Is it always possible to extend a homeomorphism from a closed ball in the plane onto a closed disk in a surface to their neighbourhoods?

Let $S$ be a two dimensional surface and $D$ a closed disk contained in $S$. Is it true that if $f$ is a homeomorphism from a closed ball $B$ in the plane onto $D$, then $f$ extends to a homeomorphism from a neighbourhood of $B$ in the plane onto a neighbourhood of $D$ in the surface?

dnd 5e – House Rule – Upcasting Enlarge / Reduce to extend duration

The party is deep into my 5e-updated classic Greyhawk Giants series.

The hill giants and their orc and goblin minions have attacked the Sterish city of Headwater and have taken about a quarter of the city. The party is about to embark on a mission to go behind enemy lines, kidnap and assassinate a stone giant who is critical to the hill giants’ city / siege offensive as being their only trained engineer. While the party’s patron recognizes that the stone giant needs to die, she also recognizes that at present the Stone Giant Thane has not joined the giant alliance and wants to keep it that way. Thus, she is requesting that they carry the body of the slain giant honorably back to his Thane in an effort to preserve a fragile peace between the humans and the stone giants. I would like the party to be able to accept this mission, without it becoming either a logistical challenge involving wagons and draft animals, or without loaning them a portable hole.

Instead, I would like to provide them with a version of the Enlarge / Reduce spell which is in all aspects identical to the original except that it can be upcast to extend the duration.

I figure the giant is 17 feet high and 1000 pounds; after reduce it would be 8.5 feet and a manageable 125 pounds.

For this version of Enlarge / Reduce I am proposing that:

When cast at 3rd level against living creatures, it lasts 10 minutes

When cast at 3rd level against objects, it lasts 1 hour

When cast at 4th level against living creatures, it lasts 1 hour (similar to polymorph)

When cast at 4th level against objects, it lasts 8 hours

When cast at 5th level against living creatures, it lasts 8 hours (much less than geas)

When cast at 5th level against objects, it lasts 24 hours

The party Wizard is currently 8th level and getting close to 9th. Requiring her to use both her fourth level slots and maintain concentration for the duration of travel every day in order to move swiftly and stealthily into the mountains with the body is just the right level of challenge for the group.

My only concern is that allowing this version of the spell to the party wizard permanently will have some unforeseen interaction with some other spell, ability, or item that I will later regret. This question, for example, asks about upcasting enlarge to permit two changes in creature size, and answers identify the interaction with levitation and grappling being problematic. I am interested in a similar troubleshooting review.

object oriented design – Is it wrong to extend an inner static class in java? (Builder pattern)

I’m working on a java project for the university. The project is a card game in which you travel around a 2D map and fight against some enemies. My part consists of creating the deck and the cards.

I made the card interface, an abstract card and an implementation that extends the abstract card and implements the interface.

Now I also want an internal static builder in the abstract class and a related builder implementation in all classes that extend the abstract class (because if the new card has a new field I also want to add it in the builder). I have extended the builder in the implementation, but I don’t know if it is correct to extend a static class.

The idea came to me from:

Effective Java, 3rd edition, Joshua Bloch, Item 2: Consider a builder when faced with many constructor parametersconstructor parameters

The question is: Is it correct to extends a static inner class?

Because what I would mainly like is to “force” all classes that extend the abstract class to have a builder, since the constructor has many parameters.

networking – Extend PC downstairs, to monitor, keyboard, mouse on second floor with low latency

So, I have a unique situation. My main PC that I use as a workstation for video editing and music production is moving to a new office in the basement. My girlfriend will never accept me being down there all day, so I’m going to have to find a way to continue working upstairs on the 2nd floor. I don’t want to buy a second computer/laptop, as I’d have to buy more licenses for software, and its a headache I don’t yet want to deal with.

So, that’s led me to the idea of extending an HDMI signal upstairs to an second monitor (or pair of monitors, if possible). The question is, how to best extend that signal.

Our modem/router is on the 1st floor, and is connected to my PC in the basement via ethernet. I have a Gigabit port on my motherboard, so that’s an option if it helps. Running ethernet to the basement was easy, but running it upstairs would be too hard to deal with.

The two ways I’ve come up with so far are, 1. Use wireless HDMI extenders, but I’m afraid those won’t extend over 2 floors, and the latency/video quality will be almost unusable. 2. KVM Switch with KVM over IP support. This seems more favorable, but now, I have no clue how to extend the Gigabit ethernet to the second floor. I know I’d need a Gigabit PoE switch, but how do I then wirelessly transmit that signal to the second floor. Wireless access point?

Any ideas, recommendations would be amazing! I know little about gigabit connections and the options available. Is it worth it in the long run, or should I just settle for HDMI extenders?


usa – Is there any way to extend my visa?

I´m a Mexican citizen and my visa has expired. I was planning on taking some marketing courses in the USA in April, but with the pandemic, the consulate is closed. Is there any way to do the paperwork?

8 – How to Extend the routing of taxonomy terms

I am trying to extend the routin of the taxonomies but I have not been able to do it, any clue of how it could be achieved?

The idea is to be able to have a route in the following way in which you can pass arguments dynamically.

For example:


Then receive in a controller those values of the taxonomy atnd the value of the node.

Is it possible to do this ?

Can a Teleconverter be used to extend a Sigma 70-300mm f/4-5.6 lens?

You can use a teleconverter with some zoom lenses — but not with your Sigma. In fact, most of Sigma’s higher-end zooms are specifically designed to work with a teleconverter (Sigma’s), and a couple of them actually perform slightly better in some respects (specifically in terms of chromatic aberration and vignetting) with the teleconverter added than without.

The main problem with your lenses (apart from any other aspect of the lens design) is the maximum aperture. A 1.4x converter increases the effective focal length of the lens by a factor of 1.4, but it also decreases the effective maximum aperture by one full stop. That means your Sigma would become a 98-420mm f/5.6-8 lens, and would not be able to autofocus on your D3100 except at its widest setting. A 2x teleconverter doubles the focal length of the lens, but at a cost of two full stops, so your lens would be a 140-600 f/8-11, and would not autofocus at all on your camera.

That said, the optical design of the lens can complicate all of this a great deal. It’s not just about the maximum aperture; the shape of the light path behind the lens has a lot to do with it as well. Not all lenses, whether zoom or prime, have an optical characteristic that plays nicely with teleconverters. Most current telephoto zooms with a maximum aperture of at least f/2.8 work well with converters (some are restricted to 1.4x converters and should not be used with 1.7x or 2x converters — see the lens manual). “Consumer grade”, slow, variable-aperture lenses usually don’t work well, or don’t work at all, with a teleconverter.