Failure of entropy condition for a singular limit

Consider a regularization of the conservation law $ $ \partial_t u + \partial_x f(u) + \epsilon \partial_{x}^3 u = 0$ $

How does one prove that the limit function $ u$ of $ \{u_\epsilon\}_\epsilon$ as $ \epsilon \to 0$ is not an entropy solution of $ $ \partial_t u + \partial_x f(u)=0 \ ?$ $

This problem was studied in a series of papers by Lax and Levermore (1983). Here I’m looking for a direct proof that $ u$ violates the entropy condition (or Olienik inequality).

Do we know the consistency strength of the Singular Cardinal Hypothesis failing on an uncountable cofinality?

Suppose that $ \kappa$ is a strong limit cardinal. The singular cardinal hypothesis states $ 2^\kappa=\kappa^+$ . We know that the failure of SCH requires large cardinals, and in fact is equiconsistent with a measurable cardinal $ \kappa$ satisfying $ o(\kappa)=\kappa^{++}$ .

But this failure is at $ \aleph_\omega$ . Suppose we wanted more.

Suppose that we wanted the failure to happen on a couple isolated points. Well, it’s not hard to redo the standard constructions and get just that. But what happens when we have limit points?

Even more, by Silver’s theorem if SCH fails at $ \kappa>\operatorname{cf}(\kappa)>\omega$ , then there is a stationary subset of $ \kappa$ where SCH failed.

What would be the consistency strength when $ \kappa$ is a singular limit of singular cardinals, and SCH fails cofinally below $ \kappa$ ? What if we require $ \kappa$ to be of uncountable cofinality?

As a side question, what if $ \kappa$ , with uncountable cofinality, does satisfy SCH, but an unbounded subset (which has to be non-stationary, of course) of it does not?

Smallest singular value distribution

Let $ G_\mathbb{R}\in\mathbb{R}^{n\times n}$ and $ G_\mathbb{C}\in\mathbb{C}^{n\times n}$ denote the real and complex Ginibre random matrices, i.e. random matrices with independent real/complex Gaussian entries of zero mean and variance $ \mathbb{E}\lvert(G_K)_{ij}\rvert^2=n^{-1}$ for $ K=\mathbb{R},\mathbb{C}$ . Denote the singular values of the matrix $ X$ by $ \sigma_1(X)\ge\dots\ge \sigma_n(X)$ . It follows from the work of Edelman that \begin{equation}\mathbb{P}\Big( \sigma_n(G_K)\le\frac{x}{n}\Big) = \begin{cases} 1-e^{-x^2/2-x}+o(1),&K=\mathbb{R}\ 1-e^{-x^2},&K=\mathbb{C}. \end{cases}\tag{*}\label{sing val}\end{equation} In particular, the smallest singular value is of order $ n^{-1}$ .

Now I am wondering about additive perturbations to $ G_K$ , say $ G_K+\lambda I$ , where $ \lambda$ is some real (or even complex) parameter and $ I$ is the identity matrix. In general the singular values of $ G_K$ give little information about the singular values of $ G_K+\lambda I$ . The most one can hope for is bounds like $ $ \sigma_n(G_K+\lambda I)\ge \lvert\lambda\rvert-\sigma_1(G_K).$ $ It is known that $ \sigma_1(G_K)$ is Tracy-Widom distributed around $ 2$ , so in particular, $ \sigma_n(G_k+\lambda I)$ is bounded away from $ 0$ as long as $ \lvert \lambda\rvert>2$ .

Question: Is the analogue of $ \eqref{sing val}$ , i.e. the distribution of $ \sigma_n(G_K+\lambda I)$ known for $ G_K+\lambda I$ ? If exact formulae are not available, I would be interested in the average scaling of $ \sigma_n(G_K+\lambda I)$ . I guess there should be some phase transition of the type $ $ \mathbb E \sigma_n(G_K+\lambda I)\sim\begin{cases}n^{-1},&\lvert\lambda\rvert<c\ n^{-?}, &\lvert\lambda\rvert=c,\ 1, &\lvert\lambda\rvert>c.\end{cases}$ $ I think $ c=1$ , but am unsure about the critical exponent.

Are pullbacks on singular cohomology unique on the nose?

Let $ C$ be the category of even-dimensional connected closed oriented topological manifolds/orientation-preserving continuous maps and $ D$ be the category of finite-dimensional graded $ \mathbb{Q}$ -algebras.

We have a functor $ H:C\rightarrow D$ given by singular cohomology.

Let us call a choice of an isomorphism of graded $ \mathbb{Q}$ -algebras $ H(M)\otimes H(M’)\rightarrow H(M\times M’)$ for every $ M$ , $ M’\in Obj(C)$ a Kunneth system. Projections $ M\times M’\rightarrow M$ , $ M’$ define a Kunneth system which is functorial with respect to $ H$ (Kunneth theorem).

