Can there be any issues in using Lumix G 25mm f/1.7 ASPH with Oly OMD10 mark 2?
Panasonic Lumix G 25mm f/1.7 ASPH vs Olympus M. Zuiko 25mm f/1.8
Evaluate $\lim_{n\to\infty} [{1\over kn}+{1\over k(n+1)}+{1\over k(n+2)}+\cdots+{1\over k(n+p-k)}]$ where $k<p$ There are four options. Which of them is correct?- (a)${p\over k}$ (b)${k\over p}$ (c)$\log({p\over k})$ (d)${\log p\over k}$ I somehow want to use the rule for evaluating this kind of sum using integration, i.e. $\int_{0}^{1} f(x) \ dx=\lim_{n\to\infty} {1\over n}\sum_{r=1}^{n}f({r\over n})$  But the given expression $\lim_{n\to\infty} [{1\over kn}+{1\over k(n+1)}+{1\over k(n+2)}+\cdots+{1\over k(n+p-k)}]={1\over k}\lim_{n\to\infty}{1\over n}\sum_{r=0}^{p-k}\frac{1}{1+{r\over n}}$, sum running from $0$ to $p-k$ instead of $n$. So, how to evaluate this limit? Thanks for assistance in advance. Android Preferences: How to load the default values when the user hasn't used the preferences-screen? Assume $x$ doesn't occur free in $\alpha$, show that: $$\vdash (\forall x \beta \to \alpha) \leftrightarrow \exists x (\beta \to \alpha)$$   This is an exercise on page 130, A Mathematical Introduction to Logic, Herbert B. Enderton（2ed).   Here's my attempt: For the '$\to$' direction, since $x$ doesn't occur free in $\alpha$, we have $α→∀x α$ and $\forall x \alpha \to \alpha$ as logical axiom(In the second formula, we have$\alpha_x^x = \alpha$). Thus it suffice to show $\vdash (\forall x \beta \to \forall x \alpha) \rightarrow \exists x (\beta \to \alpha)$. By logical axiom $∀ x(α→β)→( ∀ x α→∀x β)$, it would suffice to show $\vdash \forall x (\beta \to   \alpha) \rightarrow (\exists x (\beta \to \alpha))$. It has been shown in a previous exercise that there's a deduction $∀ x ϕ→∃x ϕ$, so we are done for the "$\to$" part.  The problem for me is how to show the "$\leftarrow$" direction.  Added: Here's logical axioms that we can employ:         Tautologies(in the sense of propositional logic);   $∀ x α →α^x_ t $, where $t$ is substitutable for $x$ in $α$ ($α^x_ t$ is the formula derived from $\alpha$ by replacing $x$ by a term $t$);   $∀ x(α→β)→( ∀ x α→∀x β)$;   $α→∀x α$, where $x$ does not occur free in $α$.      Besides, generalization theorem and deduction principle can be utilized:     If $\Gamma \vdash ϕ$ and $x$ do not occur free in any   formula in$\Gamma$, then$\Gamma \vdash ∀ x ϕ$.      If $\Gamma ; γ \vdash ϕ,$ then $\Gamma \vdash  γ \to ϕ$. Why can’t the Doctor’s companions understand babies and horses? The Doctor claimed to speak both horse and baby. The reason human companions can understand every language is because the TARDIS translates. So why can't anyone else understand babies and horses? Date and Time only partially visible in Big Sur Programming pattern to generate VGA signal with micro-controller? I want to generate VGA signal with micro-controller (like TI Tiva ARM which runs at 90/120Mhz speed).  I'm not sure how to make accurate timings with micro-controller. Which programming pattern I need to use?  Do I need any inline assembler code? How to wisely use interrupts?  Would be great if anybody show some pseudo code how to generate VGA signal.  I successfully generated VGA signal with FPGA. But I just can't figure out how to do it with MCU. Moving an already-running process to Screen I have rigged up a notch filter from . See below: -     Its the first circuit. Now I am trying to calculate its transfer function. I have calculated two equations, but I need the third one:   (Vin-V2)/(1/SC1) = V2/R3 (Vin-V1)/R1 = (V1-Vo)/(R2+(1/SC2))   V1 and V2 are the potentials at the input of the op-Amp. Overstaying Schengen Visa for 1 day I was in Germany last September, my visa said I had 14 days to stay in the Schengen area.  When I was leaving back for Egypt the passport control officer noticed that I was leaving on the 15th day. I didn't even realise that as I had written previously in my visa application that I was staying for 15 days and the embassy in Cairo had my complete airline ticket with specific dates.  Anyway the officer just filed a report on the incident but didn't stamp my passport with my exit date from the Schengen area.  