text:
# Suppose I had a hypothetical blog entry which I deemed a personal theory.

1. Feb 6, 2012

### checkitagain

Suppose I had a hypothetical blog entry which I deemed a "personal theory."

As a hypothetical example, here is the link:

http://www.boulder.swri.edu/clark/ncar799.html

It concerns asteroid/comet impacts, including 40
references.

And, hypothetically I state something along the lines of:

"My personal theory, based along the information in the
above peer-review, is that there would be an X effect
to a Y section of the Earth with a Z-type asteroid.
The following are my reasons, based on established
scientific principles."

-----------------------------

Would this be permitted in a blog?

2. Feb 7, 2012

### Drakkith

Staff Emeritus
Re: Suppose I had a hypothetical blog entry which I deemed a "personal theory."

A scientific theory is a very specific thing. If you are simply saying that an asteroid of X mass impacting on a point on the Earth would do Y amount of damage, there is nothing wrong with that and it is not a personal theory. Using established scientific principles is perfectly ok.

3. Feb 7, 2012

### AlephZero

Re: Suppose I had a hypothetical blog entry which I deemed a "personal theory."

What you have is a personal opinion, not a personal theory.

You could check how plausible your opinion is using accepted scientific theories, if you had enough knowledge and enough computing power to run the simulations.

4. Feb 7, 2012

### Ryan_m_b

Staff Emeritus
Re: Suppose I had a hypothetical blog entry which I deemed a "personal theory."

From the PF rules (emphasis mine)
Again from the rules (emphasis mine)

5. Feb 7, 2012

### checkitagain

Re: Suppose I had a hypothetical blog entry which I deemed a "personal theory."

Then, as my hypothetical example would be a personal opinion

(but not a personal theory, as posted to me by AlephZero), I would

feel free to post this personal opinion in my own blog.

6. Feb 7, 2012

### Ryan_m_b

Staff Emeritus
Re: Suppose I had a hypothetical blog entry which I deemed a "personal theory."

In the rules the term "theory" is used in the colloquial not scientific sense. If your theory, opinion, hypothesis or whatever breaches the rules regarding personal theories, non-mainstream science and overly-speculative posts then it is not allowed.

7. Feb 7, 2012

### JaredJames

Re: Suppose I had a hypothetical blog entry which I deemed a "personal theory."

I wouldn't advise being cocky using technicalities to post something which disagrees with the science behind it.

As long as the science agrees with it, then there shouldn't be a problem.

8. Feb 7, 2012

### checkitagain

Re: Suppose I had a hypothetical blog entry which I deemed a "personal theory."

No, this whole site and its extensive list of rules forces users to be

very discriminating with their ideas and terms, lest they upset

the status quo of what users, mentors, and admin deem to their liking.

So, I wouldn't claim someone to be "cocky," because they are getting

clarification on words, their meanings, and usage in the forum.

$\text{Note: JaredJames's message in post #9 is noted by me.}$

Last edited: Feb 7, 2012
9. Feb 7, 2012

### JaredJames

Re: Suppose I had a hypothetical blog entry which I deemed a "personal theory."

The site forces people to adhere to scientific standards. Period.

I'm always amused that people come here, decide they don't like it and then complain over and over or try to work around the rules rather than just going elsewhere. Private site, private rules. If you don't like it, tough.

10. Feb 7, 2012

### Ryan_m_b

Staff Emeritus
Re: Suppose I had a hypothetical blog entry which I deemed a "personal theory."

The rules of this site exist to maintain the high standard of quality teaching and discussion of science that we have. Our decision to strictly not allow personal theories or overly speculative posts exists to make sure that this site does not get bogged down and ruined with pointless, pseudo-scientific and crackpot threads. There are plenty of similar sites to ours that allow that kind of thing and they are crammed full of rubbish to the point where it is impossible to have an intelligent discussion regarding real science.

Something to bear in mind is that if you lack the knowledge to distinguish whether or not your idea, opinion or theory is in line with mainstream science then it you probably lack the knowledge to make a good theory.

11. Feb 7, 2012

### checkitagain

Re: Suppose I had a hypothetical blog entry which I deemed a "personal theory."

[itex]And I am always amused by hypocrital veteran users
who violate their own rules by attacking users with sarcasm
or uses of words such as "prattling" to squash polite, relevant
discussion, lest it hurt their egos.

The fact that this is a private site is irrelevant to all the users being
held accountable to not disparaging other users by violating the rules.

12. Feb 7, 2012

### Ryan_m_b

Staff Emeritus
Re: Suppose I had a hypothetical blog entry which I deemed a "personal theory."

If you see a post that you think breaks the rules then report it, simple as that. There is no need to talk about it more, we do not discuss mentor decisions.

13. Feb 7, 2012

### checkitagain

Re: Suppose I had a hypothetical blog entry which I deemed a "personal theory."

A personal blog defeats its purpose if user has to call into question
every minute detail as to whether or not they adhered to using
scientific methods every time. I can really try to do that.

> > Edit: I just read the post of Ryan m b to not discuss this anymore.

> > 2nd Edit: I just read the post by micromass in post # 14 below.

Last edited: Feb 7, 2012
14. Feb 7, 2012

### micromass

Re: Suppose I had a hypothetical blog entry which I deemed a "personal theory."

You know that is how science works right?? If you don't follow the scientific method in your blog, then you're not doing science. Not doing science is not allowed in this forum. If you want to discuss a personal theory/opinion, then you are free to do so but not on this forum.

I have especially created a blog on this: https://www.physicsforums.com/blog.php?b=3751 [Broken]

Last edited by a moderator: May 5, 2017
15. Feb 7, 2012

### Ryan_m_b

Staff Emeritus
Re: Suppose I had a hypothetical blog entry which I deemed a "personal theory."

This is a science forum with rules to enforce that. These rules apply to the blogs, this is not unreasonable.
I was not saying that we cannot discuss this anymore, I was saying that if you have a problem with the posts of other users then report them but we will not discuss them openly.

Last edited: Feb 7, 2012
16. Feb 7, 2012

### JaredJames

Re: Suppose I had a hypothetical blog entry which I deemed a "personal theory."

Oh, one does enjoy a good public flogging...

17. Feb 7, 2012

### micromass

Re: Suppose I had a hypothetical blog entry which I deemed a "personal theory."


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# When to use & instead of "and"?

Are there rules of usage when using the ampersand "&" instead of "and"?Are they completely interchangeable? The ampersand seems more casual, but I'm not sure.

format: plain


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# Python – All replacement combination from other list

PythonServer Side ProgrammingProgramming

When it is required to get the replacement combination from the other list, the ‘combinations’ method and the ‘list’ method is used.

## Example

Below is a demonstration of the same

from itertools import combinations

my_list = [54, 98, 11]

print("The list is :")
print(my_list)

replace_list = [8, 10]

my_result = list(combinations(my_list + replace_list, len(my_list)))

print("The result is :")
print(my_result)

## Output

The list is :
[54, 98, 11]
The result is :
[(54, 98, 11), (54, 98, 8), (54, 98, 10), (54, 11, 8), (54, 11, 10), (54, 8, 10), (98, 11, 8), (98, 11, 10), (98, 8, 10), (11, 8, 10)]

## Explanation

• The required packages are imported into the environment.

• A list is defined and is displayed on the console.

• Another replace list is defined.

• The 'combinations' method is used to concatenate the original list, the replace list, and the length of the original list.

• This is converted to a list.

• This is assigned to a variable.

• The result is displayed on the console.

Updated on 16-Sep-2021 08:51:09


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# Tag Info

2

You could set the toc headings ragged I just added one line to add fill glue as marked % Preview source code %% LyX 2.3.6 created this file. For more info, see http://www.lyx.org/. %% Do not edit unless you really know what you are doing. \documentclass[12pt,english]{article} \renewcommand{\familydefault}{\sfdefault} \usepackage[T1]{fontenc} \usepackage[...

1

Replace eqnarray* with align*. (See also eqnarray vs align). In order to uncover the individual lines of the align* environment, you could use \onslide: \documentclass{beamer} \begin{document} \begin{frame} \pause We have \pause \begin{align*} \onslide<3->{d\left(e^{bt}r\left(t\right)\right)&=e^{bt}dr\left(t\right)+r\left(t\right)de^{bt}+de^{...

0

I could not get the above to work. A (low-brow?) way to do this is with a table, ie to make: \begin{figure} \centering{}% \begin{tabular}{cc} \includegraphics[width=7cm]{image1} & \includegraphics[width=7cm]{image2}\tabularnewline (a) & (b)\tabularnewline \end{tabular}\caption{foiw} \end{figure} For that I just Insert Figure Insert 2x2 tabular ...

0

I have the same problem. Strangely, this works in 95% of the time: you get the error message "The document could not be saved..." when you click on Save, so just hit ESC (i.e. don't save it under a different name), and then press Save again (i.e. under the same name) - and Lyx saves it! In the 5% of the cases it doesn't work, just do it again: ESC, ...

0

0

For me, it helped a bit to switch of the "backup original document when saving" in LyX Tools->Settings. In LyX Version 2.3.4.3.

2

You need to specify \roman* to use the current enumeration counter, not enumi. This is the output: Note that references will match the label option, unless you specify it otherwise (through something like ref=\roman*, without the closing parenthesis )).

0

Another way to do this is with math macros (see 1, 2, 3). Go to insert > math > macro A box should popup in your document that looks like this: Change newmacroname to whatever you want. I'll call it "norm". In the two empty boxes, we need to provide the TeX and LyX code. In this case, that is \Vert\{\#1}\Vert. Type this manually into LyX, ...

