reference request – Determination of the nature of stationary values in variational calculus

In variational calculus, when we solve the Euler-Lagrange equation $frac{d}{dx}L_p(u’,u,x)-L_z(u’,u,x)$, where $L=L(p,z,x)$, to find stationary inputs of the functional
I(u)=int_0^1 L(u’,u,x)dx,

we also need to determine whether this solution $u$ is a maximum, a minimum, or a saddle point. However, the second order variation
left.frac{d^2}{dt^2}right|_{t=0} I(u+tv)= int_0^1 v’^2L_{pp}+2vv’L_{pz}+L_{zz}v^2 dx

is very hard to work with since we need to prove, for instance, this is positive for all functions $v$ to show that $u$ is a minimum. I have kept everything in one dimension for simplicity, but of course the above applies to higher dimensions as well.

I can think of two ways to confirm that it is a minimum:

  1. If the function $L$ satisfy certain convexity assumption, then a minimizer exists, and the minimizer is a solution to the differential equation. Therefore, if the differential equation has a unique solution, it is the minimizer.
  2. In some situations, we can write an inequality satisfied by $I$, and then consider the condition for the equality to be achieved in the inequality. An example is the equation for geodesic on the sphere $S^2$.

However, both methods applies only to a limited range of functions.

What are some other methods to establish that the stationary solution is a minimum/maximum? I am looking for some other ideas.

Perhaps I need some reference on this, since parts of my variational calculus and PDE books that I have read does not focus on this question.

reference request – Proving that $C_S^{infty}(M,N)$ is a Baire space

I have been reading Hirsch’s Differential topology and I am sure a lot of you know this book as a lot of typos. I believe one of them is the proof that $C_S^{infty}(M,N)$ is a Baire space. I don’t think his proof works, but I hope this is a true fact since it is used throughout the text. Now when he proves that $C_S^r(M,N)$ is a Baire space , for $0 leq r<infty$, we just use a continuous function $J^r:C^r_S(M,N)rightarrow C^0(M,J^r(M,N))$ such that the image is a weakly closed subset of $C^0(M,J^r(M,N))$ and now since $J^r(M,N)$ is complete $M$ is a manifold and $J^r$ is continuous we get that $C_S^r(M,N)$ is a Baire space. Now I think he wants to do the same type of argument for $J^{infty}:C^{infty}(M,N)rightarrow C^0(M,J^{infty}(M,N))$ but here we don’t have the fact that the function is continuous. So my question is if anyone knows a proof or a reference for seeing that $C_S^{infty}(M,N)$ will in fact be a Baire space? Thanks in advance.

unity – NullReferenceException in OnAwake but object has a reference in the inspector

The full line error is:

CinemachinePathEdit.AddLineWayPoints (LevelCurve curve, System.Collections.Generic.List`1(T) wayPoints) (at Assets/Scripts/CinemachinePathEdit.cs:39)
CinemachinePathEdit.Awake () (at Assets/Scripts/CinemachinePathEdit.cs:29)

And the context behind it is that I’m trying to get every objects “levelcurve”-script so I cant get the waypoints out of them and put them in a general list. But I just get a nullreferenceexception even though I did in the inspector.

    public List<LevelCurve> allLevelCurves;

    public List<Vector3> wayPoints;
    public List<Vector3> visibleLineWayPoints;

    private CinemachineSmoothPath.Waypoint() c_wayPoints;

    public void Awake()
        wayPoints = new List<Vector3>();

        foreach (LevelCurve curve in allLevelCurves)
            AddLineWayPoints(curve, wayPoints);

        c_wayPoints = new CinemachineSmoothPath.Waypoint(wayPoints.Count);

    private void AddLineWayPoints(LevelCurve curve, List<Vector3> wayPoints)
        foreach (Vector3 wayPoint in curve.lineWayPoints)
            wayPoints.Add(wayPoint); //error takes me to this line

            visibleLineWayPoints.Add(wayPoint); //for visibility in unity

In the inspector it looks like this:
Anybody see the problem?

Keep Cell reference in a formula fixed while adding columns to a sheet of data

I have a spreadsheet with multiple sheets, and many rows and columns of data. I like to count the number of each occurance of every value for the last “x” values entered. The problem is that when I add a new column, it changes the formula and moves the cells that it references.

This is the formula that is in Cell C10 of Sheet 6, which references the data in Sheet 3.


Every day, I add another column to Sheet 3 all the way on the left (left of Column A). I still want to perform the same formula in cell C10 of Sheet 6, but when I add the column to sheet 3, the new formula is


The new formula is fine for counting the data in the new column that I added “A”, but now it has added column “AA” to the formula. I still only want to count the data from column A to Column Z.

Please help.

foreach – Como resolver erro C# “NullReferenceException: Object reference not set to an instance of an object.”?

Estou tentando finalizar um projeto só que está apresentando erro da parte do foreach, sendo que no VSCODE não informa nenhum problema. Estou aprendendo C# agora e não estou conseguindo resolver sozinha.

esse é homecontroller:

using System;
using System.Collections.Generic;
using System.Diagnostics;
using System.Linq;
using System.Threading.Tasks;
using Microsoft.AspNetCore.Mvc;
using Microsoft.Extensions.Logging;
using Projeto_PetVet.Models;

namespace Projeto_PetVet.Controllers
public class HomeController : Controller
    private readonly ILogger<HomeController> _logger;

    public HomeController(ILogger<HomeController> logger)
        _logger = logger;

    public IActionResult Index()
        return View();

    public IActionResult Servicos()
        return View();

    public IActionResult PreAgendamento()
        return View();
    public IActionResult PreAgendamento(DadosConsulta consulta)
        return View("olhaDados");

    public IActionResult Sucesso()
        return View();

    public IActionResult olhaDados() {
    List<DadosConsulta> consulta = Consulta.olhaDados();
    return View(consulta);

