[EE-S] Kyoto Films Developed

 So, I've developed my glorious 5 rolls of films after finishing my trip. The pics were well developed than I thought, but I guess some kind of light leak is happening. A friend of mine told me that he would take care of it, thanks to him. So from the post before, I bought my first film camera at Kyoto and I took 5 rolls of films through my journey. It's about 220 shots, and as I'm a beginner using film cameras, I thought it would be OK for only about a half to be well developed, but it turned out to be that most of them (except the light exposed ones) developed in a good shape. I chose some of the best shots (in my opinion) from Kyoto rolls.

1/20 Kinkakuji

 Right after buying my camera, we headed to Kinkakuji, a marvelous temple wrapped with gold foil, which was my first sight to capture. 

 To ride a bus to Kyoto station, we headed towards the bus station in front of McDonald's and before the sun sets, I took some pictures of Kyoto streets. Kyoto gives an idyllic aesthetic, unlike from cities like Tokyo which rather gives urban aesthetics. As sunshine slowly disappeared from the sky, I stopped taking pics as the camera wasn't suitable for the low light environment. 

1/21 Fushimi-Inari Jinjya ~ Ginkakuji

 One of the sceneries of Kyoto I really wished to see was the aesthetically aligned Toriis from Fushimi-Inari Jinjya, but I was too lazy to wake up early. As tourists rush like a flood around 10am, I had to wake up early and get to there as fast as I could, thanks to Glenfiddich I kind of overslept. Well, that doesn't mean that I didn't enjoy the scenery but there were way too many heads in every shot.

 Even though it was a rush of tourists, I still enjoyed this place, but I still always think about it would be better if I had been there earlier. I ran out of shots so I changed to a new film roll.

 I took some pics of rails and trains, as I like them a lot. These are the pics of the Keihan line while heading to Kiyomizudera. 

 Kyoto has lots of shrines and temples, and this day we've been to like 5 or more of them. The next stop was Kiyomizudera, also a famous tourist attraction in Kyoto. 

 But more than Kiyomizu-dera, I was really looking forward to seeing the streets of Kyoto around Kiyomizu-dera. 

 Kyoto really gives an antique view, as if time has stopped from the Edo period. We walked a lot, about 25 kilometers this day. We headed to Yasaka Jinjya, again another shrine. 

We haven't spent a lot of time here, took some pics and headed right to Heian Jingu. 

 After having lunch ( we really needed some time to rest our legs and feet, they were literally burning ), I bought another film roll from a local film store. After sightseeing Heian Jingu we headed to Ginkakuji, but this time we took a bus. We stopped at the 'Philosopher's Path', a path along a small streamside. Before that, we've been to Ginkakuji.

 Taking some break at a coffee shop around the Philosopher's Path, we took some shots of this rustic path. Personally, I love this shot the most. Gives kind of a Showa vibe, and the winter flowers with the bicycle look really gorgeous. It looks like a postcard from 60's. 

 Walking around for about 25 kilometers, we were knocked out. Before having dinner with some beer, I stopped off at some record stores around the city hall. I'll talk about them in later posts. Well, so these are some of the shots that weren't burnt from light leakage. 

[Top] Different Definitions of Topology

 Many topology texts introduce general topological spaces with the defintion using open sets and its condition for being an open set. But we can define a topology of a certain set in differen Topological space as a mathematical object is merely a 'pair', $(X,\mathcal{T})$ equipped with a set $X$ and a collection  $\mathcal{T}\subseteq \mathcal{P}(X)$ which we call the 'topology', that satisfies some conditions. Now $\mathcal{T}$ defines the topology, and using open sets, closed sets, neighborhoods, closures, and interiors we can define in different 'words' but all equivalent.

