language design – In C++, Why do bitwise operators convert 8 or 16 bit integers to 32 bit?

Is there a logical reason why the integer is upgraded to 32+ bits?
I was trying to make an 8bit mask, and found myself a bit disappointed that the upgrade will corrupt my equations.

sizeof( quint8(0)); // 1 byte
sizeof(~quint8(0)); // 4 bytes

Usually something like this is done for a good reason, but I do not see any reason why a bitwise operator would essentially need to add more bits. It would seem to me that this would hurt performance (slightly) because now you have more bits to allocate and evaluate.

Why does C++ (Or other languages) do this?

dg.differential geometry – Anti-symmetric operators for the Dirac or Majorana spinors

In a Zoom lecture given by a mathematical physics professor, if I recalled correctly, he explained that the in 1+1 dimensional spacetime (or 2 dimensions in short), the “action” of fermions (spinors) has an anti-symmetric Dirac operator.

  1. Say, if the $psi$ is a Dirac spinor, he wrote down an action
    int d^2x sqrt{g} bar{psi} (i gamma^mu D_mu) psi

    and (I think) he claims that the operator $(i gamma^mu D_mu)$ is an anti-symmetric matrix.

  2. Say, if the $chi$ is a Majorana spinor, he wrote down an action
    int d^2x sqrt{g} {chi} (i gamma^mu D_mu) chi

    and (I think) he claims that the operator $(i gamma^mu D_mu)$ is also an anti-symmetric matrix.

Is this true that the anti-symmetric matrix has something to do with these fermions (spinors)? or fermion statistics? Why?

Pseudo-differential operators and differetial operator

Hello I am totally new to Pseudo-differential operators and I’m wondering if a differential operator is a pseudo-differential operator.
So, I want to show , using the definition of the symbol given by Hörmander definition of symbols (see Pic), that if $p(x,xi) =sum_{|alpha|le m} a_{alpha}(x) xi^{alpha}$ is a symbol of differential operator, it’s also a symbol in $S^m$ but I could not bounded the coefficients in x…


oa.operator algebras – Lower bounds in the space of compact operators

Let $H$ be a separable Hilbert space, and $K(H)$ the corresponding space of compact operators. Consider the “unit sphere” $S:={Tin K(H)|Tgeq 0text{ and }||T||=1}$. Is it true that, given any pair of operators $T_1,T_2in S$, there exists another operator $Tin S$ such that $Tleq T_1,T_2$?.

Single colo server but with direct connectivity to operators?

Hello guys,

I want to place individual servers, but in several places, both in the USA. USA As in Europe, and having direct connectivity to … Read the rest of

Functional analysis: does the space of continuous linear operators form a Banach space (under the given norm)?

Consider the following functional equation
begin {equation *}
phi (x) = x phi (a + (1-a) x) + (1-x) phi ((1-b) x), phi (0) = 0, phi (1) = 1,
end {equation *}

when $ a> frac {1} {2} $ It can be studied using shrinkage mappings.

Consider the Banach space of all limited lipschitz functions in $ (0.1) $ such that for each $ f $ in the space,
begin {equation *}
m (f) = sup_ {x neq y} frac { left vert
f (x) -f (y) right vert} { left vert x-y right vert}.
end {equation *}

We denote by $ CL 0 the subspace of all functions $ f $ for which $ f (0) = 0 $. Clearly
begin {equation *}
left Vert f right Vert _ {CL ^ {0}} = m (f)
end {equation *}

it's a norm but $ CL 0 that is, with the $ left Vert cdot right Vert _ {CL ^ {0}} $, a Banach space.

Theorem: for $ a in ( frac {1} {2}, 1) $mapping
begin {equation *}
(T phi) (x) = x phi (a + (1-a) x) + (1-x) phi ((1-b) x)
end {equation *}

is a contraction mapping in $ CL ^ {0} cap lbrace f, f in CL ^ {0}, f (1) leq 1 rbrace. $

I have two questions:

1) Can I replace the limited Lipschitz functions with the continuous linear operators? Does the space of continuous linear operators (under the given norm) form a Banach space?