1

Place your cursor in the Chapter* layout, then go to Insert > TeX Code, and paste the following: \addcontentsline{toc}{chapter}{Foreword}. The following screenshot is how it looks in LyX after doing this:

0

For the "simplified version", go to Document > Settings and in the "Custom:" field of "Class Options" put "handout". Press "OK". For more information, please see Help > Specific Manuals > Beamer Presentations

1

This answer got bumped to the front page again, so, an update. I don’t actually do this, but as of 2020, unicode-math allows you to type in many Unicode symbols directly into your source, e.g. 𝒮 (U+1D4AE) for \mathcal{S} and 𝗯 (U+1D5EF) for \mathbf{b}. If you use this style, you should load \usepackage[math-style=literal]{unicode-math}.

0

Use \cftfigindent to control the indentation. \documentclass{article} \usepackage{tocloft} \renewcommand{\cftfigpresnum}{Figure } % put this before caption number \setlength{\cftfigindent}{0.5em} % change indentation of e.g. "Figure 1" within list of figures \setlength{\cftfignumwidth}{4.5em} % increase space for augmented caption number \begin{...

2

Use the tocloft package which enables control of many aspects of the ToC, LoF and LoT layouts. % lofprob.tex SE 576535 \documentclass{article} \usepackage{tocloft} \renewcommand{\cftfigpresnum}{Figure } % put this before caption number \setlength{\cftfignumwidth}{4.5em} % increase space for augmented caption number \begin{document} \listoffigures \begin{...

Top 50 recent answers are included


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# Blowup formula for motivic cohomology

If $X$ is a smooth projective variety over a field, $Z\subset X$ a smooth closed subvariety of codimension $d$, $X'\to X$ the blowup of $X$ along $Z$, there's the blowup formula

$$H^j(X'_{et},\mathbf{Z}(n)) = H^j(X_{et},\mathbf{Z}(n))\oplus\bigoplus_{r=1}^{d-1}H^{j-2r}(Z_{et},\mathbf{Z}(j-r)).$$

I must be missing something. For $j = 2n+1$ and $n = 1$, we get

$$Br(X') = Br(X) \oplus(things)$$

and it isn't clear to me the "things" vanish. But we know the cohomological Brauer group is a birational invariant of $X$, so they must vanish.

What do I not know about $H^{j-2r}(Z_{et},\mathbf{Z}(j-r))$?

There is simply a typo in this formula, the expression $H^{j - 2r}(Z_{et},j-r)$ should be replaced by $H^{j - 2r}(Z_{et},n-r)$. This is closesly related to the fact that as motives $M(\mathbb{P}(E)) = \oplus_{r = 0}^{rank(E)} M(X)(r)[2r]$ for a vector bundle $E$ over a smooth variety $E$. In this case the sum in the RHS correspond to the exceptional divisor, which is the projectivisation of the normal bundle for a smooth blow up, the the shift in the cohomology and the weight correspond to the $"(r)"$ and the $"[2r]"$ in the formula for the motive of this vector bundle.


text:
for Journals by Title or ISSN for Articles by Keywords help
 Subjects -> PHYSICS (Total: 784 journals)     - ELECTRICITY AND MAGNETISM (9 journals)    - MECHANICS (19 journals)    - NUCLEAR PHYSICS (48 journals)    - OPTICS (90 journals)    - PHYSICS (569 journals)    - SOUND (20 journals)    - THERMODYNAMICS (29 journals) PHYSICS (569 journals)                  1 2 3 4 5 6 | Last
 Archive for Rational Mechanics and Analysis   [SJR: 3.472]   [H-I: 57]   [3 followers]  Follow         Hybrid journal (It can contain Open Access articles)    ISSN (Print) 1432-0673 - ISSN (Online) 0003-9527    Published by Springer-Verlag  [2302 journals]
• Hamiltonian Evolution of Monokinetic Measures with Rough Momentum Profile
• Abstract: Abstract Consider a monokinetic probability measure on the phase space $${{\bf R}^N_{x} \times {\bf R}^N_{\xi}}$$ , i.e. $${\mu^{\rm {in}} = \rho^{\rm {in}}(x)\delta(\xi - U^{\rm {in}}(x))}$$ where U in is a vector field on R N and ρ in a probability density on R N . Let Φ t be a Hamiltonian flow on R N ×  R N . In this paper, we study the structure of the transported measure $${\mu(t) := \Phi_t\#\mu^{\rm {in}}}$$ and of its integral in the ξ variable denoted ρ(t). In particular, we give estimates on the number of folds in $${\Phi_t({\rm graph of} U^{\rm {in}})}$$ , on which μ ( t) is concentrated. We explain how our results can be applied to investigate the classical limit of the Schrödinger equation by using the formalism of Wigner measures. Our formalism includes initial momentum profiles U in with much lower regularity than required by the WKB method. Finally, we discuss a few examples showing that our results are sharp.
PubDate: 2015-07-01

• Crystal Dislocations with Different Orientations and Collisions
• Abstract: Abstract We study a parabolic differential equation whose solution represents the atom dislocation in a crystal for a general type of Peierls-Nabarro model with possibly long range interactions and an external stress. Differently from the previous literature, we treat here the case in which such dislocation is not the superposition of transitions all occurring with the same orientations (i.e. opposite orientations are allowed as well). We show that, at a long time scale, and at a macroscopic space scale, the dislocations have the tendency to concentrate as pure jumps at points which evolve in time, driven by the external stress and by a singular potential. Due to differences in the dislocation orientations, these points may collide in finite time. More precisely, we consider the evolutionary equation $$(v_\varepsilon)_t=\frac{1}{\varepsilon}\left( \mathcal{I}v_\varepsilon-\frac{1}{\varepsilon^{2s}}W'(v_\varepsilon)+\sigma(t, x)\right),$$ where $${v_\varepsilon=v_\varepsilon(t, x)}$$ is the atom dislocation function at time t >  0 at the point $${x \in \mathbb{R}, {\mathcal{I}_{s}}}$$ is an integro-differential operator of order $${2s \in (0, 2), W}$$ is a periodic potential, $${\sigma}$$ is an external stress and $${\varepsilon > 0}$$ is a small parameter that takes into account the small periodicity scale of the crystal. We suppose that $${v_\varepsilon(0, x)}$$ is the superposition of N−K transition layers in the positive direction and K in the negative one (with $${K \in\{0,\dots,N\}}$$ ); more precisely, we fix points $${x_1^0 < \dots < x_N^0}$$ and we take $$v_\varepsilon(0, x)= \frac{\varepsilon^{2s}}{W''(0)}\sigma(0, x)+\sum_{i=1}^N u\left(\zeta_i\frac{x-x_i^0}{\varepsilon}\right).$$ Here $${\zeta_i}$$ is either −1 or 1, depending on the orientation of the transition layer u, which in turn solves the stationary equation $${\mathcal{I}_{s} u=W'(u)}$$ . We show that our problem possesses a unique solution and that, as $${\varepsilon \to 0^+}$$ , it approaches the sum of Heaviside functions H with different orientations centered at points x i (t), namely $$\sum_{i=1}^N H(\zeta_i(x-x_i(t))).$$ The point x i evolves in time from $${x_i^0}$$ , being subject to the external stress and a singular potential, which may be either attractive or repulsive, according to the different orientation of the transitions; more precisely, the speed $${\dot x_i}$$ is proportional to $$\sum_{j\neq i}\zeta_i\zeta_j\frac{x_i-x_j}{2s x_i-x_j ^{1+2s}}-\zeta_i\sigma(t, x_i).$$ The evolution of such a dynamical system may lead to collisions in finite time. We give a detailed description of such collisions when N = 2, 3 and we show that the solution itself keeps track of such collisions; indeed, at the collision time T c the two opposite dislocations have the tendency to annihilate each other and make the dislocation vanish, but only outside the coll...
PubDate: 2015-07-01

• Convex Entropy, Hopf Bifurcation, and Viscous and Inviscid Shock Stability
• Abstract: Abstract We consider by a combination of analytical and numerical techniques, some basic questions regarding the relations between inviscid and viscous stability and the existence of a convex entropy. Specifically, for a system possessing a convex entropy, in particular for the equations of gas dynamics with a convex equation of state, we ask: (1) can inviscid instability occur? (2) can viscous instability not detected by inviscid theory occur? (3) can there occur the—necessarily viscous—effect of Hopf bifurcation, or “galloping instability”? and, perhaps most important from a practical point of view, (4) as shock amplitude is increased from the (stable) weak-amplitude limit, can there occur a first transition from viscous stability to instability that is not detected by inviscid theory? We show that (1) does occur for strictly hyperbolic, genuinely nonlinear gas dynamics with certain convex equations of state, while (2) and (3) do occur for an artifically constructed system with convex viscosity-compatible entropy. We do not know of an example for which (4) occurs, leaving this as a key open question in viscous shock theory, related to the principal eigenvalue property of Sturm Liouville and related operators. In analogy with, and partly proceeding close to, the analysis of Smith on (non-)uniqueness of the Riemann problem, we obtain convenient criteria for shock (in)stability in the form of necessary and sufficient conditions on the equation of state.
PubDate: 2015-07-01

• Homogenization of the Hele-Shaw Problem in Periodic Spatiotemporal Media
• Abstract: Abstract We consider the homogenization of the Hele-Shaw problem in periodic media that are inhomogeneous both in space and time. After extending the theory of viscosity solutions into this context, we show that the solutions of the inhomogeneous problem converge in the homogenization limit to the solution of a homogeneous Hele-Shaw-type problem with a general, possibly nonlinear dependence of the free boundary velocity on the gradient. Moreover, the free boundaries converge locally uniformly in Hausdorff distance.
PubDate: 2015-07-01