 As I mentioned, most of the topology texts define a topology of set by indtroducing open sets, so defining topology by open sets are familiar to us. We define a 'collection of open sets' of a given  set, $\mathcal{T}\subseteq \mathcal{P}(X)$ and we call the elements of this collection open sets. $\mathcal{T}$ should satisfy, 

$$ \begin{split} \emptyset ,X &\in \mathcal{T} \\ \mathcal{S} \subseteq \mathcal{T} &\Rightarrow \cup \mathcal{S} \in \mathcal{T} \\ U,V \in \mathcal{T} &\Rightarrow \text{ } U \cap V \in \mathcal{T} \end{split}$$

 Now using a similar object known as the closed sets, we can define a topology in a slightly different way. Closed sets are compliments of open sets, so we define a 'collection of closed sets' $\mathcal{C} \subseteq \mathcal{P}(X)$ and the elements of this collection are closed sets. $\mathcal{C}$ should satisfy,

$$ \begin{split} \emptyset ,X &\in \mathcal{C} \\ \mathcal{S} \subseteq \mathcal{C} &\Rightarrow \cap \mathcal{S} \in \mathcal{C} \\ U,V \in \mathcal{T} &\Rightarrow \text{ } U \cup V \in \mathcal{C} \end{split}$$

 The equivalence of the first two definitions seem trivial by the nature of open sets and closed sets. As they are defined by the compliment of each other, it seems trivial. And it also makes $\emptyset$ and $X$ a trivial 'clopen' set. We call the nontrivial clopen sets as proper clopen sets, and whether there exists a proper clopen set is related with the connectedness of the given topological space. ( will be continued in later posts )

  We can also define a topology using neighborhoods. A neighborhood is defined by a mapping $\mathcal{N}:X\rightarrow\mathcal{P}(\mathcal{P}(X))$ which maps $x \in X$ as $x \mapsto \mathcal{N}_{x}$. We call $\mathcal{N}_{x}$ as the neighborhood of $x$ when it satisfies the following conditions.

$$\begin{split} \forall x \in X : N\in \mathcal{N}_{x} &\Rightarrow x \in N \\ N\in\mathcal{N}_{x} \land N\subseteq S \subseteq X &\Rightarrow S\in\mathcal{N}_{x} \\ M,N\in\mathcal{N}_{x} &\Rightarrow M\cap N \in \mathcal{N}_{x} \\ N \in \mathcal{N}_{x} &\Rightarrow \exists\mathcal{N}_{x} \text{ } s.t \text{ } \forall y\in M, N\in\mathcal{N}_{y} ,M\in\mathcal{N}_{x}\end{split}$$

 Now considering closures and interiors, someone may ask "How can we think of closures and interiors when we even don't have the notion of open/closed sets?" The answer to this question is to regard closures and interiors as a mapping, $cl$ and $int$ which makes the pair $(X,cl)$ and $(X,int)$ a topological space. The mappping $cl:\mathcal{P}(X)\rightarrow\mathcal{P}(X)$ is defined with below conditions.

$$\begin{split} cl\text{ }\emptyset &= \emptyset \\ \forall A \subseteq X &\Rightarrow A \subseteq cl \text{ } A \\ \forall A,B\subseteq X &\Rightarrow cl(A\cup B)=cl(A)\cup cl(B) \\ \forall A\subseteq X &\Rightarrow cl(cl\text{ }A)=cl\text{ }A \end{split}$$

 The mapping $int:\mathcal{P}(x)\rightarrow\mathcal{P}(x)$ is defined with below conditions.

$$\begin{split} int\text{ }X &= X \\ \forall A \subseteq X &\Rightarrow A \supseteq int \text{ } A \\ \forall A,B\subseteq X &\Rightarrow int(A\cap B)=int(A)\cap int(B) \\ \forall A\subseteq X &\Rightarrow int(int\text{ }A)=int\text{ }A \end{split}$$

 Citing nLab, the definitions mentioned above can be written again as, a set with a frame of open sets, a set with a co-frame of closed sets satisfying the dual axioms, a set with any collection of subsets, a pair $(X,int)$ where $int$ is a left exact comonad on $\mathcal{P}(X)$, and a pair $(X,cl)$ where $cl$ is a right exact Moore closure operator satisfying the dual of $int$. 

[昭和Edit] - Mads on Yukata

aesthetic as ｆｕｃｃ

 While wasting my time binging on fb, I discovered some aesthetic photos of Mads MIkkelson wearing a yukata, and on my first sight it really looked like a great Showa period single cover, so I decided to make a 'Showa Edit' on Mads.