2) How can I show that $ CL ^ {0} cap lbrace f, f in CL ^ {0}, f (1) leq 1 rbrace $ is closed?

Please guide me about it. Thank you

Fa functional analysis – Definition of Lyapunov exponents for compact operators

There is the following result well known to Goldsheid and Margulis (see Proposition 1.3) on the existence of Lyapunov exponents:

Leave $ H $ be a $ mathbb R $-Hilbert space, $ A_n in mathfrak L (H) $ be compact and $ B_n: = A_n cdots A_1 $ for $ n in mathbb N $. Leave $ | B_n |: = sqrt {B_n ^ ast B_n} $ and $ sigma_k (B_n) $ denote the $ k $th largest singular value of $ B_n $ for $ k, n in mathbb N $. Yes $$ limsup_ {n to infty} frac { ln left | A_n right | _ { mathfrak L (H)}} n le0 tag1 $$ and $$ frac1n sum_ {i = 1} ^ k ln sigma_i (B_n) xrightarrow {n to infty} gamma ^ {(k)} ; ; ; text {for everyone} k in mathbb N tag2, $$ so

  1. $$ | B_n | ^ { frac1n} xrightarrow {n to infty} B $$ for some non-negative and self-attached compact $ B in mathfrak L (H) $.
  2. $$ frac { ln sigma_k (B_n)} n xrightarrow {n to infty} Lambda_k: = left. begin {cases} gamma ^ {(k)} – gamma ^ {(k -1)} & text {, if} gamma ^ {(i)}> – infty \ – infty & text {, otherwise} end {cases} right } tag2 $$ for all $ k in mathbb N $.

Question 1: I have seen this result in many conference books, but I was wondering why it is stated in this way. First of all, it's not $ (2) $ clearly equivalent to $$ frac { sigma_k (B_n)} n xrightarrow {n to infty} lambda_i in (- infty, infty) tag3 $$ for some $ lambda_i $ for all $ k in mathbb N $ which in turn is equivalent to $$ sigma_k (B_n) ^ { frac1n} xrightarrow {n to infty} lambda_i ge0 tag4 $$ for some $ mu_i ge0 $ for all $ k in mathbb N $? $ (4) $ seems to be much more intuitive than $ (3) $since no $ lambda_i $, but $ mu_i = e ^ { lambda_i} $ are precisely the Lyapunov exponents of the limit operator $ B $. Am I missing something? The definition of $ Lambda_i $ (which is equal to $ lambda_i $) seems strange to me.

Question 2: What is the interpretation of $ B $? I'm usually seeing a discrete dynamic system $ x_n = B_nx_0 $. That makes $ B $ (or $ Bx $) tell us about the asymptotic behavior / evolution of the orbits?

Injective continuous operators between Banach spaces

Suppose $ X $ and $ Y $ they are two Banach spaces of infinite dimensions. What can we say about the set of all continuous injectable linear operators between $ X $ and $ Y $? Is it always empty?

Functional analysis of fa: Lyapunov indices of a product of operators

The deterministic part of the proof of the multiplicative ergodic theorem can be proved by using Proposition 1.3 on the paper Lyapunov indices of a product of random matrices.$ ^ 1 $

They consider a sequence of matrices. $ A_n in mathbb R ^ {d times d} $. I would like to generalize this result to $ A_n in mathfrak L (H) $ for some $ mathbb R $-Hilbert space $ H $. In section 7, the authors give some pointers on how such a generalization could be established, but I have some trouble following their arguments (and their notation).

I don't need complete generality. I am willing to assume that $ A_n $ It is compact. Please note that if $ A in mathfrak L (H) $, so $ A ^ ast A $ it is non-negative and self-attached and therefore $$ | A |: = sqrt {A ^ ast A} $$ It is well defined. So, with the assumption of compactness, we know that $ | A_n | $ It has a pleasant spectral decomposition. This could simplify the way we need to establish and test the generalization of Proposition 1.3.

$ ^ 1 $
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Functional analysis: a question about comparing positive self-attached operators

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