What’s the difference between methods for defining a matrix function ( Jordan canonical form, Hermite interpolation and Cauchy integral)?

There’s many equivalent way of defining $ f(A)$ . We focus on Jordan canonical form, Hermite interpolation and Cauchy integral.

What’s the difference between methods for defining a matrix function ( in application )? What is superiority to each other? Can you introduce a source?Thanks

Is there a canonical way to handle JSON data format changes?


Say we have a C# class with is serialized to JSON (currently serialized with Newtonsoft’s JSON.Net) and stored in a database:

public class User {     public string authInfo; } 

If the class definition changes, the old data will fail to load. Even if we try to update the database by hand, we risk data being loaded incorrectly during the conversion.

public class User {     public string username;     public string token; } 

Solution (my attempt)

We may use a callback which is run after deserialization that converts the old data to the new data format. The attribute and parameters need to be adapted based on which serialization framework is being used:

public class User {     public string username;     public string token;     [Obsolete] public string authInfo;      [OnDeserialized]     public void FixData()     {         if (username == null)         {             var parts = authInfo.Split("/");             username = parts[0];             token = parts[1];             authInfo = null;         }     } } 

If a field’s format needs to change from a list to an object (or number) or vice versa, the newer field should be called authInfo_2, and incremented when the type changes again. If a field’s format needs to change from a list of one type to a list of another type, a new field must also be created.

public class User {     [Obsolete] public List<string> address;     public List<AddressLine> address_2;     // FixData() will convert from address to address_2 } 

Problem: If null is a valid value for the old or new data, we can’t determine whether the data has been migrated to the newer format. The following is a workaround that will track whether new data has been added:

public class User {     [Obsolete] public List<string> name; // serialized old data     private string _familyName; // serialized     private bool _isFamilyNameSet; // serialized     public string familyName { get { return _familyName; } set { _familyName = value; _isFamilyNameSet = true; } } // not serialized     // FixData() will convert from name to familyName } 


This procedure is a bunch of rules I made up, and I’ve probably missed something important. Is there an accepted best practice that deals with versioning in serialized data? (Including a version number seems like it would lead to a lot of problems.)

Traffic dropped to nothing after Google’s change to canonical URL… what can I do?

Hello SEO experts,

Please, forgive me for how naive my question that will sound to professionals :)

I've been blogging for 15 years. Traffic was always steady or increasing, but it dropped to almost nothing in August 2018. This aligns with the following:

Google Search Update
An event has occurred in Google Search that might affect your site's data."

And when I look up the event, I get to:

All metrics are now assigned to the Google-selected canonical URL of the page linked to in…

Traffic dropped to nothing after Google's change to canonical URL… what can I do?

Are there any canonical correlations or known overlaps between D&D settings’ various timelines that could be used to determine when they correspond?

D&D canon has demonstrated numerous times that the various campaign settings are in fact, connected to each other. Powerful beings from Greyhawk, Forgotten Realms, and even Athas have found ways into other campaign settings in the D&D stable of worlds. (Even Earth is included in the list of alternate Primes visited, interestingly enough.)

Irrespective of the means used to cross from one campaign setting world to the other, are there any solid timeline references, correlations, or overlaps (specific dates or even a range of dates) where it can be demonstrated what year (or range of years) a cross-setting traveler would end up in when moving from any given setting to any other given setting?

For example, it has been demonstrated that both Elminster and Mordenkainen have traveled to each other’s home Prime, as well as to Earth (apparently during the 70’s – 80’s). This provides a basic frame of reference, however they both lived rather long lives, and thus only provides a common range of dates that overlap in Oerth and Abeir-Toril; namely the date range of the span of their lives, or at least after they both reached a certain level of power.

I am open to timelines from any edition of Dungeons and Dragons, and all official campaign settings in the D&D stable (Spelljammer, Maztica, Mystara, Al-Qadim, Ravenloft, Planescape, Greyhawk, Forgotten Realms, Birthright, Athas, Dragonlance, Red Steel, Odyssey, Lankhmar, Conan, and so forth). Also, as the presence of our modern Earth is explicitly included in D&D canon, using Earth-based cross-references to multiple campaign settings in order to establish a linkage is also acceptable. I would prefer to avoid time travel, unless said reference manages to establish a firm date link.

Thank you. I’m sure this will not necessarily be easy to research.

Is XSS in canonical link possible?

During regular pentesting of my site I discovered that I can close double quotes in a canonical link tag and enter an onerror attribute with a simple javascript alert(1).

enter image description here

It is visible in source code but javascript did not execute.

enter image description here

I also tried with onload event but same result.

Is there a way an attacker can use different payload to execute javascript ?

Splitting the canonical projection to the free pro-p group

Let $ \widehat F(k)$ be the free profinite group on $ k$ generators and let $ p$ be a prime. Then there is a canonical projection $ \pi\colon \widehat F(k)\to \widehat F_p(k)$ where $ \widehat F_p(k)$ is the free pro-$ p$ group on $ k$ -generators. It is well known that a free pro-$ p$ group is projective in the category of profinite groups and so $ \pi$ splits via a continuous homomorphism $ \psi\colon \widehat F_p(k)\to \widehat F(k)$ .

I would like to know if anybody knows an explicit splitting. To be more precise, let $ X_k =\{x_1,\ldots, x_k\}$ be a free generating set for $ \widehat F(k)$ . I would like explicit sequences $ \{w_{i,n}\}$ , for $ i=1,\ldots, k$ , of words over $ X_k$ such that $ w_{i,n}\to w_i$ with $ \pi(w_i) = \pi(x_i)$ and $ \overline {\langle w_1,\ldots, w_k\rangle}$ a free pro-$ p$ group.