For a contravariant functor $ G:C\rightarrow D$ such that the induced map on objects is equal to one induced by $ H$ , it is meaningful to ask if the Kunneth system above is functorial with respect to $ G$ . The question: does there exist a contravariant functor $ G:C\rightarrow D$ such that

  • $ G$ coincides with $ H$ on the level of objects but not on the level of morphisms;
  • for any non-negative even integer $ i$ , the contravariant functors from $ C$ to $ \mathbb{Q}$ -vector spaces obtained by composing either $ G$ or $ H$ with the “$ i$ -th graded piece” functor are equal;
  • the Kunneth system above is functorial with respect to $ G$ ?

P.S.: this question is inspired by this question.

Solve(covMat) retorna que system is computationally singular: reciprocal condition number

Quiero obtener la cartera de variación mínima. Los rendimientos esperados mu son:

> mu        .SXQR        .SXTR        .SXNR        .SXMR        .SXAR        .SX3R   0.100496686  0.068652744  0.065081570  0.013155820  0.086947540  0.103143934         .SX6R        .SXFR        .SXOR        .SXDR        .SX4R        .SXRR   0.054990629  0.088484620  0.085435533  0.068080455  0.098365460  0.023932074         .SXER        .SXKR        .SX7R        .SX8R        .SXIR        .SXPR   0.037525561 -0.000400454  0.024776148  0.007051037  0.042215791  0.116013074 

y la matriz de covarianza covMat esta :

> covMat            .SXQR      .SXTR      .SXNR      .SXMR      .SXAR      .SX3R      .SX6R .SXQR 0.03345763 0.03498086 0.04185753 0.03245136 0.03497776 0.02324828 0.02081399 .SXTR 0.03498086 0.04974472 0.04619830 0.04159817 0.04420657 0.02401824 0.02689768 .SXNR 0.04185753 0.04619830 0.06097311 0.04537355 0.05011720 0.02686316 0.03057380 .SXMR 0.03245136 0.04159817 0.04537355 0.04287405 0.04017447 0.02191462 0.02405812 .SXAR 0.03497776 0.04420657 0.05011720 0.04017447 0.05465295 0.02184147 0.02625680 .SX3R 0.02324828 0.02401824 0.02686316 0.02191462 0.02184147 0.02318865 0.01733667 .SX6R 0.02081399 0.02689768 0.03057380 0.02405812 0.02625680 0.01733667 0.03490965 .SXFR 0.04086303 0.05423479 0.05599539 0.04781758 0.04785298 0.03177379 0.03971822 .SXOR 0.03468033 0.04174388 0.04790487 0.03778244 0.03958789 0.02265676 0.03305845 .SXDR 0.01823918 0.02550628 0.02203189 0.02453928 0.01603731 0.01815483 0.01477641 .SX4R 0.03342966 0.03293408 0.04631396 0.03373207 0.03772380 0.02582638 0.02640921 .SXRR 0.03033003 0.03421856 0.04020604 0.03334523 0.03561357 0.01863272 0.02276777 .SXER 0.02051229 0.01525291 0.02833009 0.01743750 0.01769105 0.01464734 0.01775119 .SXKR 0.02694061 0.03196615 0.03950701 0.03682066 0.03574271 0.01394396 0.02498871 .SX7R 0.03635913 0.04453817 0.04871591 0.03789661 0.03882093 0.02804067 0.03282608 .SX8R 0.03991513 0.04959444 0.05847244 0.04934120 0.04838590 0.02522826 0.02852557 .SXIR 0.03508348 0.04831080 0.04905516 0.04555107 0.04046836 0.02949463 0.03263901 .SXPR 0.04456718 0.03223030 0.06505796 0.03031465 0.04557004 0.02648925 0.03399364            .SXFR      .SXOR       .SXDR      .SX4R      .SXRR      .SXER      .SXKR .SXQR 0.04086303 0.03468033 0.018239181 0.03342966 0.03033003 0.02051229 0.02694061 .SXTR 0.05423479 0.04174388 0.025506282 0.03293408 0.03421856 0.01525291 0.03196615 .SXNR 0.05599539 0.04790487 0.022031890 0.04631396 0.04020604 0.02833009 0.03950701 .SXMR 0.04781758 0.03778244 0.024539282 0.03373207 0.03334523 0.01743750 0.03682066 .SXAR 0.04785298 0.03958789 0.016037311 0.03772380 0.03561357 0.01769105 0.03574271 .SX3R 0.03177379 0.02265676 0.018154833 0.02582638 0.01863272 0.01464734 0.01394396 .SX6R 0.03971822 0.03305845 0.014776414 0.02640921 0.02276777 0.01775119 0.02498871 .SXFR 0.07060256 0.05433736 0.029870028 0.04370332 0.04173824 0.02543286 0.03772980 .SXOR 0.05433736 0.05018950 0.019754192 0.03724791 0.03814745 0.02642396 0.03293653 .SXDR 0.02987003 0.01975419 0.024627383 0.01758262 0.01499347 0.01045468 0.01481692 .SX4R 0.04370332 0.03724791 0.017582623 0.04093423 0.03116615 0.02429916 0.02793648 .SXRR 0.04173824 0.03814745 0.014993470 0.03116615 0.03504357 0.01910385 0.03251937 .SXER 0.02543286 0.02642396 0.010454676 0.02429916 0.01910385 0.02722964 0.01322536 .SXKR 0.03772980 0.03293653 0.014816920 0.02793648 0.03251937 0.01322536 0.04505911 .SX7R 0.05938230 0.05169770 0.023760635 0.04122183 0.03709132 0.02779035 0.02632013 .SX8R 0.05506206 0.04556889 0.027748962 0.04155255 0.03723654 0.02359730 0.04231239 .SXIR 0.05991709 0.04424346 0.033142798 0.03888777 0.03233359 0.02232765 0.03104647 .SXPR 0.04889303 0.05858437 0.006954027 0.05553596 0.04425855 0.05085459 0.02653357            .SX7R      .SX8R      .SXIR       .SXPR .SXQR 0.03635913 0.03991513 0.03508348 0.044567182 .SXTR 0.04453817 0.04959444 0.04831080 0.032230303 .SXNR 0.04871591 0.05847244 0.04905516 0.065057961 .SXMR 0.03789661 0.04934120 0.04555107 0.030314650 .SXAR 0.03882093 0.04838590 0.04046836 0.045570043 .SX3R 0.02804067 0.02522826 0.02949463 0.026489254 .SX6R 0.03282608 0.02852557 0.03263901 0.033993645 .SXFR 0.05938230 0.05506206 0.05991709 0.048893031 .SXOR 0.05169770 0.04556889 0.04424346 0.058584372 .SXDR 0.02376063 0.02774896 0.03314280 0.006954027 .SX4R 0.04122183 0.04155255 0.03888777 0.055535956 .SXRR 0.03709132 0.03723654 0.03233359 0.044258552 .SXER 0.02779035 0.02359730 0.02232765 0.050854589 .SXKR 0.02632013 0.04231239 0.03104647 0.026533566 .SX7R 0.06443470 0.04622553 0.04997693 0.062958984 .SX8R 0.04622553 0.06557348 0.05449832 0.047301440 .SXIR 0.04997693 0.05449832 0.06063113 0.032824422 .SXPR 0.06295898 0.04730144 0.03282442 0.143337184 