• On the Dirichlet Problem of Mixed Type for Lower Hybrid Waves in
Axisymmetric Cold Plasmas
• Abstract: Abstract For a class of linear second order partial differential equations of mixed elliptic-hyperbolic type, which includes a well known model for analyzing possible heating in axisymmetric cold plasmas, we give results on the weak well-posedness of the Dirichlet problem and show that such solutions are characterized by a variational principle. The weak solutions are shown to be saddle points of natural functionals suggested by the divergence form of the PDEs. Moreover, the natural domains of the functionals are the weighted Sobolev spaces to which the solutions belong. In addition, all critical levels will be characterized in terms of global extrema of the functionals restricted to suitable infinite dimensional linear subspaces. These subspaces are defined in terms of a robust spectral theory with weights which is associated to the linear operator and is developed herein. Similar characterizations for the weighted eigenvalue problem and nonlinear variants will also be given. Finally, topological methods are employed to obtain existence results for nonlinear problems including perturbations in the gradient which are then applied to the well-posedness of the linear problem with lower order terms.
PubDate: 2015-07-01

• An Analysis of Crystal Cleavage in the Passage from Atomistic Models to
Continuum Theory
• Abstract: Abstract We study the behavior of brittle atomistic models in general dimensions under uniaxial tension and investigate the system for critical fracture loads. We rigorously prove that in the discrete-to-continuum limit the minimal energy satisfies a particular cleavage law with quadratic response to small boundary displacements followed by a sharp constant cut-off beyond some critical value. Moreover, we show that the minimal energy is attained by homogeneous elastic configurations in the subcritical case and that beyond critical loading cleavage along specific crystallographic hyperplanes is energetically favorable. In particular, our results apply to mass spring models with full nearest and next-to-nearest pair interactions and provide the limiting minimal energy and minimal configurations.
PubDate: 2015-07-01

• Hyperbolic Second Order Equations with Non-Regular Time Dependent
Coefficients
• Abstract: Abstract In this paper we study weakly hyperbolic second order equations with time dependent irregular coefficients. This means assuming that the coefficients are less regular than Hölder. The characteristic roots are also allowed to have multiplicities. For such equations, we describe the notion of a ‘very weak solution’ adapted to the type of solutions that exist for regular coefficients. The construction is based on considering Friedrichs-type mollifiers of coefficients and corresponding classical solutions, and their quantitative behaviour in the regularising parameter. We show that even for distributional coefficients, the Cauchy problem does have a very weak solution, and that this notion leads to classical or to ultradistributional solutions under conditions when such solutions also exist. In concrete applications, the dependence on the regularising parameter can be traced explicitly.
PubDate: 2015-07-01

• Existence and Stability for a Non-Local Isoperimetric Model of Charged
Liquid Drops
• Abstract: Abstract We consider a variational problem related to the shape of charged liquid drops at equilibrium. We show that this problem never admits local minimizers with respect to L 1 perturbations preserving the volume. However, we prove that the ball is stable under small C 1,1 perturbations when the charge is small enough.
PubDate: 2015-07-01

• Global Small Solutions to a Complex Fluid Model in Three Dimensional
• Abstract: Abstract In this paper, we provide a much simplified proof of the main result in Lin and Zhang (Commun Pure Appl Math 67: 531–580, 2014) concerning the global existence and uniqueness of smooth solutions to the Cauchy problem for a three dimensional incompressible complex fluid model under the assumption that the initial data are close to some equilibrium states. Besides the classical energy method, the interpolating inequalities and the algebraic structure of the equations coming from the incompressibility of the fluid are crucial in our arguments. We combine the energy estimates with the L ∞ estimates for time slices to deduce the key L 1 in time estimates. The latter is responsible for the global in time existence.
PubDate: 2015-06-01

• Vanishing Shear Viscosity and Boundary Layer for the Navier–Stokes
Equations with Cylindrical Symmetry
• Abstract: Abstract Both the global well-posedness for large data and the vanishing shear viscosity limit with a boundary layer to the compressible Navier–Stokes system with cylindrical symmetry are studied under a general condition on the heat conductivity coefficient that, in particular, includes the constant coefficient. The thickness of the boundary layer is proved to be almost optimal. Moreover, the optimal L 1 convergence rate in terms of shear viscosity is obtained for the angular and axial velocity components.
PubDate: 2015-06-01

• Fick’s Law in a Random Lattice Lorentz Gas
• Abstract: Abstract We provide a proof that the stationary macroscopic current of particles in a random lattice Lorentz gas satisfies Fick’s law when connected to particles reservoirs. We consider a box on a d + 1-dimensional lattice and when $${d\geqq7}$$ , we show that under a diffusive rescaling of space and time, the probability of finding a current different from its stationary value is exponentially small in time. Its stationary value is given by the conductivity times the difference of chemical potentials of the reservoirs. The proof is based on the fact that in a high dimension, random walks have a small probability of making loops or intersecting each other when starting sufficiently far apart.
PubDate: 2015-06-01

• A Derivation of the Magnetohydrodynamic System from
Navier–Stokes–Maxwell Systems
• Abstract: Abstract We provide a full and rigorous derivation of the standard viscous magnetohydrodynamic system (MHD) as the asymptotic limit of Navier–Stokes–Maxwell systems when the speed of light is infinitely large. We work in the physical setting provided by the natural energy bounds and therefore mainly consider Leray solutions of fluid dynamical systems. Our methods are based on a direct analysis of frequencies and we are able to establish the weak stability of a crucial nonlinear term (the Lorentz force), neither assuming any strong compactness of the components nor applying standard compensated compactness methods (which actually fail in this case).
PubDate: 2015-06-01

• Optimal Shape and Location of Sensors for Parabolic Equations with Random
Initial Data
• Abstract: Abstract In this article, we consider parabolic equations on a bounded open connected subset $${\Omega}$$ of $${\mathbb{R}^n}$$ . We model and investigate the problem of optimal shape and location of the observation domain having a prescribed measure. This problem is motivated by the question of knowing how to shape and place sensors in some domain in order to maximize the quality of the observation: for instance, what is the optimal location and shape of a thermometer? We show that it is relevant to consider a spectral optimal design problem corresponding to an average of the classical observability inequality over random initial data, where the unknown ranges over the set of all possible measurable subsets of $${\Omega}$$ of fixed measure. We prove that, under appropriate sufficient spectral assumptions, this optimal design problem has a unique solution, depending only on a finite number of modes, and that the optimal domain is semi-analytic and thus has a finite number of connected components. This result is in strong contrast with hyperbolic conservative equations (wave and Schrödinger) studied in Privat et al. (J Eur Math Soc, 2015) for which relaxation does occur. We also provide examples of applications to anomalous diffusion or to the Stokes equations. In the case where the underlying operator is any positive (possible fractional) power of the negative of the Dirichlet-Laplacian, we show that, surprisingly enough, the complexity of the optimal domain may strongly depend on both the geometry of the domain and on the positive power. The results are illustrated with several numerical simulations.
PubDate: 2015-06-01

• Metric Description of Singular Defects in Isotropic Materials
• Abstract: Abstract Classical elasticity is concerned with bodies that can be modeled as smooth manifolds endowed with a reference metric that represents local equilibrium distances between neighboring material elements. The elastic energy associated with the configuration of a body in classical elasticity is the sum of local contributions that arise from a discrepancy between the actual metric and the reference metric. In contrast, the modeling of defects in solids has traditionally involved extra structure on the material manifold, notably torsion to quantify the density of dislocations and non-metricity to represent the density of point defects. We show that all the classical defects can be described within the framework of classical elasticity using tensor fields that only assume a metric structure. Specifically, bodies with singular defects can be viewed as affine manifolds; both disclinations and dislocations are captured by the monodromy that maps curves that surround the loci of the defects into affine transformations. Finally, we showthat two dimensional defectswith trivial monodromy are purely local in the sense that if we remove from the manifold a compact set that contains the locus of the defect, the punctured manifold can be isometrically embedded in a Euclidean space.
PubDate: 2015-06-01

• Uniform Lipschitz Estimates in Bumpy Half-Spaces
• Abstract: Abstract This paper is devoted to the proof of uniform Hölder and Lipschitz estimates close to oscillating boundaries, for divergence form elliptic systems with periodically oscillating coefficients. Our main point is that no structure is assumed on the oscillations of the boundary. In particular, those oscillations are neither periodic, nor quasiperiodic, nor stationary ergodic. We investigate the consequences of our estimates on the large scales of Green and Poisson kernels. Our work opens the door to the use of potential theoretic methods in problems concerned with oscillating boundaries, which is an area of active research.
PubDate: 2015-06-01

• Γ -convergence Approximation of Fracture and Cavitation in Nonlinear
Elasticity
• Abstract: Abstract Our starting point is a variational model in nonlinear elasticity that allows for cavitation and fracture that was introduced by Henao and Mora-Corral (Arch Rational Mech Anal 197:619–655, 2010). The total energy to minimize is the sum of the elastic energy plus the energy produced by crack and surface formation. It is a free discontinuity problem, since the crack set and the set of new surface are unknowns of the problem. The expression of the functional involves a volume integral and two surface integrals, and this fact makes the problem numerically intractable. In this paper we propose an approximation (in the sense of Γ-convergence) by functionals involving only volume integrals, which makes a numerical approximation by finite elements feasible. This approximation has some similarities to the Modica–Mortola approximation of the perimeter and the Ambrosio–Tortorelli approximation of the Mumford–Shah functional, but with the added difficulties typical of nonlinear elasticity, in which the deformation is assumed to be one-to-one and orientation-preserving.
PubDate: 2015-06-01

• Green’s Functions of Wave Equations in $$\mathbb{R}^n_+\times \mathbb{R}_+$$ R + n × R +
• Abstract: Abstract We study the d’Alembert equation with a boundary. We introduce the notions of Rayleigh surface wave operators, delayed/advanced mirror images, wave recombinations, and wave cancellations. This allows us to obtain the complete and simple formula of the Green’s functions for the wave equation with the presence of various boundary conditions. We are able to determine whether a Rayleigh surface wave is active or virtual, and study the lacunas of the wave equation in three dimensional with the presence of a boundary in the case of a virtual Rayleigh surface wave.
PubDate: 2015-06-01

