statistics – Fisher information of joint distribution of transformed Normal distribution

Suppose $X_1=theta+epsilon_1$ and $$X_i=sqrt{gamma}X_{i-1}+sqrt{1-gamma}epsilon_i+theta(1-sqrt{gamma})$$
Where $gamma in (0,1)$ and $theta$ is the parameter of the model. Also $epsilon_1,epsilon_2,…epsilon_n$ are iid $N(0,1)$.

What is the Fisher information of this model and for what values of $gamma$ does it tensorise. I’ve tried using the Jacobian to find the joint distribution but I’m not sure, especially when determining for which values we have tensorisation. Any help would be much appreaciated.

dnd 5e – Does a Shepherd Druid’s Mighty Summoner apply to creatures transformed by polymorph?

Does the language used in the Mighty Summoner feature, “Summoned OR Created”, allow the ability to apply to a player transformed by polymorph?

Does polymorph, which reads, “This spell transforms a creature that you can see within range into a new form”, count as creating a beast?

dnd 5e – Does a Shepherd Druids Mighty Summoner apply to creatures transformed by polymorph?

What im wondering is if the language used in the feature “Summoned OR Created” would allow it to apply to a player transformed by polymorph. Does polymorph, which reads, “This spell transforms a creature that you can see within range into a new form.” count as creating a beast?

cv.complex variables – Abscissa of convergence of transformed Dirichlet series

Let

$$F(s)=sum_{k=1}^infty frac{f(k)}{k^s} mbox{ and }F^*(s)=sum_{k=2}^infty frac{f(k)g(k)}{k^s},$$
where the infinite sum $sum f(k)$ diverges, $f(k)$ and $g(k)$ are real numbers, $s$ a complex number, and $sigma_c(F)$, the abscissa of convergence, is given by the well-known formula
$$sigma_c(F) = limsup_{nrightarrowinfty}frac{log|A(n)|}{log n},
mbox{ with } A(n)=sum_{k=1}^n f(k).$$

I am wondering what functions $g(k)$ preserves the abscissa of convergence, that is, what functions $g(k)$ result in $sigma_c(F^*)=sigma_c(F)$. I am particularly interested in the case where $f(k)=lambda(k)$ is the Liouville function, resulting in $F(s)=zeta(2s)/zeta(s)$, thus having the same roots as $zeta$ for $Re(s)>sigma_c(F)$.

Obviously, the abscissa of convergence is preserved if $g(k)=(log k)^alpha$ and $alpha$ is any positive integer: in that case, $F^*(s)$ is the $alpha$th derivative of $F(s)$, and theorem 11.12 in Apostol’s book on number theory essentially states that $sigma_c(F)=sigma_c(F^*)$ in that case. By reversing integration and derivation, it seems obvious that it must also be true if $alpha$ is any negative integer. And if it is true for (say) $alpha=2$ and $alpha=3$, there is no reason to believe it does not work with (say) $alpha=2.81$. So this has to be true for any real $alpha$. One would also easily imagine that it must work too if $g(k)=(log k)^alpha (loglog k)^beta$ for any real numbers $alpha,beta$.

My question:

Is there a reference (or can you prove / disprove) that $sigma_c(F)=sigma_c(F^*)$ assuming $A(n)$ diverges and $g(k)=(log k)^alpha$, for any real number $alpha$, or at least if $alpha$ is a negative integer?

Other questions and remarks

Let’s say $g(k)=lambda(k)$ is the Liouville function. I am wondering if $A(n)$ diverges, I am sure it does, but can’t remember seeing a proof. Also the distribution of $+1$ and $-1$ in the $(lambda(k))$ sequence is 50/50. Is that a consequence of the prime number theorem? I think I read someone saying this.

