## Covering radius of a code in the Hamming sace

A deterministic $$(2−2/(k+1))^n$$ algorithm for $$k$$-SAT based on local search

I have read this paper and I couldn’t understand how to culculate the $$covering$$ $$radius$$ of a code in the Hamming space. It is defined as follows in the paper.

We identify assignments with binary words. The set of these words of length $$n$$ is the $$Hamming$$ $$space$$ denoted by $$H_n=\{0,1\}^n$$. The $$Hamming$$ $$distance$$ between two assignments is the number of positions in which these two assignments differ. The $$ball$$ of radius $$r$$ around an assignment $$a$$ is the set of all assignments whose Hamming distance to $$a$$ is at most $$r$$.
A $$code$$ of length $$n$$ is simply a subset of $$H_n$$. The $$covering$$ $$radius$$ $$r$$ of a code $$C$$ is defined by $$r=\max_{u∈\{0,1\}^n}\min_{v∈C }d(u,v),$$ where $$d(u,v)$$ denotes the Hamming distance between $$u$$ and $$v$$.

The $$\max$$ of $$\min$$ is exactly the point I couldn’t calculate. So I considered the instance of $$C$$ and tried to calculate its covering radius.

e.g.)$$C=\{000,010,011,100,101,110\}⊆H_3$$

In this case, how can I calculate the covering radius $$r$$?

## Border Radius Preview Tool

I am running through an app idea to help improve my knowledge of Javascript. The code attached performs the following:

• Performs a regex on the input field to check it is value 0-9 If not,
• it clears the field and shows a warning (Although this only seems to work on FF)

• If it passes, updates the styles of the preview box on keyup

• Adds the styles to a text area that the user can copy when selecting the button

I am looking for suggestions where I can improve in terms of writing more efficient code. Or pointers where I have possibly used the wrong method to perform a task. One area where I struggled was getting the value of the input field.

This is the first stage for this application as in the near future I plan to add in the ability modify each corner.

Feedback is welcomed

"use strict"  const masterField = document.getElementById('masterField'); const box = document.getElementById('box'); const warn = document.getElementById('warn'); const styleSheet = document.getElementById('styleSheet');  const WARNING_TIME = 2000; var warningTimer;  masterField.addEventListener('keyup', updateCss);  function updateCss() {     let inputVal = this.value;     console.log(this.value);     const expCase = /[0-9]/;      if (expCase.test(inputVal)) {         box.style.borderRadius = inputVal + 'px';         styleSheet.textContent = 'border-radius: ' + inputVal + 'px';     } else {         this.value = '';         showWarning();     } }  function hidewarning() {     warn.classList.add('hide-warning'); }  function showWarning() {     clearTimeout(warningTimer);     warningTimer = setTimeout(hidewarning, WARNING_TIME);     warn.classList.remove('hide-warning'); }  function copyStyles() {     styleSheet.select();     document.execCommand('copy'); }
#box {             width: 300px;             height: 175px;             background-color: red;         }          #warn {             display: block;         }          #warn.hide-warning {             display: none;         }
<div id="box"></div>     <div>         <label for="masterField">All Corners</label>         <input id="masterField" class="radius-field" type="number">         <span id="warn" class="hide-warning">Only numbers 0 - 9 are allowed</span>     </div>      <button onclick="copyStyles()">Copy Styles</button>      <textarea id="styleSheet"></textarea>

## Is the injectivity radius (semi) continuous on a non-complete Riemannian manifolds?

Let $$\mathcal{M}$$ be a Riemannian manifold, and let $$\mathrm{inj} \colon \cal M \to (0, \infty]$$ be its injectivity radius function.

It is known that if $$\cal M$$ is connected and complete, then $$\mathrm{inj}$$ is a continuous function: see for example [Lee, Introduction to Riemannian Manifolds, 2018, Prop. 10.37].

What is known in the case where $$\cal M$$ is not complete? Is $$\mathrm{inj}$$ also continuous? If not, is there a known counter-example? Would $$\mathrm{inj}$$ still be semi-continuous?

This question is similar to the question “The continuity of Injectivity radius”, but the discussion there focuses on compact or complete manifolds.

