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In Similar Measure, The Invariants Catch The Pith Of Numerical Articles



Hen Lisa Piccirillo settles a decades-old secret about the “Conway tie,” she needs to beat the bunch’s extraordinary capacity to evade probably the most integral assets at any point contrived by mathematicians. Known as invariants, these devices structure the foundation of bunch hypothesis as well as numerous areas of arithmetic, extricating fundamental attributes of numerical items and finding that two articles are in a general sense not quite the same as one another.

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As the name recommends, an unchanging is a trait that doesn’t shift when the obligatory qualities of an article are changed (where “excess” signifies you really want it in a specific setting). An unchanging is the refining of some natural nature of an item, frequently as a number.


To take a model from geography, envision a ball covered with a stretchy cross section that partitions the surface into shapes like triangles and square shapes. The quantity of sizes will, obviously, rely upon the lattice you use, as well as the quantity of edges and corners. Yet, mathematicians found hundreds of years prior that a specific mix of these three numbers is dependably something similar: the quantity of shapes and the quantity of corners short the quantity of edges.

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If, for instance, your lattice isolates the circle into a swelled tetrahedron (with four triangles, four corners, and six edges), that number works out to 4 + 4 – 6 = 2. On the off chance that your net rather shapes the example of a soccer ball (32 hexagons and pentagons altogether, with 60 corners and 90 edges), you get 32 + 60 – 90 = 2 once more. In some sense, the number 2 is a natural quality of adjusted ness. This number (called the Euler normal for the circle) doesn’t change in the event that you stretch or disfigure the circle, so mathematicians call it a topological invariant.


In the event that you rather fold a cross section over a doughnut surface, you generally get an Euler normal for 0. On a two-opening doughnut, you get – 2. The Euler trademark for surfaces has a place with a progression of invariants that permit mathematicians to identify shapes even in higher aspects. This can assist topologists with recognizing two shapes that are difficult to envision, since, supposing that they have different Euler highlights, they may not be a similar topological shape.


A realistic appearance Euler trademark computations for a circle on the left and a torus on the right.

5W infographics for Quanta Magazine

The invariants are likewise used to concentrate on the 15-puzzle, an exemplary toy comprising of square tiles numbered 1 through 15 that you slide in a 4-by-4 framework. The objective is to organize a blended plan of tiles in mathematical request from left to right, beginning with the top line. If you have any desire to know whether a specific game plan is reasonable, there is an invariant that offers you the response. It yields by the same token “even” or “odd” contingent upon the amount of two numbers: the quantity of slides expected to move the unfilled square to the lower right corner and the quantity of tile coordinates that are in inverse mathematical request (the clear square with) tile addressing 16).


These two numbers change the equality (equality or peculiarity) at whatever point you slide a tile across a vacant square. So the equality of their aggregate never shows signs of change, and that implies it is an invariant of the sliding system. This is irreversible in any event, for the settled setup, since the two numbers are zero. So any design with a heterogeneous invariant is totally sad.


Deliberation explores promising thoughts in science and arithmetic. Go with us and join the discussion.

With regards to tie hypothesis, recognizing hitches is a precarious business, since you can’t make a bunch conspicuous by essentially bending it around the circle (mathematicians will generally consider hitches being in shut circles as opposed to open strings). so they can’t fix). Here, invariants are inescapable, and mathematicians have thought of handfuls that offer various qualities of bunches. However, these spineless creatures have vulnerable sides.


Take, for instance, an invariant called tricolor. A bunch outline is tricolor on the off chance that there is a method for shading its wires red, blue, and green so that at each intersection, the three strings that meet are either generally similar variety or various tones. Mathematicians have shown that in any event, when you turn the strings of a bunch, its tricolor (or scarcity in that department) continues as before. As such, tricolor is a natural quality of a bunch.


The three-crossing hitch known as the tricolor is the tricolor. In any case, the “unknot” (a circle that has no genuine bunch, despite the fact that it gives off an impression of being tangled) isn’t tricolor, giving a speedy evidence that the trefoil isn’t just a mask. In any case, while tricolor empowers us to recognize specific bunches from hitches, it’s anything but an optimal device for this reason: ties with tricolor are certainly tied, yet hitches that are not in tricolor are certainly hitched. are not totally obscure. For instance, the figure-eight bunch isn’t tricolor,But it’s really sewn. This bunch falls in the vulnerable side of tricolor – maybe the flippant saying, “The bunch of the figure of eight is, apparently, obscure.”


