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Essay about time is gold writing your case study on fundamental rights and duties

Is there any fundamental difference between BIOS and UEFI booting?

Per my understanding,

  • BIOS is the traditional boot method of Intel x86 computers. It starts by loading a short program called BIOS from somewhere on the motherboard (usually some EEPROM), which does a range of things like POST, before loading data the hard drive, including but not limited to further-stage bootloader codes.
  • UEFI is the new method of booting. It also starts with loading a small piece of code stored somewhere outside the main secondary storage (user hard drives), which also does a range of things before accessing hard drives.

The differences I’m able to tell so far:

  • BIOS loads something at a fixed location on a hard drive, specifically, the first 512 byte, called the master boot record, which loads everything further required to boot a OS.
  • UEFI doesn’t load from a fixed location on hard drive. Instead it searches the partition table (GPT) and looks for a ESP (EFI System Partition) and find. .efi files inside and executes them.

The confusions I’m currently having:

  1. I’ve heard that in BIOS and UEFI, hardware initialization order is different and UEFI is usually a bit faster in this. How exactly is it?
  2. Is there any fundamental difference between BIOS and UEFI? To me, at the very basis, they both load the initial code from a piece of special hardware (a ROM) which does the necessary things before loading data from disk, which is a “fundamental similarity”.

^[:wq

Why was noise XK fundamental pattern chosen in the transport layer (BOLT 08) of the lightning network

I am currently reviewing the transport layer of the Lightning network protocol. It builds on top of the noise protocol framework handshake patterns.

What I don’t get: Why was the fundamental pattern XK chosen?

XK   <- s   ...   -> e, es                  0                2   <- e, ee                  2                1   -> s, se                  2                5   <-                        2                5 

First of all why not KK? Nodes are announced via gossip and on my tcp socket I should see who connected to me being able to look up the static key of my peer. Was the reason so that lightning nodes could be private and in particular on tor?

Second of all why not using protocols without static keys? Is the reason that we wanted to have property 2 and 5 for initiator and recipient respectively?

Property 2:

Sender authentication resistant to key-compromise impersonation (KCI) . The sender authentication is based on an ephemeral-static DH (“es” or “se”) between the sender’s static key pair and the recipient’s ephemeral key pair. Assuming the corresponding private keys are secure, this authentication cannot be forged.

Property 5:

Encryption to a known recipient, strong forward secrecy. This payload is encrypted based on an ephemeral-ephemeral DH as well as an ephemeral-static DH with the recipient’s static key pair. Assuming the ephemeral private keys are secure, and the recipient is not being actively impersonated by an attacker that has stolen its static private key, this payload cannot be decrypted.

It feels like I have given the answer by quoting from the noise protocol framework page. But maybe I am mistaking so it would be great to get your insights.

Is the fundamental partition associated to $n$ the partition of $r_{0}(n)$ in $k_{0}(n)$ parts that maximizes entropy?

As usual, under Goldbach’s conjecture, let’s define for a large enough composite integer $ n$ the quantities $ r_{0}(n):=\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\}$ and $ k_{0}(n):=\pi(n+r_{0}(n))-\pi(n-r_{0}(n))$ .

The sequence of prime gaps in the interval $ [n-r_{0}(n),n+r_{0}(n)]$ sorted in decreasing order can be listed as $ g_{1}(n),\cdots,g_{k_{0}(n)}(n)$ and forms a partition of $ 2r_{0}(n)$ . As such, the sequence $ (h_{i}(n))_{1\leq i\leq k_{0}(n)}$ defined by $ h_{i}(n)=\frac{g_{i}(n)}{2}$ is a partition of $ r_{0}(n)$ in $ k_{0}(n)$ parts. Let’s call this partition the fundamental partition associated to $ n$ .

Among all partitions of $ r_{0}(n)$ in $ k_{0}(n)$ parts, is the fundamental partition associated to $ n$ the one that maximizes entropy in Shannon’s sense?

Computing fundamental group of a circle

I’m very new to Topology but we just covered in class an algorithm for calculating the Fundamental Group of a surface given a triangulation. The algorithm goes as follows:

1) Find a spanning tree in the triangulation

2) Add in 2-simplices where two sides are already in the tree

3) Repeat two until you have a maximal simply connected subcomplex

4) Use relations given by the 2-simplices to determine the group.

