Works $ f $ such that for every polynomial $ P $ the equation $ f (x) = P (x) $ has at most $ deg P + 1 $ solutions

I want to characterize the set $ S $ of continuous functions $ f colon I to mathbb R $ so for everyone $ n ge 0 $ and for every polynomial $ P $ of the degree $ n $, The equation $ f (x) = P (x) $ has at most $ n + 1 $ solutions.

To do this, I found it useful to recall:

lemma, To let $ g colon I to mathbb R $ be continuously differentiable. If $ g & # 39; $ has at most $ n $ Zeros in $ I $, then $ g $ has at most $ n + 1 $ Zeros in $ I $,

proof: Since $ g & # 39; $ has at most $ n $ There are zeros $ m le n $ and there are $ m + 1 $ intervals $[x_0 = inf I, x_1]. [x_1, x_2] dotsc [x_m, x_{m+1} = sup I]$ so that $ g & # 39; $ keeps the same sign $[x_i, x_{i+1}]$, That's why, $ g $ is monotone in each $[x_i, x_{i+1}]$and thus each has at most one zero $[x_i, x_{i+1}]$,

What I have found so far $ S $:

  1. If $ f in S $, then $ f $ is injective (and thus monotonous). proof: To let $ y in mathbb R $, Since $ P (x) = y $ is a polynomial of degree $ 0 $ and $ f in S $, The equation $ f (x) = y $ has at most $ 1 $ Solution.

  2. If $ f & # 39; in S $, then $ f in S $, proof: To let $ P $ be a polynomial of degree $ n $, Since $ P & # 39; $ has grad $ n-1 $ and $ f & # 39; in S $, then $ f & # 39; (x) – P & # 39; (x) $ has at most $ n $ Zeros, that is, after the lemma $ f (x) – P (x) $ has at most $ n + 1 $ Zeros.

  3. $ e ^ x in S $, proof: By induction on $ n $, The case $ n = 0 $ follows through injectivity. To let $ n> 0 $ and $ P (x) $ be a polynomial of degree $ n $, Through the inductive hypothesis, $ e ^ x – P # (x) $ has at most $ n $ Zeros, that is, after the lemma $ e ^ x – P (x) $ has at most $ n + 1 $ Zeros.

  4. $ ln x in S $, proof: To let $ n ge 0 $ and $ P (x) $ be a polynomial of degree $ n $, Then $ 1 – x P & # 39; (x) $ is a polynomial of degree $ n $ and therefore has no more than $ n $ Zeros and so on $ frac 1 x – P # (x) $ also has at most $ n $ Zeros. Again, it follows from the lemma that $ ln x – P (x) $ has at most $ n + 1 $ Zeros.

  5. If $ f in S $ and $ g (x) = a f (x) + b $ for some $ a, b in mathbb R $ With $ a neq 0 $, then $ g in S $,

  6. If $ f in S $ and $ g (x) = f (a x + b) $ for some $ a, b in mathbb R $ With $ a neq 0 $, then $ g in S $,

  7. Of course, no polynomial is included $ S $, Likewise, $ arcsin x $. $ arccos x $, and $ arctan x $ do not belong $ S $,

Is there a simple characterization of $ S $?

I do not mind limiting $ S $ to differentiable or even smooth functions, if it proves necessary to provide a simple characterization.

What is the smallest value of n, so that an algorithm with 100 * n ^ 2 works faster than 2 ^ n? [How to figure out without brute force]

Okay, so I had to find the smallest value of n so that the algorithm 100 * n ^ 2 is faster than 2 ^ n.

[what I have tried]

So I immediately thought & # 39; 0 & # 39 ;. But then I realized that it can not be 0. 0 means that no digits have been entered in the algorithm. It also means that the program will not run or exit.

I typed in 2 ^ 32 and got a figure of over 4 billion. Okay, that's good, I find numbers with 100 * n ^ 2 that are faster than 2 ^ n.