• Nonlocal Curvature Flows
• Abstract: Abstract This paper aims at building a unified framework to deal with a wide class of local and nonlocal translation-invariant geometric flows. We introduce a class of nonlocal generalized mean curvatures and prove the existence and uniqueness for the level set formulation of the corresponding geometric flows. We then introduce a class of generalized perimeters, whose first variation is an admissible generalized curvature. Within this class, we implement a minimizing movements scheme and we prove that it approximates the viscosity solution of the corresponding level set PDE. We also describe several examples and applications. Besides recovering and presenting in a unified way existence, uniqueness, and approximation results for several geometric motions already studied and scattered in the literature, the theory developed in this paper also allows us to establish new results.
PubDate: 2015-05-22

• Optimal Flux Densities for Linear Mappings and the Multiscale Geometry of
Structured Deformations
• Abstract: Abstract We establish the unexpected equality of the optimal volume density of total flux of a linear vector field $${x \longmapsto Mx}$$ and the least volume fraction that can be swept out by submacroscopic switches, separations, and interpenetrations associated with the purely submacroscopic structured deformation (i, I + M). This equality is established first by identifying a dense set $${\mathcal{S}}$$ of $${N{\times}N}$$ matrices M for which the optimal total flux density equals trM , the absolute value of the trace of M. We then use known representation formulae for relaxed energies for structured deformations to show that the desired least volume fraction associated with (i, I + M) also equals trM . We also refine the above result by showing the equality of the optimal volume density of the positive part of the flux of $${x \longmapsto Mx}$$ and the volume fraction swept out by submacroscopic separations alone, with common value (trM)+. Similarly, the optimal volume density of the negative part of the flux of $${x \longmapsto Mx}$$ and the volume fraction swept out by submacroscopic switches and interpenetrations are shown to have the common value (trM)−.
PubDate: 2015-05-21

• Structure of Helicity and Global Solutions of Incompressible
Navier–Stokes Equation
• Abstract: Abstract In this paper we derive a new energy identity for the three-dimensional incompressible Navier–Stokes equations by a special structure of helicity. The new energy functional is critical with respect to the natural scalings of the Navier–Stokes equations. Moreover, it is conditionally coercive. As an application we construct a family of finite energy smooth solutions to the Navier–Stokes equations whose critical norms can be arbitrarily large.
PubDate: 2015-05-21

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# The nature of ferromagnetism in the chiral helimagnet Cr1/3NbS2

## Abstract

The chiral helimagnet Cr1/3NbS2 hosts exotic spin textures, whose influence on the magneto-transport properties make this material an ideal candidate for future spintronic applications. To date, the interplay between macroscopic magnetic and transport degrees of freedom is believed to result from a reduction in carrier scattering following spin order. Here, we present electronic structure measurements across the helimagnetic transition temperature TC that challenges this view. We show that the Fermi surface is comprised of strongly hybridized Nb- and Cr-derived electronic states, and that spectral weight close to the Fermi level increases anomalously as the temperature is lowered below TC. These findings are rationalized on the basis of first principle density functional theory calculations, which reveal a large nearest-neighbor exchange energy, suggesting the interaction between local spin moments and hybridized Nb- and Cr-derived itinerant states to go beyond the perturbative interaction of Ruderman-Kittel-Kasuya-Yosida, suggesting instead a mechanism rooted in a Hund’s exchange interaction.

## Introduction

The development of next-generation electronic devices relies on the ability to precisely control other intrinsic degrees of freedom (DOF) beyond that of charge. In this endeavor, chiral helimagnets (CHMs) have emerged as promising materials to be used for spintronic and other information technology applications, where the intent is to actively manipulate either the individual or average spin angular momentum of itinerant carriers1. In CHMs the chiral framework of the crystal structure allows for the arrangement of spins into incommensurate periodic spirals, or helices, resulting from an anti-symmetric spin interaction known as the Dzyaloshinskii–Moriya interaction2,3,4,5,6. Such an interaction, which originates from relativistic spin–orbit coupling, not only underlies the helical ground state of CHMs but also unusual spin textures like the two-dimensional spin vortex known as a skyrmion as well7,8. Skyrmions are of particular technological interest owing to the fact that they can be manipulated at the nanoscale level by externally applied magnetic fields8 and spin polarized currents9,10, and are found to have a profound influence on electronic transport11,12,13.

Given these important functionalities, there has been a great incentive for the discovery of new CHM materials relevant to the design and fabrication of future spintronic devices14. One such material is Cr1/3NbS215, which has been shown to host a unique one-dimensional solitonic excitation known as a chiral soliton lattice (CSL)16. Much like skyrmions the CSL can be manipulated by external magnetic field16, but unlike other CHMs, Cr1/3NbS2 crystallizes in a more anisotropic, Nb3CoS6 (hp20), layered structure consisting of ferromagnetic (FM) planes of intercalated Cr atoms ordering in a (√3 × √3)R(30°) superstructure within the ab plane. Each Cr atom occupies a trigonally distorted octahedral (Oh) sites in the van der Waals gaps of 2H-NbS2 (Fig. 1a)17 and hosts a local spin moment (LSM) of ~3μB Cr−1 that order below a Curie temperature, TC = 116–132 K15,16,17. Due to such high anisotropy and the fact that inversion symmetry breaks only along the non-centrosymmetric c-axis, where the long-range (~48 nm) helimagnetic ordering develops, Cr1/3NbS2 has been recognized as an ideal system for studying spin textures in magnetic thin films, devices fabricated on a substrate, and spin–orbit coupling effects in magnetic multilayers18,19.

More generally, Cr1/3NbS2 belongs to a class of transition metal dichalcogenides (TMDs) intercalated by magnetic, 3d elements. According to a conventional description of the electronic structure for these materials, Curie–Weiss spin moments localized on the Cr site do not alter the Fermi surface (FS) or band topology of the host NbS2 compound, aside from raising the chemical potential following a donation of charge to Nb-derived bands20,21. Due to the large separation distance between Cr neighbors (√3a), long-range magnetic order is believed to be mediated by Nb conduction electrons through a Ruderman–Kittel–Kasuya–Yosida (RKKY) exchange interaction22,23,24,25. Under such an exchange mechanism, the raise in chemical potential that accompanies intercalation serves to alter the nesting condition in the FS, and in turn change the periodicity and range of interaction for the oscillatory exchange coupling between screened, LSMs26. In this regard, the identity of the intercalant atom has little impact on the resultant order or ordering temperature so long as each species possesses the same oxidation state and spacing in the lattice. Hence, both the drop in resistivity and negative magnetoresistance (MR) observed throughout this class of materials are argued to result from a reduction in spin scattering due entirely to magnetic ordering27,28. However, recent experimental and theoretical findings for Cr1/3NbS2 not only propose an alternative exchange mechanism beyond a conventional RKKY29 but also show abrupt, anisotropic changes in the magneto-transport that suggest that Cr-derived states play a more active role in the interplay between LSMs and itinerant electrons30.

The importance of these findings is underscored by a major technological appeal of Cr1/3NbS2, which lies in the interplay between transport and magnetic DOF. Namely, inter-layer magneto-transport measurements have revealed a direct correlation between the transverse, negative MR and the soliton lattice density, which is attributed to Bragg scattering of conduction electrons by the magnetic superlattice potential of the CSL27,28. However, recent in-plane transport, MR, and magnetic anisotropy measurements both in proximity to TC and at the CHM to CSL transition show abrupt changes29,31 that depend sensitively on the direction of applied magnetic field relative to the ab plane30. This includes a drop in the in-plane resistivity by more than an order of magnitude as T < TC as well as a decrease in the intra-layer MR by 6% at the CSL transition29. Such results cannot be readily explained by spin ordering arguments alone, as it is not obvious how a modulation of the CSL state along the crystallographic c-axis could have such a strong impact on the in-plane transport properties. Rather, Ghimire et al.29 and Bornstein et al.30 argue on the basis of first-principle calculations that details of the electronic structure must play a more central role in shaping magneto-transport in Cr1/3NbS2. Thus, there exists an experimental need to better elucidate the microscopic mechanisms underlying such a close interplay between in-plane transport and magnetic DOF in this material.

In this article, we provide experimental evidence showing the electronic structure to play a non-trivial role in the description of in-plane magneto-transport in Cr1/3NbS2. Through the use of resonant photoemission spectroscopy (ResPES), we find the FS of this material to be comprised of strongly hybridized Cr- and Nb-derived electronic states, while angle-resolved photoemission spectroscopy (ARPES) reveals an anomalous increase of spectral weight in proximity to the Fermi level (EF) as temperature is lowered below TC. Such behavior is inconsistent with that of a conventional itinerant ferromagnet, and can be rationalized on the basis of first-principle density functional theory (DFT) calculations to result from a strong exchange (Hund’s) interaction between itinerant electrons and LSMs on the Cr sites.

## Results

### Overview of electronic structure

Previous work focusing on how the electronic structure of NbS2 is modified with Cr intercalation for temperatures below the helimagnetic transition temperature (Fig. 1b) is reported by Sirica et al.32. To summarize these results, Fig. 1c, d shows a schematic diagram of both the FS and band dispersion along the high-symmetry ΓK-axis, where crossing points α, β1,2, γ, and δ follow the same labeling convention as used in this previous study32. As expected, Cr intercalation injects electronic charge into the NbS2 layers, resulting in a sinking of the bonding, bilayer split NbS2 band (S) below EF, while the remaining NbS2-derived, anti-bonding band has crossing points α and γ consistent with a raise in chemical potential. However, the electronic charge introduced by the Cr intercalant not only changes the total number of carriers but also causes a significant modification to the electronic structure of NbS2 through the addition of two additional bands at Γ1, β2), and one additional band at Κ (δ). Since these bands cannot be rationalized on the basis of a rigid band picture, it is natural to inquire about their origin. Here, we focus on the temperature dependence of the α and β1,2 bands in proximity to Γ, as the bands at Κ are found to exhibit no significant change with temperature across TC.