A realistic appearance, on the left, that a trefoil tie is tricolor, and on the right, that a figure-eight bunch isn’t.

5W infographics for Quanta Magazine

The Conway tie, a 11-crossing hitch that was found by John Horton Conway over quite a while back, is particularly productive at tricking the bunch’s trespassers – extraordinarily intended to distinguish quality picirillos. was called sliciness. Sliciness implies that the bunch is a piece of some smooth however sharp circle in four-layered space.


“Each time there’s another invertebrate, individuals hope to see what occurs at the Conway tie,” said Rice College’s Shelley Harvey. At this point, the Conway tie has fallen in the vulnerable side of each and every contemptuous mathematicians who have come to concentrate on sliceness.


At the point when Piccirillo was at long last ready to show that the Conway tie was not a “cut”, he did so not by making another invariant, however by tracking down a cunning method for exploiting a current one, called Rasmussen’s s-invariant. The Conway hitch tricks this contemptuous with all the others. Yet, in her paper, Piccirillo concocted an alternate bunch in that she could demonstrate the cut condition like the Conway tie. For this new bunch, Rasmussen’s s-invariant demonstrates that it’s anything but a piece. In this way, the Conway tie can’t be pieced also.


Rasmussen’s s-invariant is one of an assortment of not invariants connected with physical science found in late many years. Alisenda Grigsby of Boston School said it required some investment for mathematicians to assimilate the contribution of these innovations.


Grigsby said, Piccirillo is “essential for the new gatekeeper of low-layered topologists who grew up knowing [these invariants] in their bones.” “As far as I might be concerned, that is why is this paper energizing.


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Cannot Add Custom Object to Navigation Menu in Community Salesforce




Navigation menus in Salesforce communities provide an easy way to quickly access the most important information. However, when attempting to add a custom object to the navigation menu of a community. Many users may find that it is not possible.

This article will explain why this cannot be done. And what alternatives exist for adding custom objects to your community’s navigation menu.


Salesforce communities are designed with specific features. That allow administrators to create customized user experiences. Navigation menus are one of these features. Allowing admins to specify which items are visible on the global navigation bar. So users can quickly navigate various pages in the community. While admins can modify existing navigation items such as Home Page and Contact Us.They cannot add new ones without additional development.

Reasons Why Custom Object Cannot Be Added To Navigation Menu:

The primary reason why custom objects cannot be added to the navigation menu of a community is due to security restrictions placed on Salesforce communities by Salesforce. Each object in a community must have explicit access set for any user who will view it. Without this, users could potentially access confidential information that was not intended for them, which could lead to serious security issues. As such, custom objects are not allowed in the navigation menus of communities as they do not have the same level of access control as other components within a community.

Alternatives for Adding Custom Objects to Navigation Menu:

Although it is not possible to add custom objects directly to a community’s navigation menu, there are alternatives that can be used to achieve the same result.

One option is to add custom pages or Visualforce pages that include components such as custom objects, formulas, and other features. These pages can then be added to the navigation menu, allowing users to access them without any additional development.

Another option is to create Lightning Components that contain your custom objects and add them to the navigation menu in a similar manner as with custom pages or Visualforce pages. This can provide an even more customized experience for your users.


Adding custom objects to a navigation menu in a Salesforce Community can be tricky. Custom objects may not appear in the page layout of your Community, so it is important to understand the limitations of Salesforce Communities when it comes to custom objects. This article will explain how to add a custom object to your Community navigation menu and how you can use the Community Builder feature to customize the look and feel of your Community.

How to Add a Custom Object to Navigation Menu in Salesforce Community:

1) In Setup, navigate to App Manager > Your App > Navigation Menus.

2) select the desired menu from the drop-down list (e.g. Global Navigation).

3) Click New Item in the action bar and select Custom Object from the drop-down list that appears.

4) Enter a label for your new item, such as “My Custom Object”.

5) From the object type picklist, select your custom object.

6) Select “Active” to make it available in your Community.