## The circle is drawn by touching the x and y axis . If the point co-ordinate on the circle is (10,9),what is the circle radius?

The circle is drawn by touching the x and y axis. If the point coordinate on the circle is (10,9), what is the circle radius?

## Help with the term radius?

We had a situation come up regarding this term and how it affected spells.

In the spell Moonbeam, for example, it says a 5-foot radius cylinder appears. Does this mean that from the center of square it extends 2.5 feet into the next square hypothetically hitting up three squares or does this mean from the edge of one the target square to the five feet and five feet the other direction effectively covering only two squares for a total of 10 feet between total range of both squares?

I understand that radius means half the distance of a circle from the center point. The spell is not the issue but the understanding of radius as it pertains to the spell, as there are other spells that are worded similarly such as Fireball, etc.

Essentially we are asking, does the spell in the example effect one square or three? This would be saying the edge of the initial square is the point of origin and not the center of the square for the total distance of ten feet.

I had been working under the understanding that the radius is half of the distance and therefore dragging a ruler from the center of where I want the spell to take effect would get me the area of effect.

It’s much easier to do this with Fireball, due to the larger area to effect, but this one has such a small area and therefore sparked the debate. We are using a grid system to mark the distance, etc. We play within the digital realm of roll20.com.

## In an XAML Resource Dictionary, how can I change a border radius on a trigger event?

I’m making a combo box with a drop-down list, and I made the default state to have rounded corners. I want the bottom corners to have no rounded edges when the combo box is clicked and the list drops down.

I’ve created my own combo box control template and set the original combo box to have a border radius of 6. Then in the trigger that tracks if it’s expanded, I’ve tried to use a setter to change the border-radius.

Here’s an excessive amount of code to show my style template.

<Style x:Key="ComboBoxToggleButton" TargetType="{x:Type ToggleButton}">         <Setter Property="OverridesDefaultStyle" Value="true"/>         <Setter Property="IsTabStop" Value="false"/>         <Setter Property="Focusable" Value="false"/>         <Setter Property="ClickMode" Value="Press"/>         <Setter Property="Template">             <Setter.Value>                 <ControlTemplate TargetType="{x:Type ToggleButton}">                     <Border x:Name="templateRoot" BorderBrush="{StaticResource ComboBox.Static.Border}" BorderThickness="{TemplateBinding BorderThickness}" Background="{StaticResource ComboBox.Static.Background}" SnapsToDevicePixels="true" CornerRadius="6">                         <Border x:Name="splitBorder" BorderBrush="Transparent" BorderThickness="1" HorizontalAlignment="Right" Margin="0" SnapsToDevicePixels="true" Width="{DynamicResource {x:Static SystemParameters.VerticalScrollBarWidthKey}}">                             <Path x:Name="arrow" Data="F1 M 0,0 L 2.667,2.66665 L 5.3334,0 L 5.3334,-1.78168 L 2.6667,0.88501 L0,-1.78168 L0,0 Z" Fill="{StaticResource ComboBox.Static.Glyph}" HorizontalAlignment="Center" Margin="0" VerticalAlignment="Center"/>                         </Border>                     </Border>                     <ControlTemplate.Triggers>                         <MultiDataTrigger>                             <MultiDataTrigger.Conditions>                                 <Condition Binding="{Binding IsEditable, RelativeSource={RelativeSource AncestorType={x:Type ComboBox}}}" Value="true"/>                                 <Condition Binding="{Binding IsMouseOver, RelativeSource={RelativeSource Self}}" Value="false"/>                                 <Condition Binding="{Binding IsPressed, RelativeSource={RelativeSource Self}}" Value="false"/>                                 <Condition Binding="{Binding IsEnabled, RelativeSource={RelativeSource Self}}" Value="true"/>                             </MultiDataTrigger.Conditions>                             <Setter Property="Background" TargetName="templateRoot" Value="{StaticResource ComboBox.Static.Editable.Background}"/>                             <Setter Property="BorderBrush" TargetName="templateRoot" Value="{StaticResource ComboBox.Static.Editable.Border}"/>                             <Setter Property="Background" TargetName="splitBorder" Value="{StaticResource ComboBox.Static.Editable.Button.Background}"/>                             <Setter Property="BorderBrush" TargetName="splitBorder" Value="{StaticResource ComboBox.Static.Editable.Button.Border}"/>

I put the border-radius change here just to see if I could get it working.