### Temperature-dependent angle-resolved photoemission

Temperature-dependent ARPES is used to map the dispersion of the α and β1,2 bands in proximity to EF as the temperature is tuned across TC (Fig. 2). For T < 131 K, three dispersive bands can be identified at Γ, while only two bands are present as temperature is raised above TC. From Fig. 2, it is evident that the outermost β2 band shifts towards Γ with increasing temperature until the two β-bands can no longer be distinguished. Such behavior is more clearly demonstrated in Fig. 3, which gives the temperature dependence of the band crossing points (kF) along the two high-symmetry directions ΓM (Fig. 3a) and ΓK (Fig. 3b). Here, momentum distribution curves (MDCs) illustrating the distributions of spectral weight at a fixed energy E = EF along these two high-symmetry directions reveals a monotonic shift in kF for the β2 band to be accompanied by a progressive decrease in spectral weight for 50 K < T < 120 K. The fact that this behavior is observed along both the ΓM and ΓK directions suggests it to be isotropic about Γ. Moreover, given the close proximity to TC, the relative shift of β2 towards β1 implies the β1,2 bands arise from magnetic exchange splitting. While dedicated spin-resolved ARPES measurements are required to establish unambiguously the spin polarization of these bands, such a finding is consistent with the notion that the β1,2 bands result from intercalation of a magnetic element (Cr) into NbS2, and suggest the presence of Cr-derived states at EF. Indeed, this finding has been confirmed with ResPES, a methodology that allows photoemission spectra of the valence band (VB) to acquire elemental sensitivity33,34.

### Resonant photoemission

By tuning the incident photon energy across the Cr L3-edge, it is possible to identify Cr 3d states in the VB within 3 eV of EF (Fig. 4a, b). The increase in signal intensity at resonance, that is, with the photon energy tuned on the maximum of the Cr absorption edge, reveals that the structures at ≈2.5 eV and within ≈1 eV from EF are states possessing Cr 3d character.

The states at Γ have been shown by polarization-dependent ARPES to exhibit predominantly out-of-plane orbital character32. Therefore, in contrast to our previous ResPES measurements32, whose purpose was to identify non-dispersive Cr states remaining on the surface following sample cleave, the photon polarization used in this report was chosen to emphasize those bulk states having an out-of-plane character through ensuring a component of the photon polarization to lie perpendicular to the sample plane. In doing so, a direct comparison of ResPES spectra taken with photon energies tuned across the Cr L-edge (Fig. 4a, b) and Nb M-edge (Fig. 4c, d) reveals a clear resonance over the Nb “dz2 sub-band”21 (EB ≈0.6 eV < E < EF), indicating the presence of hybridization between Cr- and Nb-derived states in the VB. This is made possible by noting spectra taken across the Cr L3-edge to be sensitive, in these experimental conditions, to the projection of Cr 3d orbitals, whose splitting is dictated by a trigonally distorted octahedral environment22,23. Hence, our ResPES data clarify that the VB states at Γ originate from a linear combination of Nb dz2 and mixed Cr 3d orbitals ($$\sqrt 2$$dxz − dx2 − y2 and $$\sqrt 2$$dyz + dxy), having an overall out-of-plane orbital character.

In addition to taking ResPES measurements across the Cr L3 ionization edge for T > TC (Fig. 4a, b), ARPES spectra were collected at T < TC along the ΓΚ-axis using photon energies that have likewise been tuned across the Cr resonance (Fig. 5a–c). In doing so, the α band, belonging to the host NbS2 compound, and two β1,2 bands, arising from Cr intercalation, are well separated, allowing for the elemental sensitivity of ResPES to benefit from an added momentum resolution. Here, a comparison of the integrated energy distribution curves (EDCs) shown in Figs. 4b and  5d reveal a resonant enhancement of Cr 3d states occurring near EF that is independent of temperature. However, MDCs extracted from Fig. 5a–c at EF as a function of photon energy show an initial resonance of the β1,2 bands (k ~0.5 Å−1) occurring at the onset of absorption (Fig. 5e). This finding not only provides evidence for a Cr-derived elemental character of these bands but also reveals a lack of hybridization with the Nb-derived α band that is consistent with its anti-bonding orbital character32. Given the large separation distance between Cr neighbors, the formation of dispersive β1,2 bands following Cr intercalation can only occur through hybridization with Nb. This is made clear from our ResPES results, but is also indicated by the higher degree of kz dispersion exhibited by the β1,2 bands as compared to α32. Thus, by measuring ARPES across the Cr absorption edge, a band resolved picture of Cr-derived d states in the VB is obtained.

### Change in spectral weight in proximity to TC

By revealing the presence of Cr-derived d states at EF, our data show that a clear separation of magnetic and itinerant DOF does not occur in Cr1/3NbS2, as the same states forming LSMs are also participating in the formation of the FS. The consequences of this finding are illustrated in the temperature dependence of the EDCs integrated over a broad momentum range (kΓΜ = 0.24–0.8 Å−1) encompassing both crossing points of the β1,2 bands (Fig. 6a). Here, as T >50 K, the spectral weight in proximity to EF begins to drop and is transferred towards higher binding energies until T >120 K, whereupon there is no longer any change. Similarly, the temperature dependence of the α band appears to be less pronounced as compared to that of the β band, but still reveals a monotonic drop of spectral weight at EF, as well as a slight reduction in the separation between peaks at 100 and 400 meV as T > TC (Fig. 6b). In both cases, the spectral weight at EF changes in close proximity to TC (T = 130–90 K), allowing for thermal broadening effects to be excluded, suggesting instead a microscopic mechanism that links electron itinerancy to the onset of ferromagnetism.

## Discussion

For ferromagnetism being mediated by an indirect RKKY exchange interaction, as is the case for rare-earth metals like Gd, the electronic states forming the FS are distinct from those hosting LSMs35, where the interaction between local moments and itinerant electrons is treated as a weak perturbation resulting from a FS instability26. Under an RKKY exchange interaction, the VB states are exchange split by the local moment. Temperature controls the ordering of the LSMs, and affects the exchange splitting of the VB according to a behavior that is intermediate between two extreme limits. In one instance, there is no change in exchange splitting with temperature. This occurs when VB states have wavefunctions that are incoherent over the magnetic correlation length of the local moment. Such behavior is exhibited by band states that are either localized, or in systems consisting of a few magnetic impurities embedded in metals. In the opposite limit, the exchange splitting of the valence states continuously decreases with increasing temperature until vanishing all together when magnetic order is lost. This Stoner-like behavior occurs when band states are delocalized so as to remain coherent over a distance that is large compared to the magnetic correlation length of the local moment35. In a prototypical system described by an RKKY exchange, like Gd, VB states with different degrees of localization/delocalization exhibit behavior that spans these two extremes36.

As this pertains to Cr1/3NbS2, despite the merging of β1,2 bands as temperature is raised above TC, the transfer of spectral weight away from EF as T > TC runs counter to our initial first-principle, DFT calculations, showing the Stoner criterion to be well satisfied29. By revisiting these calculations using the same experimental structure as used by Ghimire et al.29 with internal coordinates optimized, we focus on an underappreciated aspect of Cr1/3NbS2, which is how a material having such a large in-plane, Cr–Cr nearest-neighbor distances (5.75 Å) can exhibit an ordering temperature as high as 131 K. While evident that such a large Cr–Cr distance discounts the possibility of a direct exchange interaction, more conventional indirect exchange interactions are expected to be weak and will therefore play a role only at considerably lower energy scales. Our calculations find an energy difference between the FM and nearest-neighbor excited anti-FM state to be in excess of 30 meV Cr−1, implying a TC ≈116 K under a mean-field approximation alone. Such a large energy scale for the exchange energetics is expected to dominate over any higher-order corrections, like the magnetic anisotropy30, that arise from spin–orbit coupling in this material. While an RKKY interaction has been experimentally established as the fundamental exchange interaction in 3d-intercalated TMDs23,25, including Cr1/3NbS237, such a significant exchange energy implies the interaction between LSMs and hybridized Nb- and Cr-derived itinerant states may go beyond this perturbative interaction, originating instead from a microscopic interaction between itinerant electrons and LSMs.

The data presented here show a clear separation of magnetic and itinerant DOF does not occur, leading to the motion of itinerant electrons influencing LSMs reciprocally, and in a self-consistent fashion, as is found for other itinerant, 3d magnetic systems38,39. In such a case, a strong Hund’s coupling establishes a connection between the onset of ferromagnetism and metallicity, as it enforces electron itinerancy on Cr sites when the spin of the majority carrier is parallel to the LSM. In this way, an on-site Hund’s correlation shapes the interaction between itinerant electrons and LSM’s in such a way that a synergistic and regenerative character is acquired. That is, FM order fosters electron itinerancy by a reduction of electron scattering from thermal fluctuations of the LSMs, while electron itinerancy promotes a decrease of the electron kinetic energy and FM order, overall resulting in a minimization of energy for the entire system. Hence, electron itinerancy acquires a fundamental role in the attainment of FM order, and the FM transition in Cr1/3NbS2 is viewed as a cooperative phenomenon driven by the mutual interaction of itinerant electrons and LSMs enforced by Hund’s coupling. The concomitant increase of electron itinerancy and alignment of the LSMs provide a route for the minimization of the total energy of the system, offering a microscopic means by which important aspects of the in-plane transport can be rationalized. This includes a natural explanation for the pronounced drop of the in-plane resistivity occurring at TC, along with the negative in-plane MR28,29. In this way, the magneto-transport properties of Cr1/3NbS2 are in some respect reminiscent of the colossal magnetoresistive manganites. Both systems exhibit the same correlation between the ordering of LSMs and the onset of metallicity, and both exhibit a negative MR, albeit quite different in order of magnitude.