  First I tried to think of a nice A-side, B-side title, ended up with A-side  "謎のキラー"(Nazo no Kira ; Mysterious Killer) and B-side "君を街角で殺したい" (Kimi wo Machikadode Koroshitai ; I Want to Kill You at the Street Corner). Obsessed with murder, huh? Whatever, I started to write the A-side kanas. 

「謎のキラー」

　

 So how I work on calligraphies is simple: Write it on a paper, scan it with a scanner or a phone ( the mobile scanning apps are powerful than I thought ), and then chop them up into single words ( kanas or kanjis ). I somehow started to manage my own digital calligraphy library starting from about 2 months ago, and I guess I started it in order to make digital work easier. 

 I erased the white umbrella that Mads is hiding as it overlapped with the letter "キ" and "ラ". Then added some label logo, this time I chose Polydor. Still looks good, but 

  I used a simple line to emphasize the cover horizontally, as the original picture seems to have a bunch of horizontal points ( the frame in the background, Mads' pose , the table ). And put his name in Japanese under the line. Hmm.. seems good but Oh! now we can add some details.

Done!

 Putting in the B-side title upper right to the A-side title, and a random label number as if it was a real 7" single. I tried to find a '45rpm' or 'stereo' logo but I guess I need to make them as Google doesn't give me any results.

 The original pic was in a good shape so that I didn't need to make the pic look older or faded through time. (I guess they took the pic with a film camera.)

[Alg] Hungerford Ch I.2 Exercises Solutions (Part 1)

Ch I.2 Homomorphisms and Subgroups

1. Homomorphisms

 For a given $f:G \rightarrow H$, assume $f$ a group homomorphism. By definition, $f(a)=f(ae_G)=f(a)f(e_G)$ and $f(a)=f(e_G a)=f(e_G)f(a)$ for both sides. Thus as $f(a)=f(a)f(e_G)=f(e_G)f(a)$, $f(e_G)$ and for any $a \in G$, $e_H=f(a)f(a)^{-1}$, $f(e_G)=e_H$. 

 Let $a$ be an arbitrary element $a \in G$. Then $e_H=f(e_G)=f(aa^{-1})=f(a)f(a^{-1})$ while also $e_H=f(a^{-1})f(a)$. As an inverse element is unique, $f(a^{-1})=f(a)^{-1}$.

 For the monoid counterexamples, consider multiplicative monoids $\mathbb{Z}_6$ and $\mathbb{Z}_3$. Define a map $f:\mathbb{Z_3}\rightarrow\mathbb{Z_6}$ by $f(\bar{0})=\bar{0}$, $f(\bar{1})=\bar{4}$ , and $f(\bar{2})=\bar{2}$. We can check that $f(\mathbb{Z}_3)=\{ \bar{0},\bar{2},\bar{4} \} $ is closed with an identity of $\bar{4}$. For all $n \in \mathbb{Z}_3$, $f(\bar{0}\bar{n})=f(\bar{0})=\bar{0}=\bar{0}f(\bar{n})=f(\bar{0})f(\bar{n})$. As $\bar{4}$ is an identity element for the image $f(\bar{n})=\bar{4}f(\bar{n})$, and $f(\bar{1}\bar{n})=f(\bar{n})$. Thus $f(\bar{1}\bar{n})=f(\bar{1})f(\bar{n})$. Moreover, $f(\bar{2}\bar{2})=f(\bar{1})=\bar{4}=\bar{2}\bar{2}=f(\bar{2})f(\bar{2})$. Therefore we've shown that $f$ is a multiplicative homomorphism while $f(\bar{1})=\bar{4}$ which show up to  be a counterexample.

2. Abelian Automorphism

 Consider $G$ an abelian group. As $G$ is abelian, $(ab)^{-1}=a^{-1}b^{-1}$. Then the map defined as $f:G \rightarrow G$ with $x \mapsto f(x)=x^{-1}$ is a homomorphism as $f(ab)=(ab)^{-1}=a^{-1}b^{-1}=f(a)f(b)$. Due to the uniqueness of inverse elements, the map $f$ is a bijection, thus $f$ is an automorphism.