<Setter Property="BorderRadius" TargetName="templateRoot" Value="0"/>

</MultiDataTrigger>                         <Trigger Property="IsMouseOver" Value="true">                             <Setter Property="Fill" TargetName="arrow" Value="{StaticResource ComboBox.MouseOver.Glyph}"/>                         </Trigger>                         <MultiDataTrigger>                             <MultiDataTrigger.Conditions>                                 <Condition Binding="{Binding IsMouseOver, RelativeSource={RelativeSource Self}}" Value="true"/>                                 <Condition Binding="{Binding IsEditable, RelativeSource={RelativeSource AncestorType={x:Type ComboBox}}}" Value="false"/>                             </MultiDataTrigger.Conditions>                             <Setter Property="Background" TargetName="templateRoot" Value="{StaticResource ComboBox.MouseOver.Background}"/>                             <Setter Property="BorderBrush" TargetName="templateRoot" Value="{StaticResource ComboBox.MouseOver.Border}"/>                         </MultiDataTrigger>                         <MultiDataTrigger>                             <MultiDataTrigger.Conditions>                                 <Condition Binding="{Binding IsMouseOver, RelativeSource={RelativeSource Self}}" Value="true"/>                                 <Condition Binding="{Binding IsEditable, RelativeSource={RelativeSource AncestorType={x:Type ComboBox}}}" Value="true"/>                             </MultiDataTrigger.Conditions>                             <Setter Property="Background" TargetName="templateRoot" Value="{StaticResource ComboBox.MouseOver.Editable.Background}"/>                             <Setter Property="BorderBrush" TargetName="templateRoot" Value="{StaticResource ComboBox.MouseOver.Editable.Border}"/>                             <Setter Property="Background" TargetName="splitBorder" Value="{StaticResource ComboBox.MouseOver.Editable.Button.Background}"/>                             <Setter Property="BorderBrush" TargetName="splitBorder" Value="{StaticResource ComboBox.MouseOver.Editable.Button.Border}"/>                         </MultiDataTrigger>                         <Trigger Property="IsPressed" Value="true">                             <Setter Property="Fill" TargetName="arrow" Value="{StaticResource ComboBox.Pressed.Glyph}"/>                         </Trigger>                         <MultiDataTrigger>                             <MultiDataTrigger.Conditions>                                 <Condition Binding="{Binding IsPressed, RelativeSource={RelativeSource Self}}" Value="true"/>                                 <Condition Binding="{Binding IsEditable, RelativeSource={RelativeSource AncestorType={x:Type ComboBox}}}" Value="false"/>                             </MultiDataTrigger.Conditions>                             <Setter Property="Background" TargetName="templateRoot" Value="{StaticResource ComboBox.Pressed.Background}"/>                             <Setter Property="BorderBrush" TargetName="templateRoot" Value="{StaticResource ComboBox.Pressed.Border}"/>                         </MultiDataTrigger>                         <MultiDataTrigger>                             <MultiDataTrigger.Conditions>                                 <Condition Binding="{Binding IsPressed, RelativeSource={RelativeSource Self}}" Value="true"/>                                 <Condition Binding="{Binding IsEditable, RelativeSource={RelativeSource AncestorType={x:Type ComboBox}}}" Value="true"/>                             </MultiDataTrigger.Conditions>                             <Setter Property="Background" TargetName="templateRoot" Value="{StaticResource ComboBox.Pressed.Editable.Background}"/>                             <Setter Property="BorderBrush" TargetName="templateRoot" Value="{StaticResource ComboBox.Pressed.Editable.Border}"/>                             <Setter Property="Background" TargetName="splitBorder" Value="{StaticResource ComboBox.Pressed.Editable.Button.Background}"/>                             <Setter Property="BorderBrush" TargetName="splitBorder" Value="{StaticResource ComboBox.Pressed.Editable.Button.Border}"/>                         </MultiDataTrigger>                         <Trigger Property="IsEnabled" Value="false">                             <Setter Property="Fill" TargetName="arrow" Value="{StaticResource ComboBox.Disabled.Glyph}"/>                         </Trigger>                         <MultiDataTrigger>                             <MultiDataTrigger.Conditions>                                 <Condition Binding="{Binding IsEnabled, RelativeSource={RelativeSource Self}}" Value="false"/>                                 <Condition Binding="{Binding IsEditable, RelativeSource={RelativeSource AncestorType={x:Type ComboBox}}}" Value="false"/>                             </MultiDataTrigger.Conditions>                             <Setter Property="Background" TargetName="templateRoot" Value="{StaticResource ComboBox.Disabled.Background}"/>                             <Setter Property="BorderBrush" TargetName="templateRoot" Value="{StaticResource ComboBox.Disabled.Border}"/>                         </MultiDataTrigger>                         <MultiDataTrigger>                             <MultiDataTrigger.Conditions>                                 <Condition Binding="{Binding IsEnabled, RelativeSource={RelativeSource Self}}" Value="false"/>                                 <Condition Binding="{Binding IsEditable, RelativeSource={RelativeSource AncestorType={x:Type ComboBox}}}" Value="true"/>                             </MultiDataTrigger.Conditions>                             <Setter Property="Background" TargetName="templateRoot" Value="{StaticResource ComboBox.Disabled.Editable.Background}"/>                             <Setter Property="BorderBrush" TargetName="templateRoot" Value="{StaticResource ComboBox.Disabled.Editable.Border}"/>                             <Setter Property="Background" TargetName="splitBorder" Value="{StaticResource ComboBox.Disabled.Editable.Button.Background}"/>                             <Setter Property="BorderBrush" TargetName="splitBorder" Value="{StaticResource ComboBox.Disabled.Editable.Button.Border}"/>                         </MultiDataTrigger>                     </ControlTemplate.Triggers>                 </ControlTemplate>             </Setter.Value>         </Setter>     </Style>  I expect to be able to change the border radius in this way, but it doesn't work. I get an error that says the property border-radius is not recognized or is not accessible.  <Setter Property="BorderRadius" TargetName="templateRoot" Value="3" />`  How might I be able to change the border radius during runtime? Any help on this would be greatly appreciated.