I cut that in half, n = 20.

I inserted values ​​that counted downwards until I came to n = 15.

I also counted from n = 10, the answer is undoubtedly n = 15, but I have a problem. , , I solved that with brute force and that's not good. What if I was given a larger number and a larger number of numbers?

[What I need]

I need a way to find the value instantly by just counting. I tried to use logarithms, but my answer was wrong. My knowledge about logs is a bit rusty and I need a little help.

Think of this as a student trying to solve a question on a SAT, or having a timer for a test.

Responsive Design – How to properly set a large title that works on the desktop and on the phone?

I am working on the main title of the website. I want to put one 100% Height wallpaper and a great title on the front. I did it that way:

In the CSS file:

Body, HTML {
Height: 100%;
}

.bg {
Wallpaper: url ("../ img / image.JPG");
Height: 100%;

Background position: middle;
Background repetition: no repetition;
Background Size: Cover;
}

bigtitle {
/ ** upholstery left: 15%; ***
Padding: 20%;

Color white;
Font family: & # 39; Trocchi & # 39 ;, serifs;
Font size: 1500%;
Font size: normal;
Edge: 0;

/ * Line height: 1.0em; * /
}

In the HTML file:

My title

This is good on the desktop, but fails completely on mobile devices or small screens. How to work on the phone and on the desktop?

Plugins – WP Rest Api GET method on the route, but the POST method works too

I do some basic API testing and find that restricting my new route to GET does not work. POST requests are also sent to Postman. (PUT, PATCH, DELETE, however, do not indicate a "no matching route / method" error):

Function displayDummyData () {
& # 39; dummy data & # 39; hand back; // responds to POST request with & # 39; dummy data & # 39;
}

Function checkIfAnyone () {
return true;
}


Function registerDummyRoute () {
register_rest_route (& # 39; wp / v2 & # 39 ;, does & # 39 ;, & # 39; [
    'methods' => 'GET',
    'callback' => 'displayDummyData',
    'permission_callback' => 'checkIfAnyone'
  ]);
}
add_action ('rest_api_init', 'registerDummyRoute');

How can I limit this route to GET requests?

Unit – Immediate animation transitions? Nothing I read about works

reimported my character today with 3 new animations (JumpStartState, JumpDownState, JumpEndState) and moved the entire animation code to the PlayerController script. All my animations are handled by transitions and their parameters in the animator. The controls are very responsive, but I can not stop the transitions immediately before an animation is played.

Enter image description here

I can jump the character immediately after hitting the ground from a previous jump (a la Mario 64), but its animation is delayed until the landing animation is finished. This is the most notable example of animation not responding.

Enter image description here

I've been reading forums from 2011 looking for a solution to this problem, and they either said "disabling the option" Hat-Exit-Time "," setting Exit Time to zero "or" moving the blue bar "close to each other (by the way the same)". None of that worked for me. Mostly there was no difference at all; Another time, my character just ran frozen like a cardboard frosty through the scene (animation stops at the start).

Enter image description here
Enter image description here

I have a link to a NEW short demo where I bounce my player through the room, while the animation diagram gives live feedback here (01:50):

Character Animation Graph Live with preview

TO EDIT: Looking at the footage, there may be an animation kink between "JumpDownState" and "JumpEndState" that takes at least half the time, but even if this is not the case, the animation must be played for the next start.