For example, if $ k=1$ , then I know how to write down a sequence converging to an element giving a splitting of $ \pi\colon \widehat {\mathbb Z}\to \mathbb Z_p$ . Namely, if $ x_1$ is the free generator of $ \widehat{\mathbb Z}$ and if $ p_1,p_2,\ldots$ is a list of the primes other than $ p$ . Then $ x_1^{(p_1p_2\cdots p_n)^{n!}}$ converges to an element generating a copy of $ \mathbb Z_p$ and mapping to $ \pi(x_1)$ .

A natural transformation between two canonical coends

I have troubles proving the following statement although I verified it for a large class of finite tensor categories.

Let $ \mathcal C$ be a finite tensor category and $ \mathcal D$ be a tensor subcategory of $ \mathcal C$ . One can consider the central Hopf monad

$ $ Z(V)=\int_{X\in \mathcal C} X\otimes V \otimes X^*.$ $

Let $ \pi_{V, X}: Z(V)\rightarrow X\otimes V \otimes X^*$ be the canonical maps associated to the coend $ Z(V)$ for all $ X\in \mathcal C$ .

It is well known that $ Z$ is a Hopf comonad and it has values in the Drinfeld center $ \mathcal Z(\mathcal C)$ . Thus each object $ Z(V)$ can be enhanced with a half-braiding $ (Z(V), \sigma_{V, -})\in \mathcal Z(\mathcal C)$ .

Moreover $ $ R:\mathcal C \rightarrow \mathcal Z(\mathcal C),\;\; V \mapsto (Z(V), \sigma_{V,-})$ $ is a right adjoint of the forgetful functor $ F: \mathcal Z(\mathcal C)\rightarrow \mathcal C$ , see the paper of Day and Street, Centres of monoidal categories of functors, in: Categories in Algebra, Geometry and Mathematical Physics, in: Contemp. Math., vol.431, Amer. Math. Soc., Providence, RI, 2007, pp.187–202. We also have $ Z=FR$ .

One can also consider the relative Hopf comonad

$ $ U(V)=\int_{X\in \mathcal D} X\otimes V \otimes X^*.$ $

Let also $ \bar{\pi}_{V, X}: U(V)\rightarrow X\otimes V \otimes X^*$ be the canonical maps associated to the coend $ U(V)$ for all $ X\in \mathcal D$ .

Similarly, one can see that $ U(V)$ can be enhanced with a relative braiding $ \bar{\sigma}_{U(V), -}$ . Thus $ (U(V), \bar{\sigma}_{U(V), -})\in \mathcal Z_{\mathcal C}(\mathcal D)$ , the relative center with respect to $ \mathcal D$ . Thus $ $ R_1: \mathcal C \rightarrow Z_{\mathcal C}(\mathcal D), \;\; V\mapsto (U(V), \bar{\sigma}_{U(V), -}) )$ $ is a right adjoint of the forgetful functor $ F_1:Z_{\mathcal C}(\mathcal D) \rightarrow \mathcal C$ and $ U=R_1F_1$ .

By the universal property of the coend $ U$ , for any $ V\in \mathcal C$ , there is a natural transformation $ $ q_V:Z(V)\rightarrow U(V)$ $ such that $ \bar{\pi}_{V, X}\circ q_V=\pi_{V, X}$ for all $ X\in \mathcal D$ . This natural transformation also appear in the paper of Shimizu, The monoidal center and the character algebra, JPAA 221 (2017) 2338–2371, see https://arxiv.org/abs/1504.01178.

One can also consider the following sequence of functors:

$ $ \mathcal Z(\mathcal C)\xrightarrow{F_2} {\mathcal Z}_{\mathcal C} (\mathcal D) \xrightarrow{F_1} \mathcal C$ $

where the first category is as above the Drinfeld center of $ \mathcal C$ and the second category is the relative center with respect to $ \mathcal D$ .

The two functors have right adjoints denoted by $ R_2$ and respectively $ R_2$ . The functor $ R_1$ was described above.

On the other hand, we have $ F=F_1F_2$ and therefore we may assume $ R=R_2R_1$ .

With these notations, it can be seen that the unit $ \eta^2:\mathrm{id}_{{\mathcal Z}_{\mathcal C} (\mathcal D)} \rightarrow F_2R_2$ of the adjunction $ (F_2,R_2)$ induces a natural transformation

$ m: Z(V)=FR(V)\rightarrow U(V)=F_1R_1(V)$ .

My question is if $ m=q$ as natural transformations?

Note: It can be relatively easy to check that this happens in the case $ \mathcal C=\mathrm{rep}(H)$ , the category of representations of a finite dimensional Hopf algebra.

google does not respect the CANONICAL tag?


This is a valid page of my site:

When I check google index for variations of the page, It shows 3-4 results:

All of them have the proper canonical tag, which tells google to index the proper page (mentioned above).

So why google does not respect the canonical tag?

In Google Search Console, under "Indexed, not submitted in sitemap" I see such pages and I want to get rid of them.

Why $\mathcal{O}(n^2)$ multiplication of coefficient required for canonical form of polynomial?

I was going to a textbook and am not able to understand the following:

Let $ F(x)$ is given as a product $ F(x) = \sum_{i=0}^{n} (x – a_i)$ . Transforming $ F(x)$ to its canonical form by consecutively multiplying the $ i$ th monomial with the product of the first $ i-1$ monomials requires $ \mathcal{O}(n^2)$ multiplications of coefficients.

A canonical form of polynomial is $ \sum_{i=0}^{n} c_i x_i$ .

Why $ \mathcal{O}(n^2)$ ?