    (ResponseCache(Duration = 0, Location = ResponseCacheLocation.None, NoStore = true))
    public IActionResult Error()
        return View(new ErrorViewModel { RequestId = Activity.Current?.Id ?? HttpContext.TraceIdentifier });

Esses são meus Models: Consulta.cs:

using System.Collections.Generic;

namespace Projeto_PetVet.Models
public class Consulta
    static List<DadosConsulta> infos = new List<DadosConsulta>();

    public static void addDados(DadosConsulta consulta) {

    public static List<DadosConsulta> olhaDados() {
        return infos;


namespace Projeto_PetVet.Models
public class DadosConsulta
    public string Nome {get; set;}
    public string Animal {get; set;}
    public string Data {get; set;}
    public string Telefone {get; set;}
    public string Horario {get; set;}
    public string Necessidade {get; set;}

E essa é a minha View, onde o erro está aparecendo no foreach:

@model List<DadosConsulta>
ViewData("Title") = "Pré agendamento";
<h1>Confirme suas informações</h1>

<p>Confirme seu pré agendamento.</p>
<p>Todos os dados estão corretos?</p>

<table class="table table-bordered">

    Consulta infos = new Consulta();

    foreach (DadosConsulta consulta in Model)

<a asp-action="Sucesso">Tudo certo!</a>
<a asp-action="PreAgendamento">Corrigir dados</a>

reference request – Book recommendation: The Gamma function

I don’t know if Math Stackexchange is for such questions (probably, I can ask such questions since there is a tag on it) but I want a book on the Gamma function, that is similar to, say, Titchmarsh’s Theory of the Riemann zeta function. I want a book that studies the gamma function in depth. So please recommend a book on the Gamma function that not just lists some properties of the gamma function. I can search Google for it, but I am not sure.

ac.commutative algebra – Reference request: functions on Stone spaces

I’m looking for references for the following closely related facts:

Given a Boolean algbera B, I denote by $mathbb{Z}(B)$ the free ring generated by symbols $e_b$ such that $e_b e_{b’} = e_{b cap b’}$ and $e_b + e_{b’} = e_{b cup b’}+ e_{e cap b’}$.


  1. The $e_b$ are the only idempotent of $mathbb{Z}(B)$.

  2. $mathbb{Z}(B)$ identifies with the algebras of continuous function from the Stone spectrum of $B$ to $mathbb{Z}$.

Ideally, I would like a proof that is constructive (using the localic Stone spectrum) and applies to non-unital” boolean algebras, but the closest approximation would already be good.

reference request – Definition of an n-category

What’s the standard definition, if any, of an $n$-category as of 2020? The literature I can tap into is quite limited, but I’ll try my best to list what I had so far.

In (Lei2001), Leinster demonstrated 10 different definitions for an $n$-cateogory, and made no comment on whether they are equivalent or not. In (BSP2011), the authors set up axioms and claimed that all (many?) definitions of an $(infty,n)$-category so far satisfy their axioms, and therefore are equivalent (up to some action). I include those definitions here for completeness:

  • (a) Charles Rezk’s complete Segal Θn-spaces,
  • (b) the n-fold complete Segal spaces,
  • (c) André Hirschowitz and Simpson’s Segal n-categories,
  • (d) the n-relative categories of Clark Barwik and Dan Kan,
  • (e) categories enriched in any internal model category whose
    underlying homotopy theory is a homotopy theory of (∞,
  • (f) when n = 1, Boardman and Vogt’s quasicategories,
  • (g) when n = 1, Lurie’s marked simplicial sets, and
  • (h) when n = 2, Lurie’s scaled simplicial sets,

However, all cases in (Lei2001) do not seem to be covered, and there are even more here. What’s the crucial difference between defining an $n$-category and an $(infty,n)$-category?


In short, there are many definitions for higher categories.. so which one should we use? Is there a list of all definitions made, and a discussion on which is equivalent to which under which sense? Are there also discussions on which definition satisfies the three hypotheses

  1. stabilization hypothesis
  2. tangle hypothesis
  3. cobordism hypothesis

postulated in (BD1995)?


  • (Lei2001): A Survey of Definitions of n-Category-(Tom Leinster)-(arXiv:math–0107188)
  • (BSP2011): On the Unicity of the Homotopy Theory of Higher Categories-(Clark Barwick and Christopher Schommer-Pries)-(arXiv:1112.0040)
  • (BD1995): Higher-dimensional Algebra and Topological Quantum Field Theory-(John C. Baez and James Dolan)-(arXiv:q-alg–9503002)

google sheets – Filter causing Circular Reference Error – Not obvious why?

I have 3 sheets. SheetA and SheetB contain data and a third sheet where I want to merge the contents of the SheetA and SheetB in it.

I type this formula in the result sheet (the 3rd sheet):

={filter(SheetA!A1:A, arrayformula(ISBLANK(A:A)=false)),
filter(SheetB!A1:A, arrayformula(ISBLANK(A:A)=false))}

However, I get circular reference error! Not sure what caused it? Each filter in the above expression works correctly.

Link to sheet

8 – Filtering by User Reference with JSON API

I try to filter some data by User Reference. My problem is, it return me an empty list

"data": []

It only works if the authenticate user own’s the data…

My request :


I replace “user_id” with the real data.
I tried a lot of synthaxes :


None of them work.

So my question is : Is it possible to filter by user reference without the current user owning the data ?