I attach my code in case:

using System.Collections;
using System.Collections.Generic;
with UnityEngine;

public class PlayerController: MonoBehaviour
{

private float charSpeed;
public float walkSpeed;
private float runSpeed;
public swimmer jumpForce;
private float coyoteTime = 0;
private const float MAXCOYOTETIME = 0.1f;
Private pose currentHeight;
Private pose previous height;
private water ride;
public character controller controller;
private animator animator;
public swimmer gravityScale;
private rigid body rb;

public static Vector3 moveDirection;

// start is called before the first frame update
void start ()
{
controller = GetComponent();
rb = GetComponent();
animator = GetComponent();
charSpeed ​​= walkSpeed;
runSpeed ​​= walkSpeed ​​* 2;
previousHeight = currentHeight;
}

// update is called once per frame
void update ()
{

currentHeight = transform.position.y;
travel = currentHeight - previousHeight;
if (Input.GetButton ("Fire3"))
{
charSpeed ​​= runSpeed;
animator.SetBool ("isRunning", true);
}
otherwise
{
charSpeed ​​= walkSpeed;
animator.SetBool ("isRunning", false);
}

// declare motion vector
moveDirection = new Vector3 (Input.GetAxis ("Horizontal") * charSpeed,
moveDirection.y,
Input.GetAxis ("Vertical") * charSpeed);

// Defining the movement conditions of the y-axis (jumping, landing, free fall)
#Region Jumping
if (IsGrounded () && Input.GetButton ("Jump"))
{
animator.SetBool ("isJumping", true);
moveDirection.y = jumpForce;

}

otherwise if (IsGrounded () == false)
{
moveDirection.y = moveDirection.y + Physics.gravity.y * gravityScale * Time.deltaTime;
if (moveDirection.y <0)
{
animator.SetBool ("isJumping", false);
animator.SetBool ("isFalling", true);
if (travel == 0 && Input.GetButton ("Jump"))
{
animator.SetBool ("isJumping", true);
animator.SetBool ("isFalling", false);
}
}
}
#endregion
if (travel == 0)
{
animator.SetBool ("isFalling", false);
animator.SetBool ("isJumping", false);
}

// turned 45 degrees to align the controls on the camera instead of space:
Vector3 rotated = quaternion.Euler (0, -45, 0) * moveDirection;
controller.Move (rotated * Time.deltaTime);

if (Input.GetAxis ("Horizontal") == 0 && Input.GetAxis ("Vertical") == 0)
{
animator.SetBool ("isMoving", false);
}
otherwise
{
animator.SetBool ("isMoving", true);
}

Vector3 lookDir = new Vector3 (rotated.x, 0, rotated.z);

if (lookDir! = Vector3.zero)
{
transform.rotation = Quaternion.LookRotation (lookDir);
}

previousHeight = currentHeight;
}

public bool IsGrounded ()
{
return coyoteTime <MAXCOYOTETIME;
}

Cancel LateUpdate ()
{
if (controller.isGrounded)
{
coyoteTime = 0;
}
otherwise
{
coyoteTime + = Time.deltaTime;
}
}
}

EDIT: IRON THE LAST WRINKLE! NEW DEMO!(6:30 or less)

Excited to say that all animations are executed immediately, except the repeated jump. Half the time, the transition to a new jump does not take place at all. New footage of improved movement and more description of the problem:

Faster movement; Iron out the last fold!

MacBook commands, tabs, and mouse clicks do not work, the rest of the keyboard and mouse movement still works

A few days ago I reported that my MacBook Pro had a glitch in which everything except volume, brightness and mouse movement did not work anymore. I now have a similar but very pronounced problem on the same device. Now the entire keyboard works, but the cmd tab to switch application, cmd search distance, and Caps Lock do not work (Caps Lock works for the other error). Strangely, I can type cmd t to open a new tab in chrome, so the key itself must work. Unlike the other glitches, I can also use the Tab key. What's happening?!

It is the thing with Forex works?

advertising

Did not you do it?

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Domain Name System – DNS records are not displayed, but whois works fine

I'm using AWS stack, including Route53. However, the domain is registered with an external registrar because Route53 does not support the TLD I need.

Until a few days ago, when the domain expired, it worked flawlessly. I deleted the old hosted zone in Route53 and created a new one. I have also renewed the domain.
The NS records of the domain are set to the current DNS entries of Route53.