In Cr1/3NbS2, elucidating the mechanisms that underlies both inter- and intra-layer magneto-transport acquires particular importance in light of the possible technological relevance that this material has for future spintronic applications. Our results suggest that arguments based entirely on magnetic scattering due to spin ordering in the CSL state may not fully capture the magneto-transport properties observed in this material. As our high-resolution photoemission results are sensitive to only the first few atomic layers, our findings are directly relevant to a microscopic description of the in-plane, magneto-transport properties. Nevertheless, it is unlikely that the microscopic interaction between itinerant electrons and LSMs evidenced by our data would be entirely irrelevant for the inter-layer magneto-transport properties as well. In fact, any mechanism proposed to explain the inter-layer MR must be rooted in a Hamiltonian that takes into account the interaction of itinerant electrons with the LSMs. Thus, despite the long length scales of the CSL, our results highlight the relevance of the electronic structure in a description of the magneto-transport properties in Cr1/3NbS2, as underscored by the strong Hund’s coupling between itinerant electrons and LSMs. For this reason, we hope this work encourages future investigations into other 3d-intercalated TMDs, where the interaction between LSMs and itinerant electrons extends beyond a rigid band interpretation and will be relevant for shaping anisotropic functionalities in this class of materials.

In conclusion, electronic structure measurements of Cr1/3NbS2, taken above and below the helimagnetic transition temperature, reveal a clear separation of magnetic DOF, and itinerant DOF does not occur, as the same states forming LSMs are also participating in the formation of the FS. An anomalous increase of spectral weight in proximity to the Fermi level (EF) as temperature is lowered below TC marks a behavior that is inconsistent with conventional itinerant ferromagnets, and is rationalized on the basis of first-principle DFT calculations to result from a strong exchange (Hund’s) interaction between itinerant electrons and LSMs on the Cr sites. Our results suggest that arguments based entirely on magnetic scattering due to spin ordering in the CSL state may not fully capture the magneto-transport properties observed in this material, as the electronic structure is clearly shown to play a non-trivial role as temperature is tuned across TC.

## Methods

### Crystal growth

Polycrystalline Cr1/3NbS2 samples were grown by heating stoichiometric ratios of Cr, Nb, and S to 950 °C for 1 week. Single crystal growth was carried out under chemical vapor transport using 0.5 g iodine transport agent per 3 g Cr1/3NbS2. The 5 mm × 5 mm plate-like crystals oriented along (0 0 1) formed across a 100 °C (950–850 °C) temperature gradient of the transport tube. Due to the deleterious effect that Cr disorder has on the appearance of helimagnetic ordering in this material40, x-ray and low-energy electron diffraction measurements were used to confirm a P6322 space group showing (√3 × √3) Cr ordering29,32. Sample quality was further verified with superconducting quantum interference device magnetometry (Fig. 1b), which reveals a prominent kink at TC = 131 K, indicating helimagnetic ordering in this sample batch.

### Photoemission spectroscopy

Temperature-dependent ARPES experiments were carried out on Cr1/3NbS2 single crystals cleaved in situ on Beamline 10.0.1 of the Advanced Light Source (ALS) and the Advanced Photoelectric Effect beamline at the Elettra Synchrotron Facility. The total instrumental energy resolution ranged from 15 to 30 meV, while an angular resolution of ±0.5° gives a momentum resolution of <0.06 Å−1 for the photon energies used in these experiments ( = 40 and 48 eV). ResPES and ResARPES experiments were performed above and below TC on the Beamline for Advanced diCHroism at Elettra with a total instrumental energy resolution being better than 300 meV.

### Ab initio calculations

First-principles DFT calculations were carried out using the linearized augmented plane-wave DFT code WIEN2K41 in the generalized gradient approximation of Perdew et al.42. The respective sphere radii of 2.01, 2.33, and 2.37 Bohr were used for S, Cr, and Nb, with a value for the product of the smallest sphere radius (S) and the largest plane-wave expansion vector being set to RKmax = 8.0. Calculations of the magnetic properties, performed with and without spin–orbit coupling, were carried out under the internal coordinates of the structure being relaxed in the FM state. A minimum of 800 k points in the full Brillouin zone were used for the FM calculations, while this number was proportionally reduced for the 2 × 2 × 1 supercell of the planar anti-FM state.

A comparison of DFT energetics with and without spin–orbit coupling is made between a FM and two possible anti-FM states denoted as AFM1 and AFM2. These states represent either anti-aligned Cr–Cr pairs along the c-axis (AFM1), or four of six anti-aligned Cr–Cr neighboring pairs in the ab plane (AFM2). From our calculation, we find an energy difference in meV/Cr for AFM1 and AFM2 relative to the FM state that shows DFT can describe the normal (i.e., non-Dzyaloshinskii–Moriya) exchange interaction without the need to consider spin–orbit coupling (Table 1). This <1 meV/Cr difference is fully consistent with the high (131 K) magnetic ordering temperature found in this material.

## Data availability

All data presented in this manuscript can be made available upon reasonable request to corresponding authors, N.S. and N.M.

## Change history

• ### 07 May 2020

An amendment to this paper has been published and can be accessed via a link at the top of the paper.

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Vilmercati, P., Cvetko, D., Cossaro, A. & Morgante, A. Heterostructured organic interfaces probed by resonant photoemission. Surf. Sci. 603, 1542 (2009).

35. 35.

Kim, B., Andrews, A. B., Erskine, J. L., Kim, K. -J. & Harmon, B. N. Temperature-dependent conduction-band exchange splitting in ferromagnetic hcp gadolinium: theoretical predictions and photoemission experiments. Phys. Rev. Lett. 68, 1931 (1992).

36. 36.

Maiti, K., Malagolim, M. C., Dallmeyer, A. & Carbone, C. Finite temperature magnetism in Gd: evidence against a stoner behavior. Phys. Rev. Lett. 88, 167205 (2002).

37. 37.

Aczel, A. A. et al. Extended exchange interactions stabilize long-period magnetic structures in Cr1/3NbS2. Appl. Phys. Lett. 113, 032404 (2018).

38. 38.

Hubbard, J. The magnetism of iron. Phys. Rev. B 19, 2626 (1979).

39. 39.

Gyorffyt, B. L., Pindors, A. J., Stauntonll, J., Stocks, G. M. & Winter, H. A first-principles theory of ferromagnetic phase transitions in metals. J. Phys. F 15, 1337 (1985).

40. 40.

Dyadkin, V. et al. Structural disorder versus chiral magnetism in Cr1/3NbS2. Phys. Rev. B 91, 184205 (2015).

41. 41.

Blaha, P. et al. WIEN2k, An Augmented Plane Wave+Local Orbitals Program for Calculating Crystal Properties ISBN 3-9501031-1-2 (Karlheinz Schwarz, Techn. Universität Wien, Austria, 2018).

42. 42.

Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865 (1996).

## Acknowledgements

This work was supported by the National Science Foundation, Division of Material Research, Grant No. DMR-1151687 (N.M.). This work was performed, in part, at the ALS, a DOE Office of Science User Facility under contract no. DE-AC02-05CH11231. N.S. acknowledges support from the Center for Integrated Nanotechnologies at Los Alamos National Laboratory (LANL), a US Department of Energy, Office of Basic Energy Science user facility. DSP was supported by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences, Materials Science and Engineering Division. Partial funding for open access to this research was provided by University of Tennessee’s Open Publishing Support Fund.

## Author information

Authors

### Contributions

Samples were provided by L.L., D.S., and D.G.M. Temperature-dependent ARPES experiments were performed by N.S. and N.M. with the help of S.-K.M., A.V.F., P.K.D., I.V., and J.F. ResPES and ResARPES experiments were carried out by N.S. with the help of F.B., I.P., and S.N. N.S. and P.V. analyzed the data, with input from N.M. and D.S.P., who provided ab initio calculations. N.S., P.V., and N.M. wrote the manuscript with input from all authors.

### Corresponding authors

Correspondence to N. Sirica or N. Mannella.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Sirica, N., Vilmercati, P., Bondino, F. et al. The nature of ferromagnetism in the chiral helimagnet Cr1/3NbS2. Commun Phys 3, 65 (2020). https://doi.org/10.1038/s42005-020-0333-3


text:
# Factoring maps of handlebodies

Any map of finite graphs (1-dimensional CW-complexes) factors as a composition of

1. a finite sequence of folds;
2. an inclusion; and
3. a finite-to-one covering map.

There should be a corresponding result for handlebodies, which presumably should say that, after a homotopy, a continuous map of handlebodies factors as:

1. a compression (by which I mean a map of a handle into the complement of its interior);
2. an inclusion; and
3. a finite-to-one covering map.

Is my intuition correct, and does anyone have a reference? I'm specifically interested in how well-behaved the homotopy can be taken to be. For instance, can it be made to respect the boundary?

Notes

A fold is a map that identifies two edges with a common endpoint. Many folds don't change the homotopy type of a graph, and one would expect not to need these in the handlebody setting. The important folds are the ones that kill a loop. In handlebody terms, you can think of this as gluing in a two-handle, or as cutting a one-handle - hence my use of the word "compression". Is this word acceptable in this context?

The graph-theoretic result is due to Stallings.

By an inclusion of handlebodies, I mean that the new one should be obtained from the old by attaching 1-handles.

EDIT (prompted by Sam's comments below) I'm not quite sure what "respect the boundary" should mean, at this point. Suggestions welcome!

