functional analysis – If $ T (t) $ is a semigroup and $ x $ satisfies $ left | T (h) xx right | to0 $ as $ h to0 $, can we conclude that $ t mapsto T (t) x $ is continuous?

Let $ E $ be a $ mathbb R $-Space Banach and $ (T (t)) _ {t ge0} $ to be a semi-group on $ E $.


  1. Yes $ x in E $then $ (T (t)) _ {t ge0} $ is called strongly continues to $ x $ Yes Yes $$[0infty)Toe\\;tmapstoT(t)xtag1$$[0infty)Toe\\;tmapstoT(t)xtag1$$[0infty)toE;;;;tmapstoT(t)xtag1$$[0infty)toE;;;;tmapstoT(t)xtag1$$ is continuous
  2. $ (T (t)) _ {t ge0} $ is called locally delineated$ ^ 1 $ Yes Yes $$ exists t> 0: sup_ {s in[0t)}left|T(s)right|_{mathfrakL(E)}<inftytag2$$[0t)}left|T(s)right|_{mathfrakL(E)}<inftytag2$$[0:t)}left|T(s)right|_{mathfrakL(E)}

Question: Let $ x in E $ with $$ left | T (h) x-x right | _E xrightarrow {h to0 +} 0 tag4. $$ Are able to show that $ (T (t)) _ {t ge0} $ is strongly continuous to $ x $?

It's easy to see $ (1) $ is just-continuous. In addition, I am able to show that $ (1) $ is left coninue, as long as I guess $ (T (t)) _ {t ge0} $ is bounded locally.

Can we abandon the hypothesis of local delimitation and still conclude continuity on the left?

My idea is: by $ (4) $, $$ exists t> 0: forall h in[0t):left|T(h)xright|_E<1+left|xright|_ETag5$$[0t):left|T(h)xright|_E<1+left|xright|_ETag5$$[0t):left|T(h)xright|_E<1+left|xright|_Etag5$$[0t):left|T(h)xright|_E<1+left|xright|_Etag5$$ and perhaps we can show in a certain way the local delimitation by the principle of uniform delimitation.

$ ^ 1 $ By the semigroup property, $ (1) $ implies that $$ sup_ {s in I} left | T (s) right | _ { mathfrak L (E)} < infty $$ for each bounded interval $ I subseteq[0infty)$[0infty)$[0infty)$[0infty)$.

mobile – How to clarify the ambiguity of the left arrow in Google search autosuggestions?

As already mentioned, this feature poses two problems:

  • low visibility
  • non-standard glyph which does not mean that the added text will be editable. This means that the selected item will be added to the search field.

Low visibility should be easy to fix with something like this in the picture below. However, Google's designers have probably wanted to stay subtle. In addition, as shown by Google Maps (not this image), the search results may also contain some lines of description. Using only the borderless icon and background color seems like a good idea to have a clean interface and less congestion.

Regarding the problem of nonstandard glyphs, it can only be tested with a different icon, which may transmit both functions.

enter the description of the image here

User interface to select different tools for right and left clicks

I'm working on a specialized painting web application (for Minecraft structure design). I would like the user to be able to assign different tools to the left and right mouse buttons. (I do not intend to override the default right-click behavior other than this one.)

I have a traditional tool palette. I have experienced a way to indicate that the user has selected a tool for the left mouse button and another for the right mouse button:

enter the description of the image here

The left and right tool indicators are differentiated in three ways:

  • Position (left vs right)
  • A little L or R.
  • Contrasting colors (although meaningless).

To select a tool, the user clicks on its icon with the left or right mouse button. This button is linked to this tool.

The functionality of the right click tool is (very) convenient, but not essential. Users who can not or do not want to click the right mouse button can use the left button exclusively. In addition, when the cursor is on the canvas, the cursor icon corresponding to the tool on the left is displayed unless the user is actively using the right tool.

Does this interface make sense? I'm not yet on the verge of testing users, but I wanted a health check because I do not think I've seen this model before. An alternative would be to have a separate tool palette for a right click, but I am afraid it will be confusing and annoying.

What will be the api query left to get the field values ​​of the site content type using api rest?

How to retrieve choice field values ​​from site content type directly>

That's what I tried:

http: //// _api / web / contenttypes

[ Politics ] Open question: When the left will finally be able to take full control of this country and implement its plans, will it just give in or?

will they defend themselves?

And if they decide to retaliate, it is unlikely that we are witnessing a civil war much more lethal than the previous one. .

MariaDB has filled the folder / tmp until there is no disk space left

Could someone here take a look at this section of the MariaDB journal and let me know what makes mysql use all the disk space of the / tmp partition? My assumption is that this was caused by a poorly optimized query, but it could also be something else …