When I launch whois through various tools, the correct information is displayed, including updated rs ns records. However, any DNS query, including NS and SOA, will fail NXDOMAIN

Networking – Ubuntu 19.04 can not connect to Ethernet. Only works if the cable was plugged in before booting up. Any lasting solutions?

I'm using Ubuntu 19.04 in my Dell laptop REALTEK Ethernet Controller and everything was alright, before I upgraded to 19.04, before I used 18.10 and when I upgraded my ethernet, it just did not work anymore Does not get an IP address It shows, however, that the wire is connected. I tried almost everything, but nothing worked.

Mouse – Bluetooth can not move the cursor after successful pairing and connection, the connection lasts only a few minutes and then never works (Kubuntu 18.04)

I'm new to Linux and have just done a fresh install of Kubuntu 18.04 LTS on my Lenovo Thinkpad T430, but I can not use my mouse (logitech m590). I'm a bit confused how Blueman works, so I did not use it.

Result of the fix listed here: https://ubuntuforums.org/showthread.php?t=2390542

$ bluetoothctl
[NEW]Controller 68: 94: 23: ED: 33: BD nicos-ThinkPad-T430 [default]
Agent registered
[bluetooth]# List
Controller 68: 94: 23: ED: 33: BD nicos-ThinkPad-T430 [default]
[bluetooth]# 68: 94: 23: ED: 33: Select BD
[bluetooth]# Show
Controller 68: 94: 23: ED: 33: BD (public)
Name: nicos-ThinkPad-T430
Alias: nicos-ThinkPad-T430
Class: 0x001c010c
Powered: yes
Recognizable: no
Pairable: yes
UUID: Headset AG (00001112-0000-1000-8000-00805f9b34fb)
UUID: generic attribute profile (00001801-0000-1000-8000-00805f9b34fb)
UUID: A / V Remote Control (0000110e-0000-1000-8000-00805f9b34fb)
UUID: OBEX file transfer (00001106-0000-1000-8000-00805f9b34fb)
UUID: Generic Access Profile (00001800-0000-1000-8000-00805f9b34fb)
UUID: OBEX Object Push (00001105-0000-1000-8000-00805f9b34fb)
UUID: PnP Information (00001200-0000-1000-8000-00805f9b34fb)
UUID: IrMC Sync (00001104-0000-1000-8000-00805f9b34fb)
UUID: A / V Remote Control Destination (0000110c-0000-1000-8000-00805f9b34fb)
UUID: audio source (0000110a-0000-1000-8000-00805f9b34fb)
UUID: Audio Sink (0000110b-0000-1000-8000-00805f9b34fb)
UUID: manufacturer-specific (00005005-0000-1000-8000-0002ee000001)
UUID: Message Notification Se .. (00001133-0000-1000-8000-00805f9b34fb)
UUID: Phone Book Access Server (0000112f-0000-1000-8000-00805f9b34fb)
UUID: Message Access Server (00001132-0000-1000-8000-00805f9b34fb)
UUID: Headset (00001108-0000-1000-8000-00805f9b34fb)
Modalias: usb: v1D6Bp0246d0530
Discover: no
[bluetooth]# on
The switch was successful
[bluetooth]# scan on
The investigation has begun
[CHG] Controller 68: 94: 23: ED: 33: BD detection: yes
[NEW] Device C8: 0F: 10: B3: D7: E0 MI_SCALE
[NEW] Device 64: B3: 14: 45: 51: 66 64-B3-14-45-51-66
[NEW] Device 5C: FF: D5: 4D: D6: E2 5C-FF-D5-4D-D6-E2
[NEW] Device 55: 35: 80: 4F: FE: BB 55-35-80-4F-FE-BB
[NEW] Device D8: 0F: 99: 30: 37: 46 D8-0F-99-30-37-46
[CHG] Device D8: 0F: 99: 30: 37: 46 TxPower: 7
[CHG] Device D8: 0F: 99: 30: 37: 46 Name: KD-43X8000E
[CHG] Device D8: 0F: 99: 30: 37: 46 Alias: KD-43X8000E
