Solving equations – Finding the double-level curve with eliminating

I want to find the double curve of the projective flat curve
$$ F (x, y, z) = (x ^ 2 + y ^ 2 + z ^ 2) x + t (x ^ 3 + y ^ 3 + z ^ 3) = 0 $$
Where $ (x, y, z) $ is a homogeneous coordinate in the projective 2-room $ mathbb P ^ 2 $. The double curve is the common algebraic equation $ G (u, v, w) = 0 $ under the condition $ F (x, y, z) = 0 $, Where
$$
begin {cases}
u = frac { partial F} { partial x} = 2x ^ 2 + (x ^ 2 + y ^ 2 + z ^ 2) + 3tx ^ 2, \
v = frac { partial F} { partial y} = 2xy ++ 3ty ^ 2, \
w = frac { partial F} { partial z} = 2xz ++ 3tz ^ 2.
end {cases}
$$

So we have to eliminate variables $ x, y, z $ and find algebraic relationship between $ u, v, w $.
I enter Mathematica:

    Eliminate({(x^2 + y^2 + z^2) x + t (x^3 + y^3 + z^3) == 0, 
    u == 2 x^2 + (x^2 + y^2 + z^2) + 3 tx^2, v == 2 xy + 3 ty^2, 
    w == 2 xz + 3 tz^2}, {x, y, z});

But the issue is

    v == 3 ty^2 + 2 xy && w == 3 tz^2 + 2 xz;

Note that they exactly match the last two equations from the input, so Mathematica doesn't solve at all! I don't understand why it doesn't eliminate variables $ x, y, z $ as directed. In this post, OP successfully finds the double curve with Eliminate with exactly the same as mine. What is wrong with my method?

linear programming – standard ILP formulation of the traveling salesman problem: purpose of the restrictions in eliminating subtours?

Consider the problem of the traveling seller:
entrance:: $ n $ Cities, distances $ c_ {ij} $ for each ordered pair $ (i, j) $ from them.

output: Find a shortest round trip that visits each city exactly once.

I came across the following ILP wording in which we introduce a variable $ x_ {ij} in {0,1 } $ for every pair of cities $ i, j $ Where $ x_ {ij} = 1 $ means bow $ i, j $ is part of the tour. Then we have:

Minimize
$$ sum_ {1 leq i, j leq n} c_ {ij} x_ {ij} $$

Subject to
$$ forall k = 1, dots, n: sum_ {1 leq i leq n} x_ {ik} = 1 $$
$$ forall k = 1, dots, n: sum_ {1 leq j leq n} x_ {kj} = 1 $$
$$ forall subsetneq S subsetneq {1, dots, n }: sum_ {i, j in S} x_ {ij} leq | S | -1 $$

Although I understand the purpose of the first two restrictions to ensure that there is only one incoming and one outgoing edge per city, I do not understand the purpose of the third limitation. Based on what I've read, this limitation is intended to ensure that solutions that consist of multiple separate tours are not possible. But how is this enforced by the third limitation and why aren't separate sub-tours already prevented by the first two restrictions? If we have two separate subtours, at least one vertex must have an incoming edge, but no outgoing edge, right?

Entity Framework – Eliminating a discrepancy between controller output and model expected from the page

I have a problem where my controller does not return the type that my page model expects.

My controller sends a kind of:

    IEnumerable <list> grouped

And my site is expecting a model of the type:

IEnumerable

The problem is, no matter what I try, I can not make them match.

Here is the error:

An unhandled exception occurred while processing the request.
InvalidOperationException: The model element passed to
ViewDataDictionary is of the type
System.Linq.Enumerable + SelectEnumerableIterator2[SystemLinqIGrouping[SystemLinqIGrouping[SystemLinqIGrouping[SystemLinqIGrouping2[System.Int32,EFGameManager.Models.GameList.GameSession], System.Collections.Generic.List1[EFGameManager.Models.GameList.GameSession]]& # 39 ;,
However, this ViewDataDictionary instance requires a model element of type
System.Collections.Generic.IEnumerable; & # 39
1[EFGameManager.Models.GameList.GameSession]& # 39 ;.
Microsoft.AspNetCore.Mvc.ViewFeatures.ViewDataDictionary.EnsureCompatible (Object
Value)

Here is the model that expects the page:

Index.cshtml:

@model IEnumerable

Here is the controller:

public async task Index (DateTime Start, DateTime End)
{
var eventGames = waitit _context.GameList
.Where (m => m.GameCatalogId! = Zero) .ToListAsync ();

var eventPlayers = waitit _context.PlayerList
.Where (p => p.PlayerType == 2) .ToListAsync ();

var query = from pl in eventPages
Join ml in eventGames on fl.GameId equals ml.GameId
Choose a new GameSession
{
Id = pl.ListingId,
GameId = ml.GameId,
SessionGameName = ml.GameDisplayName,
SessionEventName = pl.ListingDisplayName,
SessionStartTime = pl.PlayerStartTime,
SessionEndTime = pl.PlayerEndTime
};

var orderedResults = Query
.OrderBy (n => n.SessionGameName)
.DhenBy (d => d.SessionDate)
.ThenBy (t => t.SessionStartTime) .ToList ();

var ts = TimeSpan.FromMinutes (30);

var grouped = Assorted results
.OrderBy (r => r.GameId)
.ThenBy (r => r.SessionDate)
.ThenBy (r => r.SessionStartTime)
.GroupByWhile ((p, n) => p.SessionDate.Value.Date == n.SessionDate.Value.Date
&& n.SessionStartTime - p.SessionEndTime < ts)
                            .Select(rg => rg.ToList ());

return view (grouped);
}

I've tried a few things and either get a new bug or a variation of that bug.

Any insight would be very grateful!