Do any of you guys know if that would make it harder for me to be granted a visa in the future? If $\forall x \in R, x^2-x \in Z(G)$, than $R$ is commutative Ok, so at my school, everybody was supplied with a Macbook Pro. I set up the Messages app and it was all great until the county decided to block it. Now, instead of receiving text messages on my phone they go to my Mac. I have no idea how to fix this because I can't open the Messages app. Please help?! Proving replacement theorem? I want to see if I am understanding the proof of the replacement theorem correctly.   Let $V$ be a vector space that is spanned by a set $G$ containing $n$ vectors.  Let $L \subseteq V$ be a linearly independent subset containing $m$ vectors.  Then $m\leq n$ and there exists a subset H of G containing exactly $n-m$ vectors s.t. $L \cup H$ generates $V$.   The proof in the book says to start with $m=0$. In that case $L=\varnothing$, the empty set.        Let's assume this theorem is true for some integer $m \geq 0$.   Let $L = \{v_j\}_{j=1}^{m+1}$ and define it as a linearly independent subset of $V$ consisting of $m+1$ vectors.  Since any subset of a linearly independent set is linearly independent as well ($S_1 \subseteq S_2 \subseteq V$), then $\{v_j\}_{j=1}^m$ is linearly independent also.   It then says to use the induction hypothesis to say that $m \geq n$. Well, ok, but that seems sort of pulled out of nowhere to me. The next step is to say that there is therefore another subset, $\{u_k\}_{k=1}^{n-m}$ of G such that $\{v_j\}_{j=1}^m \cup \{u_k\}_{k=1}^{n-m}$ spans $V$.   That being the case there are scalars $(a_j)_{j=1}^m$ and $(b_k)_{k=1}^{n-m}$ which you can multiply by the vectors $v_j$ and $u_k$. You can then add the two sets of vectors, yielding $$\sum_{j=1}^m a_j v_j + \sum_{k=1}^{n-m} b_k u_k = v_{m+1}$$  At this point I am not sure I understand things. Because we are assuming $n-m&gt;0$ -- otherwise $v_{m+1}$ is linearly dependent. But then it says not only is $n&gt;m $ but $n&gt;m+1$, and that is where I am losing the plot to get to the next step. How do I show this $\cos^2(\arctan(x))=\frac{1}{1+x^2}$ How do I show this?  $$\cos^2(\arctan(x))=\frac{1}{1+x^2}$$  I have absolutely no idea. Thank you. Do conjugate priors just lead to a posterior that is a modification of the parameters of the prior? We know, for example, that the conjugate relationship between the classic beta-binomial is as follows:  \begin{align}  y &amp;∼ \mathcal{Bin}(n,\ θ)  \\  θ &amp;∼ \mathcal{Beta}(α,\ β)  \\ θ|y &amp;∼ \mathcal{Beta}(y + α,\  n − y + β) \end{align}  Notice how the posterior is just a change in parameterization compared to the prior. Is the posterior of conjugate priors just a change in parameters? How to disk copy with unreadable sectors? I want to clone a whole disk bytewise, something like  dd if=/dev/$SRC of=/dev/$DST bs=65536 count=$count   There are better tools for the job, which understand the file system (Windows) and work faster (I don't care), but they fail because of unreadable sectors on the old source disk.  There's actually a single invalid block I have to skip. So I thought about using dd multiple times like  dd if=/dev/$SRC of=/dev/$DST bs=65536 count=... dd if=/dev/zero of=/dev/$DST bs=65536 count=... skip=... seek=... dd if=/dev/$SRC of=/dev/$DST bs=65536 count=... skip=... seek=...   with replacing the broken sector data by zeros. Is this a valid approach? I found this  which confuses me. Text in an equation This is best explained by example. Suppose I have the following:  \[     \begin{align*}     C_3 \rightarrow 6 automorphisms\\     C_4 \rightarrow 8 automorphisms\\     C_5 \rightarrow 10 automorphisms     \end{align*} \]   How can I get the word "automorphisms" to display as if it were not enclosed by the brackets? Editing older revision (not most recent) no longer includes older title and tags, and automatic rollback note Why is $1 in a function not printing the script's first argument? Why doesn't echo $1 print $1 in this simple bash script?    #!/bin/bash # function.sh print_something () { echo $1 } print_something  $ ./function.sh 123  -&gt; why doesn't it print '123' as a result? How to work with "FIFO" in C# .NET? How to convert std::string to LPCWSTR in C++ (Unicode)