Finally, if $F(s)$ convergences for some $s=s_0$, then it is known that $F(s)$ converges for $Re(s)>Re(s_0)$, and thus $sigma_c(F)leqRe(s_0)+epsilon$ for any $epsilon>0$. In the case $f(k)=lambda(k)$, proving that $F(s)$ converges at $s=0.9$ (a real number) would imply that $zeta(s)$ has no zero in $0.9 < Re(s) < 1$. This is impossible to prove yet. When I made my computations, it really seemed to converge, and what’s more, to the correct value computed by Mathematica (Mathematica is based on the analytic continuation of $zeta$, my computations are based on the series $F(s)$). Maybe it might be easier to prove the convergence of $F^*(0.9)$ by choosing (say) $alpha=1$. It would have the same exact implications.

block editor – svg attributes not properly transformed

While svg attributes with dash in the middle are correctly transformed on edit function, they are incorrectly transformed in save.

E.g. textAnchor is transformed in text-anchor in edit, but in textanchor in save (and consequently svg is shown incorrectly in front end, but correctly in gutenberg editor). I am suspecting on webpack configuration, I am using standard @wordpress/scripts default webpack configuration. Any ideas?

These are my functions, they use svg wrapped in component

Edit:

import Krug from './images/Krug_broj.jsx';
export default function Edit({ attributes, setAttributes, context }) {
    useEffect(() => {
        attributes.buttonVariationNum = context('makeiteasy/buttonVariation');
    }, (context('makeiteasy/buttonVariation')));
    return (
        <article { ...useBlockProps() }>
            <div className="title-compound">
                <Krug number="46" variant={1}/>
                <RichText
                    aria-label={ __('Naslov elementa') }
                    placeholder={ __('Naslov...') }
                    value={ attributes.elementTitle }
                    onChange={ elementTitle => setAttributes({ elementTitle }) }
                    withoutInteractiveFormatting
                    tagName="h4"
                    multiline={ false }
                    allowedFormats={ () }
                />
            </div>
            <RichText
                aria-label={ __('Odjeljak') }
                placeholder={ __('UpiĊĦi tekst...') }
                value={ attributes.elementText }
                onChange={ elementText => setAttributes({ elementText }) }
                tagName="div"
                multiline={ true }
            />
        </article>
    );
}

save:

import Krug from './images/Krug_broj.jsx';
export default function save({ attributes }) {

    return (
        <article {...useBlockProps.save()}>
            <div className="title-compound">
                <Krug number="46" variant={1}/>
                <RichText.Content
                    aria-label={ __('Naslov elementa') }
                    value={ attributes.elementTitle }
                    tagName="h4"
                />
            </div>
            <RichText.Content
                aria-label={ __('Odjeljak') }
                value={ attributes.elementText }
                tagName="div"
            />
        </article>
    );
}

Krug_broj component:

import { memo } from '@wordpress/element';

function Krug_broj_def(props) {
    switch (props.variant) {
        case 1:
            return (
                <svg viewBox="0 0 120 120" width="2.5em" height="2.5em">
                    <path
                        d="M63.84 116C94.86 116 120 90.78 120 59.66S94.86 3.32 63.84 3.32 7.68 28.54 7.68 59.66 32.82 116 63.84 116"
                        fill="#6294bc"
                    />
                    <path
                        d="M93.72 24.86c19.58 19.64 20.01 51.06.96 70.17s-50.37 18.68-69.95-.963c-19.58-19.64-20.01-51.06-.959-70.17s50.37-18.68 69.94.962m2.308-5.196c-22.05-22.13-57.71-22.23-79.64-.225-21.93 22-21.83 57.78.224 79.9 22.05 22.13 57.71 22.23 79.64.226 21.93-22 21.83-57.78-.225-79.9"
                        fill="#b0c9de"
                    />
                    <text
                        x={60.23}
                        y={80}
                        fill="#fff"
                        fontFamily="'Fira Sans'"
                        fontSize={68}
                        strokeWidth={2.308}
                        textAnchor="middle"
                        style={{
                            fontVariantCaps: "normal",
                            fontVariantEastAsian: "normal",
                            fontVariantLigatures: "normal",
                            fontVariantNumeric: "normal",
                            lineHeight: 1.25,
                        }}
                    >
                        {props.number.toString() + '.'}
                    </text>
                </svg>
            )
            break;
    }
}

const Krug_broj = memo(Krug_broj_def);
export default Krug_broj;

Funds transformed to 10% at the recipient adress

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showing tranaaction id and the 10% fund that got transferred

integration – If a space is invariant under a transformation will the integral of a function over the transformed space be the same?