[CHG] Device D8: 0F: 99: 30: 37: 46 UUIDs: 0000110a-0000-1000-8000-00805f9b34fb
[CHG] Device D8: 0F: 99: 30: 37: 46 UUIDs: 0000110c-0000-1000-8000-00805f9b34fb
[CHG] Device D8: 0F: 99: 30: 37: 46 UUIDs: 0000110e-0000-1000-8000-00805f9b34fb
[CHG] Device D8: 0F: 99: 30: 37: 46 UUIDs: 00001200-0000-1000-8000-00805f9b34fb
[CHG] Device D8: 0F: 99: 30: 37: 46 ManufacturerData Key: 0x000f
[CHG] Device D8: 0F: 99: 30: 37: 46 ManufacturerData Value:
00 01 64 ..d             
[NEW] Device A4: E6: 15: 59: B7: 55 A4-E6-15-59-B7-55
[CHG] Device A4: E6: 15: 59: B7: 55 Name: Android TV
[CHG] Device A4: E6: 15: 59: B7: 55 Alias: Android TV
[CHG] Devices A4: E6: 15: 59: B7: 55 UUIDs: 00001105-0000-1000-8000-00805f9b34fb
[CHG] Devices A4: E6: 15: 59: B7: 55 UUIDs: 0000110a-0000-1000-8000-00805f9b34fb
[CHG] Devices A4: E6: 15: 59: B7: 55 UUIDs: 0000110c-0000-1000-8000-00805f9b34fb
[CHG] Devices A4: E6: 15: 59: B7: 55 UUIDs: 0000110e-0000-1000-8000-00805f9b34fb
[CHG] Devices A4: E6: 15: 59: B7: 55 UUIDs: 00001115-0000-1000-8000-00805f9b34fb
[CHG] Devices A4: E6: 15: 59: B7: 55 UUIDs: 00001116-0000-1000-8000-00805f9b34fb
[CHG] Devices A4: E6: 15: 59: B7: 55 UUIDs: 0000112f-0000-1000-8000-00805f9b34fb
[CHG] Devices A4: E6: 15: 59: B7: 55 UUIDs: 00001200-0000-1000-8000-00805f9b34fb
[bluetooth]# Scan off
[CHG] Device A4: E6: 15: 59: B7: 55 RSSI is zero
[CHG] Device D8: 0F: 99: 30: 37: 46 TxPower is zero
[CHG] Device D8: 0F: 99: 30: 37: 46 RSSI is zero
[CHG] Device 55: 35: 80: 4F: FE: BB RSSI is zero
[CHG] Device 5C: FF: D5: 4D: D6: E2 RSSI is zero
[CHG] Device 64: B3: 14: 45: 51: 66 RSSI is zero
[CHG] Device C8: 0F: 10: B3: D7: E0 RSSI is zero
[CHG] Controller 68: 94: 23: ED: 33: BD detection: no
Discovery stopped
[bluetooth]# scan on
The investigation has begun
[CHG] Controller 68: 94: 23: ED: 33: BD detection: yes
[CHG] Device D8: 0F: 99: 30: 37: 46 RSSI: -71
[CHG] Device D8: 0F: 99: 30: 37: 46 UUIDs: 0000110a-0000-1000-8000-00805f9b34fb
[CHG] Device D8: 0F: 99: 30: 37: 46 UUIDs: 0000110c-0000-1000-8000-00805f9b34fb
[CHG] Device D8: 0F: 99: 30: 37: 46 UUIDs: 0000110e-0000-1000-8000-00805f9b34fb
[CHG] Device D8: 0F: 99: 30: 37: 46 UUIDs: 00001200-0000-1000-8000-00805f9b34fb
[CHG] Device UUIDs D8: 0F: 99: 30: 37: 46: cbbfe0e1-f7f3-4206-84e0-84cbb3d09dfc
[CHG] Device 64: B3: 14: 45: 51: 66 RSSI: -64
[NEW] Device F6: 7E: 9D: DE: 52: 0F M585 / M590
[CHG] Device 55: 35: 80: 4F: FE: BB RSSI: -83
[CHG] Device C8: 0F: 10: B3: D7: E0 RSSI: -79
[bluetooth]# Scan off
Discovery stopped
[CHG] Controller 68: 94: 23: ED: 33: BD detection: no
[CHG] Device C8: 0F: 10: B3: D7: E0 RSSI is zero
[CHG] Device 55: 35: 80: 4F: FE: BB RSSI is zero
[CHG] Device F6: 7E: 9D: DE: 52: 0F TxPower is zero
[CHG] Device F6: 7E: 9D: DE: 52: 0F RSSI is zero
[CHG] Device 64: B3: 14: 45: 51: 66 RSSI is zero
[CHG] Device D8: 0F: 99: 30: 37: 46 RSSI is zero
[bluetooth]# agent on
Agent is already registered
[bluetooth]# Pair F6: 7E: 9D: DE: 52: 0F
An attempt was made to connect to F6: 7E: 9D: DE: 52: 0F
[CHG] Device F6: 7E: 9D: DE: 52: 0F Connected: yes
[CHG] Device F6: 7E: 9D: DE: 52: 0F Connected: No.
Pairing failed: org.bluez.Error.AuthenticationCanceled

