Gid Hanashe and Fast Days

When Yaakov was left alone to retrieve his jugs, he got embroiled in a fight with the angel of his evil brother Esav. They battled all night long, and upon seeing dawn on the horizon, the angel gave up, but not without striking Yaakov on the thigh and injuring him. Because of this, the Torah says that "על כן לא יאכלו בני ישראל את גיד הנשה - therefore, the Jews do not eat the gid hanasheh, the sciatic nerve."
Homiletically, this passuk can be interpreted to be referring to the days of the year when Jews "do not eat" at all, the public fast days:
The passuk reads as follows: "Therefore the Jews do not eat [on the days hinted to in the following words]:
"את - the ninth (תשעה) of the month of Av (אב);
"גיד - the 3rd (ג) day of the month of Tishrei;
"גיד - the 10th (י) day of the month of Teves;
"גיד - the 17th (the gematriah of גיד) day of Tammuz."
ש"ך על התורה בראשית לב, לג בסופו
See here

The Arizal and the Eruv

R. Chaim Vital writes of his esteemed master and teacher, the Arizal, that he was careful to make an eruvei chatzeiros every week before Shabbos, not like those that suffice with making one for the entire year. Indeed, the Arizal was very into the eiruv of his city, the fabled mystical town of Tzfas. Says R. Chaim Vital:
"I saw that my master would carry his tallis and chumash on Shabbos morning from his home to Shul. He would also carry it to the famous mikvah outside the city and he was not particular, nor did he bother, to verify if the eiruv for all the alleys of the city of Tzfas aw well as for the mikvah were made according to halachah."
שער הכוונות להרח"ו ז"ל ענין רחיצת פניו ורגליו ד"ה ענין העירוב
See here

Three Signs of a Scholar

The Gemara (חולין ט, א) declares that a Talmid Chacham must know three things: ksav (writing), shechitah, and milah. Rashi (ד"ה כתב) explains that "ksav" means the simple knowledge of writing, for the scholar should know how to sign his name were he to sit as a judge or a witness.
Explaining why it doesn't mean the knowledge of the laws of writing a Sefer Torah etc., the Maharsh"a explains that the Torah scholar's knowledge of all areas of halachah is a given; this cannnot be what the Gemara means when it says that he must be knowledgeable in "writing." Rather, the intent is the skill of signing his name.
Alternatively, the intent is the obligation a Talmid Chacham has to "transcribe" his Torah thoughts. One whom Hashem has bestowed with the wisdom to expand Torah knowledge is obligated to do so lest it be forgotten. Should he not do so, it is tantamount to spilling seed in vain.
פי' רוב דגן (עטייה) לחולין ט, א ד"ה ח"ח צריך שידע
ועי' בספר אברהם את עיניו(פלאג'י) למס' זו פי' יפה
See here and here