Version: 5.5.56-MariaDB & # 39; socket: /var/lib/mysql/mysql.sock&#39; port: 3306 MariaDB Server
190504 05:43:02 mysqld_safe Number of processes currently in progress: 0
190504 05:43:02 mysqld_safe mysqld restarted
190504 5:43:02 InnoDB: The InnoDB memory segment is off
190504 5:43:02 InnoDB: Mutex and rw_locks use GCC atomic coding
190504 5:43:02 InnoDB: Compressed tables use zlib 1.2.7
190504 5:43:02 InnoDB: Using the native Linux AIO
190504 5:43:02 InnoDB: Initialization of the buffer pool, size = 128.0M
190504 5:43:02 [Note] / usr / libexec / mysqld (mysqld 5.5.56-MariaDB) as a process 57441 ...
190504 5:43:02 InnoDB: Initialization Complete Buffer Pool
190504 5:43:02 InnoDB: The most supported file format is Barracuda.
190504 5:43:02 InnoDB: Launch of recovery after crash from LSN checkpoint = 83018874130
InnoDB: Restore any semi-written data pages from the double write buffer.
190504 5:43:02 InnoDB: Start of the last batch to recover 5 pages of the recovery log
190504 5:43:02 InnoDB: Waiting for the start of the threads in the background
190504 5:43:03 Percona XtraDB ( 5.5.52-MariaDB-38.3 started; log sequence number 83018875325
190504 5:43:04 [Note] The plugin & # 39; FEEDBACK & # 39; is disabled.
190504 5:43:04 [Warning] Fail to install SSL
190504 5:43:04 [Warning] SSL error: SSL_CTX_set_default_verify_paths failed
190504 5:43:04 [Note] Socket server created over IP: & # 39;
190504 5:43:04 [Note] Event planner: events loaded 0
190504 5:43:04 [Note] / usr / libexec / mysqld: ready for connections.
Version: 5.5.56-MariaDB & # 39; socket: /var/lib/mysql/mysql.sock&#39; port: 3306 MariaDB Server
190506 6:00:23 [Warning] mysqld: the disk is full by writing /tmp/#sql_e061_53.MAD&#39; (error code: 28). Wait until someone frees up space ... (wait until 60 seconds for the server to continue after freeing up disk space)
190506 6:00:23 [Warning] mysqld: try again in 60 seconds. Message reprinted in 600 seconds

This is the full diary:

Thanks in advance!

windows – A mysterious translucent rectangle appears at the bottom left of win10 after the last update

I had to move the taskbar to the side so as not to cover the box in question. you can see the box here

This area is mainly visible in full screen, such as the background, the game, etc. She is always present. I do not think it was there before the update of yesterday from Windows 10 to the latest version.

Someone knows what it is and how to get rid of it?

Thank you!

PS: I just clicked on my own link, only to realize that screen capture does not reproduce this box. I thought I was going crazy for a second. His low left side behind the search. But this is visible only in full screen and without the bar that covers it.

I've taken the snapshot with my trusty iPhone and here's what it looks like

windows 10 – Is there a way to have the file explorer (associated with the taskbar) display a context menu (drop-down list of associated folders) with a left click?

I want to see the drop-down list of my pinned folders, instead of opening the Explorer window for quick access. Similar to list view for Mac-anchored folders.

I looked around me, but I could not find a solution … surprising given the tp software designed to mimic Mac functionality.


Algebraic Geometry – Why is the direct image sheaf an exact functor on the left and not only?

This could be a pretty silly question, however, I can not understand what's wrong.

Let $ f: X to Y $ be a morphism of annular spaces, and $ mathcal {F, G, H} $ sheaves of abelian groups on $ X $. Show that the direct image functor $ f _ * $ is left exact.

assume $ 0 to mathcal F to mathcal H to mathcal H to 0 $ is a short, exact sequence of sheaves. In particular, for each $ U subset X $, the sequence $ 0 to mathcal F (U) to mathcal G (U) to mathcal H (U) to 0 $ is short exact.

Consider the sequence $ 0 to f_ * mathcal F to f_ * mathcal G to f_ * mathcal H to 0 $. This is left correct if the sequence $ 0 to f_ * mathcal F (V) to f_ * mathcal G (V) to f_ * mathcal H (V) to 0 $ is left exact for each $ V subset Y $ open. This last sequence can also be written by definition as 0 $ to mathcal F (f ^ {- 1} (V)) to mathcal G (f ^ {- 1} (V)) to mathcal H (f ^ {- 1} (V)) at $ 0. But since $ f ^ {- 1} (V) $ is open in $ X $, then the previous sequence is already short exact.

What is the problem in this argument? I think I'm missing something big.

Integration – Verification: $ left | int_0 ^ 1 f (x) dx right | leq frac {1} {2} int_0 ^ 1 | f (x) | dx. $

Let $ f (x) $ to be a function with $ f (x) $ continue on $[0,1]$. $ f (0) + f (1) = 0 $. Prove $$ left | int_0 ^ 1 f (x) dx right | leq frac {1} {2} int_0 ^ 1 | f (x) | dx. $$

Let $ x-1/2 = t $,then $ x = t + 1/2 $,$ dx = dt $. So
begin {align *}
left | int_0 ^ 1 f (x) dx right | & = left | int _ {- frac {1} {2}} ^ {~~ frac {1} {2}} f left t + frac {1} {2} right) dt right | \
& = left | left[tfleft(t+frac{1}{2}right)right]_ {- frac {1} {2}} ^ {~~ frac {1} {2}} – int _ {- frac {1} {2}} ^ {~~ frac {1} { 2}} tf & # 39; left (t + frac {1} {2} right) dt right | ~~~ & textit {integrating by parts} \
& = left | 0- int_ {0} ^ {1} left (x- frac {1} {2} right) f left (x right) dx right | ~~~ & textit {substituting $ t + 1/2 = x $} \
& leq int_0 ^ 1 left | left (x- frac {1} {2} right) f left (x right) right | dx \
& leq left | xi- frac {1} {2} right | int_0 ^ 1 | f (x) | dx ~~~ & textit {applying the first MVT to the integral} \
& leq frac {1} {2} int_0 ^ 1 | f (x) | dx
end {align *}

what is desired.