## Network Policy Server + RADIUS not listening on any ports for Windows Server 2019

Trying to setup Windows Server 2019 as a RADIUS server. However, after configuring everything, “netstat -b” shows that the machine is not listening on any of the expected RADIUS ports (1812, 1645, 1813, 1646).

1. Installed the “Network Policy and Access Services” role
2. Followed the “Configure VPN or Dial-up” wizard

3. Confirmed RADIUS configuration:

4. Confirmed “Network Policy Server” ports:

5. Confirmed the “Network Policy Server” service is running.

Result: “netstat -b” shows that the machine is not listening on any of the expected RADIUS ports (1812, 1645, 1813, 1646).

Have tried with no luck:

1. Rebooting
2. Restarting the “Network Policy Server” service

## Spectral radius of a sequence of operators

In this question $$E$$ stands for a complex Hilbert space and let $$\mathcal{L}(E)$$ the algebra of all bounded linear on $$E$$.

Let $${\bf T}=(T_1,\cdots,T_d) \in \mathcal{L}(E)^d$$ be an operator tuple.

The joint spectral radius of an operator tuple $${\bf T} = (T_1,…, T_d)$$ is defined by $$r({\bf T }):=r(T_1,\cdots,T_d)= \inf_{n\in\mathbb{N}^*}\bigg(\bigg\|\sum_{g\in {\bf \large G}(n,d)} {\bf T}_g^* {\bf T}_{g}\bigg\|^{\frac{1}{2n}} \bigg).$$ Here $${\bf \large G}(n,d)$$ denotes the set of all functions from the set $$\{1,\cdots,n\}$$ to the set $$\{1,\cdots,d\}$$ and for $$g\in {\bf \large G}$$, set $${\bf T}_g:=T_{g(1)}\cdot T_{g(2)}\cdots T_{g(n)}$$.