He visto que hay este articulo que da la ecuación y el código:

introducir la descripción de la imagen aquí

Entonces utilizo el código siguiente.

assetSymbols <- colnames(yearly_return)  mu <- colMeans(yearly_return,na.rm = TRUE) # expected returns covMat <- cov(yearly_return) # covariance matrix corMat <- cor(yearly_return) # correlation matrix   ## Minimum Variance Portfolio function #### getMinVariancePortfolio <- function(mu,covMat,assetSymbols) {   U <- rep(1, length(mu)) # vector of 1   O <- solve(covMat)     # inverse of covariance matrix   w <- O%*%U /as.numeric(t(U)%*%O%*% U)   Risk <- sqrt(t(w) %*% covMat %*% w)   ExpReturn <- t(w) %*% mu   Weights <- `names<-`(round(w, 5), assetSymbols)   list(Weights = t(Weights),        ExpReturn = round(as.numeric(ExpReturn), 5),        Risk = round(as.numeric(Risk), 5)) } 

Pero me da

 Error in solve.default(covMat) :    system is computationally singular: reciprocal condition number = 1.06734e-19  

Singular value of a partitioned matrix

all.

Let $ \mathbf{X}$ be a matrix partitioned as $ \mathbf{X} = \begin{bmatrix} \mathbf{X}_{1} \ \mathbf{X}_{2} \end{bmatrix}$ .

I want to lower bound the smallest singular value $ \sigma_{\min}(\mathbf{X}_{1})$ using $ \mathbf{X}$ . Is there any relationship between $ \sigma_{\min}(\mathbf{X}_{1})$ and $ \mathbf{X}$ ?

Thanks!

Calculate Kodaira dimension of a singular hypersurface

For a smooth projective hypersurface $ H \subseteq \mathbb{P}^n$ of degree $ d$ one can calculate its Kodaira dimension $ \kappa(H)$ , and find $ $ \kappa(H) = \begin{cases} -\infty \qquad &\mbox{if } d < n +1,\ 0 &\mbox{if } d = n+1,\ \dim H &\mbox{if } d > n+1. \end{cases}$ $

What about if $ H$ is not smooth? Can we say anything about its Kodaira dimension, or even sensibly calculate something we can call the ”Kodaira dimension”?

I’ve only ever seen canonical divisors defined over smooth varieties, so am unsure how to proceed in the singular case.

Does Singular complex determine a topological space

This question comes from thinking about the singular geometric realization adjunction $ Sing \dashv |\cdot|$ . I suspect that this adjunction is not monadic, so using the Monadicity theorem I tried to cook up two topological spaces with isomorphic singular complexes, but are not homeomorphic. However, I could not think of such a space! I feel like I am missing something. The closest I got was a torus and a Klein bottle have the same chain complex groups but different maps. So my question is, is it true that two spaces have isomorphic chain complexes iff they are homeomorphic?