to clarify the difference between this and the supposed duplicate, these two questions talk about completely different functions with completely different purposes  I was given this from a friend. They asked me to deduce what the equation is of. I played around with trying to compute alpha for some time. Plugging it into f(x), the function appeared to equal $0$ almost everywhere. I could never find the actual value of alpha. It appears to be an infinite irrational decimal number starting with $1.1973...$.  Let $$\alpha = \sum_{m=1}^{\infty} m * (2^{-(\sum_{j=2}^{m} (\lfloor \frac {(j-1)! + 1}{j} \rfloor - \lfloor \frac {(j-1)!}{j} \rfloor))^2}) \\* \left(\left\lfloor \frac {\sum_{i = 1}^{\lfloor \sqrt{m} \rfloor + 1} (\lfloor \frac {m}{i} \rfloor - \lfloor \frac {m-1}{i} \rfloor)-1}{\sqrt {(\sum_{i = 1}^{\lfloor \sqrt{m} \rfloor + 1} (\lfloor \frac {m}{i} \rfloor - \lfloor \frac {m-1}{i} \rfloor)-1)^2 + 1}} \right\rfloor + 1\right) * \left(\left\lfloor \frac {-\sum_{i = 1}^{\lfloor \sqrt{m} \rfloor + 1} (\lfloor \frac {m}{i} \rfloor - \lfloor \frac {m-1}{i} \rfloor)+1}{\sqrt {(\sum_{i = 1}^{\lfloor \sqrt{m} \rfloor + 1} (\lfloor \frac {m}{i} \rfloor - \lfloor \frac {m-1}{i} \rfloor)-1)^2 + 1}} \right\rfloor + 1\right)$$  Then let $f(x) = \lfloor 2^{x^2}* \alpha \rfloor - 2^{2x-1}*\lfloor 2^{(x-1)^2}*\alpha \rfloor$.  What is the function f(x)? Is there any way to reduce alpha to something simpler? I think this equation isn't something trivial. It doesn't appear to be $0$. It seems bizarre that alpha consists of so many sums. How do I reduce them into something simpler?
I want to simplify the following:  $\sum_{n=0}^x 1 - (1 - I[a \ne 0] * I[a \ge b]) * (1 - I[b \ne 0] * I[b \ge c]) * (1 - I[c \ne 0] * I[c \ge d]) * (1 - I[d = 0])$
I heard a lot about in-body image stabilization recently and I wonder how this technology works. I don't mean the mechanical part (I understand how the sensor moves along different axes to compensate), but the technique in principle:  When I take my 5D III and a lens like the 70-200 2.8 II IS USM with activated stabilization, the IS will constantly move one glas element within the lens in such a way that it directs the cone of light exactly onto the sensor, so that you can always use the full sensor area.  When the cone of light goes through an unstabilized lens and you need to move the sensor - how can it capture the same amount of light at all? The light enters the lens down a fixed axis right through the center of the lens. When the sensor currently moves down, for example, to compensate for a shake, I can't see how it then is still able to capture the full resolution? When a part of the sensor is below default/calm position, then there must be missing the same part on the opposite site?  There's only one way I could image IBIS to work without resolution reduction: if the sensor area was significantly bigger than the area exposed to light, so that there's still sensor available when it currently moves out of its initial position. But wouldn't this (partly) be the same as digital stabilization because the final image gets calculated from the full sensor area minus the not exposed area?  I really would like to understand this and it would be great if someone could "shed some light on this". basic geometry angle problem Two straight lines, AB and AD, lie in the same vertical plane. AB makes an angle of 25 degrees with the horizontal, and AD makes 32 degrees with the vertical. What is the angle between AB and AD?  The answer is 33 degrees or 93 degrees.   I cannot visualize what this question is asking. It says both AD and AD lie on the same vertical plane, then immediately says AD makes 32 degrees with the vertical. Isn't this a contradiction? Some people (me) would prefer the straight GNOME experience as opposed to the Unity style desktop offered in Ubuntu 17.10.  How can I get the standard GNOME desktop and interface? How to force NSLocalizedString to use a specific language "Where will you go" vs. "Where you will go" Which sentence is correct between these?      Where will you go?     Where you will go? ́╗┐' is not recognized as an internal or external command When executing certain files (mainly batch) using PsExec I get these weird ́╗┐' symbols before my command. My initial thought was, that I were using wrong encoding, but after checking, I realized that all my files were using UTF-8. Ship an application with a database I was reading a website where common English sentence errors are written -     INCORRECT - He doesn't know to swim.      CORRECT - He doesn't know how to swim.   I know He doesn't know how to swim is a correct sentence. But why is the other one wrong, is my question?  I know that know is a transitive verb. So, the following are correct -   i) He knows swimming.  Swimming is a gerund. We can also use the infinitive form here - to swim  ii) He knows to swim. Let $A$ be an infinite set. For each $a \in A$, let $x_a$ be a non-negative number. Let the value of the series $\sum_{a \in A} x_a$ be the value of the supremum of its finite partial sums, i.e.:      $$\sum_{a \in A} x_a = \sup_{n \geq 0} \sup_{(a_1, \ldots , a_n) \subset A} \left(x_{a_1} + x_{a_1} + \ldots + x_{a_n}\right)$$   (Supremum is well defined even when its value is infinite)       If $\sum_{a \in A} {x_a}$ is finite, prove that the set $A' = \left\{{a \in A | a &gt; 0}\right\}$   is countable.   