Say we have $varphi(U) = U$ for some space $U subseteq mathbb{R}^n$ and some transformation $varphi: U rightarrow mathbb{R}^n$. Is it true that
$$int_{varphi(U)} f(v) dv = int_U f(varphi(u))du$$
or perhaps
$$int_{varphi(U)} f(v) dv = int_U f(varphi(u))|text{det } varphi'(u)|du$$
This seems to me like it should be true, since we are integrating the function over the same set of values, but I’m not sure how to prove it.

transformation – How to get correct axis for scaling from transformed space?

I have scaled and non-scaled representation of model so that I don’t destroy verticis information if model will be scaled to plane, for example. Now I trying implement scaling on model elements (face, edge), I set up axis of scale tool on dislplay (scaled) model, but I stucked with produce inverse transformed axis for correct scaling in non-scaled model version. What I mean:

Here not rotated not scaled cube.

enter image description here

Then I rotate and scale by global X axis (1, 0, 0), for rotated model this will be X Y axis. Faded axis is original axis, more bright is axis that scale tool produce and with which user interact.

enter image description here

This is how scale tool axis looks after just straight full inverse transform, under “straight” I meen just muliplying on inverted matrixs in right order. That actualy expected result because they orthognal in scaled space, but not correct result for perform correct scaling.

enter image description here

I have been trying to find an answer for several weeks now, trying many combinations of transformation order, after I realize that I am just starting to randomly place the matrix I was looking at options maybe I could somehow construct or line up vector of axis but not succesefull.

Here I want emphasize that inverted axis not necessary should be line up with origin axis (1, 0, 0) (0, 1, 0) (0, 0, 1), they can point in any direction, case that I take above just for simplicity. I think that my knowlend in linear algebra just not enough and it’s actually possible tusk even if it’s not one line matrix multiplication in some tricky order. Or maybe not:).

dnd 5e – Does the Hexblade Warlock’s Hex Warrior feature apply to a magic weapon that is transformed into your Pact Weapon?

The Hexblade Warlock’s Hex Warrior feature states (emphasis mine):

(…) If you later gain the Pact of the Blade feature, this benefit extends to every pact weapon you conjure with that feature, no matter the weapon’s type.

Meanwhile, the Pact of the Blade Warlock feature states:

(…) You can transform one magic weapon into your pact weapon by performing a special ritual while you hold the weapon. You perform the ritual over the course of 1 hour, which can be done during a short rest. You can then dismiss the weapon, shunting it into an extradimensional space, and it appears whenever you create your pact weapon thereafter. (…)

Does a magic weapon that you have transformed into your pact weapon benefit from Hex Warrior?
Does it count as a weapon you have conjured using Pact of the Blade?
Does the answer change if you shunt the weapon away and then make it appear afterwards?

spellcasting – Can Spells with long casting times be disrupted by being transformed?

bit of a dense and complex question here. My character was trying to cast leomund’s tiny hut as a ritual, which because it has a long casting time, requires her to maintain concentration. The group was ambushed (though we all passed perception checks so no surprise), and the DM assumed that if he were to polymorph my character, she would lose concentration, and wouldn’t be able to finish the spell because she no longer has a spell list as a CR 0 cat.

I know concentration shouldn’t matter, but is he right about the second part? Could she finish casting the spell as a cat? Does she have to turn back into a kobold first before the spell can resolve? Does anything change if she is casting Glyph of Warding, which isn’t a ritual?

Likewise, in a similar situation, if she later gets true polymorph and turns into a Planatar, while casting raise dead would break concentration as it takes more than 1 action, would she still be able to resolve the casting at the end of 1 hour, if she maintained concentration the whole time?