-
What does "respect the boundary" mean? –  Sam Nead Nov 18 '09 at 20:21
At each "time", the homotopy should map the boundary to itself. –  HJRW Nov 18 '09 at 20:27
So the maps has to start by mapping the boundary to the boundary? For example, what about placing a trefoil knot in a three-ball, and thickening it a bit? A homotopy can unknot this (and then I guess you want to compress), but not in any boundary respecting way... –  Sam Nead Nov 18 '09 at 23:30
I'd be happy to start with any factorization, and worry about whether the homotopy can be made nice later. –  HJRW Nov 18 '09 at 23:45
What about taking a regular neighborhood of the (p,q) curve on the boundary (p times about the meridian, q times about the longitude). So this is incompressible... I suppose that you will homotope this to be a q-fold covering. I don't see how you are going to do this keeping the boundary "nice". –  Sam Nead Nov 19 '09 at 0:02

Suppose that we are given a PL map from a handlebody $W$ to a handlebody $V$. Choose a spine for $W$. Homotope the map until the image is a regular neighborhood of the image of the spine. By general position, our map is now an embedding. Fix a pants decomposition of disks $D = (D_i)$, for $V$. Suppose that $P$ is a component of $V - D$ (so $P$ is a three-ball with three distinguished disks on its boundary). Consider a component $X$ of $f(W) \cap P$. This is essentially a knotted graph. Via a homotopy (keeping $X \cap D$ fixed) unknot $X$. If the rank of $X$ is positive then another homotopy produces small lollipops which we shall compress a bit later. If $X$ meets any disk $D_i \subset \partial P$ more than once then we may homotope a leg of $X$ through $D_i$. Let $Y$ be the resulting component of $f(W) \cap P$. (Note that this reduces $f(W) \cap D$.)

Homotoping in this fashion we eventually arrive at an embedding of $W$ so that every component in every three-ball of $V - D$ is either a tripod or an interval, possibly with lollipops attached. The feet of the tripod/interval lie in distinct disks in the boundary of the containing solid pants.

Now compress all of the lollipops to get $f'(W')$ (a new handlebody, because we compressed and a new map because we have to extend it over the two-handles we added).

EDIT: This reproduces, in our context, part of Stallings paper (eg sliding the leg is a fold, arriving at only tripods and intervals produces an immersion.)

Since $f'$ is an immersion, it follows from Stallings paper that $f'$ is $\pi_1$ injective and that $W'$ embeds into a finite cover of $V$.

-
Looks good! I want to use a homotopy to make X as big as possible in P, so the complement is either empty or a ball whose boundary contains one distinguished disc. If we do so then I think we get that, after a homotopy, the map factors as: glue on some 2-handles; glue on some 1-handles; and then a finite-sheeted covering. (Note that, in the question, I mentioned that by "inclusion", I meant gluing on 1-handles.) Does that sound right? –  HJRW Nov 19 '09 at 16:27
(I mean that the complement of X in P should be a ball after X has been compressed.) –  HJRW Nov 19 '09 at 16:31
This is right. After producing f' we can 1) isotope each components of $f'(W') \cap D_i$ to be homeomorphic to its containing disk, 2) apply isotopy extension to the map $f'$, and 3) fix the boundaries of the components $f'(W') \cap P$. –  Sam Nead Nov 19 '09 at 17:54
Excellent! I think this is the best possible statement I could have hoped for. –  HJRW Nov 19 '09 at 18:21


text:
Changeset 596:293551ad254f in lemon-1.2 for lemon/gomory_hu.h

Ignore:
Timestamp:
04/15/09 09:37:51 (11 years ago)
Branch:
default
Phase:
public
Message:

Improvements and fixes for the minimum cut algorithms (#264)

File:
1 edited

Legend:

Unmodified
 r581 /// between these nodes. Moreover the components obtained by removing /// this edge from the tree determine the corresponding minimum cut. /// /// Therefore once this tree is computed, the minimum cut between any pair /// of nodes can easily be obtained. /// /// The algorithm calculates \e n-1 distinct minimum cuts (currently with /// the \ref Preflow algorithm), therefore the algorithm has /// \f$(O(n^3\sqrt{e})\f$ overall time complexity. It calculates a /// rooted Gomory-Hu tree, its structure and the weights can be obtained /// by \c predNode(), \c predValue() and \c rootDist(). /// /// The members \c minCutMap() and \c minCutValue() calculate /// the \ref Preflow algorithm), thus it has \f$O(n^3\sqrt{e})\f$ overall /// time complexity. It calculates a rooted Gomory-Hu tree. /// The structure of the tree and the edge weights can be /// obtained using \c predNode(), \c predValue() and \c rootDist(). /// The functions \c minCutMap() and \c minCutValue() calculate /// the minimum cut and the minimum cut value between any two nodes /// in the graph. You can also list (iterate on) the nodes and the /// /// \tparam GR The type of the undirected graph the algorithm runs on. /// \tparam CAP The type of the edge map describing the edge capacities. /// It is \ref concepts::Graph::EdgeMap "GR::EdgeMap" by default. /// \tparam CAP The type of the edge map containing the capacities. /// The default map type is \ref concepts::Graph::EdgeMap "GR::EdgeMap". #ifdef DOXYGEN template Value minCutMap(const Node& s, ///< The base node. Value minCutMap(const Node& s, ///< const Node& t, ///< The node you want to separate from node \c s. ///< CutMap& cutMap ///< The cut will be returned in this map. /// It must be a \c bool (or convertible) /// \ref concepts::ReadWriteMap "ReadWriteMap" /// on the graph nodes. ///< ) const { Node sn = s, tn = t; /// This iterator class lists the nodes of a minimum cut found by /// GomoryHu. Before using it, you must allocate a GomoryHu class, /// GomoryHu. Before using it, you must allocate a GomoryHu class /// and call its \ref GomoryHu::run() "run()" method. /// /// This iterator class lists the edges of a minimum cut found by /// GomoryHu. Before using it, you must allocate a GomoryHu class, /// GomoryHu. Before using it, you must allocate a GomoryHu class /// and call its \ref GomoryHu::run() "run()" method. /// ///   value+=capacities[e]; /// \endcode /// the result will be the same as it is returned by /// \ref GomoryHu::minCutValue() "gom.minCutValue(s,t)" /// The result will be the same as the value returned by /// \ref GomoryHu::minCutValue() "gom.minCutValue(s,t)". class MinCutEdgeIt { public: /// Constructor /// Constructor. /// MinCutEdgeIt(GomoryHu const &gomory, ///< The GomoryHu class. You must call its ///< If it is \c true (default) then the listed arcs ///  will be oriented from the ///  the nodes of the component containing \c s, ///  nodes of the component containing \c s, ///  otherwise they will be oriented in the opposite ///  direction.


text:
# How do you sketch the angle -(7pi)/4 in standard position?

$\left(\frac{\sqrt{2}}{2} , \frac{\sqrt{2}}{2}\right)$
Add $2 \pi$.
$2 \pi - \frac{7}{4} \pi = \frac{\pi}{4} = {45}^{o}$


text:
## type known.designs() in PowerTOST [Study Assessment]

Hi Rosy,

»
» The study had an open-label, single-dose, three-way crossover design. Subjects were randomly assigned to receive one of three treatments per period

Was it a Latin Square (ABC|BCA|CAB) or a Williams’s design (ABC|ACB|BAC|BCA|CAB|CBA)? Not mentioned anywhere in the paper. But this part is strange:

Seventeen healthy [] were enrolled.

Enrolled, how comes? It is OK to have dropouts but to start a study with incomplete sequences is bizarre. The sample size has to be a multiple of the number of sequences.

» A vs B ----> IC (84.9-106)
» A vs C ----> IC (169-211)
» B vs C ----> IC (178-222)
»
» 100*CVfromCI(lower=0.849, upper=1.06, n=17, design='2x2x3', alpha=0.05)
» it's that correct???

No, it isn’t. "2x2x3" is for the two-treatment two-sequence three-period full replicate design (TRT|RTR).
For the Latin Square use "3x3" and for the Williams’ design "3x6x3". BTW, in all functions of PowerTOST the default is alpha=0.05. As long as you deal with the 100(1–2×0.05) = 90% confidence interval you can simply leave it out.

Dif-tor heh smusma 🖖
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes


text:
# Analyzing the motion of a ball rolling without slipping inside a hemispherical bowl

Consider a solid ball of radius $r$ and mass $m$ rolling without slipping in a hemispherical bowl of radius $R$ (simple back and forth motion). Now, I assume the oscillations are small and so the small angle approximation holds. I wish to find the period of oscillation and I analyze the motion in two ways, first using conservation of energy and secondly using dynamics. However, I receive two inconsistent answers. One or both of the solutions must be wrong, but I cannot figure out which one and more importantly, I cannot figure out why.

Method 1: We write the energy conservation equation for the ball

$mgh + \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2 = Constant$

from the center of mass, we take the height as $h = R-(R-r)cos\theta$ where $\theta$ is the angle from the vertical. Applying the no slip condition $v = r\omega$ and taking the moment of inertia for a solid sphere $I = \frac{2}{5}mr^2$ we can write the energy equation as

$mg(R-(R-r)cos\theta) + \frac{7}{10}mr^2\omega^2 = Constant$

Differentiating with respect to time:

$mg(R-r)sin\theta\cdot\omega + \frac{7}{5}mr^2\omega\cdot\alpha = 0$

taking the small angle approximation $sin\theta = \theta$ we get

$g(R-r)\theta + \frac{7}{5}r^2\alpha=0$

$-\frac{5g(R-r)}{7r^2}\theta = \alpha$

from which we can get $T = 2\pi\sqrt{\frac{7r^2}{5g(R-r)}}$

Method 2: The only torque acting on the ball at any point in its motion is the friction force $f$. So we can write

$\tau = I\alpha = fr$

again using the rolling condition $a = r\alpha$ and the moment of inertia for a solid sphere,

$\frac{2}{5}ma = f$

The net force acting on the system is the tangential component of gravity and the force of friction, so

$F = ma = mgsin\theta - f$

$\frac{7}{5}a = gsin\theta$

taking the small angle approximation and converting $a$ to $\alpha$ we get

$\alpha = \frac{5g}{7r}\theta$

and a corresponding period of $T = 2\pi\sqrt{\frac{7r}{5g}}$

Now the solutions are very different and I would appreciate it if someone would point out where I went wrong.