My guess is that we have to write $A$ as a union of finitely many sets. Is it possible to model a PIN InGasAS photodiode in a SPICE simulator? I want to know if it is possible to model a InGasAs panchromatic PIN photodiode, in accord with datasheet, in simulator SPICE MicroCAP 12. Because the supplier doesn't provide a SPICE model. And if it is possible, which are those steps. Here is data-sheet  And below I will attach an electrical schematic of a transimpedance amplifier with phototdiode, used as a smoke receiver. I want to monitor graphically the impulse of photodiode in function of output of transimpedance for sensibilty. I found a type of photodiode in the Micro-CAP 12 library but it is simple photodiode, which does not match the datasheet. Can the present perfect continuous construct be used in passive voice? Can the passive voice be used for sentences in the present perfect continuous?  I understand the present perfect and its passive voice (have/has built, has been built). However, the passive voice of present perfect continuous is unclear:   Active: They have been building this house for years. Passive: ???   I have seen suggestions of “This house has been being built for years,” but it sounds awkward and I don't recall anywhere else where two be verbs follow one another. Is this the right form to use? I want to determine the set of natural numbers that can be expressed as the sum of some non-negative number of 3s and 5s.  $$S=\{3k+5j∣k,j∈\mathbb{N}∪\{0\}\}$$  I want to check whether that would be: 0,3, 5, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, and so on.  Meaning that it would include 0, 3, 5, 8. Then from 9 and on, every Natural Number. But how would I explain it as a set? or prove that these are the numbers in the set? What is the difference between "INNER JOIN" and "OUTER JOIN"? Also how do LEFT JOIN, RIGHT JOIN and FULL JOIN fit in? The no. of all possible matrices of order $3\times 3$ with each entry $0$ and $1$ and $ -1$. I can successfully tether my Mac (10.7.3) to my iPhone4 (5.0.1) via wifi, no problem, been doing it for months.  However, when I try to tether via bluetooth instead, I can't seem to see the internet from my mac.  Network Settings reports a green light for Bluetooth PAN, but my browser can't connect to anything at all.  My Mac reports very similar IP addresses for both, starting with 172.20.10 and differing only in the last octet.  Any idea why I can't successfully tether via bluetooth? Selecting features within Polygon from another layer using QGIS? I'm using QGIS 1.8 with a PostGIS database and I have three PostGIS-layers in QGIS a lines layer, a points layer and a polygon layer.  What I'm trying to do is to select a polygon in the polygon layer and mark all points within this polygon.   For me the only way to do this is to use the button "Select by Polygon" but in this way, I cannot use an existing polygon for selection. How to enumerate with small roman capitals (not lowercases)? Although I have seen many similar questions, none has been useful.   The question is quite simple: How do I reproduce the highlighted numerals in the image below?    Note that the form of the numerals addressed in questions as  is different from this one.  Thank you in advance! Why does a vertical line have a SLOPE (steepness) of undefined? The curve shows :   x goes 1 to right  y goes up by 3   the SLOPE is defined as the ratio of rise to the step  SLOPE = rise/step = 3/1 = 3  Now that :   Horizontal line has SLOPE of 0. Vertical line has SLOPE of undefined.   It makes sense for me to imagine horizontal line has SLOPE of 0 since there is no rise at all. But why the vertical has SLOPE of undefined.? Why was Gotham City changed from a mostly fictional city to New York City in the Nolan trilogy? In Batman Begins, Gotham City has a futuristic/fictional and run down look: think of Oldtown, the trains built by the Wayne Corporation etc.   In The Dark Knight Rises, Gotham City looks like plain old real world New York City — we don't see the trains anymore. Oldtown is just a standard poor neighborhood. Lots of the locations used in the movie are easily identified as real world NYC locations: the stock exchange or East River bridges for example.   My question is: Why did Nolan change Gotham city from a shady fictional more cartoon-like town to plain old NYC?   Edit: The examples from Batman Begins and The Dark Knight Rises were just for illustration; I didn't intend to limit the question to those two movies.  Edit 2: I specifically want to know why Gotham City was changed from a cartoon-like city in Begins to an almost real-world NYC in The Dark Knight Rises (via not so cartoon-like but not yet real-world like Chicago in The Dark Knight Rises). Why it's NYC or Chicago is not so important for me. But why, e.g., were the trains which featured prominently in Begins gone in The Dark Knight Rises? There was only a very short in-universe time-span between the events of the two movies, yet Gotham changed drastically. My question is in the title. I'm a little lost in how to solve this problem. There is a hint associated with the problem that states the following:   Use a line integral to show that $ |f(z)| \leq A + |z| $ where A = $\max\{1, |f(0)|\}$  Thanks for the help.