So given this and the triangulation of S1: S1 Triangulation

I get the spanning tree

Now am I right in thinking that the algorithm doesn’t work in the next step because the interior of the circle is not in the surface, so the following is invalid

Whereas the correct continuation would be to pass over step 2 (and 3) to get this

Which would mean that the fundamental group of $ S^1$ is just $ <a> = F^{1} \cong \mathbb{Z}$

When estimating camera motion from the fundamental matrix, why can translation only be recovered up to scale using epipolar geometry?

I was wondering once you have recovered the fundamental matrix F such that xFx’ = 0. If you try to recover translation and rotation from F, I am told that this is possible only up to scale? Why so?

Can the Weyl orbits of fundamental weights tell us the Cartan matrix?

Let $ \mathfrak{g}$ be a complex semisimple Lie algebra, $ \Delta$ its root system contained in $ \mathfrak{t}^{\vee}$ for a Cartan sub-algebra $ \mathfrak{t}$ of $ \mathfrak{g}$ . Let $ W$ be its Weyl group. Then we (might) have the following short exact sequence of groups: $ $ 1\rightarrow W \rightarrow Aut(\Delta) \rightarrow E \rightarrow 1$ $ where $ Aut(\Delta)\subset GL(\mathfrak{t}^{\vee})$ is the group of symmetries of $ \Delta$ , and $ E$ (should) be the group of symmetries of the associated Dynkin diagram.

It is possible to see how the map $ Aut(\Delta) \rightarrow E$ is defined from the perspective of simple roots (let them be $ \{\alpha_1,\ldots,\alpha_k\}$ ): let $ \sigma\in Aut(\Delta)$ , then $ \{\sigma(\alpha_1),\ldots,\sigma(\alpha_k)\}$ is a set of simple roots with respect to a choice of positive roots which corresponds to its dominant Weyl chamber. By composing $ \sigma$ with the unique Weyl element mapping this chamber to the original dominant Weyl chamber, we get a permutation of $ \{\alpha_1,\ldots,\alpha_k\}$ . One can see that this permutation preserves the Cartan matrix and thus defines an element of $ E$ .

What I would like to know is whether there is an alternative way to define the above map from the perspective of dominant fundamental weights, namely, let $ \{\lambda_1,\ldots,\lambda_k\}$ be the corresponding fundamental weights, then I observe that any element $ \sigma\in Aut(\Delta)$ induces a permutation of the Weyl orbits generated by the $ \lambda_i$ ‘s. So we almost get the desired map except we haven’t checked this permutation preserves the Cartan matrix. This raises the following question:

Can these Weyl orbits themselves (as polyhedra in the Euclidean space) tell us the Cartan matrix?

As an example, $ A_3$ , these orbits consist of two congruent tetrahedra and one octahedron. We know that there is no edge joining the two tetrahedra in the Dynkin diagram. Can we see this fact from their configuration?

The homotopical proof of the fundamental theorem of algebra

I am reading a homotopical proof of the fundamental theorem of algebra, and the end of the proof is as follows:

… the map $ z\mapsto r^nz^n$ (where $ r$ is a positive real number and $ z\in S^1$ ) is homotopic to a constant map $ z\mapsto a_0$ as maps from $ S^1$ to $ \mathbb R^2-0$ . But $ \mathbb R^2-0\cong S^1$ ,…

So does it follow from here that the maps from $ S^1$ to itself given by $ z\mapsto z^n$ and $ z\mapsto c$ for a constant $ c$ are homotopic, and since the fundamental group of $ S^1$ is $ \mathbb Z$ , the first map induces a nonzero map while the constant map induces the zero map which is contradictory?

Field K and integer n such that the etale fundamental group of (GL_n)_K is the profinite completion of GL_n(K)?

Prove or disprove:

Claim: there exists a field $ K$ and an integer $ n$ such that $ \pi_1^{et}((GL_n)_K)$ is isomorphic to the profinite completion of the abstract group $ GL_n(K)$ ?

Note that then $ G_K \cong \widehat{GL_n(K)}/\widehat{\mathbb{Z}}$ . In particular, $ G_K$ must have a (topological) generating set of size the cardinality of $ K$ .