This fix worked a bit, but was later accidentally disconnected

I tried reinstalling, but it just connected for a bit

Here are some things that might help:

Output of the system status Bluetooth

$ systemctl status bluetooth
● bluetooth.service - bluetooth service
Loaded: loaded (/lib/systemd/system/bluetooth.service; enabled; vendor preference: enabled)
Active: active (running) since Sat 2019-07-13 22:40:23 PST; Before 6min
Docs: man: bluetoothd (8)
Main PID: 14392 (Bluetooth)
Status: "Running"
Tasks: 1 (Limit: 4915)
Group: /system.slice/bluetooth.service
└─14392 / usr / lib / bluetooth / bluetoothd

July 13 22:40:23 nicos-ThinkPad-T430 systemd[1]: Bluetooth service is started ...
July 13 22:40:23 nicos-ThinkPad-T430 bluetoothd[14392]: Bluetooth Daemon 5.48
July 13 22:40:23 nicos-ThinkPad-T430 bluetoothd[14392]: Starting the SDP Server
July 13 22:40:23 nicos-ThinkPad-T430 systemd[1]: Bluetooth service started.
July 13 22:40:23 nicos-ThinkPad-T430 bluetoothd[14392]: Bluetooth Management Interface 1.14 initialized
July 13 22:40:23 nicos-ThinkPad-T430 bluetoothd[14392]: Endpoint registered: Sender =: 1,39 Path = / MediaEndpoint / A2DPSource
July 13 22:40:23 nicos-ThinkPad-T430 bluetoothd[14392]: Endpoint registered: Sender =: 1,39 Path = / MediaEndpoint / A2DPSink
July 13 22:42:48 nicos-ThinkPad-T430 bluetoothd[14392]: No cache for F6: 7E: 9D: DE: 52: 0F

that too could help

$ lsusb -v | grep Bluetooth | grep DeviceProtocol
Device could not be opened, some information is missing
Device could not be opened, some information is missing
Device could not be opened, some information is missing
Device could not be opened, some information is missing
Device could not be opened, some information is missing
Device could not be opened, some information is missing
Device could not be opened, some information is missing
Device could not be opened, some information is missing
Device could not be opened, some information is missing

Other things:

  • The mouse is a multi-device and seems to change the last letter of your Mac
    address
  • Mouse has both Bluetooth and receiver, both work independently. Receiver is not working
  • Running KDE Plasma 5.12.7

It's my only mouse, so I need you to pair it if I want to use Kubuntu daily. thank you in advance