It is well known that $$r({\bf T })=\lim_{n\longrightarrow+\infty}\bigg(\bigg\|\sum_{g\in {\bf \large G}(n,d)} {\bf T}_g^* {\bf T}_{g}\bigg\|^{\frac{1}{2n}} \bigg).$$

If $${\bf T }=(T_n)_{n\in \mathbb{N}}$$, is there in the literature a definition of $$r({\bf T })$$?

## How can I most clearly write a homebrew item that affects the ground below its radius after the initial explosion it creates?

This item, in essence, releases a burning chemical that deals fire damage to and ignites creatures and objects

1. within its initial sphere radius
2. everything beneath the explosion of the bomb

It then ignites the ground beneath its explosion, in the same area as the explosion. It’s basically an alchemical incendiary.

I just have no clue how to write it properly based on how I want it to work.

For example:

Paladin Pentagon is in the bomb’s blast radius. So is Square Druid in her wild shape. Star aaracokra ranger and Hexgoblin are not in the initial blast radius. Pentagon and Square are set alight because they’re in the initial explosion. Then the stuff falls in the radius of the bomb to create a burning circle on the ground. Square is not in the ground circle, but she’s still on fire because she was caught in the intial blast. Hexgoblin wasn’t in the initial blast, but he’s still set on fire because he was beneath it. Pentagon Pally would be set alight either way. Star isn’t set on fire at all.

I really have no clue how to word this area of effect, or at least something similar. So far I’ve considered:

• a cone that must be oriented downward to create the expansion/explosion effect and the circle on the ground. Problem: can spells and effects written like spells force the orientation of a spell effect? I don’t think they can.
• a sphere and a cylinder straight down, both of which use the origin point of wherever the bomb exploded. A creature can only take the damage once per turn. I think this one works, but my gut says something about rotating columns along axes that change the direction from “straight down” could be an issue.
• a sphere of effect, and then affects a circle of the same radius on the ground beneath it.
• Just be done with it and do a cylinder. Problem is, if the cylinder is thrown high enough, it won’t hit the ground at all, and for something that’s functionally napalm that doesn’t make a whole lot of sense to me.

I hate to ask stack exchange to write my stuff content for me, but I’m at a bit of a loss here and figured folks would appreciate a challenge.

## How can I most clearly write a homebrew item that affects the ground below its radius after the initial explosion it creates?

This item, in essence, releases a burning chemical that deals fire damage to and ignites creatures and objects

1. within its initial sphere radius
2. everything beneath the explosion of the bomb

It then ignites the ground beneath its explosion, in the same area as the explosion. It’s basically an alchemical incendiary.

I just have no clue how to write it properly based on how I want it to work.

For example:

Paladin Pentagon is in the bomb’s blast radius. So is Square Druid in her wild shape. Star aaracokra ranger and Hexgoblin are not in the initial blast radius. Pentagon and Square are set alight because they’re in the initial explosion. Then the stuff falls in the radius of the bomb to create a burning circle on the ground. Square is not in the ground circle, but she’s still on fire because she was caught in the intial blast. Hexgoblin wasn’t in the initial blast, but he’s still set on fire because he was beneath it. Pentagon Pally would be set alight either way. Star isn’t set on fire at all.

I really have no clue how to word this area of effect, or at least something similar. So far I’ve considered:

• a cone that must be oriented downward to create the expansion/explosion effect and the circle on the ground. Problem: can spells and effects written like spells force the orientation of a spell effect? I don’t think they can.
• a sphere and a cylinder straight down, both of which use the origin point of wherever the bomb exploded. A creature can only take the damage once per turn. I think this one works, but my gut says something about rotating columns along axes that change the direction from “straight down” could be an issue.
• a sphere of effect, and then affects a circle of the same radius on the ground beneath it.
• Just be done with it and do a cylinder. Problem is, if the cylinder is thrown high enough, it won’t hit the ground at all, and for something that’s functionally napalm that doesn’t make a whole lot of sense to me.

I hate to ask stack exchange to write my stuff content for me, but I’m at a bit of a loss here and figured folks would appreciate a challenge.