 Conversely, assume $f$ an automorphism. As automorphisms are bijective endomorphisms, $(ab)^{-1}=f(ab)=f(a)f(b)=a^{-1}b^{-1}$ and this implies that $G$ is an abelian group.

3. Quaternion Group $Q_8$

 Let $Q_8$ with the matrix multiplication a group generated by $\begin{pmatrix} 0 & 1 \\ -1 & 0  \end{pmatrix}$ and $\begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}$. Denote $A=\begin{pmatrix} 0 & 1 \\ -1 & 0  \end{pmatrix}$ and $B=\begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}$. By simple multiplication, $A^2=-I_2$. Let $P=\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ so that $B=iP$. Thus $B^2=-P^2=-I_2$, so $A^2=B^2=-I_2$. Moreover, $A^4=B^4=I_2$ followed with $A^3=-I_2A=-A$ and $B^3=-B=-I_2B=A^2B$. Through basic multiplication we can show that $PA=A^3P$, thus $BA=iPA=iA^3P=A^3iP=A^3B$. Thus any products in $Q_8$ could be expressed as $A^iB^j$ where $i,j$ are both positive integers, and using the fact that $B^3=-B=A^2B$ it could be expressed as $A^iB$. As $A$ has an order of 4, $Q_8$ can be expressed as $ Q_8 = \{ I_2, A , A^2, A^3, B, AB, A^2B, A^3B \}$. We can easily check that $Q_8$ is a group, and we've shown that it is non-abelian by expressing every product as $A^iB$. Thus $Q_8$ is an order 8 non-abelian subgroup generated by $A,B$.

[QFT] David Tong Ch1 Exercises Solutions (Part 1)

Ch1. Classical Field Theory

1. String as a set of harmonic oscillators

 1) Derive the Lagrangian.

  The partial derivatives of $y(x,t)$ are,

 $$ \frac{\partial y}{\partial t}= \sqrt{\frac{2}{a}} \sum_{n=1}^{\infty} \sin{\left( \frac{n\pi x}{a}\right)} \dot{q_n} \quad \text{,} \quad \frac{\partial y}{\partial x} = \sqrt{\frac{2}{a}} \sum_{n=1}^{\infty} \left(\frac{n \pi}{a}\right) \cos{\left( \frac{n\pi x}{a}\right)}q_n $$
 and inserting the derivatives into the given Lagrangian,

$$ \begin{split} L &= \int^{a}_{0} dx \left[ \frac{\sigma}{2} \left( \frac{\partial y}{\partial t} \right)^2 - \frac{T}{2} \left(\frac{\partial y}{\partial x} \right)^2 \right] \\ &= \int^{a}_{0} dx \left[ \frac{\sigma}{a} \left( \sum_{n=1}^{\infty} \sin{\left( \frac{n\pi x}{a}\right)} \dot{q_n} \right)^2 - \frac{T}{a} \left(\frac{n\pi}{a}\right)^2 \left( \sum_{n=1}^{\infty} \cos{\left( \frac{n\pi x}{a}\right)} q_n \right)^2\right] \\ &= \sum_{n=1}^{\infty} \left[ \frac{\sigma}{2} \dot{q_n}^2 - \frac{T}{2} \left( \frac{n \pi}{a} \right)^2 q_{n}^{2}\right] \end{split} $$
 Thus the Lagrangian shows up to be

$$ L=\sum_{n=1}^{\infty} \left[ \frac{\sigma}{2} \dot{q_n}^2 - \frac{T}{2} \left( \frac{n \pi}{a} \right)^2 q_{n}^{2}\right] $$

 2) Derive the equations of motion. 

