greatest common divisor of those two numbers? Homework EquationsThe Attempt at a Solution I found four such numbers, ##\{6,8,9,12\}##. I did this in an ad hoc way though without any real...

This is about the history of electromagnetism. I already know about the convention that defined which electric charge is the "positive" one, which ended up making the electron a negative particle. But what about the "positive" direction of the magnetic field? Does anybody here know how it was...

Homework Statement Prove that the set of positive rational numbers is is countable by showing that the function K is a 1-1 correspondence between the set of positive rational numbers and the set of positive integers if K(m/n) = p_1^{2a_1}p_2^{2a_2}...p_s^{2a_s}q_1^{2b_1-1}...q_t^{2b_t-1}...

Homework Statement A 7.2-kg mass is hanging from the ceiling of an elevator by a spring of spring constant 150N/m whose unstretched length is 80 cm. What is the overall length of the spring when the elevator: (a) starts moving upward with acceleration 0.95m/s2 ; (b) moves upward at a steady...