I Am a Fit 16-year Old Boy Looking to Get Ripped. I Have a Gym Membership and Want a Program That I

I would start by talking to the employees, since most gym memberships have one or two (if not more) meetings with a trainer you can schedule. I'd talk with that trainer to assess your goals vs. your current state, and have them help you set up something easy to start with just so you can get in the routine of it and not get overwhelmed. It's all about your approach when beginning, and it should be a relatively easy acclimation between 1 and 3 months as long as you don't overdo it. If you push yourself and bite off more than you can chew, you may get discouraged very quickly. You can also take that introductory time period to learn how to hold your body, as posture is incredibly important, as well as learn different exercises and how to get the most of each machine/weight/workout. It goes way beyond "how much" you can bench.When you start getting more comfortable with the routine and individual exercises, assuming you have a smartphone I would look at getting a free app to manage your workout schedule. I use JEFIT. You can find schedules and programs made by other users to achieve different goals, or make your own based off of what machines or weights you have available to you. Each workout listed in the app has an animation to help keep your alignment and motion correct, and the app allows you to enter your body measurements (and body goals) and what you did as far as weights and reps. It's a great tool to help you chart your progress, and you can even take "before" and "after" pics during different stages as you go.I didn't recommend a specific program because our bodies are all so different. I had to focus a lot on my back and chest; those were the weakest parts on me, but you may not have to put in the effort I did. So what got results for me will obviously be different than what works for you. Good luck, and go for the gains! I am a fit 16-year old boy looking to get ripped. I have a gym membership and want a program that I can do 3-4 days a week that will fit in with school and other activities. what would you recommend?

1. Brownian Motion - a limit

If $f:,infty) to mathbbR$ is a continuous function and $D subset ,infty)$ a dense set, then $$forall s in ,t]: f(s) geq -a iff forall s in ,t] cap D: f(s) geq -a.$$ The implication "$Rightarrow$" is obvious and the other implication follows from the continuity of $f$ (proof by contradiction: if we had $f(s) -a

ight). tag1$$Finally, note that$$mathbbP(forall s in ,t]: B_s geq -a) = mathbbP(forall s in ,t]: B_s > -a) tag2$$which follows, for instance, from the reflection principle or from the strong Markov property of Brownian motion (see this answer). Combining $(1)$ and $(2)$ proves the desired identity.

2. Function of brownian motion is a martingale

You can apply the following theorem to prove $u(t,B_t)$ a martingale. The main idea is -as Ilya wrote- to use the fact that the density $p(t,x)$ of a Gaussian distribution with mean 0, variance $t$ solves $$fracpartialpartial t p(t,x) = frac12 fracpartial^2partial x^2 p(t,x) qquad (x in mathbbR,t>0) tag1$$.Theorem Let $(B_t)_t geq 0$ a Brownian motion and $u in C^1,2 cap C(,infty) times mathbbR)$ such that $u$ and its partial derivatives are exponentially bounded. Then $$M_t^u := u(t,B_t) - u(0,B_0) - int_0^t underbraceleft( fracpartial partial t frac12 fracpartial^2partial x^2

ight) u(r,B_r)_=:Lu(r,B_r) , dr$$ is a martingale.Sketch of the proof: By the Markov property of the Brownian motion, one can show that$$beginalign* mathbbE(M_t^u-M_s^u mid mathcalF_s) &= mathbbE left( u(t,B_t)-u(s,B_s) - int_s^t Lu(r,B_r) , dr mid mathcalF_s

ight) &= mathbbE left( u(t,B_t-sz)-u(s,B_0z) - int_0^t-s Lu(rs,B_rz) , dr

ight) bigg|_z=B_s &= mathbbE bigg( underbracevarphi(t-s,B_t-s) - varphi(0,B_s) - int_0^t-s Lvarphi(r,B_r) , dr_M_t-s^varphi-M_0^varphi bigg) bigg|_z=B_s endalign*$$where $varphi(t,x) := u(ts,xz)$. To prove that this expression equals $0$, use Fubini's Theorem, partial integration and $(1)$ to show that$$mathbbE(M_t^varphi-M_varepsilon^varphi) = int (p(t,x) cdot varphi(t,x)-p(varepsilon,x) cdot varphi(varepsilon,x)) , dx - int_varepsilon^t p(r,x) cdot left( fracpartialpartial t frac12 fracpartial^2partial x^2

ight) varphi(r,x) , dx , dr = 0$$for $t > varepsilon>u$. Finally, by Doob's maximal inequality and the exponential boundedness, one can apply dominated converence to conclude that $$mathbbE(M_t^varphi-M_0^varphi) = lim_varepsilon to 0 mathbbE(M_t^varphi-M_varepsilon^varphi) = 0$$This finishs the proof.(For more details, see e.g. Ren L. Schilling/Lothar Partzsch: Brownian Motion - An Introduction to Stochastic Processes, Theorem 5.6.)Actually, the mentioned theorem is a special case of the following theorem:Theorem Let $(X_t)_t geq 0$ a Feller process with generator $A$. Then $$f(X_t)-f(X_0) - int_0^t Af(X_r) , dr$$ is a martingale for any $f$ in the domain of $A$.So, in your case the process is given by $X_t := (t,B_t)$. With some knowledge about generators, it's not that difficult to show that the generator $A$ is given by $$A = fracpartialpartial t frac12 fracpartial^2partial x^2.$$ (This leads to the topic of transition semigroup and its generator. It's also contained in the book, chapter 7.)

3. Why does the Arctic ocean not freeze when it has been -25 to -35 degrees at Alert, Nunavut for the last two months?

Nobody pointed out yet that air temperature does not translate into water temperature. Then take into account that water is always in motion, it has kinetic energy, it is not still like the water you put in an ice cube tray in your freezer

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