 The partial derivatives of the Lagrangian are,

$$ \frac{\partial L}{\partial q_n}= -T  \left( \frac{n \pi}{a} \right)^2 q_n \quad \text{,} \quad \frac{\partial L}{\partial \dot{q_n}}=\sigma \dot{q_n}$$
 and inserting the results into the Euler-Lagrange equation,

$$ \ddot{q_n} = \frac{T}{\sigma} \left( \frac{n \pi}{a} \right)^2 q_n  $$
 and solving this differential equation, it yields an infinite set of decoupled harmonic oscillators with frequencies of

$$\omega_{n} = \sqrt{\frac{T}{\sigma}} \left( \frac{n \pi}{a}\right)$$

2. Lorentz invariance and KG eq.

Under a Lorentz transformation $x^{\mu} \rightarrow \Lambda^{\mu}_{\nu} x^{\nu}$, $\phi(x)$ transforms as $\phi(x) \rightarrow \phi(\Lambda^{-1}x)$. Thus the term $\eta_{\mu\nu} \partial^{\mu}  \partial^{\nu} \phi(x)$ transforms as, 

$$\begin{split} \eta_{\mu\nu} \partial^{\mu} \partial^{\nu} \phi(x) \rightarrow \eta_{\mu\nu} \partial^{\mu} \partial^{\nu} \phi(\lambda^{-1}x) &= \eta_{\mu\nu} \left(\Lambda^{-1} \right)^{\mu}_{\alpha} \left(\Lambda^{-1} \right)^{\nu}_{\beta} \partial^{\alpha} \partial^{\beta} \phi(y) \\ &= \eta_{\alpha\beta}\partial^{\alpha} \partial^{\beta} \phi(y) \end{split} $$

Which shows that the term $\partial_{\mu}\partial^{\mu}\phi=\eta_{\mu\nu} \partial^{\mu} \partial^{\nu} \phi$ is invariant under $x^{\mu} \rightarrow \Lambda^{\mu}_{\nu} x^{\nu}$. Thus the Klein-Gordon equation $\partial_{\mu}\partial^{\mu}\phi+m^2\phi=0$ is invariant under any Lorentz transformation $\Lambda$. 

3. Complex Scalar Fields

 1)  Write down the Euler-Lagrangian field equations. 

  The E-L field equation with respect to $\psi^*$ is,

$$\frac{\partial \mathcal{L}}{\partial \psi^*}  = -m^2 \psi - \lambda \psi^2 \psi^* \quad \text{,} \quad \partial_\mu \left(\frac{\partial \mathcal{L}}{\partial \partial_{\mu}\psi^*)} \right) = \partial_{\mu} \partial^{\mu} \psi$$

$$\partial _{\mu} \partial ^{\mu} \psi + m^2 \psi + \lambda |\psi|^2 \psi = 0$$

and with respect to $\psi$, 

$$\frac{\partial \mathcal{L}}{\partial \psi} = -m^2 \psi^* - \lambda \psi (\psi^*)^2 \quad \text{,} \quad \partial_\mu \left( \frac{\partial \mathcal{L}}{\partial \partial_{\mu}\psi)}\right) = \partial_{\mu} \partial^{\mu} \psi^*$$

$$\partial _{\mu} \partial ^{\mu} \psi ^ * + m^2 \psi ^* + \lambda |\psi|^2 \psi^*= 0$$

which are complex conjugates to each other.

 2) Verify that the Lagrangian is invariant under such infinitesimal transformation.

  Under the infinitesimal transformation,

$$\delta \psi = i \alpha \psi \quad \text{,} \quad \delta\psi^{*} = -i\alpha \psi^{*}$$

 the Lagrangian transforms as, 

 $$\delta \mathcal{L} = \partial_\mu\delta \psi^* \partial^\mu \psi + \partial_\mu \psi^* \partial^\mu \delta \psi - m^2 (\delta \psi^* \psi + \psi^* \delta \psi ) - \lambda ( \psi^* |\psi| \delta \psi + \psi |\psi| \delta \psi^*)$$

As $\delta \psi = i\alpha \psi$ and $\delta \psi^* = (\delta \psi )^*$,  \delta \mathcal{L} shows up to be 0. Thus $\delta \mathcal{L} = \partial_\mu(0) = 0$, which makes the Lagrangian invariant under such infinitesimal transformation. 