7) Click Save to add the item to the menu.

Customizing Your Navigation Menu in Community Builder:

1) Navigate to Setup > Communities Settings > Community Builder.

2) Select the desired theme from the list and click Edit next to it.

3) Select Navigation Menus from the left-hand pane and choose Global Navigation from the drop-down list at the top of this page.

4) Drag and drop any items you want to add or remove from your navigation menu, including custom objects.

5) Customize the look and feel by setting colors, tokens, fonts and more.

6) Click Save to save your changes.


Adding a custom object to a navigation menu in Salesforce Community is possible with the help of App Manager and Community Builder. With these tools you can easily add or remove items from your Navigation Menu, customize its look and feel, and make sure that all users have access to the objects they need. With just a few clicks, you can ensure that your Community is up-to-date with everything it needs!


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A Step-by-Step Guide on Resolving Issues with Corrupt PST Files



Corrupt PST files

Corrupt PST files can be a major cause of concern when trying to access important data.

In the following guide, we will discuss 10 step-by-step methods to fix corrupt PST files.

1. Run Scanpst –

This is Microsoft’s inbuilt utility for scanning and repairing corrupted Outlook PST files. To run this tool, open your Control Panel and select Mail. Double-click on the Scanpst shortcut or search for it in your C:\Program Files\Microsoft Office folder. Once you have launched the tool, click “Browse” to locate your corrupted PST file, then click “Start” to begin scanning. If any errors are detected, choose “Repair” to repair them automatically.

2. Use a Third-Party Tool –

If the Scanpst utility does not resolve the issue, you can use third-party PST recovery software for repairing corrupt Outlook files. These tools are designed to repair severely corrupted PST files and can also be used to recover deleted emails. Before choosing particular software, make sure it is compatible with your version of Outlook and offers all the features you need.

3. Create a New Data File –

If all else fails, you may have to create an entirely new data file in Outlook and manually import your old messages into it. This process will require downloading Microsoft’s Import/Export Wizard or any other such tool that supports the migration of Outlook data files between different versions.

4. Run an Antivirus Scan –

Sometimes, the corruption of PST files can be caused by a virus infection. To check whether this is the case with your file, run a full system scan using antivirus software and delete all malicious items detected.

5. Use Outlook’s Inbox Repair Tool –

Outlook has its own inbuilt repair tool that helps resolve issues related to corrupt PST files. To use this tool, open the Microsoft Office folder on your computer and select “Scanpst” or “Inbox Repair Tool” (depending on your version of Outlook). Once launched, choose your corrupted PST file and click “Start” or “Repair” to begin the repair process.

6. Try Different Compression Formats –

If you are unable to open your Outlook PST file, try saving it in different compression formats such as ZIP or RAR. This can help reduce the size of the files and make them easier to access.

7. Install Updates –

Outlook periodically releases new updates that may be able to resolve issues with corrupt PST files. Therefore, make sure that you have the latest version of Microsoft Office installed on your computer before attempting any repair work.

8. Repair Windows Registry Entries –

Corrupt registry entries can also cause problems with Outlook data files so it is important to check for any damaged or missing entries in the Windows registry and repair them.

9. Use a Data Recovery Tool –

If all else fails, you may have to use a data recovery tool to recover any lost or deleted emails from your corrupted PST file. There are many such tools available on the internet and some of them can even be used to recover emails from Outlook’s Offline Storage Table (OST) files.

10. Consult a Professional –

If none of the above methods work, it is time to consult a professional for help. A certified technician will be able to analyze your computer and determine the exact cause of the issue with your PST file before recommending an appropriate solution.


The above steps should help you fix any issues with your Outlook PST files. However, if the corruption is severe and none of these methods work, it may be necessary to consult a professional for assistance. In such cases, it is important to ensure that you choose an experienced technician who can offer reliable solutions at reasonable costs.

These 10 steps should offer you assistance in resolving issues with corrupt PST files, but if you are still unable to repair them then it is best to consult a professional. The experts have the skills and knowledge required to fix your Outlook data files without further damaging them.

By following the above-mentioned steps, you should be able to resolve any issues with corrupt PST files. However, if all of these solutions fail, it’s best to consult a professional technician or IT expert for help in recovering your lost data.


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