Shows an equivalence of $ | langle textbf {z}, x rangle | <R $

Consider the following event: for a fix $ x in mathbb {R} ^ n $ With $ | x | = 1 $ and $ textbf {z} sim N (0, textbf {I} _n) $,
$$
A_x ( textbf {z}) = { exists w in mathbb {R} ^ n text {so} | w – textbf {z} | le R, text {and} text {sign} ( langle w, x rangle) ne text {sign} ( langle textbf {z}, x rangle) }.
$$

I try to show that
$$
A_x ( textbf {z}) text {happens} iff | langle textbf {z}, x rangle | <R.
$$

($ implies $) This instruction is easily executed as follows.
Accept $ A_x ( textbf {z}) $ happens. Then there is $ w in mathbb {R} ^ n $ so that $ | w – textbf {z} | R $ and $ text {sign} ( langle w, x rangle) ne text {sign} ( langle textbf {z}, x rangle) $,
That's why,
$$
R ge | w – textbf {z} | ge | langle (w – textbf {z}), x rangle | ge | langle textbf {z}, x rangle |
$$

where the third inequality is used $ text {sign} ( langle w, x rangle) ne text {sign} ( langle textbf {z}, x rangle) $,

I'm not sure why the reverse is true.
Here is my try.

($ impliedby $) Accept $ R ge | langle textbf {z}, x rangle | $,
Then our goal is to find it $ w in mathbb {R} ^ n $ satisfying
(I) $ | w – textbf {z} | R $ and (ii) $ text {sign} ( langle w, x rangle) ne text {sign} ( langle textbf {z}, x rangle) $,
I tried to construct $ w $ by setting it $ w = -a textbf {z} $ for a positive constant $ a $, This satisfies condition (ii) automatically. And I was hoping that (i) can be satisfied by careful selection $ a $,
However, this only works under the condition of $ | textbf {z} | R $Not
$ | langle textbf {z}, x rangle | R $, This is because of the attitude $ w = -a textbf {z} $, we have
$$
| w – textbf {z} | = (1 + a) | textbf {z} |.
$$

This is less than or equal to $ R $,
$$
a + 1 le frac {R} { | textbf {z} |} iff a le frac {R | textbf {z} |} { | textbf {z} |}.
$$

Since $ a $ is positive, we need $ | textbf {z} | <R $,

Suggestions / answers / comments are highly appreciated.

Third-party Nikon battery stops working

Recently, I encountered a strange problem: A non-original replacement battery (EN-EL14a) for my DSLR (Nikon D3300), which I bought about a year ago, suddenly stopped working. When inserting into the body I receive a message:
This battery can not be used. Use a battery designed for this camera.

I'm pretty shocked because it worked fine so far. By the way: I'll shoot a wedding in a few weeks: D Is there a reason for his behavior and possible solution?

Many Thanks

Picture: Original battery left, 3rd party right

Original left, third side right

ftp nat / port forward is not working

I have the following env:

  • my pc as ftp client
  • a server (called HOST) with firewall enabled and the following rules:
    public (active)
    Goal: ACCEPT
    icmp block inversion: no
    Interfaces: ens192
    Swell:
    Services: ssh dhcpv6-client ftp
    Ports: 20023 / tcp 29000-29100 / tcp 20021 / tcp 21 / tcp
    protocols:
    Masquerade: Yes
    Forwarding Ports: Port = 20023: Proto = tcp: toport = 23: toaddr = 10.1.1.2
    port = 29000-29100: proto = tcp: toport = 29000-29100: toaddr = 10.1.1.2
    port = 20021: proto = tcp: toport = 21: toaddr = 10.1.1.2
    port = 21: proto = tcp: toport = 21: toaddr = 10.1.1.2
    Source port:
    icmp-blocks:
    rich rules:
  • a virtual server (named APP) in the HOST server hosting an FTP server with a passive range between 29000 and 29100

The network configuration of Server HOST is:

ens192: flags = 4163  mtu 1492
Inet 10.10.12.6 Netmask 255.255.255.0 Broadcast 10.10.12.255

tap0: flags = 4163  mtu 1500
Inet 10.1.1.1 Netmask 255.255.255.0 Broadcast 10.1.1.255

When I try to connect from my client (using FileZilla) to the APP FTP server login, DIR, GET or others do not work, and I do not know how to properly configure my Env.

Can someone help?
Many Thanks.

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