Olympus Pen EE-S

  I've bought my first film camera at from a vintage camera shop named "第一写真店"( Dai Ichi Shashinten ) in Kyoto with a friendly owner running this shop, the Olympus Pen EE-S model from the '60s. As this model only supports ISO up to 200, I tried to practice my daylight shots with this model. It's also a half-frame model, so I was able to use two times more of the original amount of one roll. 

me and Yoshinobu holding Pen EES and EE-2

  The store was small but filled with a great selection of vintage cameras. The shop owner, Yoshinobu is a really good guy, offered us with great service. He also gave me the original leather strap, case, one roll of a film, and a lens cap from the '60s. The camera itself and its accessories were in a really good condition, plus the price was very reasonable. I've embedded the location of this store, if you have an opportunity to visit Kyoto and are interested in vintage cameras, I highly recommend this place.

Great 7" vinyls I've dug at Japan

I don't want to think about the money I've spent on this :)

 So, I always think about how great Japan is when it comes to collectables and vintage stuff. Especially for vinyl, they have a great variety of rare grooves plus condition. Compared to Korea, vinyl shops are more common and some shops are even specialized to a certain genre. I've visited about 10 vinyl stores, 4 in Fukuoka, 2 in Kyoto, and another 2 in Tokyo. I'll make another post about these stores and their locations.

 Anyways, here are the five best 7" singles I've 'excavated' at Japan.

 ( Format : Artist - A side / B side ; [ Label : Label No. , Date ] )

 1. ヒデとロザンナ－粋なうわさ／あいのひととき [ Columbia P-58 , May 1968 ]

   ( Hide and Rosanna - A Pretty Humor / Between Waves )

  This second single album released by Hide and Rosanna, a lovely Italian-Japanese couple, has a beautiful Showa-style bossa nova tune in its B side. Its dreamy melody, beautiful lyrics with their well-known harmony, made me fall in love with this single at first hearing. Just like its title, which translates into "The Moments of Love", listening to the B side just feels like falling in love.

 2. 森山良子－小さな貝がら／雨上がりのサンバ [ Philips FS-1043 , 1968 ]
   ( Moriyama Ryoko -  Chisana Kaigara / Ameagari no Samba )

 Bossa nova didn't just hit Europe and America starting from the early '60s, but it also hit Japan from the very start of it. Moriyama Ryoko, the 'Queen of College Folk', got with this flow too. The B side of this single "Ameagari no Samba", which translates into "Samba After the Rain", is one of the well known Japanese bossa nova titles.

 3. 坂本九－上を向いて歩こう／あのこの名前はなんてんかな [ Toshiba JP-5083 , 1961 ]
   ( Sakamoto Kyu - Ue wo muite arukou / Ano ko no namaewa nantenkana )

 More known as the title "Sukiyaki", which is actually not even relevant to any of the context of the song, the A side of this single was a total hit in '63 in the States. A Japanese song that topped the Billboard in '63, which was a totally unprecedented situation, actually has its own story related to the lyricist of this song. I was having a hard time trying to find this original single but thanks to a vinyl store at Kyoto I was lucky enough to find this.

 4. 小林啓子 - 比叡おろし／恋人中心世界 [ King BS-1216 , May 1970 ]
    ( Kobayashi Keiko - Hiei Oroshi / Koibito Chusin Sekai )

  I've actually heard the B side "Koibito Chusin Sekai", in English "Lover Centered World", performed by the 'Stage 101' first, but after research, this version by Keiko came first and later as she became a regular cast at the 'Stage 101', an NHK music program which was popular at the '70s, they made another version of it. Another bossa nova inspired style groove.

 5. 久美かおり－くちづけが怖い／夜明けの海 [ Columbia P-22 , 15 Jun 1968 ]
   ( Kumi Kaori - A Date Without a Kiss / On the Sands at Daybreak )

  Visiting Tokyo for the first time 2 years ago, I've tried to find this debut single of Kaori, well known for her later single "髪が揺れている／小さな鳩"(Kami ga Yureteiru/Chisana Hato), but failed to do so and ended up buying a compilation 7" which was mixed with her later singles. It was like about 5 years ago I first heard the B side "夜明けの海", when a Japanese YouTuber sent me a CD filled with Showa pops right after his copyright strike, which is still my favourite CD compilation :)