-
There are a lot of things wrong with this. The most glaring is that you take energy to be constant in the first approach, but you introduce friction into the second, which is an outside force that will reduce energy over time. So the two approaches are necessarily going to give different results because you're making different assumptions. –  Mitchell Jun 18 '11 at 5:25
@Mitchell, no, that is not a problem. The friction force does no work. I think Yuqing explained this pretty clearly. –  Mark Eichenlaub Jun 18 '11 at 5:33
Closely related: physics.stackexchange.com/questions/10798/… –  qftme Jun 18 '11 at 12:23

Your first derivation, using energy, uses two different meanings for the same symbol $\omega$. In one place, you interpret it as
$$\omega = \dot{\theta}$$
In another place, you interpret $\omega$ as the time derivative of the unnamed angle through which the ball itself has rotated.
These two angles are related to each other by the $r/(R-r)$ factor by which you are off.


text:
Volume 111, Issue 3 – May 1980

On the one-sided Wiener’s theorem for the motion group

Pages 415-422 by Yitzhak Weit

The classification of simply connected manifolds of positive scalar curvature

Pages 423-434 by Mikhael Gromov, H. Blaine Lawson, Jr.

Lattices in spaces of nonpositive curvature

Pages 435-476 by Patrick Eberlein

Bäcklund’s theorem for $n$-dimensional submanifolds of $\mathbb{R}^{2n-1}$

Pages 477-490 by Keti Tenenblat, Chuu-Lian Terng

A higher dimension generalization of the Sine-Gordon equation and its soliton theory

Pages 491-510 by Chuu-Lian Terng

Factorization of $A_p$ weights

Pages 511-530 by Peter W. Jones

Propagation of singularities for semilinear hyperbolic equations in one space variable

Pages 531-552 by Jeffrey Rauch, Michael C. Reed

The torsion of the group of $0$-cycles modulo rational equivalence

Pages 553-569 by A. A. Roitman

Special representations of reductive $p$-adic groups are not integrable

Pages 571-587 by Allan J. Silberger

The Hua operators on bounded symmetric domains of tube type

Pages 589-608 by Kenneth D. Johnson, Adam Korányi


text:
# Set frame title as \title or pdftitle

I'm preparing a series of beamer documents with different \titles, and the first slide of each document has the same title as the document \title. I want to be able to just change the content of \title for each document, and then the title of the first slide will be updated automatically.

\documentclass{beamer}

\title{Some Title}
\makeatletter
\hypersetup{pdftitle=\@title}
\makeatother

\begin{document}

\begin{frame}{Some Title}
frame content
\end{frame}

\end{document}

Since the frame title is the same as \title and pdftitle, is there a way to set the frame title to be the same as pdftitle or \title without typing it again? I tried

\begin{frame}{\@title}

Obviously this didn't work.

One solution would be to have \def\mytitle{Title}, and then \hypersetup{pdftitle={\mytitle}} and \begin{frame}{\mytitle} as well as \title{\mytitle}.


text:
# Reusing slides from Beamer presentations

I give a good number of talks and find that I can share some slides but not others based on audience, topic, etc. Powerpoint and Keynote make it pretty easy to pull slides from a deck into another deck. I'm curious about how folks deal with this situation with beamer. I can think of a few possibilities.

1. Simply cut-and-paste text from one show to the next
2. Use an IDE to improve on 1
3. Make a "supershow" that contains most of what I would want to say, but build a menu system that allows me to customize on-the-fly.

Are there other options or ideas that I should consider?

• Did you consider putting each slide in a separate file and including them according to the actual requirement? I have not tested this with beamer but would not know any reason why this should not work; for some reports I prepare regularly, I have learned to like this method. It also allows you to keep your editing limited to each slide individually, rather than the whole set. – vaettchen Mar 4 '12 at 2:44
• I had not thought of that. Certainly for some of my most common slides, this idea makes a lot of sense. – seandavi Mar 4 '12 at 12:58

I have a mountain of beamer slides that I combine in all sorts of ways for a number of different courses and presentations that I make. Here's what I do, slightly simplified:

1. I use beamer templates as containers for series of slides (sometimes a series is just a single slide, sometimes it is 40 slides):

\newcommand{\defsection}[3]{
\setbeamertemplate{slide #1 #2}{{#3}}}

I.e. in my source I put everything in "sections" a la:

\defsection{intro}{string-to-describe-what-this-is}{
\begin{frame}...\end{frame}
...
\begin{frame}...\end{frame}
} % end of \defsection

2. I use the sections of slides with the macro \usesection:

\newcommand{\usesection}[2]{
\ifbeamertemplateempty{slide #1 #2}
{\errmessage empty section: {#1}{#2}}
{\opt{texdebug}{\message{
*** Using section: (#1)(#2) }}
\usebeamertemplate{slide #1 #2}}}

3. I use the optional package to be allow conditional inclusion of stuff:

\usepackage[advanced,use-foo,dont-use-bar,do-xyz,do-frotz]{optional}

4. I include ALL stuff in a master file (main.tex):

\begin{document}
\include{stuff.tex}
\include{more-stuff.tex}
\include{even-more-stuff.tex}

\usesection{intro}{intro}

}

\opt{use-foo}{
}

\opt{use-bar}{
}

\opt{dont-use-bar}{

5. Now each "version" or "course" or "subset of stuff" (or whatever) is basically just a matter of defining the right options and then include main.tex.
I realise this may seem a bit complicated at first go, but the key idea is to separate definition of some slides (\defsection) from the use of them (\usesection). Once that separation is made, all sorts of reuse, different selections based on options, etc, etc, becomes quite trivial.


text:
# Transfer Function Magnitude, how to deal with 'j'?

I have the simple transfer function of an RC filter:

$H(s) = \frac{sRC}{1 + sRC}$

In order to find the magnitude, square the previous equation and take the square root:

$(H(s))^2 = \frac{(sRC)^2}{(sRC + 1)^2} ; s = j\omega$

$(H(s))^2 = \frac{(j\omega R C)^2}{(j\omega R C + 1)^2} \qquad(1)$

This is where I get stuck. I know that the correct magnitude should be of the form:

$|H(s)| = \sqrt{\frac{(\omega R C)^2}{(\omega R C)^2 + 1}} \qquad (2)$

However, when I actually try to calculate this out on paper I don't know how to deal with the $j$. I don't understand how to get from equation I get from equation (1) to equation (2). The $-1$ from $j^2$ causes the form to look like the following:

$(H(s))^2 = \frac{-top}{1 - bottom}$

If I were to take the square root of this I do not get equation (2).

What am I doing wrong?

• You didn't take the square of the denominator properly: it should be (1+sRC)^2. – Barry Nov 19 '17 at 23:30
• You're correct, I did not do that correctly. However, doing it again I get that: H(s)^2 = [(j^2)(w^2)(R^2)(C^2)] / [1 + 2jwRC + (j^2)(w^2)(R^2)(C^2)] ; This leaves me.. more confused than before? – Zearia Nov 19 '17 at 23:43

\begin{align*} H\left(s\right)\quad &=\quad \frac{s R C}{1 + s R C}\\\\ \mid\: H\left(s\right)\: \mid^{\:2}\quad &=\quad H\left(s\right)\cdot H\left(s^*\right)\\\\ &=\quad\frac{s R C}{1 + s R C}\cdot\frac{s^* R C}{1 + s^* R C}\\\\ &=\quad\frac{\left(\sigma^2+\omega^2\right) R^2 C^2}{1 + 2\sigma R C+\sigma^2 R^2 C^2+\omega^2 R^2 C^2} \end{align*}

Laplace uses $s=\sigma + j \omega$. But this collapses to Fourier along the imaginary axis portion, or the $j \omega$ part of that. Since you want the periodic and non-decaying part of the system response -- the frequency response -- you set $\sigma=0$ in $s$.

So the above devolves to:

\begin{align*} \mid\:H\left(j\omega\right)\:\mid\quad &=\sqrt{ \frac{\omega^2 R^2 C^2}{1 + \omega^2 R^2 C^2}}\\\\ &=\quad\frac{\omega R C}{\sqrt{1+\omega^2 R^2 C^2}} \end{align*}

I hope that helps.

• What I seem to be confusing about this whole process is that when you square the s, would you not get s^2 = (sigma^2) + [(j^2)(w^2)]? If sigma was set to 0 would that not still yield that s^2 = -(w^2)? – Zearia Nov 20 '17 at 1:27
• @Zearia By definition, the magnitude squared of a complex number is $\mathscr{C}\cdot\mathscr{C^*}$. See: Absolute value and argument. And no, we are talking about $s=\sigma+j \omega$. The magnitude squared is $\mid s\mid ^2=\left( \sigma+j \omega\right )\cdot \left( \sigma-j \omega\right)$. – jonk Nov 20 '17 at 2:03
• @Zearia If you don't see why, just plot a point on the complex plane and work out how you'd compute the hypotenuse. It should be very clear, then, I think. – jonk Nov 20 '17 at 2:16


text:
# Chapter 1 - Equations and Inequalities - 1.8 Absolute Value Equations and Inequalities - 1.8 Exercises: 3

$D$

#### Work Step by Step

RECALL: For any real number $x$, $|x| \ge 0$. Since the absolute value of any real number is never negative, then any real number satisfies $|x| \gt -7$. Therefore, the solution to the given inequality is the set of all real numbers, the graph of which is the one represented by the one in Option $D$.

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.